This document contains an unsolved mathematics paper from 2011 containing multiple choice and other types of questions. The paper is divided into 4 sections testing different skills: Section I has 8 single answer multiple choice questions about topics like functions, geometry, and algebra. Section II has 4 multi-select multiple choice questions. Section III has 6 questions where the answer is a single digit integer. Section IV has 2 matrix matching questions to connect statements with values or intervals. The paper tests a range of mathematical concepts and problem solving abilities.

1. IITJEE –Past papersMATHEMATICS- UNSOLVED PAPER - 2011

2. SECTION – ISingle Correct Answer TypeThis Section contains 8 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

3. 01ProblemIf 2bsin then the value of q isa.b.c.d.

4. Problem02Let f : [-1, 2] [0, ¥) be a continuous function such that f(x) = f(1 - x) for all x Î[-1, 2]. Let be the area of the region bounded by y = f(x), x = -1, x = 2, and the x-axis. Thena.b.c.d.

5. Problem03Let f(x) = and g(x) = sinx for all x R. Then the set of all x satisfying (f o g o g o f) (x) = (g o g o f) (x), where (f o g) (x) = f(g(x)), isa.b.c.d.

6. Problem04Let (x, y) be any point on the parabola y2 = 4x. Let P be the point that divides the line segment from (0, 0) to (x, y) in the ratio 1 : 3. Then the locus of P isa.b.c.d.

7. Problem05Let P(6, 3) be a point on the hyperbola = . If the normal at the point P intersects the x-axis at (9, 0), then the eccentricity of the hyperbola isa.b.c.d.

8. Problem06A value of b for which the equationshave one root in common isa.b.c.d.

9. Problem07Let be a cube root of unity and S be the set of all non-singular matrices of the form where each of a, b, and c is either w or w2. Then the number of distinct matrices in the set S is264 8

10. Problem08The circle passing through the point (-1, 0) and touching the y-axis at (0, 2) also passes through the pointa.b.c.d.

11. SECTION – IIMultiple Correct Answer’s TypeThis section contains 4 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONE OR MORE may be correct.

12. Problem09If f(x) is continuous at x =f(x) is not differentiable at x = 0 f(x) is differentiable at x = 1 f(x) is differentiable at x = -3/2

13. Problem10Let be defined by where be is a constant such that 0 < b < 1. Thena. f is not invertible on (0, 1) b.c.d. f-1 is differentiable on (0, 1)

14. Problem11Let L be a normal to the parabola = 4x. If L passes through the point (9, 6), then L is given byy - x + 3 = 0y + 3x - 33 = 0y + x - 15 = 0y - 2x + 12 = 0

15. Problem12Let E and F be two independent events. The probability that exactly one of them occurs is and the probability of none of them occurring is If P(T) denotes the probability of occurrence of the event T, thena.b.c.d.

16. SECTION – IIIInteger Answer TypeThis section contains 6 questions. The answer to each of the questions is a single-digit integer, ranging from 0 to 9. The bubble corresponding to the correct answer is to be darkened in the ORS..

17. Problem13Let a = and be three given vectors. If is a vector such that and then the value of is

18. Problem14The straight line 2x - 3y = 1 divides the circular region into two parts. If then the number of point(s) in S lying inside the smaller part is

19. Problem15Let = eip/3, and a, b, c, x, y, z be non-zero complex numbers such thata + b + c = xa + b = ya + = z.Then the value of is

20. Problem16The number of distinct real roots of = 0 is

21. Problem17Let where (x) denotes and g(x) is a given non constant differentiable function on with g(0) = g(2) = 0. Then the value of y(2) is

22. Problem18Let M be a 3 ´ 3 matrix satisfyingThen the sum of the diagonal entries of M is

23. SECTION – IVMatrix-Match Type This section contains 2 questions. Each question has four statements (A, B, C and D) given in Column I and five statements (p, q, r, s and t) in Column II. Any given statement in Column I can have correct matching with ONE or MORE statement(s) given in Column II. For example, if for a given question, statement B matches with the statements given q and r, then for the particular question, against statement B, darken the bubbles corresponding to q and r in the ORS.

24. Problem19Match the statements given in Column I with the values given in Column IIColumn - I Column – IIform a triangle, then the internal angle of the triangle between isb. then he value of isc. The value of section is d. The maximum value of is given By

25. Problem20Match the statements given in Column I with the intervals/union of intervals given in Column IIColumn - I Column – IIThe set isThe domain of the function is If then the setIf then is increasing in

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