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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