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Q.1) Suppose A represents the symbol 1, B represents the symbol 0, C represents the symbol 1, D

represents the symbol 0 and so on. If we divide INDIA by AGRA, then which one of the

following is the remainder in Binary Representation?

(a) (1101)2 (b) (101)2

(c) (11)2 (d) (110)2

Q.2) How many number of digits are there in 298? (Given that log102 = 0.30103)

(a) 98 (b) 99

(c) 30 (d) 29

Q.3) What is the area of the triangle on the Argand diagram formed by the complex number z, -iz,

z – iz?

(a) | z |

2

(b) 2| z |

2

(c)

2

2

z (d)

2

4

z

Q.4) In a quadratic equation, with leading coefficient 1. Sheela reads the coefficient 16 of x wrongly as

19 and obtains the roots as -15 and -4. Which of the following are the correct roots of the

equation?

(a) 8, 8 (b) 6, 10

(c) -6, -10 (d) 12, 5

Q.5) Out of 40 children 30 can swim, 27 can play chess and only 5 can do neither. How many children

can swim only?

(a) 30 (b) 22

(c) 12 (d) 8

Q.6) Both the roots of a quadratic equation x

2

– mx + 121 = 0 are greater than 10. What is the minimum

value of m?

(a) 21 (b) 22

(c) 23 (d) Cannot be determined

Q.7) What is the simplified representation of(A' ' ∩∩ ∪ ∩ ∩ B C BCAC ) ( )( ) where A, B, C are subsets

of set X?

(a) A (b) B

(c) C (d) X ∩ ∪∪ (ABC)

Q.8) Which one of the following is the domain of the relation R defined on the set N of natural numbers

as

R mn m n mn N = += ∈ { } ( ) , :2 3 30; , ?

(a) {2, 4, 6, 8} (b) {3, 7, 11, 15}

(c) {3, 6, 9, 12} (d) {3, 6, 9, 12, 15}

Q.9) How many terms are there in the expansion of ( )10

x+ +y z ?

(a) 11 (b) 33

(c) 66 (d) 310

Q.10) Which one of the following is correct?

(a) The relation R0 defined on the set of Real number as R0 = {(a, b such that a2 + b2 = 1 for all

a, b ∈ R} is an equivalence relation.

(b) The relation R0 defined on the set of Real numbers as R0 = {(a, b) such that |a-b|}<

1

3

for all

a, b ∈ R} is an equivalence relation.

(c) The relation I0 defined on the set of integers as 2 2

102 1 12 2 III I II I :3 20 − + = for all

1 2 I ,I I ∈ is an equivalence relation.

(d) We defined AEC B by the open sentence: A is cardinally equivalent to B on the family of

sets. Then the relation EC on family of sets is an equivalence relation.

Q.11) How many arrangements can be made out of the letters of the word “MOTHER” taken four at a

time so that each arrangement contains the letter ‘M’?

(a) 240 (b) 120

(c) 60 (d) 360

Q.12) What are the value of k if the term independent of x in the expansion of

10

2

k x x

is 405?

(a) +3 (b) +6

(c) +5 (d) +4

Q.13) If the equation (1) x

2

– px + q = 0 and (2) x

2

– rx + x = 0 have a root in common and the second

equation has equal roots then q + s is equal to which one of the following?

(a) 2

pr (b) 2pr

(c)

pr (d) p2

r

Q.14) If the pth, qth and r

th terms of a G.P. are again in G.P., then which one of the following is correct?

(a) p, q, r are in A.P. (b) p, q, r are in G.P.

(c) p, q, r are in H.P. (d) p, q, r are neither in A.P. nor in G.P. nor in H.P.

Q.15) In how many ways can 10 lions and 6 tigers be arranged in a row so that no two tigers are

together?

(a) 11

6 10!× P (b) 10

6 10!× P

(c) 10

7 6!× P (d) 10

6 6!× P

Q.16) The Geometric Mean of two numbers is 6. Their Arithmetic Mean A and Harmonic Mean H

satisfy the equation 90A + 5H = 918.

Which one of the following is correct?

(a) A = 10 (b) A = 1/5, A = 10

(c) A = 5, A = 10 (d) A = 1/5, A = 5

Q.17) Eighteen football teams take part in the national championship and every team meets the same

opponent twice. How many matches are played during the championship?

(a) 306 (b) 300

(c) 72 (d) 153

Q.18) If x, y, z are distinct positive numbers different from 1 such that

( )( ) log .log log log .log log log .log log 0 yz x xz y xy z xx x yy y zz z − + −+ −= ( ) , which is the value

of xyz?

(a) 2 (b) 1 (c) -1 (d) 0

Q.19) Which one of the following is the union of the closed sets 1 1 2 ,10 , 1, 2,...? n n n

⎛ ⎞ ⎜ ⎟ +− = ⎝ ⎠

(a) [2, 10] (b) (2, 10)

(c) [2, 10) (d) (2, 10]

Q.20) Indicate which one of the points given below is not a part of the solution of the inequation 3x + 4y

> 12: x > 0, y > 0?

(a) (4, 2) (b) (1, 3)

(c) (2, 1) (d) 2 11 ,

3 4

Q.21) If A is a square matrix of order m × n and k is a scalar, then adj (kA) is equal to which one of the

following?

(a) k adj A (b) k

2

adj A

(c) k

n-1 adj A (d) k

n

adj A

Q.22) For what value of k, the system of linear equations x y z x y z x y kz + += +−= + + = 2, 2 3,3 2 4

has a unique solution?

(a) k is any real number other than zero

(b) k is any real number

(c) k is any integer

(d) k = 0

Q.23) If the determinant of a non-singular matrix

111

222

333

abc

abc

abc

is denoted by Δ, what is the determinant

of the matrix

1 1 11 1

2 2 22 2

3 3 33 3

24 4

34 4

34 4

a b cb c

a b cb c

a b cb c

⎡ ⎤ + −

⎢ ⎥ + −

⎣ ⎦ + −

?

(a) Δ (b) - Δ

(c) 7 Δ (d) 4 Δ

Q.24) What is the value of the determinant

2

2

2

1

1

1

a a bc

b b ca

c c ab

−

−

−

?

(a) abc (b) ab + bc + ca

(c) 0 (d) (a – b) (b – c) (c – a)

Q.25) If A is non-singular matrix of order n × m, then which one of the following is equal to |adj A|?

(a) | A |

n+1 (b) | A |

n

(c) | A |

n-1 (d) | A |

Q.26) If

x

M y

z

and G = [3, 4, 5] are two matrices, then which one of the following is correct?

(a) In the product matrix MG = [hij], the element h32 is 5y.

(b) In the product matrix MG = [hij], the element h32 is 3x + 4y + 5z.

(c) In the product matrix MG = [hij], the element h32 is 4z. (d) In the product matrix MG = [hij], the element h32 does not exist.

Q.27) If 0 1

0 0

A

, then which one of the following is equal to ( )n

aI bA + ?

(a) n n 1 A I na bA − + (b) n n 1 a na bA − +

(c) n n 1 a I a bA − + (d) n nn aI bA +

Q.28) If A is a non-singular matrix such that 1 T A A − = , then which one of the following represents A?

(Here A’ denotes the transpose of A).

(a)

122

1 21 1

3

22 1

(b)

122

1 21 2

3

22 1

(c)

122

1 21 1

2

22 1

(d)

122

1 21 1

2

22 1

Q.29) The matrix X satisfies the following equation

13 1 1

01 0 1

Which one of the following represents X?

(a)

1 4

1 0

⎢ ⎥ ⎣ ⎦ − (b)

1 4

1 0

⎡ − ⎤ ⎢ ⎥ ⎣ ⎦

(c)

1 4

0 1

1 2

0 1

Q.30) The matrix

012

123

311

M

and its inverse N nij =. What is the element n23 of the matrix N?

(a) 2 (b) -2

(c) 1 (d) -1

Q.31) If 111 3 sin sin sin

2 xyz −−− π ++= , then what is the value of x + y + z?

(a) -3 (b) 3

(c) 1

3

− (d) 1

3

Q.32) If 11Y = 900

, then what is the value of the expression

tan tan 2 tan 3 tan 4 tan 5 tan 6 tan 7 tan 8 tan 9 tan10 ? YYYYYYYYY Y

(a) 0 (b) -1

(c) 1 (d) 2

Q.33) If sin A = p and sin B = q where | p | and | q | are both less than 1, then what are the total number

of possible (and distinct) values of sin (A + B)?

(a) 1 (b) 2

(c) 3 (d) 4

Q.34) If 3 3 x y sin cos sin cos θ + = θ θθ and x y sin cos θ = θ , then which one of the following is correct? (a) 2 2 x y + = 0 (b) 2 2 x y + =1

(c) 2 2 x y − = 0 (d) 2 2 x y − =1

Q.35) If sin C and cos C are two roots of a quadratic equation 2 2 10 x px − + = where 0

2

C π < < , then

how many possible values can p have?

(a) 1 (b) 2

(c) 3 (d) 4

Q.36) Let n be the natural number such that n > 4 and sin cos n n U XX n = + . Which one of the

following is correct?

(a) 2 2

2 4 sin cos UU U X X nn n = − − − (b) 2 3 sin cos UU U X X nn n = − − +

(c) 2 2

4 2 sin cos UU U X X nn n = − − − (d) 2

2 4 2 sin cos U U U XX nn n =− + − −

Q.37) The three angles A, B and C of a triangle ABC are in A.P. and 2 22 c ab = + . If c = 50 metres, then

what is the area of the triangle in square metres?

(a) 625

2 (b) 625 2

2

(c) 625 3

2 (d) 625

Q.38) If ( ) 4 1

4 xy n π −= + (where n is an integer) and x + y is not an odd multiple of 2

π , then what is the

value of the following expression:

sin 2 sin 2

cos 2 cos 2

x y

x y

−

+

(a) -1 (b) 0

(c) 1

2

(d) 1

Q.39) From the top of a tower 60 metres high, the angle of depression of two objects which are on the

horizontal plane and in a line with the foot of the tower are α and β with β > α. What is the

distance between the two objects in metres?

(a) 60sin cosec cosec ( ) β −α αβ (b) 60cos sec sec (β −α αβ )

(c) 60 cot cot ( ) α − β (d) 60 tan tan ( β − α)

Q.40) The points (3, 4) and (2, -6) lie on opposite sides of which one of the following lines?

(a) 10x – y = 26 (b) 5x + 3y = -9

(c) 3x – 4y = 8 (d) x – y = 10

Q.41) Two equal parabolas have the same vertex and their axes are at right angles. What is the angle

between the tangents drawn to them at their point of intersection (other than vertex)?

(a) 4

π (b) tan-12

(c) 3

π (d) 1 3

tan

4

Q.42) Which one of the following is the nearest point on the line3 4 25 x y − = from the origin?

(a) (-1, -7) (b) (3, -4) (c) (-5, -8) (d) (3, 4)

Q.43) Which one of the following is the reflection of the point (4, 3) on the line x + y = 0?

(a) (-4, 3) (b) (-3, -4)

(c) (-3, 4) (d) (4, -3)

Q.44) Which one of the following is the orthocenter of the triangle whose sides are x = 0, y = 1 and x + y

– 2 = 0?

(a) (1, 1) (b) (0, 1)

(c) (-1, 0) (d) (0, -1)

Q.45) What is the length of the focal distance from the point “t” of the parabola y

2

= 4ax?

(a) at2

(b) a(1 + t

2

)

(c)

2

1

a t

t

⎛ ⎞ ⎜ ⎟ + ⎝ ⎠ (d) at-2

Q.46) Which of the following are the equations of circles which touch the x-axis at a distance 3 from the

origin and intercept a distance 6 on the y-axis?

(a) 2 2 xy x y + − ± += 6 62 9 0 (b) 2 2 xy x y + + ± += 6 62 9 0

(c) 2 2 xy x y + ± + += 6 62 9 0 (d) 2 2 xy xy + ± − += 62 6 9 0

Q.47) Under what condition the lines x = ay + b, z = cy + d and x = a’y + b’, z = c’y + d’ are

perpendicular to each other?

(a) a/a’ + c/c’ = 0 (b) aa’ + cc’ = -1

(c) a/a’ + c/c’ = 1 (d) aa’ + cc’ = 1

Q.48) What is the range of the function () ( ) 1

f x x 3 cos 2 −

= − ?

(a) 1 ,1

4

⎡ ⎤

⎢ ⎥ ⎣ ⎦ (b) 1 1,

4 2

⎛ ⎞ ⎜ ⎟ ⎝ ⎠

(c) 1 1 ,

4 2

⎡ ⎤

− − ⎢ ⎥ ⎣ ⎦ (d) 1 1,

4 2

⎡ ⎤ ⎢ ⎥ ⎣ ⎦

Q.49) Let f : R → R be defined as f(x) = x | x |. Which one of the following is correct?

(a) f is only onto (b) f is only one-one

(c) f is neither onto nor one-one (d) f is one-one and onto

Q.50) What is the value of

3

3 lim

1

x

x

x

x

+

→∞

⎛ ⎞ + ⎜ ⎟ ⎝ ⎠ − ?

(a) e

2

(b) e

3

(c) e

4

(d) e

-4

Q.51) If x

1 lim sin x A →∞ x

⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ and x

1 lim sin x B →∞ x

⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ , then which one of the following is correct?

(a) A = 1 and B = 0 (b) A = 0 and B = 1

(c) A = 0 and B = 0 (d) A = 1 and B = 1

Q.52) What is the value of x

10 2 5 1 lim

sin

xxx

→∞ x x

−−+ ?

(a) (In 2) (In 5) (b) (In 3) (In 5)

(c) (In 10) (In 5) (d) 0

Q.53) Let f(x) = [x], where [x] denotes the greatest integer contained in x. Consider the following

statements:

1. f(x) is not onto

2. f(x) is continuous at x = 0

3. f(x) is discontinuous for all positive integral values of x.

Which of the statements given above are correct?

(a) 1 and 2 (b) 1 and 3

(c) 2 and 3 (d) 1, 2 and 3

Q.54) If( )m n m n x y xy +

+ = then what is the value of dy

dx

?

(a) x

y

(b) xy

(c) y

x

(d) 1

Q.55) For what values of ‘p’ the function ( ) 2 1 1

1

px if x f x x p if x

⎧ + ≤ = ⎨

⎩ + >

is derivable at x = 1?

(a) 1

2

(b) 2

(c) - 1

2

(d) -2

Q.56) What is the derivative of log 2x x?

(a) 1

x

(b) zero

(c) ( )

1

2x (d) 1

2

Q.57) A function f : R → R satisfies f(x + y) = f(x) f(y) for all x, y ∈ R and f(x) ≠ 0 for all x ∈ R. If f(x) is

differentiable at x = 0 and f(0) = 2 then f’(x) is equal to which one of the following?

(a) f(x) (b) - f(x)

(c) 2 f(x) (d) ( )

2

f x

Q.58) Let f(x) be a differentiable even function. Consider the following statements:

1. f’(x) is an even function.

2. f’(x) is an odd function.

3. f’(x) may be even or odd.

Which of the above statement is/are correct?

(a) 1 only (b) 2 only

(c) 1 and 3 (d) 2 and 3

Q.59) What is the derivative of |x – 1| + |x – 4 at x = 3?

(a) -3 (b) 3

(c) 0 (d) 2

Q.60) Let g(x) be the inverse of an invertible function f(x) which is differentiable at x = c. Which one of

the following is equal to g’[f(c)]? (a) f’(c) (b) ( )

1

f ' c

(c) f(c) (d) ( )

1

f c

ANSWER KEYS

1. (d) 2. (c) 3. (c) 4. (c) 5. (d) 6. (c) 7. (c) 8. (a) 9. (c) 10. (d) 11. (a) 12. (a) 13. (a) 14. (a) 15. (a) 16. (b) 17. (a) 18. (b) 19. (a) 20. (c) 21. (c) 22. (a) 23. (d) 24. (c) 25. (c) 26. (c) 27. (a) 28. (b) 29. (c) 30. (d) 31. (b) 32. (c) 33. (d) 34. (b) 35. (a) 36. (a) 37. (c) 38. (d) 39. (c) 40. (c) 41. (d) 42. (b) 43. (b) 44. (b) 45. (b) 46. (a) 47. (b) 48. (d) 49. (d) 50. (c) 51. (a) 52. (a) 53. (b) 54. (c) 55. (a) 56. (b) 57. (c) 58. (b) 59. (c) 60. (b)

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