Mathematics

Section-I

Straight Objective Type

This section contains 9 multiple choice questions. Each question has 4 choices (A), (B), (C) and (D), out of which ONLY ONE is correct.

1. A particle P starts from the point z0 = 1 + 2i, where i = √− It moves first horizontally away from origin by 5 units and then vertically away from origin by 3 units to reach a point z₁. From z₁ the particle moves √2 units in the direction of the vector  and then it moves through an angle π/2 in anticlockwise direction on a circle with centre at origin, to reach a point z₂. The point z₂ is given by

(A)  6 + 7i

(B)  −7 + 6i

(C)  7 + 6i

(D)  − 6 + 7i

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2. Let the function  be given by  Then, g is

(A)  even and is strictly increasing in (0, ∞)

(B)  odd and is strictly decreasing in (−∞, ∞)

(C)  odd and is strictly increasing in (−∞, ∞)

(D)  neither even nor odd, but is strictly increasing in (−∞, ∞)

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3. Consider a branch of the hyperbola x² − 2y² − 2√ 2 x − 4√ 2 y − 6 = 0 with vertex at the point A. Let B be one of the end points of its latus rectum. If C is the focus of the hyperbola nearest to the point A, then the area of the triangle ABC is

(A) 

(B) 

(C) 

(D) 

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4. The area of the region between the curves  bounded by the lines x = 0 and x = π/4 is

(A) 

(B) 

(C) 

(D) 

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5. Consider three points P = (−sin(β − α), − cosβ), Q = (cos(β − α), sinβ) and R = (cos(β − α + θ), sin(β − θ)), where 0 < α, β, θ < π/4. Then

(A)  P lies on the line segment RQ

(B)  Q lies on the line segment PR

(C)  R lies on the line segment QP

(D)  P, Q, R are non-collinear

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6. An experiment has 10 equally likely outcomes. Let A and B be two non-empty events of the experiment. If A consists of 4 outcomes, the number of outcomes that B must have so that A and B are independent, is

(A)  2, 4 or 8

(B)  3, 6 or 9

(C)  4 or 8

(D)  5 or 10

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7. Let two non-collinear unit vectors  form an acute angle. A point P moves so that at any time t the position vector   (where O is the origin) is given by  When P is farthest from origin O, let M be the length of  be the unit vector along  Then,

(A) 

(B) 

(C) 

(D) 

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8. Let  Then for an arbitrary constant C, the value of J – I equals

(A) 

(B) 

(C) 

(D) 

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9. Let g(x) = log(f(x)) where f(x) is a twice differentiable positive function on (0, ∞) such that f(x + 1) = x f(x). Then, for N = 1, 2, 3, …,

(A) 

(B) 

(C) 

(D) 

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SECTION – II

Reasoning Type

This section contains 4 reasoning type questions. Each question has 4 choices (A), (B), (C) and (D), out of which ONLY ONE is correct.

10. Suppose four distinct positive numbers a₁, a₂, a₃, a₄ are in G.P. Let b₁ = a₁, b₂ = b₁ + a₂, b₃ = b₂ + a₃ and b₄ = b₃ + a₄.

STATEMENT−1 : The numbers b₁, b₂, b₃, b₄ are neither in A.P. nor in G.P.

and

STATEMENT−2 : The numbers b₁, b₂, b₃, b₄ are in H.P.

(A)  STATEMENT−1 is True, STATEMENT−2 is True; STATEMENT−2 is a correct explanation for STATEMENT−1

(B)  STATEMENT−1 is True, STATEMENT−2 is True; STATEMENT−2 is NOT a correct explanation for STATEMENT−1.

(C)  STATEMENT−1 is True, STATEMENT−2 is False

(D)  STATEMENT−1 is False, STATEMENT−2 is True

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11. Let a, b, c, p, q be real numbers. Suppose α, β are the roots of the equation x² + 2px + q = 0 and α, 1/β are the roots of the equation ax² + 2bx + c = 0, where β² ∉{−1, 0, 1}.

STATEMENT−1 : (p² − q) (b² − ac) ≥ 0

and

STATEMENT−2 : b ≠ pa or c ≠ qa

(A)  STATEMENT−1 is True, STATEMENT−2 is True; STATEMENT−2 is a correct explanation for STATEMENT−1

(B)  STATEMENT−1 is True, STATEMENT−2 is True; STATEMENT−2 is NOT a correct explanation for STATEMENT−1.

(C)  STATEMENT−1 is True, STATEMENT−2 is False

(D)  STATEMENT−1 is False, STATEMENT−2 is True

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12. Consider

L₁ : 2x + 3y + p − 3 = 0

L₂ : 2x + 3y + p + 3 = 0,

where p is a real number, and C : x² + y² + 6x − 10y + 30 = 0.

STATEMENT−1 : If line L1 is a chord of circle C, then line L2 is not always a diameter of circle C.

and

STATEMENT−2 : If line L1 is a diameter of circle C, then line L₂ is not a chord of circle C.

(A)  STATEMENT−1 is True, STATEMENT−2 is True; STATEMENT−2 is a correct explanation for STATEMENT−1

(B)  STATEMENT−1 is True, STATEMENT−2 is True; STATEMENT−2 is NOT a correct explanation for STATEMENT−1.

(C)  STATEMENT−1 is True, STATEMENT−2 is False

(D)  STATEMENT−1 is False, STATEMENT−2 is True

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13. Let a solution y = y(x) of the differential equation

and

STATEMENT−2 : y(x) is given by 

(A)  STATEMENT−1 is True, STATEMENT−2 is True; STATEMENT−2 is a correct explanation for STATEMENT−1

(B)  STATEMENT−1 is True, STATEMENT−2 is True; STATEMENT−2 is NOT a correct explanation for STATEMENT−1.

(C)  STATEMENT−1 is True, STATEMENT−2 is False

(D)  STATEMENT−1 is False, STATEMENT−2 is True

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SECTION – III

Linked Comprehension Type

This section contains 2paragraphs. Based upon each paragraph, 3 multiple choice questions have to be answered. Each question has 4 choices (A), (B), (C) and (D), out of which ONLY ONE is correct.

Paragraph for Question Nos. 14 to 16

Consider the function f : (−∞, ∞) → (−∞, ∞) defined by

14. Which of the following is true?

(A)  (2 + a)² f′′(1) + (2 − a)² f′′(−1) = 0

(B)  (2 − a)² f′′(1) − (2 + a)² f′′(−1) = 0

(C)  f′(1) f′(−1) = (2 − a)²

(D)  f′(1) f′(−1) = −(2 + a)²

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15. Which of the following is true?

(A)  f(x) is decreasing on (−1, 1) and has a local minimum at x = 1

(B)  f(x) is increasing on (−1, 1) and has a local maximum at x = 1

(C)  f(x) is increasing on (−1, 1) but has neither a local maximum nor a local minimum at x = 1

(D)  f(x) is decreasing on (−1, 1) but has neither a local maximum nor a local minimum at x = 1

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16. 

which of the following is true?

(A)  g′(x) is positive on (−∞, 0) and negative on (0, ∞)

(B)  g′(x) is negative on (−∞, 0) and positive on (0, ∞)

(C)  g′(x) changes sign on both (−∞, 0) and (0, ∞)

(D)  g′(x) does not change sign on (−∞, ∞)

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Paragraph for Question Nos. 17 to 19

Consider the line 

17. The unit vector perpendicular to both L₁ and L₂ is

(A) 

(B) 

(C)   

(D) 

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18. The shortest distance between L₁ and L₂ is

(A) 0

(B) 

(C) 

(D) 

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SECTION – IV

Matrix-Match Type

This contains 3 questions. Each question contains statements given in two columns which have to be matched. Statements (A, B, C, D) in column I have to be matched with statements (p, q, r, s) in column II.

20. Consider the lines given by

L₁: x + 3y − 5 = 0

L₂ : 3x − ky − 1 = 0

L₃ : 5x + 2y − 12 = 0

Match the Statements / Expressions in Column I with the Statements / Expressions in Column II

(A)  (A) → (s); (B) → (p, q); (C) → (r); (D) → (p, q, s)

(B)  (A) → (t); (B) → (s, r); (C) → (p); (D) → (t, r, p)

(C)  (A) → (q); (B) → (r, s); (C) → (q); (D) → (q, r, t)

(D)  (A) → (p); (B) → (r, t); (C) → (t); (D) → (s, p, r)

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21. Match the Statements / Expressions in Column I with the Statements / Expressions in Column II

(A)  (A) → (s); (B) → (p, q); (C) → (r); (D) → (p, q, s)

(B)  (A) → (s); (B) → (r); (C) → (p); (D) → (s, t)

(C)  (A) → (r); (B) → (q, s); (C) → (r, s); (D) → (p, r)

(D)  (A) → (q); (B) → (t); (C) → (r); (D) → (q, r)

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22. Consider all possible permutations of the letters of the word ENDEANOEL.

Match the Statements / Expressions in Column I with the Statements / Expressions in Column II

(A)  (A) → (r); (B) → (t); (C) → (s); (D) → (p)

(B)  (A) → (p); (B) → (s); (C) → (q); (D) → (q)

(C)  (A) → (s); (B) → (r); (C) → (p); (D) → (t)

(D)  (A) → (q); (B) → (t); (C) → (r); (D) → (r)

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