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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)
2sin2θ + 4sin2θ cos2θ = 2
sin2θ + 2sin2θ(1 - sin2θ) = 1
3sin2θ - 2sin4θ - 1 = 0 => sinθ = +1/√2, +1
=> θ = Π/4 , Π/2
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 }
|a-> + b->| = √3
=> √(2+2cosα) = √3
=> 2 + 2 cosα = 3
=> α = Π/3
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IIT JEE 2009 Mathematics Paper2 Code 1 Solutions...