DEPARTMENT OF PUBLIC INSTRUCTION, HASSAN
MODEL QUESTION PAPER – 01
SUB: MATHEMATICS Max. Marks : 80
====================================================
I Four alternatives are given to each question. Choose the
appropriate answer. Write it along with its alphabet 8* 1 =8
1. If two pair of linear equations are consistent then they
A. intersects each other B. are parallel
C. coincident D. None of these
2. The 11 th term of the A.P -3,2
1 , 2 is
A. 97 B. 77 C. 22 D. 28
3. If the corresponding sides of two similar triangles are
in the ratio 3 : 5, then ratio of their areas
A. 15: 15 B. 3: 8 C. 8: 5 D. 9: 25
4. PT and QT are the tangents drawn to the circle from
an external point T such that 070POQ , then PTQ
A. 1100 B. 550 C. 1400 D. 350
5. If the roots of the equation ( a – b ) x2 + ( b – c ) x + ( c – a ) = 0 are
equal then the value of ( b + c ) is
A. 2a B. 2bc C. 2c D. 2ab
6. In the adjoining graph, number of zeros of the
polynomial p ( x ) is
A. 0 B. 1 C. 2 D. 3
7. Which of the following is not a probability of an event ?
A. 3
2 B. – 1.5 C. 15% D. 0.7
8. The value of 2cos1 is
A. sin B. sin2 C. tan2 D.
2sin
1
II. Solve the following 8* 1 = 8
9. If and are the zeros of the polynomial p (x) = x2 + 7x + 10, then
find the value of .
10. If 2x + 3y = 0 and 4x – 3y = 0 then, find the value of x + y
11. State Thales’ theorem
12. Write the standard form a quadratic equation
13. Express 156 as product of prime factors
14. If P ( x ) = 2x 2 + 3 x + 2,then find the value of p ( 1)
15. If b2 – 4ac > o , then write the nature of roots.
16. Evaluate cot( 90- ) - tan
III. Solve the following 8* 2 = 16
17. Find the 7th term of the A.P 24 ,21,18,,,,,,,,
18. In the adj. fig. if TR
PT
SQ
PS and PRQPST ,then
show that PQR is a isosceles triangle.
19. Find the distance between the points ( - 5 , 7 ) and ( - 1 , 3 )
20. The sum and product of zeros of a polynomial p ( x ) are – 3 and
2 respectively. Find the polynomial.
21. Solve 3x2 – 5x + 2 = 0 by completing square method
OR
Solve 2x2 + x + 4 = 0 by formula method
22. When a dice is thrown once, find the probability of getting
a. A odd number b. A prime number
23. Draw a circle of radius 5 cm construct two tangents from an
external point such that the angle between them is 600
24. A sphere of radius 4.2 cm is melted and recast to form a
cylinder of radius 6 cm, find the height of the cylinder
OR
Find the capacity of frustum of the cone of radius 4 cm and 2
cm with height 14 c.m.
IV. Solve the following 9* 3 = 27
25. Find the mode for the following scores
C- I 0-20 20-40 40-60 60-80 80-100 100-120
Frequency 10 35 52 61 38 29
OR
If the median of the scores is 525, with sum of the frequencies 100
find the value of ‘x’ and ‘y’
C-I 0-100 100-200 200-300 300-400 400-500 500-600 600-700 700-800 800-900 900-1000
F 2 5 X 12 17 20 Y 9 7 4
26. Prove that “length of the tangents drawn to the circle from an
external points are equal”.
27. Prove that 2)cos(cos
cos1
cos1
ec
OR
Prove that
2
2
2
2
cot1
cot
tan1
tan
= 1
28. Draw a triangle of sides 5 cm , 6 cm and 7 cm respectively.
Construct another similar triangles whose sides are 5
7 times of
the corresponding sides
OR
Construct a triangle ABC, BC = 7 cm, 00 105,45 AB , construct
another similar triangle whose sides are 3
4 times the
corresponding sides of the original triangle.
29. Show that ( 4, 5 ) ( 7, 6 ) ( 4, 3 ) and ( 1 ,2 ) forms the vertices
of a quadrilateral
30. Prove that 5 is a irrational number
31. ABC a quadrant of a circle of radius 14 cm and semi circle
drawn on BC as shown in the fig. find the area of the shaded portion.
32. Draw the less than type Ogive for the following data
C – I 100-120 120-140 140-160 160-180 180-200
F 12 14 8 6 14
33. Solve 32
11
xx
OR
If sum to n terms of an A.P is 2
133 2 nn find the 25 th term
V. Solve the following 4* 4 = 16
34. In an A.P if the 3rd term is 32 and 5th term is 40, find the sum
of first 30 terms
35. The angle of elevation of top of a cloud as observed from a 16m
tall building is 600.The angle of depression of foot of the hill is
300.Find the height of the cloud.
36. State and prove Pythagoras theorem.
37. Solve 2x+ y = 4 and 2x – y = 4 graphically.
VI. Solve the following 5*1 = 5
38. A metallic right circular cone 20 cm high and whose vertical
angle is 600 is cut into two parts at the middle of its height by a
plane parallel to its base. If the frustum so obtained be drawn
into a wire of diameter 16
1 cm, find the length of the wire.
DEPARTMENT OF PUBLIC INSTRUCTION, HASSAN
MODEL QUESTION PAPER – 02
SUB: MATHEMATICS Max. Marks : 80
====================================================
I Four alternatives are given to each question. Choose the appropriate
answer. Write it along with its alphabet 8 x 1 = 8
1. The L.C.M of is
A. 40 B. 560 C. 1120 D. 1680
2. The next term of the A.P , , , . . . . . . is
A. B. C. D.
3. The zeroes of the polynomial x 2 – 2 x – 3 are
A. -3, 2 B. -3, -1 C. 3, -1 D. 3, 1
4. Nature of the roots of the quadratic equation 2x2 – 3x +5 = 0 A.
Real and distinct B. Real and equal
C. No real roots D. None of these
5. If Sin A =5
3 , then tan A = ______
A.5
4 B.4
3 C. 3
4 D.4
5
6. Which of the following cannot be the probability of an event?
A. 3
2 B. -2.5 C. 100 D. 1
7. In the adj. fig , the value of PTQ is
A. 900 B. 1100 C. 700 D. 400
8. If yxyx 22 = 8 , then the value of y is
A. 2
3 B.3
2 C. 0 D.
II Solve the following 8 x 1 = 8
9. Express as the product of prime factors
10. Which type of solution will equation
have?
11. In the quadratic equation ax2 + bx +c =0, if a = 0, then name the
type of equation so obtained.
12. Write the co ordinates of the origin.
13. Find the distance between the points (2,3) and (4, 1)
14. Calculate the average of first ten natural numbers?
15. State Thales theorem
16. Write the formula used to find the volume of frustum of a cone.
III Solve the following 8 x 2 = 16
17. Prove that 2+ is a irrational number
18. Solve 8x+5y=19 and 3x+2y=4
19. Form a quadratic polynomial whose sum and product of the
zeroes are 2 and 3
1 respectively.
20. Prove that SecA(1-SinA)(SecA+tanA)=1
OR
Find the value of 2tan2450 cos2300 Sin2600
21. Calculate mode for the following scores
22. What is the probability of getting doublets, when two dice are
thrown simultaneously?
23. If the volume of a cube is 64cm3, then find the T.S.A of the
cube.
24. Construct two tangents to a circle of radius 5cm such that the
angle between the tangents is 600
OR
Construct two tangents to a circle of radius 3.5 cm , such that
the angle between the radii is 800
I IV. Solve the following 9 x 3=27
25. If two zeroes of the polynomial are3
5 and -3
5 , find other two
zeroes
OR
If the polynomial x3 3x2 x 2 is divided from g(x) , the
remainder and quotient are (x 2) and -2x+4 respectively ,then
find g(x).
26. If 3cot = 4, then show that
2
2
tan1
tan1
=cos2 - sin2
C – I 0-20 20-40 40-60 60-80 80-100 100-120
F 10 35 52 61 38 29
27. The shadow of a tower standing on a level ground is found to be
40m longer when the Sun’s altitude is 300 than when it is 600.
Find the height of the tower
OR
The angle of depression of the top and bottom of an 8 m tall
building from the top of a multi storeyed building are 300 and
450 respectively. Find the height of the multi storeyed building
and the distance between.
28. If A(-5,7), B(-4,-5), C(-1, -6), and D(4, 5) are the vertices of
a quadrilateral ,then find the area of the quadrilateral ABCD.
29. Draw less than type Ogive for the following data
Weight in k. g No. of
students
Less than 38 0
Less than 40 3
Less than 42 5
Less than 44 9
Less than 46 14
Less than 48 28
Less than 50 32
Less than 52 35
30. Construct a triangle of sides 5cm, 6cm and 7 cm. and then another
triangle whose sides are 5
7 of the corresponding sides of the first
triangle.
31. Prove that “length of the tangents drawn to a circle from an
external point are equal”.
32. Two poles of height 9 m and 14 m stands vertically on the
ground level ,such that the distance between them is 12 m. find
the distance between their tops.
OR
In the adj. fig POR=900, OP= 6cm , OR = 8cm ,
PQ = 24 cm and QR=26cm, then show that
PQR is a right angled triangle.
33. Find the area of the shaded regions enclosed by two
concentric circles of radius 7 cm and 14 cm , where AOC= 400
OR
The perimeter of a circle is 45cm more than its diameter, then find
the perimeter of the circle.
V. Solve the following 4 4=16
34. Prove that “ areas of two similar triangles are proportional to the
squares of their corresponding sides”.
35. Solve 0 graphically
36. An express train takes 1 hour less than a passenger train to
travel 132 km between Mysore and Bangalore. If the average
speed of the express train is 11km/h more than that of the
passenger train, find the average speed of the two trains.
37. A metallic right circular cone 20 cm high and whose vertical
angle is 600 is cut into two parts at the middle of its height by a
plane parallel to its base. If the frustum so obtained be drawn
into a wire of diameter 16
1 cm, find the length of the wire.
OR
A vessel is in the form of a hollow hemisphere mounted by a
hollow cylinder. The diameter of the hemisphere is 14 cm and
the total height of the vessel is 13 cm. find the inner surface
area of the vessel.
VI. Solve the following 5
38. The sum of 3rd and 7th term of an A.P is 6 and their product is
8, find the sum of first 16 terms.
DEPARTMENT OF PUBLIC INSTRUCTION, HASSAN
MODEL QUESTION PAPER – 03
SUB: MATHEMATICS Max. Marks : 80
====================================================
I Four alternatives are given to each question. Choose the appropriate
answer. Write it along with its alphabet 8 x 1 = 8
1. By Euclid’s division algorithm the H.C.F of 135 and 225 is
A) 45 B) 35 C) 50 D)65
2. The 30th term of the A.P 10, 7, 4…….. is
A) 97 B) 77 C)-77 D)-87
3. In the adjoining figure the zeroes of the polynomial p(x) is
A) 4 B)3 C)2 D)5
4. In the adj. fig, if DE‖‖BC then EC
A)3 B)2 C)2.5 D)4
5. The probability of getting a rain is 0.3,then the probability of not
getting a rain is
A) 0.6 B)1.7 C)1 D)0.7
6. If Sin 2A=2sinA , then A
A) 00 B) 300 C)450 D)600
7. In the adj fig if QOR=1000 then QPR=_____
A) 800 B)1800 C)400 D)900
8. The T.S.A of hemisphere of radius 7cm is
A) 616cm2 B) 154cm2 C)308cm2 D) 164cm2
II. Solve the following 8 * 1 = 8
9. Express 3825 as the product of prime factors.
10. Find the 10th term of the A.P 2,7,12……….
11. Form a quadratic polynomial whose sum and product of zeroes are 1
and 1 respectively.
12. Write the positive roots of the equation ax2+bx+c=0
13. Simplify: cos 480-sin 420
14. Find the value of 9sec2A - 9tan2A
15. Write the formula used to calculate the area of sector with
an angle of A0
16. The radius of right circular cone of slant height 14 cm is 6 cm,
find its C.S.A
III Solve the following 8 * 2 =16
17.Prove that 5+ 2 is a irrational number
18. Solve 2x+3y=46 and 3x+5y=74
19. If the roots of the equation 2x2+kx+3=0 are equal, then find the value
of ‘k’
20. If a point P divides the line segment obtained by joining the points
(4,-3), and (8,5) in the ratio 3: 1,then find the co ordinates of the point P.
21. A bag contain 3 red balls and 5 black balls, a ball is drawn randomly,
what is the probability of getting
a. a red ball b. other than a red ball ?
22. In the adj. fig if LM‖CB and LN‖CD then show that =
OR
Δ ABC is an right isosceles triangle with AC=BC , then show that
AB2=2AC2
23. Construct a pair of tangents to a circle of radius 4cm such that the
angle between the radii is 1200
24. If Sin(A-B)=2
1 , cos(A+B)= 2
1 with A>B , A and B are acute angles
then find the value of A and B
OR
If Cos A=5
4 then, find the value of Sin A and tan A
IV Solve the following 9*3 =27
25. Find the zeroes of the polynomial 4s2 - 4s+ 1, and also verify the
relationship between zeroes and its coefficients.
26. The sum of areas of two squares is 468m2 and difference of their
perimeter is 24 m, find the length of sides of the squares.
OR
The difference of squares of two numbers is 180. The square of
smaller number is twice the larger number. Find the two numbers.
27. Find the area of the triangle formed by joining the mid-points of
the sides of the triangle whose vertices are (0, –1), (2, 1) and (0,
3). Find the ratio of this area to the area of the given triangle.
OR
Find a relation between x and y such that the point (x , y) is
equidistant from the points (7, 0) and (1, 2).
28. The following distribution gives the daily income of 50 workers
of a factory. Convert the above distribution to a less than type
cumulative frequency and Draw less than type Ogive
29. The distribution below gives the weights of 30 students of a class.
Find the median weight of the students
OR
The following distribution shows that daily pocket allowance of children
of a locality. The mean pocket allowance is Rs. 18. Find the missing
frequency
X 100-120 120-140 140-160 160-180 180-200
F 12 14 8 6 10
Weight ( in Kg ) 40-45 45-50 50-55 55-60 60-65 65-70 70-75
Number of students 2 3 8 6 6 3 2
Daily pocket
allowance
( in Rs )
11-13 13-15 15-17 17-19 19-21 21-23 23-25
No. of
children 7 6 9 13 f 5 4
30. Prove that the “length of the tangents drawn to a circle from an
external points are equal”.
31. The cost of 5 pen and 3 pencils is Rs. 35 and the cost of 2 pens and 4
pencils is Rs. 28. Find the cost of each pen and pencil
OR
Solve the pair of equations + 3y = 14, - 4y = 23 by reducing them to a
pair of linear equations
32. Draw a triangle ABC with side BC = 6 cm, AB = 5 cm and
ABC = 60°.Then construct a triangle whose sides are4
3 times the
corresponding sides of the triangle ABC.
33. Find the area of the shaded region where a major sector of
radius 6 cm has been drawn with vertex ‘O’ of an equilateral
triangle OAB of side 12 cm as centre
V. Solve the following 4x4=16
34. Solve x + y = 8 and and x – y = 4 graphically
35. Sum of three numbers of an A.P is 27 and sum of their squares is 293.
Find the numbers.
OR
The sum of third and 7th term of an A.P is 6 and their product is 8,
find the A.P.
36. From a point on the ground, the angles of elevation of the bottom and
the top of a transmission tower fixed at the top of a 20 m high building are
45° and 60° respectively. Find the height of the tower.
37. A cylindrical bucket, 32 cm high and with radius of base 18 cm, is
filled with sand. This bucket is emptied on the ground and a conical heap of
sand is formed. If the height of the conical heap is 24 cm, find the radius
and slant height of the heap.
VI Answer the following question 5x1=5
38. State and prove Thale’s theorem.
DEPARTMENT OF PUBLIC INSTRUCTION, HASSAN
MODEL QUESTION PAPER – 04
SUB: MATHEMATICS Max. Marks : 80
====================================================
I Four alternatives are given to each question. Choose the
appropriate answer. Write it along with its alphabet 8*1 = 8
1. The first three terms of A.P with first term 1 and c.d – 2 is
A. – 1, 1, 3 B. 1. -1, -3 C. 0 , 1 , 3 D. – 3 , 1 , - 1
2. The value of 2cos1 is
A. 1 B. sin C. tan2 D. 2sin
1
3. Which of the following is not probability of an event?
A. 3
2 B. – 1.5 C. 0.7 D. 0.5
4. The formula used to calculate the C.S.A of a cone
A. r l B. r2l C. ( r1 +r2 ) l D. rh
5. In the adjoining figure if 0100AOB , then ACB =
A. 600 B. 700 C. 800 D. 400
6. if tan2A = cot( A - 180) , where 2A is an acute, then A =
A. 180 B. 1800 C. 3600 D. 360
7. In the adjoining graph, number of zeroes of the polynomial p ( x ) is
A. 0 B. 1 C. 2 D. 3
8. The C.S.A of a cylinder of radius 7 cm and height 10 cm is
A. 410 sq. cm B. 420 sq. cm C. 400 sq. cm D. 440 sq. cm
II. Solve the following 8 * 1 = 8
9. Express 3825 as product of prime factors
10. If q
p
is a rational number ( q o), what is the condition of ‘q’ so
that the decimal representation of q
p
is terminating?
11. If and are the zeroes of the polynomial p ( x) = x2 + 5x + 8
, then find the value of + .
12. State Thales theorem
13. If p ( x) = 2x2 + 3x + 2,then find the value of p ( 2 )
14. If b2 – 4ac = 0 then write the nature of the roots
15. If the c.d of an A.P is 4, then find the value of a15 – a10
16. Evaluate tan( 900 - )
III. Solve the following 8 * 2 = 16
17. Prove that 2+ 5 is a irrational number
18. Find the 11 th term from the last of an A.P 10,7,4…….. – 62
19. Solve x + y = 14 and x – y = 4
OR
The difference of two numbers is 26, if one number is thrice the
other, find the number.
20. If p ( x) = x3 – 3 x 2+5x – 3 and g ( x) = x2 – 2, find the
remainder and the quotient
OR
If the polynomial x3 – 3x2 +x+2 is divided by g(x) , the
remainders and quotient were x – 2 and – 2 x + 4, find g ( x).
21. Solve 2x2 – 7x + 3 = 0 by formula method.
22. A dice is thrown once, what is the probability of getting
a. A prime number b. An odd number
23. Draw a circle of radius 5 cm, construct the tangents to the circle
at the ends of radii such that the angle between radii is 600.
24. From each corner of a square of side 4 cm a
quadrant of a circle of radius 1 cm is cut and also
a circle of diameter 2 cm is cut as shown
in the figure. Find the area of the remaining portion of the square.
IV. Solve the following 9 * 3 = 27
25. A fraction becomes 1, if 1 is added to the numerator and 1 is
subtracted from the denominator. It becomes 2
1 , if 1 is added to the
numerator . find the fraction.
OR
Solve x
4 + 3y = 14 and x
3 – 4y = 23 by Elimination method
26. Prove that “the length of the tangents drawn the circle from an
external point are equal”.
27. Find the area of a quadrilateral whose vertices are ( - 4 , - 2 ) ,
( - 3 , - 3 ), ( 3, - 2 ) and ( 2,3 ).
OR
In what ratio does the point ( - 1 , 6 ) divides the line obtained by
joining the points ( - 3 ,10 ) and ( 6 , 8 )?
28. Find a cubic polynomial with the sum , sum of the product of its
zeroes taken two at a time, and the product of its zeroes are 2 ,-7 and
-14 respectively.
29. Find the area of the shaded region, where ABCD
is a square of side 10 cm and semicircles are drawn
with each side of the square as diameter ( use = 3.14)
OR
Find the area of the shaded where a circular arc of
radius 6 cm has been drawn with vertex O of an
equilateral triangle OAB of side 12 cm as centre
30. Find the median for the following scores
C-I 1-4 4-7 7-10 10-13 13-16 16-19
F 6 30 40 16 4 4
OR
If the median of the following scores is 28.5, then find the value of x
and y
31.A train covers a distance of 360 km with uniform speed , if the speed
had been 5 km / hr more, it would have taken 1 hour less for the same
journey. Find the speed of the train.
C- I 0-10 10-20 20-30 30-40 40-50 50-60
F 5 X 20 15 Y 5
32.Draw more than type Ogive for the following scores
Production
yield 50-55 55-60 60-65 65-70 70-75 75-80
No. of farms 2 8 12 24 38 16
33.Construct a triangle of sides 5cm, 6cm and 7 cm. and then another
triangle whose sides are 5
7 times the corresponding sides of the first
triangle.
V. Solve the following 4* 4 = 16
34.Solve 2x+ y = 4 and 2x – y = 4 graphically.
35.The angle of elevation of top of Hill from the top of a 16 m high
building is 600and the angle of depression of bottom of the hill is 300
. Find the height of the hill.
36.The first and last term of an A.P are 8 and 350 respectively, if the
common difference is 9, find the number of terms and the sum of the
series.
OR
The third term of an A.P is 32 and its fifth term is 40, find the sum
of first thirty terms of the A.P
37.A medicine capsule is in the shape of a cylinder with two hemispheres
stuck to each of its ends. The length of the entire capsule is 14 mm
and the diameter of the capsule is 5 mm. find its surface area.
VI. Solve the following 5 * 1 =5
38.State and Prove A. A. A. criterion of similarity triangles.
DEPARTMENT OF PUBLIC INSTRUCTION, HASSAN
MODEL QUESTION PAPER – 05
SUB: MATHEMATICS Max. Marks: 80
====================================================
I Four alternatives are given to each question. Choose the
appropriate answer. Write it along with its alphabet 8 * 1 = 8
1. The nth term of an A. P with first term ‘a’ and c.d. ‘d’ is
A. a +( n +1 ) d B. a + ( n – 1 ) d
C. a – ( n + 1 ) d D. a – ( n – 1 ) d
2. If the H.C.F and L.C.M of two positive numbers are a and b , then
H( a, b ) x L ( a, b ) = --
A. a x b B. b
a C. a – b D. a + b
3. If solution of two lines are consistent then, lines are
A. parallel and coincident B. always intersect each other
C. parallel or intersect each other D. always coincide with each other
4. The distance between origin and ( a, b ) is
A. d = 22 ba B. d = 22 ba C. d = 22 ba D. d = 22 ba
5. The sum of zeroes of the polynomial x2 – 7x +10 is
A. 7 B. – 7 C. 10 D. – 10
6. “The sum of two odd positive numbers is 290”, the algebraic form of
this statement is
A. x2 + ( x+2)2 B. (x+1)+ ( x + 3 ) = 290
C. x2 + ( x + 2 ) = 290 D. x + ( x + 2 ) 2 = 290
7. In the adj. figure if AB = 4 cm and BC = 3 cm, then tan A =
A. 4
3 B. 5
3 C. 3
4 D. 4
5
8. The C.S.A of cone of radius 7 cm and slant height 10 cm is
A. 110 cm2 B. 70 cm2 C. 440 cm2 D. 220 cm2
II. Solve the following 8 * 1 = 8
9. Find the 5th term of an A.P whose nth term is 4n2 – 1 .
10. How many tangents can be drawn to a circle from an external
point?
11. If b2 – 4ac = 0, then comment on nature of the roots .
12. If SinA = 2
1 , then find the value of A
13. Name the solid obtained by rotating a rectangle about one of its
edge.
14. Express 140 as the product of prime factors
15. If P ( E) = 0.05, then find P ( E )
16. Find the distance between the points ( 2 , 3 ) and ( 4, 1 )
III. Solve the following 8 * 2 = 16
17. Prove that 3+ 5 is a irrational number.
18. If the zeroes of the polynomial x 2 -12x+32 are a+ b and
a - b, then find the value of ‘a’ and ‘b’.
19. The sum of the reciprocal of Rehman’s age ( in years ) 3 years
ago and 5 years from now is 3
1 . Find his present age.
20. Solve x +y = 14 and x – y = 4
OR
Half the perimeter of a rectangular garden, whose length us 4 m
more than its width is 36 m. find the dimensions of the garden.
21. Two dice are thrown once, what is the probability of getting
a. Sum of the two numbers is divisible by 4
b. The product of two numbers is a perfect square ?
22. Areas of two similar triangles ABC and DEF, are 64cm2 and
121 cm2 respectively , if EF = 15.4 cm, find the length of BC.
OR
In the adj. fig LM || CB, and LN || CD, then
prove that AD
AN
AB
AM
23. In the adj. fig ABC 090ABC , AD BC
then prove that AC2 = AB2 +BC2 +2BC.BD
24. Draw a pair of tangents to a circle of radius 5 cm which are
inclined to each other at angle of 800.
IV. Solve the following 9* 3 = 27
25. If the roots of the equation 3x2 – 5x + 2 = 0 are real, then find
the roots by using formula
OR
Solve 2x2 – 7x + 3 = 0 by completing square method
26. If 3 cotA = 4, then verify A
A2
2
tan1
tan1
= cos2A – sin2 A
OR
Prove that 1cos
1cos
coscot
coscot
ecA
ecA
AA
AA
27. Draw the less than type Ogive for the following scores
Monthly
consumption
( in units)
65-85 85-105 105-125 125-145 145-165 165-185 185-205
No of
consumers 4 5 13 20 14 8 14
28. Find the median for the following scores
C – I 135-140 140-145 145-150 150-155 155-160 160-165
F 5 4 7 18 11
29. BL and MC are the medians of the right angled triangle
ABC, then prove that 4 ( BL2+MC2 ) = 5BC2
OR
Two poles of height 6 m and 11 m stand vertically on the ground level
such that the distance between them is 12 m, find the distance between
their tops.
30. Prove that “the length of the tangents drawn the circle from an
external point are equal”.
OR
Two concentric circle of radius 5 cm and 3 cm. find the length of the
chord of the larger circle which touches the smaller circle.
31. Divide 3x2 – x 3 +3x +5 by x – 1 – x 2 and verify division
algorithm.
OR
Find the zeroes of the polynomial x2 +7x +10 and verify the
relationship between the co-coefficients and the zeroes
32. Find the area of a quadrilateral whose vertices are ( - 4 , - 2 )
( - 3 , - 5 ), ( 3, -2 ) and ( 2, 3 ).
33. Construct a triangle of sides 5cm, 6cm and 7 cm. and then
another triangle whose sides are 5
7 of the corresponding sides
of the first triangle.
V. Solve the following 4*4 = 16
34. How many terms of the A.P -10,-7,-4,-1 must be added to make
the sum -104?
OR
Find three numbers in A.P whose sum is 27 and product is 648.
35. If corresponding angles of two triangles are equal, then prove
that corresponding sides are in proportion.
36. Solve x + 3y = 6 and 2x – 3y = 12 graphically
37. From a point on a bridge across a river the angles of depression
of the banks on opposite sides of the river are 300 and 450
respectively, if the bridge is at a height of 3 m from the bank,
find the width of the river.
VI. Solve the following 5 * 1 = 5
38. A cistern internally measuring 150 cm x 120 cm x 110 cm has
129600cm3 of water in it pores bricks are placed in the water
until the cistern is full to the brim each brick observes one
seventeenth of its own volume of its water. How many brick can
be put in without overflowing the water, each brick being
22.5 cm x 7.5cm x 6.5 cm.
DEPARTMENT OF PUBLIC INSTRUCTION, HASSAN
MODEL QUESTION PAPER – 06
SUB: MATHEMATICS Max. Marks : 80
====================================================
I Four alternatives are given to each question. Choose the appropriate
answer. Write it along with its alphabet 8 * 1 = 8
1. The decimal expansion of 8
3 is
A. 0.375 B. 3.75 C. 37.5 D. 375
2. If 2
1
2
1
2
1
c
c
b
b
a
a , then the lines are
A. coincident B. intersect each other
C. parallel D. none of these
3. The 30th term of the A.P 10 ,7,4,….. is
A. 61 B. 16 C. 79 D. 97
4. The value of cos480 – sin 420 is
A. 300 B. 250 C. 200 D. 0
5. The distance between the points ( 2, 3 ) and ( 4, 1) is
A. 2 units B. 2 units C. 2 2 units D. 22 units
6. The probability of winning a prize in singing competition is 0.62, then
the probability of losing is
A. 0 B. 0.038 C. 3.8 D. 38
7. Number of tangents drawn to a circle
A. 0 B. 1 C. 2 D. Infinity
8. The area of a circle of radius 7 cm is
A. 0.95 B. 0.59 C. 95 D. 15
II. Solve the following 8 * 1 = 8
9. Express 3825 as the product of prime factors
10. Write the discriminent of the equation px2 + qx + r = 0
11. If sin 5
3 and cos
5
4 , then find the value of tan
12. A man standing on building observes the a car coming towards
the building , name the angle so formed with the line of sight
and the horizontal plane
13. Write the section formula.
14. Calculate the C.S.A of a hemisphere of radius 7 cm.
15. In the adj. fig find the length of OA
16. If S = { 1,2,3,4,5,6} What is the probability of prime number?
III. Solve the following 8 * 2 = 16
17. Prove that 3 is an irrational number
18. Find the quadratic polynomial whose sum and product of the
zeroes are 2 and 3
1 .
19. If P ( x ) = x3 – 3x2 + 5x – 3 is divided from g ( x ) = x – 2
then , find the remainder and the quotient.
20. A circus artist is climbing 20 m long rope which is tightly
stretched and tide from the top of the vertical pole to the ground.
Find the height of the pole, if the angle made by the rope with
the ground level is 300.
OR
The angle of elevation of the top of the tower from a point on the
ground, which is 30 m away from the foot of the tower is 300 , find the
height of the tower.
21. In the adj. fig LM || CB, and LN || CD, then prove that AD
AN
AB
AM
OR
In the adj fig. ABC and AMP are two right triangles,
right angled at B and M respectively prove that
a. ABC ~ AMP b. MP
BC
PA
CA
22. Divide a line segment of length 8 cm in the ratio 3 : 5
23. Find the area of a sector of radius 6 cm with central angle of
600.
24. Find the volume of a hemisphere of diameter 7 cm.
IV. Solve the following 9 * 3 = 27
25. Solve the following yx 3
1
2
1 = 2 and
6
13
2
1
3
1
yx
OR
Ritu can row downstream 20 km in 2 hours, and upstream 4 km
in 2 hours. Find the speed of rowing in still water and the speed
of the current.
26. Solve x + 3y = 6 and 2x – 3y = 12 graphically
27. In an A.P, the 2nd and 3rd terms are 14 and 18 respectively, find
the 51st term of the A.P
28. Solve 5x2 – 6x – 2 = 0 using suitable formula and also discuss
the nature of the roots
29. The diagonal of rectangular field is 60m more than the shortest
side. If the longer side is 30 m more than the shorter side, find
the sides of the field.
OR
The sum of the reciprocal of Rehman’s age ( in years ) 3 years
ago and 5 years from now is 3
1 . Find his present age.
30. Prove that the points ( 3, 1 ) ( 6, 4 ) and ( 8 , 6 ) are collinear.
OR
In what ratio does the point ( - 1 , 6 ) divides the line segment
obtained by joining the points ( - 3 , 10 ) and ( 6 , - 8 )?
31. The following shows the ages of the patients admitted in the
hospital during the year.
Ages
( in years) 5-15 15-25 25-35 35-45 45-55 55-65
No. of
patients 6 11 21 23 14 5
Find the mean and the mode for the above data
32. TP and TQ are the tangents drawn to the circle
with centre O from an external point T. Prove
that OPQPTQ 2
33. A drinking glass in the shape of a frustum of a cone of height
14 cm. the diameter of its two circular ends are 4 cm and 2 cm .
find the capacity of glass.
IV. Solve the following 4 * 4 = 16
34. Three numbers are in A.P whose sum is 27 and product is 648.
find the numbers
OR
The 2nd and 12th term of an A.P are 14 and 18 respectively, find the
sum of 51st terms of the A.P
35. If cosec2 A = 1+cot2 A , then prove that
1sincos
1sincos
AA
AA = cosecA +cotA
36. Draw more than type Ogive for the following scores
Production
yield 50-55 55-60 60-65 65-70 70-75 75-80
No. of
farms. 2 8 12 24 38 16
37. Construct a triangle of sides 5cm, 6cm and 7 cm. and then
another triangle whose sides are 5
7 of the corresponding sides of the
first triangle
VI. Solve the following 5 * 1 = 5
38. State and prove pythagoreas theorem
DEPARTMENT OF PUBLIC INSTRUCTION, HASSAN
MODEL QUESTION PAPER – 07
SUB: MATHEMATICS Max. Marks : 80
====================================================
I Four alternatives are given to each question. Choose the appropriate
answer. Write it along with its alphabet 8 * 1 = 8
1. The H.C.F of the smallest prime number and the composite number
A. 4 B.3 C. 2 D. 1
2. The sum of first n natural numbers is
A. 2
)1(
nnSn B. Sn = an (n +1 ) C. Sn = n ( n + 1 ) D.
2
)1(
nnSn
3. The zeroes of the polynomial 4x2 – 12x + 9 are
A. 2
3,
2
3 B. 2
3,
2
3 C. 3,-4 D. -3,-4
4. The discriminent of the quadratic equation 2x2 – 4x + 3 = 0 is
A. – 4 B. – 8 C. 4 D. 6
5. If the distance between the points ( 4, p ) and ( 1 , 0 ) is 5 units, then
the value of ‘p’ is
A. 4 B. – 4 C. 1 D. None of these
6. The value of sin300 is
A. 0 B. 2
1 C. 2
3 D. 2
1
7. The coordinates of the origin is
A. ( 0 , 0 ) B. ( 1,1) C. ( 2,2) D. ( 3,3)
8. Total number of possible outcomes, when a coin is tossed thrice
repeatedly is
A. 4 B. 8 C. 5 D. 7
II. Solve the following 8* 1 = 8
9. What is the measure of the angle subtended in a semicircle?
10. Find the distance between the points (2, 3) and the origin.
11. Write the formula used to find the C.S.A of frustum of a cone
12. Simplify A
A2
2
cot1
tan1
13. If the volume of a cube is 64 cm3, then find the length of each
side of the cube
14. If One root of the equation x2 = 3 x – 10 is 5 find the other root
15. Express 15 as the product of prime factors
16. Find the 11th term of the A.P 3,2
1 , -2 …………….
III. Solve the following 8* 2 = 16
17. Prove that 5 - 3 is irrational number
18. Solve 2x+ y = 5 and 3x + 2 y = 8
19. Solve 2x2 – 3x – 5 = 0 by formula method
20. If tan2A = cot ( A – 18) and 2A is an acute angle, then find the
value of A
OR
Prove that AAA
Atansec
sin1
sin1
21. Find the area of a triangle, whose vertices are ( 2, 3 ) , ( - 1 , 0 )
and ( 2 , - 4 )
22. When a dice is thrown once, what is the probability of getting
a. A prime number b. A number between 2 and 6 ?
23. ABCD is a trapezium in which AB || CD, diagonals BD and CD
intersects at O, then show that OD
CO
CO
AO
OR
A ladder 10 m long reaches a window 8 m above the ground.
Find the distance of the foot of the ladder from the base of the wall
24. Construct two tangents to a circle of radius 5 cm from an
external point such that the angle between them is 600.
IV. Solve the following 9 * 3 = 27
25. Ritu can row downstream 20 km in 2 hours, and upstream 4 km
in 2 hours. Find the speed of rowing in still water and the speed
of the current.
OR
The ratio of incomes of two persons is 9 : 7 and the ratio of their
expenditures is 4 : 3. if each of them manages to save Rs. 2000
per month, find their monthly incomes.
26. Find the coordinates of the points of trisection of the line
segment joining the points A ( 2, 2) and B ( - 7 , 4 )
OR
Show that the points ( 1, 5 )( 2, 3 ) and ( - 2 , -11) are collinear.
27. Find the remainder and quotient by dividing the polynomial
p ( x ) = 3x4 +5x3- 7x2+2x+2 by x2 +3x+1, and also verify
division algorithm.
28. Prove that “length of the tangents drawn from the external point
to the circle are equal”.
29. Construct a triangle of sides 4cm, 5cm and 6 cm. and then
another triangle whose sides are 3
2 of the corresponding sides of
the first triangle
30. A train travels a distance of 480 km at a uniform speed. If the
speed had been 8 km/hr less, then it would have taken 3 hours more to
cover the same distance. Find the speed of the train.
31. Find the mean for the following scores
C – I 10-20 20-30 30-40 40-50
F 1 2 3 4
32. The annual profits earned by 30 shops of a shopping complex in
a locality give rise to the following distribution, Draw the less than
type Ogive
Classes 5-10 10-15 15-20 20-25 25-30 30-35 35-40
No of
shops 2 12 2 4 3 4 3
33. From each corner of a square of side 4 cm a
quadrant of a circle of radius 1 cm is cut and also a
circle of diameter 2 cm is cut as shown in the figure.
Find the area of the remaining portion of the square.
OR
Find the area of the shaded region in the adj. fig
where ABCD is a square of side 14 cm
V. Solve the following 4 * 4= 16
34. Solve x+3y = 6 and 2x – 3y = 12 graphically
35. Sum of first 7 terms of an A.P is 49 and sum of first 17 terms is
289, find the sum of first n terms
36. The angles of depression of top and bottom of an 8 m tall
building from the top of a multi storeyed building are 300 and
450 respectively. Find the height of the multi storeyed building
and the distance between the two buildings.
37. The radii of the ends of a frustum of a cone 45 cm and high are
28 cm and 7 cm . find its volume, the curved surface are and the
total surface area ( use 7
22 )
VI. Solve the following 5*1 = 5
38. State and prove Pythagoras theorem
DEPARTMENT OF PUBLIC INSTRUCTION, HASSAN
MODEL QUESTION PAPER – 08
SUB: MATHEMATICS Max. Marks : 80
====================================================
I Four alternatives are given to each question. Choose the appropriate
answer. Write it along with its alphabet 8 * 1 = 8
1. The H.C.F of 36 and 18 is
A. 6 B. 2 C. 9 D. 18
2. The ratio of corresponding sides of two similar triangles are 4 : 9 ,
then ratio of their areas is
A. 1 : 2 B. 2 : 3 C. 16 : 81 D. 81 : 16
3. In the adj fig the value of ‘x’ is
A. 600 B. 1200 C. 900 D. 300
4. The 10 the term f the A.P 2,5,8,….. is
A. 30 B. 25 C. 20 D. 29
5. Sum of the zeroes of the polynomial x2 – 2x – 8 is
A. – 2 B. 1 C. 2 D. 0
6. The root of the quadratic equation 6x2 – x – 2 = 0 are
A. 3
2 and 2
1 B. 3
2 and 2
1 C. 3
2 and 2
1 D. 3
2 and 2
1
7. If Sin A = 2
1 , then A =
A. 800 B. 600 C. 300 D. 450
8. If P ( E ) = 0.05, then P ( E ) =
A. 0.95 B. 0.59 C. 95 D. 59
II. Solve the following 8 * 1 = 8
9. Find the common difference of the A.P 3,1,-1,-3,……………
10. State Thale’s theorem
11. Express 140 as the product of prime factors
12. Form a quadratic polynomial whose sum and product of zeroes
are - 3 and 2 respectively.
13. Evaluate sin600 +cos300
14. In the adj. figure find the value of ‘x’
15. Write the formula used to calculate the C.S.A of frustum of a
cone
16. Calculate the C.S.A of a hemisphere of radius 3 cm.
III. Solve the following 8 * 2 = 16
17. Areas of two similar triangles ABC and DEF, are 64cm2 and
121 cm2 , if EF = 15.4 cm, find the length of BC.
OR
ABCD is a trapezium in which AB || CD, diagonals BD and CD
intersects at O, then show that OD
CO
CO
AO
18. Solve 2x + y = 6 and 4x – 2y – 4 = 0
19. Construct two tangents to a circle of radius 5 cm, such that the
angle between the tangents is 600.
20. Find the distance between the points ( – 5 , 7 ) and ( - 3 , 3 )
21. Prove that 2 + 3 is irrational number
22. Solve 2x2 – 7x + 3 = 0 using formula method
23. If tan 2A = cot ( A – 180), where 2A is an acute angle, find ‘A’.
OR
Prove that A
A
A
A
cos
sin1
sin1
cos
= 2 secA
24. When a dice is thrown twice, what is the probability of getting a
number whose product is 12 ?
IV. Solve the following 9 * 3 = 27
25. A fraction becomes 11
9 , if 2 is added to both the numerator and
the denominator if 3 is added to both the numerator and the
denominator it becomes 6
5 . Find the fraction.
OR
If we add 1 to the numerator and subtract 1 from the denominator a
fraction reduces to 1. it becomes 2
1 if we only add 1 to the denominator.
What is the fraction?
26. Divide 3x3 – x 2 + 2x + 5 by 1 +2x + x2
27. Sum of two numbers is 27 and their product is 182. Find the
numbers.
OR
The height of a right angled triangle is 11 cm longer than its shortest
sides. If the area of the triangle is 13 cm2. find the sides of the triangle
28. Find the area of a triangle whose vertices are ( 5, 2 ), ( 4, 7 ) and
( 7 , - 4 )
OR
If A ( - 5 , 7 ) , B ( - 4 , - 5 ) , c ( -1 , - 6 ) and D (4, 5 ) are the vertices
of a quadrilateral, find the area of the quadrilateral ABCD.
29. A survey conducted on 20 households on a locality by a group of
students resulted in the following frequency table for the number of
family members in a household, find the mode of this data
Family
Size 1-3 3-5 5-7 7-9 9-11
Number of
families 7 8 2 2 1
30. The following distribution gives the daily income of 50 workers
of a factory
Daily
income
( in Rs. )
100-120 120-140 140-160 160-180 180-200
Number of
workers 12 14 8 6 10
Convert the above distribution to a less than type cumulative frequency
distribution and draw its Ogive.
31. Prove that the “length of the tangents drawn to a circle from an
external points are equal”.
32. Construct a triangle similar to a given triangle ABC with its sides
equal to 4
3 of the corresponding sides of the triangle ABC.
33. Find the area of the shaded region in the adj. fig
where ABCD is a square of side 14 cm
OR
Find the area of the shaded region in the adj fig.
if ABCD is a square of side 14 cm and APD and
BPC are semicircles.
V. Solve the following 4 * 4 = 16
34. The pth , qth and rth term of an A.P are a, b, and c respectively,
then show that a ( q- r ) +b ( r – p ) +c ( p – q ) = 0
OR
The sum of 3rd and 7th term of an A.P is 6 and their product is 8, find
the sum of first 16 terms.
35. Solve x + 3y = 6 and 2x – 3y = 12 graphically
36. Prove that area of similar triangles are proportional to squares of
their corresponding sides.
37. From the top of a 7 m high building, the angle of elevation of top
of a cable tower is 600 and the angle of depression of its foot is 450.
determine the height of the tower.
VI. Solve the following 5* 1 = 5
38. A metallic right circular cone 20 cm high and whose vertical
angle is 600 is cut into two parts at the middle of its height by a plane
parallel to its base. If the frustum so obtained be drawn into a wire of
diameter 16
1 cm, find the length of the wire.
DEPARTMENT OF PUBLIC INSTRUCTION, HASSAN
MODEL QUESTION PAPER – 09
SUB: MATHEMATICS Max. Marks : 80
====================================================
I Four alternatives are given to each question. Choose the appropriate
answer. Write it along with its alphabet 8 * 1 = 8
1. If the H.C.F of 6 and 20 is 2, then their L.C.M is
A. 40 B. 120 C. 60 D. 240
2. The pair of linear equations 2x + 3y – 9 = 0 and 4x + 6y – 18 = 0 has
A. No solution B one solution C.
Only two solutions D. Many solutions
3. The common difference of the A.P 2
3 , 2
1 ,2
1 ,……. Is
A. -1 B. 2
1 C. 2
1 D. 3
2
4. The degree of the polynomial 4x3 – 2x2 + 3x5 – 4 is
A. 2 B. 3 C. 4 D. 5
5. The discriminent of the quadratic equation ax2 + bx + c = 0 is
A. b2 – ac B. b2 – 4ac C. acb 42 D. b2 +4ac
6. In the adj. fig. A quadrilateral ABCD is circumscribe a circle with AB
= 3 cm, and CD= 8 cm , then AD+BC= ---- cm
A. 11 B. 8 C. 3 D. 5
7. Area of a sector of angle ‘p’ of a circle of radius p is
A. 360
2pR B. 270
2pR C. 360
2Rp D. 270
2Rp
8. Formula used to calculate the C.S.A of frustum of a cone is
A. ( r1 +r2) l B. ( r1 +r2) h C. ( r1 - r2) l D. ( r1 - r2) h
II. Solve the following 8 * 1 = 8
9. State fundamental theorem of Arithmetic
10. If the nth term of an A.P is 3 + 2n, then find its fourth term
11. If SinA = 5
4 , then find cosecA
12. What is the probability of a sure event?
13. State Thale’s theorem
14. What is the sum of angles between tangent and radii of a
circle?
15. The angle subtended by the arc of a circle of radius ‘r’ units
is 900. Find the area of a sector.
16. Name the combination of solids of a petrol tanker.
III. Solve the following 8* 2 = 16
17. Prove that 6 - 3 is a irrational number
18. Solve x – y = 26 and x – 3y = 0
19. Solve 3x2 – 5x +2 = 0 using suitable formula
20. Find he distance between the point ( 8 , 3 ) and ( 6 , - 2 )
21. A lot of 20 bulbs contain 4 defective ones. One bulb is drawn
at random from the lot. What is the probability that this bulb is
a. Defective b. Non defective?
22. In the adj fig, DE || BC, find AD .
OR
In the adj. fig , altitudes AD and CE of ABC
intersect each other at the point P . Show that AEP ~ CDP
23. Construct tangents to a circle of radius 3.5 cm such that the angle
between the tangents is 700.
24. From each corner of a square of side 4 cm a quadrant
of a circle of radius 1 cm is cut and also a circle of
diameter 2 cm is cut as shown in the figure. Find the area of the
remaining portion of the square.
OR
In the adj. fig OACB is a quadrant of a circle with centre O and
radius 3.5 cm. if OD = 2 cm find the area of the shaded region
IV. Solve the following 9 * 3 = 27
25. Find the zeroes of the polynomial 6x2 – 3 – 7x and verify the
relationship between the co-coefficients and the zeroes
26. If the polynomial p ( x) = x2 – 3x2 +5x -3 is divided from
g ( x) = x 2 – 2, find the remainder and quotient and hence verify
Euclid’s division algorithm.
27. A train covers a distance of 360 km with uniform speed , if the
speed had been 5 km / hr more, it would have taken 1 hour less for the
same journey. Find the speed of the train.
OR
The perimeter of a rectangular field is 24 m and its area is 400 m2,
find length and breadth of the rectangular field.
28. Prove that
2
2
2
2
cot1
cot
tan1
tan
= 1
29. If A ( 6, 1 ) , B ( 8 , 2 ) , C ( 9 , 4 ) and D ( p , 3 ) are the vertices
of a parallelogram taken in an order, then find the value of ‘p’.
OR
Show that the points ( - 3 , 1 2 ) , ( 7 , 6 ) and ( 8 , 9 ) are collinear.
30. Draw the less than type Ogive for the following scores
Class
interval 0-3 3-6 6-9 9-12 12-15
Frequency 9 3 5 3 1
31. A class teacher has the following absentee record of 40 students
of a class for the whole term. Find the mean number of days a student
was absent.
Number
of days 0-5 5-10 10-15 15-20 20-25 25-30
Number
of
students
11 10 7 4 5 3
OR
Calculate mode for the following scores
Class
interval 1-3 3-5 5-7 7-9 9-11
Frequency 7 8 2 2 1
32. Prove that “the length of the tangents drawn from the external
point to the circle are equal”.
OR
In the adj fig TP and TQ are the tangents
drawn to the circle with centre O from an external point T.
Prove that OPQPTQ 2
33. Construct a triangle of sides 5cm, 6cm and 7 cm. and then
another triangle whose sides are 3
2 of the corresponding sides of the
first triangle.
V. Solve the following 4 * 4 = 16
34. Solve x – 2y = 0 and x + y = 6 graphically
35. The first term and last term of an A.P are 17 and 350
respectively with the common difference 9. How many terms
are there in the A.P ? and also find the sum of the series.
36. Three numbers are in A.P whose sum is 27 and product is 648.
find the numbers
37. A vessel is in the form of a hollow hemisphere mounted by a
hollow cylinder. The diameter of the hemisphere is 14 cm and
the total height of the vessel is 13 cm. find the inner surface
area of he vessel
VI. Solve the following 5* 1 = 5
38. A tower stands vertically on a bank of a canal. From a point on
the other bank directly opposite to the tower, the angle of elevation of
the top of the tower is 600. from another point 20 m away from this
point on the line joining this point the foot of the tower is 300. find the
height to the tower and the width of this canal.
( use tan 580 = 1.6003)