Mathematics

Section-I

(Single Correct Answer Type)

This section contains 8 multiple choice questions. Each question has four choices (A), (B) (C) and (D) out of which ONLY ONE is correct.

1. Let P(6, 3) be a point on the hyperbola  If the normal at the point P intersects the x-axis at (9, 0), then the eccentricity of the hyperbola is

(A)    

(B)   

(C)   

(D)    

Show Answer

2. A value of b for which the equations

x² + bx – 1 = 0

x² + x + b = 0,

have one root in common is

(A)  −√2

(B)  −i√3

(C)  i√5

(D)  √2

Show Answer

3. Let ω ≠ 1 be a cube root of unity and S be the set of all non-singulr matrices of the form

where each of a, b and c is either ω or ω². Then the number of distinct matrices in the set S is

(A)  2

(B)  6

(C)  4

(D)  8

Show Answer

4. The circle passing through the point (−1, 0) and touching the y-axis at (0, 2) also passes through the point

(A)   

(B)    

(C)   

(D)    

Show Answer

45. If  b > 0 and θ ∈ (−π, π], then the value of θ is

(A)  ±π/4

(B)  ±π/3

(C)  ±π/6

(D)  ±π/2

Show Answer

6. Let f : [−1, 2] → [0, ∞) be a continuous function such that f(x) = f(1 – x) for all x ∈[−1, 2]. Let  and R₂ be the area of the region bounded by y = f(x), x = −1, x = 2, and the x-axis. Then

(A)  R₁ = 2R₂

(B)  R₁ = 3R₂

(C)  2R₁ = R₂

(D)  3R₁ = R₂

Show Answer

7. Let f(x) = x² and g(x) = sin x for all x ∈ ℝ. Then the set of all x satisfying (f ∘ g ∘ g ∘ f) (x) = (g ∘ g ∘ f) (x), where (f ∘ g) (x) = f(g(x)), is

(A)    

(B)   

(C)   

(D)    

Show Answer

8. Let (x, y) be any point on the parabola y² = 4x. Let P be the point that divides the line segment from (0, 0) to (x, y) in the ratio 1 : 3. Then the locus of P is

(A)  x² = y

(B)  y² = 2x

(C)  y² = x

(D)  x² = 2y

Show Answer

Section-II

Multiple Correct Answer(s) Type

This section  contains 4 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONE or MORE may be correct.

9. If

then

(A)  f(x) is continuous at x = −π/2

(B)  f(x) is not differentiable at x = 0

(C)  f(x) is differentiable at x = 1

(D)  f(x) is differentiable at x = 3/2

Show Answer

11. Let L be a normal to the parabola y² = 4x. If L passes through the point (9, 6), then L is given by

(A)  y – x + 3 = 0

(B)  y + 3x – 33 = 0

(C)  y + x – 15 = 0

(D)  y – 2x + 12 = 0

Show Answer

12. Let f : (0, 1) → ℝ be defined by

where b is a constant such that 0 < b < 1. Then

(A)  f is not invertible on (0, 1)

(B)  f ≠ f⁻¹ on (0, 1) and   

(C)  f = f⁻¹ on (0, 1) and 

(D)  f⁻¹ is differentiable on (0, 1)

Show Answer

Section-III

(Integer Answer Type)

This section contains 6 questions. The answer to each of the questions is a single-digit integer, ranging from 0 to 9.

13. Let ω = ^iπ/3, and a, b, c x, y, z be non-zero complex numbers such that

a + b + c = x

a + bω + cω² = y

a + bω² + cω = z.

Then the value of   is

Show Answer

14. The number of distinct real roots of x⁴ – 4x³ + 12x² + x – 1 = 0 is

Show Answer

15. Let yʹ(x) + y(x) gʹ(x) = g(x)gʹ(x), y(0) = 0, x ∈ ℝ, where fʹ(x) denotes. and g(x) is a given non-constant differentiable function on ℝ with g(0) = g(2) = 0. Then the value of y(2) is

Show Answer

16. Let M be a 3 × 3 matrix satisfying

Then the sum of the diagonal entries of M is

Show Answer

17. Let  be three given vectors, If   is a vector such that  then the value of  is

Show Answer

18. The straight line 2x – 3y = 1 divides the circular region x² + y² ≤ 6 into two parts. If

then the number of point(s) in S lying inside the smaller part is

Show Answer

Section-IV

(Matrix-Match Type)

This section contains 2 questions. Each question has four statements (A, B, C and D) given in Column I and five statements (p, q, r, s and t) in Column II. Any given statement in Column I can have correct matching with ONE or MORE statement(s) given in Column II. For example, if for a given question, statement B matches with the statements given in q and r, then for the particular question, against statement B.

19. Match the statements given in Column I with the values given in Column II

(A)  A – q ; B – p or p, q, r, s and t ; C – s ; D – t

(B)  A – r ; B – s and p ;  C – r ; D – q

(C)  A – q ; B – t and r ;  C – p ; D – s

(D)  A – p ; B – q and t ;  C – s ; D – t

Show Answer

20. Match the statements given in Column I with the intervals/union of intervals given in Column II

(A)  A – s ; B  – t ; C – r ; D – r

(B)  A – p ; B  – s ; C – p ; D – q

(C)  A – p ; B  – q ; C – s ; D – t

(D)  A – r ; B  – s ; C – p ; D – q

Show Answer

Latest Govt Job & Exam Updates:

View Full List ...