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```IIT JEE 2009 Mathematics Paper2 Code 1 Solutions

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

Column I

Column II

(A)

Root(s) of the expression 2sin2θ + sin22θ = 2

(p)

Π/6

(B)

Points of discontinuity of the function f(x) = [6x/Π]cos[3x/Π], where [y] denotes the largest integer less than or equal to y

(q)

Π/4

(C)

Volume of the parallelepiped with its edges represented by the vectors i+j , i+2j and i+j+Πk

(r)

Π/3

(D)

Angle between vectors a and b where vectors a, b and c are unit vectors satisfying a + b + √3c = 0

(s)

Π/2

(t)

Π

Sol.   (A --> q, s); (B --> p, r, s, t); (C --> t); (D --> r)

(A)

2sin2θ + 4sin2θ cos2θ = 2

sin2θ + 2sin2θ(1 - sin2θ) = 1

3sin2θ - 2sin4θ - 1 = 0 => sinθ = +1/√2, +1

=> θ = Π/4 , Π/2

(B)

Let y = 3x/Π

=> 1/2 < y < 3           for all x € [Π/6 , Π]

Now f(y) = [2y] cos[y]

Critical points are y = 1/2, y = 1, y = 3/2, y = 3

=> points of discontinuity {Π/6, Π/3, Π/2, Π }

(C)

(D)

|a-> + b->| = √3

=> √(2+2cosα) = √3

=> 2 + 2 cosα = 3

=> α = Π/3

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