# 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 x^{9} in the expansion of (1 + x) (1 + x^{2}) (1 + x^{3})…..(1 + x^{100}) is

2. Suppose that the foci of ellipse are (f_{1}, 0) and (f_{2}, 0) where f_{1} > 0 and f_{2} < 0. Let P_{1} and P_{2} be two parabolas with a common vertex at (0, 0) and with foci at (f_{1}, 0) and (2f_{2}, 0), respectively. Let T_{1} be a tangent to P_{1} which passes through (2f_{2}, 0) and T_{2} be a tangent to P_{2} which passes through (f_{1}, 0). If m_{1} is the slope of T_{1} and m_{2} is the slope of T_{2}, then the value of is

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

4. If where tan^{−1} x takes only principal values, then the value of is

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

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

7. For any integer k, let where . The value of the expression is

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

**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 :

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)

10. Let f(x) = 7tan^{8}x + 7tan^{6}x – 3tan^{4}x – 3tan^{2}x all Then the correct expression(s) is (are)

(A)

(B)

(C)

(D)

11. Let for all x ∈ R with . If 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

12. Let S be the set of all non-zero real numbers α such that the quadratic equation αX^{2} – X + α = 0 has two distinct real roots x_{1} and x_{2} satisfying the inequality |x_{1} – x_{2}| < 1. Which of the following intervals is(are) a subset() of S?

(A)

(B)

(C)

(D)

13. If where 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

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

(A)

(B)

(C)

(D)

15. Consider the hyperbola H : x^{2} – y^{2} = 1 and a circle S with center N(x_{2}, 0). Suppose that H and S touch each other at a point P(x_{1}, y_{1}) with x_{1} > 1 and y_{1} > 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)

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

(A)

(B)

(C)

(D)

**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 n_{1} and n_{2} be the number of red and black balls, respectively, in box I. Let n_{3} and n_{4} 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 n_{1}, n_{2}, n_{3} and n_{4} is(are)

(A)

(B)

(C)

(D)

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 n_{1} and n_{2} is(are)

(A)

(B)

(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)

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

(A)

(B)

(C)

(D)