This document contains an unsolved mathematics paper from 2008 with 40 multiple choice questions. The questions cover topics in vectors, geometry, trigonometry, calculus, differential equations, probability, and matrices. The correct answers to each question are labeled with letters a-d.

1. VIT – PAST PAPERS
MATHEMATICS - UNSOLVED PAPER - 2008

2. 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

3. 01 Problem
      1   
If a, b, c be three unit vectors such that a x b x c b, b and c being non-
2
 
parallel. If 1 is the angle between is a and b and 2 the angle between
 
a and c , then
a. 1 , 2
6 3
b. 1 , 2
3 6
c. 1 , 2
2 3
d. 1 , 2
3 2

4. 02 Problem
2   
The r r . c h 0, c h , represents
a. circle
b. ellipse
c. cone
d. sphere

5. 03 Problem
The simplified expression of sin tan 1 x , for any real number x is given by
1
a.
1 x2
b.
1 x2
1
c.
1 x2
1
d.
1 x2

6. 04 Problem
If z 25 5 , find the value of
z .
z 1
a. 3
b. 4
c. 5
d. 6

7. 05 Problem
Argument of the complex number 1 3i is:
2 i
a. 45
b. 135
c. 225
d. 240

8. 06 Problem
In a triangle ABC, the sides band c are the roots of the equation
4
x2 61x 820 0 and A tan 1
, then a2 is equal to:
3
a. 1098
b. 1096
c. 1097
d. 1095

9. 07 Problem
The shortest distance between the straight lines through the points
A1 6, 2, 2 and A2 4, 0, 1 , in the directions of (1,-2, 2) and
(3, - 2,- 2) is
a. (a) 6
b. (b) 8
c. (c) 12
d. (d) 9

10. 08 Problem
2
The centre and radius of the sphere x y2 z2 3x 4z 1 0 are:
a. 3 21
, 0, 2 ;
2 2
3
b. , 0, 2 ; 21
2
3 21
c. , 0, 2 ;
2 2
3 21
, 2, 0 ;
d. 2 2

11. 09 Problem
Let A and B are two fixed points in a plane, then locus of another point C on the
same plane such that CA+ CB = constant, (> AB) is
a. circle
b. ellipse
c. parabola
d. nyperbola

12. 10 Problem
The directrix of the parabola y2 4x 3 0 is
a. 4
x 0
3
4
b. x 0
3
3
c. x 0
4
1
d. x 0
4

13. 11 Problem
If g(x) is a polynomial satisfying g(x) g(y) = g(x) + g(y)+ g(xy) - 2 for all real x and y
and g(2) = 5, then lim g x is:
x 3
a. 9
b. 10
c. 25
d. 20

14. 12 Problem
The value of f(0) so that ex 2x may be x continuous at x = 0 is
x
a. log 1
2
b. 0
c. 4
d. - 1 + log 2

15. 13 Problem
Let [ ] denotes the greatest integer function and f x tan2 x Then,
a. lim f x does not exist
x 0
b. f(x) is continuous at x = 0
c. f (x) is not differentiable at x = 0
d. f(x) = 1

16. 14 Problem
A spherical balloon is expanding. If the radius is increasing at the rate of 2
cm/min, the rate at which the volume increases (in cubic centimeters per minute)
when the radius, is 5 cm, is
a. 10
b. 100
c. 200
d. 50

17. 15 Problem
2
The length of the parabola y 12x cut off by the latus rectum is
a. 6 2 + log 1 2
b. 3 2 log 1 2
c. 6 2 - log 1 2
3 2 - log 1 2
d.

18. 16 Problem
If I x5
dx , then I is equal to
1 x3
5 3
a. 2 1 x 3 2 2
1 x 3 2
c
9 3
b. log x 1 x3 c
c. log x 1 x3 c
3 1
2 3 2 2 3 2
d. 1 x 1 x c
9 3

19. 17 Problem
2
Area enclosed by the curve 4 x 2 y2 8 is:
a. sq unit
b. 2 sq unit
c. 3 sq unit
d. 4 sq unit

20. 18 Problem
a a x
The value of dx is:
0 x
a. a
2
a
b.
4
a
c.
2
a
d.
4

21. 19 Problem
Lety be the number of people in a village at time t. Assume that the rate of
change of the population is proportional to the number of people in the village
at any time, and further assume that the population never increases in time.
Then, the population of the village at any fixed time t is given by
a. y ekt c, for some constants, c 0 and k 0
b. y ce ct , for some constants c 0 and k 0
c. y ect k, for some constants c 0 and k 0
y ke ct , for some constants c 0 and k 0
d.

22. 20 Problem
The differential equation of all straight lines touching the circle x 2 y2 a2
is:
2 2
dy 2 dy
a. y a 1
dx dx
2 2
dy 2 dy
y a 1
b. dx dx
dy dy
c. y x a2 1
dx dx
dy dy
d. y a2 1
dx dx

23. 21 Problem
The differential equation dy admits
y 3 0
dx
a. infinite number of solutions
b. no solutions
c. a unique solution
d. many solutions

24. 22 Problem
Solution of the differential equation x dy y dx x2 y2 dx 0 is:
a. y x2 y2 cx2
b. y+ x2 y2 cx2
c. y+ x2 y2 cy2
d. x x2 y2 cy2

25. 23 Problem
Let p, q, rand s be statements and suppose that p q r p If ~s r , then
a. s ~q
b. ~q ~s
c. ~s ~q
d. q ~s

26. 24 Problem
In how many number of ways can 10 students be divided into three teams, one
containing four students and the other three?
a. 400
b. 700
c. 1050
d. 2100

27. 25 Problem
If R be a relation defined as aRb iff a b 0 ,then the relation is
a. reflexive
b. symmetric
c. transitive
d. symmetric and transitive

28. 26 Problem
Let S be a finite set containing n elements. Then the total number of
commutative binary operation on S is
n n 1
a. 2
n
nn 1
b. 2
n
c. n2
n
n2
d. 2

29. 27 Problem
A manufacturer of cotter pins knows that 5% of his product is defective. He sells
pins in boxes of 100and guarantees that not more than one pin will be defective
in a box. In order to find the probability that a box will fail to meet theguaranteed
quality, the probability distribution one has to employ is
a. binomial
b. poisson
c. normal
d. exponential

30. 28 Problem
The probability that a certain kind ot component will survive a given shock test
3
is 4
.The probability that exactly 2 of the next 4 components tested survive is
9
a.
41
25
b. 128
1
c. 5
27
d. 128

31. 29 Problem
Mean and standard deviation from the following observations of marks of 5
students of a tutorial group (marks out of 25) 8 12 13 15 22 are
a. 14, 4.604
b. 15, 4.604
c. 14, 5.604
d. None of these

32. 30 Problem
A random variable X follows binomial distribution with mean
and variance . Then
a. 0
b. 0
c. 0
d. 0

33. 31 Problem
31.The system of equations
x + y + z=0
2x + 3y + z = 0
And x + 2y = 0
Has
a. a unique solution; x = 0,y =0, z = 0
b. infinite solutions
c. no solution
d. finite number of non-zero solutions

34. 32 Problem
4
0 a , then
b 0
a. a = 1 = 2b
b. a = b
c. a b2
d. ab = 1

35. 33 Problem
If D diag d1 , d2 , dn , where di 0 , for i = 1, 2, ... , n, then is equal to
a.
b. D
c. adj (D)
d. diag (d 1 , d 1 ,........d 1)
1 2 n

36. 34 Problem
a b-y c-z
If x,y, z are different from zero and a-x b c-z
=0,,then the value of
a b c a-x b-y c
expression is:
x y z
a. 0
b. -1
c. 1
d. 2

37. 35 Problem
Probability of getting positive integral roots of the equation x2 n 0 for the
integer n 1 n 40 , is:
1
a. 5
1
b. 10
3
c. 20
1
d. 20

38. 36 Problem
The number of real roots of the equation x 4 x4 20 22 is:
a. 4
b. 2
c. 0
d. 1

39. 37 Problem
Let , be the roots of the equation
x2 ax b 0 and A n n n
then A n+1 -aA n +bA n-1 is equal to:
a. - a
b. b
c. 0
d. a - b

40. 38 Problem
If the sides of a right angle triangle form an AP, the 'sin' of the acute angles are
a. 3 4
,
5 5
1
b. 3,
3
5 1 5 1
,
c. 2 2
3 1 3 1
,
d. 2 2

41. 39 Problem
The plane through the point (- 1, - 1, - 1) and containing the line of intersection of
 
r ˆ
i ˆ ˆ
3j – k 0 and r ˆ
j ˆ
2k 0
the planes is:

i ˆ ˆ
r. ˆ 2j + 3k 0
a.

i j ˆ
r. ˆ 4ˆ + k 0
b.

c. i j ˆ
r. ˆ 5ˆ 5k 0

d. r. ˆ ˆ ˆ
i j+3k 0

42. 40 Problem
 
a ˆ
i ˆ
j ˆ and b
k ˆ
2i ˆ
4j ˆ
3k are one of the sides and medians
respectively, of a triangle through the same vertex, then area of the triangle is
1
a. 83
2
b. 83
1
c. 85
2
d. 86

43. FOR SOLUTION VISIT WWW.VASISTA.NET