IIT-JEE 2009 Mathematics Paper - I I Questions and Solutions

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PART � II: MATHEMATICSSECTION � I Straight Objective Type

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This section contains 4 multiple choice questions. Each question has 4 choices (A), (B), (C) and (D), out of which ONLY ONE is correct.

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*20. If the sum of first n terms of an A.P. is cn², then the sum of squares of these n terms is

21. A 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
equals
(A) 1 (B) √2
(C) √3 (D) 2
Key: (C)
Sol.: Let the line make the angle α with the axes, then we have
3cos2α = 1 [sum of the square's of DC's = 1]
cosα = 1/ √3
x- 2/√3 + y+ 1/√3+ z- 2/√3 = r
By solving we get r =√3

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*22. The 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 points
(A) (�3√5/2,�2/7) (B)(�3√5/2,�√19/4)

(C)(�2√3,�1/7) (D)(�2√3,�4√3/7)
Key (C)

23. The locus of the orthocentre 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, is
(A) a hyperbola (B) a parabola
(C) an ellipse (D) a straight line

Key : (D)

SECTION � II Multiple Correct Answer Type

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This section contains 5 multiple correct answer's type questions. Each question has 4 choices (A), (B), (C) and (D), out of which ONE OR MORE is/are correct.

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24. If In = _-Π∫^-Π sin nx/(1+Πx)*sin xdx, n = 0, 1, 2, .... , then

*25. An 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 coordinate axes, then
(A) equation of ellipse is x² + 2y² = 2 (B) the foci of ellipse are (� 1 , 0)
(C) equation of ellipse is x² + 2y² = 4 (D) the foci of ellipse are (� 2 , 0)
Key (A, B)

SOl: x² - y² = 1
(rectangular hyperbola)
eccentricity of rectangular hyperbola = √2
eccentricity of ellipse = 1/√2
Let equation of an ellipse is x²/a2+y²/b2 = 1
Let the ellipse and hyperbola intersect at (α,β)

(dy/dx )at _(α,β) for hyperbola =α/ β

(dy/dx )at _(α,β) for ellipse = -b2/a2*α/ β

26. For the function f(x) = x cos 1/x, x ≥ 1,
(A) for at least one x in the interval [1, ∞), f(x + 2) - f(x) < 2
(B)lim_x→∞f′(x) = 1
(C) for all x in the interval [1, ∞), f(x + 2) - f(x) > 2
(D) f′(x) is strictly decreasing in the interval [1, ∞)
Key (B, C, D)
Sol.:

=>g(x)>2 for every x ≥1

*27. The 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 whose
(A) vertex is 2a ,0 (B) directrix is x = 0
(C) latus rectum is 2a3 (D) focus is (a, 0)
Key (A, D)
Sol.:

*28. For 0 < θ < π/2, the solution(s) _m=1Σ⁶ cos ec(θ+(m-1)*π/4)cos ec(θ+m*π/4) = 4√2 is are

By solving above we get θ = π/12 or 5π/12 SECTION � IIIMatrix Match Type

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This section contains 2 questions. Each question contains statements given in two columns, which have to be matched. The statements in Column I are labeled 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:

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29. Match the statements/expressions given in Column I with the values given in Column II.

(A) solution of the equation 2sin2θ + sin22θ = 2	(p) π/6
(B) Points of discontinuity of f(x)= 6x/π cos3x/π	(q) π/4
(C) Volume of the parallelepiped with its edges represented by the vectors � i +j,i+ 2j and � i +j+2 k	(r) π/3
(D) Angle between vectors a and b where a, b and c are unit vectors satisfying a+ b+√ 3c =0	(s) π/2 (t) π

30. Match the statements/expressions given in Column I with the values given in Column II.

(A) The number of solutions of the equation xesinx-cox=0 over (0, π/2)	(p) 1
(B)values of k for which planes kx+4y+z=0 , 4x+ky + 2z = 0 and 2x + 2y + z = 0 intersect in a straight line	(q) 2
(C) Value(s) of k for which |x - 1| + |x - 2| + |x + 1| + |x +2| =4k has integer solution	(r) 3
(D)If y′ = y + 1 and y(0) = 1, then values(s) of y (ln 2)	(s) 4 (t) 5

SECTION � IVInteger Answer Type

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This section contains 8 questions. The answer to each of the question 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:

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31. The maximum value of the function f(x) = 2x³ - 15x² + 36x - 48 on the set A = {x |x² + 20 ≤ 9x} is
Key 7
Sol.: A = {x |x² + 20 - 9x ≤ 0}
x² - 9x + 20 ≤ 0
(x - 5) (x - 4) ≤ 0
x ε [4, 5]
f(x) = 2x³ - 15x² + 36x - 48
f′(x) = 6(x - 2) (x - 3)
f(x) is strictly increasing in (4, 5)
So, f(5) = 7

32. Let (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
Key 7
Sol.: 3x � y � z = 0 �..(i)
−3x + z = 0 �..(ii)
−3x + 2y + z = 0 �..(iii)
Solving (i) & (ii)
y = 0
So 3x � z = 0
z = 3x
Now x² + y² + z² ≤ 100
x² + 9x² ≤ 100
| x | ≤√10
√10 ≤ x ≤ √10
Integral values of �x� are −3, −2, −1, 0, 1, 2, 3
So �7� points are there.

*33. Let 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

Key 4

Sol.:

34. Let p(x) be a polynomial of degree 4 having extremum at x = 1, 2 and LIM _{x→ 0}(1+p(x)/x2) =2 then the value of p(2)

Key 0
SOl:

35. Let f : R → R be a continuous function which satisfies
x
∫ f (t)dt Then the value of f(ln 5) is
0
Sol.: f ′(x) = f(x)
f(x) = ce^x
c = 0 because f(0) = 0
f(ln 5) = 0

*36. The centres of two circles C1 and C2 each of unit radius are at a distance of 6 units from each other. Let P bethe mid point of the line segment joining the centres of C1 and C2 and C be a circle touching circles C1 and C2 extrenally. 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
Key 8
Sol.:

*37. The smallest value of k, fow which both the roots of the equation
x2 - 8kx + 16 (k2 - k + 1) = 0
are real, distinct and have values at least 4, is
Key 2
Sol.: For the root to be real & distinct and having minimum value, following points should hold :

38. If the function f(x) = x3 + e^x/2 and g(x) = f^-1 (x), then the value of g′(1) is

Key 2