JEE Advanced Exam 2012 Paper-I Mathematics Question Paper With Answer Key

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Part III : Mathematics

This section contains 10 multiple questions. Each question has four choices (A), (B) (C) and (D) out of which ONLY ONE is correct.

1. The total number of ways in which 5 balls of different colours can be distributed among 3 persons so that each person gets at least one ball is

(A)  75

(B)  150

(C)  210

(D)  243

Answer: (B)

2. Let 

then f is

(A)  differentiable both at x = 0 but not differentiable at x = 2

(B)  differentiable at x = 0 but not differentiable at x = 2

(C)  not differentiable at x = 0 but differentiable at x = 2

(D)  differentiable neither at x = 0 nor at x = 2

Answer: (B)

3. The function f : [0, 3] → [1, 29], defined by f(x) = 2x3 – 15x2 + 36x + 1, is

(A)  one-one and onto.

(B)  onto but not one-one.

(C)  one-one but not onto.

(D)  neither one-one nor onto.

Answer: (B)

4. If then

(A)  a = 1, b = 4

(B)  a = 1, b = −4

(C)  a = 2, b = −3

(D)  a = 2, b = 3

Answer: (B)

5. Let z be a complex number such that the imaginary part of z is nonzero and a = z2 + z + 1 is real. Then a cannot take the value

(A)  −1

(B)  1/3

(C)  1/2

(D)  3/4

Answer: (D)

6. The ellipse  is inscribed in a rectangle R whose sides are parallel to the coordinate axes. Another ellipse E2 passing through the point (0, 4) circumscribes the rectangle R. The eccentricity of the ellipse E2 is

(A)  √2/2

(B)  √3/2

(C)  1/2

(D)  3/4

Answer: (C)

7. Let P = [aij] be a 3 × 3 matrix and let Q = [bij], where bij = 2i+jaij for 1 ≤ i, j ≤ If the determinant of P is 2, then the determinant of the matrix Q is

(A)  210

(B)  211

(C)  212

(D)  213

Answer: (D)

8. The integral  equals (for some arbitrary constant K)

(A)   

(B)   

(C)   

(D)   

Answer: (C)

9. The point P is the intersection of the straight line joining the points Q(2, 3, 5) and R(1, −1, 4) with the plane 5x – 4y – z = 1. If S is the foot of the perpendicular drawn from the point T(2, 1, 4) to QR, then the length of the line segment PS is

(A)  1/√2

(B)  √2

(C)  2

(D)  2√2

Answer: (A)

10. The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line 4x – 5y = 20 to the circle x2 + y2 = 9 is

(A)  20(x2 + y2) – 36x + 45y = 0

(B)  20(x2 + y2) + 36x – 45y = 0

(C)  36(x2 + y2) – 20x + 45y = 0

(D)  36(x2 + y2) + 20x – 45y = 0

Answer: (A)

Section II : Multiple Correct Answer(s) Type

The section contains 5 multiple choice questions. Each question has hour choices (A), (B), (C) and (D) out of which ONE or MORE are correct.

11. Let θ, φ ∈[0, 2π] be such that

Then φ cannot satisfy

(A)   

(B)   

(C)   

(D)   

Answer: (A, C, D)

12. Let S be the area of the region enclosed by  y = 0,  x = 0, and x = 1. Then

(A)   

(B)   

(C)   

(D)   

Answer: (A, B, D)

13. A ship is fitted with three engines E1, E2 and E3. The engines function independently of each other with respective probabilities  For the ship to be operational at least two of its engines must function. Let X denote the event that the ship is operational and let X1, X2 and X3 denote respectively the events that the engines E1, E2 and E3 are functioning. Which of the following is (are) true ?

(A)   

(B)  P[Exactly two engines of the ship are functioning] 

(C)   

(D)   

Answer: (B, D)

14. Tangents are drawn to the hyperbola  parallel to the straight line 2x – y = 1. The points of contact of the tangents on the hyperbola are

(A)   

(B)   

(C)   

(D)   

Answer: (A, B)

15. If y(x) satisfies the differential equation y’ – y tan x = 2x sec x and y(0) = 0, then

(A)   

(B)   

(C)   

(D)   

Answer: (A, D)

SECTION III : Integer Answer Type

This section contains 5 questions. The answer to each question is a single digit integer, ranging from 0 to 9 (both inclusive).

16. Let f : IR → IR be defined as f(x) = |x| + |x2 – 1|. The total number of points at which f attains either a local maximum or a local minimum is

Answer: (5)

17. The value of  is

Answer: (4)

18. Let p(x) be a real polynomial of least degree which has a local maximum at x = 1 and local minimum at x = 3. If p(1) = 6 and p(3) = 2, then p'(0) is

Answer: (9)

19. If  are unit vectors satisfying  then  is

Answer: (3)

20. Let S be the focus of the parabola y2 = 8x and let PQ be the common chord of the circle x2 + y2 – 2x – 4y = 0 and the given parabola. The area of the triangle PQS is

Answer: (4)