IITJEE - 2009 ii - mathematics

IITJEE –Past papersMATHEMATICS - UNSOLVED PAPER - 2009

SECTION – ISingle Correct Answer TypeThis section contains 4 multiple choice questions. Each question has 4 choices (A), (B), (C) and (D) for its answer, out of which ONLY ONE is correct.

01ProblemThe normal at a point P on the ellipse x2 + 4y2 = 16 meets the x-axis at Q. If M is the mid point of the line segment PQ, then the locus of M intersects the latus rectums of the given ellipse at the pointsa.b.c.d.

Problem02The locus of the orthocenter of the triangle formed by the lines (1 + p)x – py + p(1 + p) = 0, (1 + q)x – qy + q(1 + q) = 0 and y = 0, where p ≠ q, isa hyperbola a parabola an ellipse a straight line

Problem03A line with positive direction cosines passes through the point P(2, – 1, 2) and makes equal angles with the coordinate axes. The line meets the plane 2x + y + z = 9 at point Q. The length of the line segment PQ equals1√2√3 2

Problem04If the sum of first n terms of an A.P. is cn2, then the sum of squares of these n terms isa.b.c.d.

SECTION – IIMultiple Correct Choice Type This section contains 5 multiple choice questions. Each question has 4 choices (A), (B), (C) and (D) for its answer, out of which ONE OR MORE is/are correct.

Problem05The tangent PT and the normal PN to the parabola y2 = 4ax at a point P on it meet its axis at points T and N, respectively. The locus of the centroid of the triangle PTN is a parabola whosevertex is 2a , 0directrix is x = 0latus rectum is 2afocus is (a, 0)

Problem06For function f(x) = x cos , x ≥ 1,for atleast one x in interval [1, ∞), f(x + 2) – f(x) < 2lim f x 1 (x ) ->∞for all x in the interval [1, ∞), f(x + 2) – f(x) > 2f′(x) is strictly decreasing in the interval [1, ∞)

Problem07For o < Ѳ < , the solution (s) of Cosec Cosec=4 √2 ‘s (are)a.b.c.d.

Problem08An ellipse intersects the hyperbola 2x2 – 2y2 = 1 orthogonally. The eccentricity of the ellipse is reciprocal of that of the hyperbola. If the axes of the ellipse are along the coordinates axes, thenequation of ellipse is x2 + 2y2 = 2 the foci of ellipse are (± 1, 0)equation of ellipse is x2 + 2y2 = 4 the foci of ellipse are (± √2 , 0)

Problem09If n = 0, 1, 2, …, thena.b.c.d.

SECTION – IIIMatch Type This section contains 2 questions. Each question contains statements given in two columns, which have to be matched. The statement in Column I are labelled A, B, C and D, while the statements in Column II are labelled p, q, r, s and t. Any given statement in Column I can have correct matching with ONE OR MORE statement (s) in Column II. The appropriate bubbles corresponding to the answers to these questions have to be darkened as illustrated in the following example: If the correct matches are A − p, s and t; B − q and r; C − p and q; and D −s and t; then the correct darkening of bubbles will look like the following

Problem10Match the statements/expressions in Column I with the values given in Column II.Column I Column IIThe number of solutions of the equation xesinx – (p) 1cosx = 0 in the interval 0,(B) Value(s) of k for which the planes kx + 4y + z (q) 2= 0, 4x + ky + 2z = 0 and 2x + 2y + z = 0 intersect in a straight line(C) Value(s) of k for which |x – 1| + |x – 2| + |x + 1 (r) 3| + |x + 2| = 4k has integer solution(s)(D) If y′ = y + 1 and y(0) = 1 then value(s) of y(ln2) (s) 4 (t) 5

Problem11Match the statements/expressions in Column I with the values given in Column II.Column I Column IIa. Root(s) of the expression 2sin2θ + sin22θ = 2 (p)b. (Points of discontinuity of the function (q)where [y] denotes the largest integer less than or equal to yc. Volume of the parallelopiped with its edges represented by (r)the vectorsd. Angle between vectors a and b where a, b and c are (s)unit vectors satisfying (t)

SECTION – IVInteger Answer Type This section contains 8 questions. The answer to each of the questions is a single digit integer, ranging from 0 to 9. The appropriate bubbles below the respective question numbers in the ORS have to be darkened. For example, if the correct answers to question numbers X, Y, Z and W (say) are 6, 0, 9 and 2, respectively, then the correct darkening of bubbles will look like the following:

Problem12Let f: R -> R be a continuous function which satisfies dt . Then the value of f(ln5) is

Problem13The centres of two circles C1 and C2 each of unit radius are at a distance of 6 units from each other. Let P be the mid point of the line segment joining the centres of C1 and C2 and C be a circle touching circles C1 and C2 externally. If a common tangent to C1 and C passing through P is also a common tangent to C2 and C, then the radius of the circle C is

Problem14The smallest value of k, for which both the roots of the equation x2 − 8kx + 16(k2 − k + 1 ) = 0 are real, distinct and have values at least 4, is

Problem15The maximum value of the function f(x) = 2x3 − 15 x2 + 36x − 48 on the set A = {x|x2 + 20 ≤ 9x|} is

Problem16Let ABC and ABC′ be two non−congruent triangles with sides AB = 4, AC = AC′ = 2√ 2 and angle B = 30°. The absolute value of the difference between the areas of these triangles is

Problem17If the function f(x) = x3 + ex/2 and g(x) = f−1(x), then the value of g′(1) is

Problem18Let p(x) be a polynomial of degree 4 having extremum at x = 1, 2 and x 0 2 . Then the value of p(2) is

Problem19Let (x, y, z) be points with integer coordinates satisfying the system of homogeneous equations:3x − y − z = 0− 3x + z = 0−3x + 2y +z = 0.Then the number of such points for which x2 + y2 + z2 ≤ 100 is

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IITJEE - 2009 ii - mathematics

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