SSLC EXAM- TARGET 40 Study notes for Revision 2014

SSLC EXAM- TARGET 40 Study notes for Revision 2014

first1 Yakub SGHS Nada Belthangady TalukDK Ph9008983286 Email yhokkilagmailcom For MSTF Mangalore(Belthangady)

1Chapter ndash Real Numbers ( Total marks 3)

SlNo Chapters MCQ 1-Marks 2-Marks 3-Marks 4-marks Total K U A S K U A S K U A S 1 Real Numbers 1 2 3

Euclidrsquos Division Lemma a = bq + r [ 0 le r lt 푞 ] a ndash Dividend b ndash Divisor Q ndash quotient r ndash Remainder

If lsquoarsquo and lsquobrsquo are any two positive integers HCF of ( a b ) x LCM of ( a b ) = a x b

Euclidrsquos Division Lemma Question Answer

a = bq + r [ 0 le r lt 푞 ] Divide 735 by 40 and write quotient and remainder 735 and 40

40]735 [18 -40 335 -320 15 By Euclidrsquos Lemma 735 = ( 40 x 18 ) + 15

Find the HCF and LCM of 18 and 45 by prime factorization method

18 = 2 x 3 x 3 = 2 x 32

45 = 3 x 3 x 5 = 32 x 5 there4 HCF of 18 and 45 = 3 x 3 = 9 LCM of 18 and 45 = 3 x 3 x 2 x 5 = 90

Express the 120 as a product of prime factors 2 x 2 x 2 x 3 x 5 Divide120 by prime numbers Note The number which is divisible by 1 and itself is called prime numbers [ Example 2 3 5 7 11 13 17 19helliphelliphelliphelliphellipetc



Chapter -2 Sets (Total 3 Marks)

SLNo Chapter MCQ 1-mark 2-marks 3-marks 4-marks Total K U A S K U A S K U A S 2 Sets 1 2 3

U ndash Universal Set A B and C are non-empty sets

ರೂಪ ಾಜಕ ಯಮ- Union of sets is distributive over intersection of sets

Intersection of sets is distributive over union of sets Example U = 0123456789 A=1237 B=378 C=1238 Note Complimentary of sets ndash A1 = 0145689 B1 = 0124569 C1 = 045679

Properties Union Intersection

Commutative

AUB = 1237 U 378 = 12378helliphelliphellip(1) BUA = 378 U 1237 = 12378helliphelliphellip(2)

From (1) and (2) AUB = BUA

AcapB = 1237 cap 378 = 37 helliphelliphelliphelliphellip(1) BcapA = 378cap1237 = 37 helliphelliphelliphelliphellip(2)

From (1) and (2) AcapB = BcapA

Properties Union Intersection Commutative AUB = BUA AcapB = BcapA

Associative AU(BUC) = (AUB)UC (AcapB)capC = (AcapBcapC) Distributive AU(BcapC) = (AUB)cap(AUC) Acap(BUC) = (AcapB)U(AcapC) DrsquoMorganrsquos (AUB)1 = A1capB1 (AcapB)1 = A1UB1



Associative

AU(BUC) = 1237 U [378U1238] AU(BUC) = 1237 U 12378 AU(BUC) = 12378helliphelliphelliphelliphelliphelliphelliphelliphellip (1) (AUB)UC = [ 1237U378] U 1238 (AUB)UC = 12378U1238 (AUB)UC = 12378helliphelliphelliphelliphelliphelliphelliphelliphellip(2)

From (1) and (2) AU(BUC)=(AUB)UC

Acap(BcapC) = 1237cap[378cap1238] Acap(BcapC) = 1237cap38 AU(BUC) = 3helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip (1) (AcapB)capC = [ 1237cap378]cap1238 (AUB)UC = 37cap1238 (AcapB)capC = 3helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip(2)

From (1)and (2) (AcapB)capC= (AcapBcapC)

Distributive

AU(BcapC) = 1237U[378cap1238] AU(BcapC) = 1237U38 AU(BcapC) = 12378helliphelliphelliphelliphelliphelliphelliphelliphellip(1) (AUB)cap(AUC) = [1237U378]cap[1237U1238] (AUB)cap(AUC) = 12378cap12378 (AUB)cap(AUC) = 12378 helliphelliphelliphelliphelliphelliphelliphellip(2)

From (1) and (2) AU(BcapC)=(AUB)cap(AUC)

Acap(BUC) = 1237cap[378U1238] Acap(BUC) = 1237cap12378 Acap(BUC) = 1237helliphelliphelliphelliphelliphelliphelliphelliphellip(1) (AcapB)U(AcapC) = [1237cap378]U[1237cap1238] (AcapB)U(AcapC) = 37U123 (AcapB)U(AcapC) = 1237helliphelliphelliphelliphelliphelliphelliphelliphelliphellip(2)

From (1) and (2) Acap(BUC)=(AcapB)U(AcapC)

Drsquomorgans Law

(AUB)1 = [1237U378]1

(AUB)1 = 123781 (AUB)1 = 04569helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip(1) A1capB1 = 123781cap3781

A1capB1 = 04569cap0124569

A1capB1 = 04569 helliphelliphelliphelliphelliphelliphelliphelliphellip helliphelliphellip(2) From (1)and (2)

(AUB)1 = A1capB1

(AcapB)1 = [1237cap378]1 (AcapB)1 = 371

(AcapB)1 = 01245689helliphelliphelliphelliphelliphelliphelliphellip(1) A1UB1 = 12371U3781

A1UB1 = 045689U0124569

A1UB1 = 012345689helliphelliphelliphelliphelliphelliphellip(2)

From (1) and (2) (AcapB)1 = A1UB1



Cardinality of sets

Disjoint sets Non Disjoint set n(AUB) = n(A) + n(B) n(AUB) = n(A) + n(B) ndash n(AcapB)

A = 01234 there4 n(A) = 5 B = 56789 there4 n(B) = 5 AUB = 0123456789 there4 n(AUB) = 10 AcapB = there4 n(AcapB) = 0 n(AUB) = n(A) + n(B) ndash n(AcapB) 10 = 5 + 5 10 = 10

A = 01234 there4 n(A) = 5 B = 23456 there4 n(B) = 5 AUB = 0123456 there4 n(AUB) = 7 AcapB = 234 there4 n(AcapB) = 3 n(AUB) = n(A) + n(B) ndash n(AcapB) 7 = 5 + 5 ndash 3 7 = 10 -3 7 = 7

A group of 100 passengers 100 know Kannada 50 know English and 25 know both If the passenger know either Kannada or English How many passengers are in the group

n(AUB) = n(A) + n(B) ndash n(AcapB) A ndash Know Kannada B ndash Know English there4 n(A) = 100 n(B) = 50 n(AcapB) = 25 there4 n(AUB) = 100 + 50 ndash 25 there4 n(AUB) = 125

In a class 50 students offered Mathematics 42 offered Biology and 24 offered both the subjects Find the number of students who offer 1) Mathematics only 2) Biology only and also find the total number of students

n(AUB) = n(A) + n(B) ndash n(AcapB) A ndash Offer Mathematics B ndash Offer biology there4 n(A) = 50 n(B) = 42 n(AcapB) = 24 Total number of students = there4 n(AUB) = 50 + 42 ndash 24 = 68 Number of students Offer Mathematics only = 50-24 =26 Number of students Offer Biology only= 42-24=18



AB and BA

A = 123456 B = 14578 AB = 236 BA = 78

AUB AcapB A1 (AUB)capC



Acap(BUC) A1UB1 A1capB1 AB

Chapter3 Progressions(Total Marks-8)

SlNo Chapter MCQ 1-Mark 2-Marks 3-Marks 4-Marks Total K U A S K U A S K U A S 3 Progressions 1 1 1 8

Arithmetic rogression

Formularsquos

Standard form of Arithmetic Progression a a+d a+2d a+3dhelliphelliphelliphelliphellipa + (n-1)d a ndashFirst term d ndash Common difference nth term of AP Tn = a + (n ndash 1)d a ndashFirst termd- c d n ndash Number of terms (n+1)th term of AP Tn+1 = Tn + d d ndash Common difference (n-1)th term of AP Tn-1 = Tn ndash d d ndash Common difference Given a term in AP find another term Tp = Tq + (p-q)d Tq ndashGiven term d ndash Common difference

Find Common difference of AP d = 퐓퐩minus 퐓퐪퐩 minus 퐪

Tp and Tq ndashterms of AP d ndash Common difference

If [T = T and T = a ] d = 퐓퐧minus퐚 퐧minusퟏ

T ndashLast term a ndashFirst term n ndash Number of terms

Sum to nth term of an AP Sn = 퐧ퟐ

[ퟐ퐚 + (퐧 minus ퟏ)퐝] a ndashFirst term n ndash Number of terms d ndash Common difference

If first term (a) and last term ( Tn) Given Sn = 퐧ퟐ

[풂 + 푻풏] a ndashFirst term n ndash number of terms T ndashLast term

The Sum of first lsquonrsquo Natural numbers Sn = 풏(풏+ퟏ)ퟐ

n ndash Number of terms

NoteAn arithmetic is a sequence in which each term is obtained by adding a fixed number to the proceeding term (exept the first term)



The sum of first lsquonrsquo terms of an AP is equal to the everage of its first and last term SLNo Question Answer

1 Find the 3rd term of 2n + 3 T3 = 2x3 + 3 = 6 + 3 = 9 2 If Tn = 3n ndash 10 then the 20th term is T20 = 3x20 -10 = 60-10 =50

3 If Tn = n3 ndash 1 Tn = 26 then lsquonrsquo =

n3 ndash 1 = 26 n3 = 26 + 1 n3 = 27 n3 = 33

there4 n = 3

4 If Tn = 2n2 + 5 then T3 = T3 = 2x32 + 5 = 2x9 + 5 = 18+5 =23 5 If Tn = 5 ndash 4n then 3term is Tn = 5 ndash 4x3 = 5 ndash 12 = -7

6 If Tn = n2 ndash 1 then Tn+1 = Tn+1 = (n+1)2 ndash 1 =n2+2n+1-1 = n2+2n OR n(n+2)

7 If Tn = n2 + 1 then find S2 Tn = n2 + 1 T1 = 12 +1 = 2 T2 = 22 + 1 = 5 S2 = T1 + T2 = 2 + 5 = 7

Formula SlNo Questions Answer

Tn = a + (n ndash 1)d 1 Find the 15th term of 12 19 26helliphelliphelliphelliphellip T15 = 12 + (15 ndash 1)7 T15 = 12 + 14x7 T15 = 12+ 98 T15 = 110


Tn = a + (n ndash 1)d

2 Find the number of terms of the AP 71319 helliphelliphelliphellip151

a=7 d=6 Tn =151 n= 151 = 7 + (n ndash 1)6 151 = 7 + 6n ndash 6 151 = 6n + 1 6n = 151 ndash 1 6n = 150 n = = 25



Tn = a + (n ndash 1)d 3 If d = -2 T22 = -39 then find lsquoarsquo

d = -2 T22 = -39 n = 22 a = -39 = a + (22 ndash 1)-2 -39 = a + 21 x-2 -39 = a - 42 a = -39 + 42 a = 3

4 If a = 13 T15 = 55then find lsquodrsquo =

a = 13 T15 = 55 n=15 lsquodrsquo = 55 = 13 + (15 ndash 1)d 55 = 13 + 14d 14d = 55 ndash 13 14d = 42 d = d = 3

Sn = 퐧ퟐ

[ퟐ퐚 + (퐧 minus ퟏ)퐝] What is the sum of first 21 terms of 1 + 4 + 7 + helliphelliphelliphellip

n = 21 a = 1 d = 3Sn = S21 = [2x1 +(21-1)3]

S21 = [2 +20x3]

S21 = [2 +60] S21 = x62 S21 = 21x31 S21 = 651

Exercise 1)3 + 7 + 11 + ----------- Find the sum of first 15 terms

Exercise 2)2 + 5 + 8 + ----------------- -- Find the sum of first 25 terms

Exercise 3)3+ 5 + 7 + ------------find the sum of 30 terms



Sn = 퐧ퟐ

[퐚 + 퐓퐧] The First and 25th term of an AP is 4 and 76 respectively Find the sum of 25 terms

a = 4 Tn = 76 n = 25 Sn = S25 = 25

2[4 + 76]

S25 = 252

[80] S25 = 25x40 S25 = 1000

Sn = 풏(풏+ퟏ)ퟐ

Find the sum of all natural numbers from 1 to 201 which are divisible by 5 Exercise Find the sum of all natural numbers from 200 to 300 which are dividible by 6

5 + 10 + 15 + ------------- + 200 rArr5x1 + 5x2 + 5x3 + --------- + 5x 40 rArr5[1 + 2 + 3 + -----------------40] rArr5xS40 n = 40 rArr5x40(40+1)

2

rArr5x20x41 rArr 4100

Harmonic ProgressionA sequence in which the reciprocals of the terms from an arithmetic progression is called a harmonic progression n term of HP Tn = ퟏ

풂 + (풏 ndash ퟏ)풅 a ndashFirst term d ndash Common difference

n ndash Number of terms Tn = ퟏ

풂 + (풏 ndash ퟏ)풅 1

2 1

4 1

6 -------Find the 21st term

Exercise 1 -1-------Find the 10th term

T21 = ퟏퟐ + (ퟐퟏ ndash ퟏ)ퟐ

rArr ퟏퟐ + (ퟐퟎ)ퟐ

rArr ퟏ ퟐ + ퟒퟎ

rArr ퟏퟒퟐ

In HP T3 = 17 and

T7 = then Find T15

AnswerIn HP T3 = 17 T7 = 1

5

rArrIn AP T3 = 7 T7 = 5 d = Tpminus Tq

p minus q Tp = T7 = 5 Tq = T3 = 7

d = T7minus T37 minus 3

d = 5minus 77 minus 3

rArr d = minus24

rArr d = minus12

a + (n ndash 1)d = Tn rArr a + (7 ndash 1)x minus12

= T7 rArr a + 6xminus12

= 5

Exercise 1)In HP T5 = 1

12 and

T11 = 115

then FindT25

2)In HP T4 = 111

and

T14 = then find T7



rArr a ndash 3 = 5 rArr a = 8 there4 T15 = 8 + (15 ndash 1)xminus1

2

rArr T15 = 8 + (14)xminus12

rArr T15 = 8 ndash 7 rArrT15 = 1 there4 Reciprocal of the 15th term 1 = 1

Geometric Progression

Formulas

Standard form of GP a ar ar2 ar3helliphelliphelliphelliphelliparn-1 a ndashFirst term r ndash Common ratio nth term of GP Tn = a rn-1 a ndashFirst term r ndash Common ratio n ndash number of terms (n+1)th term Tn+1 = Tn xr r ndashCommon ratio (n-1)th term Tn-1 = 퐓퐧

퐫 r ndash Common ratio

Sum to nrsquoterm of GP Sn = 퐚 퐫퐧minusퟏ퐫minusퟏ

if r gt 1 a ndash First term n ndash number of terms r ndash Common ratio

Sum to nrsquoterm of GP Sn = 퐚 ퟏminus 퐫퐧

ퟏminus퐫 if r lt 1 a ndash First term n ndash number of terms r ndash Common ratio

Sum to nrsquoterm of GP Sn = 퐧퐚 if r = 1 a ndash First term n ndash number of terms

Sum to infinite series of GP 퐬infin = 퐚ퟏminus퐫

a ndash First term r ndash Common ratio

ಕ ಗಳ

Tn = a rn-1

If a = 4 and r = 2 then find the 3rd term of GP T3 = 4x 23-1

rArr T3 = 4x 22

rArr T3 = 4x 4

rArr T3 = 16

Tn = a rn-1 If first term is 3 and common ratio is 2 of the GP then find the 8th term

T8 = 3x 28-1

rArr T8 = 3x 27

rArr T8 = 3x 128

rArr T8 = 384



Tn+1 = Tn xr The 3rd term of GP is 18 and common ratio is 3 find the 4th term

T4 = T3x 3 rArr 18x3 = 54

Tn-1 = 퐓퐧퐫

The fifth term of a GP is 32common ratio is 2 find the 4th term T4= T5

r rArr T4= 32

2 = 16

Sn = 퐚 퐫퐧minusퟏ퐫minusퟏ

if r gt 1

1 + 2 + 4 +------10 Sum to 10th term

Exercise How many terms of the series 1 + 4 + 16+ ----

------make the sum 1365

a = 1 r = 2 S10=

S10 = 1 (210minus12minus1

)

S10 = 1 (1024minus11

) S10 = 1023

Sn = 퐚 ퟏminus 퐫퐧

ퟏminus퐫 if r lt 1 + + +--------------- find the sum of this

series

Sn = a ( 1minus rn

1minusr) a = 1

2 n = 10 r = 1

2

Sn = 12

[ 1minus( 12)10

1minus12

]

Sn = 12

[ 1minus 1

210

12]

Sn = 12

x 21

[1024minus11024

]

Sn = [10231024

]

퐬infin = 퐚ퟏminus퐫

Find the infinite terms of the series 2 + 23 + 2

9---

a = 2 r = 13

퐬infin = ퟐퟏminusퟏퟑ

= ퟐퟐퟑ

= 2x32 = 3

Find the 3 terms of AP whose sum and products are 21 and 231 respectively

Find the three terms of GP whose sum and product s are 21 and 216 respectively

Consider a ndash d a a + d are the three terms a ndash d + a + a + d = 21 3a = 21 a = 7 (a ndash d) a (a + d) = 231 (7 ndash d) 7 (7 + d) = 231

ar a ar - are the three terms

ar x a x ar = 216

a3 = 216 a = 6 6r + 6 + 6r = 21



(7 ndash d)(7 + d) = 2317

72 - d2 = 33 d2 = 49 ndash 33 d2 = 16 d = 4 Three terms 7-4 7 7+4 = 3 7 11

6r2 + 6r + 6 = 21r 6r2 - 15r + 6 = 0 6r2 ndash 12 -3r + 6 = 0 6r(r ndash 2) -3(r - 2) = 0 6r-3 = 0 or r ndash 2 = 0 r = 1

2 or r = 2

there4 Three terms - 3 6 12

Means

Arithmetic Mean Geometric Mean Harmonic Mean

A = 풂 + 풃ퟐ

G = radic풂풃 H = ퟐ풂풃풂+ 풃

If a A b are in AP A ndash a = b ndash A A + A = a + b 2A = a + b

A = 푎 + 푏2

If a G b are in GP G a

= bG

GxG = ab

G2 = ab G = radicab

If a H b are in HP then 1푎 1

H 1

b are in AP

1H

- 1푎 = 1

b - 1

H

1H

+ 1 H

= 1b

+ 1푎

1+1H

+ = a+bab

2H

+ = a+bab

rArr H = 2푎푏푎+푏



If 12 X 1

8 are in AP find the value of X

A = 푎 + 푏2

X = 12 +

18

2

X = 4+18 2

X = 58 2

rArr X = 516

The GM of 9 and 18 G = radic푎푏 G = radic9x18 G = radic162 G = radic81x2 G = 9radic2

If 5 8 X are in HP X = H = 2푎푏

푎+푏

8 = 25푥5+푥

8(5+x) = 10x 40 +8x = 10x 40 = 2x X = 20

Chapter 4 Permutation and Combination(5 marks)

SLNo Chapter MCQ 1-Mark 2-Marks 3-Marks 4-Marks Total K U A S K U A S K U A S 4 Permutation and

Combination 1 1 1 5

Fundamental principle of counting If one activity can be done in lsquomrsquo number of different ways and corresponding to each of these

ways of the first activitysecond activity(independent of first activity) can be done in (mxn) number of ways

Permutation Combination



5 different books are to be arranged on a shelf A committee of 5 members to be choosen from a group of 8 people

In a committee of seven persions a chairpersion a secretary and a treasurer are to be choosen

In a question paper having 12 questions students must answer the first 2 questions but may select any eight of the remaining ones

Forming 3 letters word from the letters of ARITHMETIC assuming that no letter is repeated

A box contains 5 black and 7 white balls The 3 balls to be picked in which 2 are black and is white

8 persions to be seated in 8 chairs A collection of 10 toys are to be divided equally between two children

How many 3 digit numbers can be formed using the digits 13579 without repeatation

The triangles and straight lines are to be drawn from joining eight points no three points are collinear

Five keys are to be arranged in a circular key ring Number of diagonals to be drawn in a polygon

Factorial notation n = n(n-1)(n-2)(n-3)helliphelliphelliphelliphelliphellip321 Note 0 = 1

Example 1x2x3x4x5x6 = 6 1x2x3x4x5x6x7x8x9x10 = 10 8 = 8x7x6x5x4x3x2x1


Formula nPr = 푛(푛minus푟)

nCr = 푛(푛minus푟)푟

The value of 7P3 is ExerciseFind the values of 1) 8P5 2) 6P3

7P3= 7(7minus3)

7P3= 7

4

7P3= 7x6x5x4x3x2x14x3x2x1

7P3= 7x6x5 7P3= 210

The value of 7C3 is ExerciseFind the vaues of

1) 8C5 2) 6C3

7C3 = 7(7minus3)3

7C3 = 7

43

7C3 = 7x6x53x2x1

7C3 = 210

6

7C3 = 35 nP0 = 1 nP1 = n nPn = n nPr = nCr xr nC0 = 1 nC1 = n nCn = 1 nCr = nCn-r

If nP2 = 90 then the value of lsquonrsquo n(n-1) = 90 10(10-1) =90 rArr n = 10

If nC2 = 10 then the value of lsquonrsquo

푛(푛minus1)2

= 10 rArr n(n-1) = 20 rArr 5(5-1) =20 rArr n = 5

If nPn=5040 then what is the value nPn=5040 If 6Pr = 360 and 6Cr = 15 6Pr = 6Cr x r



of nrsquo n = 5040 1x2x3x4x5x6x7 = 5040 rArr n = 7

then find the value of rrsquo 360 = 15xr r = 360

15

r = 24 = 4 rArr r = 4 If 11Pr =990 then the value of rrsquo is 11Pr =990

11 x 10 x 9 = 990 rArr r = 3 IfnP8 = nP12 then the value of lsquorrsquo

r = 8 + 12 = 20

Note The number of diagonals to be drawn in a polygon - nC2 -n

Some questions

Pemutation Combination

1 In how many ways 7 different books be arranged on a shelf such that 3 particular books are always together

5P5x3P3 1 How many diagonals can be drawn in a hexagon

6C2 -6

2 How many 2-digit numbers are there 10P2-9+9 2 10 friends are shake hand mutuallyFind the number of handshakes

10C2

3 1)How many 3 digits number to be formed from the digits 12356 2) In which how many numbers are even

1) 5P3 2) 4P2x2P1

3 There are 8 points such that any 3 of them are non collinear

a) How many triangles can be formed b) How many straight lines can be formed

1) 8C2 2) 8C3

4 LASER How many 3 letters word can be made from the letters of the word LASER without repeat any letter

5P3 4 There are 3 white and 4 red roses are in a garden In how many ways can 4 flowers of which 2 red b picked

3C2 x 4C2

Problems on Combination continued

1 There are 8 teachers in a school including the Headmaster 1) How many 5 members committee can be formed 2) With headmaster as a member 3) Without head master

1) 8C5 2) 7C4 3) 7C5

2 A committee of 5 is to be formed out of 6 men and 4 ladies In how many ways can this be done when a) At least 2 ladies are included b) at most 2 ladies are included

1) 6C3x4C2 +6C2x4C3 +6C1x4C4 2) 6C3x4C2 +6C4x4C1 +6C5x4C0



Chapter 5 Probability (Marks -3)

SLNo Chapter MCQ 1-Mark 2-Marks 3-Marks 4-Marks Total K U A S K U A S K U A S 5 Probability 1 1 3

Random experiment 1) It has more than one possible outcome 2) It is not possible to predict the outcome in advance Example 1) Tossing a coin 2) Tossing two coins at a time 3) Throwing a die Elementary events Each outcomes of the Random Experiment Example Two coins are tossed Sample space = HH HT TH TT ndash E1 = HH E2 =HT E3 = TH E4 = TT These are elementary events Compound events It is the association of two or more elementary events Example Two coins are tossed 1) Getting atleast one head ndash E1 = HT TH HH 2) Getting one head E2 = HT TH

The sample spaces of Random experiment

1 Tossing a coin S= H T n(S) = 2 2 Tossing two coins ata time or tossing a coin twice S = HH HT TH TT n(S) = 4 3 Tossing a coin thrice S = HHH HHT HTH THH TTH THT HTTTTT n(S) = 8 4 Throwing an unbiased die S = 1 2 3 4 5 6 n(S) = 6

5 Throwing two dice at a time

S = (11)(12)(13)(14)(15)(16)(21)(22)(23) (24) (25)(26)(31)(32)(33)(34)(35)(36)(41) (42)(43)(44)(45)(46)(51)(52)(53) (54)(55) (56)(61)(62) (63)(64)(65)(66)

n(S) = 36



Formula to find probability and some problems

P(A) = n(A)n(S)

1) Getting even numberswhen a die is thrown P(A) = 36

2)Getting headwhen a coin is tossed P(A) = 12

3)Getting atleast one head when a coin is tossed twice P(A) = 34

4)Getting all heads when a coin is tossed thrice P(A) = 18

5)Getting sum is 6 when two dice are thrown at a time P(A) = 536

Certain(Sure) event Impossible event Complimentary event Mutually exclusive event

The event surely occur in any trail of the experiment

An Event will not occur in any tail of the Random

experiment

An Event A occurs only when A1 does not occur and vice versa

The occurance of one event prevents the other

Probability= 1 Probability = 0 P(A1) = 1 ndash P(A) P(E1UE2) = P(E1) + P(E2) Getting head or tail when a coin is

tossed Getting 7 when a die is

thrown Getting even number and getting

odd numbers when a die is thrown

Getting Head or Tail when a coin is tossed

Note 1) 0le 퐏(퐀) le ퟏ 2) P(E1UE2) = P(E1) + P(E2) ndash P(E1capE2)

1 If the probability of winning a game is 03 what is the probability of loosing it 07 2 The probability that it will rain on a particular day is 064what is the probability that

it will not rain on that day 036

3 There are 8 teachers in a school including the HeadmasterWhat is the probability that 5 members committee can be formed a) With headmaster as a member b) Without head master

n(S) = 8C5 1) n(A) = 7C4 P(A) = 푛(퐴)

푛(푆)

2)n(B) =7C5 P(B) = 푛(퐵)푛(푆)



4 A committee of 5 is to be formed out of 6 men and 4 ladies What is the probility of the committee can be done a) At least 2 ladies are included b) at most 2 ladies are included

n(S) = 10C5

1) n(A) = 6C3x4C2 +6C2x4C3 +6C1x4C4 P(A) = 푛(퐴)

푛(푆)

2)n(B) = 6C3x4C2 +6C4x4C1 +6C5x4C0 P(B) = 푛(퐵)

푛(푆)

Chapter 6Statistics(4marks)

SLNo Chapter MCQ 1-Mark 2-Marks 3-Marks 4-Marks Total K U A S K U A S K U A S 6 Statistics 1 1 4

The formulas to find Standard deviation

Un grouped data

Direct method Acutal Mean Method Assumed Mean Method Step-Deviation Method

흈 = sum푿ퟐ

풏 ndash ( sum푿

풏)ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

The formulas to find Standard deviation Grouped data


흈 = sum풇풙ퟐ

풏 minus sum풇풙

풏

ퟐ 흈 = sum풇풅

ퟐ

풏 흈 = sum풇풅

ퟐ

풏 minus sum풇풅

풏

ퟐ 흈 = sum풇풅

ퟐ

풏 minus sum풇풅

풏

ퟐ 퐱퐂



For ungrouped data

Direct Method Actual Mean Method Assumed Mean Method Step deviation method x X2 x d=x-퐱 d2 x d=x - A d2 x X - A d = (퐱minus퐀)

퐂 d2

sumx= sumx2 = sumx= sumd2 = sumx= sumd= sumd2 = sumx= sumd= sumd2 =

Actual Mean 푿 = sum푿풏

For grouped data

Direct Method Actual Mean Method X f fx X2 fx2 X f fx d=X -

풙 d2 fd2

n = sumfx = sumfx2

= n= sumfx = sumfd2=

Actual Mean 푿 = sum 풇푿풏



Assumed Mean Method Step deviation MEthod

x f d=x-A fd d2 fd2 x f x-A d = (퐱minus퐀)퐂

fd d2 fd2

n = sumfd = sumfd2

= n= sumfd

= sumfd2=

For Ungrouped data Example

Direct Method Actual Mean Method Assumed Mean Method Step deviation Mehod x X2 x d=x-퐱 d2 x d=x - A d2 x X - A d = (퐱minus퐀)

퐂 d2

23 529 23 -11 121 23 -12 124 23 31 961 31 -3 9 31 -4 16 31 If data having common factorthen we use this

formula 32 1024 32 -2 4 32 -3 9 32 34 1156 34 0 0 34 -1 1 34 35 1225 35 1 1 35 0 0 35 36 1296 36 2 4 36 1 1 36 39 1521 39 5 25 39 4 16 39 42 1764 42 8 64 42 7 49 42

272 9476 272 228 -8 216 sumd= sumd2 =




rArr ퟐퟕퟐퟖ

=34 Assumed Mean 35

Direct Method Actual Mean Method Assumed Mean Method Step deviation Mehod

흈 = sum푿ퟐ

풏 ndash ( sum푿

풏)ퟐ 흈 =

sum퐝ퟐ

퐧

흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

흈 = ퟗퟒퟕퟔퟖ

ndash ( ퟐퟕퟐퟖ

)ퟐ

휎 = 11845 ndash 1156

휎 = radic285

휎 = radic285

휎 = 534

흈 = ퟐퟐퟖퟖ

흈 = radicퟐퟖퟓ

흈 = ퟓퟑퟒ

흈 =

ퟐퟏퟔퟖ

ndash ( ퟖퟖ

)ퟐ

흈 = ퟐퟕ ndash (minusퟏ)ퟐ

흈 = radicퟐퟕ + ퟏ

흈 = radicퟐퟖ

흈 = ퟓퟐퟗ

We use when the factors are equal

Direct Method Actual Mean Method CI f X fx X2 fx2 CI f X fx d=X - 푿 d2 fd2

1-5 2 3 6 9 18 1-5 2 3 6 -7 49 98 6-10 3 8 24 64 192 6-10 3 8 24 -2 4 12

11-15 4 13 52 169 676 11-15 4 13 52 3 9 36 16-20 1 18 18 324 324 16-20 1 18 18 8 64 64

10 100 1210 10 100 210



Assumed Mean Methdo Step Deviation Method CI f X d=x-A fd d2 fd2 CI f X x-A d = (퐱minus퐀)

퐂 fd d2 fd2

1-5 2 3 -10 -20 100 200 1-5 2 3 -10 -2 -4 4 8 6-10 3 8 -5 -15 25 75 6-10 3 8 -5 -1 -3 1 3

11-15 4 13 0 0 0 0 11-15 4 13 0 0 0 0 0 16-20 1 18 5 5 25 25 16-20 1 18 5 1 1 1 1

10 -30 300 10 -6 12

Actual mean 푿 = sum 풇푿풏

rArr ퟏퟎퟎퟏퟎ

rArr 푿 = 10 Assumed MeanA=13

Direct Method Actual Mean Method Assumed mean Method Step deviation Method

흈 = sum풇풙ퟐ

풏 minus sum풇풙

풏

ퟐ

흈 = ퟏퟐퟏퟎퟏퟎ

minus ퟏퟎퟎퟏퟎ

ퟐ

흈 = radic ퟏퟐퟏ minus ퟏퟎퟐ 흈 = radic ퟏퟐퟏ minus ퟏퟎퟎ 흈 = radic ퟐퟏ 흈 = ퟒퟔ

흈 = sum 풇풅ퟐ

풏

흈 = ퟐퟏퟎퟏퟎ

흈 = radic ퟐퟏ 흈 = ퟒퟔ

흈 = sum풇풅ퟐ

풏 minus sum풇풅

풏

ퟐ

흈 = ퟑퟎퟎퟏퟎ

minus minusퟑퟎퟏퟎ

ퟐ

흈 = ퟑퟎ minus (minusퟑ)ퟐ 흈 = radic ퟐퟏ 흈 = ퟒퟔ

흈 = sum풇풅ퟐ

풏 minus sum풇풅

풏

ퟐ 퐱퐂

흈 = ퟏퟐퟏퟎ

minus minusퟔퟏퟎ

ퟐ 퐱ퟓ

흈 = ퟏퟐ minus (minusퟎퟔ)ퟐ 퐱ퟓ

흈 = ퟏퟐ ndashퟎퟑퟔ 퐱ퟓ

흈 = radic ퟎퟖퟒ 퐱ퟓ 흈 = ퟎퟗퟏx 5 흈 = 455



Coefficient of variation CV= 푺풕풂풏풅풂풓풅 푫풆풗풊풂풕풊풐풏

푴풆풂풏x 100 rArr CV = 훔

퐗x100

Some problems on Statisticcs

Find the standard deviation for the following data 1 9 12 15 18 20 22 23 24 26 31 632 2 50 56 59 60 63 67 68 583 3 2 4 6 8 10 12 14 16 458 4 14 16 21 9 16 17 14 12 11 20 36 5 58 55 57 42 50 47 48 48 50 58 586

Find the standard deviation for the following data Rain(in mm) 35 40 45 50 55 67 Number of places 6 8 12 5 9

CI 0-10 10-20 20-30 30-40 40-50 131 Freequency (f) 7 10 15 8 10

CI 5-15 15-25 25-35 35-45 45-55 55-65 134 Freequency (f) 8 12 20 10 7 3



Find the standard deviation for the following data Marks 10 20 30 40 50 푥 =29

휎 = 261 CV=4348

Number of Students 4 3 6 5 2

How the

students come to school

Number of students

Central Angle

Walk 12 1236

x3600 = 1200

Cycle 8 836

x3600 = 800 Bus 3 3

36x3600 = 300

Car 4 436

x3600 = 400 School Van 9 9

36x3600 = 900

36 3600

Chapter 6Surds(4 Marks) SLNo Chapter MCQ 1-Mark 2-Marks 3-Marks 4-Marks Total K U A S K U A S K U A S

7 Surds 2 4

Addition of Surds Simplify 4radic63 + 5radic7 minus 8radic28 4radic9x 7 + 5radic7 minus 8radic4x7

= 4x3radic7 + 5radic7 - 8x2radic7



Addition of Surds

= 12radic7 + 5radic7 - 16radic7 = (12+5-16)radic7 = radic7

Simplify 2radic163 + radic813 - radic1283 +radic1923

2radic163 + radic813 - radic1283 +radic1923 =2radic8x23 + radic27x33 - radic64x23 +radic64x33 =2radic8x23 + radic27x33 - radic64x23 +radic64x33 =4radic23 +3 radic33 -4 radic23 +4 radic33 =(4-4)radic23 +(3+4) radic33 =7radic33

Exercise 1Simplifyradic75 + radic108 - radic192

Exercise 2Simplify4radic12 - radic50 - 7radic48

Exercise 1Simplifyradic45 - 3radic20 - 3radic5

NOTE The surds having same order and same radicand is called like surds Only like surds can be added and substracted We can multiply the surds of same order only(Radicand can either be same or different)

Simplify Soln Exercise

radic2xradic43 radic2 = 2

12 rArr 2

12x3

3 rArr 236 rArr radic236 rArr radic86

radic43 = 413 rArr 4

13x2


radic86 xradic166 = radic1286

1 radic23 x radic34 2 radic5 x radic33 3 radic43 xradic25



(3radic2 + 2radic3 )(2radic3 -4radic3 )

(3radic2 + 2radic3 )(2radic3 -4radic3 ) =(3radic2 + 2radic3 ) 2radic3 minus(3radic2 + 2radic3 ) 4radic3 =3radic2X2radic3 +2radic3 X2radic3 -3radic2X4radic3 -2radic3 X4radic3 =6radic6 + 4radic9 - 12radic6 -8radic9 =6radic6 + 4x3 - 12radic6 -8x3 =radic6 + 12 - 12radic6 -24 =-6radic6 -12

1 (6radic2-7radic3)( 6radic2 -7radic3) 2 (3radic18 +2radic12)( radic50 -radic27)

Rationalising the denominator 3

radic5minusradic3

3radic5minusradic3

xradic5+radic3radic5+radic3

= 3(radic5+radic3)(radic5)2minus(radic3)2

= 3(radic5+radic3)2

1 radic6+radic3radic6minusradic3


3 3 + radic6radic3+ 6

4 5radic2minusradic33radic2minusradic5

Chapter 8 Polynomials(4 Marks)

SlNo Chapter MCQ 1-Mark 2-Marks 3-Marks 4-Marks Total K U A S K U A S K U A S 8 Polynomials 1 1 1 4



Problems Soln Exercise

The degree of the polynomial 푥 +17x -21 -푥 3 The degree of the polynomial 2x + 4 + 6x2 is

If f(x) = 2x3 + 3x2 -11x + 6 then f(-1) f(-1) = 2(-1)3 + 3(-1)2 ndash 11(-1) + 6 = -2 + 3 + 11 +6 = 18

1 If x = 1 then the value of g(x) = 7x2 +2x +14

2 If f(x) =2x3 + 3x2 -11x + 6 then find the value of f(0)

Find the zeros of x2 + 4x + 4

X2 + 4x + 4 =x2 + 2x +2x +4 =(x + 2)(x+2) rArrx = -2 there4 Zero of the polynomial = -2

Find the zeros of the following 1 x2 -2x -15 2 x2 +14x +48 3 4a2 -49

Find the reminder of P(x) = x3 -4x2 +3x +1 divided by (x ndash 1) using reminder theorem

P(x) =12 ndash 4 x 1 + 3 x 1 = 1 =1 - 4 + 3 + 1 = 1

Find the reminder of g(x) = x3 + 3x2 - 5x + 8 is divided by (x ndash 3) using reminder theorem

Show that (x + 2) is the factor of (x3 ndash 4x2 -2x + 20)

If (x + 2) is the factor of p(x) = (x3 ndash 4x2 -2x + 20) then P(-2) =0 P(-2)= (-2)3 ndash 4(-2)2 ndash 2(-2) +20 = -8 -16 + 4 + 20 = 0 there4(x + 2) is the factor of (x3 ndash 4x2 -2x + 20)

1 (x ndash 2) ಇದು x3 -3x2 +6x -8

ೕ ೂೕ ಯ ಅಪವತ ನ ಂದು

ೂೕ

Divide 3x3 +11x2 31x +106 by x-3 by Synthetic division

Quotient = 3x2 +20x + 94 Reminder = 388

Find the quotient and the reminder by Synthetic division

1 (X3 + x2 -3x +5) divide (x-1) 2 (3x3 -2x2 +7x -5)divide(x+3)

Note Linear polynomial having 1 zero Quadratic Polynomial having 2 zeros



Chapter 9 Quadratic equations(Marks 9)

SlNo Chapter MCQ 1-Mark 2-Marks 3-Marks 4-Marks Total K U A S K U A S K U A S 9 Quadratic equations 1 1 1 9

Standard form ax2 + bx + c = 0 x ndash variable a b and c are real numbers a ne 0

In a quadratic equation if b = 0 then it is pure quadratic equation

If b ne 0 thenit is called adfected quadratic equation

Pure quadratic equations Adfected quadratic equations Verify the given values of xrsquo are the roots of the quadratic equations or not

x2 = 144 x2 ndash x = 0 x2 + 14x + 13 = 0 (x = -1) (x = -13)

4x = 81푥

x2 + 3 = 2x 7x2 -12x = 0 ( x = 13 )

7x = 647푥

x + 1x = 5 2m2 ndash 6m + 3 = 0 ( m = 1

2 )

Solving pure quadratic equations

If K = m푣 then solve for lsquovrsquo and find the value of vrsquo when K = 100and m = 2

K = 12m푣2

푣2=2퐾푚

v = plusmn 2퐾푚

K = 100 m = 2 there4 v = plusmn 2x100

2

there4 v = plusmn radic100 there4 v = plusmn 10

ಅ ಾ ಸ 1 If r2 = l2 + d2 then solve for drsquo

and find the value of drsquo when r = 5 l = 4

2 If 푣2 = 푢2 + 2asthen solve for vrsquo and find the value of vrsquo when u = 0 a = 2 and s =100 ಆದ lsquovrsquo ಯ ಕಂಡು



Roots of the Quadratic equation ( ax2 + bx + c = 0) are 풙 = 풃plusmn 풃ퟐ ퟒ풂풄ퟐ풂

Solving the quadratic equations

Facterisation Method Completing the square methood Solve using formula

3x2 ndash 5x + 2 = 0

3x2 ndash 5x + 2 = 0

3x2 ndash 3x - 2x + 2 = 0 3x(x -1) ndash 2 (x ndash1) = 0 (x-1)(3x-2) = 0 rArrx - 1 = 0 or 3x ndash 2 = 0 rArr x = 1 or x = 2

3

3x2 ndash 5x + 2 = 0 hellipdivide(3) x2 ndash 5

3x = minus ퟐ

ퟑ

x2 - 53x = - 2

3

x2 - 53x +(5

6)2 = minus 2

3 + (5

6)2

(푥 minus 5 6

)2 minus 2436

+ 2536

(푥 minus 5 6

)2 = 136

(푥 minus 5 6

) = plusmn 16

x = 56 plusmn 1

6 rArr x = 6

6 or x = 4

6

rArr x = 1 or x = 23

3x2 ndash 5x + 2 = 0 a=3 b= -5 c = 2

푥 =minus(minus5) plusmn (minus5)2 minus 4(3)(2)

2(3)

푥 =5 plusmn radic25 minus 24

6

푥 =5 plusmn radic1

6

푥 =5 plusmn 1

6

푥 = 66 or x = 4

6

x = 1 or x = 23

ퟏퟐ of the coefficient of lsquob is to be added both side of the quadratic equation



Exercise


6x2 ndash x -2 =0 x2 - 3x + 1 =0 x2 ndash 4x +2 = 0 x2 ndash 15x + 50 = 0 2x2 + 5x -3 = 0 x2 ndash 2x + 4 = 0

6 ndash p = p2 X2 + 16x ndash 9 = 0 x2 ndash 7x + 12 = 0

b2 ndash 4ac determines the nature of the roots of a quadratic equation ax2 + bx + c = 0 Therefor it is called the discriminant of the quadratic equation and denoted by the symbol ∆

∆ = 0 Roots are real and equal ∆ gt 0 Roots are real and distinct ∆ lt 0 No real roots( roots are imaginary)

Nature of the Roots

Discuss the nature of the roots of y2 -7y +2 = 0

∆ = 푏2 ndash 4푎푐 ∆ = (minus7)2 ndash 4(1)(2) ∆ = 49ndash 8 ∆ = 41 ∆ gt 0 rArrRoots are real and distinct

Exercise 1 x2 - 2x + 3 = 0 2 a2 + 4a + 4 = 0 3 x2 + 3x ndash 4 = 0

Sum and Product of a quadratic equation

Sum of the roots m + n =

ಮೂಲಗಳ ಗುಣಲಬ m x n =

Find the sum and product of the roots of the Sum of the roots (m+n) = minus푏

푎 = minus2

1 = -2 Exercise Find the sum and product of



equation x2 + 2x + 1 = 0 Product of the roots (mn) = 푐푎 = 1

1 = 1

the roots of the following equations 1 3x2 + 5 = 0 2 x2 ndash 5x + 8 3 8m2 ndash m = 2

Forming a quadratic equation when the sum and product of the roots are given

Formula x2 ndash (m+n)x + mn = 0 [x2 ndash (Sum of the roots)x + Product of the roots = 0 ]

Form the quadratic equation whose roots are 3+2radic5 and 3-2radic5

m = 3+2radic5 n = 3-2radic5 m+n = 3+3 = 6 mn = 33 - (2radic5)2 mn = 9 - 4x5 mn = 9 -20 = -11 Quadratic equation x2 ndash(m+n) + mn = 0 X2 ndash 6x -11 = 0

ExerciseForm the quadratic equations for the following sum and product of the roots

1 2 ಮತು 3

2 6 ಮತು -5

3 2 + radic3 ಮತು 2 - radic3

4 -3 ಮತು 32

Graph of the quadratic equation

y = x2 x 0 +1 -1 +2 -2 +3 -3 1 Draw the graph of y = x2 ndash 2x

2 Draw the graph of y = x2 ndash 8x + 7 3Solve graphically y = x2 ndash x - 2 4Draw the graphs of y = x2 y = 2x2 y = x2 and hence find the values of radic3radic5 radic10

y

y = 2x2 x 0 +1 -1 +2 -2 +3 -3

y

y =ퟏퟐx2

x 0 +1 -1 +2 -2 +3 -3

y



Y=2x2 ನ ನ y = x2 ನ ನ y = ퟏퟐ풙ퟐ ನ ನ

Details of Solving Quadratic equation by graph is given in GET 12 WITH SKILL ndash Exercise Papers 1 to 10



10Similar triangles

ಕಮ ಸಂ ಅ ಾಯ MCQ 1-ಅಂಕ 2-ಅಂಕ 3-ಅಂಕ 4-ಅಂಕ ಒಟು

K U A S K U A S K U A S 10 ಸಮರೂಪ ಭುಜಗಳ 1 1 1 6

If two triangles are similar Their corresponding angles are equal or Their corresponding sides are proportional

In the fig angA =angDangB=angEangC= angF Or 퐴퐵

퐷퐸= 퐴퐶퐷퐹

= 퐵퐶퐸퐹

there4 ∆ABC ~ ∆DEF

1 If ∆ABC ಯ XY BC XY = 3cmAY = 2cmAC = 6cm then BC

2 At a certain time of the daya pole10m heightcasts his shadow 8m long Find the length of the shadow cast by a building

nearby 110m highat the same time 3 At a certain time of the daya man6ft tallcasts his shadow 8ft long Find the length of the shadow cast by a building nearby 45ft

highat the same time 4



4 ∆ABC ಯ DE BC AD=57cmBD=95cmEC=6cmAE=

5 In ∆ABC DE BC퐴퐷퐷퐵

=23 AE=37 find

EC

6 In ∆ABC ಯ DE ABAD =7cm CD= 5cm and BC=18cm find BE and CE

Theorem -1( Thales theorem If a straight line is drawn parallel to a side of a trianglethen it divides the other two sides proportionally Given ∆ABC ಯ DEBC

To prove ADDB

= AEEC

Construction 1 Join DE and EB 2Draw EL ⟘ AB and DN⟘ AC

Proof ∆ABC∆BDE

= 12 12

xADxELxDBxEL

[∵ A = 12

xbxh

∆ABC∆BDE

= ADDB

∆ADE∆CDE

= 12 12

xAExDNxDBxDN

[∵ A = 12

xbxh

∆ADE∆CDE

= AEEC

there4 퐀퐃

퐃퐁 = 퐀퐄

퐄퐂 [∵∆BDE equiv ∆퐶퐷퐸



Theorem ldquo If two triangles are equiangularthen their corresponding sides are proportionalrdquo

Given In ∆ABC and ∆DEF ( i) angBAC = angEDF (ii) angABC = angDEF To prove AB

DE = BC

EF = CA

FD

Construction i) Mark points Grsquo and Hrsquo on AB and AC such that ProofIn ∆AGH and ∆DEF AG = DE [ ∵ Construction angBAC = angEDF [ ∵ Given AH = DF [ ∵ Construdtion there4 ∆AGH equiv ∆DEF [ ∵ SAS postulates there4 angAGH = angDEF [∵ Corresponding angles] ಆದ angABC = angDEF [ ∵ Given rArr angAGH = angABC [ ∵ Axioms there4 GH BC

there4 ABAG

= BCGH

= CA HA

[∵ converse of thales Theorem

there4 퐀퐁퐃퐄

= 퐁퐂퐄퐅

= 퐂퐀 퐅퐃

[∵ ∆AGH equiv ∆DEF



Theorem ldquoThe areas of similar triangles are proportional to squares on the corresponding sidesrdquo

Given ∆ABC ~ ∆DEF ABDE

= BCEF

= CA DF

To prove Area of ∆ABCArea of ∆DEF

= 퐁퐂ퟐ

퐄퐅ퟐ

Construction Draw AL ⟘ BC and DM ⟘ EF Proof In ∆ALB and ∆DME angABL = angDEM [ ∵ Given

angALB = angDME = 900 [ ∵ Construction ∆ALB ~ ∆DME [∵AA criteria rArr AL

DM = AB

DE and BC

EF = AB

DE [ ∵ Given

there4 ALDM

= BCEF

helliphellip(1)

Area of ∆ABCArea of ∆DEF

= 1212

xBCxALxEFxDM

rArr Area of ∆ABCArea of∆DEF

= BCxALEFxDM

[ ∵ ( 1)

= BCxBCEFxEF

= 퐁퐂ퟐ

퐄퐅ퟐ

But ABDE

= BCEF

= CA DF

[ ∵ Given

there4 Area of ∆ABCArea of ∆DEF

= AB2

DE2 = BC2

EF2 = CA2

DF2



11Phythagoras Theorem- (4 Marks)

SLNo Chapter MCQ 1-Mark 2-Marks 3-Marks 4-Marks Total K U A S K U A S K U A S 11 Phythagoras Theorem 1 4

TheoremPhythagoras Therem In a right angled trianglethe square of the hypotenuse is equal to the sum of the square of the other two sides Given ∆ABC In which angABC = 900 To Prove AB2 + BC2 = CA2 Construction Draw BD ⟘ AC Proof In ∆ABC and ∆ADB angABC = angADB = 900 [ ∵ Given and Construction angBAD =angBAD [∵ Common angle there4 ∆ABC ~ ∆ADB [∵ AA criteria

rArr ABAD

= ACAB

rArr AB2 = ACADhelliphellip(1) In ∆ABC and ∆BDC angABC = angBDC = 900 [ ∵ Given and construction angACB = angACB [∵ Common angle there4 ∆ABC ~ ∆BDC [∵ AA criteria

rArr BCDC

= ACBC

rArr BC2 = ACDChelliphellip(2) (1) + (2) AB2+ BC2 = (ACAD) + (ACDC) AB2+ BC2 = AC(AD + DC) AB2+ BC2 = ACAC AB2+ BC2 = AC2 [ ∵AD + DC = AC]



Converse of Phythagoras Theorem In triangleIf a square of a side is equal to the sum of the squares of the other two sidesthen it will be a reight angled triangle Given In the ∆ABC AB2+ BC2 = AC2 To prove angABC = 900 Construction At B draw AB⟘BC extend BC to D such that DB = BC Join lsquoArsquo and lsquoDrsquo Proof ∆ABD ಯ angABC = 900 [ ∵ Construction there4 AD2 = AB2 + BC2 [∵Phythagoras theorem But In ∆ABC AC2 = AB2 + BC2 [ ∵ Given

rArr AD2 = AC2 there4 AD = AC In ∆ABD and ∆ABC AD = AC [ ∵ Proved BD = BC [ ∵ Construction AB = AB [ ∵ Common ∆ABD equiv ∆ABC [ ∵ SSS Axiom rArr angABD = angABC But angABD +angABC =1800 [ ∵ BDC is straight line rArr angABD = angABC = 900



12Trigonometry

SlNo Chapter MCQ 1-Mark 2-Marks 3-Marks 4-Marks Total K U A S K U A S K U A S 12 Trigonometry 1 1 1 6

Trigonometric Ratios

휽

Note 흅 = ퟏퟖퟎ0

퐬퐢퐧 휽 = ퟏ퐜퐨퐬퐜 휽

퐭퐚퐧휽 = 퐬퐢퐧 휽퐜퐨퐬 휽

퐜퐨퐬휽 = ퟏ

퐬퐞퐜 휽

퐭퐚퐧휽 = ퟏ퐜퐨퐭 휽

퐜퐨퐭 휽 =퐜퐨퐬 휽퐬퐢퐧휽

퐬퐢퐧 휽 푶풑풑풐풔풊풕풆푯풚풑풐풕풆풏풖풔풆

퐴퐵퐴퐶

퐬퐢퐧(ퟗퟎ minus 휽) = 퐜퐨퐬휽

퐜퐨퐬휽 푨풅풋풖풄풆풏풕푯풚풑풐풕풆풏풖풔풆

퐵퐶퐴퐶 퐜퐨퐬(ퟗퟎ minus 휽) = 퐬퐢퐧휽

퐭퐚퐧휽 푶풑풑풐풔풊풕풆푨풅풋풖풄풆풏풕

퐴퐵퐵퐶 퐭퐚퐧(ퟗퟎ minus 휽) = 퐜퐨퐭 휽

퐜퐨퐬풆퐜 휽 푯풚풑풐풕풆풏풖풔풆푶풑풑풐풔풊풕풆

퐴퐶퐴퐵 퐜퐨퐬퐞퐜(ퟗퟎ minus 휽 )= 퐬퐞퐜 휽

퐬퐞퐜휽 푯풚풑풐풕풆풏풖풔풆푨풅풋풂풄풆풏풕

퐴퐶퐵퐶 퐬퐞퐜(ퟗퟎ minus 휽) = 퐜퐨퐬퐞퐜 휽

퐜퐨퐭 휽 푨풅풋풂풄풆풏풕푶풑풑풐풔풊풕풆

퐵퐶퐴퐵 퐜퐨퐭(ퟗퟎ minus 휽) = 퐭퐚퐧휽



Values 00 300 450 600 900

퐬퐢퐧 휽 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1

퐜퐨퐬휽 1 radicퟑퟐ

ퟏradicퟐ

ퟏퟐ 0

퐭퐚퐧휽 0 ퟏradicퟑ

1 radicퟑ ND

퐜퐬퐜 휽 ND 2 radicퟐ ퟐradicퟑ

1

퐬퐞퐜 휽 1 ퟐradicퟑ

radicퟐ 2 ND

퐜퐨퐭 휽 ND radicퟑ 1 ퟏradicퟑ

0

Trigonometric identities 퐬퐢퐧ퟐ 휽+ 퐜퐨퐬ퟐ 휽 = 1 ퟏ + 풄풐풕ퟐ휽 = 풄풐풔풆풄ퟐ 휽 퐭퐚퐧ퟐ 휽 + 1 = 퐬퐞퐜ퟐ 휽

If sin 휃 = write the remaining ratio

In ∆ABC angABC = 900

there4 BC2 = 132 ndash 52 = 169 ndash 25 = 144 there4 BC = 12 rArrcos휃 =12

13 tan 휃 = 5

12

Cosec휃 = 135

sec휃 = 1312

cot휃 = 125

What is the value of tan2600 + 2tan2450

tan600 = radic3 tan450= 1 there4 tan2600 + 2tan2450 = (radic3)2+ 2 x 12

rArr 3+2 = 5



Exercise 1 Write all the trigonometric ratios

2 Find the value of 퐜퐨퐬퐞퐜 ퟔퟎ0 - 퐬퐞퐜 ퟒퟓ0 +퐜퐨퐭 ퟑퟎ0 3 Find the value of 퐬퐢퐧ퟐ 흅

ퟒ + 풄풐풔 ퟐ 흅

ퟒ - 퐭퐚퐧ퟐ 흅

ퟑ

13Coordinate Geometry(4 Marks)

SlNo Chapter MCQ 1-Mark 2-Marks 3-Marks 4-Marks Total K U A S K U A S K U A S 13 Coordinate Geometry 2 1 4

Inclination The angle formed by a positive direction with X- axis Represented by 휃

If the Slope of a line 1radic3

then the inclination ----- tan휃 = 1

radic3

tan300= 1radic3

rArr 휃 = 300



Slope The ratio of the vertical distance to the horizontal distance is called slope Slope = 푉푒푟푡푖푐푎푙 퐷푖푠푡푎푛푐푒

퐻표푟푖푧표푛푡푎푙 푑푖푠푡푎푛푐푒 = 퐵퐶

퐴퐵

= Gradient m = tan휃

The slope of a line whose inclination is 600---- m = tan휃 m = tan600 m = radic3

Slope of a line passing throw two given points tan휃 = 푦2minus 푦1

푥2minus푥1

A(x1y1) and B(x2y2)

Find the slope of a line joining the points (3-2) and (45) tan 휃 = 푦2minus 푦1

푥2minus푥1

tan 휃 = 5minus(minus2)4minus3

tan 휃 = 7



Parallel lines have equal slopes 푡푎푛 휃1 = tan휃2 m1 = m2

m1 = Slope of AB m1 = Slope of AC

Find whether the lines drawn through the points (52)(05) and(00)(-53) parallel or not m1 = tan휃 = 푦2minus 푦1

푥2minus푥1

m1 = 5minus20minus5

= 3minus5

m2 = 3minus0minus5minus0

= 3minus5

there4 m1 = m2 there4 Lines are parallel

Slope of mutually perpendicular lines m1 = m2

m1 = slope of AB m1 = slope of AC

휃 훼

Verify whether the line through the points (45)(0-2) and (2-3)(-51) are parallel or mutually perpendicular m1 = tan휃 = 푦2minus 푦1

푥2minus푥1

m1 = minus2minus50minus4

= minus7minus4

= 74

m2 = 1minus(minus3)minus5minus2

= 4minus7

m1 x m2 = 74 x 4

minus7 = -1

there4 Line are mutually perpendicular



The equation of a line with slope lsquomrsquo and whose

y-intercept is lsquocrsquo is given by y = mx +c

The slope of a line is 12 and

y ndash intercept is -3 Find the equation m = 1

2 c = -3

there4 y = mx + c y = 1

2x -3rArr2y = x -6

rArr x -2y -6 =0

The distance between two points d = (푥 minus 푥 ) + (푦 minus 푦 )

Find the distance between the points(23) and (66) d = (푥2 minus 푥1)2 + (푦2 minus 푦1)2 d = (6 minus 2)2 + (6 minus 3)2 d = radic42 + 32 d = radic16 + 9 rArrd = radic25 d = 5units

Distance of a point in a plan from the Origin d = 푥2 + 푦2

Find the distance between the point (12-5) and the Origin d = 푥2 + 푦2 d = 122 + (minus5)2 d = radic144 + 25 rArr d = radic169 d = 13 Units



The Point P(xy) divides the line AB joining the points A(x1y1) and B(x2y2) in the ratio mnThen the coordinates of P(xy) is P (xy) = [푚푥2+푚푥1

푚+푛푚푦2+푚푦1

푚+푛]

If mn = 11 P (xy) = this is called the Mid-Point formula

Find the coordinates of the midpoint of a line segment joining the points (23) and (47) Coordinates of the Midpoint = [푥2+푥1

2 푦2+푦1

2]

= [4+22

7+32

]

= [62

102

] = (35)

Exercise 1 The slope of the line of inclination 450 ------- The inclination of a line having slope 1--------- Find the slope of a line joining the points (4-8) and(5-2) Verify whether the lines passing through the points(47)(35) and (-16)(17) are parallel or perpendicular Write the equation of a line of inclination 450 and y ndash intercept is 2 Find the distance between the points(28) and (68) Find the distance from the origin to a point (-815) If a point P divides the line joining the points (4-5) and(63) in the ratio 25 then find the cocordinates of P Find the coordinates of the midpoint of a line segment joining the points (-310) and (6-8)



14amp15Circles ndash Chord-Tangent properties

SlNo Chapter MCQ 1-Mark 2-Marks 3-Marks 4-Marks Total K U A S K U A S K U A S 14amp15 Circles and its properties 1 1 1 1 10

Draw a circle of radius 3cm and construct a chord of length 5cm Draw a circle of radius 22cm and construct a chord of length 4cm in it Mesure the distance between the centre and the

chord Construct two chords of length 9cm and 7cm on either side of the centre of a circle of radius 5cm

Note

Equal chods of a circle are equidistance from the centre If the chords of a circle are at equal distance from the centre then they are equal length If the length of the chord increasesits perpendicular distance from the centre discreases If the length of the chord decreasesits perpendicular distance from the centre increases The largest chord always passing through the centre(Diametre) All angles in the same segments are equal Angles in the minor segment are abtuse angles Angles in the major segment are acute angles Circles having the same centre but different radii are called concentric circles Circles having same radii but different centres are called congruent circles A straight line which intersects a circle at two distinct points is called a Secant A straight line which touches the circle at only one point is called Tangent In any circle the radius drawn at the point of contact is perpendicular to the tangent In a circle the perpendicular to the radius at its non-centre end is the tangent to the circle Only two tangents can be drawn from an external poit to a circle Tangents drawn from an external point to a circle are equal Two circles having only one common point of contact are called touching circles



If two circles touch each other externally the distance between their centres is d = R + r ( Rampr Radius) If two circles touch each other internally the distance between their centres is d = R - r ( Rampr Radius) If both the cicles lie on the same side of a common tangent then the common tangent is called Direct

common tangent(DCT) If both the circles lie on either side of a common tangent then the common tangent is called Transverse

common tangent(TCT) Three common tangents can be drawn to the circles touches externally Only one common tanget can be drawn to the circles touches internally





Steps of construction are given in GET 12 WITH SKILL



1 Construct a tangent at any point on a circle of radius 4cm 2 Draw a circle of radius 45cm and construct a pair of tangents at the non-centre end of two radii such that the

angle between the is 700 3 Draw a circle of radius 3cm and construct a pair of tangents such that the angle between them is 400 4 In a circle of radius 35 cm draw a chord of 5cmConstruct tangents at the end of the chord 5 Draw a circle of radius 5cm and construct tangents to it from an external point 8cm away from the centre 6 Draw a pair of tangents to a circle of radius 4cmfrom an external point 4cm away from the circle 7 Construct two direct common tangents to two circles of radii 4cm and 3cm and whose centres are 9cm

apart 8 Construct two tranverse common tangents to two circles of radii 45cm and 3cm and their centres are 95 cm

apart



Theorem The tangent drawn from an external point to a circle

(a) are equal (b) subtend equal angles at the centre (c) are equally inclined to the line joining the centre and the external point GivenA is the centreB is an external point BP and BQ are the tangentsAP AQ and AB are joined To prove (a) BP = BQ (b) angPAB = angQAB (c) angPBA = angQBA Proof In ∆APB and ∆AQB AP = AQ [ ∵ Radius of the same circle angAPB = angAQB =900 [ ∵ Radius drawn at the point of contact is perpendicular to the tangent ಕಣ AB = ಕಣ AB there4 ∆APB equiv ∆AQB [ ∵ RHS postulates there4 (a) BP = BQ (b) angPAB = angQAB [ ∵ CPCT (c) angPBA = angQBA

Theorem

If two circles touch each other the centres and the point of contact are collinear



Case-1) If two circles touch each other externally thecentres and the point of contact are collinear GivenA and B are the centres of touching circles P is the point of contact To prove APand B are collinear Construction Draw the tangent XPY ProofIn the figure angAPX = 900helliphelliphelliphelliphellip(1) ∵Radius drawn at the point of contact is angBPX = 900 helliphelliphelliphellip (2) perpendicular to the tangent angAPX + angBPX = 900 +900 [ by adding (1) and (2) angAPB = 1800 [ APB is a straight line there4 APB is a straight line there4 A P andB are collinear Theorem

Case-2 ) If two circles touch each other internally the centres and the point of contact are collinear GivenA and B are centres of touching circles P is point of contact To prove APand B are collinear Construction Draw the common tangent XPY Join AP and BP ProofIn the figure angAPX = 900helliphelliphelliphelliphellip(1) ∵Radius drawn at the point of contact angBPX = 900 helliphelliphelliphellip (2) is perpendicular to the tangent angAPX = angBPX = 900 [ From (1) and (2) AP and BP lie on the same line there4 APB is a straight line there4 A P and B are collinear



16Mensuration(5 Marks) Slno Chapter MCQ 1-Mark 2-Marks 3-Marks 4-Marks Total K U A S K U A S K U A S

16 Mensuration 1 1 1 5

Formulas

Name of the solid Curved surface area Total surface area Volume Cylinder ퟐ흅풓풉 ퟐ흅풓(풓+ 풉) 흅풓ퟐ풉

Cone 흅풓풍 흅풓(풓 + 풍) ퟏퟑ흅풓

ퟐ풉

Sphere ퟒ흅풓ퟐ ퟒ흅풓ퟐ ퟒퟑ흅풓

ퟑ

Hemisphere ퟑ흅풓ퟐ ퟐ흅풓ퟐ ퟐퟑ흅풓

ퟑ

흅 = ퟐퟐퟕ

풓 minus 푹풂풅풊풖풔 풍 minus 푺풍풂풏풕 풉풊품풉풕 풍 = radic풓ퟐ + 풉ퟐ

Volume of a frustum of a cone = ퟏퟑ흅풉(풓ퟏퟐ + 풓ퟐퟐ + 풓ퟏ풓ퟐ)



Find the curved surface area Total surface area and volume of a cylinderconesphere and hemisphere having hight= 10cm and diameter of the Base = 14 cm

d =14cm

r= 7cm

흅 = ퟐퟐퟕ

h=10cm

l= 풓ퟐ + 풉ퟐ

l= ퟕퟐ + ퟏퟎퟐ

l=radicퟒퟗ+ ퟏퟎퟎ

l=radicퟏퟒퟗ

l=122

Name of the Solid Curved surface area Total surface area Volume

Cylinder 2휋푟ℎ =2 x 22

7 x 7 x 10

=440 sqcm

2휋푟(푟 + ℎ) =2 x 22

7 x 7(7+10)

=44 x 17 =748 sqcm

휋푟2ℎ =22

7 x 72 x 10

=1540cm3

Cone 휋푟푙 =22

7 x 7 x 122

=2684 sqcm

휋푟(푟 + 푙) =22

7 x 7 x ( 7 + 122 )

=22 x 192= 4224

13휋푟2ℎ

=13 x 22

7 x 72 x 10

=13 x 22

7 x 72 x 10

=5133 cm3

Sphere

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

43휋푟3

= 43

x 227

x 73 =14373 cm3

Hemisphere 3휋푟2 = 3 x 22

7 x 72

=462 sqcm

2휋푟2 =2 x 22

7 x 72

=308 sqcm

23휋푟3

= 23

x 22x 7

x 73 = 7186 cm3

Find the curved surface area Total surface area and Volume of a cylinder and a cone of hight = 9cm Radius of the base = 7 cm and also find the Lateral surface area toal surface area and volume of a sphere and hemi sphere of 14cm diameter

If the circumference of a cylinder is 44cm and the height is 10cm then find the curved surface area and total surface area Find the Lateral Surfac areaTotal surface area and volume of a cylinder and conehaving radius 7cm and height 24cm



Sketch the plan for the given data

TO D

80To E

150

100

80

30

70to C

40To B

From A

Ans Scale 1cm = 20m rArr 1m = cm

30m = 30 x = 15cm

70m = 70x = 35cm

80m = 80 x = 4cm

100m = 100x = 5cm

150m = 150x =75cm



Sketch the plan for the following

Scale 1 cm = 40m Scale 1cm= = 50m Scale 1cm = 25m

To C E To D

120to D

E to E

220

210

120

80

40 to B

120toD

75to C

50to B

350

300

250

150

50

F 150toF

100to G

100toE

50toF

25toG

225

175

125

100

75

50

25toC

75toB

From A A ಂದ From A

The solved problems for this are given in GET 12 WITH SKILL Exercise Papers 1-10



17Graphs and Polyhedra(2 ಅಂಕಗಳ )

SLNo Chapter MCQ 1-Mark 2-Marks 3-Marks 4-Marks Total K U A S K U A S K U A S 17 Graphs and Polyhedra 1 2

Graph Graph is a set of points joined by pairs of lines

Node(N) A vertex in a graph

Arc(A) A line joining two points Region(R) The area surrounded by arcs(Including outside) Traversable graph The graph which can be traced without lifting the pencil from the paper without retracing any arc Order of the nodeIn a graph the number of arcs at a node

Verify Eulerrsquos formula for the following graph

N + R = A + 2

N = 3 R = 4 A = 5 N+R = 3 +4 = 7 A+2 = 5 +2 = 7 there4 N+R = A+2

Exercise



N = 8 R = 6 A = 12 N+R = 8 +6 = 14 A+2 = 12 +2 = 14 there4 N+R = A+2

N = 3 R = 5 A = 6 N+R = 3 +5 = 8 A+2 = 6 +2 = 8 there4 N+R = A+2

Note NIRA rArrN + R = A + 2

Condition on traversability of graph 1 A graph should have only even nodes 2 A graph should have only two odd nodes

Verify the traversability

Even nodes ndash 8 Odd nodes - 0 All nodes are even there4 This is travesable

ExerciseVerify traversability



Even Nodes ndash 2 Odd nodes ndash 4 Odd nodes are more than 2 there4 The graphs are non-traversable

Eulerrsquos Formula for polyhedra F + V = E + 2

SLNo Polyhedra F- Faces V- Vertices E- Edges F + V = E + 2

1

4 4 6 4 +4 = 6 +2

2

3



4

5

Platonic Solids Number of faces Shape of the face Tetrahedraon 4 Isocels triangle Hexahedron 6 Square octahedron 8 Isocels triangle Dodacahedraon 12 Regular pentagon Icosahedron 20 Isocels triangle



1Real Numbers

lsquoarsquo Dividend lsquobrsquo Divisor lsquoqrsquo Quotient and lsquorrsquo Remainder Then the Euclidrsquos Division Lemma a = bq + r ( 0 le r lt q ) 2Set theory

Commutative property Union of Sets Intersection of Sets

AUB=BUA AcapB=BcapA

Associative Property Union of Sets Intersection of Sets

( Acup B)cup C=Acup (Bcup C) ( AcapB)capC=Acap(BcapC)

Distributive Law Union of sets is distributive over intersection of

sets Acup(BcapC)=( AcupB)cap( AcupC)

Intersection of sets is distributive over union of sets

Acap(BcupC)=( AcapB)cup( AcapC)

De Morganrsquos Law

I - Law ( Acup B)1=A1capB1 II- Law ( AcapB)1=A1UB1

Cardinality of sets Disjoint sets

n( Acup B) = n(A ) + n(B) Non-Disjoint sets

n( Acup B) = n(A ) + n(B) - n( AcapB)

For three sets n( AcupBcupC) = n(A ) + n(B) + n(C) - n( AcapB) - n(BcapC)minusn( AcapC)+n( AcapBcapC)



tandard form of Arithmetic progression

If lsquoarsquo First term lsquodrsquo Common difference then the standard form is a a + d a + 2d a + 3 a + (n-1)d Formula to find nth term of AP Tn = a + (n ndash 1)d [ a-First term n ndash Number of terms d ndash Common difference] Tn+1 = Tn + d Tn-1 = Tn ndash d

d = 퐓퐩 퐓퐧퐩 퐪

[If 푇 = 푇 and 푇 = 푎] d = 푻풏 풂풏 ퟏ

The sum to nth term of an AP Sn = 풏ퟐ[2a + (n-1)d] [ Sn ndash Sum of nth term a ndash First term n ndash Number of terms d ndash Common difference]

The Sum of first lsquonrsquo natural numbers Sn = 풏(풏+ퟏ)ퟐ

Given First term lsquoarsquo and last term lsquoTnrsquo and common difference lsquodrsquo not given The sum to nth term of an AP Sn = 풏

ퟐ[풂 + 푻풏]

The standard form of the Harmonic Progression ퟏ풂

ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅

ퟏ풂+(풏minusퟏ)풅

a ndash First term d ndash Common difference nth Term of HP Tn = ퟏ풂+(풏minusퟏ)풅

The Standard form of Geometric progression a ar ar2 ar3 helliphelliphellip ar(n-1) [ a ndash First term r ndash Common difference] nth term of the GP Tn = ar(n-1) The sum to nth term of the GP Sn = a ( 풓

풏minusퟏ풓minusퟏ

) [ r gt 1 ] Sn = a ( ퟏminus풓풏

ퟏminus풏 ) [ r lt 1 ] Sn = na [ r = 1 ]

The sum of an infinite Geometric Series Sn = 풂ퟏminus풓

Arithmetic Mean(AM) 퐀 = 퐚 + 퐛ퟐ

Harmonic Mean(HM) 퐇 = ퟐ퐚퐛퐚 + 퐛

Geometric Mean(GM) 퐆 = radic퐚퐛

Permutation and Combination



Fundamental principle of countingIf one activity can be done in lsquomrsquo number of different waysand corresponding to each of these ways of the first activities second activity can be done in lsquonrsquo number of different ways then both the activitiesone after the other can be done in (mxn) number of ways

1 0 = 1 ퟐ풏푷풓= 풏(풏minus풓)

ퟑ풏푷ퟎ= 1 ퟒ풏푪ퟎ= 1 ퟓ풏푪ퟎ= 1

ퟔ풏푷풏= n ퟕ풏푷ퟏ= n ퟖ풏푪풓= 풏(풏minus풓)풓

ퟗ풏푷풓= 풏푪풓x r ퟏퟎ풏푪ퟏ= n

1n = n(n-1)(n-2)(n-3) helliphelliphelliphellip3x2x1 ퟏퟐ풏푪풓= 풏푪풏minus풓 or 풏푪풓- 풏푪풏minus풓= 0 Number of diagonals can be drawn in a polygon = 퐧퐂ퟐ- n

The number of straight lines can be drawn (3 of them are non collinear) - 퐧퐂ퟐ Number of Triangles - 퐧퐂ퟑ Probability

Probabilty of an Event P(A) = 퐧(퐄)퐧(퐒)

[ n(E) = E Number of elementary events favourable to the eventn(S) = Total number of elementary events in sample space] a) Probability of Certain event or Sure event = 1 b) Probability of impossible event = 0

Complimentary of P(A) P(A1) = 1 ndash P(A) Addition Rule of Probability [P(E1UE2)= P(E1)+P(E2) ndash P(E1capE2)]

5Statistics

To Find standard deviation

Direct Method Actual method Assumed Mean Method Step Deviation Method

Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ

풏 ndash ( sum풇푿

풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ

풏 ndash ( sum풇풅

풏)ퟐ 흈 =

sum 풇풅ퟐ

풏 ndash ( sum 풇풅

풏)ퟐ 퐱퐂

d = (X - X ) amp 푋 = sum

d = x ndash A d =

[ C ndash The Class intervals should be equal]

Coefficient of Variation = 푺풕풂풏풅풂풓풅 푫풆풗풊풂풕풊풐풏

푴풆풂풏x 100 rArr CV =

훔퐗x100

6Quadratic Equations

Standard for of quadratic equation The roots of quadratic equation Discriminant of quadratic equation

aX2 + bX + c = 0 풙 =minus풃plusmn radic풃ퟐ minus ퟒ풂풄

ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0

Roots are real and equal Roots are real and distinct Roots are imaginary

Sum of the roots Product of roots Form the quadratic equation when roots are given

m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry

sin 휃 cos 휃 tan휃 cosec휃 sec휃 cot휃 Opposite

Hypotenuse Adjacent

Hypotenuse OppositeAdjacent

HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1

cos휃 1 radicퟑퟐ

ퟏradicퟐ

ퟏퟐ 0

tan 휃 0 ퟏradicퟑ

1 radicퟑ ND

csc휃 ND 2 radicퟐ ퟐradicퟑ

1

sec휃 1 ퟐradicퟑ

radicퟐ 2 ND

cot휃 ND radicퟑ 1 ퟏradicퟑ

0

퐬퐢퐧ퟐ 휽+퐜퐨퐬ퟐ휽 = 1 1 + 퐜퐨퐭ퟐ 휽 = 퐜퐨퐬퐞퐜ퟐ 휽 퐭퐚퐧ퟐ 휽 + 1 = 퐬퐞퐜ퟐ 휽 Coordinates geometry

Slopem tan휽 The slope of a straight line passing through two given points m = 풚ퟐminus풚ퟏ

풙ퟐminus풙ퟏ

Distance between two points d = (풙ퟐ minus 풙ퟏ)ퟐ + ( 풚ퟐ minus 풚ퟏ)ퟐ Distance of a line in a plane from the orgin d = 풙ퟐ + 풚ퟐ If y-intercept =c Slope =m are given y=mx =c



Section formula P(xy) devides the line joining the pointsA(x1y1)B(x2y2) then the coordinates of point P

P(xy) =[ 풎풙ퟐ+풏풙ퟏ풎+풏

풎풚ퟐ+풏풚ퟏ풎+풏

]

If P is the midpoint of AB (Mid point formula) P(xy) = [ 풙ퟐ+풙ퟏ

ퟐ 풚ퟐ+풚ퟏퟐ

]

Circles

Find the length of a tangent drawn from an external point to a given circle T = 풅ퟐ minus 풓ퟐ

[d-distance from the centre to an external point)r-Radius] The distance of the centers of two circles touches externally d = R + r Touches internally d = R ndash r The Length of direct common tangents

DCT = 퐝ퟐ ndash (퐑minus 퐫)ퟐ

The length of transverse common tangents TCT = 퐝ퟐ ndash (퐑 + 퐫)ퟐ

Mensuration Curved Surface

area Total Surfac area Volume

cylinder 2흅풓풉 2흅풓(풉 + 풓) 흅풓ퟐ풉

Cone 흅풓풍 흅풓(풓 + 풍) ퟏퟑ 흅풓ퟐ풉



Graph and polyhedra Eulerrsquos Formula for Graphs N + R = A + 2 N - Nodes R - Regions A ndash Arcs Eulerrsquos Formula for Polyhedrs F + V = E + 2 F ndash number of faces V ndash number of vertices

E ndash Edg

Sphere 4흅풓ퟐ 4흅풓ퟐ ퟒퟑ흅풓

ퟑ

Hemisphere 2흅풓ퟐ 3흅풓ퟐ

ퟐퟑ흅풓

ퟑ

Volume of frustum of cone V = ퟏퟑ흅풉(풓ퟏퟐ + 풓ퟏퟐ + 풓ퟏ풓ퟐ)



Chapter -2 Sets (Total 3 Marks)

SLNo Chapter MCQ 1-mark 2-marks 3-marks 4-marks Total K U A S K U A S K U A S 2 Sets 1 2 3

U ndash Universal Set A B and C are non-empty sets

ರೂಪ ಾಜಕ ಯಮ- Union of sets is distributive over intersection of sets

Intersection of sets is distributive over union of sets Example U = 0123456789 A=1237 B=378 C=1238 Note Complimentary of sets ndash A1 = 0145689 B1 = 0124569 C1 = 045679

Properties Union Intersection

Commutative

AUB = 1237 U 378 = 12378helliphelliphellip(1) BUA = 378 U 1237 = 12378helliphelliphellip(2)

From (1) and (2) AUB = BUA

AcapB = 1237 cap 378 = 37 helliphelliphelliphelliphellip(1) BcapA = 378cap1237 = 37 helliphelliphelliphelliphellip(2)

From (1) and (2) AcapB = BcapA

Properties Union Intersection Commutative AUB = BUA AcapB = BcapA

Associative AU(BUC) = (AUB)UC (AcapB)capC = (AcapBcapC) Distributive AU(BcapC) = (AUB)cap(AUC) Acap(BUC) = (AcapB)U(AcapC) DrsquoMorganrsquos (AUB)1 = A1capB1 (AcapB)1 = A1UB1



Associative





Distributive





Drsquomorgans Law

(AUB)1 = [1237U378]1


A1capB1 = 04569cap0124569


(AUB)1 = A1capB1

(AcapB)1 = [1237cap378]1 (AcapB)1 = 371


A1UB1 = 045689U0124569





Cardinality of sets










AB and BA

A = 123456 B = 14578 AB = 236 BA = 78








Formularsquos


















n3 ndash 1 = 26 n3 = 26 + 1 n3 = 27 n3 = 33

there4 n = 3
















Sn = 퐧ퟐ


n = 21 a = 1 d = 3Sn = S21 = [2x1 +(21-1)3]

S21 = [2 +20x3]

S21 = [2 +60] S21 = x62 S21 = 21x31 S21 = 651






Sn = 퐧ퟐ


a = 4 Tn = 76 n = 25 Sn = S25 = 25

2[4 + 76]

S25 = 252

[80] S25 = 25x40 S25 = 1000




2






2 1

4 1






rArr ퟏퟒퟐ

In HP T3 = 17 and

T7 = then Find T15


5


p minus q Tp = T7 = 5 Tq = T3 = 7



rArr d = minus24

rArr d = minus12



= 5


12 and

T11 = 115

then FindT25

2)In HP T4 = 111

and

T14 = then find T7




2




Formulas










ಕ ಗಳ

Tn = a rn-1


rArr T3 = 4x 22

rArr T3 = 4x 4

rArr T3 = 16


T8 = 3x 28-1

rArr T8 = 3x 27

rArr T8 = 3x 128

rArr T8 = 384




T4 = T3x 3 rArr 18x3 = 54

Tn-1 = 퐓퐧퐫


r rArr T4= 32

2 = 16


if r gt 1

1 + 2 + 4 +------10 Sum to 10th term



a = 1 r = 2 S10=


)

S10 = 1 (1024minus11

) S10 = 1023



series

Sn = a ( 1minus rn

1minusr) a = 1

2 n = 10 r = 1

2

Sn = 12

[ 1minus( 12)10

1minus12

]

Sn = 12

[ 1minus 1

210

12]

Sn = 12

x 21

[1024minus11024

]

Sn = [10231024

]



9---

a = 2 r = 13


= ퟐퟐퟑ

= 2x32 = 3





ar x a x ar = 216

a3 = 216 a = 6 6r + 6 + 6r = 21



(7 ndash d)(7 + d) = 2317



2 or r = 2


Means


A = 풂 + 풃ퟐ



A = 푎 + 푏2


= bG

GxG = ab

G2 = ab G = radicab


H 1

b are in AP

1H

- 1푎 = 1

b - 1

H

1H

+ 1 H

= 1b

+ 1푎

1+1H

+ = a+bab

2H

+ = a+bab




If 12 X 1


A = 푎 + 푏2

X = 12 +

18

2

X = 4+18 2

X = 58 2

rArr X = 516



푎+푏

8 = 25푥5+푥

8(5+x) = 10x 40 +8x = 10x 40 = 2x X = 20



Combination 1 1 1 5





















7P3= 7(7minus3)

7P3= 7

4


7P3= 7x6x5 7P3= 210


1) 8C5 2) 6C3

7C3 = 7(7minus3)3

7C3 = 7

43

7C3 = 7x6x53x2x1

7C3 = 210

6




푛(푛minus1)2







15



r = 8 + 12 = 20


Some questions




6C2 -6


10C2


1) 5P3 2) 4P2x2P1



1) 8C2 2) 8C3



3C2 x 4C2



1) 8C5 2) 7C4 3) 7C5











S = (11)(12)(13)(14)(15)(16)(21)(22)(23) (24) (25)(26)(31)(32)(33)(34)(35)(36)(41) (42)(43)(44)(45)(46)(51)(52)(53) (54)(55) (56)(61)(62) (63)(64)(65)(66)

n(S) = 36




P(A) = n(A)n(S)









experiment












n(S) = 8C5 1) n(A) = 7C4 P(A) = 푛(퐴)

푛(푆)

2)n(B) =7C5 P(B) = 푛(퐵)푛(푆)




n(S) = 10C5


푛(푆)


푛(푆)




Un grouped data


흈 = sum푿ퟐ

풏 ndash ( sum푿

풏)ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂



흈 = sum풇풙ퟐ

풏 minus sum풇풙

풏

ퟐ 흈 = sum풇풅

ퟐ

풏 흈 = sum풇풅

ퟐ

풏 minus sum풇풅

풏

ퟐ 흈 = sum풇풅

ퟐ

풏 minus sum풇풅

풏

ퟐ 퐱퐂



For ungrouped data


퐂 d2



For grouped data


풙 d2 fd2

n = sumfx = sumfx2







fd d2 fd2

n = sumfd = sumfd2

= n= sumfd

= sumfd2=



퐂 d2


formula 32 1024 32 -2 4 32 -3 9 32 34 1156 34 0 0 34 -1 1 34 35 1225 35 1 1 35 0 0 35 36 1296 36 2 4 36 1 1 36 39 1521 39 5 25 39 4 16 39 42 1764 42 8 64 42 7 49 42

272 9476 272 228 -8 216 sumd= sumd2 =




rArr ퟐퟕퟐퟖ

=34 Assumed Mean 35


흈 = sum푿ퟐ

풏 ndash ( sum푿

풏)ퟐ 흈 =

sum퐝ퟐ

퐧

흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂



)ퟐ

휎 = 11845 ndash 1156

휎 = radic285

휎 = radic285

휎 = 534

흈 = ퟐퟐퟖퟖ


흈 = ퟓퟑퟒ

흈 =

ퟐퟏퟔퟖ

ndash ( ퟖퟖ

)ퟐ



흈 = radicퟐퟖ

흈 = ퟓퟐퟗ



1-5 2 3 6 9 18 1-5 2 3 6 -7 49 98 6-10 3 8 24 64 192 6-10 3 8 24 -2 4 12

11-15 4 13 52 169 676 11-15 4 13 52 3 9 36 16-20 1 18 18 324 324 16-20 1 18 18 8 64 64

10 100 1210 10 100 210




퐂 fd d2 fd2

1-5 2 3 -10 -20 100 200 1-5 2 3 -10 -2 -4 4 8 6-10 3 8 -5 -15 25 75 6-10 3 8 -5 -1 -3 1 3

11-15 4 13 0 0 0 0 11-15 4 13 0 0 0 0 0 16-20 1 18 5 5 25 25 16-20 1 18 5 1 1 1 1

10 -30 300 10 -6 12





흈 = sum풇풙ퟐ

풏 minus sum풇풙

풏

ퟐ



ퟐ


흈 = sum 풇풅ퟐ

풏



흈 = sum풇풅ퟐ

풏 minus sum풇풅

풏

ퟐ



ퟐ


흈 = sum풇풅ퟐ

풏 minus sum풇풅

풏

ퟐ 퐱퐂

흈 = ퟏퟐퟏퟎ


ퟐ 퐱ퟓ








퐗x100




CI 0-10 10-20 20-30 30-40 40-50 131 Freequency (f) 7 10 15 8 10

CI 5-15 15-25 25-35 35-45 45-55 55-65 134 Freequency (f) 8 12 20 10 7 3




휎 = 261 CV=4348


How the


Number of students

Central Angle

Walk 12 1236

x3600 = 1200

Cycle 8 836

x3600 = 800 Bus 3 3

36x3600 = 300

Car 4 436

x3600 = 400 School Van 9 9

36x3600 = 900

36 3600


7 Surds 2 4





Addition of Surds










12 rArr 2

12x3



13x2










radic5minusradic3

3radic5minusradic3



= 3(radic5+radic3)2


















P(x) =12 ndash 4 x 1 + 3 x 1 = 1 =1 - 4 + 3 + 1 = 1




1 (x ndash 2) ಇದು x3 -3x2 +6x -8


ೂೕ














x2 = 144 x2 ndash x = 0 x2 + 14x + 13 = 0 (x = -1) (x = -13)

4x = 81푥

x2 + 3 = 2x 7x2 -12x = 0 ( x = 13 )

7x = 647푥

x + 1x = 5 2m2 ndash 6m + 3 = 0 ( m = 1

2 )



K = 12m푣2

푣2=2퐾푚

v = plusmn 2퐾푚


2










3x2 ndash 5x + 2 = 0

3x2 ndash 5x + 2 = 0


3


3x = minus ퟐ

ퟑ

x2 - 53x = - 2

3

x2 - 53x +(5

6)2 = minus 2

3 + (5

6)2

(푥 minus 5 6

)2 minus 2436

+ 2536

(푥 minus 5 6

)2 = 136

(푥 minus 5 6

) = plusmn 16

x = 56 plusmn 1

6 rArr x = 6

6 or x = 4

6


3x2 ndash 5x + 2 = 0 a=3 b= -5 c = 2


2(3)


6


6

푥 =5 plusmn 1

6

푥 = 66 or x = 4

6

x = 1 or x = 23




Exercise






Nature of the Roots








푎 = minus2





1 = 1







1 2 ಮತು 3

2 6 ಮತು -5


4 -3 ಮತು 32




y

y = 2x2 x 0 +1 -1 +2 -2 +3 -3

y

y =ퟏퟐx2

x 0 +1 -1 +2 -2 +3 -3

y







10Similar triangles






= 퐵퐶퐸퐹










=23 AE=37 find

EC



To prove ADDB

= AEEC


Proof ∆ABC∆BDE

= 12 12

xADxELxDBxEL

[∵ A = 12

xbxh

∆ABC∆BDE

= ADDB

∆ADE∆CDE

= 12 12

xAExDNxDBxDN

[∵ A = 12

xbxh

∆ADE∆CDE

= AEEC

there4 퐀퐃

퐃퐁 = 퐀퐄






DE = BC

EF = CA

FD


there4 ABAG

= BCGH

= CA HA


there4 퐀퐁퐃퐄

= 퐁퐂퐄퐅

= 퐂퐀 퐅퐃






= BCEF

= CA DF


= 퐁퐂ퟐ

퐄퐅ퟐ



DM = AB

DE and BC

EF = AB

DE [ ∵ Given

there4 ALDM

= BCEF

helliphellip(1)


= 1212

xBCxALxEFxDM


= BCxALEFxDM

[ ∵ ( 1)

= BCxBCEFxEF

= 퐁퐂ퟐ

퐄퐅ퟐ

But ABDE

= BCEF

= CA DF

[ ∵ Given


= AB2

DE2 = BC2

EF2 = CA2

DF2






rArr ABAD

= ACAB


rArr BCDC

= ACBC








12Trigonometry



휽




퐜퐨퐬휽 = ퟏ

퐬퐞퐜 휽




퐴퐵퐴퐶














Values 00 300 450 600 900


ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0





13 tan 휃 = 5

12

Cosec휃 = 135

sec휃 = 1312

cot휃 = 125



rArr 3+2 = 5







ퟑ






radic3

tan300= 1radic3

rArr 휃 = 300





퐴퐵




푥2minus푥1

A(x1y1) and B(x2y2)


푥2minus푥1


tan 휃 = 7






푥2minus푥1

m1 = 5minus20minus5

= 3minus5


= 3minus5




휃 훼


푥2minus푥1


= minus7minus4

= 74


= 4minus7

m1 x m2 = 74 x 4

minus7 = -1








2 c = -3


2x -3rArr2y = x -6

rArr x -2y -6 =0









푚+푛]



2 푦2+푦1

2]

= [4+22

7+32

]

= [62

102

] = (35)








Note

















apart





Theorem










Formulas



ퟐ풉


ퟑ


ퟑ

흅 = ퟐퟐퟕ






d =14cm

r= 7cm

흅 = ퟐퟐퟕ

h=10cm

l= 풓ퟐ + 풉ퟐ



l=radicퟏퟒퟗ

l=122



7 x 7 x 10

=440 sqcm

2휋푟(푟 + ℎ) =2 x 22

7 x 7(7+10)

=44 x 17 =748 sqcm

휋푟2ℎ =22

7 x 72 x 10

=1540cm3

Cone 휋푟푙 =22

7 x 7 x 122

=2684 sqcm

휋푟(푟 + 푙) =22

7 x 7 x ( 7 + 122 )

=22 x 192= 4224

13휋푟2ℎ

=13 x 22

7 x 72 x 10

=13 x 22

7 x 72 x 10

=5133 cm3

Sphere

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

43휋푟3

= 43

x 227

x 73 =14373 cm3


7 x 72

=462 sqcm

2휋푟2 =2 x 22

7 x 72

=308 sqcm

23휋푟3

= 23

x 22x 7

x 73 = 7186 cm3






TO D

80To E

150

100

80

30

70to C

40To B

From A


30m = 30 x = 15cm

70m = 70x = 35cm

80m = 80 x = 4cm

100m = 100x = 5cm

150m = 150x =75cm





To C E To D

120to D

E to E

220

210

120

80

40 to B

120toD

75to C

50to B

350

300

250

150

50

F 150toF

100to G

100toE

50toF

25toG

225

175

125

100

75

50

25toC

75toB











N + R = A + 2

N = 3 R = 4 A = 5 N+R = 3 +4 = 7 A+2 = 5 +2 = 7 there4 N+R = A+2

Exercise



N = 8 R = 6 A = 12 N+R = 8 +6 = 14 A+2 = 12 +2 = 14 there4 N+R = A+2

N = 3 R = 5 A = 6 N+R = 3 +5 = 8 A+2 = 6 +2 = 8 there4 N+R = A+2











1

4 4 6 4 +4 = 6 +2

2

3



4

5




1Real Numbers



AUB=BUA AcapB=BcapA







De Morganrsquos Law















ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ




Associative





Distributive





Drsquomorgans Law

(AUB)1 = [1237U378]1


A1capB1 = 04569cap0124569


(AUB)1 = A1capB1

(AcapB)1 = [1237cap378]1 (AcapB)1 = 371


A1UB1 = 045689U0124569





Cardinality of sets










AB and BA

A = 123456 B = 14578 AB = 236 BA = 78








Formularsquos


















n3 ndash 1 = 26 n3 = 26 + 1 n3 = 27 n3 = 33

there4 n = 3
















Sn = 퐧ퟐ


n = 21 a = 1 d = 3Sn = S21 = [2x1 +(21-1)3]

S21 = [2 +20x3]

S21 = [2 +60] S21 = x62 S21 = 21x31 S21 = 651






Sn = 퐧ퟐ


a = 4 Tn = 76 n = 25 Sn = S25 = 25

2[4 + 76]

S25 = 252

[80] S25 = 25x40 S25 = 1000




2






2 1

4 1






rArr ퟏퟒퟐ

In HP T3 = 17 and

T7 = then Find T15


5


p minus q Tp = T7 = 5 Tq = T3 = 7



rArr d = minus24

rArr d = minus12



= 5


12 and

T11 = 115

then FindT25

2)In HP T4 = 111

and

T14 = then find T7




2




Formulas










ಕ ಗಳ

Tn = a rn-1


rArr T3 = 4x 22

rArr T3 = 4x 4

rArr T3 = 16


T8 = 3x 28-1

rArr T8 = 3x 27

rArr T8 = 3x 128

rArr T8 = 384




T4 = T3x 3 rArr 18x3 = 54

Tn-1 = 퐓퐧퐫


r rArr T4= 32

2 = 16


if r gt 1

1 + 2 + 4 +------10 Sum to 10th term



a = 1 r = 2 S10=


)

S10 = 1 (1024minus11

) S10 = 1023



series

Sn = a ( 1minus rn

1minusr) a = 1

2 n = 10 r = 1

2

Sn = 12

[ 1minus( 12)10

1minus12

]

Sn = 12

[ 1minus 1

210

12]

Sn = 12

x 21

[1024minus11024

]

Sn = [10231024

]



9---

a = 2 r = 13


= ퟐퟐퟑ

= 2x32 = 3





ar x a x ar = 216

a3 = 216 a = 6 6r + 6 + 6r = 21



(7 ndash d)(7 + d) = 2317



2 or r = 2


Means


A = 풂 + 풃ퟐ



A = 푎 + 푏2


= bG

GxG = ab

G2 = ab G = radicab


H 1

b are in AP

1H

- 1푎 = 1

b - 1

H

1H

+ 1 H

= 1b

+ 1푎

1+1H

+ = a+bab

2H

+ = a+bab




If 12 X 1


A = 푎 + 푏2

X = 12 +

18

2

X = 4+18 2

X = 58 2

rArr X = 516



푎+푏

8 = 25푥5+푥

8(5+x) = 10x 40 +8x = 10x 40 = 2x X = 20



Combination 1 1 1 5





















7P3= 7(7minus3)

7P3= 7

4


7P3= 7x6x5 7P3= 210


1) 8C5 2) 6C3

7C3 = 7(7minus3)3

7C3 = 7

43

7C3 = 7x6x53x2x1

7C3 = 210

6




푛(푛minus1)2







15



r = 8 + 12 = 20


Some questions




6C2 -6


10C2


1) 5P3 2) 4P2x2P1



1) 8C2 2) 8C3



3C2 x 4C2



1) 8C5 2) 7C4 3) 7C5











S = (11)(12)(13)(14)(15)(16)(21)(22)(23) (24) (25)(26)(31)(32)(33)(34)(35)(36)(41) (42)(43)(44)(45)(46)(51)(52)(53) (54)(55) (56)(61)(62) (63)(64)(65)(66)

n(S) = 36




P(A) = n(A)n(S)









experiment












n(S) = 8C5 1) n(A) = 7C4 P(A) = 푛(퐴)

푛(푆)

2)n(B) =7C5 P(B) = 푛(퐵)푛(푆)




n(S) = 10C5


푛(푆)


푛(푆)




Un grouped data


흈 = sum푿ퟐ

풏 ndash ( sum푿

풏)ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂



흈 = sum풇풙ퟐ

풏 minus sum풇풙

풏

ퟐ 흈 = sum풇풅

ퟐ

풏 흈 = sum풇풅

ퟐ

풏 minus sum풇풅

풏

ퟐ 흈 = sum풇풅

ퟐ

풏 minus sum풇풅

풏

ퟐ 퐱퐂



For ungrouped data


퐂 d2



For grouped data


풙 d2 fd2

n = sumfx = sumfx2







fd d2 fd2

n = sumfd = sumfd2

= n= sumfd

= sumfd2=



퐂 d2


formula 32 1024 32 -2 4 32 -3 9 32 34 1156 34 0 0 34 -1 1 34 35 1225 35 1 1 35 0 0 35 36 1296 36 2 4 36 1 1 36 39 1521 39 5 25 39 4 16 39 42 1764 42 8 64 42 7 49 42

272 9476 272 228 -8 216 sumd= sumd2 =




rArr ퟐퟕퟐퟖ

=34 Assumed Mean 35


흈 = sum푿ퟐ

풏 ndash ( sum푿

풏)ퟐ 흈 =

sum퐝ퟐ

퐧

흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂



)ퟐ

휎 = 11845 ndash 1156

휎 = radic285

휎 = radic285

휎 = 534

흈 = ퟐퟐퟖퟖ


흈 = ퟓퟑퟒ

흈 =

ퟐퟏퟔퟖ

ndash ( ퟖퟖ

)ퟐ



흈 = radicퟐퟖ

흈 = ퟓퟐퟗ



1-5 2 3 6 9 18 1-5 2 3 6 -7 49 98 6-10 3 8 24 64 192 6-10 3 8 24 -2 4 12

11-15 4 13 52 169 676 11-15 4 13 52 3 9 36 16-20 1 18 18 324 324 16-20 1 18 18 8 64 64

10 100 1210 10 100 210




퐂 fd d2 fd2

1-5 2 3 -10 -20 100 200 1-5 2 3 -10 -2 -4 4 8 6-10 3 8 -5 -15 25 75 6-10 3 8 -5 -1 -3 1 3

11-15 4 13 0 0 0 0 11-15 4 13 0 0 0 0 0 16-20 1 18 5 5 25 25 16-20 1 18 5 1 1 1 1

10 -30 300 10 -6 12





흈 = sum풇풙ퟐ

풏 minus sum풇풙

풏

ퟐ



ퟐ


흈 = sum 풇풅ퟐ

풏



흈 = sum풇풅ퟐ

풏 minus sum풇풅

풏

ퟐ



ퟐ


흈 = sum풇풅ퟐ

풏 minus sum풇풅

풏

ퟐ 퐱퐂

흈 = ퟏퟐퟏퟎ


ퟐ 퐱ퟓ








퐗x100




CI 0-10 10-20 20-30 30-40 40-50 131 Freequency (f) 7 10 15 8 10

CI 5-15 15-25 25-35 35-45 45-55 55-65 134 Freequency (f) 8 12 20 10 7 3




휎 = 261 CV=4348


How the


Number of students

Central Angle

Walk 12 1236

x3600 = 1200

Cycle 8 836

x3600 = 800 Bus 3 3

36x3600 = 300

Car 4 436

x3600 = 400 School Van 9 9

36x3600 = 900

36 3600


7 Surds 2 4





Addition of Surds










12 rArr 2

12x3



13x2










radic5minusradic3

3radic5minusradic3



= 3(radic5+radic3)2


















P(x) =12 ndash 4 x 1 + 3 x 1 = 1 =1 - 4 + 3 + 1 = 1




1 (x ndash 2) ಇದು x3 -3x2 +6x -8


ೂೕ














x2 = 144 x2 ndash x = 0 x2 + 14x + 13 = 0 (x = -1) (x = -13)

4x = 81푥

x2 + 3 = 2x 7x2 -12x = 0 ( x = 13 )

7x = 647푥

x + 1x = 5 2m2 ndash 6m + 3 = 0 ( m = 1

2 )



K = 12m푣2

푣2=2퐾푚

v = plusmn 2퐾푚


2










3x2 ndash 5x + 2 = 0

3x2 ndash 5x + 2 = 0


3


3x = minus ퟐ

ퟑ

x2 - 53x = - 2

3

x2 - 53x +(5

6)2 = minus 2

3 + (5

6)2

(푥 minus 5 6

)2 minus 2436

+ 2536

(푥 minus 5 6

)2 = 136

(푥 minus 5 6

) = plusmn 16

x = 56 plusmn 1

6 rArr x = 6

6 or x = 4

6


3x2 ndash 5x + 2 = 0 a=3 b= -5 c = 2


2(3)


6


6

푥 =5 plusmn 1

6

푥 = 66 or x = 4

6

x = 1 or x = 23




Exercise






Nature of the Roots








푎 = minus2





1 = 1







1 2 ಮತು 3

2 6 ಮತು -5


4 -3 ಮತು 32




y

y = 2x2 x 0 +1 -1 +2 -2 +3 -3

y

y =ퟏퟐx2

x 0 +1 -1 +2 -2 +3 -3

y







10Similar triangles






= 퐵퐶퐸퐹










=23 AE=37 find

EC



To prove ADDB

= AEEC


Proof ∆ABC∆BDE

= 12 12

xADxELxDBxEL

[∵ A = 12

xbxh

∆ABC∆BDE

= ADDB

∆ADE∆CDE

= 12 12

xAExDNxDBxDN

[∵ A = 12

xbxh

∆ADE∆CDE

= AEEC

there4 퐀퐃

퐃퐁 = 퐀퐄






DE = BC

EF = CA

FD


there4 ABAG

= BCGH

= CA HA


there4 퐀퐁퐃퐄

= 퐁퐂퐄퐅

= 퐂퐀 퐅퐃






= BCEF

= CA DF


= 퐁퐂ퟐ

퐄퐅ퟐ



DM = AB

DE and BC

EF = AB

DE [ ∵ Given

there4 ALDM

= BCEF

helliphellip(1)


= 1212

xBCxALxEFxDM


= BCxALEFxDM

[ ∵ ( 1)

= BCxBCEFxEF

= 퐁퐂ퟐ

퐄퐅ퟐ

But ABDE

= BCEF

= CA DF

[ ∵ Given


= AB2

DE2 = BC2

EF2 = CA2

DF2






rArr ABAD

= ACAB


rArr BCDC

= ACBC








12Trigonometry



휽




퐜퐨퐬휽 = ퟏ

퐬퐞퐜 휽




퐴퐵퐴퐶














Values 00 300 450 600 900


ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0





13 tan 휃 = 5

12

Cosec휃 = 135

sec휃 = 1312

cot휃 = 125



rArr 3+2 = 5







ퟑ






radic3

tan300= 1radic3

rArr 휃 = 300





퐴퐵




푥2minus푥1

A(x1y1) and B(x2y2)


푥2minus푥1


tan 휃 = 7






푥2minus푥1

m1 = 5minus20minus5

= 3minus5


= 3minus5




휃 훼


푥2minus푥1


= minus7minus4

= 74


= 4minus7

m1 x m2 = 74 x 4

minus7 = -1








2 c = -3


2x -3rArr2y = x -6

rArr x -2y -6 =0









푚+푛]



2 푦2+푦1

2]

= [4+22

7+32

]

= [62

102

] = (35)








Note

















apart





Theorem










Formulas



ퟐ풉


ퟑ


ퟑ

흅 = ퟐퟐퟕ






d =14cm

r= 7cm

흅 = ퟐퟐퟕ

h=10cm

l= 풓ퟐ + 풉ퟐ



l=radicퟏퟒퟗ

l=122



7 x 7 x 10

=440 sqcm

2휋푟(푟 + ℎ) =2 x 22

7 x 7(7+10)

=44 x 17 =748 sqcm

휋푟2ℎ =22

7 x 72 x 10

=1540cm3

Cone 휋푟푙 =22

7 x 7 x 122

=2684 sqcm

휋푟(푟 + 푙) =22

7 x 7 x ( 7 + 122 )

=22 x 192= 4224

13휋푟2ℎ

=13 x 22

7 x 72 x 10

=13 x 22

7 x 72 x 10

=5133 cm3

Sphere

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

43휋푟3

= 43

x 227

x 73 =14373 cm3


7 x 72

=462 sqcm

2휋푟2 =2 x 22

7 x 72

=308 sqcm

23휋푟3

= 23

x 22x 7

x 73 = 7186 cm3






TO D

80To E

150

100

80

30

70to C

40To B

From A


30m = 30 x = 15cm

70m = 70x = 35cm

80m = 80 x = 4cm

100m = 100x = 5cm

150m = 150x =75cm





To C E To D

120to D

E to E

220

210

120

80

40 to B

120toD

75to C

50to B

350

300

250

150

50

F 150toF

100to G

100toE

50toF

25toG

225

175

125

100

75

50

25toC

75toB











N + R = A + 2

N = 3 R = 4 A = 5 N+R = 3 +4 = 7 A+2 = 5 +2 = 7 there4 N+R = A+2

Exercise



N = 8 R = 6 A = 12 N+R = 8 +6 = 14 A+2 = 12 +2 = 14 there4 N+R = A+2

N = 3 R = 5 A = 6 N+R = 3 +5 = 8 A+2 = 6 +2 = 8 there4 N+R = A+2











1

4 4 6 4 +4 = 6 +2

2

3



4

5




1Real Numbers



AUB=BUA AcapB=BcapA







De Morganrsquos Law















ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ




Cardinality of sets










AB and BA

A = 123456 B = 14578 AB = 236 BA = 78








Formularsquos


















n3 ndash 1 = 26 n3 = 26 + 1 n3 = 27 n3 = 33

there4 n = 3
















Sn = 퐧ퟐ


n = 21 a = 1 d = 3Sn = S21 = [2x1 +(21-1)3]

S21 = [2 +20x3]

S21 = [2 +60] S21 = x62 S21 = 21x31 S21 = 651






Sn = 퐧ퟐ


a = 4 Tn = 76 n = 25 Sn = S25 = 25

2[4 + 76]

S25 = 252

[80] S25 = 25x40 S25 = 1000




2






2 1

4 1






rArr ퟏퟒퟐ

In HP T3 = 17 and

T7 = then Find T15


5


p minus q Tp = T7 = 5 Tq = T3 = 7



rArr d = minus24

rArr d = minus12



= 5


12 and

T11 = 115

then FindT25

2)In HP T4 = 111

and

T14 = then find T7




2




Formulas










ಕ ಗಳ

Tn = a rn-1


rArr T3 = 4x 22

rArr T3 = 4x 4

rArr T3 = 16


T8 = 3x 28-1

rArr T8 = 3x 27

rArr T8 = 3x 128

rArr T8 = 384




T4 = T3x 3 rArr 18x3 = 54

Tn-1 = 퐓퐧퐫


r rArr T4= 32

2 = 16


if r gt 1

1 + 2 + 4 +------10 Sum to 10th term



a = 1 r = 2 S10=


)

S10 = 1 (1024minus11

) S10 = 1023



series

Sn = a ( 1minus rn

1minusr) a = 1

2 n = 10 r = 1

2

Sn = 12

[ 1minus( 12)10

1minus12

]

Sn = 12

[ 1minus 1

210

12]

Sn = 12

x 21

[1024minus11024

]

Sn = [10231024

]



9---

a = 2 r = 13


= ퟐퟐퟑ

= 2x32 = 3





ar x a x ar = 216

a3 = 216 a = 6 6r + 6 + 6r = 21



(7 ndash d)(7 + d) = 2317



2 or r = 2


Means


A = 풂 + 풃ퟐ



A = 푎 + 푏2


= bG

GxG = ab

G2 = ab G = radicab


H 1

b are in AP

1H

- 1푎 = 1

b - 1

H

1H

+ 1 H

= 1b

+ 1푎

1+1H

+ = a+bab

2H

+ = a+bab




If 12 X 1


A = 푎 + 푏2

X = 12 +

18

2

X = 4+18 2

X = 58 2

rArr X = 516



푎+푏

8 = 25푥5+푥

8(5+x) = 10x 40 +8x = 10x 40 = 2x X = 20



Combination 1 1 1 5





















7P3= 7(7minus3)

7P3= 7

4


7P3= 7x6x5 7P3= 210


1) 8C5 2) 6C3

7C3 = 7(7minus3)3

7C3 = 7

43

7C3 = 7x6x53x2x1

7C3 = 210

6




푛(푛minus1)2







15



r = 8 + 12 = 20


Some questions




6C2 -6


10C2


1) 5P3 2) 4P2x2P1



1) 8C2 2) 8C3



3C2 x 4C2



1) 8C5 2) 7C4 3) 7C5











S = (11)(12)(13)(14)(15)(16)(21)(22)(23) (24) (25)(26)(31)(32)(33)(34)(35)(36)(41) (42)(43)(44)(45)(46)(51)(52)(53) (54)(55) (56)(61)(62) (63)(64)(65)(66)

n(S) = 36




P(A) = n(A)n(S)









experiment












n(S) = 8C5 1) n(A) = 7C4 P(A) = 푛(퐴)

푛(푆)

2)n(B) =7C5 P(B) = 푛(퐵)푛(푆)




n(S) = 10C5


푛(푆)


푛(푆)




Un grouped data


흈 = sum푿ퟐ

풏 ndash ( sum푿

풏)ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂



흈 = sum풇풙ퟐ

풏 minus sum풇풙

풏

ퟐ 흈 = sum풇풅

ퟐ

풏 흈 = sum풇풅

ퟐ

풏 minus sum풇풅

풏

ퟐ 흈 = sum풇풅

ퟐ

풏 minus sum풇풅

풏

ퟐ 퐱퐂



For ungrouped data


퐂 d2



For grouped data


풙 d2 fd2

n = sumfx = sumfx2







fd d2 fd2

n = sumfd = sumfd2

= n= sumfd

= sumfd2=



퐂 d2


formula 32 1024 32 -2 4 32 -3 9 32 34 1156 34 0 0 34 -1 1 34 35 1225 35 1 1 35 0 0 35 36 1296 36 2 4 36 1 1 36 39 1521 39 5 25 39 4 16 39 42 1764 42 8 64 42 7 49 42

272 9476 272 228 -8 216 sumd= sumd2 =




rArr ퟐퟕퟐퟖ

=34 Assumed Mean 35


흈 = sum푿ퟐ

풏 ndash ( sum푿

풏)ퟐ 흈 =

sum퐝ퟐ

퐧

흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂



)ퟐ

휎 = 11845 ndash 1156

휎 = radic285

휎 = radic285

휎 = 534

흈 = ퟐퟐퟖퟖ


흈 = ퟓퟑퟒ

흈 =

ퟐퟏퟔퟖ

ndash ( ퟖퟖ

)ퟐ



흈 = radicퟐퟖ

흈 = ퟓퟐퟗ



1-5 2 3 6 9 18 1-5 2 3 6 -7 49 98 6-10 3 8 24 64 192 6-10 3 8 24 -2 4 12

11-15 4 13 52 169 676 11-15 4 13 52 3 9 36 16-20 1 18 18 324 324 16-20 1 18 18 8 64 64

10 100 1210 10 100 210




퐂 fd d2 fd2

1-5 2 3 -10 -20 100 200 1-5 2 3 -10 -2 -4 4 8 6-10 3 8 -5 -15 25 75 6-10 3 8 -5 -1 -3 1 3

11-15 4 13 0 0 0 0 11-15 4 13 0 0 0 0 0 16-20 1 18 5 5 25 25 16-20 1 18 5 1 1 1 1

10 -30 300 10 -6 12





흈 = sum풇풙ퟐ

풏 minus sum풇풙

풏

ퟐ



ퟐ


흈 = sum 풇풅ퟐ

풏



흈 = sum풇풅ퟐ

풏 minus sum풇풅

풏

ퟐ



ퟐ


흈 = sum풇풅ퟐ

풏 minus sum풇풅

풏

ퟐ 퐱퐂

흈 = ퟏퟐퟏퟎ


ퟐ 퐱ퟓ








퐗x100




CI 0-10 10-20 20-30 30-40 40-50 131 Freequency (f) 7 10 15 8 10

CI 5-15 15-25 25-35 35-45 45-55 55-65 134 Freequency (f) 8 12 20 10 7 3




휎 = 261 CV=4348


How the


Number of students

Central Angle

Walk 12 1236

x3600 = 1200

Cycle 8 836

x3600 = 800 Bus 3 3

36x3600 = 300

Car 4 436

x3600 = 400 School Van 9 9

36x3600 = 900

36 3600


7 Surds 2 4





Addition of Surds










12 rArr 2

12x3



13x2










radic5minusradic3

3radic5minusradic3



= 3(radic5+radic3)2


















P(x) =12 ndash 4 x 1 + 3 x 1 = 1 =1 - 4 + 3 + 1 = 1




1 (x ndash 2) ಇದು x3 -3x2 +6x -8


ೂೕ














x2 = 144 x2 ndash x = 0 x2 + 14x + 13 = 0 (x = -1) (x = -13)

4x = 81푥

x2 + 3 = 2x 7x2 -12x = 0 ( x = 13 )

7x = 647푥

x + 1x = 5 2m2 ndash 6m + 3 = 0 ( m = 1

2 )



K = 12m푣2

푣2=2퐾푚

v = plusmn 2퐾푚


2










3x2 ndash 5x + 2 = 0

3x2 ndash 5x + 2 = 0


3


3x = minus ퟐ

ퟑ

x2 - 53x = - 2

3

x2 - 53x +(5

6)2 = minus 2

3 + (5

6)2

(푥 minus 5 6

)2 minus 2436

+ 2536

(푥 minus 5 6

)2 = 136

(푥 minus 5 6

) = plusmn 16

x = 56 plusmn 1

6 rArr x = 6

6 or x = 4

6


3x2 ndash 5x + 2 = 0 a=3 b= -5 c = 2


2(3)


6


6

푥 =5 plusmn 1

6

푥 = 66 or x = 4

6

x = 1 or x = 23




Exercise






Nature of the Roots








푎 = minus2





1 = 1







1 2 ಮತು 3

2 6 ಮತು -5


4 -3 ಮತು 32




y

y = 2x2 x 0 +1 -1 +2 -2 +3 -3

y

y =ퟏퟐx2

x 0 +1 -1 +2 -2 +3 -3

y







10Similar triangles






= 퐵퐶퐸퐹










=23 AE=37 find

EC



To prove ADDB

= AEEC


Proof ∆ABC∆BDE

= 12 12

xADxELxDBxEL

[∵ A = 12

xbxh

∆ABC∆BDE

= ADDB

∆ADE∆CDE

= 12 12

xAExDNxDBxDN

[∵ A = 12

xbxh

∆ADE∆CDE

= AEEC

there4 퐀퐃

퐃퐁 = 퐀퐄






DE = BC

EF = CA

FD


there4 ABAG

= BCGH

= CA HA


there4 퐀퐁퐃퐄

= 퐁퐂퐄퐅

= 퐂퐀 퐅퐃






= BCEF

= CA DF


= 퐁퐂ퟐ

퐄퐅ퟐ



DM = AB

DE and BC

EF = AB

DE [ ∵ Given

there4 ALDM

= BCEF

helliphellip(1)


= 1212

xBCxALxEFxDM


= BCxALEFxDM

[ ∵ ( 1)

= BCxBCEFxEF

= 퐁퐂ퟐ

퐄퐅ퟐ

But ABDE

= BCEF

= CA DF

[ ∵ Given


= AB2

DE2 = BC2

EF2 = CA2

DF2






rArr ABAD

= ACAB


rArr BCDC

= ACBC








12Trigonometry



휽




퐜퐨퐬휽 = ퟏ

퐬퐞퐜 휽




퐴퐵퐴퐶














Values 00 300 450 600 900


ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0





13 tan 휃 = 5

12

Cosec휃 = 135

sec휃 = 1312

cot휃 = 125



rArr 3+2 = 5







ퟑ






radic3

tan300= 1radic3

rArr 휃 = 300





퐴퐵




푥2minus푥1

A(x1y1) and B(x2y2)


푥2minus푥1


tan 휃 = 7






푥2minus푥1

m1 = 5minus20minus5

= 3minus5


= 3minus5




휃 훼


푥2minus푥1


= minus7minus4

= 74


= 4minus7

m1 x m2 = 74 x 4

minus7 = -1








2 c = -3


2x -3rArr2y = x -6

rArr x -2y -6 =0









푚+푛]



2 푦2+푦1

2]

= [4+22

7+32

]

= [62

102

] = (35)








Note

















apart





Theorem










Formulas



ퟐ풉


ퟑ


ퟑ

흅 = ퟐퟐퟕ






d =14cm

r= 7cm

흅 = ퟐퟐퟕ

h=10cm

l= 풓ퟐ + 풉ퟐ



l=radicퟏퟒퟗ

l=122



7 x 7 x 10

=440 sqcm

2휋푟(푟 + ℎ) =2 x 22

7 x 7(7+10)

=44 x 17 =748 sqcm

휋푟2ℎ =22

7 x 72 x 10

=1540cm3

Cone 휋푟푙 =22

7 x 7 x 122

=2684 sqcm

휋푟(푟 + 푙) =22

7 x 7 x ( 7 + 122 )

=22 x 192= 4224

13휋푟2ℎ

=13 x 22

7 x 72 x 10

=13 x 22

7 x 72 x 10

=5133 cm3

Sphere

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

43휋푟3

= 43

x 227

x 73 =14373 cm3


7 x 72

=462 sqcm

2휋푟2 =2 x 22

7 x 72

=308 sqcm

23휋푟3

= 23

x 22x 7

x 73 = 7186 cm3






TO D

80To E

150

100

80

30

70to C

40To B

From A


30m = 30 x = 15cm

70m = 70x = 35cm

80m = 80 x = 4cm

100m = 100x = 5cm

150m = 150x =75cm





To C E To D

120to D

E to E

220

210

120

80

40 to B

120toD

75to C

50to B

350

300

250

150

50

F 150toF

100to G

100toE

50toF

25toG

225

175

125

100

75

50

25toC

75toB











N + R = A + 2

N = 3 R = 4 A = 5 N+R = 3 +4 = 7 A+2 = 5 +2 = 7 there4 N+R = A+2

Exercise



N = 8 R = 6 A = 12 N+R = 8 +6 = 14 A+2 = 12 +2 = 14 there4 N+R = A+2

N = 3 R = 5 A = 6 N+R = 3 +5 = 8 A+2 = 6 +2 = 8 there4 N+R = A+2











1

4 4 6 4 +4 = 6 +2

2

3



4

5




1Real Numbers



AUB=BUA AcapB=BcapA







De Morganrsquos Law















ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ




AB and BA

A = 123456 B = 14578 AB = 236 BA = 78








Formularsquos


















n3 ndash 1 = 26 n3 = 26 + 1 n3 = 27 n3 = 33

there4 n = 3
















Sn = 퐧ퟐ


n = 21 a = 1 d = 3Sn = S21 = [2x1 +(21-1)3]

S21 = [2 +20x3]

S21 = [2 +60] S21 = x62 S21 = 21x31 S21 = 651






Sn = 퐧ퟐ


a = 4 Tn = 76 n = 25 Sn = S25 = 25

2[4 + 76]

S25 = 252

[80] S25 = 25x40 S25 = 1000




2






2 1

4 1






rArr ퟏퟒퟐ

In HP T3 = 17 and

T7 = then Find T15


5


p minus q Tp = T7 = 5 Tq = T3 = 7



rArr d = minus24

rArr d = minus12



= 5


12 and

T11 = 115

then FindT25

2)In HP T4 = 111

and

T14 = then find T7




2




Formulas










ಕ ಗಳ

Tn = a rn-1


rArr T3 = 4x 22

rArr T3 = 4x 4

rArr T3 = 16


T8 = 3x 28-1

rArr T8 = 3x 27

rArr T8 = 3x 128

rArr T8 = 384




T4 = T3x 3 rArr 18x3 = 54

Tn-1 = 퐓퐧퐫


r rArr T4= 32

2 = 16


if r gt 1

1 + 2 + 4 +------10 Sum to 10th term



a = 1 r = 2 S10=


)

S10 = 1 (1024minus11

) S10 = 1023



series

Sn = a ( 1minus rn

1minusr) a = 1

2 n = 10 r = 1

2

Sn = 12

[ 1minus( 12)10

1minus12

]

Sn = 12

[ 1minus 1

210

12]

Sn = 12

x 21

[1024minus11024

]

Sn = [10231024

]



9---

a = 2 r = 13


= ퟐퟐퟑ

= 2x32 = 3





ar x a x ar = 216

a3 = 216 a = 6 6r + 6 + 6r = 21



(7 ndash d)(7 + d) = 2317



2 or r = 2


Means


A = 풂 + 풃ퟐ



A = 푎 + 푏2


= bG

GxG = ab

G2 = ab G = radicab


H 1

b are in AP

1H

- 1푎 = 1

b - 1

H

1H

+ 1 H

= 1b

+ 1푎

1+1H

+ = a+bab

2H

+ = a+bab




If 12 X 1


A = 푎 + 푏2

X = 12 +

18

2

X = 4+18 2

X = 58 2

rArr X = 516



푎+푏

8 = 25푥5+푥

8(5+x) = 10x 40 +8x = 10x 40 = 2x X = 20



Combination 1 1 1 5





















7P3= 7(7minus3)

7P3= 7

4


7P3= 7x6x5 7P3= 210


1) 8C5 2) 6C3

7C3 = 7(7minus3)3

7C3 = 7

43

7C3 = 7x6x53x2x1

7C3 = 210

6




푛(푛minus1)2







15



r = 8 + 12 = 20


Some questions




6C2 -6


10C2


1) 5P3 2) 4P2x2P1



1) 8C2 2) 8C3



3C2 x 4C2



1) 8C5 2) 7C4 3) 7C5











S = (11)(12)(13)(14)(15)(16)(21)(22)(23) (24) (25)(26)(31)(32)(33)(34)(35)(36)(41) (42)(43)(44)(45)(46)(51)(52)(53) (54)(55) (56)(61)(62) (63)(64)(65)(66)

n(S) = 36




P(A) = n(A)n(S)









experiment












n(S) = 8C5 1) n(A) = 7C4 P(A) = 푛(퐴)

푛(푆)

2)n(B) =7C5 P(B) = 푛(퐵)푛(푆)




n(S) = 10C5


푛(푆)


푛(푆)




Un grouped data


흈 = sum푿ퟐ

풏 ndash ( sum푿

풏)ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂



흈 = sum풇풙ퟐ

풏 minus sum풇풙

풏

ퟐ 흈 = sum풇풅

ퟐ

풏 흈 = sum풇풅

ퟐ

풏 minus sum풇풅

풏

ퟐ 흈 = sum풇풅

ퟐ

풏 minus sum풇풅

풏

ퟐ 퐱퐂



For ungrouped data


퐂 d2



For grouped data


풙 d2 fd2

n = sumfx = sumfx2







fd d2 fd2

n = sumfd = sumfd2

= n= sumfd

= sumfd2=



퐂 d2


formula 32 1024 32 -2 4 32 -3 9 32 34 1156 34 0 0 34 -1 1 34 35 1225 35 1 1 35 0 0 35 36 1296 36 2 4 36 1 1 36 39 1521 39 5 25 39 4 16 39 42 1764 42 8 64 42 7 49 42

272 9476 272 228 -8 216 sumd= sumd2 =




rArr ퟐퟕퟐퟖ

=34 Assumed Mean 35


흈 = sum푿ퟐ

풏 ndash ( sum푿

풏)ퟐ 흈 =

sum퐝ퟐ

퐧

흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂



)ퟐ

휎 = 11845 ndash 1156

휎 = radic285

휎 = radic285

휎 = 534

흈 = ퟐퟐퟖퟖ


흈 = ퟓퟑퟒ

흈 =

ퟐퟏퟔퟖ

ndash ( ퟖퟖ

)ퟐ



흈 = radicퟐퟖ

흈 = ퟓퟐퟗ



1-5 2 3 6 9 18 1-5 2 3 6 -7 49 98 6-10 3 8 24 64 192 6-10 3 8 24 -2 4 12

11-15 4 13 52 169 676 11-15 4 13 52 3 9 36 16-20 1 18 18 324 324 16-20 1 18 18 8 64 64

10 100 1210 10 100 210




퐂 fd d2 fd2

1-5 2 3 -10 -20 100 200 1-5 2 3 -10 -2 -4 4 8 6-10 3 8 -5 -15 25 75 6-10 3 8 -5 -1 -3 1 3

11-15 4 13 0 0 0 0 11-15 4 13 0 0 0 0 0 16-20 1 18 5 5 25 25 16-20 1 18 5 1 1 1 1

10 -30 300 10 -6 12





흈 = sum풇풙ퟐ

풏 minus sum풇풙

풏

ퟐ



ퟐ


흈 = sum 풇풅ퟐ

풏



흈 = sum풇풅ퟐ

풏 minus sum풇풅

풏

ퟐ



ퟐ


흈 = sum풇풅ퟐ

풏 minus sum풇풅

풏

ퟐ 퐱퐂

흈 = ퟏퟐퟏퟎ


ퟐ 퐱ퟓ








퐗x100




CI 0-10 10-20 20-30 30-40 40-50 131 Freequency (f) 7 10 15 8 10

CI 5-15 15-25 25-35 35-45 45-55 55-65 134 Freequency (f) 8 12 20 10 7 3




휎 = 261 CV=4348


How the


Number of students

Central Angle

Walk 12 1236

x3600 = 1200

Cycle 8 836

x3600 = 800 Bus 3 3

36x3600 = 300

Car 4 436

x3600 = 400 School Van 9 9

36x3600 = 900

36 3600


7 Surds 2 4





Addition of Surds










12 rArr 2

12x3



13x2










radic5minusradic3

3radic5minusradic3



= 3(radic5+radic3)2


















P(x) =12 ndash 4 x 1 + 3 x 1 = 1 =1 - 4 + 3 + 1 = 1




1 (x ndash 2) ಇದು x3 -3x2 +6x -8


ೂೕ














x2 = 144 x2 ndash x = 0 x2 + 14x + 13 = 0 (x = -1) (x = -13)

4x = 81푥

x2 + 3 = 2x 7x2 -12x = 0 ( x = 13 )

7x = 647푥

x + 1x = 5 2m2 ndash 6m + 3 = 0 ( m = 1

2 )



K = 12m푣2

푣2=2퐾푚

v = plusmn 2퐾푚


2










3x2 ndash 5x + 2 = 0

3x2 ndash 5x + 2 = 0


3


3x = minus ퟐ

ퟑ

x2 - 53x = - 2

3

x2 - 53x +(5

6)2 = minus 2

3 + (5

6)2

(푥 minus 5 6

)2 minus 2436

+ 2536

(푥 minus 5 6

)2 = 136

(푥 minus 5 6

) = plusmn 16

x = 56 plusmn 1

6 rArr x = 6

6 or x = 4

6


3x2 ndash 5x + 2 = 0 a=3 b= -5 c = 2


2(3)


6


6

푥 =5 plusmn 1

6

푥 = 66 or x = 4

6

x = 1 or x = 23




Exercise






Nature of the Roots








푎 = minus2





1 = 1







1 2 ಮತು 3

2 6 ಮತು -5


4 -3 ಮತು 32




y

y = 2x2 x 0 +1 -1 +2 -2 +3 -3

y

y =ퟏퟐx2

x 0 +1 -1 +2 -2 +3 -3

y







10Similar triangles






= 퐵퐶퐸퐹










=23 AE=37 find

EC



To prove ADDB

= AEEC


Proof ∆ABC∆BDE

= 12 12

xADxELxDBxEL

[∵ A = 12

xbxh

∆ABC∆BDE

= ADDB

∆ADE∆CDE

= 12 12

xAExDNxDBxDN

[∵ A = 12

xbxh

∆ADE∆CDE

= AEEC

there4 퐀퐃

퐃퐁 = 퐀퐄






DE = BC

EF = CA

FD


there4 ABAG

= BCGH

= CA HA


there4 퐀퐁퐃퐄

= 퐁퐂퐄퐅

= 퐂퐀 퐅퐃






= BCEF

= CA DF


= 퐁퐂ퟐ

퐄퐅ퟐ



DM = AB

DE and BC

EF = AB

DE [ ∵ Given

there4 ALDM

= BCEF

helliphellip(1)


= 1212

xBCxALxEFxDM


= BCxALEFxDM

[ ∵ ( 1)

= BCxBCEFxEF

= 퐁퐂ퟐ

퐄퐅ퟐ

But ABDE

= BCEF

= CA DF

[ ∵ Given


= AB2

DE2 = BC2

EF2 = CA2

DF2






rArr ABAD

= ACAB


rArr BCDC

= ACBC








12Trigonometry



휽




퐜퐨퐬휽 = ퟏ

퐬퐞퐜 휽




퐴퐵퐴퐶














Values 00 300 450 600 900


ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0





13 tan 휃 = 5

12

Cosec휃 = 135

sec휃 = 1312

cot휃 = 125



rArr 3+2 = 5







ퟑ






radic3

tan300= 1radic3

rArr 휃 = 300





퐴퐵




푥2minus푥1

A(x1y1) and B(x2y2)


푥2minus푥1


tan 휃 = 7






푥2minus푥1

m1 = 5minus20minus5

= 3minus5


= 3minus5




휃 훼


푥2minus푥1


= minus7minus4

= 74


= 4minus7

m1 x m2 = 74 x 4

minus7 = -1








2 c = -3


2x -3rArr2y = x -6

rArr x -2y -6 =0









푚+푛]



2 푦2+푦1

2]

= [4+22

7+32

]

= [62

102

] = (35)








Note

















apart





Theorem










Formulas



ퟐ풉


ퟑ


ퟑ

흅 = ퟐퟐퟕ






d =14cm

r= 7cm

흅 = ퟐퟐퟕ

h=10cm

l= 풓ퟐ + 풉ퟐ



l=radicퟏퟒퟗ

l=122



7 x 7 x 10

=440 sqcm

2휋푟(푟 + ℎ) =2 x 22

7 x 7(7+10)

=44 x 17 =748 sqcm

휋푟2ℎ =22

7 x 72 x 10

=1540cm3

Cone 휋푟푙 =22

7 x 7 x 122

=2684 sqcm

휋푟(푟 + 푙) =22

7 x 7 x ( 7 + 122 )

=22 x 192= 4224

13휋푟2ℎ

=13 x 22

7 x 72 x 10

=13 x 22

7 x 72 x 10

=5133 cm3

Sphere

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

43휋푟3

= 43

x 227

x 73 =14373 cm3


7 x 72

=462 sqcm

2휋푟2 =2 x 22

7 x 72

=308 sqcm

23휋푟3

= 23

x 22x 7

x 73 = 7186 cm3






TO D

80To E

150

100

80

30

70to C

40To B

From A


30m = 30 x = 15cm

70m = 70x = 35cm

80m = 80 x = 4cm

100m = 100x = 5cm

150m = 150x =75cm





To C E To D

120to D

E to E

220

210

120

80

40 to B

120toD

75to C

50to B

350

300

250

150

50

F 150toF

100to G

100toE

50toF

25toG

225

175

125

100

75

50

25toC

75toB











N + R = A + 2

N = 3 R = 4 A = 5 N+R = 3 +4 = 7 A+2 = 5 +2 = 7 there4 N+R = A+2

Exercise



N = 8 R = 6 A = 12 N+R = 8 +6 = 14 A+2 = 12 +2 = 14 there4 N+R = A+2

N = 3 R = 5 A = 6 N+R = 3 +5 = 8 A+2 = 6 +2 = 8 there4 N+R = A+2











1

4 4 6 4 +4 = 6 +2

2

3



4

5




1Real Numbers



AUB=BUA AcapB=BcapA







De Morganrsquos Law















ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ








Formularsquos


















n3 ndash 1 = 26 n3 = 26 + 1 n3 = 27 n3 = 33

there4 n = 3
















Sn = 퐧ퟐ


n = 21 a = 1 d = 3Sn = S21 = [2x1 +(21-1)3]

S21 = [2 +20x3]

S21 = [2 +60] S21 = x62 S21 = 21x31 S21 = 651






Sn = 퐧ퟐ


a = 4 Tn = 76 n = 25 Sn = S25 = 25

2[4 + 76]

S25 = 252

[80] S25 = 25x40 S25 = 1000




2






2 1

4 1






rArr ퟏퟒퟐ

In HP T3 = 17 and

T7 = then Find T15


5


p minus q Tp = T7 = 5 Tq = T3 = 7



rArr d = minus24

rArr d = minus12



= 5


12 and

T11 = 115

then FindT25

2)In HP T4 = 111

and

T14 = then find T7




2




Formulas










ಕ ಗಳ

Tn = a rn-1


rArr T3 = 4x 22

rArr T3 = 4x 4

rArr T3 = 16


T8 = 3x 28-1

rArr T8 = 3x 27

rArr T8 = 3x 128

rArr T8 = 384




T4 = T3x 3 rArr 18x3 = 54

Tn-1 = 퐓퐧퐫


r rArr T4= 32

2 = 16


if r gt 1

1 + 2 + 4 +------10 Sum to 10th term



a = 1 r = 2 S10=


)

S10 = 1 (1024minus11

) S10 = 1023



series

Sn = a ( 1minus rn

1minusr) a = 1

2 n = 10 r = 1

2

Sn = 12

[ 1minus( 12)10

1minus12

]

Sn = 12

[ 1minus 1

210

12]

Sn = 12

x 21

[1024minus11024

]

Sn = [10231024

]



9---

a = 2 r = 13


= ퟐퟐퟑ

= 2x32 = 3





ar x a x ar = 216

a3 = 216 a = 6 6r + 6 + 6r = 21



(7 ndash d)(7 + d) = 2317



2 or r = 2


Means


A = 풂 + 풃ퟐ



A = 푎 + 푏2


= bG

GxG = ab

G2 = ab G = radicab


H 1

b are in AP

1H

- 1푎 = 1

b - 1

H

1H

+ 1 H

= 1b

+ 1푎

1+1H

+ = a+bab

2H

+ = a+bab




If 12 X 1


A = 푎 + 푏2

X = 12 +

18

2

X = 4+18 2

X = 58 2

rArr X = 516



푎+푏

8 = 25푥5+푥

8(5+x) = 10x 40 +8x = 10x 40 = 2x X = 20



Combination 1 1 1 5





















7P3= 7(7minus3)

7P3= 7

4


7P3= 7x6x5 7P3= 210


1) 8C5 2) 6C3

7C3 = 7(7minus3)3

7C3 = 7

43

7C3 = 7x6x53x2x1

7C3 = 210

6




푛(푛minus1)2







15



r = 8 + 12 = 20


Some questions




6C2 -6


10C2


1) 5P3 2) 4P2x2P1



1) 8C2 2) 8C3



3C2 x 4C2



1) 8C5 2) 7C4 3) 7C5











S = (11)(12)(13)(14)(15)(16)(21)(22)(23) (24) (25)(26)(31)(32)(33)(34)(35)(36)(41) (42)(43)(44)(45)(46)(51)(52)(53) (54)(55) (56)(61)(62) (63)(64)(65)(66)

n(S) = 36




P(A) = n(A)n(S)









experiment












n(S) = 8C5 1) n(A) = 7C4 P(A) = 푛(퐴)

푛(푆)

2)n(B) =7C5 P(B) = 푛(퐵)푛(푆)




n(S) = 10C5


푛(푆)


푛(푆)




Un grouped data


흈 = sum푿ퟐ

풏 ndash ( sum푿

풏)ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂



흈 = sum풇풙ퟐ

풏 minus sum풇풙

풏

ퟐ 흈 = sum풇풅

ퟐ

풏 흈 = sum풇풅

ퟐ

풏 minus sum풇풅

풏

ퟐ 흈 = sum풇풅

ퟐ

풏 minus sum풇풅

풏

ퟐ 퐱퐂



For ungrouped data


퐂 d2



For grouped data


풙 d2 fd2

n = sumfx = sumfx2







fd d2 fd2

n = sumfd = sumfd2

= n= sumfd

= sumfd2=



퐂 d2


formula 32 1024 32 -2 4 32 -3 9 32 34 1156 34 0 0 34 -1 1 34 35 1225 35 1 1 35 0 0 35 36 1296 36 2 4 36 1 1 36 39 1521 39 5 25 39 4 16 39 42 1764 42 8 64 42 7 49 42

272 9476 272 228 -8 216 sumd= sumd2 =




rArr ퟐퟕퟐퟖ

=34 Assumed Mean 35


흈 = sum푿ퟐ

풏 ndash ( sum푿

풏)ퟐ 흈 =

sum퐝ퟐ

퐧

흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂



)ퟐ

휎 = 11845 ndash 1156

휎 = radic285

휎 = radic285

휎 = 534

흈 = ퟐퟐퟖퟖ


흈 = ퟓퟑퟒ

흈 =

ퟐퟏퟔퟖ

ndash ( ퟖퟖ

)ퟐ



흈 = radicퟐퟖ

흈 = ퟓퟐퟗ



1-5 2 3 6 9 18 1-5 2 3 6 -7 49 98 6-10 3 8 24 64 192 6-10 3 8 24 -2 4 12

11-15 4 13 52 169 676 11-15 4 13 52 3 9 36 16-20 1 18 18 324 324 16-20 1 18 18 8 64 64

10 100 1210 10 100 210




퐂 fd d2 fd2

1-5 2 3 -10 -20 100 200 1-5 2 3 -10 -2 -4 4 8 6-10 3 8 -5 -15 25 75 6-10 3 8 -5 -1 -3 1 3

11-15 4 13 0 0 0 0 11-15 4 13 0 0 0 0 0 16-20 1 18 5 5 25 25 16-20 1 18 5 1 1 1 1

10 -30 300 10 -6 12





흈 = sum풇풙ퟐ

풏 minus sum풇풙

풏

ퟐ



ퟐ


흈 = sum 풇풅ퟐ

풏



흈 = sum풇풅ퟐ

풏 minus sum풇풅

풏

ퟐ



ퟐ


흈 = sum풇풅ퟐ

풏 minus sum풇풅

풏

ퟐ 퐱퐂

흈 = ퟏퟐퟏퟎ


ퟐ 퐱ퟓ








퐗x100




CI 0-10 10-20 20-30 30-40 40-50 131 Freequency (f) 7 10 15 8 10

CI 5-15 15-25 25-35 35-45 45-55 55-65 134 Freequency (f) 8 12 20 10 7 3




휎 = 261 CV=4348


How the


Number of students

Central Angle

Walk 12 1236

x3600 = 1200

Cycle 8 836

x3600 = 800 Bus 3 3

36x3600 = 300

Car 4 436

x3600 = 400 School Van 9 9

36x3600 = 900

36 3600


7 Surds 2 4





Addition of Surds










12 rArr 2

12x3



13x2










radic5minusradic3

3radic5minusradic3



= 3(radic5+radic3)2


















P(x) =12 ndash 4 x 1 + 3 x 1 = 1 =1 - 4 + 3 + 1 = 1




1 (x ndash 2) ಇದು x3 -3x2 +6x -8


ೂೕ














x2 = 144 x2 ndash x = 0 x2 + 14x + 13 = 0 (x = -1) (x = -13)

4x = 81푥

x2 + 3 = 2x 7x2 -12x = 0 ( x = 13 )

7x = 647푥

x + 1x = 5 2m2 ndash 6m + 3 = 0 ( m = 1

2 )



K = 12m푣2

푣2=2퐾푚

v = plusmn 2퐾푚


2










3x2 ndash 5x + 2 = 0

3x2 ndash 5x + 2 = 0


3


3x = minus ퟐ

ퟑ

x2 - 53x = - 2

3

x2 - 53x +(5

6)2 = minus 2

3 + (5

6)2

(푥 minus 5 6

)2 minus 2436

+ 2536

(푥 minus 5 6

)2 = 136

(푥 minus 5 6

) = plusmn 16

x = 56 plusmn 1

6 rArr x = 6

6 or x = 4

6


3x2 ndash 5x + 2 = 0 a=3 b= -5 c = 2


2(3)


6


6

푥 =5 plusmn 1

6

푥 = 66 or x = 4

6

x = 1 or x = 23




Exercise






Nature of the Roots








푎 = minus2





1 = 1







1 2 ಮತು 3

2 6 ಮತು -5


4 -3 ಮತು 32




y

y = 2x2 x 0 +1 -1 +2 -2 +3 -3

y

y =ퟏퟐx2

x 0 +1 -1 +2 -2 +3 -3

y







10Similar triangles






= 퐵퐶퐸퐹










=23 AE=37 find

EC



To prove ADDB

= AEEC


Proof ∆ABC∆BDE

= 12 12

xADxELxDBxEL

[∵ A = 12

xbxh

∆ABC∆BDE

= ADDB

∆ADE∆CDE

= 12 12

xAExDNxDBxDN

[∵ A = 12

xbxh

∆ADE∆CDE

= AEEC

there4 퐀퐃

퐃퐁 = 퐀퐄






DE = BC

EF = CA

FD


there4 ABAG

= BCGH

= CA HA


there4 퐀퐁퐃퐄

= 퐁퐂퐄퐅

= 퐂퐀 퐅퐃






= BCEF

= CA DF


= 퐁퐂ퟐ

퐄퐅ퟐ



DM = AB

DE and BC

EF = AB

DE [ ∵ Given

there4 ALDM

= BCEF

helliphellip(1)


= 1212

xBCxALxEFxDM


= BCxALEFxDM

[ ∵ ( 1)

= BCxBCEFxEF

= 퐁퐂ퟐ

퐄퐅ퟐ

But ABDE

= BCEF

= CA DF

[ ∵ Given


= AB2

DE2 = BC2

EF2 = CA2

DF2






rArr ABAD

= ACAB


rArr BCDC

= ACBC








12Trigonometry



휽




퐜퐨퐬휽 = ퟏ

퐬퐞퐜 휽




퐴퐵퐴퐶














Values 00 300 450 600 900


ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0





13 tan 휃 = 5

12

Cosec휃 = 135

sec휃 = 1312

cot휃 = 125



rArr 3+2 = 5







ퟑ






radic3

tan300= 1radic3

rArr 휃 = 300





퐴퐵




푥2minus푥1

A(x1y1) and B(x2y2)


푥2minus푥1


tan 휃 = 7






푥2minus푥1

m1 = 5minus20minus5

= 3minus5


= 3minus5




휃 훼


푥2minus푥1


= minus7minus4

= 74


= 4minus7

m1 x m2 = 74 x 4

minus7 = -1








2 c = -3


2x -3rArr2y = x -6

rArr x -2y -6 =0









푚+푛]



2 푦2+푦1

2]

= [4+22

7+32

]

= [62

102

] = (35)








Note

















apart





Theorem










Formulas



ퟐ풉


ퟑ


ퟑ

흅 = ퟐퟐퟕ






d =14cm

r= 7cm

흅 = ퟐퟐퟕ

h=10cm

l= 풓ퟐ + 풉ퟐ



l=radicퟏퟒퟗ

l=122



7 x 7 x 10

=440 sqcm

2휋푟(푟 + ℎ) =2 x 22

7 x 7(7+10)

=44 x 17 =748 sqcm

휋푟2ℎ =22

7 x 72 x 10

=1540cm3

Cone 휋푟푙 =22

7 x 7 x 122

=2684 sqcm

휋푟(푟 + 푙) =22

7 x 7 x ( 7 + 122 )

=22 x 192= 4224

13휋푟2ℎ

=13 x 22

7 x 72 x 10

=13 x 22

7 x 72 x 10

=5133 cm3

Sphere

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

43휋푟3

= 43

x 227

x 73 =14373 cm3


7 x 72

=462 sqcm

2휋푟2 =2 x 22

7 x 72

=308 sqcm

23휋푟3

= 23

x 22x 7

x 73 = 7186 cm3






TO D

80To E

150

100

80

30

70to C

40To B

From A


30m = 30 x = 15cm

70m = 70x = 35cm

80m = 80 x = 4cm

100m = 100x = 5cm

150m = 150x =75cm





To C E To D

120to D

E to E

220

210

120

80

40 to B

120toD

75to C

50to B

350

300

250

150

50

F 150toF

100to G

100toE

50toF

25toG

225

175

125

100

75

50

25toC

75toB











N + R = A + 2

N = 3 R = 4 A = 5 N+R = 3 +4 = 7 A+2 = 5 +2 = 7 there4 N+R = A+2

Exercise



N = 8 R = 6 A = 12 N+R = 8 +6 = 14 A+2 = 12 +2 = 14 there4 N+R = A+2

N = 3 R = 5 A = 6 N+R = 3 +5 = 8 A+2 = 6 +2 = 8 there4 N+R = A+2











1

4 4 6 4 +4 = 6 +2

2

3



4

5




1Real Numbers



AUB=BUA AcapB=BcapA







De Morganrsquos Law















ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ







n3 ndash 1 = 26 n3 = 26 + 1 n3 = 27 n3 = 33

there4 n = 3
















Sn = 퐧ퟐ


n = 21 a = 1 d = 3Sn = S21 = [2x1 +(21-1)3]

S21 = [2 +20x3]

S21 = [2 +60] S21 = x62 S21 = 21x31 S21 = 651






Sn = 퐧ퟐ


a = 4 Tn = 76 n = 25 Sn = S25 = 25

2[4 + 76]

S25 = 252

[80] S25 = 25x40 S25 = 1000




2






2 1

4 1






rArr ퟏퟒퟐ

In HP T3 = 17 and

T7 = then Find T15


5


p minus q Tp = T7 = 5 Tq = T3 = 7



rArr d = minus24

rArr d = minus12



= 5


12 and

T11 = 115

then FindT25

2)In HP T4 = 111

and

T14 = then find T7




2




Formulas










ಕ ಗಳ

Tn = a rn-1


rArr T3 = 4x 22

rArr T3 = 4x 4

rArr T3 = 16


T8 = 3x 28-1

rArr T8 = 3x 27

rArr T8 = 3x 128

rArr T8 = 384




T4 = T3x 3 rArr 18x3 = 54

Tn-1 = 퐓퐧퐫


r rArr T4= 32

2 = 16


if r gt 1

1 + 2 + 4 +------10 Sum to 10th term



a = 1 r = 2 S10=


)

S10 = 1 (1024minus11

) S10 = 1023



series

Sn = a ( 1minus rn

1minusr) a = 1

2 n = 10 r = 1

2

Sn = 12

[ 1minus( 12)10

1minus12

]

Sn = 12

[ 1minus 1

210

12]

Sn = 12

x 21

[1024minus11024

]

Sn = [10231024

]



9---

a = 2 r = 13


= ퟐퟐퟑ

= 2x32 = 3





ar x a x ar = 216

a3 = 216 a = 6 6r + 6 + 6r = 21



(7 ndash d)(7 + d) = 2317



2 or r = 2


Means


A = 풂 + 풃ퟐ



A = 푎 + 푏2


= bG

GxG = ab

G2 = ab G = radicab


H 1

b are in AP

1H

- 1푎 = 1

b - 1

H

1H

+ 1 H

= 1b

+ 1푎

1+1H

+ = a+bab

2H

+ = a+bab




If 12 X 1


A = 푎 + 푏2

X = 12 +

18

2

X = 4+18 2

X = 58 2

rArr X = 516



푎+푏

8 = 25푥5+푥

8(5+x) = 10x 40 +8x = 10x 40 = 2x X = 20



Combination 1 1 1 5





















7P3= 7(7minus3)

7P3= 7

4


7P3= 7x6x5 7P3= 210


1) 8C5 2) 6C3

7C3 = 7(7minus3)3

7C3 = 7

43

7C3 = 7x6x53x2x1

7C3 = 210

6




푛(푛minus1)2







15



r = 8 + 12 = 20


Some questions




6C2 -6


10C2


1) 5P3 2) 4P2x2P1



1) 8C2 2) 8C3



3C2 x 4C2



1) 8C5 2) 7C4 3) 7C5











S = (11)(12)(13)(14)(15)(16)(21)(22)(23) (24) (25)(26)(31)(32)(33)(34)(35)(36)(41) (42)(43)(44)(45)(46)(51)(52)(53) (54)(55) (56)(61)(62) (63)(64)(65)(66)

n(S) = 36




P(A) = n(A)n(S)









experiment












n(S) = 8C5 1) n(A) = 7C4 P(A) = 푛(퐴)

푛(푆)

2)n(B) =7C5 P(B) = 푛(퐵)푛(푆)




n(S) = 10C5


푛(푆)


푛(푆)




Un grouped data


흈 = sum푿ퟐ

풏 ndash ( sum푿

풏)ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂



흈 = sum풇풙ퟐ

풏 minus sum풇풙

풏

ퟐ 흈 = sum풇풅

ퟐ

풏 흈 = sum풇풅

ퟐ

풏 minus sum풇풅

풏

ퟐ 흈 = sum풇풅

ퟐ

풏 minus sum풇풅

풏

ퟐ 퐱퐂



For ungrouped data


퐂 d2



For grouped data


풙 d2 fd2

n = sumfx = sumfx2







fd d2 fd2

n = sumfd = sumfd2

= n= sumfd

= sumfd2=



퐂 d2


formula 32 1024 32 -2 4 32 -3 9 32 34 1156 34 0 0 34 -1 1 34 35 1225 35 1 1 35 0 0 35 36 1296 36 2 4 36 1 1 36 39 1521 39 5 25 39 4 16 39 42 1764 42 8 64 42 7 49 42

272 9476 272 228 -8 216 sumd= sumd2 =




rArr ퟐퟕퟐퟖ

=34 Assumed Mean 35


흈 = sum푿ퟐ

풏 ndash ( sum푿

풏)ퟐ 흈 =

sum퐝ퟐ

퐧

흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂



)ퟐ

휎 = 11845 ndash 1156

휎 = radic285

휎 = radic285

휎 = 534

흈 = ퟐퟐퟖퟖ


흈 = ퟓퟑퟒ

흈 =

ퟐퟏퟔퟖ

ndash ( ퟖퟖ

)ퟐ



흈 = radicퟐퟖ

흈 = ퟓퟐퟗ



1-5 2 3 6 9 18 1-5 2 3 6 -7 49 98 6-10 3 8 24 64 192 6-10 3 8 24 -2 4 12

11-15 4 13 52 169 676 11-15 4 13 52 3 9 36 16-20 1 18 18 324 324 16-20 1 18 18 8 64 64

10 100 1210 10 100 210




퐂 fd d2 fd2

1-5 2 3 -10 -20 100 200 1-5 2 3 -10 -2 -4 4 8 6-10 3 8 -5 -15 25 75 6-10 3 8 -5 -1 -3 1 3

11-15 4 13 0 0 0 0 11-15 4 13 0 0 0 0 0 16-20 1 18 5 5 25 25 16-20 1 18 5 1 1 1 1

10 -30 300 10 -6 12





흈 = sum풇풙ퟐ

풏 minus sum풇풙

풏

ퟐ



ퟐ


흈 = sum 풇풅ퟐ

풏



흈 = sum풇풅ퟐ

풏 minus sum풇풅

풏

ퟐ



ퟐ


흈 = sum풇풅ퟐ

풏 minus sum풇풅

풏

ퟐ 퐱퐂

흈 = ퟏퟐퟏퟎ


ퟐ 퐱ퟓ








퐗x100




CI 0-10 10-20 20-30 30-40 40-50 131 Freequency (f) 7 10 15 8 10

CI 5-15 15-25 25-35 35-45 45-55 55-65 134 Freequency (f) 8 12 20 10 7 3




휎 = 261 CV=4348


How the


Number of students

Central Angle

Walk 12 1236

x3600 = 1200

Cycle 8 836

x3600 = 800 Bus 3 3

36x3600 = 300

Car 4 436

x3600 = 400 School Van 9 9

36x3600 = 900

36 3600


7 Surds 2 4





Addition of Surds










12 rArr 2

12x3



13x2










radic5minusradic3

3radic5minusradic3



= 3(radic5+radic3)2


















P(x) =12 ndash 4 x 1 + 3 x 1 = 1 =1 - 4 + 3 + 1 = 1




1 (x ndash 2) ಇದು x3 -3x2 +6x -8


ೂೕ














x2 = 144 x2 ndash x = 0 x2 + 14x + 13 = 0 (x = -1) (x = -13)

4x = 81푥

x2 + 3 = 2x 7x2 -12x = 0 ( x = 13 )

7x = 647푥

x + 1x = 5 2m2 ndash 6m + 3 = 0 ( m = 1

2 )



K = 12m푣2

푣2=2퐾푚

v = plusmn 2퐾푚


2










3x2 ndash 5x + 2 = 0

3x2 ndash 5x + 2 = 0


3


3x = minus ퟐ

ퟑ

x2 - 53x = - 2

3

x2 - 53x +(5

6)2 = minus 2

3 + (5

6)2

(푥 minus 5 6

)2 minus 2436

+ 2536

(푥 minus 5 6

)2 = 136

(푥 minus 5 6

) = plusmn 16

x = 56 plusmn 1

6 rArr x = 6

6 or x = 4

6


3x2 ndash 5x + 2 = 0 a=3 b= -5 c = 2


2(3)


6


6

푥 =5 plusmn 1

6

푥 = 66 or x = 4

6

x = 1 or x = 23




Exercise






Nature of the Roots








푎 = minus2





1 = 1







1 2 ಮತು 3

2 6 ಮತು -5


4 -3 ಮತು 32




y

y = 2x2 x 0 +1 -1 +2 -2 +3 -3

y

y =ퟏퟐx2

x 0 +1 -1 +2 -2 +3 -3

y







10Similar triangles






= 퐵퐶퐸퐹










=23 AE=37 find

EC



To prove ADDB

= AEEC


Proof ∆ABC∆BDE

= 12 12

xADxELxDBxEL

[∵ A = 12

xbxh

∆ABC∆BDE

= ADDB

∆ADE∆CDE

= 12 12

xAExDNxDBxDN

[∵ A = 12

xbxh

∆ADE∆CDE

= AEEC

there4 퐀퐃

퐃퐁 = 퐀퐄






DE = BC

EF = CA

FD


there4 ABAG

= BCGH

= CA HA


there4 퐀퐁퐃퐄

= 퐁퐂퐄퐅

= 퐂퐀 퐅퐃






= BCEF

= CA DF


= 퐁퐂ퟐ

퐄퐅ퟐ



DM = AB

DE and BC

EF = AB

DE [ ∵ Given

there4 ALDM

= BCEF

helliphellip(1)


= 1212

xBCxALxEFxDM


= BCxALEFxDM

[ ∵ ( 1)

= BCxBCEFxEF

= 퐁퐂ퟐ

퐄퐅ퟐ

But ABDE

= BCEF

= CA DF

[ ∵ Given


= AB2

DE2 = BC2

EF2 = CA2

DF2






rArr ABAD

= ACAB


rArr BCDC

= ACBC








12Trigonometry



휽




퐜퐨퐬휽 = ퟏ

퐬퐞퐜 휽




퐴퐵퐴퐶














Values 00 300 450 600 900


ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0





13 tan 휃 = 5

12

Cosec휃 = 135

sec휃 = 1312

cot휃 = 125



rArr 3+2 = 5







ퟑ






radic3

tan300= 1radic3

rArr 휃 = 300





퐴퐵




푥2minus푥1

A(x1y1) and B(x2y2)


푥2minus푥1


tan 휃 = 7






푥2minus푥1

m1 = 5minus20minus5

= 3minus5


= 3minus5




휃 훼


푥2minus푥1


= minus7minus4

= 74


= 4minus7

m1 x m2 = 74 x 4

minus7 = -1








2 c = -3


2x -3rArr2y = x -6

rArr x -2y -6 =0









푚+푛]



2 푦2+푦1

2]

= [4+22

7+32

]

= [62

102

] = (35)








Note

















apart





Theorem










Formulas



ퟐ풉


ퟑ


ퟑ

흅 = ퟐퟐퟕ






d =14cm

r= 7cm

흅 = ퟐퟐퟕ

h=10cm

l= 풓ퟐ + 풉ퟐ



l=radicퟏퟒퟗ

l=122



7 x 7 x 10

=440 sqcm

2휋푟(푟 + ℎ) =2 x 22

7 x 7(7+10)

=44 x 17 =748 sqcm

휋푟2ℎ =22

7 x 72 x 10

=1540cm3

Cone 휋푟푙 =22

7 x 7 x 122

=2684 sqcm

휋푟(푟 + 푙) =22

7 x 7 x ( 7 + 122 )

=22 x 192= 4224

13휋푟2ℎ

=13 x 22

7 x 72 x 10

=13 x 22

7 x 72 x 10

=5133 cm3

Sphere

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

43휋푟3

= 43

x 227

x 73 =14373 cm3


7 x 72

=462 sqcm

2휋푟2 =2 x 22

7 x 72

=308 sqcm

23휋푟3

= 23

x 22x 7

x 73 = 7186 cm3






TO D

80To E

150

100

80

30

70to C

40To B

From A


30m = 30 x = 15cm

70m = 70x = 35cm

80m = 80 x = 4cm

100m = 100x = 5cm

150m = 150x =75cm





To C E To D

120to D

E to E

220

210

120

80

40 to B

120toD

75to C

50to B

350

300

250

150

50

F 150toF

100to G

100toE

50toF

25toG

225

175

125

100

75

50

25toC

75toB











N + R = A + 2

N = 3 R = 4 A = 5 N+R = 3 +4 = 7 A+2 = 5 +2 = 7 there4 N+R = A+2

Exercise



N = 8 R = 6 A = 12 N+R = 8 +6 = 14 A+2 = 12 +2 = 14 there4 N+R = A+2

N = 3 R = 5 A = 6 N+R = 3 +5 = 8 A+2 = 6 +2 = 8 there4 N+R = A+2











1

4 4 6 4 +4 = 6 +2

2

3



4

5




1Real Numbers



AUB=BUA AcapB=BcapA







De Morganrsquos Law















ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ








Sn = 퐧ퟐ


n = 21 a = 1 d = 3Sn = S21 = [2x1 +(21-1)3]

S21 = [2 +20x3]

S21 = [2 +60] S21 = x62 S21 = 21x31 S21 = 651






Sn = 퐧ퟐ


a = 4 Tn = 76 n = 25 Sn = S25 = 25

2[4 + 76]

S25 = 252

[80] S25 = 25x40 S25 = 1000




2






2 1

4 1






rArr ퟏퟒퟐ

In HP T3 = 17 and

T7 = then Find T15


5


p minus q Tp = T7 = 5 Tq = T3 = 7



rArr d = minus24

rArr d = minus12



= 5


12 and

T11 = 115

then FindT25

2)In HP T4 = 111

and

T14 = then find T7




2




Formulas










ಕ ಗಳ

Tn = a rn-1


rArr T3 = 4x 22

rArr T3 = 4x 4

rArr T3 = 16


T8 = 3x 28-1

rArr T8 = 3x 27

rArr T8 = 3x 128

rArr T8 = 384




T4 = T3x 3 rArr 18x3 = 54

Tn-1 = 퐓퐧퐫


r rArr T4= 32

2 = 16


if r gt 1

1 + 2 + 4 +------10 Sum to 10th term



a = 1 r = 2 S10=


)

S10 = 1 (1024minus11

) S10 = 1023



series

Sn = a ( 1minus rn

1minusr) a = 1

2 n = 10 r = 1

2

Sn = 12

[ 1minus( 12)10

1minus12

]

Sn = 12

[ 1minus 1

210

12]

Sn = 12

x 21

[1024minus11024

]

Sn = [10231024

]



9---

a = 2 r = 13


= ퟐퟐퟑ

= 2x32 = 3





ar x a x ar = 216

a3 = 216 a = 6 6r + 6 + 6r = 21



(7 ndash d)(7 + d) = 2317



2 or r = 2


Means


A = 풂 + 풃ퟐ



A = 푎 + 푏2


= bG

GxG = ab

G2 = ab G = radicab


H 1

b are in AP

1H

- 1푎 = 1

b - 1

H

1H

+ 1 H

= 1b

+ 1푎

1+1H

+ = a+bab

2H

+ = a+bab




If 12 X 1


A = 푎 + 푏2

X = 12 +

18

2

X = 4+18 2

X = 58 2

rArr X = 516



푎+푏

8 = 25푥5+푥

8(5+x) = 10x 40 +8x = 10x 40 = 2x X = 20



Combination 1 1 1 5





















7P3= 7(7minus3)

7P3= 7

4


7P3= 7x6x5 7P3= 210


1) 8C5 2) 6C3

7C3 = 7(7minus3)3

7C3 = 7

43

7C3 = 7x6x53x2x1

7C3 = 210

6




푛(푛minus1)2







15



r = 8 + 12 = 20


Some questions




6C2 -6


10C2


1) 5P3 2) 4P2x2P1



1) 8C2 2) 8C3



3C2 x 4C2



1) 8C5 2) 7C4 3) 7C5











S = (11)(12)(13)(14)(15)(16)(21)(22)(23) (24) (25)(26)(31)(32)(33)(34)(35)(36)(41) (42)(43)(44)(45)(46)(51)(52)(53) (54)(55) (56)(61)(62) (63)(64)(65)(66)

n(S) = 36




P(A) = n(A)n(S)









experiment












n(S) = 8C5 1) n(A) = 7C4 P(A) = 푛(퐴)

푛(푆)

2)n(B) =7C5 P(B) = 푛(퐵)푛(푆)




n(S) = 10C5


푛(푆)


푛(푆)




Un grouped data


흈 = sum푿ퟐ

풏 ndash ( sum푿

풏)ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂



흈 = sum풇풙ퟐ

풏 minus sum풇풙

풏

ퟐ 흈 = sum풇풅

ퟐ

풏 흈 = sum풇풅

ퟐ

풏 minus sum풇풅

풏

ퟐ 흈 = sum풇풅

ퟐ

풏 minus sum풇풅

풏

ퟐ 퐱퐂



For ungrouped data


퐂 d2



For grouped data


풙 d2 fd2

n = sumfx = sumfx2







fd d2 fd2

n = sumfd = sumfd2

= n= sumfd

= sumfd2=



퐂 d2


formula 32 1024 32 -2 4 32 -3 9 32 34 1156 34 0 0 34 -1 1 34 35 1225 35 1 1 35 0 0 35 36 1296 36 2 4 36 1 1 36 39 1521 39 5 25 39 4 16 39 42 1764 42 8 64 42 7 49 42

272 9476 272 228 -8 216 sumd= sumd2 =




rArr ퟐퟕퟐퟖ

=34 Assumed Mean 35


흈 = sum푿ퟐ

풏 ndash ( sum푿

풏)ퟐ 흈 =

sum퐝ퟐ

퐧

흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂



)ퟐ

휎 = 11845 ndash 1156

휎 = radic285

휎 = radic285

휎 = 534

흈 = ퟐퟐퟖퟖ


흈 = ퟓퟑퟒ

흈 =

ퟐퟏퟔퟖ

ndash ( ퟖퟖ

)ퟐ



흈 = radicퟐퟖ

흈 = ퟓퟐퟗ



1-5 2 3 6 9 18 1-5 2 3 6 -7 49 98 6-10 3 8 24 64 192 6-10 3 8 24 -2 4 12

11-15 4 13 52 169 676 11-15 4 13 52 3 9 36 16-20 1 18 18 324 324 16-20 1 18 18 8 64 64

10 100 1210 10 100 210




퐂 fd d2 fd2

1-5 2 3 -10 -20 100 200 1-5 2 3 -10 -2 -4 4 8 6-10 3 8 -5 -15 25 75 6-10 3 8 -5 -1 -3 1 3

11-15 4 13 0 0 0 0 11-15 4 13 0 0 0 0 0 16-20 1 18 5 5 25 25 16-20 1 18 5 1 1 1 1

10 -30 300 10 -6 12





흈 = sum풇풙ퟐ

풏 minus sum풇풙

풏

ퟐ



ퟐ


흈 = sum 풇풅ퟐ

풏



흈 = sum풇풅ퟐ

풏 minus sum풇풅

풏

ퟐ



ퟐ


흈 = sum풇풅ퟐ

풏 minus sum풇풅

풏

ퟐ 퐱퐂

흈 = ퟏퟐퟏퟎ


ퟐ 퐱ퟓ








퐗x100




CI 0-10 10-20 20-30 30-40 40-50 131 Freequency (f) 7 10 15 8 10

CI 5-15 15-25 25-35 35-45 45-55 55-65 134 Freequency (f) 8 12 20 10 7 3




휎 = 261 CV=4348


How the


Number of students

Central Angle

Walk 12 1236

x3600 = 1200

Cycle 8 836

x3600 = 800 Bus 3 3

36x3600 = 300

Car 4 436

x3600 = 400 School Van 9 9

36x3600 = 900

36 3600


7 Surds 2 4





Addition of Surds










12 rArr 2

12x3



13x2










radic5minusradic3

3radic5minusradic3



= 3(radic5+radic3)2


















P(x) =12 ndash 4 x 1 + 3 x 1 = 1 =1 - 4 + 3 + 1 = 1




1 (x ndash 2) ಇದು x3 -3x2 +6x -8


ೂೕ














x2 = 144 x2 ndash x = 0 x2 + 14x + 13 = 0 (x = -1) (x = -13)

4x = 81푥

x2 + 3 = 2x 7x2 -12x = 0 ( x = 13 )

7x = 647푥

x + 1x = 5 2m2 ndash 6m + 3 = 0 ( m = 1

2 )



K = 12m푣2

푣2=2퐾푚

v = plusmn 2퐾푚


2










3x2 ndash 5x + 2 = 0

3x2 ndash 5x + 2 = 0


3


3x = minus ퟐ

ퟑ

x2 - 53x = - 2

3

x2 - 53x +(5

6)2 = minus 2

3 + (5

6)2

(푥 minus 5 6

)2 minus 2436

+ 2536

(푥 minus 5 6

)2 = 136

(푥 minus 5 6

) = plusmn 16

x = 56 plusmn 1

6 rArr x = 6

6 or x = 4

6


3x2 ndash 5x + 2 = 0 a=3 b= -5 c = 2


2(3)


6


6

푥 =5 plusmn 1

6

푥 = 66 or x = 4

6

x = 1 or x = 23




Exercise






Nature of the Roots








푎 = minus2





1 = 1







1 2 ಮತು 3

2 6 ಮತು -5


4 -3 ಮತು 32




y

y = 2x2 x 0 +1 -1 +2 -2 +3 -3

y

y =ퟏퟐx2

x 0 +1 -1 +2 -2 +3 -3

y







10Similar triangles






= 퐵퐶퐸퐹










=23 AE=37 find

EC



To prove ADDB

= AEEC


Proof ∆ABC∆BDE

= 12 12

xADxELxDBxEL

[∵ A = 12

xbxh

∆ABC∆BDE

= ADDB

∆ADE∆CDE

= 12 12

xAExDNxDBxDN

[∵ A = 12

xbxh

∆ADE∆CDE

= AEEC

there4 퐀퐃

퐃퐁 = 퐀퐄






DE = BC

EF = CA

FD


there4 ABAG

= BCGH

= CA HA


there4 퐀퐁퐃퐄

= 퐁퐂퐄퐅

= 퐂퐀 퐅퐃






= BCEF

= CA DF


= 퐁퐂ퟐ

퐄퐅ퟐ



DM = AB

DE and BC

EF = AB

DE [ ∵ Given

there4 ALDM

= BCEF

helliphellip(1)


= 1212

xBCxALxEFxDM


= BCxALEFxDM

[ ∵ ( 1)

= BCxBCEFxEF

= 퐁퐂ퟐ

퐄퐅ퟐ

But ABDE

= BCEF

= CA DF

[ ∵ Given


= AB2

DE2 = BC2

EF2 = CA2

DF2






rArr ABAD

= ACAB


rArr BCDC

= ACBC








12Trigonometry



휽




퐜퐨퐬휽 = ퟏ

퐬퐞퐜 휽




퐴퐵퐴퐶














Values 00 300 450 600 900


ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0





13 tan 휃 = 5

12

Cosec휃 = 135

sec휃 = 1312

cot휃 = 125



rArr 3+2 = 5







ퟑ






radic3

tan300= 1radic3

rArr 휃 = 300





퐴퐵




푥2minus푥1

A(x1y1) and B(x2y2)


푥2minus푥1


tan 휃 = 7






푥2minus푥1

m1 = 5minus20minus5

= 3minus5


= 3minus5




휃 훼


푥2minus푥1


= minus7minus4

= 74


= 4minus7

m1 x m2 = 74 x 4

minus7 = -1








2 c = -3


2x -3rArr2y = x -6

rArr x -2y -6 =0









푚+푛]



2 푦2+푦1

2]

= [4+22

7+32

]

= [62

102

] = (35)








Note

















apart





Theorem










Formulas



ퟐ풉


ퟑ


ퟑ

흅 = ퟐퟐퟕ






d =14cm

r= 7cm

흅 = ퟐퟐퟕ

h=10cm

l= 풓ퟐ + 풉ퟐ



l=radicퟏퟒퟗ

l=122



7 x 7 x 10

=440 sqcm

2휋푟(푟 + ℎ) =2 x 22

7 x 7(7+10)

=44 x 17 =748 sqcm

휋푟2ℎ =22

7 x 72 x 10

=1540cm3

Cone 휋푟푙 =22

7 x 7 x 122

=2684 sqcm

휋푟(푟 + 푙) =22

7 x 7 x ( 7 + 122 )

=22 x 192= 4224

13휋푟2ℎ

=13 x 22

7 x 72 x 10

=13 x 22

7 x 72 x 10

=5133 cm3

Sphere

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

43휋푟3

= 43

x 227

x 73 =14373 cm3


7 x 72

=462 sqcm

2휋푟2 =2 x 22

7 x 72

=308 sqcm

23휋푟3

= 23

x 22x 7

x 73 = 7186 cm3






TO D

80To E

150

100

80

30

70to C

40To B

From A


30m = 30 x = 15cm

70m = 70x = 35cm

80m = 80 x = 4cm

100m = 100x = 5cm

150m = 150x =75cm





To C E To D

120to D

E to E

220

210

120

80

40 to B

120toD

75to C

50to B

350

300

250

150

50

F 150toF

100to G

100toE

50toF

25toG

225

175

125

100

75

50

25toC

75toB











N + R = A + 2

N = 3 R = 4 A = 5 N+R = 3 +4 = 7 A+2 = 5 +2 = 7 there4 N+R = A+2

Exercise



N = 8 R = 6 A = 12 N+R = 8 +6 = 14 A+2 = 12 +2 = 14 there4 N+R = A+2

N = 3 R = 5 A = 6 N+R = 3 +5 = 8 A+2 = 6 +2 = 8 there4 N+R = A+2











1

4 4 6 4 +4 = 6 +2

2

3



4

5




1Real Numbers



AUB=BUA AcapB=BcapA







De Morganrsquos Law















ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ




Sn = 퐧ퟐ


a = 4 Tn = 76 n = 25 Sn = S25 = 25

2[4 + 76]

S25 = 252

[80] S25 = 25x40 S25 = 1000




2






2 1

4 1






rArr ퟏퟒퟐ

In HP T3 = 17 and

T7 = then Find T15


5


p minus q Tp = T7 = 5 Tq = T3 = 7



rArr d = minus24

rArr d = minus12



= 5


12 and

T11 = 115

then FindT25

2)In HP T4 = 111

and

T14 = then find T7




2




Formulas










ಕ ಗಳ

Tn = a rn-1


rArr T3 = 4x 22

rArr T3 = 4x 4

rArr T3 = 16


T8 = 3x 28-1

rArr T8 = 3x 27

rArr T8 = 3x 128

rArr T8 = 384




T4 = T3x 3 rArr 18x3 = 54

Tn-1 = 퐓퐧퐫


r rArr T4= 32

2 = 16


if r gt 1

1 + 2 + 4 +------10 Sum to 10th term



a = 1 r = 2 S10=


)

S10 = 1 (1024minus11

) S10 = 1023



series

Sn = a ( 1minus rn

1minusr) a = 1

2 n = 10 r = 1

2

Sn = 12

[ 1minus( 12)10

1minus12

]

Sn = 12

[ 1minus 1

210

12]

Sn = 12

x 21

[1024minus11024

]

Sn = [10231024

]



9---

a = 2 r = 13


= ퟐퟐퟑ

= 2x32 = 3





ar x a x ar = 216

a3 = 216 a = 6 6r + 6 + 6r = 21



(7 ndash d)(7 + d) = 2317



2 or r = 2


Means


A = 풂 + 풃ퟐ



A = 푎 + 푏2


= bG

GxG = ab

G2 = ab G = radicab


H 1

b are in AP

1H

- 1푎 = 1

b - 1

H

1H

+ 1 H

= 1b

+ 1푎

1+1H

+ = a+bab

2H

+ = a+bab




If 12 X 1


A = 푎 + 푏2

X = 12 +

18

2

X = 4+18 2

X = 58 2

rArr X = 516



푎+푏

8 = 25푥5+푥

8(5+x) = 10x 40 +8x = 10x 40 = 2x X = 20



Combination 1 1 1 5





















7P3= 7(7minus3)

7P3= 7

4


7P3= 7x6x5 7P3= 210


1) 8C5 2) 6C3

7C3 = 7(7minus3)3

7C3 = 7

43

7C3 = 7x6x53x2x1

7C3 = 210

6




푛(푛minus1)2







15



r = 8 + 12 = 20


Some questions




6C2 -6


10C2


1) 5P3 2) 4P2x2P1



1) 8C2 2) 8C3



3C2 x 4C2



1) 8C5 2) 7C4 3) 7C5











S = (11)(12)(13)(14)(15)(16)(21)(22)(23) (24) (25)(26)(31)(32)(33)(34)(35)(36)(41) (42)(43)(44)(45)(46)(51)(52)(53) (54)(55) (56)(61)(62) (63)(64)(65)(66)

n(S) = 36




P(A) = n(A)n(S)









experiment












n(S) = 8C5 1) n(A) = 7C4 P(A) = 푛(퐴)

푛(푆)

2)n(B) =7C5 P(B) = 푛(퐵)푛(푆)




n(S) = 10C5


푛(푆)


푛(푆)




Un grouped data


흈 = sum푿ퟐ

풏 ndash ( sum푿

풏)ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂



흈 = sum풇풙ퟐ

풏 minus sum풇풙

풏

ퟐ 흈 = sum풇풅

ퟐ

풏 흈 = sum풇풅

ퟐ

풏 minus sum풇풅

풏

ퟐ 흈 = sum풇풅

ퟐ

풏 minus sum풇풅

풏

ퟐ 퐱퐂



For ungrouped data


퐂 d2



For grouped data


풙 d2 fd2

n = sumfx = sumfx2







fd d2 fd2

n = sumfd = sumfd2

= n= sumfd

= sumfd2=



퐂 d2


formula 32 1024 32 -2 4 32 -3 9 32 34 1156 34 0 0 34 -1 1 34 35 1225 35 1 1 35 0 0 35 36 1296 36 2 4 36 1 1 36 39 1521 39 5 25 39 4 16 39 42 1764 42 8 64 42 7 49 42

272 9476 272 228 -8 216 sumd= sumd2 =




rArr ퟐퟕퟐퟖ

=34 Assumed Mean 35


흈 = sum푿ퟐ

풏 ndash ( sum푿

풏)ퟐ 흈 =

sum퐝ퟐ

퐧

흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂



)ퟐ

휎 = 11845 ndash 1156

휎 = radic285

휎 = radic285

휎 = 534

흈 = ퟐퟐퟖퟖ


흈 = ퟓퟑퟒ

흈 =

ퟐퟏퟔퟖ

ndash ( ퟖퟖ

)ퟐ



흈 = radicퟐퟖ

흈 = ퟓퟐퟗ



1-5 2 3 6 9 18 1-5 2 3 6 -7 49 98 6-10 3 8 24 64 192 6-10 3 8 24 -2 4 12

11-15 4 13 52 169 676 11-15 4 13 52 3 9 36 16-20 1 18 18 324 324 16-20 1 18 18 8 64 64

10 100 1210 10 100 210




퐂 fd d2 fd2

1-5 2 3 -10 -20 100 200 1-5 2 3 -10 -2 -4 4 8 6-10 3 8 -5 -15 25 75 6-10 3 8 -5 -1 -3 1 3

11-15 4 13 0 0 0 0 11-15 4 13 0 0 0 0 0 16-20 1 18 5 5 25 25 16-20 1 18 5 1 1 1 1

10 -30 300 10 -6 12





흈 = sum풇풙ퟐ

풏 minus sum풇풙

풏

ퟐ



ퟐ


흈 = sum 풇풅ퟐ

풏



흈 = sum풇풅ퟐ

풏 minus sum풇풅

풏

ퟐ



ퟐ


흈 = sum풇풅ퟐ

풏 minus sum풇풅

풏

ퟐ 퐱퐂

흈 = ퟏퟐퟏퟎ


ퟐ 퐱ퟓ








퐗x100




CI 0-10 10-20 20-30 30-40 40-50 131 Freequency (f) 7 10 15 8 10

CI 5-15 15-25 25-35 35-45 45-55 55-65 134 Freequency (f) 8 12 20 10 7 3




휎 = 261 CV=4348


How the


Number of students

Central Angle

Walk 12 1236

x3600 = 1200

Cycle 8 836

x3600 = 800 Bus 3 3

36x3600 = 300

Car 4 436

x3600 = 400 School Van 9 9

36x3600 = 900

36 3600


7 Surds 2 4





Addition of Surds










12 rArr 2

12x3



13x2










radic5minusradic3

3radic5minusradic3



= 3(radic5+radic3)2


















P(x) =12 ndash 4 x 1 + 3 x 1 = 1 =1 - 4 + 3 + 1 = 1




1 (x ndash 2) ಇದು x3 -3x2 +6x -8


ೂೕ














x2 = 144 x2 ndash x = 0 x2 + 14x + 13 = 0 (x = -1) (x = -13)

4x = 81푥

x2 + 3 = 2x 7x2 -12x = 0 ( x = 13 )

7x = 647푥

x + 1x = 5 2m2 ndash 6m + 3 = 0 ( m = 1

2 )



K = 12m푣2

푣2=2퐾푚

v = plusmn 2퐾푚


2










3x2 ndash 5x + 2 = 0

3x2 ndash 5x + 2 = 0


3


3x = minus ퟐ

ퟑ

x2 - 53x = - 2

3

x2 - 53x +(5

6)2 = minus 2

3 + (5

6)2

(푥 minus 5 6

)2 minus 2436

+ 2536

(푥 minus 5 6

)2 = 136

(푥 minus 5 6

) = plusmn 16

x = 56 plusmn 1

6 rArr x = 6

6 or x = 4

6


3x2 ndash 5x + 2 = 0 a=3 b= -5 c = 2


2(3)


6


6

푥 =5 plusmn 1

6

푥 = 66 or x = 4

6

x = 1 or x = 23




Exercise






Nature of the Roots








푎 = minus2





1 = 1







1 2 ಮತು 3

2 6 ಮತು -5


4 -3 ಮತು 32




y

y = 2x2 x 0 +1 -1 +2 -2 +3 -3

y

y =ퟏퟐx2

x 0 +1 -1 +2 -2 +3 -3

y







10Similar triangles






= 퐵퐶퐸퐹










=23 AE=37 find

EC



To prove ADDB

= AEEC


Proof ∆ABC∆BDE

= 12 12

xADxELxDBxEL

[∵ A = 12

xbxh

∆ABC∆BDE

= ADDB

∆ADE∆CDE

= 12 12

xAExDNxDBxDN

[∵ A = 12

xbxh

∆ADE∆CDE

= AEEC

there4 퐀퐃

퐃퐁 = 퐀퐄






DE = BC

EF = CA

FD


there4 ABAG

= BCGH

= CA HA


there4 퐀퐁퐃퐄

= 퐁퐂퐄퐅

= 퐂퐀 퐅퐃






= BCEF

= CA DF


= 퐁퐂ퟐ

퐄퐅ퟐ



DM = AB

DE and BC

EF = AB

DE [ ∵ Given

there4 ALDM

= BCEF

helliphellip(1)


= 1212

xBCxALxEFxDM


= BCxALEFxDM

[ ∵ ( 1)

= BCxBCEFxEF

= 퐁퐂ퟐ

퐄퐅ퟐ

But ABDE

= BCEF

= CA DF

[ ∵ Given


= AB2

DE2 = BC2

EF2 = CA2

DF2






rArr ABAD

= ACAB


rArr BCDC

= ACBC








12Trigonometry



휽




퐜퐨퐬휽 = ퟏ

퐬퐞퐜 휽




퐴퐵퐴퐶














Values 00 300 450 600 900


ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0





13 tan 휃 = 5

12

Cosec휃 = 135

sec휃 = 1312

cot휃 = 125



rArr 3+2 = 5







ퟑ






radic3

tan300= 1radic3

rArr 휃 = 300





퐴퐵




푥2minus푥1

A(x1y1) and B(x2y2)


푥2minus푥1


tan 휃 = 7






푥2minus푥1

m1 = 5minus20minus5

= 3minus5


= 3minus5




휃 훼


푥2minus푥1


= minus7minus4

= 74


= 4minus7

m1 x m2 = 74 x 4

minus7 = -1








2 c = -3


2x -3rArr2y = x -6

rArr x -2y -6 =0









푚+푛]



2 푦2+푦1

2]

= [4+22

7+32

]

= [62

102

] = (35)








Note

















apart





Theorem










Formulas



ퟐ풉


ퟑ


ퟑ

흅 = ퟐퟐퟕ






d =14cm

r= 7cm

흅 = ퟐퟐퟕ

h=10cm

l= 풓ퟐ + 풉ퟐ



l=radicퟏퟒퟗ

l=122



7 x 7 x 10

=440 sqcm

2휋푟(푟 + ℎ) =2 x 22

7 x 7(7+10)

=44 x 17 =748 sqcm

휋푟2ℎ =22

7 x 72 x 10

=1540cm3

Cone 휋푟푙 =22

7 x 7 x 122

=2684 sqcm

휋푟(푟 + 푙) =22

7 x 7 x ( 7 + 122 )

=22 x 192= 4224

13휋푟2ℎ

=13 x 22

7 x 72 x 10

=13 x 22

7 x 72 x 10

=5133 cm3

Sphere

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

43휋푟3

= 43

x 227

x 73 =14373 cm3


7 x 72

=462 sqcm

2휋푟2 =2 x 22

7 x 72

=308 sqcm

23휋푟3

= 23

x 22x 7

x 73 = 7186 cm3






TO D

80To E

150

100

80

30

70to C

40To B

From A


30m = 30 x = 15cm

70m = 70x = 35cm

80m = 80 x = 4cm

100m = 100x = 5cm

150m = 150x =75cm





To C E To D

120to D

E to E

220

210

120

80

40 to B

120toD

75to C

50to B

350

300

250

150

50

F 150toF

100to G

100toE

50toF

25toG

225

175

125

100

75

50

25toC

75toB











N + R = A + 2

N = 3 R = 4 A = 5 N+R = 3 +4 = 7 A+2 = 5 +2 = 7 there4 N+R = A+2

Exercise



N = 8 R = 6 A = 12 N+R = 8 +6 = 14 A+2 = 12 +2 = 14 there4 N+R = A+2

N = 3 R = 5 A = 6 N+R = 3 +5 = 8 A+2 = 6 +2 = 8 there4 N+R = A+2











1

4 4 6 4 +4 = 6 +2

2

3



4

5




1Real Numbers



AUB=BUA AcapB=BcapA







De Morganrsquos Law















ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ





2




Formulas










ಕ ಗಳ

Tn = a rn-1


rArr T3 = 4x 22

rArr T3 = 4x 4

rArr T3 = 16


T8 = 3x 28-1

rArr T8 = 3x 27

rArr T8 = 3x 128

rArr T8 = 384




T4 = T3x 3 rArr 18x3 = 54

Tn-1 = 퐓퐧퐫


r rArr T4= 32

2 = 16


if r gt 1

1 + 2 + 4 +------10 Sum to 10th term



a = 1 r = 2 S10=


)

S10 = 1 (1024minus11

) S10 = 1023



series

Sn = a ( 1minus rn

1minusr) a = 1

2 n = 10 r = 1

2

Sn = 12

[ 1minus( 12)10

1minus12

]

Sn = 12

[ 1minus 1

210

12]

Sn = 12

x 21

[1024minus11024

]

Sn = [10231024

]



9---

a = 2 r = 13


= ퟐퟐퟑ

= 2x32 = 3





ar x a x ar = 216

a3 = 216 a = 6 6r + 6 + 6r = 21



(7 ndash d)(7 + d) = 2317



2 or r = 2


Means


A = 풂 + 풃ퟐ



A = 푎 + 푏2


= bG

GxG = ab

G2 = ab G = radicab


H 1

b are in AP

1H

- 1푎 = 1

b - 1

H

1H

+ 1 H

= 1b

+ 1푎

1+1H

+ = a+bab

2H

+ = a+bab




If 12 X 1


A = 푎 + 푏2

X = 12 +

18

2

X = 4+18 2

X = 58 2

rArr X = 516



푎+푏

8 = 25푥5+푥

8(5+x) = 10x 40 +8x = 10x 40 = 2x X = 20



Combination 1 1 1 5





















7P3= 7(7minus3)

7P3= 7

4


7P3= 7x6x5 7P3= 210


1) 8C5 2) 6C3

7C3 = 7(7minus3)3

7C3 = 7

43

7C3 = 7x6x53x2x1

7C3 = 210

6




푛(푛minus1)2







15



r = 8 + 12 = 20


Some questions




6C2 -6


10C2


1) 5P3 2) 4P2x2P1



1) 8C2 2) 8C3



3C2 x 4C2



1) 8C5 2) 7C4 3) 7C5











S = (11)(12)(13)(14)(15)(16)(21)(22)(23) (24) (25)(26)(31)(32)(33)(34)(35)(36)(41) (42)(43)(44)(45)(46)(51)(52)(53) (54)(55) (56)(61)(62) (63)(64)(65)(66)

n(S) = 36




P(A) = n(A)n(S)









experiment












n(S) = 8C5 1) n(A) = 7C4 P(A) = 푛(퐴)

푛(푆)

2)n(B) =7C5 P(B) = 푛(퐵)푛(푆)




n(S) = 10C5


푛(푆)


푛(푆)




Un grouped data


흈 = sum푿ퟐ

풏 ndash ( sum푿

풏)ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂



흈 = sum풇풙ퟐ

풏 minus sum풇풙

풏

ퟐ 흈 = sum풇풅

ퟐ

풏 흈 = sum풇풅

ퟐ

풏 minus sum풇풅

풏

ퟐ 흈 = sum풇풅

ퟐ

풏 minus sum풇풅

풏

ퟐ 퐱퐂



For ungrouped data


퐂 d2



For grouped data


풙 d2 fd2

n = sumfx = sumfx2







fd d2 fd2

n = sumfd = sumfd2

= n= sumfd

= sumfd2=



퐂 d2


formula 32 1024 32 -2 4 32 -3 9 32 34 1156 34 0 0 34 -1 1 34 35 1225 35 1 1 35 0 0 35 36 1296 36 2 4 36 1 1 36 39 1521 39 5 25 39 4 16 39 42 1764 42 8 64 42 7 49 42

272 9476 272 228 -8 216 sumd= sumd2 =




rArr ퟐퟕퟐퟖ

=34 Assumed Mean 35


흈 = sum푿ퟐ

풏 ndash ( sum푿

풏)ퟐ 흈 =

sum퐝ퟐ

퐧

흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂



)ퟐ

휎 = 11845 ndash 1156

휎 = radic285

휎 = radic285

휎 = 534

흈 = ퟐퟐퟖퟖ


흈 = ퟓퟑퟒ

흈 =

ퟐퟏퟔퟖ

ndash ( ퟖퟖ

)ퟐ



흈 = radicퟐퟖ

흈 = ퟓퟐퟗ



1-5 2 3 6 9 18 1-5 2 3 6 -7 49 98 6-10 3 8 24 64 192 6-10 3 8 24 -2 4 12

11-15 4 13 52 169 676 11-15 4 13 52 3 9 36 16-20 1 18 18 324 324 16-20 1 18 18 8 64 64

10 100 1210 10 100 210




퐂 fd d2 fd2

1-5 2 3 -10 -20 100 200 1-5 2 3 -10 -2 -4 4 8 6-10 3 8 -5 -15 25 75 6-10 3 8 -5 -1 -3 1 3

11-15 4 13 0 0 0 0 11-15 4 13 0 0 0 0 0 16-20 1 18 5 5 25 25 16-20 1 18 5 1 1 1 1

10 -30 300 10 -6 12





흈 = sum풇풙ퟐ

풏 minus sum풇풙

풏

ퟐ



ퟐ


흈 = sum 풇풅ퟐ

풏



흈 = sum풇풅ퟐ

풏 minus sum풇풅

풏

ퟐ



ퟐ


흈 = sum풇풅ퟐ

풏 minus sum풇풅

풏

ퟐ 퐱퐂

흈 = ퟏퟐퟏퟎ


ퟐ 퐱ퟓ








퐗x100




CI 0-10 10-20 20-30 30-40 40-50 131 Freequency (f) 7 10 15 8 10

CI 5-15 15-25 25-35 35-45 45-55 55-65 134 Freequency (f) 8 12 20 10 7 3




휎 = 261 CV=4348


How the


Number of students

Central Angle

Walk 12 1236

x3600 = 1200

Cycle 8 836

x3600 = 800 Bus 3 3

36x3600 = 300

Car 4 436

x3600 = 400 School Van 9 9

36x3600 = 900

36 3600


7 Surds 2 4





Addition of Surds










12 rArr 2

12x3



13x2










radic5minusradic3

3radic5minusradic3



= 3(radic5+radic3)2


















P(x) =12 ndash 4 x 1 + 3 x 1 = 1 =1 - 4 + 3 + 1 = 1




1 (x ndash 2) ಇದು x3 -3x2 +6x -8


ೂೕ














x2 = 144 x2 ndash x = 0 x2 + 14x + 13 = 0 (x = -1) (x = -13)

4x = 81푥

x2 + 3 = 2x 7x2 -12x = 0 ( x = 13 )

7x = 647푥

x + 1x = 5 2m2 ndash 6m + 3 = 0 ( m = 1

2 )



K = 12m푣2

푣2=2퐾푚

v = plusmn 2퐾푚


2










3x2 ndash 5x + 2 = 0

3x2 ndash 5x + 2 = 0


3


3x = minus ퟐ

ퟑ

x2 - 53x = - 2

3

x2 - 53x +(5

6)2 = minus 2

3 + (5

6)2

(푥 minus 5 6

)2 minus 2436

+ 2536

(푥 minus 5 6

)2 = 136

(푥 minus 5 6

) = plusmn 16

x = 56 plusmn 1

6 rArr x = 6

6 or x = 4

6


3x2 ndash 5x + 2 = 0 a=3 b= -5 c = 2


2(3)


6


6

푥 =5 plusmn 1

6

푥 = 66 or x = 4

6

x = 1 or x = 23




Exercise






Nature of the Roots








푎 = minus2





1 = 1







1 2 ಮತು 3

2 6 ಮತು -5


4 -3 ಮತು 32




y

y = 2x2 x 0 +1 -1 +2 -2 +3 -3

y

y =ퟏퟐx2

x 0 +1 -1 +2 -2 +3 -3

y







10Similar triangles






= 퐵퐶퐸퐹










=23 AE=37 find

EC



To prove ADDB

= AEEC


Proof ∆ABC∆BDE

= 12 12

xADxELxDBxEL

[∵ A = 12

xbxh

∆ABC∆BDE

= ADDB

∆ADE∆CDE

= 12 12

xAExDNxDBxDN

[∵ A = 12

xbxh

∆ADE∆CDE

= AEEC

there4 퐀퐃

퐃퐁 = 퐀퐄






DE = BC

EF = CA

FD


there4 ABAG

= BCGH

= CA HA


there4 퐀퐁퐃퐄

= 퐁퐂퐄퐅

= 퐂퐀 퐅퐃






= BCEF

= CA DF


= 퐁퐂ퟐ

퐄퐅ퟐ



DM = AB

DE and BC

EF = AB

DE [ ∵ Given

there4 ALDM

= BCEF

helliphellip(1)


= 1212

xBCxALxEFxDM


= BCxALEFxDM

[ ∵ ( 1)

= BCxBCEFxEF

= 퐁퐂ퟐ

퐄퐅ퟐ

But ABDE

= BCEF

= CA DF

[ ∵ Given


= AB2

DE2 = BC2

EF2 = CA2

DF2






rArr ABAD

= ACAB


rArr BCDC

= ACBC








12Trigonometry



휽




퐜퐨퐬휽 = ퟏ

퐬퐞퐜 휽




퐴퐵퐴퐶














Values 00 300 450 600 900


ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0





13 tan 휃 = 5

12

Cosec휃 = 135

sec휃 = 1312

cot휃 = 125



rArr 3+2 = 5







ퟑ






radic3

tan300= 1radic3

rArr 휃 = 300





퐴퐵




푥2minus푥1

A(x1y1) and B(x2y2)


푥2minus푥1


tan 휃 = 7






푥2minus푥1

m1 = 5minus20minus5

= 3minus5


= 3minus5




휃 훼


푥2minus푥1


= minus7minus4

= 74


= 4minus7

m1 x m2 = 74 x 4

minus7 = -1








2 c = -3


2x -3rArr2y = x -6

rArr x -2y -6 =0









푚+푛]



2 푦2+푦1

2]

= [4+22

7+32

]

= [62

102

] = (35)








Note

















apart





Theorem










Formulas



ퟐ풉


ퟑ


ퟑ

흅 = ퟐퟐퟕ






d =14cm

r= 7cm

흅 = ퟐퟐퟕ

h=10cm

l= 풓ퟐ + 풉ퟐ



l=radicퟏퟒퟗ

l=122



7 x 7 x 10

=440 sqcm

2휋푟(푟 + ℎ) =2 x 22

7 x 7(7+10)

=44 x 17 =748 sqcm

휋푟2ℎ =22

7 x 72 x 10

=1540cm3

Cone 휋푟푙 =22

7 x 7 x 122

=2684 sqcm

휋푟(푟 + 푙) =22

7 x 7 x ( 7 + 122 )

=22 x 192= 4224

13휋푟2ℎ

=13 x 22

7 x 72 x 10

=13 x 22

7 x 72 x 10

=5133 cm3

Sphere

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

43휋푟3

= 43

x 227

x 73 =14373 cm3


7 x 72

=462 sqcm

2휋푟2 =2 x 22

7 x 72

=308 sqcm

23휋푟3

= 23

x 22x 7

x 73 = 7186 cm3






TO D

80To E

150

100

80

30

70to C

40To B

From A


30m = 30 x = 15cm

70m = 70x = 35cm

80m = 80 x = 4cm

100m = 100x = 5cm

150m = 150x =75cm





To C E To D

120to D

E to E

220

210

120

80

40 to B

120toD

75to C

50to B

350

300

250

150

50

F 150toF

100to G

100toE

50toF

25toG

225

175

125

100

75

50

25toC

75toB











N + R = A + 2

N = 3 R = 4 A = 5 N+R = 3 +4 = 7 A+2 = 5 +2 = 7 there4 N+R = A+2

Exercise



N = 8 R = 6 A = 12 N+R = 8 +6 = 14 A+2 = 12 +2 = 14 there4 N+R = A+2

N = 3 R = 5 A = 6 N+R = 3 +5 = 8 A+2 = 6 +2 = 8 there4 N+R = A+2











1

4 4 6 4 +4 = 6 +2

2

3



4

5




1Real Numbers



AUB=BUA AcapB=BcapA







De Morganrsquos Law















ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ





T4 = T3x 3 rArr 18x3 = 54

Tn-1 = 퐓퐧퐫


r rArr T4= 32

2 = 16


if r gt 1

1 + 2 + 4 +------10 Sum to 10th term



a = 1 r = 2 S10=


)

S10 = 1 (1024minus11

) S10 = 1023



series

Sn = a ( 1minus rn

1minusr) a = 1

2 n = 10 r = 1

2

Sn = 12

[ 1minus( 12)10

1minus12

]

Sn = 12

[ 1minus 1

210

12]

Sn = 12

x 21

[1024minus11024

]

Sn = [10231024

]



9---

a = 2 r = 13


= ퟐퟐퟑ

= 2x32 = 3





ar x a x ar = 216

a3 = 216 a = 6 6r + 6 + 6r = 21



(7 ndash d)(7 + d) = 2317



2 or r = 2


Means


A = 풂 + 풃ퟐ



A = 푎 + 푏2


= bG

GxG = ab

G2 = ab G = radicab


H 1

b are in AP

1H

- 1푎 = 1

b - 1

H

1H

+ 1 H

= 1b

+ 1푎

1+1H

+ = a+bab

2H

+ = a+bab




If 12 X 1


A = 푎 + 푏2

X = 12 +

18

2

X = 4+18 2

X = 58 2

rArr X = 516



푎+푏

8 = 25푥5+푥

8(5+x) = 10x 40 +8x = 10x 40 = 2x X = 20



Combination 1 1 1 5





















7P3= 7(7minus3)

7P3= 7

4


7P3= 7x6x5 7P3= 210


1) 8C5 2) 6C3

7C3 = 7(7minus3)3

7C3 = 7

43

7C3 = 7x6x53x2x1

7C3 = 210

6




푛(푛minus1)2







15



r = 8 + 12 = 20


Some questions




6C2 -6


10C2


1) 5P3 2) 4P2x2P1



1) 8C2 2) 8C3



3C2 x 4C2



1) 8C5 2) 7C4 3) 7C5











S = (11)(12)(13)(14)(15)(16)(21)(22)(23) (24) (25)(26)(31)(32)(33)(34)(35)(36)(41) (42)(43)(44)(45)(46)(51)(52)(53) (54)(55) (56)(61)(62) (63)(64)(65)(66)

n(S) = 36




P(A) = n(A)n(S)









experiment












n(S) = 8C5 1) n(A) = 7C4 P(A) = 푛(퐴)

푛(푆)

2)n(B) =7C5 P(B) = 푛(퐵)푛(푆)




n(S) = 10C5


푛(푆)


푛(푆)




Un grouped data


흈 = sum푿ퟐ

풏 ndash ( sum푿

풏)ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂



흈 = sum풇풙ퟐ

풏 minus sum풇풙

풏

ퟐ 흈 = sum풇풅

ퟐ

풏 흈 = sum풇풅

ퟐ

풏 minus sum풇풅

풏

ퟐ 흈 = sum풇풅

ퟐ

풏 minus sum풇풅

풏

ퟐ 퐱퐂



For ungrouped data


퐂 d2



For grouped data


풙 d2 fd2

n = sumfx = sumfx2







fd d2 fd2

n = sumfd = sumfd2

= n= sumfd

= sumfd2=



퐂 d2


formula 32 1024 32 -2 4 32 -3 9 32 34 1156 34 0 0 34 -1 1 34 35 1225 35 1 1 35 0 0 35 36 1296 36 2 4 36 1 1 36 39 1521 39 5 25 39 4 16 39 42 1764 42 8 64 42 7 49 42

272 9476 272 228 -8 216 sumd= sumd2 =




rArr ퟐퟕퟐퟖ

=34 Assumed Mean 35


흈 = sum푿ퟐ

풏 ndash ( sum푿

풏)ퟐ 흈 =

sum퐝ퟐ

퐧

흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂



)ퟐ

휎 = 11845 ndash 1156

휎 = radic285

휎 = radic285

휎 = 534

흈 = ퟐퟐퟖퟖ


흈 = ퟓퟑퟒ

흈 =

ퟐퟏퟔퟖ

ndash ( ퟖퟖ

)ퟐ



흈 = radicퟐퟖ

흈 = ퟓퟐퟗ



1-5 2 3 6 9 18 1-5 2 3 6 -7 49 98 6-10 3 8 24 64 192 6-10 3 8 24 -2 4 12

11-15 4 13 52 169 676 11-15 4 13 52 3 9 36 16-20 1 18 18 324 324 16-20 1 18 18 8 64 64

10 100 1210 10 100 210




퐂 fd d2 fd2

1-5 2 3 -10 -20 100 200 1-5 2 3 -10 -2 -4 4 8 6-10 3 8 -5 -15 25 75 6-10 3 8 -5 -1 -3 1 3

11-15 4 13 0 0 0 0 11-15 4 13 0 0 0 0 0 16-20 1 18 5 5 25 25 16-20 1 18 5 1 1 1 1

10 -30 300 10 -6 12





흈 = sum풇풙ퟐ

풏 minus sum풇풙

풏

ퟐ



ퟐ


흈 = sum 풇풅ퟐ

풏



흈 = sum풇풅ퟐ

풏 minus sum풇풅

풏

ퟐ



ퟐ


흈 = sum풇풅ퟐ

풏 minus sum풇풅

풏

ퟐ 퐱퐂

흈 = ퟏퟐퟏퟎ


ퟐ 퐱ퟓ








퐗x100




CI 0-10 10-20 20-30 30-40 40-50 131 Freequency (f) 7 10 15 8 10

CI 5-15 15-25 25-35 35-45 45-55 55-65 134 Freequency (f) 8 12 20 10 7 3




휎 = 261 CV=4348


How the


Number of students

Central Angle

Walk 12 1236

x3600 = 1200

Cycle 8 836

x3600 = 800 Bus 3 3

36x3600 = 300

Car 4 436

x3600 = 400 School Van 9 9

36x3600 = 900

36 3600


7 Surds 2 4





Addition of Surds










12 rArr 2

12x3



13x2










radic5minusradic3

3radic5minusradic3



= 3(radic5+radic3)2


















P(x) =12 ndash 4 x 1 + 3 x 1 = 1 =1 - 4 + 3 + 1 = 1




1 (x ndash 2) ಇದು x3 -3x2 +6x -8


ೂೕ














x2 = 144 x2 ndash x = 0 x2 + 14x + 13 = 0 (x = -1) (x = -13)

4x = 81푥

x2 + 3 = 2x 7x2 -12x = 0 ( x = 13 )

7x = 647푥

x + 1x = 5 2m2 ndash 6m + 3 = 0 ( m = 1

2 )



K = 12m푣2

푣2=2퐾푚

v = plusmn 2퐾푚


2










3x2 ndash 5x + 2 = 0

3x2 ndash 5x + 2 = 0


3


3x = minus ퟐ

ퟑ

x2 - 53x = - 2

3

x2 - 53x +(5

6)2 = minus 2

3 + (5

6)2

(푥 minus 5 6

)2 minus 2436

+ 2536

(푥 minus 5 6

)2 = 136

(푥 minus 5 6

) = plusmn 16

x = 56 plusmn 1

6 rArr x = 6

6 or x = 4

6


3x2 ndash 5x + 2 = 0 a=3 b= -5 c = 2


2(3)


6


6

푥 =5 plusmn 1

6

푥 = 66 or x = 4

6

x = 1 or x = 23




Exercise






Nature of the Roots








푎 = minus2





1 = 1







1 2 ಮತು 3

2 6 ಮತು -5


4 -3 ಮತು 32




y

y = 2x2 x 0 +1 -1 +2 -2 +3 -3

y

y =ퟏퟐx2

x 0 +1 -1 +2 -2 +3 -3

y







10Similar triangles






= 퐵퐶퐸퐹










=23 AE=37 find

EC



To prove ADDB

= AEEC


Proof ∆ABC∆BDE

= 12 12

xADxELxDBxEL

[∵ A = 12

xbxh

∆ABC∆BDE

= ADDB

∆ADE∆CDE

= 12 12

xAExDNxDBxDN

[∵ A = 12

xbxh

∆ADE∆CDE

= AEEC

there4 퐀퐃

퐃퐁 = 퐀퐄






DE = BC

EF = CA

FD


there4 ABAG

= BCGH

= CA HA


there4 퐀퐁퐃퐄

= 퐁퐂퐄퐅

= 퐂퐀 퐅퐃






= BCEF

= CA DF


= 퐁퐂ퟐ

퐄퐅ퟐ



DM = AB

DE and BC

EF = AB

DE [ ∵ Given

there4 ALDM

= BCEF

helliphellip(1)


= 1212

xBCxALxEFxDM


= BCxALEFxDM

[ ∵ ( 1)

= BCxBCEFxEF

= 퐁퐂ퟐ

퐄퐅ퟐ

But ABDE

= BCEF

= CA DF

[ ∵ Given


= AB2

DE2 = BC2

EF2 = CA2

DF2






rArr ABAD

= ACAB


rArr BCDC

= ACBC








12Trigonometry



휽




퐜퐨퐬휽 = ퟏ

퐬퐞퐜 휽




퐴퐵퐴퐶














Values 00 300 450 600 900


ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0





13 tan 휃 = 5

12

Cosec휃 = 135

sec휃 = 1312

cot휃 = 125



rArr 3+2 = 5







ퟑ






radic3

tan300= 1radic3

rArr 휃 = 300





퐴퐵




푥2minus푥1

A(x1y1) and B(x2y2)


푥2minus푥1


tan 휃 = 7






푥2minus푥1

m1 = 5minus20minus5

= 3minus5


= 3minus5




휃 훼


푥2minus푥1


= minus7minus4

= 74


= 4minus7

m1 x m2 = 74 x 4

minus7 = -1








2 c = -3


2x -3rArr2y = x -6

rArr x -2y -6 =0









푚+푛]



2 푦2+푦1

2]

= [4+22

7+32

]

= [62

102

] = (35)








Note

















apart





Theorem










Formulas



ퟐ풉


ퟑ


ퟑ

흅 = ퟐퟐퟕ






d =14cm

r= 7cm

흅 = ퟐퟐퟕ

h=10cm

l= 풓ퟐ + 풉ퟐ



l=radicퟏퟒퟗ

l=122



7 x 7 x 10

=440 sqcm

2휋푟(푟 + ℎ) =2 x 22

7 x 7(7+10)

=44 x 17 =748 sqcm

휋푟2ℎ =22

7 x 72 x 10

=1540cm3

Cone 휋푟푙 =22

7 x 7 x 122

=2684 sqcm

휋푟(푟 + 푙) =22

7 x 7 x ( 7 + 122 )

=22 x 192= 4224

13휋푟2ℎ

=13 x 22

7 x 72 x 10

=13 x 22

7 x 72 x 10

=5133 cm3

Sphere

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

43휋푟3

= 43

x 227

x 73 =14373 cm3


7 x 72

=462 sqcm

2휋푟2 =2 x 22

7 x 72

=308 sqcm

23휋푟3

= 23

x 22x 7

x 73 = 7186 cm3






TO D

80To E

150

100

80

30

70to C

40To B

From A


30m = 30 x = 15cm

70m = 70x = 35cm

80m = 80 x = 4cm

100m = 100x = 5cm

150m = 150x =75cm





To C E To D

120to D

E to E

220

210

120

80

40 to B

120toD

75to C

50to B

350

300

250

150

50

F 150toF

100to G

100toE

50toF

25toG

225

175

125

100

75

50

25toC

75toB











N + R = A + 2

N = 3 R = 4 A = 5 N+R = 3 +4 = 7 A+2 = 5 +2 = 7 there4 N+R = A+2

Exercise



N = 8 R = 6 A = 12 N+R = 8 +6 = 14 A+2 = 12 +2 = 14 there4 N+R = A+2

N = 3 R = 5 A = 6 N+R = 3 +5 = 8 A+2 = 6 +2 = 8 there4 N+R = A+2











1

4 4 6 4 +4 = 6 +2

2

3



4

5




1Real Numbers



AUB=BUA AcapB=BcapA







De Morganrsquos Law















ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ




(7 ndash d)(7 + d) = 2317



2 or r = 2


Means


A = 풂 + 풃ퟐ



A = 푎 + 푏2


= bG

GxG = ab

G2 = ab G = radicab


H 1

b are in AP

1H

- 1푎 = 1

b - 1

H

1H

+ 1 H

= 1b

+ 1푎

1+1H

+ = a+bab

2H

+ = a+bab




If 12 X 1


A = 푎 + 푏2

X = 12 +

18

2

X = 4+18 2

X = 58 2

rArr X = 516



푎+푏

8 = 25푥5+푥

8(5+x) = 10x 40 +8x = 10x 40 = 2x X = 20



Combination 1 1 1 5





















7P3= 7(7minus3)

7P3= 7

4


7P3= 7x6x5 7P3= 210


1) 8C5 2) 6C3

7C3 = 7(7minus3)3

7C3 = 7

43

7C3 = 7x6x53x2x1

7C3 = 210

6




푛(푛minus1)2







15



r = 8 + 12 = 20


Some questions




6C2 -6


10C2


1) 5P3 2) 4P2x2P1



1) 8C2 2) 8C3



3C2 x 4C2



1) 8C5 2) 7C4 3) 7C5











S = (11)(12)(13)(14)(15)(16)(21)(22)(23) (24) (25)(26)(31)(32)(33)(34)(35)(36)(41) (42)(43)(44)(45)(46)(51)(52)(53) (54)(55) (56)(61)(62) (63)(64)(65)(66)

n(S) = 36




P(A) = n(A)n(S)









experiment












n(S) = 8C5 1) n(A) = 7C4 P(A) = 푛(퐴)

푛(푆)

2)n(B) =7C5 P(B) = 푛(퐵)푛(푆)




n(S) = 10C5


푛(푆)


푛(푆)




Un grouped data


흈 = sum푿ퟐ

풏 ndash ( sum푿

풏)ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂



흈 = sum풇풙ퟐ

풏 minus sum풇풙

풏

ퟐ 흈 = sum풇풅

ퟐ

풏 흈 = sum풇풅

ퟐ

풏 minus sum풇풅

풏

ퟐ 흈 = sum풇풅

ퟐ

풏 minus sum풇풅

풏

ퟐ 퐱퐂



For ungrouped data


퐂 d2



For grouped data


풙 d2 fd2

n = sumfx = sumfx2







fd d2 fd2

n = sumfd = sumfd2

= n= sumfd

= sumfd2=



퐂 d2


formula 32 1024 32 -2 4 32 -3 9 32 34 1156 34 0 0 34 -1 1 34 35 1225 35 1 1 35 0 0 35 36 1296 36 2 4 36 1 1 36 39 1521 39 5 25 39 4 16 39 42 1764 42 8 64 42 7 49 42

272 9476 272 228 -8 216 sumd= sumd2 =




rArr ퟐퟕퟐퟖ

=34 Assumed Mean 35


흈 = sum푿ퟐ

풏 ndash ( sum푿

풏)ퟐ 흈 =

sum퐝ퟐ

퐧

흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂



)ퟐ

휎 = 11845 ndash 1156

휎 = radic285

휎 = radic285

휎 = 534

흈 = ퟐퟐퟖퟖ


흈 = ퟓퟑퟒ

흈 =

ퟐퟏퟔퟖ

ndash ( ퟖퟖ

)ퟐ



흈 = radicퟐퟖ

흈 = ퟓퟐퟗ



1-5 2 3 6 9 18 1-5 2 3 6 -7 49 98 6-10 3 8 24 64 192 6-10 3 8 24 -2 4 12

11-15 4 13 52 169 676 11-15 4 13 52 3 9 36 16-20 1 18 18 324 324 16-20 1 18 18 8 64 64

10 100 1210 10 100 210




퐂 fd d2 fd2

1-5 2 3 -10 -20 100 200 1-5 2 3 -10 -2 -4 4 8 6-10 3 8 -5 -15 25 75 6-10 3 8 -5 -1 -3 1 3

11-15 4 13 0 0 0 0 11-15 4 13 0 0 0 0 0 16-20 1 18 5 5 25 25 16-20 1 18 5 1 1 1 1

10 -30 300 10 -6 12





흈 = sum풇풙ퟐ

풏 minus sum풇풙

풏

ퟐ



ퟐ


흈 = sum 풇풅ퟐ

풏



흈 = sum풇풅ퟐ

풏 minus sum풇풅

풏

ퟐ



ퟐ


흈 = sum풇풅ퟐ

풏 minus sum풇풅

풏

ퟐ 퐱퐂

흈 = ퟏퟐퟏퟎ


ퟐ 퐱ퟓ








퐗x100




CI 0-10 10-20 20-30 30-40 40-50 131 Freequency (f) 7 10 15 8 10

CI 5-15 15-25 25-35 35-45 45-55 55-65 134 Freequency (f) 8 12 20 10 7 3




휎 = 261 CV=4348


How the


Number of students

Central Angle

Walk 12 1236

x3600 = 1200

Cycle 8 836

x3600 = 800 Bus 3 3

36x3600 = 300

Car 4 436

x3600 = 400 School Van 9 9

36x3600 = 900

36 3600


7 Surds 2 4





Addition of Surds










12 rArr 2

12x3



13x2










radic5minusradic3

3radic5minusradic3



= 3(radic5+radic3)2


















P(x) =12 ndash 4 x 1 + 3 x 1 = 1 =1 - 4 + 3 + 1 = 1




1 (x ndash 2) ಇದು x3 -3x2 +6x -8


ೂೕ














x2 = 144 x2 ndash x = 0 x2 + 14x + 13 = 0 (x = -1) (x = -13)

4x = 81푥

x2 + 3 = 2x 7x2 -12x = 0 ( x = 13 )

7x = 647푥

x + 1x = 5 2m2 ndash 6m + 3 = 0 ( m = 1

2 )



K = 12m푣2

푣2=2퐾푚

v = plusmn 2퐾푚


2










3x2 ndash 5x + 2 = 0

3x2 ndash 5x + 2 = 0


3


3x = minus ퟐ

ퟑ

x2 - 53x = - 2

3

x2 - 53x +(5

6)2 = minus 2

3 + (5

6)2

(푥 minus 5 6

)2 minus 2436

+ 2536

(푥 minus 5 6

)2 = 136

(푥 minus 5 6

) = plusmn 16

x = 56 plusmn 1

6 rArr x = 6

6 or x = 4

6


3x2 ndash 5x + 2 = 0 a=3 b= -5 c = 2


2(3)


6


6

푥 =5 plusmn 1

6

푥 = 66 or x = 4

6

x = 1 or x = 23




Exercise






Nature of the Roots








푎 = minus2





1 = 1







1 2 ಮತು 3

2 6 ಮತು -5


4 -3 ಮತು 32




y

y = 2x2 x 0 +1 -1 +2 -2 +3 -3

y

y =ퟏퟐx2

x 0 +1 -1 +2 -2 +3 -3

y







10Similar triangles






= 퐵퐶퐸퐹










=23 AE=37 find

EC



To prove ADDB

= AEEC


Proof ∆ABC∆BDE

= 12 12

xADxELxDBxEL

[∵ A = 12

xbxh

∆ABC∆BDE

= ADDB

∆ADE∆CDE

= 12 12

xAExDNxDBxDN

[∵ A = 12

xbxh

∆ADE∆CDE

= AEEC

there4 퐀퐃

퐃퐁 = 퐀퐄






DE = BC

EF = CA

FD


there4 ABAG

= BCGH

= CA HA


there4 퐀퐁퐃퐄

= 퐁퐂퐄퐅

= 퐂퐀 퐅퐃






= BCEF

= CA DF


= 퐁퐂ퟐ

퐄퐅ퟐ



DM = AB

DE and BC

EF = AB

DE [ ∵ Given

there4 ALDM

= BCEF

helliphellip(1)


= 1212

xBCxALxEFxDM


= BCxALEFxDM

[ ∵ ( 1)

= BCxBCEFxEF

= 퐁퐂ퟐ

퐄퐅ퟐ

But ABDE

= BCEF

= CA DF

[ ∵ Given


= AB2

DE2 = BC2

EF2 = CA2

DF2






rArr ABAD

= ACAB


rArr BCDC

= ACBC








12Trigonometry



휽




퐜퐨퐬휽 = ퟏ

퐬퐞퐜 휽




퐴퐵퐴퐶














Values 00 300 450 600 900


ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0





13 tan 휃 = 5

12

Cosec휃 = 135

sec휃 = 1312

cot휃 = 125



rArr 3+2 = 5







ퟑ






radic3

tan300= 1radic3

rArr 휃 = 300





퐴퐵




푥2minus푥1

A(x1y1) and B(x2y2)


푥2minus푥1


tan 휃 = 7






푥2minus푥1

m1 = 5minus20minus5

= 3minus5


= 3minus5




휃 훼


푥2minus푥1


= minus7minus4

= 74


= 4minus7

m1 x m2 = 74 x 4

minus7 = -1








2 c = -3


2x -3rArr2y = x -6

rArr x -2y -6 =0









푚+푛]



2 푦2+푦1

2]

= [4+22

7+32

]

= [62

102

] = (35)








Note

















apart





Theorem










Formulas



ퟐ풉


ퟑ


ퟑ

흅 = ퟐퟐퟕ






d =14cm

r= 7cm

흅 = ퟐퟐퟕ

h=10cm

l= 풓ퟐ + 풉ퟐ



l=radicퟏퟒퟗ

l=122



7 x 7 x 10

=440 sqcm

2휋푟(푟 + ℎ) =2 x 22

7 x 7(7+10)

=44 x 17 =748 sqcm

휋푟2ℎ =22

7 x 72 x 10

=1540cm3

Cone 휋푟푙 =22

7 x 7 x 122

=2684 sqcm

휋푟(푟 + 푙) =22

7 x 7 x ( 7 + 122 )

=22 x 192= 4224

13휋푟2ℎ

=13 x 22

7 x 72 x 10

=13 x 22

7 x 72 x 10

=5133 cm3

Sphere

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

43휋푟3

= 43

x 227

x 73 =14373 cm3


7 x 72

=462 sqcm

2휋푟2 =2 x 22

7 x 72

=308 sqcm

23휋푟3

= 23

x 22x 7

x 73 = 7186 cm3






TO D

80To E

150

100

80

30

70to C

40To B

From A


30m = 30 x = 15cm

70m = 70x = 35cm

80m = 80 x = 4cm

100m = 100x = 5cm

150m = 150x =75cm





To C E To D

120to D

E to E

220

210

120

80

40 to B

120toD

75to C

50to B

350

300

250

150

50

F 150toF

100to G

100toE

50toF

25toG

225

175

125

100

75

50

25toC

75toB











N + R = A + 2

N = 3 R = 4 A = 5 N+R = 3 +4 = 7 A+2 = 5 +2 = 7 there4 N+R = A+2

Exercise



N = 8 R = 6 A = 12 N+R = 8 +6 = 14 A+2 = 12 +2 = 14 there4 N+R = A+2

N = 3 R = 5 A = 6 N+R = 3 +5 = 8 A+2 = 6 +2 = 8 there4 N+R = A+2











1

4 4 6 4 +4 = 6 +2

2

3



4

5




1Real Numbers



AUB=BUA AcapB=BcapA







De Morganrsquos Law















ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ




If 12 X 1


A = 푎 + 푏2

X = 12 +

18

2

X = 4+18 2

X = 58 2

rArr X = 516



푎+푏

8 = 25푥5+푥

8(5+x) = 10x 40 +8x = 10x 40 = 2x X = 20



Combination 1 1 1 5





















7P3= 7(7minus3)

7P3= 7

4


7P3= 7x6x5 7P3= 210


1) 8C5 2) 6C3

7C3 = 7(7minus3)3

7C3 = 7

43

7C3 = 7x6x53x2x1

7C3 = 210

6




푛(푛minus1)2







15



r = 8 + 12 = 20


Some questions




6C2 -6


10C2


1) 5P3 2) 4P2x2P1



1) 8C2 2) 8C3



3C2 x 4C2



1) 8C5 2) 7C4 3) 7C5











S = (11)(12)(13)(14)(15)(16)(21)(22)(23) (24) (25)(26)(31)(32)(33)(34)(35)(36)(41) (42)(43)(44)(45)(46)(51)(52)(53) (54)(55) (56)(61)(62) (63)(64)(65)(66)

n(S) = 36




P(A) = n(A)n(S)









experiment












n(S) = 8C5 1) n(A) = 7C4 P(A) = 푛(퐴)

푛(푆)

2)n(B) =7C5 P(B) = 푛(퐵)푛(푆)




n(S) = 10C5


푛(푆)


푛(푆)




Un grouped data


흈 = sum푿ퟐ

풏 ndash ( sum푿

풏)ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂



흈 = sum풇풙ퟐ

풏 minus sum풇풙

풏

ퟐ 흈 = sum풇풅

ퟐ

풏 흈 = sum풇풅

ퟐ

풏 minus sum풇풅

풏

ퟐ 흈 = sum풇풅

ퟐ

풏 minus sum풇풅

풏

ퟐ 퐱퐂



For ungrouped data


퐂 d2



For grouped data


풙 d2 fd2

n = sumfx = sumfx2







fd d2 fd2

n = sumfd = sumfd2

= n= sumfd

= sumfd2=



퐂 d2


formula 32 1024 32 -2 4 32 -3 9 32 34 1156 34 0 0 34 -1 1 34 35 1225 35 1 1 35 0 0 35 36 1296 36 2 4 36 1 1 36 39 1521 39 5 25 39 4 16 39 42 1764 42 8 64 42 7 49 42

272 9476 272 228 -8 216 sumd= sumd2 =




rArr ퟐퟕퟐퟖ

=34 Assumed Mean 35


흈 = sum푿ퟐ

풏 ndash ( sum푿

풏)ퟐ 흈 =

sum퐝ퟐ

퐧

흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂



)ퟐ

휎 = 11845 ndash 1156

휎 = radic285

휎 = radic285

휎 = 534

흈 = ퟐퟐퟖퟖ


흈 = ퟓퟑퟒ

흈 =

ퟐퟏퟔퟖ

ndash ( ퟖퟖ

)ퟐ



흈 = radicퟐퟖ

흈 = ퟓퟐퟗ



1-5 2 3 6 9 18 1-5 2 3 6 -7 49 98 6-10 3 8 24 64 192 6-10 3 8 24 -2 4 12

11-15 4 13 52 169 676 11-15 4 13 52 3 9 36 16-20 1 18 18 324 324 16-20 1 18 18 8 64 64

10 100 1210 10 100 210




퐂 fd d2 fd2

1-5 2 3 -10 -20 100 200 1-5 2 3 -10 -2 -4 4 8 6-10 3 8 -5 -15 25 75 6-10 3 8 -5 -1 -3 1 3

11-15 4 13 0 0 0 0 11-15 4 13 0 0 0 0 0 16-20 1 18 5 5 25 25 16-20 1 18 5 1 1 1 1

10 -30 300 10 -6 12





흈 = sum풇풙ퟐ

풏 minus sum풇풙

풏

ퟐ



ퟐ


흈 = sum 풇풅ퟐ

풏



흈 = sum풇풅ퟐ

풏 minus sum풇풅

풏

ퟐ



ퟐ


흈 = sum풇풅ퟐ

풏 minus sum풇풅

풏

ퟐ 퐱퐂

흈 = ퟏퟐퟏퟎ


ퟐ 퐱ퟓ








퐗x100




CI 0-10 10-20 20-30 30-40 40-50 131 Freequency (f) 7 10 15 8 10

CI 5-15 15-25 25-35 35-45 45-55 55-65 134 Freequency (f) 8 12 20 10 7 3




휎 = 261 CV=4348


How the


Number of students

Central Angle

Walk 12 1236

x3600 = 1200

Cycle 8 836

x3600 = 800 Bus 3 3

36x3600 = 300

Car 4 436

x3600 = 400 School Van 9 9

36x3600 = 900

36 3600


7 Surds 2 4





Addition of Surds










12 rArr 2

12x3



13x2










radic5minusradic3

3radic5minusradic3



= 3(radic5+radic3)2


















P(x) =12 ndash 4 x 1 + 3 x 1 = 1 =1 - 4 + 3 + 1 = 1




1 (x ndash 2) ಇದು x3 -3x2 +6x -8


ೂೕ














x2 = 144 x2 ndash x = 0 x2 + 14x + 13 = 0 (x = -1) (x = -13)

4x = 81푥

x2 + 3 = 2x 7x2 -12x = 0 ( x = 13 )

7x = 647푥

x + 1x = 5 2m2 ndash 6m + 3 = 0 ( m = 1

2 )



K = 12m푣2

푣2=2퐾푚

v = plusmn 2퐾푚


2










3x2 ndash 5x + 2 = 0

3x2 ndash 5x + 2 = 0


3


3x = minus ퟐ

ퟑ

x2 - 53x = - 2

3

x2 - 53x +(5

6)2 = minus 2

3 + (5

6)2

(푥 minus 5 6

)2 minus 2436

+ 2536

(푥 minus 5 6

)2 = 136

(푥 minus 5 6

) = plusmn 16

x = 56 plusmn 1

6 rArr x = 6

6 or x = 4

6


3x2 ndash 5x + 2 = 0 a=3 b= -5 c = 2


2(3)


6


6

푥 =5 plusmn 1

6

푥 = 66 or x = 4

6

x = 1 or x = 23




Exercise






Nature of the Roots








푎 = minus2





1 = 1







1 2 ಮತು 3

2 6 ಮತು -5


4 -3 ಮತು 32




y

y = 2x2 x 0 +1 -1 +2 -2 +3 -3

y

y =ퟏퟐx2

x 0 +1 -1 +2 -2 +3 -3

y







10Similar triangles






= 퐵퐶퐸퐹










=23 AE=37 find

EC



To prove ADDB

= AEEC


Proof ∆ABC∆BDE

= 12 12

xADxELxDBxEL

[∵ A = 12

xbxh

∆ABC∆BDE

= ADDB

∆ADE∆CDE

= 12 12

xAExDNxDBxDN

[∵ A = 12

xbxh

∆ADE∆CDE

= AEEC

there4 퐀퐃

퐃퐁 = 퐀퐄






DE = BC

EF = CA

FD


there4 ABAG

= BCGH

= CA HA


there4 퐀퐁퐃퐄

= 퐁퐂퐄퐅

= 퐂퐀 퐅퐃






= BCEF

= CA DF


= 퐁퐂ퟐ

퐄퐅ퟐ



DM = AB

DE and BC

EF = AB

DE [ ∵ Given

there4 ALDM

= BCEF

helliphellip(1)


= 1212

xBCxALxEFxDM


= BCxALEFxDM

[ ∵ ( 1)

= BCxBCEFxEF

= 퐁퐂ퟐ

퐄퐅ퟐ

But ABDE

= BCEF

= CA DF

[ ∵ Given


= AB2

DE2 = BC2

EF2 = CA2

DF2






rArr ABAD

= ACAB


rArr BCDC

= ACBC








12Trigonometry



휽




퐜퐨퐬휽 = ퟏ

퐬퐞퐜 휽




퐴퐵퐴퐶














Values 00 300 450 600 900


ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0





13 tan 휃 = 5

12

Cosec휃 = 135

sec휃 = 1312

cot휃 = 125



rArr 3+2 = 5







ퟑ






radic3

tan300= 1radic3

rArr 휃 = 300





퐴퐵




푥2minus푥1

A(x1y1) and B(x2y2)


푥2minus푥1


tan 휃 = 7






푥2minus푥1

m1 = 5minus20minus5

= 3minus5


= 3minus5




휃 훼


푥2minus푥1


= minus7minus4

= 74


= 4minus7

m1 x m2 = 74 x 4

minus7 = -1








2 c = -3


2x -3rArr2y = x -6

rArr x -2y -6 =0









푚+푛]



2 푦2+푦1

2]

= [4+22

7+32

]

= [62

102

] = (35)








Note

















apart





Theorem










Formulas



ퟐ풉


ퟑ


ퟑ

흅 = ퟐퟐퟕ






d =14cm

r= 7cm

흅 = ퟐퟐퟕ

h=10cm

l= 풓ퟐ + 풉ퟐ



l=radicퟏퟒퟗ

l=122



7 x 7 x 10

=440 sqcm

2휋푟(푟 + ℎ) =2 x 22

7 x 7(7+10)

=44 x 17 =748 sqcm

휋푟2ℎ =22

7 x 72 x 10

=1540cm3

Cone 휋푟푙 =22

7 x 7 x 122

=2684 sqcm

휋푟(푟 + 푙) =22

7 x 7 x ( 7 + 122 )

=22 x 192= 4224

13휋푟2ℎ

=13 x 22

7 x 72 x 10

=13 x 22

7 x 72 x 10

=5133 cm3

Sphere

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

43휋푟3

= 43

x 227

x 73 =14373 cm3


7 x 72

=462 sqcm

2휋푟2 =2 x 22

7 x 72

=308 sqcm

23휋푟3

= 23

x 22x 7

x 73 = 7186 cm3






TO D

80To E

150

100

80

30

70to C

40To B

From A


30m = 30 x = 15cm

70m = 70x = 35cm

80m = 80 x = 4cm

100m = 100x = 5cm

150m = 150x =75cm





To C E To D

120to D

E to E

220

210

120

80

40 to B

120toD

75to C

50to B

350

300

250

150

50

F 150toF

100to G

100toE

50toF

25toG

225

175

125

100

75

50

25toC

75toB











N + R = A + 2

N = 3 R = 4 A = 5 N+R = 3 +4 = 7 A+2 = 5 +2 = 7 there4 N+R = A+2

Exercise



N = 8 R = 6 A = 12 N+R = 8 +6 = 14 A+2 = 12 +2 = 14 there4 N+R = A+2

N = 3 R = 5 A = 6 N+R = 3 +5 = 8 A+2 = 6 +2 = 8 there4 N+R = A+2











1

4 4 6 4 +4 = 6 +2

2

3



4

5




1Real Numbers



AUB=BUA AcapB=BcapA







De Morganrsquos Law















ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ



















7P3= 7(7minus3)

7P3= 7

4


7P3= 7x6x5 7P3= 210


1) 8C5 2) 6C3

7C3 = 7(7minus3)3

7C3 = 7

43

7C3 = 7x6x53x2x1

7C3 = 210

6




푛(푛minus1)2







15



r = 8 + 12 = 20


Some questions




6C2 -6


10C2


1) 5P3 2) 4P2x2P1



1) 8C2 2) 8C3



3C2 x 4C2



1) 8C5 2) 7C4 3) 7C5











S = (11)(12)(13)(14)(15)(16)(21)(22)(23) (24) (25)(26)(31)(32)(33)(34)(35)(36)(41) (42)(43)(44)(45)(46)(51)(52)(53) (54)(55) (56)(61)(62) (63)(64)(65)(66)

n(S) = 36




P(A) = n(A)n(S)









experiment












n(S) = 8C5 1) n(A) = 7C4 P(A) = 푛(퐴)

푛(푆)

2)n(B) =7C5 P(B) = 푛(퐵)푛(푆)




n(S) = 10C5


푛(푆)


푛(푆)




Un grouped data


흈 = sum푿ퟐ

풏 ndash ( sum푿

풏)ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂



흈 = sum풇풙ퟐ

풏 minus sum풇풙

풏

ퟐ 흈 = sum풇풅

ퟐ

풏 흈 = sum풇풅

ퟐ

풏 minus sum풇풅

풏

ퟐ 흈 = sum풇풅

ퟐ

풏 minus sum풇풅

풏

ퟐ 퐱퐂



For ungrouped data


퐂 d2



For grouped data


풙 d2 fd2

n = sumfx = sumfx2







fd d2 fd2

n = sumfd = sumfd2

= n= sumfd

= sumfd2=



퐂 d2


formula 32 1024 32 -2 4 32 -3 9 32 34 1156 34 0 0 34 -1 1 34 35 1225 35 1 1 35 0 0 35 36 1296 36 2 4 36 1 1 36 39 1521 39 5 25 39 4 16 39 42 1764 42 8 64 42 7 49 42

272 9476 272 228 -8 216 sumd= sumd2 =




rArr ퟐퟕퟐퟖ

=34 Assumed Mean 35


흈 = sum푿ퟐ

풏 ndash ( sum푿

풏)ퟐ 흈 =

sum퐝ퟐ

퐧

흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂



)ퟐ

휎 = 11845 ndash 1156

휎 = radic285

휎 = radic285

휎 = 534

흈 = ퟐퟐퟖퟖ


흈 = ퟓퟑퟒ

흈 =

ퟐퟏퟔퟖ

ndash ( ퟖퟖ

)ퟐ



흈 = radicퟐퟖ

흈 = ퟓퟐퟗ



1-5 2 3 6 9 18 1-5 2 3 6 -7 49 98 6-10 3 8 24 64 192 6-10 3 8 24 -2 4 12

11-15 4 13 52 169 676 11-15 4 13 52 3 9 36 16-20 1 18 18 324 324 16-20 1 18 18 8 64 64

10 100 1210 10 100 210




퐂 fd d2 fd2

1-5 2 3 -10 -20 100 200 1-5 2 3 -10 -2 -4 4 8 6-10 3 8 -5 -15 25 75 6-10 3 8 -5 -1 -3 1 3

11-15 4 13 0 0 0 0 11-15 4 13 0 0 0 0 0 16-20 1 18 5 5 25 25 16-20 1 18 5 1 1 1 1

10 -30 300 10 -6 12





흈 = sum풇풙ퟐ

풏 minus sum풇풙

풏

ퟐ



ퟐ


흈 = sum 풇풅ퟐ

풏



흈 = sum풇풅ퟐ

풏 minus sum풇풅

풏

ퟐ



ퟐ


흈 = sum풇풅ퟐ

풏 minus sum풇풅

풏

ퟐ 퐱퐂

흈 = ퟏퟐퟏퟎ


ퟐ 퐱ퟓ








퐗x100




CI 0-10 10-20 20-30 30-40 40-50 131 Freequency (f) 7 10 15 8 10

CI 5-15 15-25 25-35 35-45 45-55 55-65 134 Freequency (f) 8 12 20 10 7 3




휎 = 261 CV=4348


How the


Number of students

Central Angle

Walk 12 1236

x3600 = 1200

Cycle 8 836

x3600 = 800 Bus 3 3

36x3600 = 300

Car 4 436

x3600 = 400 School Van 9 9

36x3600 = 900

36 3600


7 Surds 2 4





Addition of Surds










12 rArr 2

12x3



13x2










radic5minusradic3

3radic5minusradic3



= 3(radic5+radic3)2


















P(x) =12 ndash 4 x 1 + 3 x 1 = 1 =1 - 4 + 3 + 1 = 1




1 (x ndash 2) ಇದು x3 -3x2 +6x -8


ೂೕ














x2 = 144 x2 ndash x = 0 x2 + 14x + 13 = 0 (x = -1) (x = -13)

4x = 81푥

x2 + 3 = 2x 7x2 -12x = 0 ( x = 13 )

7x = 647푥

x + 1x = 5 2m2 ndash 6m + 3 = 0 ( m = 1

2 )



K = 12m푣2

푣2=2퐾푚

v = plusmn 2퐾푚


2










3x2 ndash 5x + 2 = 0

3x2 ndash 5x + 2 = 0


3


3x = minus ퟐ

ퟑ

x2 - 53x = - 2

3

x2 - 53x +(5

6)2 = minus 2

3 + (5

6)2

(푥 minus 5 6

)2 minus 2436

+ 2536

(푥 minus 5 6

)2 = 136

(푥 minus 5 6

) = plusmn 16

x = 56 plusmn 1

6 rArr x = 6

6 or x = 4

6


3x2 ndash 5x + 2 = 0 a=3 b= -5 c = 2


2(3)


6


6

푥 =5 plusmn 1

6

푥 = 66 or x = 4

6

x = 1 or x = 23




Exercise






Nature of the Roots








푎 = minus2





1 = 1







1 2 ಮತು 3

2 6 ಮತು -5


4 -3 ಮತು 32




y

y = 2x2 x 0 +1 -1 +2 -2 +3 -3

y

y =ퟏퟐx2

x 0 +1 -1 +2 -2 +3 -3

y







10Similar triangles






= 퐵퐶퐸퐹










=23 AE=37 find

EC



To prove ADDB

= AEEC


Proof ∆ABC∆BDE

= 12 12

xADxELxDBxEL

[∵ A = 12

xbxh

∆ABC∆BDE

= ADDB

∆ADE∆CDE

= 12 12

xAExDNxDBxDN

[∵ A = 12

xbxh

∆ADE∆CDE

= AEEC

there4 퐀퐃

퐃퐁 = 퐀퐄






DE = BC

EF = CA

FD


there4 ABAG

= BCGH

= CA HA


there4 퐀퐁퐃퐄

= 퐁퐂퐄퐅

= 퐂퐀 퐅퐃






= BCEF

= CA DF


= 퐁퐂ퟐ

퐄퐅ퟐ



DM = AB

DE and BC

EF = AB

DE [ ∵ Given

there4 ALDM

= BCEF

helliphellip(1)


= 1212

xBCxALxEFxDM


= BCxALEFxDM

[ ∵ ( 1)

= BCxBCEFxEF

= 퐁퐂ퟐ

퐄퐅ퟐ

But ABDE

= BCEF

= CA DF

[ ∵ Given


= AB2

DE2 = BC2

EF2 = CA2

DF2






rArr ABAD

= ACAB


rArr BCDC

= ACBC








12Trigonometry



휽




퐜퐨퐬휽 = ퟏ

퐬퐞퐜 휽




퐴퐵퐴퐶














Values 00 300 450 600 900


ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0





13 tan 휃 = 5

12

Cosec휃 = 135

sec휃 = 1312

cot휃 = 125



rArr 3+2 = 5







ퟑ






radic3

tan300= 1radic3

rArr 휃 = 300





퐴퐵




푥2minus푥1

A(x1y1) and B(x2y2)


푥2minus푥1


tan 휃 = 7






푥2minus푥1

m1 = 5minus20minus5

= 3minus5


= 3minus5




휃 훼


푥2minus푥1


= minus7minus4

= 74


= 4minus7

m1 x m2 = 74 x 4

minus7 = -1








2 c = -3


2x -3rArr2y = x -6

rArr x -2y -6 =0









푚+푛]



2 푦2+푦1

2]

= [4+22

7+32

]

= [62

102

] = (35)








Note

















apart





Theorem










Formulas



ퟐ풉


ퟑ


ퟑ

흅 = ퟐퟐퟕ






d =14cm

r= 7cm

흅 = ퟐퟐퟕ

h=10cm

l= 풓ퟐ + 풉ퟐ



l=radicퟏퟒퟗ

l=122



7 x 7 x 10

=440 sqcm

2휋푟(푟 + ℎ) =2 x 22

7 x 7(7+10)

=44 x 17 =748 sqcm

휋푟2ℎ =22

7 x 72 x 10

=1540cm3

Cone 휋푟푙 =22

7 x 7 x 122

=2684 sqcm

휋푟(푟 + 푙) =22

7 x 7 x ( 7 + 122 )

=22 x 192= 4224

13휋푟2ℎ

=13 x 22

7 x 72 x 10

=13 x 22

7 x 72 x 10

=5133 cm3

Sphere

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

43휋푟3

= 43

x 227

x 73 =14373 cm3


7 x 72

=462 sqcm

2휋푟2 =2 x 22

7 x 72

=308 sqcm

23휋푟3

= 23

x 22x 7

x 73 = 7186 cm3






TO D

80To E

150

100

80

30

70to C

40To B

From A


30m = 30 x = 15cm

70m = 70x = 35cm

80m = 80 x = 4cm

100m = 100x = 5cm

150m = 150x =75cm





To C E To D

120to D

E to E

220

210

120

80

40 to B

120toD

75to C

50to B

350

300

250

150

50

F 150toF

100to G

100toE

50toF

25toG

225

175

125

100

75

50

25toC

75toB











N + R = A + 2

N = 3 R = 4 A = 5 N+R = 3 +4 = 7 A+2 = 5 +2 = 7 there4 N+R = A+2

Exercise



N = 8 R = 6 A = 12 N+R = 8 +6 = 14 A+2 = 12 +2 = 14 there4 N+R = A+2

N = 3 R = 5 A = 6 N+R = 3 +5 = 8 A+2 = 6 +2 = 8 there4 N+R = A+2











1

4 4 6 4 +4 = 6 +2

2

3



4

5




1Real Numbers



AUB=BUA AcapB=BcapA







De Morganrsquos Law















ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ






15



r = 8 + 12 = 20


Some questions




6C2 -6


10C2


1) 5P3 2) 4P2x2P1



1) 8C2 2) 8C3



3C2 x 4C2



1) 8C5 2) 7C4 3) 7C5











S = (11)(12)(13)(14)(15)(16)(21)(22)(23) (24) (25)(26)(31)(32)(33)(34)(35)(36)(41) (42)(43)(44)(45)(46)(51)(52)(53) (54)(55) (56)(61)(62) (63)(64)(65)(66)

n(S) = 36




P(A) = n(A)n(S)









experiment












n(S) = 8C5 1) n(A) = 7C4 P(A) = 푛(퐴)

푛(푆)

2)n(B) =7C5 P(B) = 푛(퐵)푛(푆)




n(S) = 10C5


푛(푆)


푛(푆)




Un grouped data


흈 = sum푿ퟐ

풏 ndash ( sum푿

풏)ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂



흈 = sum풇풙ퟐ

풏 minus sum풇풙

풏

ퟐ 흈 = sum풇풅

ퟐ

풏 흈 = sum풇풅

ퟐ

풏 minus sum풇풅

풏

ퟐ 흈 = sum풇풅

ퟐ

풏 minus sum풇풅

풏

ퟐ 퐱퐂



For ungrouped data


퐂 d2



For grouped data


풙 d2 fd2

n = sumfx = sumfx2







fd d2 fd2

n = sumfd = sumfd2

= n= sumfd

= sumfd2=



퐂 d2


formula 32 1024 32 -2 4 32 -3 9 32 34 1156 34 0 0 34 -1 1 34 35 1225 35 1 1 35 0 0 35 36 1296 36 2 4 36 1 1 36 39 1521 39 5 25 39 4 16 39 42 1764 42 8 64 42 7 49 42

272 9476 272 228 -8 216 sumd= sumd2 =




rArr ퟐퟕퟐퟖ

=34 Assumed Mean 35


흈 = sum푿ퟐ

풏 ndash ( sum푿

풏)ퟐ 흈 =

sum퐝ퟐ

퐧

흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂



)ퟐ

휎 = 11845 ndash 1156

휎 = radic285

휎 = radic285

휎 = 534

흈 = ퟐퟐퟖퟖ


흈 = ퟓퟑퟒ

흈 =

ퟐퟏퟔퟖ

ndash ( ퟖퟖ

)ퟐ



흈 = radicퟐퟖ

흈 = ퟓퟐퟗ



1-5 2 3 6 9 18 1-5 2 3 6 -7 49 98 6-10 3 8 24 64 192 6-10 3 8 24 -2 4 12

11-15 4 13 52 169 676 11-15 4 13 52 3 9 36 16-20 1 18 18 324 324 16-20 1 18 18 8 64 64

10 100 1210 10 100 210




퐂 fd d2 fd2

1-5 2 3 -10 -20 100 200 1-5 2 3 -10 -2 -4 4 8 6-10 3 8 -5 -15 25 75 6-10 3 8 -5 -1 -3 1 3

11-15 4 13 0 0 0 0 11-15 4 13 0 0 0 0 0 16-20 1 18 5 5 25 25 16-20 1 18 5 1 1 1 1

10 -30 300 10 -6 12





흈 = sum풇풙ퟐ

풏 minus sum풇풙

풏

ퟐ



ퟐ


흈 = sum 풇풅ퟐ

풏



흈 = sum풇풅ퟐ

풏 minus sum풇풅

풏

ퟐ



ퟐ


흈 = sum풇풅ퟐ

풏 minus sum풇풅

풏

ퟐ 퐱퐂

흈 = ퟏퟐퟏퟎ


ퟐ 퐱ퟓ








퐗x100




CI 0-10 10-20 20-30 30-40 40-50 131 Freequency (f) 7 10 15 8 10

CI 5-15 15-25 25-35 35-45 45-55 55-65 134 Freequency (f) 8 12 20 10 7 3




휎 = 261 CV=4348


How the


Number of students

Central Angle

Walk 12 1236

x3600 = 1200

Cycle 8 836

x3600 = 800 Bus 3 3

36x3600 = 300

Car 4 436

x3600 = 400 School Van 9 9

36x3600 = 900

36 3600


7 Surds 2 4





Addition of Surds










12 rArr 2

12x3



13x2










radic5minusradic3

3radic5minusradic3



= 3(radic5+radic3)2


















P(x) =12 ndash 4 x 1 + 3 x 1 = 1 =1 - 4 + 3 + 1 = 1




1 (x ndash 2) ಇದು x3 -3x2 +6x -8


ೂೕ














x2 = 144 x2 ndash x = 0 x2 + 14x + 13 = 0 (x = -1) (x = -13)

4x = 81푥

x2 + 3 = 2x 7x2 -12x = 0 ( x = 13 )

7x = 647푥

x + 1x = 5 2m2 ndash 6m + 3 = 0 ( m = 1

2 )



K = 12m푣2

푣2=2퐾푚

v = plusmn 2퐾푚


2










3x2 ndash 5x + 2 = 0

3x2 ndash 5x + 2 = 0


3


3x = minus ퟐ

ퟑ

x2 - 53x = - 2

3

x2 - 53x +(5

6)2 = minus 2

3 + (5

6)2

(푥 minus 5 6

)2 minus 2436

+ 2536

(푥 minus 5 6

)2 = 136

(푥 minus 5 6

) = plusmn 16

x = 56 plusmn 1

6 rArr x = 6

6 or x = 4

6


3x2 ndash 5x + 2 = 0 a=3 b= -5 c = 2


2(3)


6


6

푥 =5 plusmn 1

6

푥 = 66 or x = 4

6

x = 1 or x = 23




Exercise






Nature of the Roots








푎 = minus2





1 = 1







1 2 ಮತು 3

2 6 ಮತು -5


4 -3 ಮತು 32




y

y = 2x2 x 0 +1 -1 +2 -2 +3 -3

y

y =ퟏퟐx2

x 0 +1 -1 +2 -2 +3 -3

y







10Similar triangles






= 퐵퐶퐸퐹










=23 AE=37 find

EC



To prove ADDB

= AEEC


Proof ∆ABC∆BDE

= 12 12

xADxELxDBxEL

[∵ A = 12

xbxh

∆ABC∆BDE

= ADDB

∆ADE∆CDE

= 12 12

xAExDNxDBxDN

[∵ A = 12

xbxh

∆ADE∆CDE

= AEEC

there4 퐀퐃

퐃퐁 = 퐀퐄






DE = BC

EF = CA

FD


there4 ABAG

= BCGH

= CA HA


there4 퐀퐁퐃퐄

= 퐁퐂퐄퐅

= 퐂퐀 퐅퐃






= BCEF

= CA DF


= 퐁퐂ퟐ

퐄퐅ퟐ



DM = AB

DE and BC

EF = AB

DE [ ∵ Given

there4 ALDM

= BCEF

helliphellip(1)


= 1212

xBCxALxEFxDM


= BCxALEFxDM

[ ∵ ( 1)

= BCxBCEFxEF

= 퐁퐂ퟐ

퐄퐅ퟐ

But ABDE

= BCEF

= CA DF

[ ∵ Given


= AB2

DE2 = BC2

EF2 = CA2

DF2






rArr ABAD

= ACAB


rArr BCDC

= ACBC








12Trigonometry



휽




퐜퐨퐬휽 = ퟏ

퐬퐞퐜 휽




퐴퐵퐴퐶














Values 00 300 450 600 900


ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0





13 tan 휃 = 5

12

Cosec휃 = 135

sec휃 = 1312

cot휃 = 125



rArr 3+2 = 5







ퟑ






radic3

tan300= 1radic3

rArr 휃 = 300





퐴퐵




푥2minus푥1

A(x1y1) and B(x2y2)


푥2minus푥1


tan 휃 = 7






푥2minus푥1

m1 = 5minus20minus5

= 3minus5


= 3minus5




휃 훼


푥2minus푥1


= minus7minus4

= 74


= 4minus7

m1 x m2 = 74 x 4

minus7 = -1








2 c = -3


2x -3rArr2y = x -6

rArr x -2y -6 =0









푚+푛]



2 푦2+푦1

2]

= [4+22

7+32

]

= [62

102

] = (35)








Note

















apart





Theorem










Formulas



ퟐ풉


ퟑ


ퟑ

흅 = ퟐퟐퟕ






d =14cm

r= 7cm

흅 = ퟐퟐퟕ

h=10cm

l= 풓ퟐ + 풉ퟐ



l=radicퟏퟒퟗ

l=122



7 x 7 x 10

=440 sqcm

2휋푟(푟 + ℎ) =2 x 22

7 x 7(7+10)

=44 x 17 =748 sqcm

휋푟2ℎ =22

7 x 72 x 10

=1540cm3

Cone 휋푟푙 =22

7 x 7 x 122

=2684 sqcm

휋푟(푟 + 푙) =22

7 x 7 x ( 7 + 122 )

=22 x 192= 4224

13휋푟2ℎ

=13 x 22

7 x 72 x 10

=13 x 22

7 x 72 x 10

=5133 cm3

Sphere

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

43휋푟3

= 43

x 227

x 73 =14373 cm3


7 x 72

=462 sqcm

2휋푟2 =2 x 22

7 x 72

=308 sqcm

23휋푟3

= 23

x 22x 7

x 73 = 7186 cm3






TO D

80To E

150

100

80

30

70to C

40To B

From A


30m = 30 x = 15cm

70m = 70x = 35cm

80m = 80 x = 4cm

100m = 100x = 5cm

150m = 150x =75cm





To C E To D

120to D

E to E

220

210

120

80

40 to B

120toD

75to C

50to B

350

300

250

150

50

F 150toF

100to G

100toE

50toF

25toG

225

175

125

100

75

50

25toC

75toB











N + R = A + 2

N = 3 R = 4 A = 5 N+R = 3 +4 = 7 A+2 = 5 +2 = 7 there4 N+R = A+2

Exercise



N = 8 R = 6 A = 12 N+R = 8 +6 = 14 A+2 = 12 +2 = 14 there4 N+R = A+2

N = 3 R = 5 A = 6 N+R = 3 +5 = 8 A+2 = 6 +2 = 8 there4 N+R = A+2











1

4 4 6 4 +4 = 6 +2

2

3



4

5




1Real Numbers



AUB=BUA AcapB=BcapA







De Morganrsquos Law















ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ










S = (11)(12)(13)(14)(15)(16)(21)(22)(23) (24) (25)(26)(31)(32)(33)(34)(35)(36)(41) (42)(43)(44)(45)(46)(51)(52)(53) (54)(55) (56)(61)(62) (63)(64)(65)(66)

n(S) = 36




P(A) = n(A)n(S)









experiment












n(S) = 8C5 1) n(A) = 7C4 P(A) = 푛(퐴)

푛(푆)

2)n(B) =7C5 P(B) = 푛(퐵)푛(푆)




n(S) = 10C5


푛(푆)


푛(푆)




Un grouped data


흈 = sum푿ퟐ

풏 ndash ( sum푿

풏)ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂



흈 = sum풇풙ퟐ

풏 minus sum풇풙

풏

ퟐ 흈 = sum풇풅

ퟐ

풏 흈 = sum풇풅

ퟐ

풏 minus sum풇풅

풏

ퟐ 흈 = sum풇풅

ퟐ

풏 minus sum풇풅

풏

ퟐ 퐱퐂



For ungrouped data


퐂 d2



For grouped data


풙 d2 fd2

n = sumfx = sumfx2







fd d2 fd2

n = sumfd = sumfd2

= n= sumfd

= sumfd2=



퐂 d2


formula 32 1024 32 -2 4 32 -3 9 32 34 1156 34 0 0 34 -1 1 34 35 1225 35 1 1 35 0 0 35 36 1296 36 2 4 36 1 1 36 39 1521 39 5 25 39 4 16 39 42 1764 42 8 64 42 7 49 42

272 9476 272 228 -8 216 sumd= sumd2 =




rArr ퟐퟕퟐퟖ

=34 Assumed Mean 35


흈 = sum푿ퟐ

풏 ndash ( sum푿

풏)ퟐ 흈 =

sum퐝ퟐ

퐧

흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂



)ퟐ

휎 = 11845 ndash 1156

휎 = radic285

휎 = radic285

휎 = 534

흈 = ퟐퟐퟖퟖ


흈 = ퟓퟑퟒ

흈 =

ퟐퟏퟔퟖ

ndash ( ퟖퟖ

)ퟐ



흈 = radicퟐퟖ

흈 = ퟓퟐퟗ



1-5 2 3 6 9 18 1-5 2 3 6 -7 49 98 6-10 3 8 24 64 192 6-10 3 8 24 -2 4 12

11-15 4 13 52 169 676 11-15 4 13 52 3 9 36 16-20 1 18 18 324 324 16-20 1 18 18 8 64 64

10 100 1210 10 100 210




퐂 fd d2 fd2

1-5 2 3 -10 -20 100 200 1-5 2 3 -10 -2 -4 4 8 6-10 3 8 -5 -15 25 75 6-10 3 8 -5 -1 -3 1 3

11-15 4 13 0 0 0 0 11-15 4 13 0 0 0 0 0 16-20 1 18 5 5 25 25 16-20 1 18 5 1 1 1 1

10 -30 300 10 -6 12





흈 = sum풇풙ퟐ

풏 minus sum풇풙

풏

ퟐ



ퟐ


흈 = sum 풇풅ퟐ

풏



흈 = sum풇풅ퟐ

풏 minus sum풇풅

풏

ퟐ



ퟐ


흈 = sum풇풅ퟐ

풏 minus sum풇풅

풏

ퟐ 퐱퐂

흈 = ퟏퟐퟏퟎ


ퟐ 퐱ퟓ








퐗x100




CI 0-10 10-20 20-30 30-40 40-50 131 Freequency (f) 7 10 15 8 10

CI 5-15 15-25 25-35 35-45 45-55 55-65 134 Freequency (f) 8 12 20 10 7 3




휎 = 261 CV=4348


How the


Number of students

Central Angle

Walk 12 1236

x3600 = 1200

Cycle 8 836

x3600 = 800 Bus 3 3

36x3600 = 300

Car 4 436

x3600 = 400 School Van 9 9

36x3600 = 900

36 3600


7 Surds 2 4





Addition of Surds










12 rArr 2

12x3



13x2










radic5minusradic3

3radic5minusradic3



= 3(radic5+radic3)2


















P(x) =12 ndash 4 x 1 + 3 x 1 = 1 =1 - 4 + 3 + 1 = 1




1 (x ndash 2) ಇದು x3 -3x2 +6x -8


ೂೕ














x2 = 144 x2 ndash x = 0 x2 + 14x + 13 = 0 (x = -1) (x = -13)

4x = 81푥

x2 + 3 = 2x 7x2 -12x = 0 ( x = 13 )

7x = 647푥

x + 1x = 5 2m2 ndash 6m + 3 = 0 ( m = 1

2 )



K = 12m푣2

푣2=2퐾푚

v = plusmn 2퐾푚


2










3x2 ndash 5x + 2 = 0

3x2 ndash 5x + 2 = 0


3


3x = minus ퟐ

ퟑ

x2 - 53x = - 2

3

x2 - 53x +(5

6)2 = minus 2

3 + (5

6)2

(푥 minus 5 6

)2 minus 2436

+ 2536

(푥 minus 5 6

)2 = 136

(푥 minus 5 6

) = plusmn 16

x = 56 plusmn 1

6 rArr x = 6

6 or x = 4

6


3x2 ndash 5x + 2 = 0 a=3 b= -5 c = 2


2(3)


6


6

푥 =5 plusmn 1

6

푥 = 66 or x = 4

6

x = 1 or x = 23




Exercise






Nature of the Roots








푎 = minus2





1 = 1







1 2 ಮತು 3

2 6 ಮತು -5


4 -3 ಮತು 32




y

y = 2x2 x 0 +1 -1 +2 -2 +3 -3

y

y =ퟏퟐx2

x 0 +1 -1 +2 -2 +3 -3

y







10Similar triangles






= 퐵퐶퐸퐹










=23 AE=37 find

EC



To prove ADDB

= AEEC


Proof ∆ABC∆BDE

= 12 12

xADxELxDBxEL

[∵ A = 12

xbxh

∆ABC∆BDE

= ADDB

∆ADE∆CDE

= 12 12

xAExDNxDBxDN

[∵ A = 12

xbxh

∆ADE∆CDE

= AEEC

there4 퐀퐃

퐃퐁 = 퐀퐄






DE = BC

EF = CA

FD


there4 ABAG

= BCGH

= CA HA


there4 퐀퐁퐃퐄

= 퐁퐂퐄퐅

= 퐂퐀 퐅퐃






= BCEF

= CA DF


= 퐁퐂ퟐ

퐄퐅ퟐ



DM = AB

DE and BC

EF = AB

DE [ ∵ Given

there4 ALDM

= BCEF

helliphellip(1)


= 1212

xBCxALxEFxDM


= BCxALEFxDM

[ ∵ ( 1)

= BCxBCEFxEF

= 퐁퐂ퟐ

퐄퐅ퟐ

But ABDE

= BCEF

= CA DF

[ ∵ Given


= AB2

DE2 = BC2

EF2 = CA2

DF2






rArr ABAD

= ACAB


rArr BCDC

= ACBC








12Trigonometry



휽




퐜퐨퐬휽 = ퟏ

퐬퐞퐜 휽




퐴퐵퐴퐶














Values 00 300 450 600 900


ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0





13 tan 휃 = 5

12

Cosec휃 = 135

sec휃 = 1312

cot휃 = 125



rArr 3+2 = 5







ퟑ






radic3

tan300= 1radic3

rArr 휃 = 300





퐴퐵




푥2minus푥1

A(x1y1) and B(x2y2)


푥2minus푥1


tan 휃 = 7






푥2minus푥1

m1 = 5minus20minus5

= 3minus5


= 3minus5




휃 훼


푥2minus푥1


= minus7minus4

= 74


= 4minus7

m1 x m2 = 74 x 4

minus7 = -1








2 c = -3


2x -3rArr2y = x -6

rArr x -2y -6 =0









푚+푛]



2 푦2+푦1

2]

= [4+22

7+32

]

= [62

102

] = (35)








Note

















apart





Theorem










Formulas



ퟐ풉


ퟑ


ퟑ

흅 = ퟐퟐퟕ






d =14cm

r= 7cm

흅 = ퟐퟐퟕ

h=10cm

l= 풓ퟐ + 풉ퟐ



l=radicퟏퟒퟗ

l=122



7 x 7 x 10

=440 sqcm

2휋푟(푟 + ℎ) =2 x 22

7 x 7(7+10)

=44 x 17 =748 sqcm

휋푟2ℎ =22

7 x 72 x 10

=1540cm3

Cone 휋푟푙 =22

7 x 7 x 122

=2684 sqcm

휋푟(푟 + 푙) =22

7 x 7 x ( 7 + 122 )

=22 x 192= 4224

13휋푟2ℎ

=13 x 22

7 x 72 x 10

=13 x 22

7 x 72 x 10

=5133 cm3

Sphere

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

43휋푟3

= 43

x 227

x 73 =14373 cm3


7 x 72

=462 sqcm

2휋푟2 =2 x 22

7 x 72

=308 sqcm

23휋푟3

= 23

x 22x 7

x 73 = 7186 cm3






TO D

80To E

150

100

80

30

70to C

40To B

From A


30m = 30 x = 15cm

70m = 70x = 35cm

80m = 80 x = 4cm

100m = 100x = 5cm

150m = 150x =75cm





To C E To D

120to D

E to E

220

210

120

80

40 to B

120toD

75to C

50to B

350

300

250

150

50

F 150toF

100to G

100toE

50toF

25toG

225

175

125

100

75

50

25toC

75toB











N + R = A + 2

N = 3 R = 4 A = 5 N+R = 3 +4 = 7 A+2 = 5 +2 = 7 there4 N+R = A+2

Exercise



N = 8 R = 6 A = 12 N+R = 8 +6 = 14 A+2 = 12 +2 = 14 there4 N+R = A+2

N = 3 R = 5 A = 6 N+R = 3 +5 = 8 A+2 = 6 +2 = 8 there4 N+R = A+2











1

4 4 6 4 +4 = 6 +2

2

3



4

5




1Real Numbers



AUB=BUA AcapB=BcapA







De Morganrsquos Law















ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ





P(A) = n(A)n(S)









experiment












n(S) = 8C5 1) n(A) = 7C4 P(A) = 푛(퐴)

푛(푆)

2)n(B) =7C5 P(B) = 푛(퐵)푛(푆)




n(S) = 10C5


푛(푆)


푛(푆)




Un grouped data


흈 = sum푿ퟐ

풏 ndash ( sum푿

풏)ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂



흈 = sum풇풙ퟐ

풏 minus sum풇풙

풏

ퟐ 흈 = sum풇풅

ퟐ

풏 흈 = sum풇풅

ퟐ

풏 minus sum풇풅

풏

ퟐ 흈 = sum풇풅

ퟐ

풏 minus sum풇풅

풏

ퟐ 퐱퐂



For ungrouped data


퐂 d2



For grouped data


풙 d2 fd2

n = sumfx = sumfx2







fd d2 fd2

n = sumfd = sumfd2

= n= sumfd

= sumfd2=



퐂 d2


formula 32 1024 32 -2 4 32 -3 9 32 34 1156 34 0 0 34 -1 1 34 35 1225 35 1 1 35 0 0 35 36 1296 36 2 4 36 1 1 36 39 1521 39 5 25 39 4 16 39 42 1764 42 8 64 42 7 49 42

272 9476 272 228 -8 216 sumd= sumd2 =




rArr ퟐퟕퟐퟖ

=34 Assumed Mean 35


흈 = sum푿ퟐ

풏 ndash ( sum푿

풏)ퟐ 흈 =

sum퐝ퟐ

퐧

흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂



)ퟐ

휎 = 11845 ndash 1156

휎 = radic285

휎 = radic285

휎 = 534

흈 = ퟐퟐퟖퟖ


흈 = ퟓퟑퟒ

흈 =

ퟐퟏퟔퟖ

ndash ( ퟖퟖ

)ퟐ



흈 = radicퟐퟖ

흈 = ퟓퟐퟗ



1-5 2 3 6 9 18 1-5 2 3 6 -7 49 98 6-10 3 8 24 64 192 6-10 3 8 24 -2 4 12

11-15 4 13 52 169 676 11-15 4 13 52 3 9 36 16-20 1 18 18 324 324 16-20 1 18 18 8 64 64

10 100 1210 10 100 210




퐂 fd d2 fd2

1-5 2 3 -10 -20 100 200 1-5 2 3 -10 -2 -4 4 8 6-10 3 8 -5 -15 25 75 6-10 3 8 -5 -1 -3 1 3

11-15 4 13 0 0 0 0 11-15 4 13 0 0 0 0 0 16-20 1 18 5 5 25 25 16-20 1 18 5 1 1 1 1

10 -30 300 10 -6 12





흈 = sum풇풙ퟐ

풏 minus sum풇풙

풏

ퟐ



ퟐ


흈 = sum 풇풅ퟐ

풏



흈 = sum풇풅ퟐ

풏 minus sum풇풅

풏

ퟐ



ퟐ


흈 = sum풇풅ퟐ

풏 minus sum풇풅

풏

ퟐ 퐱퐂

흈 = ퟏퟐퟏퟎ


ퟐ 퐱ퟓ








퐗x100




CI 0-10 10-20 20-30 30-40 40-50 131 Freequency (f) 7 10 15 8 10

CI 5-15 15-25 25-35 35-45 45-55 55-65 134 Freequency (f) 8 12 20 10 7 3




휎 = 261 CV=4348


How the


Number of students

Central Angle

Walk 12 1236

x3600 = 1200

Cycle 8 836

x3600 = 800 Bus 3 3

36x3600 = 300

Car 4 436

x3600 = 400 School Van 9 9

36x3600 = 900

36 3600


7 Surds 2 4





Addition of Surds










12 rArr 2

12x3



13x2










radic5minusradic3

3radic5minusradic3



= 3(radic5+radic3)2


















P(x) =12 ndash 4 x 1 + 3 x 1 = 1 =1 - 4 + 3 + 1 = 1




1 (x ndash 2) ಇದು x3 -3x2 +6x -8


ೂೕ














x2 = 144 x2 ndash x = 0 x2 + 14x + 13 = 0 (x = -1) (x = -13)

4x = 81푥

x2 + 3 = 2x 7x2 -12x = 0 ( x = 13 )

7x = 647푥

x + 1x = 5 2m2 ndash 6m + 3 = 0 ( m = 1

2 )



K = 12m푣2

푣2=2퐾푚

v = plusmn 2퐾푚


2










3x2 ndash 5x + 2 = 0

3x2 ndash 5x + 2 = 0


3


3x = minus ퟐ

ퟑ

x2 - 53x = - 2

3

x2 - 53x +(5

6)2 = minus 2

3 + (5

6)2

(푥 minus 5 6

)2 minus 2436

+ 2536

(푥 minus 5 6

)2 = 136

(푥 minus 5 6

) = plusmn 16

x = 56 plusmn 1

6 rArr x = 6

6 or x = 4

6


3x2 ndash 5x + 2 = 0 a=3 b= -5 c = 2


2(3)


6


6

푥 =5 plusmn 1

6

푥 = 66 or x = 4

6

x = 1 or x = 23




Exercise






Nature of the Roots








푎 = minus2





1 = 1







1 2 ಮತು 3

2 6 ಮತು -5


4 -3 ಮತು 32




y

y = 2x2 x 0 +1 -1 +2 -2 +3 -3

y

y =ퟏퟐx2

x 0 +1 -1 +2 -2 +3 -3

y







10Similar triangles






= 퐵퐶퐸퐹










=23 AE=37 find

EC



To prove ADDB

= AEEC


Proof ∆ABC∆BDE

= 12 12

xADxELxDBxEL

[∵ A = 12

xbxh

∆ABC∆BDE

= ADDB

∆ADE∆CDE

= 12 12

xAExDNxDBxDN

[∵ A = 12

xbxh

∆ADE∆CDE

= AEEC

there4 퐀퐃

퐃퐁 = 퐀퐄






DE = BC

EF = CA

FD


there4 ABAG

= BCGH

= CA HA


there4 퐀퐁퐃퐄

= 퐁퐂퐄퐅

= 퐂퐀 퐅퐃






= BCEF

= CA DF


= 퐁퐂ퟐ

퐄퐅ퟐ



DM = AB

DE and BC

EF = AB

DE [ ∵ Given

there4 ALDM

= BCEF

helliphellip(1)


= 1212

xBCxALxEFxDM


= BCxALEFxDM

[ ∵ ( 1)

= BCxBCEFxEF

= 퐁퐂ퟐ

퐄퐅ퟐ

But ABDE

= BCEF

= CA DF

[ ∵ Given


= AB2

DE2 = BC2

EF2 = CA2

DF2






rArr ABAD

= ACAB


rArr BCDC

= ACBC








12Trigonometry



휽




퐜퐨퐬휽 = ퟏ

퐬퐞퐜 휽




퐴퐵퐴퐶














Values 00 300 450 600 900


ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0





13 tan 휃 = 5

12

Cosec휃 = 135

sec휃 = 1312

cot휃 = 125



rArr 3+2 = 5







ퟑ






radic3

tan300= 1radic3

rArr 휃 = 300





퐴퐵




푥2minus푥1

A(x1y1) and B(x2y2)


푥2minus푥1


tan 휃 = 7






푥2minus푥1

m1 = 5minus20minus5

= 3minus5


= 3minus5




휃 훼


푥2minus푥1


= minus7minus4

= 74


= 4minus7

m1 x m2 = 74 x 4

minus7 = -1








2 c = -3


2x -3rArr2y = x -6

rArr x -2y -6 =0









푚+푛]



2 푦2+푦1

2]

= [4+22

7+32

]

= [62

102

] = (35)








Note

















apart





Theorem










Formulas



ퟐ풉


ퟑ


ퟑ

흅 = ퟐퟐퟕ






d =14cm

r= 7cm

흅 = ퟐퟐퟕ

h=10cm

l= 풓ퟐ + 풉ퟐ



l=radicퟏퟒퟗ

l=122



7 x 7 x 10

=440 sqcm

2휋푟(푟 + ℎ) =2 x 22

7 x 7(7+10)

=44 x 17 =748 sqcm

휋푟2ℎ =22

7 x 72 x 10

=1540cm3

Cone 휋푟푙 =22

7 x 7 x 122

=2684 sqcm

휋푟(푟 + 푙) =22

7 x 7 x ( 7 + 122 )

=22 x 192= 4224

13휋푟2ℎ

=13 x 22

7 x 72 x 10

=13 x 22

7 x 72 x 10

=5133 cm3

Sphere

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

43휋푟3

= 43

x 227

x 73 =14373 cm3


7 x 72

=462 sqcm

2휋푟2 =2 x 22

7 x 72

=308 sqcm

23휋푟3

= 23

x 22x 7

x 73 = 7186 cm3






TO D

80To E

150

100

80

30

70to C

40To B

From A


30m = 30 x = 15cm

70m = 70x = 35cm

80m = 80 x = 4cm

100m = 100x = 5cm

150m = 150x =75cm





To C E To D

120to D

E to E

220

210

120

80

40 to B

120toD

75to C

50to B

350

300

250

150

50

F 150toF

100to G

100toE

50toF

25toG

225

175

125

100

75

50

25toC

75toB











N + R = A + 2

N = 3 R = 4 A = 5 N+R = 3 +4 = 7 A+2 = 5 +2 = 7 there4 N+R = A+2

Exercise



N = 8 R = 6 A = 12 N+R = 8 +6 = 14 A+2 = 12 +2 = 14 there4 N+R = A+2

N = 3 R = 5 A = 6 N+R = 3 +5 = 8 A+2 = 6 +2 = 8 there4 N+R = A+2











1

4 4 6 4 +4 = 6 +2

2

3



4

5




1Real Numbers



AUB=BUA AcapB=BcapA







De Morganrsquos Law















ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ





n(S) = 10C5


푛(푆)


푛(푆)




Un grouped data


흈 = sum푿ퟐ

풏 ndash ( sum푿

풏)ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂



흈 = sum풇풙ퟐ

풏 minus sum풇풙

풏

ퟐ 흈 = sum풇풅

ퟐ

풏 흈 = sum풇풅

ퟐ

풏 minus sum풇풅

풏

ퟐ 흈 = sum풇풅

ퟐ

풏 minus sum풇풅

풏

ퟐ 퐱퐂



For ungrouped data


퐂 d2



For grouped data


풙 d2 fd2

n = sumfx = sumfx2







fd d2 fd2

n = sumfd = sumfd2

= n= sumfd

= sumfd2=



퐂 d2


formula 32 1024 32 -2 4 32 -3 9 32 34 1156 34 0 0 34 -1 1 34 35 1225 35 1 1 35 0 0 35 36 1296 36 2 4 36 1 1 36 39 1521 39 5 25 39 4 16 39 42 1764 42 8 64 42 7 49 42

272 9476 272 228 -8 216 sumd= sumd2 =




rArr ퟐퟕퟐퟖ

=34 Assumed Mean 35


흈 = sum푿ퟐ

풏 ndash ( sum푿

풏)ퟐ 흈 =

sum퐝ퟐ

퐧

흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂



)ퟐ

휎 = 11845 ndash 1156

휎 = radic285

휎 = radic285

휎 = 534

흈 = ퟐퟐퟖퟖ


흈 = ퟓퟑퟒ

흈 =

ퟐퟏퟔퟖ

ndash ( ퟖퟖ

)ퟐ



흈 = radicퟐퟖ

흈 = ퟓퟐퟗ



1-5 2 3 6 9 18 1-5 2 3 6 -7 49 98 6-10 3 8 24 64 192 6-10 3 8 24 -2 4 12

11-15 4 13 52 169 676 11-15 4 13 52 3 9 36 16-20 1 18 18 324 324 16-20 1 18 18 8 64 64

10 100 1210 10 100 210




퐂 fd d2 fd2

1-5 2 3 -10 -20 100 200 1-5 2 3 -10 -2 -4 4 8 6-10 3 8 -5 -15 25 75 6-10 3 8 -5 -1 -3 1 3

11-15 4 13 0 0 0 0 11-15 4 13 0 0 0 0 0 16-20 1 18 5 5 25 25 16-20 1 18 5 1 1 1 1

10 -30 300 10 -6 12





흈 = sum풇풙ퟐ

풏 minus sum풇풙

풏

ퟐ



ퟐ


흈 = sum 풇풅ퟐ

풏



흈 = sum풇풅ퟐ

풏 minus sum풇풅

풏

ퟐ



ퟐ


흈 = sum풇풅ퟐ

풏 minus sum풇풅

풏

ퟐ 퐱퐂

흈 = ퟏퟐퟏퟎ


ퟐ 퐱ퟓ








퐗x100




CI 0-10 10-20 20-30 30-40 40-50 131 Freequency (f) 7 10 15 8 10

CI 5-15 15-25 25-35 35-45 45-55 55-65 134 Freequency (f) 8 12 20 10 7 3




휎 = 261 CV=4348


How the


Number of students

Central Angle

Walk 12 1236

x3600 = 1200

Cycle 8 836

x3600 = 800 Bus 3 3

36x3600 = 300

Car 4 436

x3600 = 400 School Van 9 9

36x3600 = 900

36 3600


7 Surds 2 4





Addition of Surds










12 rArr 2

12x3



13x2










radic5minusradic3

3radic5minusradic3



= 3(radic5+radic3)2


















P(x) =12 ndash 4 x 1 + 3 x 1 = 1 =1 - 4 + 3 + 1 = 1




1 (x ndash 2) ಇದು x3 -3x2 +6x -8


ೂೕ














x2 = 144 x2 ndash x = 0 x2 + 14x + 13 = 0 (x = -1) (x = -13)

4x = 81푥

x2 + 3 = 2x 7x2 -12x = 0 ( x = 13 )

7x = 647푥

x + 1x = 5 2m2 ndash 6m + 3 = 0 ( m = 1

2 )



K = 12m푣2

푣2=2퐾푚

v = plusmn 2퐾푚


2










3x2 ndash 5x + 2 = 0

3x2 ndash 5x + 2 = 0


3


3x = minus ퟐ

ퟑ

x2 - 53x = - 2

3

x2 - 53x +(5

6)2 = minus 2

3 + (5

6)2

(푥 minus 5 6

)2 minus 2436

+ 2536

(푥 minus 5 6

)2 = 136

(푥 minus 5 6

) = plusmn 16

x = 56 plusmn 1

6 rArr x = 6

6 or x = 4

6


3x2 ndash 5x + 2 = 0 a=3 b= -5 c = 2


2(3)


6


6

푥 =5 plusmn 1

6

푥 = 66 or x = 4

6

x = 1 or x = 23




Exercise






Nature of the Roots








푎 = minus2





1 = 1







1 2 ಮತು 3

2 6 ಮತು -5


4 -3 ಮತು 32




y

y = 2x2 x 0 +1 -1 +2 -2 +3 -3

y

y =ퟏퟐx2

x 0 +1 -1 +2 -2 +3 -3

y







10Similar triangles






= 퐵퐶퐸퐹










=23 AE=37 find

EC



To prove ADDB

= AEEC


Proof ∆ABC∆BDE

= 12 12

xADxELxDBxEL

[∵ A = 12

xbxh

∆ABC∆BDE

= ADDB

∆ADE∆CDE

= 12 12

xAExDNxDBxDN

[∵ A = 12

xbxh

∆ADE∆CDE

= AEEC

there4 퐀퐃

퐃퐁 = 퐀퐄






DE = BC

EF = CA

FD


there4 ABAG

= BCGH

= CA HA


there4 퐀퐁퐃퐄

= 퐁퐂퐄퐅

= 퐂퐀 퐅퐃






= BCEF

= CA DF


= 퐁퐂ퟐ

퐄퐅ퟐ



DM = AB

DE and BC

EF = AB

DE [ ∵ Given

there4 ALDM

= BCEF

helliphellip(1)


= 1212

xBCxALxEFxDM


= BCxALEFxDM

[ ∵ ( 1)

= BCxBCEFxEF

= 퐁퐂ퟐ

퐄퐅ퟐ

But ABDE

= BCEF

= CA DF

[ ∵ Given


= AB2

DE2 = BC2

EF2 = CA2

DF2






rArr ABAD

= ACAB


rArr BCDC

= ACBC








12Trigonometry



휽




퐜퐨퐬휽 = ퟏ

퐬퐞퐜 휽




퐴퐵퐴퐶














Values 00 300 450 600 900


ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0





13 tan 휃 = 5

12

Cosec휃 = 135

sec휃 = 1312

cot휃 = 125



rArr 3+2 = 5







ퟑ






radic3

tan300= 1radic3

rArr 휃 = 300





퐴퐵




푥2minus푥1

A(x1y1) and B(x2y2)


푥2minus푥1


tan 휃 = 7






푥2minus푥1

m1 = 5minus20minus5

= 3minus5


= 3minus5




휃 훼


푥2minus푥1


= minus7minus4

= 74


= 4minus7

m1 x m2 = 74 x 4

minus7 = -1








2 c = -3


2x -3rArr2y = x -6

rArr x -2y -6 =0









푚+푛]



2 푦2+푦1

2]

= [4+22

7+32

]

= [62

102

] = (35)








Note

















apart





Theorem










Formulas



ퟐ풉


ퟑ


ퟑ

흅 = ퟐퟐퟕ






d =14cm

r= 7cm

흅 = ퟐퟐퟕ

h=10cm

l= 풓ퟐ + 풉ퟐ



l=radicퟏퟒퟗ

l=122



7 x 7 x 10

=440 sqcm

2휋푟(푟 + ℎ) =2 x 22

7 x 7(7+10)

=44 x 17 =748 sqcm

휋푟2ℎ =22

7 x 72 x 10

=1540cm3

Cone 휋푟푙 =22

7 x 7 x 122

=2684 sqcm

휋푟(푟 + 푙) =22

7 x 7 x ( 7 + 122 )

=22 x 192= 4224

13휋푟2ℎ

=13 x 22

7 x 72 x 10

=13 x 22

7 x 72 x 10

=5133 cm3

Sphere

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

43휋푟3

= 43

x 227

x 73 =14373 cm3


7 x 72

=462 sqcm

2휋푟2 =2 x 22

7 x 72

=308 sqcm

23휋푟3

= 23

x 22x 7

x 73 = 7186 cm3






TO D

80To E

150

100

80

30

70to C

40To B

From A


30m = 30 x = 15cm

70m = 70x = 35cm

80m = 80 x = 4cm

100m = 100x = 5cm

150m = 150x =75cm





To C E To D

120to D

E to E

220

210

120

80

40 to B

120toD

75to C

50to B

350

300

250

150

50

F 150toF

100to G

100toE

50toF

25toG

225

175

125

100

75

50

25toC

75toB











N + R = A + 2

N = 3 R = 4 A = 5 N+R = 3 +4 = 7 A+2 = 5 +2 = 7 there4 N+R = A+2

Exercise



N = 8 R = 6 A = 12 N+R = 8 +6 = 14 A+2 = 12 +2 = 14 there4 N+R = A+2

N = 3 R = 5 A = 6 N+R = 3 +5 = 8 A+2 = 6 +2 = 8 there4 N+R = A+2











1

4 4 6 4 +4 = 6 +2

2

3



4

5




1Real Numbers



AUB=BUA AcapB=BcapA







De Morganrsquos Law















ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ




For ungrouped data


퐂 d2



For grouped data


풙 d2 fd2

n = sumfx = sumfx2







fd d2 fd2

n = sumfd = sumfd2

= n= sumfd

= sumfd2=



퐂 d2


formula 32 1024 32 -2 4 32 -3 9 32 34 1156 34 0 0 34 -1 1 34 35 1225 35 1 1 35 0 0 35 36 1296 36 2 4 36 1 1 36 39 1521 39 5 25 39 4 16 39 42 1764 42 8 64 42 7 49 42

272 9476 272 228 -8 216 sumd= sumd2 =




rArr ퟐퟕퟐퟖ

=34 Assumed Mean 35


흈 = sum푿ퟐ

풏 ndash ( sum푿

풏)ퟐ 흈 =

sum퐝ퟐ

퐧

흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂



)ퟐ

휎 = 11845 ndash 1156

휎 = radic285

휎 = radic285

휎 = 534

흈 = ퟐퟐퟖퟖ


흈 = ퟓퟑퟒ

흈 =

ퟐퟏퟔퟖ

ndash ( ퟖퟖ

)ퟐ



흈 = radicퟐퟖ

흈 = ퟓퟐퟗ



1-5 2 3 6 9 18 1-5 2 3 6 -7 49 98 6-10 3 8 24 64 192 6-10 3 8 24 -2 4 12

11-15 4 13 52 169 676 11-15 4 13 52 3 9 36 16-20 1 18 18 324 324 16-20 1 18 18 8 64 64

10 100 1210 10 100 210




퐂 fd d2 fd2

1-5 2 3 -10 -20 100 200 1-5 2 3 -10 -2 -4 4 8 6-10 3 8 -5 -15 25 75 6-10 3 8 -5 -1 -3 1 3

11-15 4 13 0 0 0 0 11-15 4 13 0 0 0 0 0 16-20 1 18 5 5 25 25 16-20 1 18 5 1 1 1 1

10 -30 300 10 -6 12





흈 = sum풇풙ퟐ

풏 minus sum풇풙

풏

ퟐ



ퟐ


흈 = sum 풇풅ퟐ

풏



흈 = sum풇풅ퟐ

풏 minus sum풇풅

풏

ퟐ



ퟐ


흈 = sum풇풅ퟐ

풏 minus sum풇풅

풏

ퟐ 퐱퐂

흈 = ퟏퟐퟏퟎ


ퟐ 퐱ퟓ








퐗x100




CI 0-10 10-20 20-30 30-40 40-50 131 Freequency (f) 7 10 15 8 10

CI 5-15 15-25 25-35 35-45 45-55 55-65 134 Freequency (f) 8 12 20 10 7 3




휎 = 261 CV=4348


How the


Number of students

Central Angle

Walk 12 1236

x3600 = 1200

Cycle 8 836

x3600 = 800 Bus 3 3

36x3600 = 300

Car 4 436

x3600 = 400 School Van 9 9

36x3600 = 900

36 3600


7 Surds 2 4





Addition of Surds










12 rArr 2

12x3



13x2










radic5minusradic3

3radic5minusradic3



= 3(radic5+radic3)2


















P(x) =12 ndash 4 x 1 + 3 x 1 = 1 =1 - 4 + 3 + 1 = 1




1 (x ndash 2) ಇದು x3 -3x2 +6x -8


ೂೕ














x2 = 144 x2 ndash x = 0 x2 + 14x + 13 = 0 (x = -1) (x = -13)

4x = 81푥

x2 + 3 = 2x 7x2 -12x = 0 ( x = 13 )

7x = 647푥

x + 1x = 5 2m2 ndash 6m + 3 = 0 ( m = 1

2 )



K = 12m푣2

푣2=2퐾푚

v = plusmn 2퐾푚


2










3x2 ndash 5x + 2 = 0

3x2 ndash 5x + 2 = 0


3


3x = minus ퟐ

ퟑ

x2 - 53x = - 2

3

x2 - 53x +(5

6)2 = minus 2

3 + (5

6)2

(푥 minus 5 6

)2 minus 2436

+ 2536

(푥 minus 5 6

)2 = 136

(푥 minus 5 6

) = plusmn 16

x = 56 plusmn 1

6 rArr x = 6

6 or x = 4

6


3x2 ndash 5x + 2 = 0 a=3 b= -5 c = 2


2(3)


6


6

푥 =5 plusmn 1

6

푥 = 66 or x = 4

6

x = 1 or x = 23




Exercise






Nature of the Roots








푎 = minus2





1 = 1







1 2 ಮತು 3

2 6 ಮತು -5


4 -3 ಮತು 32




y

y = 2x2 x 0 +1 -1 +2 -2 +3 -3

y

y =ퟏퟐx2

x 0 +1 -1 +2 -2 +3 -3

y







10Similar triangles






= 퐵퐶퐸퐹










=23 AE=37 find

EC



To prove ADDB

= AEEC


Proof ∆ABC∆BDE

= 12 12

xADxELxDBxEL

[∵ A = 12

xbxh

∆ABC∆BDE

= ADDB

∆ADE∆CDE

= 12 12

xAExDNxDBxDN

[∵ A = 12

xbxh

∆ADE∆CDE

= AEEC

there4 퐀퐃

퐃퐁 = 퐀퐄






DE = BC

EF = CA

FD


there4 ABAG

= BCGH

= CA HA


there4 퐀퐁퐃퐄

= 퐁퐂퐄퐅

= 퐂퐀 퐅퐃






= BCEF

= CA DF


= 퐁퐂ퟐ

퐄퐅ퟐ



DM = AB

DE and BC

EF = AB

DE [ ∵ Given

there4 ALDM

= BCEF

helliphellip(1)


= 1212

xBCxALxEFxDM


= BCxALEFxDM

[ ∵ ( 1)

= BCxBCEFxEF

= 퐁퐂ퟐ

퐄퐅ퟐ

But ABDE

= BCEF

= CA DF

[ ∵ Given


= AB2

DE2 = BC2

EF2 = CA2

DF2






rArr ABAD

= ACAB


rArr BCDC

= ACBC








12Trigonometry



휽




퐜퐨퐬휽 = ퟏ

퐬퐞퐜 휽




퐴퐵퐴퐶














Values 00 300 450 600 900


ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0





13 tan 휃 = 5

12

Cosec휃 = 135

sec휃 = 1312

cot휃 = 125



rArr 3+2 = 5







ퟑ






radic3

tan300= 1radic3

rArr 휃 = 300





퐴퐵




푥2minus푥1

A(x1y1) and B(x2y2)


푥2minus푥1


tan 휃 = 7






푥2minus푥1

m1 = 5minus20minus5

= 3minus5


= 3minus5




휃 훼


푥2minus푥1


= minus7minus4

= 74


= 4minus7

m1 x m2 = 74 x 4

minus7 = -1








2 c = -3


2x -3rArr2y = x -6

rArr x -2y -6 =0









푚+푛]



2 푦2+푦1

2]

= [4+22

7+32

]

= [62

102

] = (35)








Note

















apart





Theorem










Formulas



ퟐ풉


ퟑ


ퟑ

흅 = ퟐퟐퟕ






d =14cm

r= 7cm

흅 = ퟐퟐퟕ

h=10cm

l= 풓ퟐ + 풉ퟐ



l=radicퟏퟒퟗ

l=122



7 x 7 x 10

=440 sqcm

2휋푟(푟 + ℎ) =2 x 22

7 x 7(7+10)

=44 x 17 =748 sqcm

휋푟2ℎ =22

7 x 72 x 10

=1540cm3

Cone 휋푟푙 =22

7 x 7 x 122

=2684 sqcm

휋푟(푟 + 푙) =22

7 x 7 x ( 7 + 122 )

=22 x 192= 4224

13휋푟2ℎ

=13 x 22

7 x 72 x 10

=13 x 22

7 x 72 x 10

=5133 cm3

Sphere

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

43휋푟3

= 43

x 227

x 73 =14373 cm3


7 x 72

=462 sqcm

2휋푟2 =2 x 22

7 x 72

=308 sqcm

23휋푟3

= 23

x 22x 7

x 73 = 7186 cm3






TO D

80To E

150

100

80

30

70to C

40To B

From A


30m = 30 x = 15cm

70m = 70x = 35cm

80m = 80 x = 4cm

100m = 100x = 5cm

150m = 150x =75cm





To C E To D

120to D

E to E

220

210

120

80

40 to B

120toD

75to C

50to B

350

300

250

150

50

F 150toF

100to G

100toE

50toF

25toG

225

175

125

100

75

50

25toC

75toB











N + R = A + 2

N = 3 R = 4 A = 5 N+R = 3 +4 = 7 A+2 = 5 +2 = 7 there4 N+R = A+2

Exercise



N = 8 R = 6 A = 12 N+R = 8 +6 = 14 A+2 = 12 +2 = 14 there4 N+R = A+2

N = 3 R = 5 A = 6 N+R = 3 +5 = 8 A+2 = 6 +2 = 8 there4 N+R = A+2











1

4 4 6 4 +4 = 6 +2

2

3



4

5




1Real Numbers



AUB=BUA AcapB=BcapA







De Morganrsquos Law















ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ






fd d2 fd2

n = sumfd = sumfd2

= n= sumfd

= sumfd2=



퐂 d2


formula 32 1024 32 -2 4 32 -3 9 32 34 1156 34 0 0 34 -1 1 34 35 1225 35 1 1 35 0 0 35 36 1296 36 2 4 36 1 1 36 39 1521 39 5 25 39 4 16 39 42 1764 42 8 64 42 7 49 42

272 9476 272 228 -8 216 sumd= sumd2 =




rArr ퟐퟕퟐퟖ

=34 Assumed Mean 35


흈 = sum푿ퟐ

풏 ndash ( sum푿

풏)ퟐ 흈 =

sum퐝ퟐ

퐧

흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂



)ퟐ

휎 = 11845 ndash 1156

휎 = radic285

휎 = radic285

휎 = 534

흈 = ퟐퟐퟖퟖ


흈 = ퟓퟑퟒ

흈 =

ퟐퟏퟔퟖ

ndash ( ퟖퟖ

)ퟐ



흈 = radicퟐퟖ

흈 = ퟓퟐퟗ



1-5 2 3 6 9 18 1-5 2 3 6 -7 49 98 6-10 3 8 24 64 192 6-10 3 8 24 -2 4 12

11-15 4 13 52 169 676 11-15 4 13 52 3 9 36 16-20 1 18 18 324 324 16-20 1 18 18 8 64 64

10 100 1210 10 100 210




퐂 fd d2 fd2

1-5 2 3 -10 -20 100 200 1-5 2 3 -10 -2 -4 4 8 6-10 3 8 -5 -15 25 75 6-10 3 8 -5 -1 -3 1 3

11-15 4 13 0 0 0 0 11-15 4 13 0 0 0 0 0 16-20 1 18 5 5 25 25 16-20 1 18 5 1 1 1 1

10 -30 300 10 -6 12





흈 = sum풇풙ퟐ

풏 minus sum풇풙

풏

ퟐ



ퟐ


흈 = sum 풇풅ퟐ

풏



흈 = sum풇풅ퟐ

풏 minus sum풇풅

풏

ퟐ



ퟐ


흈 = sum풇풅ퟐ

풏 minus sum풇풅

풏

ퟐ 퐱퐂

흈 = ퟏퟐퟏퟎ


ퟐ 퐱ퟓ








퐗x100




CI 0-10 10-20 20-30 30-40 40-50 131 Freequency (f) 7 10 15 8 10

CI 5-15 15-25 25-35 35-45 45-55 55-65 134 Freequency (f) 8 12 20 10 7 3




휎 = 261 CV=4348


How the


Number of students

Central Angle

Walk 12 1236

x3600 = 1200

Cycle 8 836

x3600 = 800 Bus 3 3

36x3600 = 300

Car 4 436

x3600 = 400 School Van 9 9

36x3600 = 900

36 3600


7 Surds 2 4





Addition of Surds










12 rArr 2

12x3



13x2










radic5minusradic3

3radic5minusradic3



= 3(radic5+radic3)2


















P(x) =12 ndash 4 x 1 + 3 x 1 = 1 =1 - 4 + 3 + 1 = 1




1 (x ndash 2) ಇದು x3 -3x2 +6x -8


ೂೕ














x2 = 144 x2 ndash x = 0 x2 + 14x + 13 = 0 (x = -1) (x = -13)

4x = 81푥

x2 + 3 = 2x 7x2 -12x = 0 ( x = 13 )

7x = 647푥

x + 1x = 5 2m2 ndash 6m + 3 = 0 ( m = 1

2 )



K = 12m푣2

푣2=2퐾푚

v = plusmn 2퐾푚


2










3x2 ndash 5x + 2 = 0

3x2 ndash 5x + 2 = 0


3


3x = minus ퟐ

ퟑ

x2 - 53x = - 2

3

x2 - 53x +(5

6)2 = minus 2

3 + (5

6)2

(푥 minus 5 6

)2 minus 2436

+ 2536

(푥 minus 5 6

)2 = 136

(푥 minus 5 6

) = plusmn 16

x = 56 plusmn 1

6 rArr x = 6

6 or x = 4

6


3x2 ndash 5x + 2 = 0 a=3 b= -5 c = 2


2(3)


6


6

푥 =5 plusmn 1

6

푥 = 66 or x = 4

6

x = 1 or x = 23




Exercise






Nature of the Roots








푎 = minus2





1 = 1







1 2 ಮತು 3

2 6 ಮತು -5


4 -3 ಮತು 32




y

y = 2x2 x 0 +1 -1 +2 -2 +3 -3

y

y =ퟏퟐx2

x 0 +1 -1 +2 -2 +3 -3

y







10Similar triangles






= 퐵퐶퐸퐹










=23 AE=37 find

EC



To prove ADDB

= AEEC


Proof ∆ABC∆BDE

= 12 12

xADxELxDBxEL

[∵ A = 12

xbxh

∆ABC∆BDE

= ADDB

∆ADE∆CDE

= 12 12

xAExDNxDBxDN

[∵ A = 12

xbxh

∆ADE∆CDE

= AEEC

there4 퐀퐃

퐃퐁 = 퐀퐄






DE = BC

EF = CA

FD


there4 ABAG

= BCGH

= CA HA


there4 퐀퐁퐃퐄

= 퐁퐂퐄퐅

= 퐂퐀 퐅퐃






= BCEF

= CA DF


= 퐁퐂ퟐ

퐄퐅ퟐ



DM = AB

DE and BC

EF = AB

DE [ ∵ Given

there4 ALDM

= BCEF

helliphellip(1)


= 1212

xBCxALxEFxDM


= BCxALEFxDM

[ ∵ ( 1)

= BCxBCEFxEF

= 퐁퐂ퟐ

퐄퐅ퟐ

But ABDE

= BCEF

= CA DF

[ ∵ Given


= AB2

DE2 = BC2

EF2 = CA2

DF2






rArr ABAD

= ACAB


rArr BCDC

= ACBC








12Trigonometry



휽




퐜퐨퐬휽 = ퟏ

퐬퐞퐜 휽




퐴퐵퐴퐶














Values 00 300 450 600 900


ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0





13 tan 휃 = 5

12

Cosec휃 = 135

sec휃 = 1312

cot휃 = 125



rArr 3+2 = 5







ퟑ






radic3

tan300= 1radic3

rArr 휃 = 300





퐴퐵




푥2minus푥1

A(x1y1) and B(x2y2)


푥2minus푥1


tan 휃 = 7






푥2minus푥1

m1 = 5minus20minus5

= 3minus5


= 3minus5




휃 훼


푥2minus푥1


= minus7minus4

= 74


= 4minus7

m1 x m2 = 74 x 4

minus7 = -1








2 c = -3


2x -3rArr2y = x -6

rArr x -2y -6 =0









푚+푛]



2 푦2+푦1

2]

= [4+22

7+32

]

= [62

102

] = (35)








Note

















apart





Theorem










Formulas



ퟐ풉


ퟑ


ퟑ

흅 = ퟐퟐퟕ






d =14cm

r= 7cm

흅 = ퟐퟐퟕ

h=10cm

l= 풓ퟐ + 풉ퟐ



l=radicퟏퟒퟗ

l=122



7 x 7 x 10

=440 sqcm

2휋푟(푟 + ℎ) =2 x 22

7 x 7(7+10)

=44 x 17 =748 sqcm

휋푟2ℎ =22

7 x 72 x 10

=1540cm3

Cone 휋푟푙 =22

7 x 7 x 122

=2684 sqcm

휋푟(푟 + 푙) =22

7 x 7 x ( 7 + 122 )

=22 x 192= 4224

13휋푟2ℎ

=13 x 22

7 x 72 x 10

=13 x 22

7 x 72 x 10

=5133 cm3

Sphere

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

43휋푟3

= 43

x 227

x 73 =14373 cm3


7 x 72

=462 sqcm

2휋푟2 =2 x 22

7 x 72

=308 sqcm

23휋푟3

= 23

x 22x 7

x 73 = 7186 cm3






TO D

80To E

150

100

80

30

70to C

40To B

From A


30m = 30 x = 15cm

70m = 70x = 35cm

80m = 80 x = 4cm

100m = 100x = 5cm

150m = 150x =75cm





To C E To D

120to D

E to E

220

210

120

80

40 to B

120toD

75to C

50to B

350

300

250

150

50

F 150toF

100to G

100toE

50toF

25toG

225

175

125

100

75

50

25toC

75toB











N + R = A + 2

N = 3 R = 4 A = 5 N+R = 3 +4 = 7 A+2 = 5 +2 = 7 there4 N+R = A+2

Exercise



N = 8 R = 6 A = 12 N+R = 8 +6 = 14 A+2 = 12 +2 = 14 there4 N+R = A+2

N = 3 R = 5 A = 6 N+R = 3 +5 = 8 A+2 = 6 +2 = 8 there4 N+R = A+2











1

4 4 6 4 +4 = 6 +2

2

3



4

5




1Real Numbers



AUB=BUA AcapB=BcapA







De Morganrsquos Law















ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ





rArr ퟐퟕퟐퟖ

=34 Assumed Mean 35


흈 = sum푿ퟐ

풏 ndash ( sum푿

풏)ퟐ 흈 =

sum퐝ퟐ

퐧

흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂



)ퟐ

휎 = 11845 ndash 1156

휎 = radic285

휎 = radic285

휎 = 534

흈 = ퟐퟐퟖퟖ


흈 = ퟓퟑퟒ

흈 =

ퟐퟏퟔퟖ

ndash ( ퟖퟖ

)ퟐ



흈 = radicퟐퟖ

흈 = ퟓퟐퟗ



1-5 2 3 6 9 18 1-5 2 3 6 -7 49 98 6-10 3 8 24 64 192 6-10 3 8 24 -2 4 12

11-15 4 13 52 169 676 11-15 4 13 52 3 9 36 16-20 1 18 18 324 324 16-20 1 18 18 8 64 64

10 100 1210 10 100 210




퐂 fd d2 fd2

1-5 2 3 -10 -20 100 200 1-5 2 3 -10 -2 -4 4 8 6-10 3 8 -5 -15 25 75 6-10 3 8 -5 -1 -3 1 3

11-15 4 13 0 0 0 0 11-15 4 13 0 0 0 0 0 16-20 1 18 5 5 25 25 16-20 1 18 5 1 1 1 1

10 -30 300 10 -6 12





흈 = sum풇풙ퟐ

풏 minus sum풇풙

풏

ퟐ



ퟐ


흈 = sum 풇풅ퟐ

풏



흈 = sum풇풅ퟐ

풏 minus sum풇풅

풏

ퟐ



ퟐ


흈 = sum풇풅ퟐ

풏 minus sum풇풅

풏

ퟐ 퐱퐂

흈 = ퟏퟐퟏퟎ


ퟐ 퐱ퟓ








퐗x100




CI 0-10 10-20 20-30 30-40 40-50 131 Freequency (f) 7 10 15 8 10

CI 5-15 15-25 25-35 35-45 45-55 55-65 134 Freequency (f) 8 12 20 10 7 3




휎 = 261 CV=4348


How the


Number of students

Central Angle

Walk 12 1236

x3600 = 1200

Cycle 8 836

x3600 = 800 Bus 3 3

36x3600 = 300

Car 4 436

x3600 = 400 School Van 9 9

36x3600 = 900

36 3600


7 Surds 2 4





Addition of Surds










12 rArr 2

12x3



13x2










radic5minusradic3

3radic5minusradic3



= 3(radic5+radic3)2


















P(x) =12 ndash 4 x 1 + 3 x 1 = 1 =1 - 4 + 3 + 1 = 1




1 (x ndash 2) ಇದು x3 -3x2 +6x -8


ೂೕ














x2 = 144 x2 ndash x = 0 x2 + 14x + 13 = 0 (x = -1) (x = -13)

4x = 81푥

x2 + 3 = 2x 7x2 -12x = 0 ( x = 13 )

7x = 647푥

x + 1x = 5 2m2 ndash 6m + 3 = 0 ( m = 1

2 )



K = 12m푣2

푣2=2퐾푚

v = plusmn 2퐾푚


2










3x2 ndash 5x + 2 = 0

3x2 ndash 5x + 2 = 0


3


3x = minus ퟐ

ퟑ

x2 - 53x = - 2

3

x2 - 53x +(5

6)2 = minus 2

3 + (5

6)2

(푥 minus 5 6

)2 minus 2436

+ 2536

(푥 minus 5 6

)2 = 136

(푥 minus 5 6

) = plusmn 16

x = 56 plusmn 1

6 rArr x = 6

6 or x = 4

6


3x2 ndash 5x + 2 = 0 a=3 b= -5 c = 2


2(3)


6


6

푥 =5 plusmn 1

6

푥 = 66 or x = 4

6

x = 1 or x = 23




Exercise






Nature of the Roots








푎 = minus2





1 = 1







1 2 ಮತು 3

2 6 ಮತು -5


4 -3 ಮತು 32




y

y = 2x2 x 0 +1 -1 +2 -2 +3 -3

y

y =ퟏퟐx2

x 0 +1 -1 +2 -2 +3 -3

y







10Similar triangles






= 퐵퐶퐸퐹










=23 AE=37 find

EC



To prove ADDB

= AEEC


Proof ∆ABC∆BDE

= 12 12

xADxELxDBxEL

[∵ A = 12

xbxh

∆ABC∆BDE

= ADDB

∆ADE∆CDE

= 12 12

xAExDNxDBxDN

[∵ A = 12

xbxh

∆ADE∆CDE

= AEEC

there4 퐀퐃

퐃퐁 = 퐀퐄






DE = BC

EF = CA

FD


there4 ABAG

= BCGH

= CA HA


there4 퐀퐁퐃퐄

= 퐁퐂퐄퐅

= 퐂퐀 퐅퐃






= BCEF

= CA DF


= 퐁퐂ퟐ

퐄퐅ퟐ



DM = AB

DE and BC

EF = AB

DE [ ∵ Given

there4 ALDM

= BCEF

helliphellip(1)


= 1212

xBCxALxEFxDM


= BCxALEFxDM

[ ∵ ( 1)

= BCxBCEFxEF

= 퐁퐂ퟐ

퐄퐅ퟐ

But ABDE

= BCEF

= CA DF

[ ∵ Given


= AB2

DE2 = BC2

EF2 = CA2

DF2






rArr ABAD

= ACAB


rArr BCDC

= ACBC








12Trigonometry



휽




퐜퐨퐬휽 = ퟏ

퐬퐞퐜 휽




퐴퐵퐴퐶














Values 00 300 450 600 900


ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0





13 tan 휃 = 5

12

Cosec휃 = 135

sec휃 = 1312

cot휃 = 125



rArr 3+2 = 5







ퟑ






radic3

tan300= 1radic3

rArr 휃 = 300





퐴퐵




푥2minus푥1

A(x1y1) and B(x2y2)


푥2minus푥1


tan 휃 = 7






푥2minus푥1

m1 = 5minus20minus5

= 3minus5


= 3minus5




휃 훼


푥2minus푥1


= minus7minus4

= 74


= 4minus7

m1 x m2 = 74 x 4

minus7 = -1








2 c = -3


2x -3rArr2y = x -6

rArr x -2y -6 =0









푚+푛]



2 푦2+푦1

2]

= [4+22

7+32

]

= [62

102

] = (35)








Note

















apart





Theorem










Formulas



ퟐ풉


ퟑ


ퟑ

흅 = ퟐퟐퟕ






d =14cm

r= 7cm

흅 = ퟐퟐퟕ

h=10cm

l= 풓ퟐ + 풉ퟐ



l=radicퟏퟒퟗ

l=122



7 x 7 x 10

=440 sqcm

2휋푟(푟 + ℎ) =2 x 22

7 x 7(7+10)

=44 x 17 =748 sqcm

휋푟2ℎ =22

7 x 72 x 10

=1540cm3

Cone 휋푟푙 =22

7 x 7 x 122

=2684 sqcm

휋푟(푟 + 푙) =22

7 x 7 x ( 7 + 122 )

=22 x 192= 4224

13휋푟2ℎ

=13 x 22

7 x 72 x 10

=13 x 22

7 x 72 x 10

=5133 cm3

Sphere

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

43휋푟3

= 43

x 227

x 73 =14373 cm3


7 x 72

=462 sqcm

2휋푟2 =2 x 22

7 x 72

=308 sqcm

23휋푟3

= 23

x 22x 7

x 73 = 7186 cm3






TO D

80To E

150

100

80

30

70to C

40To B

From A


30m = 30 x = 15cm

70m = 70x = 35cm

80m = 80 x = 4cm

100m = 100x = 5cm

150m = 150x =75cm





To C E To D

120to D

E to E

220

210

120

80

40 to B

120toD

75to C

50to B

350

300

250

150

50

F 150toF

100to G

100toE

50toF

25toG

225

175

125

100

75

50

25toC

75toB











N + R = A + 2

N = 3 R = 4 A = 5 N+R = 3 +4 = 7 A+2 = 5 +2 = 7 there4 N+R = A+2

Exercise



N = 8 R = 6 A = 12 N+R = 8 +6 = 14 A+2 = 12 +2 = 14 there4 N+R = A+2

N = 3 R = 5 A = 6 N+R = 3 +5 = 8 A+2 = 6 +2 = 8 there4 N+R = A+2











1

4 4 6 4 +4 = 6 +2

2

3



4

5




1Real Numbers



AUB=BUA AcapB=BcapA







De Morganrsquos Law















ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ





퐂 fd d2 fd2

1-5 2 3 -10 -20 100 200 1-5 2 3 -10 -2 -4 4 8 6-10 3 8 -5 -15 25 75 6-10 3 8 -5 -1 -3 1 3

11-15 4 13 0 0 0 0 11-15 4 13 0 0 0 0 0 16-20 1 18 5 5 25 25 16-20 1 18 5 1 1 1 1

10 -30 300 10 -6 12





흈 = sum풇풙ퟐ

풏 minus sum풇풙

풏

ퟐ



ퟐ


흈 = sum 풇풅ퟐ

풏



흈 = sum풇풅ퟐ

풏 minus sum풇풅

풏

ퟐ



ퟐ


흈 = sum풇풅ퟐ

풏 minus sum풇풅

풏

ퟐ 퐱퐂

흈 = ퟏퟐퟏퟎ


ퟐ 퐱ퟓ








퐗x100




CI 0-10 10-20 20-30 30-40 40-50 131 Freequency (f) 7 10 15 8 10

CI 5-15 15-25 25-35 35-45 45-55 55-65 134 Freequency (f) 8 12 20 10 7 3




휎 = 261 CV=4348


How the


Number of students

Central Angle

Walk 12 1236

x3600 = 1200

Cycle 8 836

x3600 = 800 Bus 3 3

36x3600 = 300

Car 4 436

x3600 = 400 School Van 9 9

36x3600 = 900

36 3600


7 Surds 2 4





Addition of Surds










12 rArr 2

12x3



13x2










radic5minusradic3

3radic5minusradic3



= 3(radic5+radic3)2


















P(x) =12 ndash 4 x 1 + 3 x 1 = 1 =1 - 4 + 3 + 1 = 1




1 (x ndash 2) ಇದು x3 -3x2 +6x -8


ೂೕ














x2 = 144 x2 ndash x = 0 x2 + 14x + 13 = 0 (x = -1) (x = -13)

4x = 81푥

x2 + 3 = 2x 7x2 -12x = 0 ( x = 13 )

7x = 647푥

x + 1x = 5 2m2 ndash 6m + 3 = 0 ( m = 1

2 )



K = 12m푣2

푣2=2퐾푚

v = plusmn 2퐾푚


2










3x2 ndash 5x + 2 = 0

3x2 ndash 5x + 2 = 0


3


3x = minus ퟐ

ퟑ

x2 - 53x = - 2

3

x2 - 53x +(5

6)2 = minus 2

3 + (5

6)2

(푥 minus 5 6

)2 minus 2436

+ 2536

(푥 minus 5 6

)2 = 136

(푥 minus 5 6

) = plusmn 16

x = 56 plusmn 1

6 rArr x = 6

6 or x = 4

6


3x2 ndash 5x + 2 = 0 a=3 b= -5 c = 2


2(3)


6


6

푥 =5 plusmn 1

6

푥 = 66 or x = 4

6

x = 1 or x = 23




Exercise






Nature of the Roots








푎 = minus2





1 = 1







1 2 ಮತು 3

2 6 ಮತು -5


4 -3 ಮತು 32




y

y = 2x2 x 0 +1 -1 +2 -2 +3 -3

y

y =ퟏퟐx2

x 0 +1 -1 +2 -2 +3 -3

y







10Similar triangles






= 퐵퐶퐸퐹










=23 AE=37 find

EC



To prove ADDB

= AEEC


Proof ∆ABC∆BDE

= 12 12

xADxELxDBxEL

[∵ A = 12

xbxh

∆ABC∆BDE

= ADDB

∆ADE∆CDE

= 12 12

xAExDNxDBxDN

[∵ A = 12

xbxh

∆ADE∆CDE

= AEEC

there4 퐀퐃

퐃퐁 = 퐀퐄






DE = BC

EF = CA

FD


there4 ABAG

= BCGH

= CA HA


there4 퐀퐁퐃퐄

= 퐁퐂퐄퐅

= 퐂퐀 퐅퐃






= BCEF

= CA DF


= 퐁퐂ퟐ

퐄퐅ퟐ



DM = AB

DE and BC

EF = AB

DE [ ∵ Given

there4 ALDM

= BCEF

helliphellip(1)


= 1212

xBCxALxEFxDM


= BCxALEFxDM

[ ∵ ( 1)

= BCxBCEFxEF

= 퐁퐂ퟐ

퐄퐅ퟐ

But ABDE

= BCEF

= CA DF

[ ∵ Given


= AB2

DE2 = BC2

EF2 = CA2

DF2






rArr ABAD

= ACAB


rArr BCDC

= ACBC








12Trigonometry



휽




퐜퐨퐬휽 = ퟏ

퐬퐞퐜 휽




퐴퐵퐴퐶














Values 00 300 450 600 900


ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0





13 tan 휃 = 5

12

Cosec휃 = 135

sec휃 = 1312

cot휃 = 125



rArr 3+2 = 5







ퟑ






radic3

tan300= 1radic3

rArr 휃 = 300





퐴퐵




푥2minus푥1

A(x1y1) and B(x2y2)


푥2minus푥1


tan 휃 = 7






푥2minus푥1

m1 = 5minus20minus5

= 3minus5


= 3minus5




휃 훼


푥2minus푥1


= minus7minus4

= 74


= 4minus7

m1 x m2 = 74 x 4

minus7 = -1








2 c = -3


2x -3rArr2y = x -6

rArr x -2y -6 =0









푚+푛]



2 푦2+푦1

2]

= [4+22

7+32

]

= [62

102

] = (35)








Note

















apart





Theorem










Formulas



ퟐ풉


ퟑ


ퟑ

흅 = ퟐퟐퟕ






d =14cm

r= 7cm

흅 = ퟐퟐퟕ

h=10cm

l= 풓ퟐ + 풉ퟐ



l=radicퟏퟒퟗ

l=122



7 x 7 x 10

=440 sqcm

2휋푟(푟 + ℎ) =2 x 22

7 x 7(7+10)

=44 x 17 =748 sqcm

휋푟2ℎ =22

7 x 72 x 10

=1540cm3

Cone 휋푟푙 =22

7 x 7 x 122

=2684 sqcm

휋푟(푟 + 푙) =22

7 x 7 x ( 7 + 122 )

=22 x 192= 4224

13휋푟2ℎ

=13 x 22

7 x 72 x 10

=13 x 22

7 x 72 x 10

=5133 cm3

Sphere

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

43휋푟3

= 43

x 227

x 73 =14373 cm3


7 x 72

=462 sqcm

2휋푟2 =2 x 22

7 x 72

=308 sqcm

23휋푟3

= 23

x 22x 7

x 73 = 7186 cm3






TO D

80To E

150

100

80

30

70to C

40To B

From A


30m = 30 x = 15cm

70m = 70x = 35cm

80m = 80 x = 4cm

100m = 100x = 5cm

150m = 150x =75cm





To C E To D

120to D

E to E

220

210

120

80

40 to B

120toD

75to C

50to B

350

300

250

150

50

F 150toF

100to G

100toE

50toF

25toG

225

175

125

100

75

50

25toC

75toB











N + R = A + 2

N = 3 R = 4 A = 5 N+R = 3 +4 = 7 A+2 = 5 +2 = 7 there4 N+R = A+2

Exercise



N = 8 R = 6 A = 12 N+R = 8 +6 = 14 A+2 = 12 +2 = 14 there4 N+R = A+2

N = 3 R = 5 A = 6 N+R = 3 +5 = 8 A+2 = 6 +2 = 8 there4 N+R = A+2











1

4 4 6 4 +4 = 6 +2

2

3



4

5




1Real Numbers



AUB=BUA AcapB=BcapA







De Morganrsquos Law















ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ






퐗x100




CI 0-10 10-20 20-30 30-40 40-50 131 Freequency (f) 7 10 15 8 10

CI 5-15 15-25 25-35 35-45 45-55 55-65 134 Freequency (f) 8 12 20 10 7 3




휎 = 261 CV=4348


How the


Number of students

Central Angle

Walk 12 1236

x3600 = 1200

Cycle 8 836

x3600 = 800 Bus 3 3

36x3600 = 300

Car 4 436

x3600 = 400 School Van 9 9

36x3600 = 900

36 3600


7 Surds 2 4





Addition of Surds










12 rArr 2

12x3



13x2










radic5minusradic3

3radic5minusradic3



= 3(radic5+radic3)2


















P(x) =12 ndash 4 x 1 + 3 x 1 = 1 =1 - 4 + 3 + 1 = 1




1 (x ndash 2) ಇದು x3 -3x2 +6x -8


ೂೕ














x2 = 144 x2 ndash x = 0 x2 + 14x + 13 = 0 (x = -1) (x = -13)

4x = 81푥

x2 + 3 = 2x 7x2 -12x = 0 ( x = 13 )

7x = 647푥

x + 1x = 5 2m2 ndash 6m + 3 = 0 ( m = 1

2 )



K = 12m푣2

푣2=2퐾푚

v = plusmn 2퐾푚


2










3x2 ndash 5x + 2 = 0

3x2 ndash 5x + 2 = 0


3


3x = minus ퟐ

ퟑ

x2 - 53x = - 2

3

x2 - 53x +(5

6)2 = minus 2

3 + (5

6)2

(푥 minus 5 6

)2 minus 2436

+ 2536

(푥 minus 5 6

)2 = 136

(푥 minus 5 6

) = plusmn 16

x = 56 plusmn 1

6 rArr x = 6

6 or x = 4

6


3x2 ndash 5x + 2 = 0 a=3 b= -5 c = 2


2(3)


6


6

푥 =5 plusmn 1

6

푥 = 66 or x = 4

6

x = 1 or x = 23




Exercise






Nature of the Roots








푎 = minus2





1 = 1







1 2 ಮತು 3

2 6 ಮತು -5


4 -3 ಮತು 32




y

y = 2x2 x 0 +1 -1 +2 -2 +3 -3

y

y =ퟏퟐx2

x 0 +1 -1 +2 -2 +3 -3

y







10Similar triangles






= 퐵퐶퐸퐹










=23 AE=37 find

EC



To prove ADDB

= AEEC


Proof ∆ABC∆BDE

= 12 12

xADxELxDBxEL

[∵ A = 12

xbxh

∆ABC∆BDE

= ADDB

∆ADE∆CDE

= 12 12

xAExDNxDBxDN

[∵ A = 12

xbxh

∆ADE∆CDE

= AEEC

there4 퐀퐃

퐃퐁 = 퐀퐄






DE = BC

EF = CA

FD


there4 ABAG

= BCGH

= CA HA


there4 퐀퐁퐃퐄

= 퐁퐂퐄퐅

= 퐂퐀 퐅퐃






= BCEF

= CA DF


= 퐁퐂ퟐ

퐄퐅ퟐ



DM = AB

DE and BC

EF = AB

DE [ ∵ Given

there4 ALDM

= BCEF

helliphellip(1)


= 1212

xBCxALxEFxDM


= BCxALEFxDM

[ ∵ ( 1)

= BCxBCEFxEF

= 퐁퐂ퟐ

퐄퐅ퟐ

But ABDE

= BCEF

= CA DF

[ ∵ Given


= AB2

DE2 = BC2

EF2 = CA2

DF2






rArr ABAD

= ACAB


rArr BCDC

= ACBC








12Trigonometry



휽




퐜퐨퐬휽 = ퟏ

퐬퐞퐜 휽




퐴퐵퐴퐶














Values 00 300 450 600 900


ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0





13 tan 휃 = 5

12

Cosec휃 = 135

sec휃 = 1312

cot휃 = 125



rArr 3+2 = 5







ퟑ






radic3

tan300= 1radic3

rArr 휃 = 300





퐴퐵




푥2minus푥1

A(x1y1) and B(x2y2)


푥2minus푥1


tan 휃 = 7






푥2minus푥1

m1 = 5minus20minus5

= 3minus5


= 3minus5




휃 훼


푥2minus푥1


= minus7minus4

= 74


= 4minus7

m1 x m2 = 74 x 4

minus7 = -1








2 c = -3


2x -3rArr2y = x -6

rArr x -2y -6 =0









푚+푛]



2 푦2+푦1

2]

= [4+22

7+32

]

= [62

102

] = (35)








Note

















apart





Theorem










Formulas



ퟐ풉


ퟑ


ퟑ

흅 = ퟐퟐퟕ






d =14cm

r= 7cm

흅 = ퟐퟐퟕ

h=10cm

l= 풓ퟐ + 풉ퟐ



l=radicퟏퟒퟗ

l=122



7 x 7 x 10

=440 sqcm

2휋푟(푟 + ℎ) =2 x 22

7 x 7(7+10)

=44 x 17 =748 sqcm

휋푟2ℎ =22

7 x 72 x 10

=1540cm3

Cone 휋푟푙 =22

7 x 7 x 122

=2684 sqcm

휋푟(푟 + 푙) =22

7 x 7 x ( 7 + 122 )

=22 x 192= 4224

13휋푟2ℎ

=13 x 22

7 x 72 x 10

=13 x 22

7 x 72 x 10

=5133 cm3

Sphere

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

43휋푟3

= 43

x 227

x 73 =14373 cm3


7 x 72

=462 sqcm

2휋푟2 =2 x 22

7 x 72

=308 sqcm

23휋푟3

= 23

x 22x 7

x 73 = 7186 cm3






TO D

80To E

150

100

80

30

70to C

40To B

From A


30m = 30 x = 15cm

70m = 70x = 35cm

80m = 80 x = 4cm

100m = 100x = 5cm

150m = 150x =75cm





To C E To D

120to D

E to E

220

210

120

80

40 to B

120toD

75to C

50to B

350

300

250

150

50

F 150toF

100to G

100toE

50toF

25toG

225

175

125

100

75

50

25toC

75toB











N + R = A + 2

N = 3 R = 4 A = 5 N+R = 3 +4 = 7 A+2 = 5 +2 = 7 there4 N+R = A+2

Exercise



N = 8 R = 6 A = 12 N+R = 8 +6 = 14 A+2 = 12 +2 = 14 there4 N+R = A+2

N = 3 R = 5 A = 6 N+R = 3 +5 = 8 A+2 = 6 +2 = 8 there4 N+R = A+2











1

4 4 6 4 +4 = 6 +2

2

3



4

5




1Real Numbers



AUB=BUA AcapB=BcapA







De Morganrsquos Law















ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ





휎 = 261 CV=4348


How the


Number of students

Central Angle

Walk 12 1236

x3600 = 1200

Cycle 8 836

x3600 = 800 Bus 3 3

36x3600 = 300

Car 4 436

x3600 = 400 School Van 9 9

36x3600 = 900

36 3600


7 Surds 2 4





Addition of Surds










12 rArr 2

12x3



13x2










radic5minusradic3

3radic5minusradic3



= 3(radic5+radic3)2


















P(x) =12 ndash 4 x 1 + 3 x 1 = 1 =1 - 4 + 3 + 1 = 1




1 (x ndash 2) ಇದು x3 -3x2 +6x -8


ೂೕ














x2 = 144 x2 ndash x = 0 x2 + 14x + 13 = 0 (x = -1) (x = -13)

4x = 81푥

x2 + 3 = 2x 7x2 -12x = 0 ( x = 13 )

7x = 647푥

x + 1x = 5 2m2 ndash 6m + 3 = 0 ( m = 1

2 )



K = 12m푣2

푣2=2퐾푚

v = plusmn 2퐾푚


2










3x2 ndash 5x + 2 = 0

3x2 ndash 5x + 2 = 0


3


3x = minus ퟐ

ퟑ

x2 - 53x = - 2

3

x2 - 53x +(5

6)2 = minus 2

3 + (5

6)2

(푥 minus 5 6

)2 minus 2436

+ 2536

(푥 minus 5 6

)2 = 136

(푥 minus 5 6

) = plusmn 16

x = 56 plusmn 1

6 rArr x = 6

6 or x = 4

6


3x2 ndash 5x + 2 = 0 a=3 b= -5 c = 2


2(3)


6


6

푥 =5 plusmn 1

6

푥 = 66 or x = 4

6

x = 1 or x = 23




Exercise






Nature of the Roots








푎 = minus2





1 = 1







1 2 ಮತು 3

2 6 ಮತು -5


4 -3 ಮತು 32




y

y = 2x2 x 0 +1 -1 +2 -2 +3 -3

y

y =ퟏퟐx2

x 0 +1 -1 +2 -2 +3 -3

y







10Similar triangles






= 퐵퐶퐸퐹










=23 AE=37 find

EC



To prove ADDB

= AEEC


Proof ∆ABC∆BDE

= 12 12

xADxELxDBxEL

[∵ A = 12

xbxh

∆ABC∆BDE

= ADDB

∆ADE∆CDE

= 12 12

xAExDNxDBxDN

[∵ A = 12

xbxh

∆ADE∆CDE

= AEEC

there4 퐀퐃

퐃퐁 = 퐀퐄






DE = BC

EF = CA

FD


there4 ABAG

= BCGH

= CA HA


there4 퐀퐁퐃퐄

= 퐁퐂퐄퐅

= 퐂퐀 퐅퐃






= BCEF

= CA DF


= 퐁퐂ퟐ

퐄퐅ퟐ



DM = AB

DE and BC

EF = AB

DE [ ∵ Given

there4 ALDM

= BCEF

helliphellip(1)


= 1212

xBCxALxEFxDM


= BCxALEFxDM

[ ∵ ( 1)

= BCxBCEFxEF

= 퐁퐂ퟐ

퐄퐅ퟐ

But ABDE

= BCEF

= CA DF

[ ∵ Given


= AB2

DE2 = BC2

EF2 = CA2

DF2






rArr ABAD

= ACAB


rArr BCDC

= ACBC








12Trigonometry



휽




퐜퐨퐬휽 = ퟏ

퐬퐞퐜 휽




퐴퐵퐴퐶














Values 00 300 450 600 900


ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0





13 tan 휃 = 5

12

Cosec휃 = 135

sec휃 = 1312

cot휃 = 125



rArr 3+2 = 5







ퟑ






radic3

tan300= 1radic3

rArr 휃 = 300





퐴퐵




푥2minus푥1

A(x1y1) and B(x2y2)


푥2minus푥1


tan 휃 = 7






푥2minus푥1

m1 = 5minus20minus5

= 3minus5


= 3minus5




휃 훼


푥2minus푥1


= minus7minus4

= 74


= 4minus7

m1 x m2 = 74 x 4

minus7 = -1








2 c = -3


2x -3rArr2y = x -6

rArr x -2y -6 =0









푚+푛]



2 푦2+푦1

2]

= [4+22

7+32

]

= [62

102

] = (35)








Note

















apart





Theorem










Formulas



ퟐ풉


ퟑ


ퟑ

흅 = ퟐퟐퟕ






d =14cm

r= 7cm

흅 = ퟐퟐퟕ

h=10cm

l= 풓ퟐ + 풉ퟐ



l=radicퟏퟒퟗ

l=122



7 x 7 x 10

=440 sqcm

2휋푟(푟 + ℎ) =2 x 22

7 x 7(7+10)

=44 x 17 =748 sqcm

휋푟2ℎ =22

7 x 72 x 10

=1540cm3

Cone 휋푟푙 =22

7 x 7 x 122

=2684 sqcm

휋푟(푟 + 푙) =22

7 x 7 x ( 7 + 122 )

=22 x 192= 4224

13휋푟2ℎ

=13 x 22

7 x 72 x 10

=13 x 22

7 x 72 x 10

=5133 cm3

Sphere

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

43휋푟3

= 43

x 227

x 73 =14373 cm3


7 x 72

=462 sqcm

2휋푟2 =2 x 22

7 x 72

=308 sqcm

23휋푟3

= 23

x 22x 7

x 73 = 7186 cm3






TO D

80To E

150

100

80

30

70to C

40To B

From A


30m = 30 x = 15cm

70m = 70x = 35cm

80m = 80 x = 4cm

100m = 100x = 5cm

150m = 150x =75cm





To C E To D

120to D

E to E

220

210

120

80

40 to B

120toD

75to C

50to B

350

300

250

150

50

F 150toF

100to G

100toE

50toF

25toG

225

175

125

100

75

50

25toC

75toB











N + R = A + 2

N = 3 R = 4 A = 5 N+R = 3 +4 = 7 A+2 = 5 +2 = 7 there4 N+R = A+2

Exercise



N = 8 R = 6 A = 12 N+R = 8 +6 = 14 A+2 = 12 +2 = 14 there4 N+R = A+2

N = 3 R = 5 A = 6 N+R = 3 +5 = 8 A+2 = 6 +2 = 8 there4 N+R = A+2











1

4 4 6 4 +4 = 6 +2

2

3



4

5




1Real Numbers



AUB=BUA AcapB=BcapA







De Morganrsquos Law















ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ




Addition of Surds










12 rArr 2

12x3



13x2










radic5minusradic3

3radic5minusradic3



= 3(radic5+radic3)2


















P(x) =12 ndash 4 x 1 + 3 x 1 = 1 =1 - 4 + 3 + 1 = 1




1 (x ndash 2) ಇದು x3 -3x2 +6x -8


ೂೕ














x2 = 144 x2 ndash x = 0 x2 + 14x + 13 = 0 (x = -1) (x = -13)

4x = 81푥

x2 + 3 = 2x 7x2 -12x = 0 ( x = 13 )

7x = 647푥

x + 1x = 5 2m2 ndash 6m + 3 = 0 ( m = 1

2 )



K = 12m푣2

푣2=2퐾푚

v = plusmn 2퐾푚


2










3x2 ndash 5x + 2 = 0

3x2 ndash 5x + 2 = 0


3


3x = minus ퟐ

ퟑ

x2 - 53x = - 2

3

x2 - 53x +(5

6)2 = minus 2

3 + (5

6)2

(푥 minus 5 6

)2 minus 2436

+ 2536

(푥 minus 5 6

)2 = 136

(푥 minus 5 6

) = plusmn 16

x = 56 plusmn 1

6 rArr x = 6

6 or x = 4

6


3x2 ndash 5x + 2 = 0 a=3 b= -5 c = 2


2(3)


6


6

푥 =5 plusmn 1

6

푥 = 66 or x = 4

6

x = 1 or x = 23




Exercise






Nature of the Roots








푎 = minus2





1 = 1







1 2 ಮತು 3

2 6 ಮತು -5


4 -3 ಮತು 32




y

y = 2x2 x 0 +1 -1 +2 -2 +3 -3

y

y =ퟏퟐx2

x 0 +1 -1 +2 -2 +3 -3

y







10Similar triangles






= 퐵퐶퐸퐹










=23 AE=37 find

EC



To prove ADDB

= AEEC


Proof ∆ABC∆BDE

= 12 12

xADxELxDBxEL

[∵ A = 12

xbxh

∆ABC∆BDE

= ADDB

∆ADE∆CDE

= 12 12

xAExDNxDBxDN

[∵ A = 12

xbxh

∆ADE∆CDE

= AEEC

there4 퐀퐃

퐃퐁 = 퐀퐄






DE = BC

EF = CA

FD


there4 ABAG

= BCGH

= CA HA


there4 퐀퐁퐃퐄

= 퐁퐂퐄퐅

= 퐂퐀 퐅퐃






= BCEF

= CA DF


= 퐁퐂ퟐ

퐄퐅ퟐ



DM = AB

DE and BC

EF = AB

DE [ ∵ Given

there4 ALDM

= BCEF

helliphellip(1)


= 1212

xBCxALxEFxDM


= BCxALEFxDM

[ ∵ ( 1)

= BCxBCEFxEF

= 퐁퐂ퟐ

퐄퐅ퟐ

But ABDE

= BCEF

= CA DF

[ ∵ Given


= AB2

DE2 = BC2

EF2 = CA2

DF2






rArr ABAD

= ACAB


rArr BCDC

= ACBC








12Trigonometry



휽




퐜퐨퐬휽 = ퟏ

퐬퐞퐜 휽




퐴퐵퐴퐶














Values 00 300 450 600 900


ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0





13 tan 휃 = 5

12

Cosec휃 = 135

sec휃 = 1312

cot휃 = 125



rArr 3+2 = 5







ퟑ






radic3

tan300= 1radic3

rArr 휃 = 300





퐴퐵




푥2minus푥1

A(x1y1) and B(x2y2)


푥2minus푥1


tan 휃 = 7






푥2minus푥1

m1 = 5minus20minus5

= 3minus5


= 3minus5




휃 훼


푥2minus푥1


= minus7minus4

= 74


= 4minus7

m1 x m2 = 74 x 4

minus7 = -1








2 c = -3


2x -3rArr2y = x -6

rArr x -2y -6 =0









푚+푛]



2 푦2+푦1

2]

= [4+22

7+32

]

= [62

102

] = (35)








Note

















apart





Theorem










Formulas



ퟐ풉


ퟑ


ퟑ

흅 = ퟐퟐퟕ






d =14cm

r= 7cm

흅 = ퟐퟐퟕ

h=10cm

l= 풓ퟐ + 풉ퟐ



l=radicퟏퟒퟗ

l=122



7 x 7 x 10

=440 sqcm

2휋푟(푟 + ℎ) =2 x 22

7 x 7(7+10)

=44 x 17 =748 sqcm

휋푟2ℎ =22

7 x 72 x 10

=1540cm3

Cone 휋푟푙 =22

7 x 7 x 122

=2684 sqcm

휋푟(푟 + 푙) =22

7 x 7 x ( 7 + 122 )

=22 x 192= 4224

13휋푟2ℎ

=13 x 22

7 x 72 x 10

=13 x 22

7 x 72 x 10

=5133 cm3

Sphere

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

43휋푟3

= 43

x 227

x 73 =14373 cm3


7 x 72

=462 sqcm

2휋푟2 =2 x 22

7 x 72

=308 sqcm

23휋푟3

= 23

x 22x 7

x 73 = 7186 cm3






TO D

80To E

150

100

80

30

70to C

40To B

From A


30m = 30 x = 15cm

70m = 70x = 35cm

80m = 80 x = 4cm

100m = 100x = 5cm

150m = 150x =75cm





To C E To D

120to D

E to E

220

210

120

80

40 to B

120toD

75to C

50to B

350

300

250

150

50

F 150toF

100to G

100toE

50toF

25toG

225

175

125

100

75

50

25toC

75toB











N + R = A + 2

N = 3 R = 4 A = 5 N+R = 3 +4 = 7 A+2 = 5 +2 = 7 there4 N+R = A+2

Exercise



N = 8 R = 6 A = 12 N+R = 8 +6 = 14 A+2 = 12 +2 = 14 there4 N+R = A+2

N = 3 R = 5 A = 6 N+R = 3 +5 = 8 A+2 = 6 +2 = 8 there4 N+R = A+2











1

4 4 6 4 +4 = 6 +2

2

3



4

5




1Real Numbers



AUB=BUA AcapB=BcapA







De Morganrsquos Law















ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ








radic5minusradic3

3radic5minusradic3



= 3(radic5+radic3)2


















P(x) =12 ndash 4 x 1 + 3 x 1 = 1 =1 - 4 + 3 + 1 = 1




1 (x ndash 2) ಇದು x3 -3x2 +6x -8


ೂೕ














x2 = 144 x2 ndash x = 0 x2 + 14x + 13 = 0 (x = -1) (x = -13)

4x = 81푥

x2 + 3 = 2x 7x2 -12x = 0 ( x = 13 )

7x = 647푥

x + 1x = 5 2m2 ndash 6m + 3 = 0 ( m = 1

2 )



K = 12m푣2

푣2=2퐾푚

v = plusmn 2퐾푚


2










3x2 ndash 5x + 2 = 0

3x2 ndash 5x + 2 = 0


3


3x = minus ퟐ

ퟑ

x2 - 53x = - 2

3

x2 - 53x +(5

6)2 = minus 2

3 + (5

6)2

(푥 minus 5 6

)2 minus 2436

+ 2536

(푥 minus 5 6

)2 = 136

(푥 minus 5 6

) = plusmn 16

x = 56 plusmn 1

6 rArr x = 6

6 or x = 4

6


3x2 ndash 5x + 2 = 0 a=3 b= -5 c = 2


2(3)


6


6

푥 =5 plusmn 1

6

푥 = 66 or x = 4

6

x = 1 or x = 23




Exercise






Nature of the Roots








푎 = minus2





1 = 1







1 2 ಮತು 3

2 6 ಮತು -5


4 -3 ಮತು 32




y

y = 2x2 x 0 +1 -1 +2 -2 +3 -3

y

y =ퟏퟐx2

x 0 +1 -1 +2 -2 +3 -3

y







10Similar triangles






= 퐵퐶퐸퐹










=23 AE=37 find

EC



To prove ADDB

= AEEC


Proof ∆ABC∆BDE

= 12 12

xADxELxDBxEL

[∵ A = 12

xbxh

∆ABC∆BDE

= ADDB

∆ADE∆CDE

= 12 12

xAExDNxDBxDN

[∵ A = 12

xbxh

∆ADE∆CDE

= AEEC

there4 퐀퐃

퐃퐁 = 퐀퐄






DE = BC

EF = CA

FD


there4 ABAG

= BCGH

= CA HA


there4 퐀퐁퐃퐄

= 퐁퐂퐄퐅

= 퐂퐀 퐅퐃






= BCEF

= CA DF


= 퐁퐂ퟐ

퐄퐅ퟐ



DM = AB

DE and BC

EF = AB

DE [ ∵ Given

there4 ALDM

= BCEF

helliphellip(1)


= 1212

xBCxALxEFxDM


= BCxALEFxDM

[ ∵ ( 1)

= BCxBCEFxEF

= 퐁퐂ퟐ

퐄퐅ퟐ

But ABDE

= BCEF

= CA DF

[ ∵ Given


= AB2

DE2 = BC2

EF2 = CA2

DF2






rArr ABAD

= ACAB


rArr BCDC

= ACBC








12Trigonometry



휽




퐜퐨퐬휽 = ퟏ

퐬퐞퐜 휽




퐴퐵퐴퐶














Values 00 300 450 600 900


ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0





13 tan 휃 = 5

12

Cosec휃 = 135

sec휃 = 1312

cot휃 = 125



rArr 3+2 = 5







ퟑ






radic3

tan300= 1radic3

rArr 휃 = 300





퐴퐵




푥2minus푥1

A(x1y1) and B(x2y2)


푥2minus푥1


tan 휃 = 7






푥2minus푥1

m1 = 5minus20minus5

= 3minus5


= 3minus5




휃 훼


푥2minus푥1


= minus7minus4

= 74


= 4minus7

m1 x m2 = 74 x 4

minus7 = -1








2 c = -3


2x -3rArr2y = x -6

rArr x -2y -6 =0









푚+푛]



2 푦2+푦1

2]

= [4+22

7+32

]

= [62

102

] = (35)








Note

















apart





Theorem










Formulas



ퟐ풉


ퟑ


ퟑ

흅 = ퟐퟐퟕ






d =14cm

r= 7cm

흅 = ퟐퟐퟕ

h=10cm

l= 풓ퟐ + 풉ퟐ



l=radicퟏퟒퟗ

l=122



7 x 7 x 10

=440 sqcm

2휋푟(푟 + ℎ) =2 x 22

7 x 7(7+10)

=44 x 17 =748 sqcm

휋푟2ℎ =22

7 x 72 x 10

=1540cm3

Cone 휋푟푙 =22

7 x 7 x 122

=2684 sqcm

휋푟(푟 + 푙) =22

7 x 7 x ( 7 + 122 )

=22 x 192= 4224

13휋푟2ℎ

=13 x 22

7 x 72 x 10

=13 x 22

7 x 72 x 10

=5133 cm3

Sphere

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

43휋푟3

= 43

x 227

x 73 =14373 cm3


7 x 72

=462 sqcm

2휋푟2 =2 x 22

7 x 72

=308 sqcm

23휋푟3

= 23

x 22x 7

x 73 = 7186 cm3






TO D

80To E

150

100

80

30

70to C

40To B

From A


30m = 30 x = 15cm

70m = 70x = 35cm

80m = 80 x = 4cm

100m = 100x = 5cm

150m = 150x =75cm





To C E To D

120to D

E to E

220

210

120

80

40 to B

120toD

75to C

50to B

350

300

250

150

50

F 150toF

100to G

100toE

50toF

25toG

225

175

125

100

75

50

25toC

75toB











N + R = A + 2

N = 3 R = 4 A = 5 N+R = 3 +4 = 7 A+2 = 5 +2 = 7 there4 N+R = A+2

Exercise



N = 8 R = 6 A = 12 N+R = 8 +6 = 14 A+2 = 12 +2 = 14 there4 N+R = A+2

N = 3 R = 5 A = 6 N+R = 3 +5 = 8 A+2 = 6 +2 = 8 there4 N+R = A+2











1

4 4 6 4 +4 = 6 +2

2

3



4

5




1Real Numbers



AUB=BUA AcapB=BcapA







De Morganrsquos Law















ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ













P(x) =12 ndash 4 x 1 + 3 x 1 = 1 =1 - 4 + 3 + 1 = 1




1 (x ndash 2) ಇದು x3 -3x2 +6x -8


ೂೕ














x2 = 144 x2 ndash x = 0 x2 + 14x + 13 = 0 (x = -1) (x = -13)

4x = 81푥

x2 + 3 = 2x 7x2 -12x = 0 ( x = 13 )

7x = 647푥

x + 1x = 5 2m2 ndash 6m + 3 = 0 ( m = 1

2 )



K = 12m푣2

푣2=2퐾푚

v = plusmn 2퐾푚


2










3x2 ndash 5x + 2 = 0

3x2 ndash 5x + 2 = 0


3


3x = minus ퟐ

ퟑ

x2 - 53x = - 2

3

x2 - 53x +(5

6)2 = minus 2

3 + (5

6)2

(푥 minus 5 6

)2 minus 2436

+ 2536

(푥 minus 5 6

)2 = 136

(푥 minus 5 6

) = plusmn 16

x = 56 plusmn 1

6 rArr x = 6

6 or x = 4

6


3x2 ndash 5x + 2 = 0 a=3 b= -5 c = 2


2(3)


6


6

푥 =5 plusmn 1

6

푥 = 66 or x = 4

6

x = 1 or x = 23




Exercise






Nature of the Roots








푎 = minus2





1 = 1







1 2 ಮತು 3

2 6 ಮತು -5


4 -3 ಮತು 32




y

y = 2x2 x 0 +1 -1 +2 -2 +3 -3

y

y =ퟏퟐx2

x 0 +1 -1 +2 -2 +3 -3

y







10Similar triangles






= 퐵퐶퐸퐹










=23 AE=37 find

EC



To prove ADDB

= AEEC


Proof ∆ABC∆BDE

= 12 12

xADxELxDBxEL

[∵ A = 12

xbxh

∆ABC∆BDE

= ADDB

∆ADE∆CDE

= 12 12

xAExDNxDBxDN

[∵ A = 12

xbxh

∆ADE∆CDE

= AEEC

there4 퐀퐃

퐃퐁 = 퐀퐄






DE = BC

EF = CA

FD


there4 ABAG

= BCGH

= CA HA


there4 퐀퐁퐃퐄

= 퐁퐂퐄퐅

= 퐂퐀 퐅퐃






= BCEF

= CA DF


= 퐁퐂ퟐ

퐄퐅ퟐ



DM = AB

DE and BC

EF = AB

DE [ ∵ Given

there4 ALDM

= BCEF

helliphellip(1)


= 1212

xBCxALxEFxDM


= BCxALEFxDM

[ ∵ ( 1)

= BCxBCEFxEF

= 퐁퐂ퟐ

퐄퐅ퟐ

But ABDE

= BCEF

= CA DF

[ ∵ Given


= AB2

DE2 = BC2

EF2 = CA2

DF2






rArr ABAD

= ACAB


rArr BCDC

= ACBC








12Trigonometry



휽




퐜퐨퐬휽 = ퟏ

퐬퐞퐜 휽




퐴퐵퐴퐶














Values 00 300 450 600 900


ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0





13 tan 휃 = 5

12

Cosec휃 = 135

sec휃 = 1312

cot휃 = 125



rArr 3+2 = 5







ퟑ






radic3

tan300= 1radic3

rArr 휃 = 300





퐴퐵




푥2minus푥1

A(x1y1) and B(x2y2)


푥2minus푥1


tan 휃 = 7






푥2minus푥1

m1 = 5minus20minus5

= 3minus5


= 3minus5




휃 훼


푥2minus푥1


= minus7minus4

= 74


= 4minus7

m1 x m2 = 74 x 4

minus7 = -1








2 c = -3


2x -3rArr2y = x -6

rArr x -2y -6 =0









푚+푛]



2 푦2+푦1

2]

= [4+22

7+32

]

= [62

102

] = (35)








Note

















apart





Theorem










Formulas



ퟐ풉


ퟑ


ퟑ

흅 = ퟐퟐퟕ






d =14cm

r= 7cm

흅 = ퟐퟐퟕ

h=10cm

l= 풓ퟐ + 풉ퟐ



l=radicퟏퟒퟗ

l=122



7 x 7 x 10

=440 sqcm

2휋푟(푟 + ℎ) =2 x 22

7 x 7(7+10)

=44 x 17 =748 sqcm

휋푟2ℎ =22

7 x 72 x 10

=1540cm3

Cone 휋푟푙 =22

7 x 7 x 122

=2684 sqcm

휋푟(푟 + 푙) =22

7 x 7 x ( 7 + 122 )

=22 x 192= 4224

13휋푟2ℎ

=13 x 22

7 x 72 x 10

=13 x 22

7 x 72 x 10

=5133 cm3

Sphere

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

43휋푟3

= 43

x 227

x 73 =14373 cm3


7 x 72

=462 sqcm

2휋푟2 =2 x 22

7 x 72

=308 sqcm

23휋푟3

= 23

x 22x 7

x 73 = 7186 cm3






TO D

80To E

150

100

80

30

70to C

40To B

From A


30m = 30 x = 15cm

70m = 70x = 35cm

80m = 80 x = 4cm

100m = 100x = 5cm

150m = 150x =75cm





To C E To D

120to D

E to E

220

210

120

80

40 to B

120toD

75to C

50to B

350

300

250

150

50

F 150toF

100to G

100toE

50toF

25toG

225

175

125

100

75

50

25toC

75toB











N + R = A + 2

N = 3 R = 4 A = 5 N+R = 3 +4 = 7 A+2 = 5 +2 = 7 there4 N+R = A+2

Exercise



N = 8 R = 6 A = 12 N+R = 8 +6 = 14 A+2 = 12 +2 = 14 there4 N+R = A+2

N = 3 R = 5 A = 6 N+R = 3 +5 = 8 A+2 = 6 +2 = 8 there4 N+R = A+2











1

4 4 6 4 +4 = 6 +2

2

3



4

5




1Real Numbers



AUB=BUA AcapB=BcapA







De Morganrsquos Law















ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ










x2 = 144 x2 ndash x = 0 x2 + 14x + 13 = 0 (x = -1) (x = -13)

4x = 81푥

x2 + 3 = 2x 7x2 -12x = 0 ( x = 13 )

7x = 647푥

x + 1x = 5 2m2 ndash 6m + 3 = 0 ( m = 1

2 )



K = 12m푣2

푣2=2퐾푚

v = plusmn 2퐾푚


2










3x2 ndash 5x + 2 = 0

3x2 ndash 5x + 2 = 0


3


3x = minus ퟐ

ퟑ

x2 - 53x = - 2

3

x2 - 53x +(5

6)2 = minus 2

3 + (5

6)2

(푥 minus 5 6

)2 minus 2436

+ 2536

(푥 minus 5 6

)2 = 136

(푥 minus 5 6

) = plusmn 16

x = 56 plusmn 1

6 rArr x = 6

6 or x = 4

6


3x2 ndash 5x + 2 = 0 a=3 b= -5 c = 2


2(3)


6


6

푥 =5 plusmn 1

6

푥 = 66 or x = 4

6

x = 1 or x = 23




Exercise






Nature of the Roots








푎 = minus2





1 = 1







1 2 ಮತು 3

2 6 ಮತು -5


4 -3 ಮತು 32




y

y = 2x2 x 0 +1 -1 +2 -2 +3 -3

y

y =ퟏퟐx2

x 0 +1 -1 +2 -2 +3 -3

y







10Similar triangles






= 퐵퐶퐸퐹










=23 AE=37 find

EC



To prove ADDB

= AEEC


Proof ∆ABC∆BDE

= 12 12

xADxELxDBxEL

[∵ A = 12

xbxh

∆ABC∆BDE

= ADDB

∆ADE∆CDE

= 12 12

xAExDNxDBxDN

[∵ A = 12

xbxh

∆ADE∆CDE

= AEEC

there4 퐀퐃

퐃퐁 = 퐀퐄






DE = BC

EF = CA

FD


there4 ABAG

= BCGH

= CA HA


there4 퐀퐁퐃퐄

= 퐁퐂퐄퐅

= 퐂퐀 퐅퐃






= BCEF

= CA DF


= 퐁퐂ퟐ

퐄퐅ퟐ



DM = AB

DE and BC

EF = AB

DE [ ∵ Given

there4 ALDM

= BCEF

helliphellip(1)


= 1212

xBCxALxEFxDM


= BCxALEFxDM

[ ∵ ( 1)

= BCxBCEFxEF

= 퐁퐂ퟐ

퐄퐅ퟐ

But ABDE

= BCEF

= CA DF

[ ∵ Given


= AB2

DE2 = BC2

EF2 = CA2

DF2






rArr ABAD

= ACAB


rArr BCDC

= ACBC








12Trigonometry



휽




퐜퐨퐬휽 = ퟏ

퐬퐞퐜 휽




퐴퐵퐴퐶














Values 00 300 450 600 900


ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0





13 tan 휃 = 5

12

Cosec휃 = 135

sec휃 = 1312

cot휃 = 125



rArr 3+2 = 5







ퟑ






radic3

tan300= 1radic3

rArr 휃 = 300





퐴퐵




푥2minus푥1

A(x1y1) and B(x2y2)


푥2minus푥1


tan 휃 = 7






푥2minus푥1

m1 = 5minus20minus5

= 3minus5


= 3minus5




휃 훼


푥2minus푥1


= minus7minus4

= 74


= 4minus7

m1 x m2 = 74 x 4

minus7 = -1








2 c = -3


2x -3rArr2y = x -6

rArr x -2y -6 =0









푚+푛]



2 푦2+푦1

2]

= [4+22

7+32

]

= [62

102

] = (35)








Note

















apart





Theorem










Formulas



ퟐ풉


ퟑ


ퟑ

흅 = ퟐퟐퟕ






d =14cm

r= 7cm

흅 = ퟐퟐퟕ

h=10cm

l= 풓ퟐ + 풉ퟐ



l=radicퟏퟒퟗ

l=122



7 x 7 x 10

=440 sqcm

2휋푟(푟 + ℎ) =2 x 22

7 x 7(7+10)

=44 x 17 =748 sqcm

휋푟2ℎ =22

7 x 72 x 10

=1540cm3

Cone 휋푟푙 =22

7 x 7 x 122

=2684 sqcm

휋푟(푟 + 푙) =22

7 x 7 x ( 7 + 122 )

=22 x 192= 4224

13휋푟2ℎ

=13 x 22

7 x 72 x 10

=13 x 22

7 x 72 x 10

=5133 cm3

Sphere

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

43휋푟3

= 43

x 227

x 73 =14373 cm3


7 x 72

=462 sqcm

2휋푟2 =2 x 22

7 x 72

=308 sqcm

23휋푟3

= 23

x 22x 7

x 73 = 7186 cm3






TO D

80To E

150

100

80

30

70to C

40To B

From A


30m = 30 x = 15cm

70m = 70x = 35cm

80m = 80 x = 4cm

100m = 100x = 5cm

150m = 150x =75cm





To C E To D

120to D

E to E

220

210

120

80

40 to B

120toD

75to C

50to B

350

300

250

150

50

F 150toF

100to G

100toE

50toF

25toG

225

175

125

100

75

50

25toC

75toB











N + R = A + 2

N = 3 R = 4 A = 5 N+R = 3 +4 = 7 A+2 = 5 +2 = 7 there4 N+R = A+2

Exercise



N = 8 R = 6 A = 12 N+R = 8 +6 = 14 A+2 = 12 +2 = 14 there4 N+R = A+2

N = 3 R = 5 A = 6 N+R = 3 +5 = 8 A+2 = 6 +2 = 8 there4 N+R = A+2











1

4 4 6 4 +4 = 6 +2

2

3



4

5




1Real Numbers



AUB=BUA AcapB=BcapA







De Morganrsquos Law















ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ







3x2 ndash 5x + 2 = 0

3x2 ndash 5x + 2 = 0


3


3x = minus ퟐ

ퟑ

x2 - 53x = - 2

3

x2 - 53x +(5

6)2 = minus 2

3 + (5

6)2

(푥 minus 5 6

)2 minus 2436

+ 2536

(푥 minus 5 6

)2 = 136

(푥 minus 5 6

) = plusmn 16

x = 56 plusmn 1

6 rArr x = 6

6 or x = 4

6


3x2 ndash 5x + 2 = 0 a=3 b= -5 c = 2


2(3)


6


6

푥 =5 plusmn 1

6

푥 = 66 or x = 4

6

x = 1 or x = 23




Exercise






Nature of the Roots








푎 = minus2





1 = 1







1 2 ಮತು 3

2 6 ಮತು -5


4 -3 ಮತು 32




y

y = 2x2 x 0 +1 -1 +2 -2 +3 -3

y

y =ퟏퟐx2

x 0 +1 -1 +2 -2 +3 -3

y







10Similar triangles






= 퐵퐶퐸퐹










=23 AE=37 find

EC



To prove ADDB

= AEEC


Proof ∆ABC∆BDE

= 12 12

xADxELxDBxEL

[∵ A = 12

xbxh

∆ABC∆BDE

= ADDB

∆ADE∆CDE

= 12 12

xAExDNxDBxDN

[∵ A = 12

xbxh

∆ADE∆CDE

= AEEC

there4 퐀퐃

퐃퐁 = 퐀퐄






DE = BC

EF = CA

FD


there4 ABAG

= BCGH

= CA HA


there4 퐀퐁퐃퐄

= 퐁퐂퐄퐅

= 퐂퐀 퐅퐃






= BCEF

= CA DF


= 퐁퐂ퟐ

퐄퐅ퟐ



DM = AB

DE and BC

EF = AB

DE [ ∵ Given

there4 ALDM

= BCEF

helliphellip(1)


= 1212

xBCxALxEFxDM


= BCxALEFxDM

[ ∵ ( 1)

= BCxBCEFxEF

= 퐁퐂ퟐ

퐄퐅ퟐ

But ABDE

= BCEF

= CA DF

[ ∵ Given


= AB2

DE2 = BC2

EF2 = CA2

DF2






rArr ABAD

= ACAB


rArr BCDC

= ACBC








12Trigonometry



휽




퐜퐨퐬휽 = ퟏ

퐬퐞퐜 휽




퐴퐵퐴퐶














Values 00 300 450 600 900


ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0





13 tan 휃 = 5

12

Cosec휃 = 135

sec휃 = 1312

cot휃 = 125



rArr 3+2 = 5







ퟑ






radic3

tan300= 1radic3

rArr 휃 = 300





퐴퐵




푥2minus푥1

A(x1y1) and B(x2y2)


푥2minus푥1


tan 휃 = 7






푥2minus푥1

m1 = 5minus20minus5

= 3minus5


= 3minus5




휃 훼


푥2minus푥1


= minus7minus4

= 74


= 4minus7

m1 x m2 = 74 x 4

minus7 = -1








2 c = -3


2x -3rArr2y = x -6

rArr x -2y -6 =0









푚+푛]



2 푦2+푦1

2]

= [4+22

7+32

]

= [62

102

] = (35)








Note

















apart





Theorem










Formulas



ퟐ풉


ퟑ


ퟑ

흅 = ퟐퟐퟕ






d =14cm

r= 7cm

흅 = ퟐퟐퟕ

h=10cm

l= 풓ퟐ + 풉ퟐ



l=radicퟏퟒퟗ

l=122



7 x 7 x 10

=440 sqcm

2휋푟(푟 + ℎ) =2 x 22

7 x 7(7+10)

=44 x 17 =748 sqcm

휋푟2ℎ =22

7 x 72 x 10

=1540cm3

Cone 휋푟푙 =22

7 x 7 x 122

=2684 sqcm

휋푟(푟 + 푙) =22

7 x 7 x ( 7 + 122 )

=22 x 192= 4224

13휋푟2ℎ

=13 x 22

7 x 72 x 10

=13 x 22

7 x 72 x 10

=5133 cm3

Sphere

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

43휋푟3

= 43

x 227

x 73 =14373 cm3


7 x 72

=462 sqcm

2휋푟2 =2 x 22

7 x 72

=308 sqcm

23휋푟3

= 23

x 22x 7

x 73 = 7186 cm3






TO D

80To E

150

100

80

30

70to C

40To B

From A


30m = 30 x = 15cm

70m = 70x = 35cm

80m = 80 x = 4cm

100m = 100x = 5cm

150m = 150x =75cm





To C E To D

120to D

E to E

220

210

120

80

40 to B

120toD

75to C

50to B

350

300

250

150

50

F 150toF

100to G

100toE

50toF

25toG

225

175

125

100

75

50

25toC

75toB











N + R = A + 2

N = 3 R = 4 A = 5 N+R = 3 +4 = 7 A+2 = 5 +2 = 7 there4 N+R = A+2

Exercise



N = 8 R = 6 A = 12 N+R = 8 +6 = 14 A+2 = 12 +2 = 14 there4 N+R = A+2

N = 3 R = 5 A = 6 N+R = 3 +5 = 8 A+2 = 6 +2 = 8 there4 N+R = A+2











1

4 4 6 4 +4 = 6 +2

2

3



4

5




1Real Numbers



AUB=BUA AcapB=BcapA







De Morganrsquos Law















ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ




Exercise






Nature of the Roots








푎 = minus2





1 = 1







1 2 ಮತು 3

2 6 ಮತು -5


4 -3 ಮತು 32




y

y = 2x2 x 0 +1 -1 +2 -2 +3 -3

y

y =ퟏퟐx2

x 0 +1 -1 +2 -2 +3 -3

y







10Similar triangles






= 퐵퐶퐸퐹










=23 AE=37 find

EC



To prove ADDB

= AEEC


Proof ∆ABC∆BDE

= 12 12

xADxELxDBxEL

[∵ A = 12

xbxh

∆ABC∆BDE

= ADDB

∆ADE∆CDE

= 12 12

xAExDNxDBxDN

[∵ A = 12

xbxh

∆ADE∆CDE

= AEEC

there4 퐀퐃

퐃퐁 = 퐀퐄






DE = BC

EF = CA

FD


there4 ABAG

= BCGH

= CA HA


there4 퐀퐁퐃퐄

= 퐁퐂퐄퐅

= 퐂퐀 퐅퐃






= BCEF

= CA DF


= 퐁퐂ퟐ

퐄퐅ퟐ



DM = AB

DE and BC

EF = AB

DE [ ∵ Given

there4 ALDM

= BCEF

helliphellip(1)


= 1212

xBCxALxEFxDM


= BCxALEFxDM

[ ∵ ( 1)

= BCxBCEFxEF

= 퐁퐂ퟐ

퐄퐅ퟐ

But ABDE

= BCEF

= CA DF

[ ∵ Given


= AB2

DE2 = BC2

EF2 = CA2

DF2






rArr ABAD

= ACAB


rArr BCDC

= ACBC








12Trigonometry



휽




퐜퐨퐬휽 = ퟏ

퐬퐞퐜 휽




퐴퐵퐴퐶














Values 00 300 450 600 900


ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0





13 tan 휃 = 5

12

Cosec휃 = 135

sec휃 = 1312

cot휃 = 125



rArr 3+2 = 5







ퟑ






radic3

tan300= 1radic3

rArr 휃 = 300





퐴퐵




푥2minus푥1

A(x1y1) and B(x2y2)


푥2minus푥1


tan 휃 = 7






푥2minus푥1

m1 = 5minus20minus5

= 3minus5


= 3minus5




휃 훼


푥2minus푥1


= minus7minus4

= 74


= 4minus7

m1 x m2 = 74 x 4

minus7 = -1








2 c = -3


2x -3rArr2y = x -6

rArr x -2y -6 =0









푚+푛]



2 푦2+푦1

2]

= [4+22

7+32

]

= [62

102

] = (35)








Note

















apart





Theorem










Formulas



ퟐ풉


ퟑ


ퟑ

흅 = ퟐퟐퟕ






d =14cm

r= 7cm

흅 = ퟐퟐퟕ

h=10cm

l= 풓ퟐ + 풉ퟐ



l=radicퟏퟒퟗ

l=122



7 x 7 x 10

=440 sqcm

2휋푟(푟 + ℎ) =2 x 22

7 x 7(7+10)

=44 x 17 =748 sqcm

휋푟2ℎ =22

7 x 72 x 10

=1540cm3

Cone 휋푟푙 =22

7 x 7 x 122

=2684 sqcm

휋푟(푟 + 푙) =22

7 x 7 x ( 7 + 122 )

=22 x 192= 4224

13휋푟2ℎ

=13 x 22

7 x 72 x 10

=13 x 22

7 x 72 x 10

=5133 cm3

Sphere

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

43휋푟3

= 43

x 227

x 73 =14373 cm3


7 x 72

=462 sqcm

2휋푟2 =2 x 22

7 x 72

=308 sqcm

23휋푟3

= 23

x 22x 7

x 73 = 7186 cm3






TO D

80To E

150

100

80

30

70to C

40To B

From A


30m = 30 x = 15cm

70m = 70x = 35cm

80m = 80 x = 4cm

100m = 100x = 5cm

150m = 150x =75cm





To C E To D

120to D

E to E

220

210

120

80

40 to B

120toD

75to C

50to B

350

300

250

150

50

F 150toF

100to G

100toE

50toF

25toG

225

175

125

100

75

50

25toC

75toB











N + R = A + 2

N = 3 R = 4 A = 5 N+R = 3 +4 = 7 A+2 = 5 +2 = 7 there4 N+R = A+2

Exercise



N = 8 R = 6 A = 12 N+R = 8 +6 = 14 A+2 = 12 +2 = 14 there4 N+R = A+2

N = 3 R = 5 A = 6 N+R = 3 +5 = 8 A+2 = 6 +2 = 8 there4 N+R = A+2











1

4 4 6 4 +4 = 6 +2

2

3



4

5




1Real Numbers



AUB=BUA AcapB=BcapA







De Morganrsquos Law















ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ





1 = 1







1 2 ಮತು 3

2 6 ಮತು -5


4 -3 ಮತು 32




y

y = 2x2 x 0 +1 -1 +2 -2 +3 -3

y

y =ퟏퟐx2

x 0 +1 -1 +2 -2 +3 -3

y







10Similar triangles






= 퐵퐶퐸퐹










=23 AE=37 find

EC



To prove ADDB

= AEEC


Proof ∆ABC∆BDE

= 12 12

xADxELxDBxEL

[∵ A = 12

xbxh

∆ABC∆BDE

= ADDB

∆ADE∆CDE

= 12 12

xAExDNxDBxDN

[∵ A = 12

xbxh

∆ADE∆CDE

= AEEC

there4 퐀퐃

퐃퐁 = 퐀퐄






DE = BC

EF = CA

FD


there4 ABAG

= BCGH

= CA HA


there4 퐀퐁퐃퐄

= 퐁퐂퐄퐅

= 퐂퐀 퐅퐃






= BCEF

= CA DF


= 퐁퐂ퟐ

퐄퐅ퟐ



DM = AB

DE and BC

EF = AB

DE [ ∵ Given

there4 ALDM

= BCEF

helliphellip(1)


= 1212

xBCxALxEFxDM


= BCxALEFxDM

[ ∵ ( 1)

= BCxBCEFxEF

= 퐁퐂ퟐ

퐄퐅ퟐ

But ABDE

= BCEF

= CA DF

[ ∵ Given


= AB2

DE2 = BC2

EF2 = CA2

DF2






rArr ABAD

= ACAB


rArr BCDC

= ACBC








12Trigonometry



휽




퐜퐨퐬휽 = ퟏ

퐬퐞퐜 휽




퐴퐵퐴퐶














Values 00 300 450 600 900


ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0





13 tan 휃 = 5

12

Cosec휃 = 135

sec휃 = 1312

cot휃 = 125



rArr 3+2 = 5







ퟑ






radic3

tan300= 1radic3

rArr 휃 = 300





퐴퐵




푥2minus푥1

A(x1y1) and B(x2y2)


푥2minus푥1


tan 휃 = 7






푥2minus푥1

m1 = 5minus20minus5

= 3minus5


= 3minus5




휃 훼


푥2minus푥1


= minus7minus4

= 74


= 4minus7

m1 x m2 = 74 x 4

minus7 = -1








2 c = -3


2x -3rArr2y = x -6

rArr x -2y -6 =0









푚+푛]



2 푦2+푦1

2]

= [4+22

7+32

]

= [62

102

] = (35)








Note

















apart





Theorem










Formulas



ퟐ풉


ퟑ


ퟑ

흅 = ퟐퟐퟕ






d =14cm

r= 7cm

흅 = ퟐퟐퟕ

h=10cm

l= 풓ퟐ + 풉ퟐ



l=radicퟏퟒퟗ

l=122



7 x 7 x 10

=440 sqcm

2휋푟(푟 + ℎ) =2 x 22

7 x 7(7+10)

=44 x 17 =748 sqcm

휋푟2ℎ =22

7 x 72 x 10

=1540cm3

Cone 휋푟푙 =22

7 x 7 x 122

=2684 sqcm

휋푟(푟 + 푙) =22

7 x 7 x ( 7 + 122 )

=22 x 192= 4224

13휋푟2ℎ

=13 x 22

7 x 72 x 10

=13 x 22

7 x 72 x 10

=5133 cm3

Sphere

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

43휋푟3

= 43

x 227

x 73 =14373 cm3


7 x 72

=462 sqcm

2휋푟2 =2 x 22

7 x 72

=308 sqcm

23휋푟3

= 23

x 22x 7

x 73 = 7186 cm3






TO D

80To E

150

100

80

30

70to C

40To B

From A


30m = 30 x = 15cm

70m = 70x = 35cm

80m = 80 x = 4cm

100m = 100x = 5cm

150m = 150x =75cm





To C E To D

120to D

E to E

220

210

120

80

40 to B

120toD

75to C

50to B

350

300

250

150

50

F 150toF

100to G

100toE

50toF

25toG

225

175

125

100

75

50

25toC

75toB











N + R = A + 2

N = 3 R = 4 A = 5 N+R = 3 +4 = 7 A+2 = 5 +2 = 7 there4 N+R = A+2

Exercise



N = 8 R = 6 A = 12 N+R = 8 +6 = 14 A+2 = 12 +2 = 14 there4 N+R = A+2

N = 3 R = 5 A = 6 N+R = 3 +5 = 8 A+2 = 6 +2 = 8 there4 N+R = A+2











1

4 4 6 4 +4 = 6 +2

2

3



4

5




1Real Numbers



AUB=BUA AcapB=BcapA







De Morganrsquos Law















ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ








10Similar triangles






= 퐵퐶퐸퐹










=23 AE=37 find

EC



To prove ADDB

= AEEC


Proof ∆ABC∆BDE

= 12 12

xADxELxDBxEL

[∵ A = 12

xbxh

∆ABC∆BDE

= ADDB

∆ADE∆CDE

= 12 12

xAExDNxDBxDN

[∵ A = 12

xbxh

∆ADE∆CDE

= AEEC

there4 퐀퐃

퐃퐁 = 퐀퐄






DE = BC

EF = CA

FD


there4 ABAG

= BCGH

= CA HA


there4 퐀퐁퐃퐄

= 퐁퐂퐄퐅

= 퐂퐀 퐅퐃






= BCEF

= CA DF


= 퐁퐂ퟐ

퐄퐅ퟐ



DM = AB

DE and BC

EF = AB

DE [ ∵ Given

there4 ALDM

= BCEF

helliphellip(1)


= 1212

xBCxALxEFxDM


= BCxALEFxDM

[ ∵ ( 1)

= BCxBCEFxEF

= 퐁퐂ퟐ

퐄퐅ퟐ

But ABDE

= BCEF

= CA DF

[ ∵ Given


= AB2

DE2 = BC2

EF2 = CA2

DF2






rArr ABAD

= ACAB


rArr BCDC

= ACBC








12Trigonometry



휽




퐜퐨퐬휽 = ퟏ

퐬퐞퐜 휽




퐴퐵퐴퐶














Values 00 300 450 600 900


ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0





13 tan 휃 = 5

12

Cosec휃 = 135

sec휃 = 1312

cot휃 = 125



rArr 3+2 = 5







ퟑ






radic3

tan300= 1radic3

rArr 휃 = 300





퐴퐵




푥2minus푥1

A(x1y1) and B(x2y2)


푥2minus푥1


tan 휃 = 7






푥2minus푥1

m1 = 5minus20minus5

= 3minus5


= 3minus5




휃 훼


푥2minus푥1


= minus7minus4

= 74


= 4minus7

m1 x m2 = 74 x 4

minus7 = -1








2 c = -3


2x -3rArr2y = x -6

rArr x -2y -6 =0









푚+푛]



2 푦2+푦1

2]

= [4+22

7+32

]

= [62

102

] = (35)








Note

















apart





Theorem










Formulas



ퟐ풉


ퟑ


ퟑ

흅 = ퟐퟐퟕ






d =14cm

r= 7cm

흅 = ퟐퟐퟕ

h=10cm

l= 풓ퟐ + 풉ퟐ



l=radicퟏퟒퟗ

l=122



7 x 7 x 10

=440 sqcm

2휋푟(푟 + ℎ) =2 x 22

7 x 7(7+10)

=44 x 17 =748 sqcm

휋푟2ℎ =22

7 x 72 x 10

=1540cm3

Cone 휋푟푙 =22

7 x 7 x 122

=2684 sqcm

휋푟(푟 + 푙) =22

7 x 7 x ( 7 + 122 )

=22 x 192= 4224

13휋푟2ℎ

=13 x 22

7 x 72 x 10

=13 x 22

7 x 72 x 10

=5133 cm3

Sphere

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

43휋푟3

= 43

x 227

x 73 =14373 cm3


7 x 72

=462 sqcm

2휋푟2 =2 x 22

7 x 72

=308 sqcm

23휋푟3

= 23

x 22x 7

x 73 = 7186 cm3






TO D

80To E

150

100

80

30

70to C

40To B

From A


30m = 30 x = 15cm

70m = 70x = 35cm

80m = 80 x = 4cm

100m = 100x = 5cm

150m = 150x =75cm





To C E To D

120to D

E to E

220

210

120

80

40 to B

120toD

75to C

50to B

350

300

250

150

50

F 150toF

100to G

100toE

50toF

25toG

225

175

125

100

75

50

25toC

75toB











N + R = A + 2

N = 3 R = 4 A = 5 N+R = 3 +4 = 7 A+2 = 5 +2 = 7 there4 N+R = A+2

Exercise



N = 8 R = 6 A = 12 N+R = 8 +6 = 14 A+2 = 12 +2 = 14 there4 N+R = A+2

N = 3 R = 5 A = 6 N+R = 3 +5 = 8 A+2 = 6 +2 = 8 there4 N+R = A+2











1

4 4 6 4 +4 = 6 +2

2

3



4

5




1Real Numbers



AUB=BUA AcapB=BcapA







De Morganrsquos Law















ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ






=23 AE=37 find

EC



To prove ADDB

= AEEC


Proof ∆ABC∆BDE

= 12 12

xADxELxDBxEL

[∵ A = 12

xbxh

∆ABC∆BDE

= ADDB

∆ADE∆CDE

= 12 12

xAExDNxDBxDN

[∵ A = 12

xbxh

∆ADE∆CDE

= AEEC

there4 퐀퐃

퐃퐁 = 퐀퐄






DE = BC

EF = CA

FD


there4 ABAG

= BCGH

= CA HA


there4 퐀퐁퐃퐄

= 퐁퐂퐄퐅

= 퐂퐀 퐅퐃






= BCEF

= CA DF


= 퐁퐂ퟐ

퐄퐅ퟐ



DM = AB

DE and BC

EF = AB

DE [ ∵ Given

there4 ALDM

= BCEF

helliphellip(1)


= 1212

xBCxALxEFxDM


= BCxALEFxDM

[ ∵ ( 1)

= BCxBCEFxEF

= 퐁퐂ퟐ

퐄퐅ퟐ

But ABDE

= BCEF

= CA DF

[ ∵ Given


= AB2

DE2 = BC2

EF2 = CA2

DF2






rArr ABAD

= ACAB


rArr BCDC

= ACBC








12Trigonometry



휽




퐜퐨퐬휽 = ퟏ

퐬퐞퐜 휽




퐴퐵퐴퐶














Values 00 300 450 600 900


ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0





13 tan 휃 = 5

12

Cosec휃 = 135

sec휃 = 1312

cot휃 = 125



rArr 3+2 = 5







ퟑ






radic3

tan300= 1radic3

rArr 휃 = 300





퐴퐵




푥2minus푥1

A(x1y1) and B(x2y2)


푥2minus푥1


tan 휃 = 7






푥2minus푥1

m1 = 5minus20minus5

= 3minus5


= 3minus5




휃 훼


푥2minus푥1


= minus7minus4

= 74


= 4minus7

m1 x m2 = 74 x 4

minus7 = -1








2 c = -3


2x -3rArr2y = x -6

rArr x -2y -6 =0









푚+푛]



2 푦2+푦1

2]

= [4+22

7+32

]

= [62

102

] = (35)








Note

















apart





Theorem










Formulas



ퟐ풉


ퟑ


ퟑ

흅 = ퟐퟐퟕ






d =14cm

r= 7cm

흅 = ퟐퟐퟕ

h=10cm

l= 풓ퟐ + 풉ퟐ



l=radicퟏퟒퟗ

l=122



7 x 7 x 10

=440 sqcm

2휋푟(푟 + ℎ) =2 x 22

7 x 7(7+10)

=44 x 17 =748 sqcm

휋푟2ℎ =22

7 x 72 x 10

=1540cm3

Cone 휋푟푙 =22

7 x 7 x 122

=2684 sqcm

휋푟(푟 + 푙) =22

7 x 7 x ( 7 + 122 )

=22 x 192= 4224

13휋푟2ℎ

=13 x 22

7 x 72 x 10

=13 x 22

7 x 72 x 10

=5133 cm3

Sphere

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

43휋푟3

= 43

x 227

x 73 =14373 cm3


7 x 72

=462 sqcm

2휋푟2 =2 x 22

7 x 72

=308 sqcm

23휋푟3

= 23

x 22x 7

x 73 = 7186 cm3






TO D

80To E

150

100

80

30

70to C

40To B

From A


30m = 30 x = 15cm

70m = 70x = 35cm

80m = 80 x = 4cm

100m = 100x = 5cm

150m = 150x =75cm





To C E To D

120to D

E to E

220

210

120

80

40 to B

120toD

75to C

50to B

350

300

250

150

50

F 150toF

100to G

100toE

50toF

25toG

225

175

125

100

75

50

25toC

75toB











N + R = A + 2

N = 3 R = 4 A = 5 N+R = 3 +4 = 7 A+2 = 5 +2 = 7 there4 N+R = A+2

Exercise



N = 8 R = 6 A = 12 N+R = 8 +6 = 14 A+2 = 12 +2 = 14 there4 N+R = A+2

N = 3 R = 5 A = 6 N+R = 3 +5 = 8 A+2 = 6 +2 = 8 there4 N+R = A+2











1

4 4 6 4 +4 = 6 +2

2

3



4

5




1Real Numbers



AUB=BUA AcapB=BcapA







De Morganrsquos Law















ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ






DE = BC

EF = CA

FD


there4 ABAG

= BCGH

= CA HA


there4 퐀퐁퐃퐄

= 퐁퐂퐄퐅

= 퐂퐀 퐅퐃






= BCEF

= CA DF


= 퐁퐂ퟐ

퐄퐅ퟐ



DM = AB

DE and BC

EF = AB

DE [ ∵ Given

there4 ALDM

= BCEF

helliphellip(1)


= 1212

xBCxALxEFxDM


= BCxALEFxDM

[ ∵ ( 1)

= BCxBCEFxEF

= 퐁퐂ퟐ

퐄퐅ퟐ

But ABDE

= BCEF

= CA DF

[ ∵ Given


= AB2

DE2 = BC2

EF2 = CA2

DF2






rArr ABAD

= ACAB


rArr BCDC

= ACBC








12Trigonometry



휽




퐜퐨퐬휽 = ퟏ

퐬퐞퐜 휽




퐴퐵퐴퐶














Values 00 300 450 600 900


ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0





13 tan 휃 = 5

12

Cosec휃 = 135

sec휃 = 1312

cot휃 = 125



rArr 3+2 = 5







ퟑ






radic3

tan300= 1radic3

rArr 휃 = 300





퐴퐵




푥2minus푥1

A(x1y1) and B(x2y2)


푥2minus푥1


tan 휃 = 7






푥2minus푥1

m1 = 5minus20minus5

= 3minus5


= 3minus5




휃 훼


푥2minus푥1


= minus7minus4

= 74


= 4minus7

m1 x m2 = 74 x 4

minus7 = -1








2 c = -3


2x -3rArr2y = x -6

rArr x -2y -6 =0









푚+푛]



2 푦2+푦1

2]

= [4+22

7+32

]

= [62

102

] = (35)








Note

















apart





Theorem










Formulas



ퟐ풉


ퟑ


ퟑ

흅 = ퟐퟐퟕ






d =14cm

r= 7cm

흅 = ퟐퟐퟕ

h=10cm

l= 풓ퟐ + 풉ퟐ



l=radicퟏퟒퟗ

l=122



7 x 7 x 10

=440 sqcm

2휋푟(푟 + ℎ) =2 x 22

7 x 7(7+10)

=44 x 17 =748 sqcm

휋푟2ℎ =22

7 x 72 x 10

=1540cm3

Cone 휋푟푙 =22

7 x 7 x 122

=2684 sqcm

휋푟(푟 + 푙) =22

7 x 7 x ( 7 + 122 )

=22 x 192= 4224

13휋푟2ℎ

=13 x 22

7 x 72 x 10

=13 x 22

7 x 72 x 10

=5133 cm3

Sphere

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

43휋푟3

= 43

x 227

x 73 =14373 cm3


7 x 72

=462 sqcm

2휋푟2 =2 x 22

7 x 72

=308 sqcm

23휋푟3

= 23

x 22x 7

x 73 = 7186 cm3






TO D

80To E

150

100

80

30

70to C

40To B

From A


30m = 30 x = 15cm

70m = 70x = 35cm

80m = 80 x = 4cm

100m = 100x = 5cm

150m = 150x =75cm





To C E To D

120to D

E to E

220

210

120

80

40 to B

120toD

75to C

50to B

350

300

250

150

50

F 150toF

100to G

100toE

50toF

25toG

225

175

125

100

75

50

25toC

75toB











N + R = A + 2

N = 3 R = 4 A = 5 N+R = 3 +4 = 7 A+2 = 5 +2 = 7 there4 N+R = A+2

Exercise



N = 8 R = 6 A = 12 N+R = 8 +6 = 14 A+2 = 12 +2 = 14 there4 N+R = A+2

N = 3 R = 5 A = 6 N+R = 3 +5 = 8 A+2 = 6 +2 = 8 there4 N+R = A+2











1

4 4 6 4 +4 = 6 +2

2

3



4

5




1Real Numbers



AUB=BUA AcapB=BcapA







De Morganrsquos Law















ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ






= BCEF

= CA DF


= 퐁퐂ퟐ

퐄퐅ퟐ



DM = AB

DE and BC

EF = AB

DE [ ∵ Given

there4 ALDM

= BCEF

helliphellip(1)


= 1212

xBCxALxEFxDM


= BCxALEFxDM

[ ∵ ( 1)

= BCxBCEFxEF

= 퐁퐂ퟐ

퐄퐅ퟐ

But ABDE

= BCEF

= CA DF

[ ∵ Given


= AB2

DE2 = BC2

EF2 = CA2

DF2






rArr ABAD

= ACAB


rArr BCDC

= ACBC








12Trigonometry



휽




퐜퐨퐬휽 = ퟏ

퐬퐞퐜 휽




퐴퐵퐴퐶














Values 00 300 450 600 900


ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0





13 tan 휃 = 5

12

Cosec휃 = 135

sec휃 = 1312

cot휃 = 125



rArr 3+2 = 5







ퟑ






radic3

tan300= 1radic3

rArr 휃 = 300





퐴퐵




푥2minus푥1

A(x1y1) and B(x2y2)


푥2minus푥1


tan 휃 = 7






푥2minus푥1

m1 = 5minus20minus5

= 3minus5


= 3minus5




휃 훼


푥2minus푥1


= minus7minus4

= 74


= 4minus7

m1 x m2 = 74 x 4

minus7 = -1








2 c = -3


2x -3rArr2y = x -6

rArr x -2y -6 =0









푚+푛]



2 푦2+푦1

2]

= [4+22

7+32

]

= [62

102

] = (35)








Note

















apart





Theorem










Formulas



ퟐ풉


ퟑ


ퟑ

흅 = ퟐퟐퟕ






d =14cm

r= 7cm

흅 = ퟐퟐퟕ

h=10cm

l= 풓ퟐ + 풉ퟐ



l=radicퟏퟒퟗ

l=122



7 x 7 x 10

=440 sqcm

2휋푟(푟 + ℎ) =2 x 22

7 x 7(7+10)

=44 x 17 =748 sqcm

휋푟2ℎ =22

7 x 72 x 10

=1540cm3

Cone 휋푟푙 =22

7 x 7 x 122

=2684 sqcm

휋푟(푟 + 푙) =22

7 x 7 x ( 7 + 122 )

=22 x 192= 4224

13휋푟2ℎ

=13 x 22

7 x 72 x 10

=13 x 22

7 x 72 x 10

=5133 cm3

Sphere

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

43휋푟3

= 43

x 227

x 73 =14373 cm3


7 x 72

=462 sqcm

2휋푟2 =2 x 22

7 x 72

=308 sqcm

23휋푟3

= 23

x 22x 7

x 73 = 7186 cm3






TO D

80To E

150

100

80

30

70to C

40To B

From A


30m = 30 x = 15cm

70m = 70x = 35cm

80m = 80 x = 4cm

100m = 100x = 5cm

150m = 150x =75cm





To C E To D

120to D

E to E

220

210

120

80

40 to B

120toD

75to C

50to B

350

300

250

150

50

F 150toF

100to G

100toE

50toF

25toG

225

175

125

100

75

50

25toC

75toB











N + R = A + 2

N = 3 R = 4 A = 5 N+R = 3 +4 = 7 A+2 = 5 +2 = 7 there4 N+R = A+2

Exercise



N = 8 R = 6 A = 12 N+R = 8 +6 = 14 A+2 = 12 +2 = 14 there4 N+R = A+2

N = 3 R = 5 A = 6 N+R = 3 +5 = 8 A+2 = 6 +2 = 8 there4 N+R = A+2











1

4 4 6 4 +4 = 6 +2

2

3



4

5




1Real Numbers



AUB=BUA AcapB=BcapA







De Morganrsquos Law















ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ







rArr ABAD

= ACAB


rArr BCDC

= ACBC








12Trigonometry



휽




퐜퐨퐬휽 = ퟏ

퐬퐞퐜 휽




퐴퐵퐴퐶














Values 00 300 450 600 900


ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0





13 tan 휃 = 5

12

Cosec휃 = 135

sec휃 = 1312

cot휃 = 125



rArr 3+2 = 5







ퟑ






radic3

tan300= 1radic3

rArr 휃 = 300





퐴퐵




푥2minus푥1

A(x1y1) and B(x2y2)


푥2minus푥1


tan 휃 = 7






푥2minus푥1

m1 = 5minus20minus5

= 3minus5


= 3minus5




휃 훼


푥2minus푥1


= minus7minus4

= 74


= 4minus7

m1 x m2 = 74 x 4

minus7 = -1








2 c = -3


2x -3rArr2y = x -6

rArr x -2y -6 =0









푚+푛]



2 푦2+푦1

2]

= [4+22

7+32

]

= [62

102

] = (35)








Note

















apart





Theorem










Formulas



ퟐ풉


ퟑ


ퟑ

흅 = ퟐퟐퟕ






d =14cm

r= 7cm

흅 = ퟐퟐퟕ

h=10cm

l= 풓ퟐ + 풉ퟐ



l=radicퟏퟒퟗ

l=122



7 x 7 x 10

=440 sqcm

2휋푟(푟 + ℎ) =2 x 22

7 x 7(7+10)

=44 x 17 =748 sqcm

휋푟2ℎ =22

7 x 72 x 10

=1540cm3

Cone 휋푟푙 =22

7 x 7 x 122

=2684 sqcm

휋푟(푟 + 푙) =22

7 x 7 x ( 7 + 122 )

=22 x 192= 4224

13휋푟2ℎ

=13 x 22

7 x 72 x 10

=13 x 22

7 x 72 x 10

=5133 cm3

Sphere

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

43휋푟3

= 43

x 227

x 73 =14373 cm3


7 x 72

=462 sqcm

2휋푟2 =2 x 22

7 x 72

=308 sqcm

23휋푟3

= 23

x 22x 7

x 73 = 7186 cm3






TO D

80To E

150

100

80

30

70to C

40To B

From A


30m = 30 x = 15cm

70m = 70x = 35cm

80m = 80 x = 4cm

100m = 100x = 5cm

150m = 150x =75cm





To C E To D

120to D

E to E

220

210

120

80

40 to B

120toD

75to C

50to B

350

300

250

150

50

F 150toF

100to G

100toE

50toF

25toG

225

175

125

100

75

50

25toC

75toB











N + R = A + 2

N = 3 R = 4 A = 5 N+R = 3 +4 = 7 A+2 = 5 +2 = 7 there4 N+R = A+2

Exercise



N = 8 R = 6 A = 12 N+R = 8 +6 = 14 A+2 = 12 +2 = 14 there4 N+R = A+2

N = 3 R = 5 A = 6 N+R = 3 +5 = 8 A+2 = 6 +2 = 8 there4 N+R = A+2











1

4 4 6 4 +4 = 6 +2

2

3



4

5




1Real Numbers



AUB=BUA AcapB=BcapA







De Morganrsquos Law















ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ








12Trigonometry



휽




퐜퐨퐬휽 = ퟏ

퐬퐞퐜 휽




퐴퐵퐴퐶














Values 00 300 450 600 900


ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0





13 tan 휃 = 5

12

Cosec휃 = 135

sec휃 = 1312

cot휃 = 125



rArr 3+2 = 5







ퟑ






radic3

tan300= 1radic3

rArr 휃 = 300





퐴퐵




푥2minus푥1

A(x1y1) and B(x2y2)


푥2minus푥1


tan 휃 = 7






푥2minus푥1

m1 = 5minus20minus5

= 3minus5


= 3minus5




휃 훼


푥2minus푥1


= minus7minus4

= 74


= 4minus7

m1 x m2 = 74 x 4

minus7 = -1








2 c = -3


2x -3rArr2y = x -6

rArr x -2y -6 =0









푚+푛]



2 푦2+푦1

2]

= [4+22

7+32

]

= [62

102

] = (35)








Note

















apart





Theorem










Formulas



ퟐ풉


ퟑ


ퟑ

흅 = ퟐퟐퟕ






d =14cm

r= 7cm

흅 = ퟐퟐퟕ

h=10cm

l= 풓ퟐ + 풉ퟐ



l=radicퟏퟒퟗ

l=122



7 x 7 x 10

=440 sqcm

2휋푟(푟 + ℎ) =2 x 22

7 x 7(7+10)

=44 x 17 =748 sqcm

휋푟2ℎ =22

7 x 72 x 10

=1540cm3

Cone 휋푟푙 =22

7 x 7 x 122

=2684 sqcm

휋푟(푟 + 푙) =22

7 x 7 x ( 7 + 122 )

=22 x 192= 4224

13휋푟2ℎ

=13 x 22

7 x 72 x 10

=13 x 22

7 x 72 x 10

=5133 cm3

Sphere

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

43휋푟3

= 43

x 227

x 73 =14373 cm3


7 x 72

=462 sqcm

2휋푟2 =2 x 22

7 x 72

=308 sqcm

23휋푟3

= 23

x 22x 7

x 73 = 7186 cm3






TO D

80To E

150

100

80

30

70to C

40To B

From A


30m = 30 x = 15cm

70m = 70x = 35cm

80m = 80 x = 4cm

100m = 100x = 5cm

150m = 150x =75cm





To C E To D

120to D

E to E

220

210

120

80

40 to B

120toD

75to C

50to B

350

300

250

150

50

F 150toF

100to G

100toE

50toF

25toG

225

175

125

100

75

50

25toC

75toB











N + R = A + 2

N = 3 R = 4 A = 5 N+R = 3 +4 = 7 A+2 = 5 +2 = 7 there4 N+R = A+2

Exercise



N = 8 R = 6 A = 12 N+R = 8 +6 = 14 A+2 = 12 +2 = 14 there4 N+R = A+2

N = 3 R = 5 A = 6 N+R = 3 +5 = 8 A+2 = 6 +2 = 8 there4 N+R = A+2











1

4 4 6 4 +4 = 6 +2

2

3



4

5




1Real Numbers



AUB=BUA AcapB=BcapA







De Morganrsquos Law















ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ




Values 00 300 450 600 900


ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0





13 tan 휃 = 5

12

Cosec휃 = 135

sec휃 = 1312

cot휃 = 125



rArr 3+2 = 5







ퟑ






radic3

tan300= 1radic3

rArr 휃 = 300





퐴퐵




푥2minus푥1

A(x1y1) and B(x2y2)


푥2minus푥1


tan 휃 = 7






푥2minus푥1

m1 = 5minus20minus5

= 3minus5


= 3minus5




휃 훼


푥2minus푥1


= minus7minus4

= 74


= 4minus7

m1 x m2 = 74 x 4

minus7 = -1








2 c = -3


2x -3rArr2y = x -6

rArr x -2y -6 =0









푚+푛]



2 푦2+푦1

2]

= [4+22

7+32

]

= [62

102

] = (35)








Note

















apart





Theorem










Formulas



ퟐ풉


ퟑ


ퟑ

흅 = ퟐퟐퟕ






d =14cm

r= 7cm

흅 = ퟐퟐퟕ

h=10cm

l= 풓ퟐ + 풉ퟐ



l=radicퟏퟒퟗ

l=122



7 x 7 x 10

=440 sqcm

2휋푟(푟 + ℎ) =2 x 22

7 x 7(7+10)

=44 x 17 =748 sqcm

휋푟2ℎ =22

7 x 72 x 10

=1540cm3

Cone 휋푟푙 =22

7 x 7 x 122

=2684 sqcm

휋푟(푟 + 푙) =22

7 x 7 x ( 7 + 122 )

=22 x 192= 4224

13휋푟2ℎ

=13 x 22

7 x 72 x 10

=13 x 22

7 x 72 x 10

=5133 cm3

Sphere

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

43휋푟3

= 43

x 227

x 73 =14373 cm3


7 x 72

=462 sqcm

2휋푟2 =2 x 22

7 x 72

=308 sqcm

23휋푟3

= 23

x 22x 7

x 73 = 7186 cm3






TO D

80To E

150

100

80

30

70to C

40To B

From A


30m = 30 x = 15cm

70m = 70x = 35cm

80m = 80 x = 4cm

100m = 100x = 5cm

150m = 150x =75cm





To C E To D

120to D

E to E

220

210

120

80

40 to B

120toD

75to C

50to B

350

300

250

150

50

F 150toF

100to G

100toE

50toF

25toG

225

175

125

100

75

50

25toC

75toB











N + R = A + 2

N = 3 R = 4 A = 5 N+R = 3 +4 = 7 A+2 = 5 +2 = 7 there4 N+R = A+2

Exercise



N = 8 R = 6 A = 12 N+R = 8 +6 = 14 A+2 = 12 +2 = 14 there4 N+R = A+2

N = 3 R = 5 A = 6 N+R = 3 +5 = 8 A+2 = 6 +2 = 8 there4 N+R = A+2











1

4 4 6 4 +4 = 6 +2

2

3



4

5




1Real Numbers



AUB=BUA AcapB=BcapA







De Morganrsquos Law















ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ








ퟑ






radic3

tan300= 1radic3

rArr 휃 = 300





퐴퐵




푥2minus푥1

A(x1y1) and B(x2y2)


푥2minus푥1


tan 휃 = 7






푥2minus푥1

m1 = 5minus20minus5

= 3minus5


= 3minus5




휃 훼


푥2minus푥1


= minus7minus4

= 74


= 4minus7

m1 x m2 = 74 x 4

minus7 = -1








2 c = -3


2x -3rArr2y = x -6

rArr x -2y -6 =0









푚+푛]



2 푦2+푦1

2]

= [4+22

7+32

]

= [62

102

] = (35)








Note

















apart





Theorem










Formulas



ퟐ풉


ퟑ


ퟑ

흅 = ퟐퟐퟕ






d =14cm

r= 7cm

흅 = ퟐퟐퟕ

h=10cm

l= 풓ퟐ + 풉ퟐ



l=radicퟏퟒퟗ

l=122



7 x 7 x 10

=440 sqcm

2휋푟(푟 + ℎ) =2 x 22

7 x 7(7+10)

=44 x 17 =748 sqcm

휋푟2ℎ =22

7 x 72 x 10

=1540cm3

Cone 휋푟푙 =22

7 x 7 x 122

=2684 sqcm

휋푟(푟 + 푙) =22

7 x 7 x ( 7 + 122 )

=22 x 192= 4224

13휋푟2ℎ

=13 x 22

7 x 72 x 10

=13 x 22

7 x 72 x 10

=5133 cm3

Sphere

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

43휋푟3

= 43

x 227

x 73 =14373 cm3


7 x 72

=462 sqcm

2휋푟2 =2 x 22

7 x 72

=308 sqcm

23휋푟3

= 23

x 22x 7

x 73 = 7186 cm3






TO D

80To E

150

100

80

30

70to C

40To B

From A


30m = 30 x = 15cm

70m = 70x = 35cm

80m = 80 x = 4cm

100m = 100x = 5cm

150m = 150x =75cm





To C E To D

120to D

E to E

220

210

120

80

40 to B

120toD

75to C

50to B

350

300

250

150

50

F 150toF

100to G

100toE

50toF

25toG

225

175

125

100

75

50

25toC

75toB











N + R = A + 2

N = 3 R = 4 A = 5 N+R = 3 +4 = 7 A+2 = 5 +2 = 7 there4 N+R = A+2

Exercise



N = 8 R = 6 A = 12 N+R = 8 +6 = 14 A+2 = 12 +2 = 14 there4 N+R = A+2

N = 3 R = 5 A = 6 N+R = 3 +5 = 8 A+2 = 6 +2 = 8 there4 N+R = A+2











1

4 4 6 4 +4 = 6 +2

2

3



4

5




1Real Numbers



AUB=BUA AcapB=BcapA







De Morganrsquos Law















ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ






퐴퐵




푥2minus푥1

A(x1y1) and B(x2y2)


푥2minus푥1


tan 휃 = 7






푥2minus푥1

m1 = 5minus20minus5

= 3minus5


= 3minus5




휃 훼


푥2minus푥1


= minus7minus4

= 74


= 4minus7

m1 x m2 = 74 x 4

minus7 = -1








2 c = -3


2x -3rArr2y = x -6

rArr x -2y -6 =0









푚+푛]



2 푦2+푦1

2]

= [4+22

7+32

]

= [62

102

] = (35)








Note

















apart





Theorem










Formulas



ퟐ풉


ퟑ


ퟑ

흅 = ퟐퟐퟕ






d =14cm

r= 7cm

흅 = ퟐퟐퟕ

h=10cm

l= 풓ퟐ + 풉ퟐ



l=radicퟏퟒퟗ

l=122



7 x 7 x 10

=440 sqcm

2휋푟(푟 + ℎ) =2 x 22

7 x 7(7+10)

=44 x 17 =748 sqcm

휋푟2ℎ =22

7 x 72 x 10

=1540cm3

Cone 휋푟푙 =22

7 x 7 x 122

=2684 sqcm

휋푟(푟 + 푙) =22

7 x 7 x ( 7 + 122 )

=22 x 192= 4224

13휋푟2ℎ

=13 x 22

7 x 72 x 10

=13 x 22

7 x 72 x 10

=5133 cm3

Sphere

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

43휋푟3

= 43

x 227

x 73 =14373 cm3


7 x 72

=462 sqcm

2휋푟2 =2 x 22

7 x 72

=308 sqcm

23휋푟3

= 23

x 22x 7

x 73 = 7186 cm3






TO D

80To E

150

100

80

30

70to C

40To B

From A


30m = 30 x = 15cm

70m = 70x = 35cm

80m = 80 x = 4cm

100m = 100x = 5cm

150m = 150x =75cm





To C E To D

120to D

E to E

220

210

120

80

40 to B

120toD

75to C

50to B

350

300

250

150

50

F 150toF

100to G

100toE

50toF

25toG

225

175

125

100

75

50

25toC

75toB











N + R = A + 2

N = 3 R = 4 A = 5 N+R = 3 +4 = 7 A+2 = 5 +2 = 7 there4 N+R = A+2

Exercise



N = 8 R = 6 A = 12 N+R = 8 +6 = 14 A+2 = 12 +2 = 14 there4 N+R = A+2

N = 3 R = 5 A = 6 N+R = 3 +5 = 8 A+2 = 6 +2 = 8 there4 N+R = A+2











1

4 4 6 4 +4 = 6 +2

2

3



4

5




1Real Numbers



AUB=BUA AcapB=BcapA







De Morganrsquos Law















ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ







푥2minus푥1

m1 = 5minus20minus5

= 3minus5


= 3minus5




휃 훼


푥2minus푥1


= minus7minus4

= 74


= 4minus7

m1 x m2 = 74 x 4

minus7 = -1








2 c = -3


2x -3rArr2y = x -6

rArr x -2y -6 =0









푚+푛]



2 푦2+푦1

2]

= [4+22

7+32

]

= [62

102

] = (35)








Note

















apart





Theorem










Formulas



ퟐ풉


ퟑ


ퟑ

흅 = ퟐퟐퟕ






d =14cm

r= 7cm

흅 = ퟐퟐퟕ

h=10cm

l= 풓ퟐ + 풉ퟐ



l=radicퟏퟒퟗ

l=122



7 x 7 x 10

=440 sqcm

2휋푟(푟 + ℎ) =2 x 22

7 x 7(7+10)

=44 x 17 =748 sqcm

휋푟2ℎ =22

7 x 72 x 10

=1540cm3

Cone 휋푟푙 =22

7 x 7 x 122

=2684 sqcm

휋푟(푟 + 푙) =22

7 x 7 x ( 7 + 122 )

=22 x 192= 4224

13휋푟2ℎ

=13 x 22

7 x 72 x 10

=13 x 22

7 x 72 x 10

=5133 cm3

Sphere

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

43휋푟3

= 43

x 227

x 73 =14373 cm3


7 x 72

=462 sqcm

2휋푟2 =2 x 22

7 x 72

=308 sqcm

23휋푟3

= 23

x 22x 7

x 73 = 7186 cm3






TO D

80To E

150

100

80

30

70to C

40To B

From A


30m = 30 x = 15cm

70m = 70x = 35cm

80m = 80 x = 4cm

100m = 100x = 5cm

150m = 150x =75cm





To C E To D

120to D

E to E

220

210

120

80

40 to B

120toD

75to C

50to B

350

300

250

150

50

F 150toF

100to G

100toE

50toF

25toG

225

175

125

100

75

50

25toC

75toB











N + R = A + 2

N = 3 R = 4 A = 5 N+R = 3 +4 = 7 A+2 = 5 +2 = 7 there4 N+R = A+2

Exercise



N = 8 R = 6 A = 12 N+R = 8 +6 = 14 A+2 = 12 +2 = 14 there4 N+R = A+2

N = 3 R = 5 A = 6 N+R = 3 +5 = 8 A+2 = 6 +2 = 8 there4 N+R = A+2











1

4 4 6 4 +4 = 6 +2

2

3



4

5




1Real Numbers



AUB=BUA AcapB=BcapA







De Morganrsquos Law















ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ








2 c = -3


2x -3rArr2y = x -6

rArr x -2y -6 =0









푚+푛]



2 푦2+푦1

2]

= [4+22

7+32

]

= [62

102

] = (35)








Note

















apart





Theorem










Formulas



ퟐ풉


ퟑ


ퟑ

흅 = ퟐퟐퟕ






d =14cm

r= 7cm

흅 = ퟐퟐퟕ

h=10cm

l= 풓ퟐ + 풉ퟐ



l=radicퟏퟒퟗ

l=122



7 x 7 x 10

=440 sqcm

2휋푟(푟 + ℎ) =2 x 22

7 x 7(7+10)

=44 x 17 =748 sqcm

휋푟2ℎ =22

7 x 72 x 10

=1540cm3

Cone 휋푟푙 =22

7 x 7 x 122

=2684 sqcm

휋푟(푟 + 푙) =22

7 x 7 x ( 7 + 122 )

=22 x 192= 4224

13휋푟2ℎ

=13 x 22

7 x 72 x 10

=13 x 22

7 x 72 x 10

=5133 cm3

Sphere

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

43휋푟3

= 43

x 227

x 73 =14373 cm3


7 x 72

=462 sqcm

2휋푟2 =2 x 22

7 x 72

=308 sqcm

23휋푟3

= 23

x 22x 7

x 73 = 7186 cm3






TO D

80To E

150

100

80

30

70to C

40To B

From A


30m = 30 x = 15cm

70m = 70x = 35cm

80m = 80 x = 4cm

100m = 100x = 5cm

150m = 150x =75cm





To C E To D

120to D

E to E

220

210

120

80

40 to B

120toD

75to C

50to B

350

300

250

150

50

F 150toF

100to G

100toE

50toF

25toG

225

175

125

100

75

50

25toC

75toB











N + R = A + 2

N = 3 R = 4 A = 5 N+R = 3 +4 = 7 A+2 = 5 +2 = 7 there4 N+R = A+2

Exercise



N = 8 R = 6 A = 12 N+R = 8 +6 = 14 A+2 = 12 +2 = 14 there4 N+R = A+2

N = 3 R = 5 A = 6 N+R = 3 +5 = 8 A+2 = 6 +2 = 8 there4 N+R = A+2











1

4 4 6 4 +4 = 6 +2

2

3



4

5




1Real Numbers



AUB=BUA AcapB=BcapA







De Morganrsquos Law















ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ






푚+푛]



2 푦2+푦1

2]

= [4+22

7+32

]

= [62

102

] = (35)








Note

















apart





Theorem










Formulas



ퟐ풉


ퟑ


ퟑ

흅 = ퟐퟐퟕ






d =14cm

r= 7cm

흅 = ퟐퟐퟕ

h=10cm

l= 풓ퟐ + 풉ퟐ



l=radicퟏퟒퟗ

l=122



7 x 7 x 10

=440 sqcm

2휋푟(푟 + ℎ) =2 x 22

7 x 7(7+10)

=44 x 17 =748 sqcm

휋푟2ℎ =22

7 x 72 x 10

=1540cm3

Cone 휋푟푙 =22

7 x 7 x 122

=2684 sqcm

휋푟(푟 + 푙) =22

7 x 7 x ( 7 + 122 )

=22 x 192= 4224

13휋푟2ℎ

=13 x 22

7 x 72 x 10

=13 x 22

7 x 72 x 10

=5133 cm3

Sphere

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

43휋푟3

= 43

x 227

x 73 =14373 cm3


7 x 72

=462 sqcm

2휋푟2 =2 x 22

7 x 72

=308 sqcm

23휋푟3

= 23

x 22x 7

x 73 = 7186 cm3






TO D

80To E

150

100

80

30

70to C

40To B

From A


30m = 30 x = 15cm

70m = 70x = 35cm

80m = 80 x = 4cm

100m = 100x = 5cm

150m = 150x =75cm





To C E To D

120to D

E to E

220

210

120

80

40 to B

120toD

75to C

50to B

350

300

250

150

50

F 150toF

100to G

100toE

50toF

25toG

225

175

125

100

75

50

25toC

75toB











N + R = A + 2

N = 3 R = 4 A = 5 N+R = 3 +4 = 7 A+2 = 5 +2 = 7 there4 N+R = A+2

Exercise



N = 8 R = 6 A = 12 N+R = 8 +6 = 14 A+2 = 12 +2 = 14 there4 N+R = A+2

N = 3 R = 5 A = 6 N+R = 3 +5 = 8 A+2 = 6 +2 = 8 there4 N+R = A+2











1

4 4 6 4 +4 = 6 +2

2

3



4

5




1Real Numbers



AUB=BUA AcapB=BcapA







De Morganrsquos Law















ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ








Note

















apart





Theorem










Formulas



ퟐ풉


ퟑ


ퟑ

흅 = ퟐퟐퟕ






d =14cm

r= 7cm

흅 = ퟐퟐퟕ

h=10cm

l= 풓ퟐ + 풉ퟐ



l=radicퟏퟒퟗ

l=122



7 x 7 x 10

=440 sqcm

2휋푟(푟 + ℎ) =2 x 22

7 x 7(7+10)

=44 x 17 =748 sqcm

휋푟2ℎ =22

7 x 72 x 10

=1540cm3

Cone 휋푟푙 =22

7 x 7 x 122

=2684 sqcm

휋푟(푟 + 푙) =22

7 x 7 x ( 7 + 122 )

=22 x 192= 4224

13휋푟2ℎ

=13 x 22

7 x 72 x 10

=13 x 22

7 x 72 x 10

=5133 cm3

Sphere

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

43휋푟3

= 43

x 227

x 73 =14373 cm3


7 x 72

=462 sqcm

2휋푟2 =2 x 22

7 x 72

=308 sqcm

23휋푟3

= 23

x 22x 7

x 73 = 7186 cm3






TO D

80To E

150

100

80

30

70to C

40To B

From A


30m = 30 x = 15cm

70m = 70x = 35cm

80m = 80 x = 4cm

100m = 100x = 5cm

150m = 150x =75cm





To C E To D

120to D

E to E

220

210

120

80

40 to B

120toD

75to C

50to B

350

300

250

150

50

F 150toF

100to G

100toE

50toF

25toG

225

175

125

100

75

50

25toC

75toB











N + R = A + 2

N = 3 R = 4 A = 5 N+R = 3 +4 = 7 A+2 = 5 +2 = 7 there4 N+R = A+2

Exercise



N = 8 R = 6 A = 12 N+R = 8 +6 = 14 A+2 = 12 +2 = 14 there4 N+R = A+2

N = 3 R = 5 A = 6 N+R = 3 +5 = 8 A+2 = 6 +2 = 8 there4 N+R = A+2











1

4 4 6 4 +4 = 6 +2

2

3



4

5




1Real Numbers



AUB=BUA AcapB=BcapA







De Morganrsquos Law















ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ

















apart





Theorem










Formulas



ퟐ풉


ퟑ


ퟑ

흅 = ퟐퟐퟕ






d =14cm

r= 7cm

흅 = ퟐퟐퟕ

h=10cm

l= 풓ퟐ + 풉ퟐ



l=radicퟏퟒퟗ

l=122



7 x 7 x 10

=440 sqcm

2휋푟(푟 + ℎ) =2 x 22

7 x 7(7+10)

=44 x 17 =748 sqcm

휋푟2ℎ =22

7 x 72 x 10

=1540cm3

Cone 휋푟푙 =22

7 x 7 x 122

=2684 sqcm

휋푟(푟 + 푙) =22

7 x 7 x ( 7 + 122 )

=22 x 192= 4224

13휋푟2ℎ

=13 x 22

7 x 72 x 10

=13 x 22

7 x 72 x 10

=5133 cm3

Sphere

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

43휋푟3

= 43

x 227

x 73 =14373 cm3


7 x 72

=462 sqcm

2휋푟2 =2 x 22

7 x 72

=308 sqcm

23휋푟3

= 23

x 22x 7

x 73 = 7186 cm3






TO D

80To E

150

100

80

30

70to C

40To B

From A


30m = 30 x = 15cm

70m = 70x = 35cm

80m = 80 x = 4cm

100m = 100x = 5cm

150m = 150x =75cm





To C E To D

120to D

E to E

220

210

120

80

40 to B

120toD

75to C

50to B

350

300

250

150

50

F 150toF

100to G

100toE

50toF

25toG

225

175

125

100

75

50

25toC

75toB











N + R = A + 2

N = 3 R = 4 A = 5 N+R = 3 +4 = 7 A+2 = 5 +2 = 7 there4 N+R = A+2

Exercise



N = 8 R = 6 A = 12 N+R = 8 +6 = 14 A+2 = 12 +2 = 14 there4 N+R = A+2

N = 3 R = 5 A = 6 N+R = 3 +5 = 8 A+2 = 6 +2 = 8 there4 N+R = A+2











1

4 4 6 4 +4 = 6 +2

2

3



4

5




1Real Numbers



AUB=BUA AcapB=BcapA







De Morganrsquos Law















ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ






Theorem










Formulas



ퟐ풉


ퟑ


ퟑ

흅 = ퟐퟐퟕ






d =14cm

r= 7cm

흅 = ퟐퟐퟕ

h=10cm

l= 풓ퟐ + 풉ퟐ



l=radicퟏퟒퟗ

l=122



7 x 7 x 10

=440 sqcm

2휋푟(푟 + ℎ) =2 x 22

7 x 7(7+10)

=44 x 17 =748 sqcm

휋푟2ℎ =22

7 x 72 x 10

=1540cm3

Cone 휋푟푙 =22

7 x 7 x 122

=2684 sqcm

휋푟(푟 + 푙) =22

7 x 7 x ( 7 + 122 )

=22 x 192= 4224

13휋푟2ℎ

=13 x 22

7 x 72 x 10

=13 x 22

7 x 72 x 10

=5133 cm3

Sphere

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

43휋푟3

= 43

x 227

x 73 =14373 cm3


7 x 72

=462 sqcm

2휋푟2 =2 x 22

7 x 72

=308 sqcm

23휋푟3

= 23

x 22x 7

x 73 = 7186 cm3






TO D

80To E

150

100

80

30

70to C

40To B

From A


30m = 30 x = 15cm

70m = 70x = 35cm

80m = 80 x = 4cm

100m = 100x = 5cm

150m = 150x =75cm





To C E To D

120to D

E to E

220

210

120

80

40 to B

120toD

75to C

50to B

350

300

250

150

50

F 150toF

100to G

100toE

50toF

25toG

225

175

125

100

75

50

25toC

75toB











N + R = A + 2

N = 3 R = 4 A = 5 N+R = 3 +4 = 7 A+2 = 5 +2 = 7 there4 N+R = A+2

Exercise



N = 8 R = 6 A = 12 N+R = 8 +6 = 14 A+2 = 12 +2 = 14 there4 N+R = A+2

N = 3 R = 5 A = 6 N+R = 3 +5 = 8 A+2 = 6 +2 = 8 there4 N+R = A+2











1

4 4 6 4 +4 = 6 +2

2

3



4

5




1Real Numbers



AUB=BUA AcapB=BcapA







De Morganrsquos Law















ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ










Formulas



ퟐ풉


ퟑ


ퟑ

흅 = ퟐퟐퟕ






d =14cm

r= 7cm

흅 = ퟐퟐퟕ

h=10cm

l= 풓ퟐ + 풉ퟐ



l=radicퟏퟒퟗ

l=122



7 x 7 x 10

=440 sqcm

2휋푟(푟 + ℎ) =2 x 22

7 x 7(7+10)

=44 x 17 =748 sqcm

휋푟2ℎ =22

7 x 72 x 10

=1540cm3

Cone 휋푟푙 =22

7 x 7 x 122

=2684 sqcm

휋푟(푟 + 푙) =22

7 x 7 x ( 7 + 122 )

=22 x 192= 4224

13휋푟2ℎ

=13 x 22

7 x 72 x 10

=13 x 22

7 x 72 x 10

=5133 cm3

Sphere

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

43휋푟3

= 43

x 227

x 73 =14373 cm3


7 x 72

=462 sqcm

2휋푟2 =2 x 22

7 x 72

=308 sqcm

23휋푟3

= 23

x 22x 7

x 73 = 7186 cm3






TO D

80To E

150

100

80

30

70to C

40To B

From A


30m = 30 x = 15cm

70m = 70x = 35cm

80m = 80 x = 4cm

100m = 100x = 5cm

150m = 150x =75cm





To C E To D

120to D

E to E

220

210

120

80

40 to B

120toD

75to C

50to B

350

300

250

150

50

F 150toF

100to G

100toE

50toF

25toG

225

175

125

100

75

50

25toC

75toB











N + R = A + 2

N = 3 R = 4 A = 5 N+R = 3 +4 = 7 A+2 = 5 +2 = 7 there4 N+R = A+2

Exercise



N = 8 R = 6 A = 12 N+R = 8 +6 = 14 A+2 = 12 +2 = 14 there4 N+R = A+2

N = 3 R = 5 A = 6 N+R = 3 +5 = 8 A+2 = 6 +2 = 8 there4 N+R = A+2











1

4 4 6 4 +4 = 6 +2

2

3



4

5




1Real Numbers



AUB=BUA AcapB=BcapA







De Morganrsquos Law















ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ





d =14cm

r= 7cm

흅 = ퟐퟐퟕ

h=10cm

l= 풓ퟐ + 풉ퟐ



l=radicퟏퟒퟗ

l=122



7 x 7 x 10

=440 sqcm

2휋푟(푟 + ℎ) =2 x 22

7 x 7(7+10)

=44 x 17 =748 sqcm

휋푟2ℎ =22

7 x 72 x 10

=1540cm3

Cone 휋푟푙 =22

7 x 7 x 122

=2684 sqcm

휋푟(푟 + 푙) =22

7 x 7 x ( 7 + 122 )

=22 x 192= 4224

13휋푟2ℎ

=13 x 22

7 x 72 x 10

=13 x 22

7 x 72 x 10

=5133 cm3

Sphere

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

4휋푟2 = 4 x 22

7 x 72

=616 ಚ ಂ ೕ

43휋푟3

= 43

x 227

x 73 =14373 cm3


7 x 72

=462 sqcm

2휋푟2 =2 x 22

7 x 72

=308 sqcm

23휋푟3

= 23

x 22x 7

x 73 = 7186 cm3






TO D

80To E

150

100

80

30

70to C

40To B

From A


30m = 30 x = 15cm

70m = 70x = 35cm

80m = 80 x = 4cm

100m = 100x = 5cm

150m = 150x =75cm





To C E To D

120to D

E to E

220

210

120

80

40 to B

120toD

75to C

50to B

350

300

250

150

50

F 150toF

100to G

100toE

50toF

25toG

225

175

125

100

75

50

25toC

75toB











N + R = A + 2

N = 3 R = 4 A = 5 N+R = 3 +4 = 7 A+2 = 5 +2 = 7 there4 N+R = A+2

Exercise



N = 8 R = 6 A = 12 N+R = 8 +6 = 14 A+2 = 12 +2 = 14 there4 N+R = A+2

N = 3 R = 5 A = 6 N+R = 3 +5 = 8 A+2 = 6 +2 = 8 there4 N+R = A+2











1

4 4 6 4 +4 = 6 +2

2

3



4

5




1Real Numbers



AUB=BUA AcapB=BcapA







De Morganrsquos Law















ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ





TO D

80To E

150

100

80

30

70to C

40To B

From A


30m = 30 x = 15cm

70m = 70x = 35cm

80m = 80 x = 4cm

100m = 100x = 5cm

150m = 150x =75cm





To C E To D

120to D

E to E

220

210

120

80

40 to B

120toD

75to C

50to B

350

300

250

150

50

F 150toF

100to G

100toE

50toF

25toG

225

175

125

100

75

50

25toC

75toB











N + R = A + 2

N = 3 R = 4 A = 5 N+R = 3 +4 = 7 A+2 = 5 +2 = 7 there4 N+R = A+2

Exercise



N = 8 R = 6 A = 12 N+R = 8 +6 = 14 A+2 = 12 +2 = 14 there4 N+R = A+2

N = 3 R = 5 A = 6 N+R = 3 +5 = 8 A+2 = 6 +2 = 8 there4 N+R = A+2











1

4 4 6 4 +4 = 6 +2

2

3



4

5




1Real Numbers



AUB=BUA AcapB=BcapA







De Morganrsquos Law















ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ






To C E To D

120to D

E to E

220

210

120

80

40 to B

120toD

75to C

50to B

350

300

250

150

50

F 150toF

100to G

100toE

50toF

25toG

225

175

125

100

75

50

25toC

75toB











N + R = A + 2

N = 3 R = 4 A = 5 N+R = 3 +4 = 7 A+2 = 5 +2 = 7 there4 N+R = A+2

Exercise



N = 8 R = 6 A = 12 N+R = 8 +6 = 14 A+2 = 12 +2 = 14 there4 N+R = A+2

N = 3 R = 5 A = 6 N+R = 3 +5 = 8 A+2 = 6 +2 = 8 there4 N+R = A+2











1

4 4 6 4 +4 = 6 +2

2

3



4

5




1Real Numbers



AUB=BUA AcapB=BcapA







De Morganrsquos Law















ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ










N + R = A + 2

N = 3 R = 4 A = 5 N+R = 3 +4 = 7 A+2 = 5 +2 = 7 there4 N+R = A+2

Exercise



N = 8 R = 6 A = 12 N+R = 8 +6 = 14 A+2 = 12 +2 = 14 there4 N+R = A+2

N = 3 R = 5 A = 6 N+R = 3 +5 = 8 A+2 = 6 +2 = 8 there4 N+R = A+2











1

4 4 6 4 +4 = 6 +2

2

3



4

5




1Real Numbers



AUB=BUA AcapB=BcapA







De Morganrsquos Law















ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ




N = 8 R = 6 A = 12 N+R = 8 +6 = 14 A+2 = 12 +2 = 14 there4 N+R = A+2

N = 3 R = 5 A = 6 N+R = 3 +5 = 8 A+2 = 6 +2 = 8 there4 N+R = A+2











1

4 4 6 4 +4 = 6 +2

2

3



4

5




1Real Numbers



AUB=BUA AcapB=BcapA







De Morganrsquos Law















ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ







1

4 4 6 4 +4 = 6 +2

2

3



4

5




1Real Numbers



AUB=BUA AcapB=BcapA







De Morganrsquos Law















ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ




4

5




1Real Numbers



AUB=BUA AcapB=BcapA







De Morganrsquos Law















ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ




1Real Numbers



AUB=BUA AcapB=BcapA







De Morganrsquos Law















ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ











ퟐ[풂 + 푻풏]


ퟏ풂 + 풅

ퟏ풂 + ퟐ풅

ퟏ풂 + ퟑ풅
























5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ














5Statistics



Un Grouped data

흈 =sum퐗ퟐ

퐧 minus ( sum푿

풏) ퟐ 흈 =

sum퐝ퟐ

퐧 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 흈 =

sum풅ퟐ

풏 ndash ( sum풅

풏)ퟐ 퐱퐂

Grouped Data



흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ




흈 = sum풇푿ퟐ


풏)ퟐ 흈 =

sum 퐟퐝ퟐ

퐧 흈 =

sum 풇풅ퟐ


풏)ퟐ 흈 =

sum 풇풅ퟐ


풏)ퟐ 퐱퐂


d = x ndash A d =




훔퐗x100




ퟐ풂 ∆ = b2 - 4ac

∆ = 0 ∆ gt 0 ∆ lt 0



m + n = minus퐛퐚

mn = 퐜퐚 x2 - (m + n)x + mn = 0

Trigonometry


Hypotenuse Adjacent


HypotenuseOpposite

AdjacentOpposite



=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ




=

=

=

=

=

=

00 300 450 600 900

sin휃 0 ퟏퟐ

ퟏradicퟐ

radicퟑퟐ

1


ퟏradicퟐ

ퟏퟐ 0


1 radicퟑ ND


1


radicퟐ 2 ND


0



풙ퟐminus풙ퟏ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ







]



]

Circles












E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ





E ndash Edg


ퟑ


ퟐퟑ흅풓

ퟑ


SSLC EXAM- TARGET 40 Study notes for Revision 2014

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Transcript of SSLC EXAM- TARGET 40 Study notes for Revision 2014