1

2

3

BASIC MATHEMATICS DPP - 1

1. log 25 + log 4 – log 5 is equal to (A) log 20 (B) log 25 (C) log 15 (D) log 10

2. loge 8 is equal to

(A) loge 2 (B) 2loge 2

(C) 3loge 2 (D) 4loge 2

3. loge 15 is equal to

(A) loge 3 + loge 5

(B) loge 5 – loge 3

(C) loge 10 + loge 5

(D) loge 10 – loge 5

4. Find the sum of first 20 Number.

(A) 210 (B) 200 (C) 220 (D) 230

5. Find sum of 1 + 1

3 + 1

9 +

1

27 + .... up to

. (A) 3/2 (B) 2/3 (C) 4/3 (D) ¾

6. If

1 2

1 2

5

3–, then

1

2

=

(A) 16 (B) 8 (C) 32 (D) 64

7. If 1

2

A 7

A 8, then

1 2

1 2

A A

A – A =

(A) 30 (B) –30 (C) –15 (D) 15

8. If

2 21 22 21 2

a a 5

3a – a, then 1

2

a

a=

(A) 0 (B) 1 (C) 2 (D) 4

9. If

sin cos 7

sin – cos 3 , then tan =

(A) 3/5 (B) 5/2

(C) 3/5 (D) 2/5

10. 1 sin is equal to

(A) (sin + cos ) (B) sin – cos

(C) sin 2

+ cos2

(D) sin 2

– cos 2

11. The value of cos–1

3

5 is

(A) – cos–1

3

5 (B)

2

– cos–1

3

5

(C) 2

+ cos–1

3

5 (D) None of these

12. A car is moving towards a building with

speed 10 m/s. At any instant the angle of elevation of the building is 30º after 5 sec. the angle of elevation of the building becomes 45º, then height of building is.

(A)

50h

3 1 (B)

20h

3 1

(C)

20h

3 1 (D)

50h

3 1

13. Find 1 – 1

2 +

1

4– 1

8 + 1

16 – 1

32 ...... .

(A) 3/4 (B) 4/3 (C) 2/3 (D) 3/2

14. Initial velocity of particle is v and final

velocity of particle is

v

2 then find

percentage change in kinetic energy. (A) 75% decrease (B) 75% increase (C) 35% decrease (D) 35% increase

15. Change in area of square is 4% then find change in length of square. (A) 8% increase

4

(B) 8% decrease (C) 2% increase (D) 6% decrease

True /False 16. The binomial theorem is applicable

below 5% variation

17. K.E. = 2p

2m if momentum is doubled then

% change in kinetic energy is 300%.

18. Length of cube is increased by 1.5% then

% change in volume is 4%.

19. We can write cos = cos (–)

20. cos (A + B) = cos A cos B + sin A sin B

5

DPP – 2

1. The equation of straight line having slope 3 and y intercept of –2 will be :

(A) y = 3x 2 (B) y = 3x 2

(C) y = 3x 2 (D) y = 3x 2

2. The equation of line is 2y = 3x – 6 its x and y intercepts are (A) –3, 2 (B) 3, –2 (C) 2, –3 (D) –2, 3

3. The equation of line making an angle 135º with the positive x-axis and passing through a point (2, 3) will be: (A) y – x = 5 (B) y = x – 5

(C) y – x = 1 (D) y = –x + 5

4. The points where the line y = x intersect

the curve x2 + y2 = 32 are

(A) 4 2 , 4 2 (B) 4, 4

(C) 4 2 , 4 2 (D) –2, –2

5. 2d2x 1

dx

(A)2x(2x2 + 1)1/2

(B)2x(2x2 + 1)–1/2

(C)(2x2 + 1)1/2

(D)(2x2 + 1)–1/2

6. x = at2; y = 2at, then dydx

(A) t (B) 1t

(C) 1 (D) None of these

7. 2xde

dx

(A) 2xe

2x (B) 2x2xe

(C) 2xe (D)–1/2(2x)e

8.

2d (x 1)

dx x 1

(A)

2

2

x 2x – 1

(x 1) (B)

2

2

x – 2x 1

(x 1)

(C)

2x 2x – 1

x 1 (D)

2

2

x 2x 1

(x 1)

9 2 3

d 1 11

dx x x

(A) 2 3

1 1x

x x (B)

3 4

–2 3–

x x

(C)2 3

1 3x – –

x x (D)

2

–2 3–

x x

10. If y = sin2 x – 2 tan2 x, then dydx

at x = 4

is : (A) –11 (B) –7 (C) –13 (D) –15

11. d

dx [log (cos x)] is :

(A) – tan x (B) tan x

(C) cot x (D) – cot x

12. d

dx (sin x cosec x) is :

(A) sin2 x – cosec2 x (B) x

(C) 0 (D) 1

13.

d 1x logx tanx

dx x =

(A) 1 – 2

1

x + sec2 x

(B) 1 + 1

x + sec2 x

(C) 1 +2

1

x +

1

x + sec2 x

(D) 1 – 2

1

x+

1

x+ sec2 x

14. d

dx (sin 30°) is equal to :

6

(A) cos 30° (B) cosec 30° (C) 0 (D) sin 30°

15.

2d 1

xdx x

is equal to :

(A) 1 + 2

1

x (B)–1 +

2

1

x

(C)1 – 2

1

x (D)x2 – 1

True /False

16. The curve y2 = 2x is symmetric about x-axis.

17. The line 2y – 3x = 1 is perpendicular to the line 3y + 2x = 5.

18. If the temperature of an object varies

with x as T = 2x2 + 3 then the temperature gradient at x = 2 will be 8.

19. Double differentiation of displacement of w.r.t. time is called acceleration.

20. Force is directly proportional to double differentiation of displacement w.r.t. time.

7

DPP – 3

1. V = 2t – 4t2 + 6 here t is time and v is velocity then find value of time where velocity is maximum or minimum and value of velocity.

(A) 20/3 Maximum at t = ½ (B) 20/4 Minimum at t = 1/3

(C) 25/4 Maximum at t = 1/4 (D) 25/4 Minimum at t = ¼

2. A cattle is moving in a field in spiral path such that the length of the string tying the cattle is changing with time as = 2t + 3, At what rate the area covered by the cattle will be changing at t = 2.

(A) 28 (unit)2/ sec (B) 26 (unit)2/ sec

(C) 24 (unit)2/ sec (D) 22 (unit)2/ sec

3. Charge flowing in conductor as q = 2t2 + 3t + 1, here q is charge in coulomb and t is time in sec. Find the current flowing in conductor at t = 3 sec. (Current is rate of flow of charge)

(A) 5 Coulomb/sec (B) 10 Coulomb/sec

(C) 12 Coulomb/sec (D) 15 Coulomb/sec

4. Liquid is flowing in a pipe, such that the position of any fluid particle is x

= 4t2 + 3t here x is distance in meter and

t is time in sec, if area of pipe is 5 m2 then find rate of fluid flow at t = 4 sec.

(A) 25 meter/sec (B)35 meter/sec (C) 30 meter/sec (D)33 meter/sec 5. Find the rate of temperature change of

body at t = 10 sec if temperature of body vary according to T = 6t2 + 4, here T temperature in kelvin and t time in sec.

(A) 120 Kel/sec (B) 140 Kel/sec (C) 30 Kel/sec (D) 80 Kel/sec

6. What is rate of change in volume at t = 1 sec if radius of sphere varies as r = 3t2 + 2t + 1, here r in meter and t time in sec.

(A) 1152 m3/sec (B) 1512 m3/sec (C) 2152 m3/sec

(D) 152 m3/sec

7. 1

3

0

(x 1) dx

(A) 1

4 (B)

3

4

(C) 5

4 (D)

7

4

8.

/2

0

sin2x dx

(A) 2 (B) 0 (C) 1 (D) –2

9. x1

e dxx

(A) log x + x + c

(B) log x + ex + c (C) log x – x + c

(D) log x – ex + c

10.

2

0sec x dx is :

(A) 1 (B) 2 (C) 3 (D) 4

11.

2

0[sin2 cos2 ]d is

(A) 0 (B) 1 (C) 2 (D) 3

12. Value of dx

5

3

1

2x 3 is :

(A) In 139

(B) 1

2 In 13

9

(C) 1

2 In 15

9

(D) None of these

8

13. Value of 1

20

1dx

(3 – 2x) is :

(A) – 19

(B) – 2

9

(C) – 4

9 (D) None of these

14. 2(4cos t t ) dt is equal to:

(A) –4 sin t + 3t

3 + C (B) –4 sin t+t2 + C

(C) 4 sin t + 3t

3 + C (D) 4 sin t + 2t3+ C

15.

3

3

x 2dx

x is :

(A) x – 2

1

x+ c (B) x +

2

1

x + c

(C) x2 + 2x + c (D) –x + 2

1

x + c

True /False 16. A curve with positive slope will have

increasing nature 17. A curve with negative slope of slope will

be concave down in shape 18. Area under velocity line curve on time

axis is displacement 19. Direction of average velocity is always in

direction in displacement 20. Average value of sin t in it’s half time

period is 2

9

Power Booster 1 A car is moving towards a building with

speed 10 m/s. At any instant the angle of elevation of the building is 30º after 5 sec. the angle of elevation of the building becomes 45°, then height of building is.

(A)

5h

3 1 (B)

2h

3 1

(C)

2h

3 1 (D)

5h

3 1

2 A person at any instant finds angle of elevation of 40 m high building as 30° after some time the angle of elevation becomes 60° then value of distance the person moved towards the building will be :

(A) 40

d3

(B) 80

d3

(C) 60

d3

(D) 120

d3

3 What will be the displacement of a bike which moves in a circular curve of radius 30 m by 2°.

(A) 1.05 m (B) 60 m (C) 30 m (D) 2.1 m

4 V = 2t – 4t2 + 6 here t is time and v is velocity then find value of time where velocity is maximum or minimum and value of velocity.

(A) 20/3 Maximum at t = 1/2 (B) 20/4 Minimum at t = 1/3 (C) 25/4 Maximum at t = 1/4 (D) 25/4 Minimum at t = 1/4

5 A cattle is moving in a field in spiral path such that the length of the string tying the cattle is changing with time as = 2t

+ 3, At what rate the area covered by the cattle will be changing at t = 2.

(A) 28 (unit)2/ sec

(B) 26 (unit)2/ sec

(C) 24 (unit)2/ sec

(D) 22 (unit)2/ sec

6 Charge flowing in conductor as q = 2t2 + 3t + 1, here q is charge in coulomb and t is time in sec. Find the current flowing in conductor at t = 3 sec. (Current is rate of flow of charge)

(A) 5 Coulomb/sec (B) 10Coulomb/sec

(C) 12 Coulomb/sec (D)15 Coulomb/sec

7. Liquid is flowing in a pipe, such that the

position of any fluid particle is x = 4t2 + 3t here x is distance in meter and t is

time in sec, if area of pipe is 5 m2 then find rate of fluid flow at t = 4 sec.

(A) 25 meter/sec (B) 35 meter/sec (C) 30 meter/sec (D) 33 meter/sec 8. Find the rate of temperature change of

body at t = 10 sec if temperature of body

vary according to T = 6t2 + 4, here T temperature in kelvin and t time in sec.

(A) 120 Kel/sec (B) 140 Kel/sec (C) 30 Kel/sec (D) 80 Kel/sec 9. What is rate of change in volume at t = 1

sec if radius of sphere varies as r = 3t2 + 2t + 1, here r in meter and t time in sec.

(A) 1152 m3/sec (B) 1512 m3/sec

(C) 2152 m3/sec (D) 152 m3/sec

10. If + r = 12 here is length of cylinder

and r is radius of cylinder then find maximum value of volume of cylinder

(A) 156 (B) 350 (C) 256 (D) 250

11. If r = 36 here is length of cylinder and

r is radius of cylinder the n find minimum total surface area of cylinder ?

(A) 700 (B) 720

(C) 750 (D) 800

12. If × b = 32 here is length of rectangle

and b is width of rectangle then find minimum perhaps of rectangle

(A) 8 2 (B) 2 16

(C) 32 2 (D) 16 2

10

13. If y = sin2 x – 2 tan2 x, then dy

dx at x =

4

is :

(A) –11 (B) –7

(C) –13 (D) –15

14. The position of a particle moving along x-

axis varies with time t as x = 4t– t2 + 1. Find the time interval (s) during which the particle is moving along positive x-direction.

(A) 0 < t < 2 (B) t = 2 (C) 0 < t 2 (D) 0 t 2

15. An ideal gas is subjected to a

thermodynamic process PV–1 = 0.4 where P is pressure in pascal and V is

volume in (metre)3. What is the slope of the P-V curve with volume plotted

against x-axis at V = 1m3? (A) –1 (B) + 1 (C) 0.4 (D) – 0.4

16. A stone is e dropped into a quiet lake and waves move in circles spreading out radially at the speed of 0.25 m/s. At the instant when the radius of the circular wave is 4/ m, how fast is the enclosed area increasing?

(A) 4

m2/s (B)2

m2/s

(C) 2m2/s (D)4

m2/s

17. The equation of straight line having slope and y intercept of –2 will be :

(A)y = 3x 2 (B) y = 3x 2

(C)y = 3x 2 (D)y = 3x 2

18. The minimum distance of origin from line 3x – 4y + 5 = 0 is

(A) 1 (B) 2

(C) 3 (D) 4

19. What will be the slope of a line perpendicular to the line 2y + 3x = 5

(A) 3

2 (B)

3

2

(C) 2

3 (D)

2

3

20. A car is moving towards a building with speed 10 m/s. At any instant the angle of elevation of the building is 30º after 5 sec. the angle of elevation of the building becomes 45º, then height of building is.

(A)

5h

3 1 (B)

2h

3 1

(C)

2h

3 1 (D)

5h

3 1

11

ANSWER KEY

DPP 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

A C A A A A C C B C A D C A C T T F T F

DPP 2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

B C D B B B A A B B A C D C C T T T T T

POWER BOOSTER

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

D B A C A D B A A C B D B B C C B A D D

DPP 3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

C A D B A A C C B A B B D C A T T T F T

12

VECTOR (DPP – 1)

1. Which uantity can be added with a velocity

vector ? (A) Displacement (B) acceleration (C) Speed (D) none of the above

2. If B nA and

A makes an angle with

B ,

then n is: (A) a vector (B) negative and dimensionless number (C) a scalar (D) a negative scalar 3. A vector gets changed when we : (A) rotate the coordinate axis (B) slide the vector without changing its orientation (C) rotate vector by some angle other than 2 (D) all of the above

4. If A is a vector having magnitude 6 units

due west. What is the magnitude and

direction of a vector 4A .

(A) 4 units due east (B) 6 units due west (C) 24 units due west (D) 24 units due east

5. A vector A is rotated through an angle 2,

the magnitude of new vector is : (A) 2A (B) A (C) A/2 (D) 2A/3 6. A vector may change if : (A) frame of reference is translated (B) vector is rotated (C) frame of reference is rotated (D) vector is translated parallel to itself 7. A vector is not changed if: (A) it is displaced parallel to itself (B) it is rotated through an arbitrary angle

(C) it is multiplied vectorialy by a unit vector (D) it is multiplied by an arbitrary scalar 8. A particle at (1, 3, 4) moves with 10 m/s to

reach on x axis in minimum time. What is its velocity vector?

(A) 6 j + 8 k (B) 8 j + 6 k

(C) – 8 j – 6 k (D) –6 j – 8 k

9. A force ˆ ˆ ˆ(10i – 3j 6k) acts on a particle of

mass 2 kg. The magnitude of acceleration of the particle is:

(A) 12 m/ s2 (B) 4 m/ s2 (C) 6 m/ s2 (D) 8 m/ s2

10. The vectors a i + b j will be in same

direction as that of pi + j if (A) ap = b (B) bp = a (C) ap = – b (D) bp= – a

11. Three forces ˆ ˆ ˆ2i –3j 4k , ˆ ˆ ˆ8i –7j 6k and

ˆ ˆ ˆm(i – j k) produces zero resultant when

m is: (A) 10 (B) –10

(C) 20 (D) –20

12. In the given diagram, what will be

| a 2b |

| a | 3,| b | 4

(A) 97 (B) 98

(C) 99 (D) 95 13. The ratio of maximum and minimum

magnitudes of the resultant of two vectors a and

b is 2 : 1. then

| a | is equal to :

(A)

| b | (B)

2|b |

(C)

3|b | (D)

4|b|

13

14. The resultant of vectors A = 3 units and B

= 6 units is perpendicular to A. The angle between A and B is

(A) 60° (B) 120° (C) 135° (D) 145°

15. Four forces of magnitude P, 6P, 3P and 4P

act along the four sides of a suare ABCD in cyclic order. Find the resultant force :

(A) 2P (B) 3 2P

(C) 0 (D) 2 2P

True / False 16. A physical uantity cannot be called a

vector, if its magnitude is zero. 17. Magnitude and directions are necessary to

define vector uantity. 18. Pressure and surface tension are scaler.

19. Temperature gradient is a vector uantity.

20. Any physical uantity having direction is a

vector.

14

DPP – 2 1. The resultant of two vectors of magnitude

A and B inclined at an angle , becomes twice, when the inclination is changed to(180° – ). If initially their resultant was A , find the ratio of A to B :

(A) 1 / 2 (B) 5 / 7

(C) 2 / 3 (D) 7 / 2 2. ABCDEF is a regular hexagon with point O

as centre. The value of

AB AC AD AE AF is: (A)

2AO (B)

4 AO

(C)

6 AO (D) 0

3. A boy moves 20 m along a vector 3 i + 4 j and then moves “x” meter along –ve x direction, such that his final displacement is along y direction. What is the value of x ?

(A) 16 (B) 24 (C) 12 (D) 60

4. The sum of the magnitudes of two forces

acting at a point is 18N. The resultant of these forces is perpendicular to the smaller force and has magnitude of 12N. Then the smaller force is: (A) 3 (B) 5 (C) 8 (D) 4

5. Which of the following is not true for the

given geometry:

(A)

A B E 0 (B)

C–D –A

(C)

B –E – C –D (D) All of these

6. Two vectors A & B are

ˆ ˆA 3i 4jand

ˆ ˆB 7i 24j . Then a vector having the

same magnitude as B and parallel to

A

will be

(A) ˆ ˆ5i 10j (B) ˆ ˆ15i 10j

(C) ˆ ˆ15i 20j (D) ˆ ˆ5i 20j

7. What is the projection of

a over

b , If

ˆ ˆ ˆa 2i 3j k and

ˆ ˆ ˆb i 3j 4k .

(A) 3

26 (B)

3

24

(C)2

26 (D)

2

24

8. Find the component (projection) of ˆ ˆi j in

the direction of i . (A) 0 (B)1

(C) –2 (D) –1 9. Two different vectors

a and

b of same

magnitude are lying in x-y plane. Projection of

a on

b .

(A) ˆb.a (B)

ˆa.b

(C)

a.b (D) Zero

10. The projection of a vector ˆ ˆ ˆr 3i j 2k on

the x-y plane has magnitude? (A) 3 (B) 4

(C) 14 (D) 10 11. Three children Ajay, Chintu & Micky are

pulling a bat with forces of magnitudes 2N, 3 N and 2 as shown in the figure. What is the resultant force acting on the bat.

(A) 1 iN (B) 2 iN

(C) 3 iN (D) 4 iN

12. Determine the component of the vector A

= i – 3 j + 2 k along the line which passes through the points P(–1, 2, 3) and (1, –1, 2).

(A) 9

6 (B)

9

14

(C)14

9 (D) 6

9

15

13. The x and y components of vector A are

3 and 5 m respectively. The x and y

components of vector A +

B are 7 and 9

m respectively. Then, the angle made by B with x-axis is:

(A) tan–10 (B) tan–1 2

(C) tan–11 (D) tan–1 1

2 14. For what value of p, the vectors

ˆ ˆ ˆ2i j k , ˆ ˆ ˆi 2 j k and ˆ ˆ ˆ3i pj 5k are

coplanar

(A) 223

(B) 183

(C) 22

3 (D) 2

15. The angle which a vector ˆ ˆ ˆi j 2k makes with X, Y and Z axes respectively are

(A) 60°, 60°, 60° (B) 45°, 45°, 45°

(C) 60°, 60°, 45° (D) 45°, 45°, 60°

True / False 16. The resultant of two forces, each P,

acting at angle is 2P sin 2

17. If vector addition of three vectors is zero,

then they must lie in a plane. 18. The difference & sum of two vectors can

never be same 19. Three colinear vector can’t produce a null

vector 20. If three vectors of equal magnitude have

zero resultant then the angle between

any two vectors is3

16

DPP – 3

1. A force of ˆ ˆ ˆ(10i – 3j 6k) newton acts on a

block of mass 10g and displaces it from

ˆ ˆ ˆ(6i 5j – 3k) meter to ˆ ˆ ˆ(10i – 2j 7k) meter.

The work done on the block is: (A) 21 J (B) -43 J

(C) 43 J (D) 85 J

2. The value offor which vector ˆ ˆ ˆ2i 3j 8k

is perpendicular to the vector ˆ ˆ ˆ4 j 3i k ,

is : (A) – 9/4 (B) 9/4

(C) 3 (D) –3

3. For two vectors A & B, A.B = AB, So, what

will be the value of (A B)·(A –B) ?

(A) 0 (B) A2 – B2 (C) A2 + B2 (D) zero vector

4. A particle moves in the x – y plane under

the action of a force F such that the value

of its linear momentum (P) at anytime t is

Px = 2 cos t, Py = 2 sin t. The angle

between F and

P at a given time t will be

dPF

dt

(A) = 0° (B) = 30° (C) = 90° (D)= 180°

5. For two vectorsA and

B , The value of

the triple product A·(B A) is

(A) A2B (B) Zero (C) A2B sin (D) A2B cos

6. The angle that the vector ˆ ˆA 2i 3j

makes

with y-axis is : (A) tan–1 (3/2) (B) tan–1 (2/3) (C) sin–1 (2/3) (D) cos–1 (3/2)

7. If

A B C , which of the following option is wrong:

(A) Cis to

A

(B) C is to

B

(C) C is to (

A +

B)

(D) Cis to (

A×

B)

8. If

ˆ ˆ ˆA 3 i j 2k and

ˆ ˆ ˆB –2i – 2 j 4k , then

value of

| A B | will be

(A) 4 21

(B) 8 3

(C) 8 21 (D) 4 3

9. If ˆ ˆa i j ,

ˆ ˆb i – j then find a vector of

length 5 unit and perpendicular to both a

and b

(A) ˆ5k (B) ˆ5 j

(C) ˆ5 i (D) ˆ ˆ ˆ5( i j k)

3 10. For two vectors, the value of

(A B) (A – B) is always perpendicular to:

(A) A + 2

B (B)

A +

B

(C) B A (D) Both A & B

11. If ˆ ˆa i j ,

ˆ ˆb i – j then find area of the

triangle formed by vectors a and

b

(A) 0 (B) 2 (C) 1 (D) 4

12. The vectors from origin to the points A

and B are

ˆ ˆ ˆA 3 i – 6 j 2k and

ˆ ˆ ˆB 2i j – 2k

respectively. The area of the triangle OAB be

(A) 5

172

s. units (B) 2

175

s. units

(C) 3

175

s. units (D) 5

173

s. units

13. Two adjacent sides of a parallelogram are

represented by the two vectors ˆ ˆ ˆi 2 j 3k

and ˆ ˆ ˆ3i – 2 j k . What is the area of

parallelogram?

(A) 8 (B) 8 3

(C) 3 8 (D) 192

17

14. If two vectors A & B follows 2

AB|BA|

then angle between BandA is -

(A) 30° (B) 60° (C) 90° (D) 45°

15. If ˆ ˆ3 4 A i j and ˆ ˆ6 8

B i j and A and B are

the magnitudes of A and

B , then which of

the following is not true ?

(A) 0 A B (B)

1

2

A

B

(C) . 48 AB (D) A = 5

True / False 16. The angle between vectors A B

andB A

is radian.

17. Two null vectors A A and

A A have

same directions.

18. A B is perpendicular to

3A 4B .

19. Scalar product of two vectors can be a negative uantity.

20. The dot product of one vector with another vector having different dimensions may be a scalar or a vector.

18

Power Booster

1. ˆˆ ˆa b c 0 where a, b, c are unit vector

and ˆa.b – ˆ ˆb.c – ˆ ˆc.a = x. Then 4x is equal

to (A) 0 (B) 2 (C) 4 (D) – 4

2. Acceleration of a particle in a magnetic

field is given by q

a V Bm

if a merged

particle is projected in a magnetic field

ˆ ˆ ˆ2i 2j 2k tesla, then acceleration of

the particle at an instant is ˆ ˆ ˆxi 2j 6k

m/s2. value of x is (A) 4 (B) 2

(C) 3 (D) 1 3. Select incorrect statement for three

vectors

ˆ ˆ ˆa 3i 2 j k ;

ˆ ˆ ˆb i 3 j 5k and

ˆ ˆ ˆc 2i j 4k

(A) Angle between vectors a and

b is

obtuce

(B) Vector

a,b,c form right angled

triangle

(C) Vector

a,b,c form acute angled

triangle

(D)

a .b 0 and a2 c2 = b2

4. If ˆ ˆa 2i 5 j and

ˆ ˆ ˆb 5i 5 j 4k ,

then find a vector of same magnitude a

and parallel to

a b

(A) ˆ ˆ7i 2 5 j 4k

3 (B) – ˆ ˆ3i 4k

(C) ˆ ˆ9i 12k

5 (D) ˆ ˆ9i 12k

5. If a and

b are two unit vector such that

a 2b and

5a 4b are perpendicular to

each other then angular between a and

b is

(A) 45º (B) 60º

(C) cos–1

1

3 (D) cos–1

2

7

6. Two vector ˆ ˆ ˆa 3i 8 j 2k and

ˆ ˆ ˆb 6i 16 j xk are such that the

component of b perpendicular to

a is 0.

Then the value of x is (A) 8 (B) – 4

(C) +4 (D) –8 7. The displacement vector of a particle is

given as B = (t2 – 2t + 12) i + t2 j . The

time after which velocity vector and acceleration vector becomes perpendicular to each other is equal to

(A) 1 (B) 2 (C) 3 (D) 4

8. If A and

B are the components of

C ,

then:

(A) B = 3

C2

(B) A = C

2

(C) B = C

2 (D) A =

3C

2 9. The position vectors of points A, B, C and

D are A= ˆ ˆ ˆ3 i 4 j 5k , B= ˆ ˆ ˆ4 i 5 j 6k , C =

ˆ ˆ ˆ7 i 9 j 3k and D = ˆ ˆ4i 6 j , then the

displacement vectors AB and CD are (A) perpendicular (B) Parallel (C) Antiparallel (D) Inclined at an angle of 60º

10. A vector ˆOA 3i is rotated by an angle q about its starting point O in x-z plane in clockwise sense, as seen by an observer located at a point on +y -axis. The new vector will be :

19

(A) 3 cos i + 3sin i

(B) 3[cos i + 3sin k]

(C) 3[cos i – 3sin k]

(D) 3[sin i + 3cos k]

11. Let there be two vectors a and

b such

that

a b is in same direction as

a b . Select the correct alternative.

(A)

a b 0

(B)

| a | | b |

(C) Both (A) & (B) must be simultaneously true

(D)

a.b = 0

12. There are three vectors P ,

Q , and

R .

The angle between P and

Q is 60º and

R is perpendicular to the plane

containing the vectors P and

Q .

Consider the following relations.

(i)

P Q R 0

(ii)

P Q R

(iii)

P R Q The possible relations are (A) (i) & (ii) (B) (i) & (iii) (C) (ii) & (iii) (D) Only (ii) 13. Displacement versus time plot for two

particles A and B is shown below. XA XB

and YA YB refer to x and y coordinates of

particles A and B.

Velocity of particle A with respect to

particle B is

(A) ˆ ˆ0 i 0 j

(B) Dependent of time t

(C) 2 2ˆ ˆi j3 3

(D) 2 2ˆ ˆi j3 3

14. A body moving with a constant speed describes a circular path whose radius

vector is given by r 15 (cos pt i + sin

pt j ) m, where p is in rad/s, and t is in

second. What is its centripetal acceleration at t = 3 s ?

(A) 45p2 m/s2 (B) 5p2 m/s2

(C) 15p m/s2 (D) 15p2 m/s2 15. A ladder 10 m long rests against a

vertical wall with the lower end on the horizontal ground. The lower end of the ladder is pulled along the ground away from the wall at the rate of 3cm/sec. The height of the upper end while it is descending at the rate of 4 cm/sec. is

(A) 4 3 m (B) 6m

(C) 5 2 m (D) 8 m

16. If xm.yn = (x + y)m + n then dy

dx is

(A) y

x (B)

x y

xy

(C) xy (D) x

y

17. Following forces start acting on a particle at rest at the origin of the co-ordinate system simultaneously

1

ˆ ˆ ˆF 4 i 5 j 5k , 2

ˆ ˆ ˆF 5i 8 j 6k ,

3

ˆ ˆ ˆF 3i 4 j 7k and 4

ˆ ˆ ˆF 2i 3 j 2k

then the particle will move (A) In x-y plane (B) In y-z plane (C) In x-z plane (D) Along x-axis 18. A body is at rest under the action of

three forces two of which are 1

ˆF 4i ,

2

ˆF 6 j , the third force is

(A) ˆ ˆ4i 6 j (B) ˆ ˆ4i 6 j

(C) ˆ ˆ4 i 6 j (D) ˆ ˆ4 i 6 j

19. The component of

ˆ ˆ ˆA i j 5k

perpendicular to ˆ ˆ3i 4 j is

(A)

4 3ˆ ˆ ˆi j 5k

25 25 (B)

3 6ˆ ˆ ˆi j 5k25 25

(C) 4 3ˆ ˆ ˆi j 5k25 25

(D) 8 3ˆ ˆ ˆi j 5k25 25

20

20. Find the resultant of three vectors OA ,

OB and OC shown in the following figure. Radius of the circle is R.

(A) 2R (B) R(1 + 2 )

(C) R 2 (D) R( 2 1)

21

ANSWER KEY OF DPP – 1 1 2 3 4 5 6 7 8 9 10 D D C C B B A D C B 11 12 13 14 15 16 17 18 19 20 B A C B D F T T T F

ANSWER KEY OF DPP – 2 1 2 3 4 5 6 7 8 9 10 C C C B C C A D B D 11 12 13 14 15 16 17 18 19 20 A B D C C F F F F T

ANSWER KEY OF DPP – 3 1 2 3 4 5 6 7 8 9 10 C A B C B B D A A D 11 12 13 14 15 16 17 18 19 20 C A B A C T F T T F

POWER BOOSTER

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

B A C C B B B B D B C D C D B A B D C B

22

UNIT & DIMENSIONS DPP-1

1. kg m/s2 stand for the unit of :

(A) Energy (B) Acceleration (C) Force (D) Momentum

2. In the euation

2

aP

V (V – b) = constant,

the unit(s) of a is/are:

(A) N m2 (B) N m

4

(C) N m3 (D) N m

2

3. The velocity u of particles is given in terms

of time t by the euation. u = at + 2

b

t c .

The dimension of a, b and c are:

(A) L2, T, LT

2 (B) LT

2, LT, L

(C) LT–2

, LT, T2 (D) L, LT, T

2

4. The freuency (n) of vibration of a string is

given as n =l

1 T2 m

, where T is tension and

l is the length of vibration string, then the dimensional formula for m is:

(A) [M0L

1T

1] (B) [M

0L

2T]

(C) [M–1L

–1T

0] (D) [ML

–1T

0]

5. Dimensions of velocity gradient are same

as that of: (A) time period (B) freuency (C) angular acceleration (D) linear acceleration

6. Select the pair whose dimensions are same

(A) Pressure and stress (B) Stress and strain (C) Pressure and force (D) Power and force

7. Dimensional formula [ML–1T

–2] does not

represent the physical uantity (A) Young’s modulus of elasticity (B) Stress (C) Strain (D) Pressure

8. Which pair has the same dimensions?

(A) Work and power (B) Density and relative density (C) Momentum and impulse (D) Stress and strain

9. The time period of a body under S.H.M. is

represented by T = PDS

where P is pressure, D is density and S is surface tension, then values of , and are:

(Surface tension S = l

F)

(A) 3 1

, , 12 2

(B) 1

1,2,3

(C) –1, –2, 3 (D) 1 3 1

, ,2 2 2

10. The resistive force on a very-high velocity

object of area A passing through a gas of density r at a velocity v is :

Force = Ar r

s vt Then:

(A)r = 2, s = 1, t = 1 (B)r = 2, s = 1, t = 2 (C) r = 1, s = 1, t = 2 (D) r = 1, s = 1, t = 1

11. The freuency of vibration f if a mass m suspended from a spring of spring

constant k is given by relation f = cmxk

–y,

where c is a dimensionless constant. The values of x and y are:

(A) 1 1,

2 2 (B)

1 1,

2 2

(C) 1 1,

2 2 (D)

1 1,

2 2

12. Which of the following euation is dimensionally incorrect

(A) v = u + at

(B) s= ut + 12

at2

(C) v2 = u

2 – 2gs

23

(D) snth

= u+12

a(2t–1)

13. Convert gravitational constant (G) from

MKS to CGS system. (G = 6.67 × 10–11 N

m2/kg

2)

(A) 6.67 × 10–8

(B) 6.67 × 1011

(C) 6.67 × 108 (D) 6.67 × 10

–11

14. Which of the following euation can not be derived dimensionally

(A) F = 6r (B) = t

(C)

dAv

dt (D) P = gh

15. Which of the following euation can be

derived dimensionally

(A)

2

qp

v(v-b)= nRT

(B) s = ut + 12

at2

(C) v = u + at

(D) F = YAL

L True/False 16. Dimensionally correct euation may or

may not be physically correct 17. The relation v = u + at is dimensionally

correct.

18. The relation 21S ut at

2 is

dimensionally correct if S, u, t, have their usual meaning.

19. Dimensional euation is an expression for physical uantity in term of the fundamental uantities.

20. Erg is unit in MKS system

24

DPP-2 1. Find round off value of x = 7.82. (A) 7 (B) 7.9 (C) 7.8 (D) 7.7 2. Find round off value of x = 3.250 (A) 3.2 (B) 3.25 (C) 3.5 (D) 3 3. The mean length of an object is 5 cm.

Which of the following measurement is most accurate ?

(A) 4.9 cm (B) 4.805 cm (C) 5.25 cm (D) 5.4 cm 4. Which of the following measurement is

most precise ? (A) 5.00 mm (B) 5.00 cm (C) 5.00 m (D) 5.00 km 5. Error in the measurement of radius of

sphere is 1%. Then error in the measurement of volume is:

(A) 2% (B) 4% (C) 1% (D) 3%

6. If x = a2b, the maximum percentage error

in the measurement of x will be:

(A)

2 a b100% 100%

a b

(B)

2 a b100% 100%

a b

(C)

2 a b–

a b × 100%

(D)

2 a ba b

× 100%

7. A rectangular plate has length (2 ±

0.02)cm and width (1 ± 0.01)cm. The maximum percentage error in the measurement of its area is:

(A) 1% (B) 2% (C) 3% (D) 5%

8. If f = x2, then the relative error in f is :

(A) 2 xx

(B) 2( x)x

(C)

xx

(D) (x)2

9. The number of significant figure in 243.4 (A) 2 (B) 1 (C) 3 (D) 4

10. 3.1421 + 0.241 + 0.09 is eual to (after rounding off two decimal place)

(A) 3.43 (B) 3.47 (C) 3.48 (D) 3.46

11. 142.06 0.23 is eual to (Two significant figures)

(A) 35 (B) 37 (C) 39 (D) 33

12. When 96.54 is divided by 2.40, the correct result is :

(A) 40.2250 (B) 40.225 (C) 40.23 (D) 40.2

13. Given P = 0.0030 m, = 2.40 m and R = 3000m, the number of significant figures in P, , R are respectively :

(A) 1, 2, 1 (B) 2, 3, 1 (C) 4, 2, 1 (D) 4, 2, 4

14. The number of significant figures in (i) 0.03800 and (ii) 90.00 is :

(A) (i) 4 (ii) 4 (B) (i) 2 (ii) 1 (C) (i) 3 (ii) 3 (D) (i) 2 (ii) 4

15. The number of significant figures in 3.04 ×

1023

is: (A) 2 (B) 3 (C) 23 (D) 25

True/False

16. Errors are always additive in nature. 17. For greater accuracy, the uantity with

higher power should have least error. 18. Absolute error is not dimensionless

uantity. 19. Relative error is a dimensionless uantity. 20. Smaller the least count, higher is the

accuracy of measurement.

25

DPP-3 1. The dimensional formula for impulse is

same is the dimensional formula for (A) Momentum (B) Force (C) Rate of change of momentum (D) Torue

2. Which of the following euation can be derived dimensionally

(A) s = vt – 12

at2 (B) v

2 = u

2 – 2as

(C) h = 2 2r2g

(D) d

vt

3. Convert 1 joule to ergs.

(A) 105 erg (B) 10

7 erg

(C) 108 erg (D)10

2 erg

4. If force F, acceleration A and time T are basic physical uantities, the dimensions of energy are:

(A) [F2A

–1T] (B) [FAT

2]

(C) [FAT–2

] (D) [FA–1T]

5. Kilowatt-hour is unit of: (A) electrical charge (B) energy (C) power (D) force

6. The dimensional formula of latent heat is :

(A) M0L

2T

–2 (B) ML

2T

–1

(C) MLT–1 (D) M

0L

2T

–1

7. If force (F), acceleration (a) and time (T)

are used as the fundamental units, the dimensional formula for length will be :

(A) F0aT

2 (B) Fa

0T

2

(C) Fa2T

0 (D) FaT

8. If x = a – b, the maximum percentage error in the measurement of x will be :

(A)

a ba b

× 100%

(B)

a b–

a b × 100%

(C)

a ba –b a –b

× 100%

(D)

a b–

a –b a –b × 100%

9. The number of significant figure in 343.00 (A) 2 (B) 3 (C) 5 (D) 6

10. 33.3 + 3.11 + 0.313 is eual to (after rounding off one decimal place)

(A) 36.3 (B) 36.5 (C) 36.7 (D) 36.8

11. What is the fractional error in g calculated from T = 2p? Given fraction errors of T and L are ±x and ±y respectively :

(A) x + y (B) 2x – y (C) 2x + y (D) x – 2y

12. 51.028 × 1.31 is eual to (Three significant figures)

(A) 66.4 (B) 66.8 (C) 66.6 (D) 66.0

13. Subtract 0.2 J from 5.27 and express the result with correct number of significant figures :

(A) 5.1J (B) 5.06J (C) 5.0J (D) 5J

14. The number of significant figures in 4.004 (A) 4 (B) 3 (C) 2 (D) 1

15. The percentage errors in measurement of mass and speed are 1% and 2% respectively. The error in kinetic energy will be :

(A) 6% (B) 5% (C) 10% (D) 12%

True/False 16. The magnitude of physical uantity and

units are directly proportional to each other

17. Ratio of two similar physical uantity has units

18. Work = Torue is dimensionally correct but not physically.

19. In the error in measurement of distance and time are 3% and 2% respectively, error in calculation of speed is 5%.

20. In every measurement only the last significant digit is not accurately known last observation is more precise

26

Power Booster 1. If the speed of light ( ), acceleration due

to gravity ( ) and pressure ( ) are taken as the fundamental quantities, then the dimension of gravitational constant is (A) (B) (C) (D)

2. The quantity = : is the

permittivity of free space, is length, is potential difference and is time. The dimensions of are same as that of (A) Resistance (B) Charge (C) Voltage (D) Current

3. The value of Planck’s constant is

(A) 6.63 × 10 - / (B) 6.63 × 10 - /m (C) 6.63 × 10 - (D) 6.63 × 10 -

4. Given = ( / ) / , where , and are

gravitational constant, Planck’s constant and the velocity of light respectively. Dimensions of are the same as those of (A) Mass (B) Time (C) Length (D) Acceleration

5. A vernier callipers has marks on the main scale. It has equal divisions on the Vernier scale which match with

main scale divisions. For this Vernier callipers , the least count is (A) 0.02 (B) 0.05 (C) 0.1 (D) 0.2

6. If the value of the resistance is .

and the value of the current is 3.23 A, then the potential difference is 35.02935 V. its value in correct significant figures would be (A) 35 V (B) 35.0 V (C) 35.03 V (D) 35.029 V

7. = ( − ), where is speed and

is displacement. The unit of is (A) (B) (C) (D)

8. Find out dimension of 2ehc

where, e =

charge of electron , 0 = permittivity of

free space, h = plank’s constant , c = speed of light

(A) 0 0 0(M L T ) (B) 1 4 3(ML T )

(C) 1 2 1(M L T ) (D) 1 6 3(ML T )

9. The relative density of a metal may be

found by hanging a block of the metal from a spring balance and noting that in air the balance reads (3.00± 0.03) N while in water it reads (2.00±0.03)N. The Relative density would be quoted as (A) (3.00± 0.03%) (B) (3.00± 11%) (C) (3.00± 0.10%) (D) (3.00± 6%)

10. In an experiment refractive Index of glass

was observed to be 1.26.1.27, 1.28, 1.32, 1.33 and 1.34 the mean absolute error in the experiment is (A) ± 0.025 (B) ± 0.03 (C) - 0.025 (D) + 0.03

11. In the Relation ; z/kp e ,

p is

distance , k is boltzman constant and is the temperature. The dimensional formula of will be (A) [M0L0T0] (B) [M1L2T1] (C) [M1L0T-1] (D) [M0L2T0]

12. A quantity x is given by L v

t

where

is the permittivity of free space. L is

length, v is a potential difference and t is a time interval. The dimensional

formula for x is the same as that of (A) Resistance (B) Charge (C) Voltage (D) Current

13. The period of oscillation of a non linear

oscillator depends on the mass ‘m’, with dimensions of M, a restoring force constant ‘K’, with dimensions of ML–2T–2 and the amplitude A, with dimensions of L. Dimensional analysis shows that the period of oscillation should be proportional to -

(A) m

AK

(B) A2m/K

27

(C) 1 mA

K

(D) A2K3/m

14. If the speed of light (c), acceleration due

to gravity (g) and pressure (p) are taken as fundamental units, the dimensions of gravitational constant (G) are (A) c0gp–3 (B) c2g3p–2

(C) c0g2p–1 (D) c2g2p–2

15. The SI unit of the inductance, can be

written as

(A) weber/ampere2

(B) volt-second/ ampere

(C) joule/ampere

(D) ohm/second2

16. The ratio of dimensional formula of 0 to

0 where 0 denote the permittivity of the

vacuum and µ0 permeability of the

vacuum.

(A) M–2L–4T6I4 (B) M–1L–4T6I4

(C) M–2L–5T6I4 (D) M–2L–4T2I4

17. Dimensions of 4b is MxLyTzw. Find the

positive value of x + y + z + w is (Here

is Stefan constant and b is wien’s

constant)

(A) 2 (B) 4

(C) –1 (D) 0

18. If the velocity of light c, acceleration due

to gravity g, and the atmosphere

pressure P are taken as the fundamental

units. Then the unit of mass will be n2

1034 kg. Then the value of n.

(A) 8 (B) 64

(C) 9 (D) 81

19. Assuming that the largest mass that can

be moved by a flowing river depends on

velocity of flow, density of river water

and on acceleration due to gravity. The

dimension of mass is vxygz . Find the

value of xzy

(A) 18 (B) 16

(C) 14 (D) 12

20. The refractive index (µ) of water in an

experiment is recorded as 1.29, 1.33, 1.34,

1.31, 1.33 and 1.36 respectively. The mean

absolute error will be

(A) 0.06 (B) 0.04

(C) 0.02 (D) 0.01

28

ANSWER KEY OF DPP – 1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

C B C D D A C C A C A D A A D T T T T F

ANSWER KEY OF DPP – 2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

C A A A D D B A D B D D B A B T T T T T

ANSWER KEY OF DPP – 3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

A D B B B A A C C C C B A A B F F T T T

POWER BOOSTER

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

B D A C D B A A B A D D C C B A A C A C