20 9 B. 20 4

1. What is the value of log7 log7 7 7 7 equal

to?

A. 3 log2 7

B. 1 ? 3 log2 7

7

20 A. 9

9 C. 4

9 B. 20

4 D. 9

C. 1 ? 3 log7 2

D. 8

Consider the information given below and

2. If an infinite GP has the first term x and answer the two (02) items that follow:

the sum 5, then which one of the following

is correct?

A. x < - 10

B. - 10 < x < 0

C. 0 < x < 10

D. x > 10

A survey was conducted among 300 students. It was found that 125 students like to play cricket, 145 students like to play football and 90 students like to play

3. Consider the following expression :

tennis. 32 students like to play exactly two

1.

x

x2

1 x

games out of the three games. 8. How many student like to play ail the three

2.

ax2

bx

x

c

d x

e x2

3. 3x2 ? 5x + ab

2

4. x2 ax b3

game?

A. 14

B. 21

C. 28

D. 35

9. How many student like to play exactly only

one game?

A. 196

B. 228

12 5. x x 5 Which of the above are rational expressions? A. 1, 4 and 5 only B. 1, 3, 4 and 5 only C. 2, 4 and 5 only

C. 254

D. 268

10. If and ( 0) are the roots of the quadratic equation x2 + x - = 0, then the quadratic expression ?x2 + x + where x R has

1 A. Least value 4

D. 1 and 2 only

4. A square matrix A is called orthogonal if

A. A = A2

B. A' = A-1

C. A = A-1

D. A = A'

Where A' is the transpose of A.

9 B. Least value 4

1 C. Greatest value 4

5. If A, B and C are subsets of a Universal

9

set, then which one of the following is not

D. Greatest value 4

correct?

11. What is the coefficient of the middle term

A. A (B C) = (A B) (A C)

in the binomial expansion of (2 + 3x)4?

B. A' (A B) = (B' A)' A

A. 6

B. 12

C. A' (B C) = (C' B)' A'

C. 108

D. 216

D. (A B) C = (A C) (B C)

12. For a square matrix A, which of the

Where A' is the complement of A.

following properties hold?

6. Let x be the number of intgers laying

1. (A-1)-1 = A

between 2999 and 8001 which have at least two digits equal. Then x is equal to

A. 2480

B. 2481

C. 2482

D. 2483

7.

The

sum

of

the

series

3

1

1 3

1 9

......

is

2.

det A1

1 det A

3. (A)-1 = A-1 where is a scalar

Select the correct answer using the code

given below :

A. 1 and 2 only

B. 2 and 3 only

equal to

C. 1 and 3 only

D. 1, 2 and 3

13. Which one of the following factors does the

expansion

of

the

determinant

xy3

x2 5y3 9

x3 10y5 27 contain?

A. x ? 3

B. x ? y

C. y ? 3

D. x ? 3y

14. What is the adjoint of the matrix

cos sin sin cos ?

cos sin

cos sin

A.

sin

cos

B.

sin

cos

cos sin

C.

sin

cos

cos sin

D.

sin

cos

15. What

is

the

value

of

1 i 2

3

3n

1 1i 2

3

3n

,

where i

1 ?

A. 3

B. 2

C. 1

D. 0

16. There are 17 cricket players, out or which 5

players can bowl. In how many ways can a

team of 11 players be selected so a Lo

Include 3 bowlers?

A. C (17, 11)

B. C (12, 8)

C. C (17, 5) ? C(5, 3)

D. C(5, 3) ? C (12, 8)

17. What is the value of log9 27 + log8 32?

7

19

A. 2

B. 6

C. 4

D. 7

18. If A and B are two invertible square

matrices of same order, then what is (AB)-1

equal to?

A. B-1 A-1

B. A-1 B-1

C. B-1 A

D. A-1 B

19. IF a + b + c = 0, then one of the solutions

ax c

c bx

of b

a

A. x = a

b

a 0

cx

is

B.

x

3 a2 b2 c2 2

C.

x

2 a2 b2 c2 3

D. x = 0

20. What should be the value of x so that the

2 4

matrix

8

x

does not have an inverse?

A. 16

B. -16

C. 8

D. -8

21. The system of equations

2x + y ? 3z = 5,

3x - 2y + 2z = 5 and

5x ? 3y - z = 16

A. is inconsistent

B. is consistent, with a unique solution

C. is consistent, with infinitely many

solutions

D. has its solution tying along z-axis in

three-dimensional space

22. Which one of the following is correct in

respect of the cube roots of unity?

A. They are collinear

B. They lie on a circle of radius 3

C. They form an equilateral triangle

D. None of the above

23. If u, v and w (all positive) are the pth, qth and rth terms of a GP, then the determinant

lnu p 1

ln v

q 1

of the matrix ln w r 1 is

A. 0

B. 1

C. (p ? q) (q ? r) (r ? q)

D. ln u ? ln v ? ln w

24. Let the coefficient of the middle term of the

binomial expansion of (1 + x)2n be and

those of two middle terms of the binomial

expansion of (1 + x)2n ? 1 be and . Which

one of the following relations is correct?

A. > +

B. < +

C. = +

D. =

25. Let A = {x R : - 1 x 1}.

B = {y R : - 1 y 1} and S be the

subset of A ? B, defined by

S = [(x, y) A ? B : x2 + y2 = 1].

Which one of the following is correct?

A. S is a one-one function from A into B

B. S is a many-one function from A into B

C. S is a bijective mapping from A into B

D. S is not a function

26. Let Tr be the rth term of an AP for r = 1, 2, 3, ...... . If for same distinct positive

integers m and n we have Tm = 1/n and

Tn = 1/m, then what is Tmn equal to?

A. (mn)-1

B. m-1 + n-1

C. 1

D. 0

27. Suppose f(x) is such a quadratic expression

that it, is positive for all real x.

If g(x) = f(x) + f'(x) + f"(x), then for any

real x

A. g(x) < 0

B. g(x) > 0

C. g(x) = 0

D. g(x) 0

28. Consider the following in respect of

matrices A, B and C of same order :

1. (A + B + C)' = A' + B' + C'

2. (AB)' = A'B'

3. (ABC)' = C'B'A'

where A' is the transpose of the matrix A.

Which of the above are correct?

A. 1 and 2 only

B. 2 and 3 only

C. 1 and 3 only

D. 1, 2 and 3

29. The sum of the binary numbers (11011)2, (10110110)2 and (10011x0y)2 is the binary number (101101101)2. What are the values of x and y?

A. x = 1, y = 1

B. x = 1, y = 0

C. x = 0, y = 1

D. x = 0, y = 0

30. Let matrix B be the adjoint of a square

matrix A, l be the identity matrix of same

order as A. If k ( 0) is the determinant of

the matrix A, then what is AB equal to?

A. l

B. kl

C. k2l

D. (1/k)l

31. If (0.2)x = 2 and log10 2 = 0.3010, then what is the value of x to the nearest tenth?

A. -10.0

B. -0.5

C. -0.4

D. -0.2

32. The total number of 5-digit numbers that

can be composed of distinct digits from 0

to 9 is

A. 45360

B. 30240

C. 27216

D. 15120

33. What is the determinant of the matrix

x y y z

z

x

z x

y z x y ?

A. (x ? y) (y ? z) (z ? x)

B. (x ? y) (y ? z)

C. (y ? z) (z ? x)

D. (z ? x)2 (x + y + z) 34. If A, B and C are the angles of a triangle

1

1

1

1 sin A

1 sinB

1 sin C 0,

and sin A sin2 A sinB sin2 B sin C sin2 C

then which one of the following is correct?

A. The triangle ABC is isosceles

B. The triangle ABC is equilateral

C. The triangle ABC is scalene

D. No conclusion can be drawn with regard to the nature of the triangle

35. Consider the following in respect of matrices A and B of same order :

1. A2 ? B2 = (A + B) (A ? B)

2. (A ? I) (I + A) = O A2 = I

Where I is the identity matrix and O is the null matrix.

Which of the above is/are correct?

A. 1 only

B. 2 only

C. Both 1 and 2

D. Neither 1 nor 2

2 tan 36. What is 1 tan2 equal to?

A. cos 2

B. tan 2

C. sin 2

D. cosec 2

37. If sec ( - ), sec and sec ( + ) are in AP, where cos 1, then what is the value of sin2 + cos ?

A. 0

B. 1

1

C. -1

D. 2

38. If A + B + C = 180o, then what is sin 2A ?

sin 2B ? sin 2C equal to?

A. - 4 sin A sin B sin C

B. - 4 cos A sin B cos C

C. - 4 cos A cos B sin C

D. - 4 sin A cos B cos C

39. A balloon is directly above one end of a

bridge. The angle of depression of the

order end of the bridge from the balloon is

48o. If the height of the balloon above the

bridge is 122 m, then what is the length of

the bridge?

A. 122 sin 48o mB. 122 tan 42o m

C. 122 cos 48o m D. 122 tan 48o m

40. A is an angle in the fourth quadrant. It

satisfies the trigonometric equation

3(3 ? tan2 A ? cot A)2 = 1. Which one of

the following is a value of A?

A. 300o

B. 315o

C. 330o

D. 345o

41. The top of a hill observed from the top and bottom of a building of height h is at

angles of elevation 6 and 3 respectively.

What is the height of the hill?

3h

A. 2h

B. 2

h

C. h

D. 2

42. What is/are the solution(s) of the

trigonometric equation cosec x cot x 3, where 0 < x < 2?

5 A. 3 only

B. 3 only

C. only

D.

,

3

,

5 3

43.

If

8

,

then

what

is

the value of (2

cos

+ 1)10 (2 cos 2 ? 1)10 (2 cos ? 1)10 (2

cos 4 ? 1)10?

A. 0

B. 1

C. 2

D. 4

44. If cos and cos (0 < < < ) are the roots of the quadratic equation 4x2 ? 3 =0, then what is the value of sec ? sec ?

4 A. 3

4 B. 3

3 C. 4

3 D. 4

45. Consider the following values of x :

1. 8

2. -4

1

3. 6

1 4. 4

Which of the above values of x is/are the solution(s) of the equation

tan1 2x

tan1 3x

4

?

A. 3 only

B. 2 and 3 only

C. 1 and 4 only

D. 4 only

46. If the second term of a GP is 2 and the sum of its infinite terms is 8, then the GP is

A.

8,

2,

1 2

,

1 8

,....

B.

10,

2,

2 5

,

2 25

,.....

C.

4, 2,1,

1 2

,

1 2 2

,....

D.

6, 3,

3 2

,

3 4

,....

ab 47. If a, b, c are in AP or GP or HP, then b c

is equal to

b

b

cc

A. a or 1 or c

B. a or b or 1

aa

ac

C. 1 or b or c

D. 1 or b or a

48. What is the sum of all three-digit numbers that can be formed using all the digits 3,4 and 5, when repetition of digits is not allowed?

A. 2664

B. 3882

C. 4044

D. 4444

49. The ratio of roots of the equations ax2 + bx + c = 0 and px2 + qx + r = 0 are equal. If D1 and D2 are respective discriminants, then what is equal to?

a2

A. p2

b2

B. q2

c2

C. r2

D. None of the above

50. If A = sin2 + cos4 , then for all real , which one of the following is correct?

A. 1 A 2

B.

3 4

A

1

C.

13 16

A

1

D.

3 4

A

13 16

51. The equation of a circle whose end points of a diameter are (x1, y1) and (x2, y2) is

A. (x ? x1)(x ? x2) + (y ? y1)(y ? y2) = x2 + y2

B. (x ? x1)2 + (y ? y1)2 = x2y2 C. x2 + y2 + 2x1x2 + 2y1y2 = 0 D. (x ? x1)(x ? x2) + (y ? y1)(y ? y2) = 0 52. The second degree equation x2 + 4y2 ? 4y + 2 = 0 represents

A. A point

B. An ellipse of semi-major axis 1

3 C. An ellipse with eccentricity 2 D. None of the above 53. The angle between the two lines lx + my + n = 0 and l'x + m'y + n' = 0 is given by tan-1 . What is equal to?

lm ' l ' m A. ll ' mm'

lm' l ' m B. ll ' mm'

lm ' l ' m

lm' l ' m

C. ll ' mm'

D. ll ' mm'

54. Consider the following statements :

1. The distance between the lines y = mx

c1 c2 . + c1 and y = mx + c2 is 1 m2 2. The distance between the lines ax + by

+ c1 = 0 and ax + by + c2 = 0 is

c1 c2

a2 b2 .

3. The distance between the lines x = c1 and x = c2 is |c1 ? c2|.

Which of the above statements are correct?

A. 1 and 2 only

B. 2 and 3 only

C. 1 and 3 only

D. 1, 2 and 3

55. What is the equation of straight line

passing through the point of intersection of

the lines

x 2

y 3

1

and

x 3

y 2

1,

and

parallel to the line 4x + 5y ? 6 = 0?

A. 20x + 25y ? 54 = 0

B. 25x + 20y ? 54 = 0

C. 4x + 5y ? 54 = 0

D. 4x + 5y ? 45 = 0

56. What is the distance of the point (2, 3, 4)

from the plane 3x - 6y + 2z + 11 = 0?

A. 1 unit

B. 2 units

C. 3 units

D. 4 units

57. Coordinates of the points O, P, Q and R are

respectively (0, 0, 0), (4, 6, 2m), (2, 0,

2n) and (2, 4, 6). Let L, M, N and K be

points on the sides OR, OP, PQ and QR

respectively such that LMNK is a

parallelogram whose two adjacent sides LK

and LM are each of length 2. What are

the values of m and n respectively?

A. 6, 2

B. 1, 3

C. 3, 1

D. None of the above

58.

The line

x 1 2

y 2 3

z 3 4

is given by

A. x + y + z = 6, x + 2y ? 3z = - 4

B. x + 2y ? 2z = -1, 4x + 4y ? 5z ? 3 = 0

C. 3x + 2y ? 3z = 0, 3x ? 6y + 3z = - 2

D. 3x + 2y ? 3z = - 2, 3x ? 6y + 3z = 0

59. Consider the following statements :

1. The angle between the planes 2x ? y +

z = 1 and x + y + 2z = 3 is 3 .

2. The distance between the planes 6x ? 3y + 6z + 2 = 0 and 2x ? y + 2z + 4

10

0 is 9 .

Which of the above statements is/are correct?

A. 1 only

B. 2 only

C. Both 1 and 2

D. Neither 1 nor 2

60. Consider the following statements :

Statement I : If the line segment joining the points P(m, n) and Q(r, s) subtends an angle at the origin, then

cos

ms nr

m2 n2 r2 s2 .

Statement II : In any triangle ABC, it is

true that a2 = b2 + c2 ? 2bc cos A.

Which one of the following is correct in

respect of the above two statements?

A. Both Statement I and Statement II are

true and Statement II is the correct

explanation of Statement I

B. Both Statement I and Statement II are

true, but Statement II is not the

correct explanation of Statement I

C. Statement I is true, but Statement II is

false

D. Statement I is false, but Statement II

is true

61. What is the area of the triangle with

vertices ?

A. |(x1 ? x2)(x2 ? x3)(x3 ? x1)|

B. 0

x1 x2 x2 x3 x3 x1

C.

x1x2x3

x1 x2 x2 x3 x3 x1

D.

2x1x2x3

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