Mathematics

SECTION – I

Straight Objective Type

This section contains 6 multiple choice questions. Each question has 4 choices (A), (B), (C) and (D), out of which ONLY ONE is correct..

1. Consider the two curves C₁ : y² = 4x, C² : x² + y² – 6x + 1 = 0. Then,

(A) C₁ and C₂ touch each other only at one point

(B) C₁ and C₂ touch each other exactly at two points

(D) C₁ and C₂ neither intersect nor touch each other

Answer: (B)

2. If 0 < x < 1, then is equal to

(A)

(B)

(C)

(D)

Answer: (C)

3. The edges of a parallelopiped are of unit length and are parallel to non-coplanar unit vectors such that Then the volume of the parallelopiped is

(A)

(B)

(C)

(D)

Answer: (A)

4. Let a and b be non-zero real numbers. Then, the equation (ax² + by² + c) (x² – 5xy + 6y²) = 0 represents

(A) four straight lines, when c = 0 and a, b are of the same sign

(B) two straight lines and a circle, when a = b, and c is of sign opposite to that of a

(D) a circle and an ellipse, when a and b are of the same sign and c is of sign opposite to that of a

Answer: (B)

5. Let 0 < x < 2, m and n are integers, m ≠ 0, n > 0, and let p be the left hand derivative of |x – 1| at x = 1. If then

(A) n = 1, m = 1

(B) n = 1, m = −1

(D) n > 2, m = n

Answer: (C)

6. The total number of local maxima and local minima of the function is

(A) 0

(B) 1

(D) 3

Answer: (C)

SECTION – II

Multiple Correct Answers Type

This section contains 4 multiple correct answer(s) type questions. Each question has 4 choices (A), (B), (C) and (D), out of which ONE OR MORE is/are correct.

7. A straight line through the vertex P of a triangle PQR intersects the side QR at the point S and the circumcircle of the triangle PQR at the point T. If S is not the centre of the circumcircle, then

(A)

(B)

(C)

(D)

Answer: (B, D)

8. Let P (x₁, y₁) and Q (x₂, y₂), y₁ < 0, y₂ < 0, be the end points of the latus rectum of the ellipse x² + 4y² = 4. The equations of parabolas with latus rectum PQ are

(A) x² + 2√3y = 3 + √3

(B) x² – 2√3y = 3 + √3

(D) x² – 2√3y = 3 – √3

Answer: (B, C)

9. Let and for n = 1, 2, 3, …. Then,

(A)

(B)

(C)

(D)

Answer: (A, D)

10. Let f (x) be a non-constant twice differentiable function defined on (– ∞, ∞) such that f (x) = f (1 – x) and Then

(A)

(B)

(C)

(D)

Answer: (A, B, C, D)

SECTION – III

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.

11. Let f and g be real valued functions defined on interval (−1, 1) such that g″(x) is continuous, g(0) ≠ 0, g′(0) = 0, g″(0) ≠ 0, and f(x) = g(x) sinx.

STATEMENT-1 :

and

STATEMENT -2 : f′(0) = g(0).

(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

(D) Statement -1 is False, Statement -2 is True

Answer: (A)

12. Consider three planes

P₁ : x − y + z = 1

P₂ : x + y − z = −1

P₃ : x − 3y + 3z = 2.

Let L₁, L₂, L₃ be the lines of intersection of the planes P₂ and P₃, P₃ and P₁, and P₁ and P₂, respectively.

STATEMENT -1 : At least two of the lines L₁, L₂ and L₃ are non-parallel.

and

STATEMENT -2 : The three planes do not have a common point.

(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

(D) Statement -1 is False, Statement -2 is True

Answer: (D)

13. Consider the system of equations

x − 2y + 3z = −1

−x + y − 2z = k

x − 3y + 4z = 1

STATEMENT -1 : The system of equations has no solution for k ≠ 3.

and

STATEMENT -2 : The determinant for k ≠ 3.

(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

(D) Statement -1 is False, Statement -2 is True

Answer: (A)

14. Consider the system of equations ax + by = 0, cx + dy = 0, where a, b, c, d ∈ {0, 1}.

STATEMENT – 1: The probability that the system of equations has a unique solution is 3/8.

and

STATEMENT – 2: The probability that the system of equations has a solution is 1.

(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

(D) Statement -1 is False, Statement -2 is True

Answer: (B)

SECTION − IV

Linked Comprehension Type

This section contains 3 paragraphs. 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. 15 to 17

A circle C of radius 1 is inscribed in an equilateral triangle PQR. The points of contact of C with the sides PQ, QR, RP are D, E, F, respectively. The line PQ is given by the equation √3x + y − 6 = 0 and the point D is Further, it is given that the origin and the centre of C are on the same side of the line PQ.

15. The equation of circle C is

(A)

(B)

(C)

(D)

Answer: (D)

16. Points E and F are given by

(A)

(B)

(C)

(D)

Answer: (A)

17. Equation of the sides QR, RP are

(A)

(B)

(C)

(D)

Answer: (D)

Paragraph for Question 18 to 20

Consider the functions defined implicitly by the equation y3 − 3y + x = 0 on various intervals in the real line. If x ∈ (−∞, −2) ∪ (2, ∞), the equation implicitly defines a unique real valued differentiable function y = f(x).

If x ∈(−2, 2), the equation implicitly defines a unique real valued differentiable function y = g(x) satisfying g(0) = 0.

18. If f(−10√2) = 2√2, then fʹʹ(−10√2) =

(A)