JEE Advanced Exam 2015 Paper-II Mathematics Question Paper With Answer Key

Mathematics

SECTION – 1 : (Maximum Marks : 32)

• This section contains EIGHT questions

• The answer to each question is a SINGLE DIGIT INTEGER ranging from 0 to 9, both inclusive

1. The coefficient of x9 in the expansion of (1 + x) (1 + x2) (1 + x3)…..(1 + x100) is

Answer: (8)

2. Suppose that the foci of ellipse 42 are (f1, 0) and (f2, 0) where f1 > 0 and f2 < 0. Let P1 and P2 be two parabolas with a common vertex at (0, 0) and with foci at (f­1, 0) and (2f­2, 0), respectively. Let T1 be a tangent to P1 which passes through (2f2, 0) and T2 be a tangent to P2 which passes through (f1, 0). If m1 is the slope of T1 and m2 is the slope of T2, then the value of 42-1 is

Answer: (4)

3. Let m and n be two positive integers greater than 1. If 43 then the value of 43-1is

Answer: (2)

4. If 44where tan−1 x takes only principal values, then the value of 44-1 is

Answer: (9)

5. Let f: R → R be a continuous odd function, which vanishes exactly at one point and f(1) = 1/2. Suppose that 45 for all x ∈ [−1, 2] and 45-1 for all x ∈ [−1, 2]. If 45-2 then the value of f(1/2) is

Answer: (7)

6. Suppose that 46 are three non-coplanar vectors in R3. Let the components of a vector 46-1 along 46-2 be 4, 3 and 5, respectively. If the components of this vector 46-3 along 46-4 are x, y and z, respectively, then the value of 2x + y + z is

Answer: (Bonus)

7. For any integer k, let 47 where 47-1 . The value of the expression 47-2 is

Answer: (4)

8. Suppose that all the terms of an arithmetic progression (A.P.) are natural numbers. If the ratio of the sum of the first seven terms to the sum of the first eleven terms is 6 : 11 and the seventh term lies in between 130 and 140, then the common difference of this A.P. is

Answer: (9)

SECTION – 2 : (Maximum Marks : 32)

• This section contains EIGHT questions

• Each question has FOUR options (A), (B), (C) and (D). ONE OR MORE THAN ONE of these four option(s) is(are) correct

9. Let f, g : [–1, 2] → R be continuous function which are twice differentiable on the interval (–1, 2). Let the values of f and g at the points –1, 0 and 2 be as given in the following table :

49

In each of the intervals (–1, 0) and (0, 2) the function (f – 3g)” never vanishes. Then the correct statement(s) is (are)

(A) f'(x) – 3g'(x) = 0 has exactly three solutions in (–1, 0) ∪ (0, 2) 

(B) f'(x) – 3g'(x) = 0 has exactly one solution in (–1, 0) 

(C) f'(x) – 3g'(x) = 0 has exactly one solution in (0, 2) 

(D)  f'(x) – 3g'(x) = 0 has exactly two solutions in (–1, 0) and exactly two solutions in (0, 2) 

Answer: (B, C)

10. Let f(x) = 7tan8x + 7tan6x – 3tan4x – 3tan2x all 50  Then the correct expression(s) is (are)

(A) 

(B) 

(C) 

(D) 

Answer: (A, B)

11. Let 51 for all x ∈ R with 51-1. If 51-2 then the possible values of m and M are

(A) m = 13, M = 24

(B) m = 1/4, M = 1/2 

(C) m = −11, M = 0 

(D) m = 1, M = 12 

Answer: (D)

12. Let S be the set of all non-zero real numbers α such that the quadratic equation αX2 – X + α = 0 has two distinct real roots x1 and x2 satisfying the inequality |x1 – x2| < 1. Which of the following intervals is(are) a subset() of S?

(A) 

(B) 

(C) 

(D) 

Answer: (A, D)

13. If 53where the inverse trigonometric functions take only the principal values, then the correct option(s) is(are)

(A) cos β > 0

(B) sin β < 0 

(C)  cos(α + β) > 0

(D) cos α < 0 

Answer: (B, C, D)

14. Let E1 and E2 be two ellipses whose centers are at the origin. The major axes of E1 and E2 lie along the x-axis and the y-axis, respectively. Let S be the circle x2+ (y – 1)2 = 2. The straight line x + y = 3 touches the curves S, E1 and E2 at P, Q and R, respectively. Suppose that54If e1 and e2 are the eccentricities of E1 and E2, respectively, then the correct expression(s) is(are)

(A) 

(B) 

(C) 

(D) 

Answer: (A, B)

15. Consider the hyperbola H : x2 – y2 = 1 and a circle S with center N(x2, 0). Suppose that H and S touch each other at a point P(x1, y1) with x1 > 1 and y1 > 0. The common tangent to H and S at P intersects the x-axis at point M. If (l, m) is the centroid of the triangle ∆PMN, then the correct expression(s) is(are)

(A) 

(B) 

(C) 

(D) 

Answer: (A, B, D)

16. The option(s) with the values of a and L that satisfy the following equation is(are) 

(A) 

(B) 

(C) 

(D) 

Answer: (A, C)

SECTION – 3 : (Maximum Marks : 16)

• This section contains TWO paragraphs

• Based on each paragraph, there will be TWO questions.

• Each equation has FOUR options (A), (B), (C) and (D). ONE OR MORE THAN ONE of these four option(s) is(are) correct

PARAGRAPH – 1

(17 & 18)

Let n1 and n2 be the number of red and black balls, respectively, in box I. Let n3 and n4 be the number of red and black balls, respectively, in box II.

17. One of the two boxes, box I and box II, was selected at random and a ball was drawn randomly out of this box. The ball was found to be red. If the probability that this red ball was drawn from box II is 1/3, then the correct option(s) with the possible values of n1, n2, n3 and n4 is(are)

(A) 

(B) 

(C) 

(D) 

Answer: (A, B)

18. A ball is drawn at random from box I and transferred to box II. If the probability of drawing a red ball from box I, after this transfer, is 1/3, then the correct option(s) with the possible values of n1 and n2 is(are)

(A) 

(B) 

(C) 

(D) 

Answer: (C, D)

PARAGRAPH-2

(19 and 20)

Let F : R → R be a thrice differentiable function. Suppose that F(1) = 0, F(3) = −4 and F′(x) < 0 for all x ∈ (1/2, 3). Let f(x) = xF(x) for all x ∈ R.

19. The correct statement(s) is(are)

(A) f ‘(1) < 0 

(B) f (2) < 0 

(C) f ‘(x) ≠ 0 for any x ∈ (1, 3) 

(D) f ‘(x) = 0 for some x ∈ (1, 3) 

Answer: (A, B, C)

20. If 60 then the correct expression(s) is(are)

(A) 

(B) 

(C) 

(D) 

Answer: (C, D)