vit - mathematics -2007 unsolved paper

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MATHEMATICS - UNSOLVED PAPER - 2007

SECTION – I

Single Correct Answer Type There are five parts in this question. Four choices are given for each part and one of them is

correct. Indicate you choice of the correct answer for each part in your answer-book by

writing the letter (a), (b), (c) or (d) whichever is appropriate

01If the normal to the curve y = f(x) at (3, 4) makes an angle with the positive x-

axis, then f' (3) is equal to :

a. -1

b.

c. 1

d.

Problem

34

34

02The function . Then the maximum value of f(x) is:

a.

b.

c.

d.

Problem

2 2xf x x e , x 0

1e

12e

2

1e

2

1e

Problem03if is equal to :

a. sin u

b. cosec u

c. 2 tan u

d. 3 tan u

2 2 u ux y sin u x y ,then x y

x y

Problem04The angle between the tangents at those points on the curve

where it meets x-axis is :

a.

b.

c.

d.

2 2x t 1 and y t t 6

1 4 tan

29

1 10 tan

49

1 5 tan

49

1 5 tan

49

Problem05The value of is equal to:

a. 2

b.

c.

d.

4

1x 3 dx

52

12

32

Problem06The area of the region bounded by the straight lines x = 0 and x = 2, and the

curves is equal to :

a.

b.

c.

d.

x 2y 2 and y 2x x

2 4log 2 3

3 4log 2 3

1 4log 2 3

4 3log 2 2

Problem07The value of is equal to:

a.

b.

c.

d.

2 20

dx

a x

2

2a

a

12a

Problem08The value of the integral is:

a.

b.

c.

d.

2X

2

1 xe dx

1 x

x2

1 xe c

1 x

x2

1 xe c

1 x

x

2

ec

1 x

xe 1 x c

Problem09If , then the value of is equal

to:

a. x

b.

c. log x

d.

y yx sin dy y sin x dx and y 1

x x 2

ycos

x

1x

xe

Problem10The differential equation of the system of all circles of radius r in the xy plane is :

a.

b.

c.

d.

2 23 22

2

dy d y1 r

dx dx

2 33 22

2

dy d y1 r

dx dx

3 22 22

2

dy d y1 r

dx dx

3 32 22

2

dy d y1 r

dx dx

Problem11The general solution of the differential equation

is given by :

a.

b.

c.

d.

23x

2

d y dy 2 y 2e

dxdx

3x

x1 2

ey c c x e

8

-3x

-x1 2

ey c c x e

8

3x

-x1 2

ey c c x e

8

3x

-x1 2

ey c c x e

8

Problem12The solution of the differential equation is :

a.

b.

c.

d.

3ydx x y dy 0

31xy y c

3

4xy y c

4 y 4xy c

34y y c

Problem13The number of integral solutions of , is :

a.

b.

c.

d.

1 2 3 ix x x 0, with x   5

152C

162C

172C

182C

Problem14Let A = {1, 2, 3, ... , n} and B = {a, b, c}, then the number of functions from A to

B that are onto is :

a.

b.

c.

d.

E

n n3 – 2

n n3 – 2 1

n3 2 1

n n3 3 2 1

Problem15Everybody in a room shakes hands with everybody else. The total number of

hand shakes is 66. The total number of persons in the room is :

a. 9

b. 12

c. 10

d. 14

Problem16In a group G = {1, 3,7, 9} under multiplication modulo 10, the inverse of 7 is :

a. 7

b. 3

c. 9

d. 1

Problem17A box contains 9 tickets numbered 1 to 9 inclusive. If 3 tickets are drawn from

the box one at a time, the probability that they are alternatively either {odd,

even, odd} or {even, odd, even} is :

a.

b.

c.

d.

517

417

516

518

Problem18 is equal to :

a.

b.

c.

d.

1 5 B 1If P A     ,P B   and P   , then P A B

12 12 A 15

89180

90180

92180

91180

Problem19If the probability density function of a random variable X is

,then P(X > 1.5 | X > 1) is equal to :

a.

b.

c.

d.

xf x  in a 0 x 2

2

716

34

712

2164

Problem20If X follows a binomial distribution with parameters n = 100 and , then P(X

=r) is maximum when r is equal to

a. 16

b. 32

c. 33

d. none of these

1p

3

Problem21If and AB =1, then B is equal to:

a.

b.

c.

d.

1 tan A

-tan 1

2sec

A

A2

A

A2

Problem22If x=-5 is a root of then the other roots are :

a. 3, 3.5

b. 1, 3.5

c. 1, 7

d. 2, 7

2x 1 4 8

2 2x 2 0

7 6 2x

Problem23The simultaneous equations Kx + 2y - z = 1,(K -I)y - 2z = 2and (K + 2) z =3 have

only one solution when:

a. k = -2

b. k = -1

c. k = 0

d. k = 1

Problem24If the rank of the matrix is 1,then the value of a is:

a. -1

b. 2

c. -6

d. 4

1 2 5

2 -4 a-2

1 -2 a+1

Problem25If ,then all the roots of the equation

will be real if:

a. b>0,a<0,c>0

b. b<0,a>0,c>0

c. b>0,a>0,c>0

d. b>0,a>0,c<0

2 4 2b 4ac for the equation ax bx c 0

Problem26If x>0 and ,then x

equals:

a. 9

b. 81

c. 1

d. 27

8 1643 3 3 3 3log x log x +log x +log x +log x . 4

Problem27the number of real roots of equation is:

a. 0

b. 2

c. 4

d. 6

31 1

x x 0x x

Problem28If H is the harmonic mean between P and Q, then the value of is:

a. 2

b.

c.

d.

H HP Q

PQP Q

12

P QPQ

Problem29If is a unit vector, then is:

a.

b.

c.

d.

a . b b b x a x b

b

2

a b

a.b a

a

b

Problem30If is the angle between the lines AB and AC where A, Band C are the three

points with coordinates (1, 2, -1), (2, 0, 3), (3, -1, 2) respectively, then

is equal to:

a. 20

b. 10

c. 30

d. 40

462 cos

Problem31Let the pairs each determine a plane. Then the planes are

parallel, if:

a.

b.

c.

d.

a,b and c, d

a x c x b x d 0

a x c . b x d 0

a x b x c x d 0

a x b . c x d 0

Problem32The area of a parallelogram with as diagonals

is :

a.

b.

c.

d.

ˆ ˆˆ ˆ ˆ ˆ3i j 2k and i 3j 4k

72

73

74

75

Problem33 If

is equal to:

a. 2

b. (b)1

c. (c) -1

d. (d)0

2 12 10 8 6cos x cos x 1, then the value of sin x 3 sin x 3 sin x sin x 1

Problem34The product of all values of is:

a. 1

b.

c.

d.

cos i sin

cos 3 i sin 3

cos 3 i sin 3

3/5cos i sin

Problem35The imaginary part of is:

a.

b. 0

c.

d.

21 i

i 2i 1

45

25

45

Problem36If , then is equal to :

a.

b.

c.

d.

1 1sin x sin y2

1 1cos x cos y

2

4

34

Problem37The equation of a directrix of the ellipse is:

a. 3y = 5

b. y = 5

c. 3y = 25

d. y = 3

2 2x y + 1

16 25

Problem38If the normal at on the parabola , meets the

parabola again at , then:

a.

b.

c.

d.

2ap , 2ap 2y 4ax

2aq , 2aq

2p pq 2 0

2p - pq 2 0

2q pq 2 0

2p pq 1 0

Problem39The length of the straight line x - 3y = 1 intercepted by the hyperbola

is :

a.

b.

c.

d.

2 2x 4y 1

10

65

1

10

610

5

Problem40The curve described parametrically by , y = 3t + 5

represents:

a. an ellipse

b. a hyperbola

c. a parabola

d. a circle

2x t 2t 1

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