KVPY 2019 SB SX Exam Previous Papers

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Question : 85 of 120

Marks: +1, -0

The number of solutions to

\sin(π\sin^2 (θ)) + \sin(π\cos^2(θ)) = 2\cos (π/2\cos(θ))

satisfying

0 ≤θ≤ 2π

is

1
2
4
7

Solution:

\sin(π-π\cos^2 θ ) + \sin(π\cos^2 θ)

= 2\cos ({π \cosθ}/2)

⇒

2\sin(π\cos^2 θ) =2\cos({π\cosθ}/2)

⇒

\cos(π/2-π\cos^2θ)= \cos({π\cosθ}/2)

⇒

π/2-π\cos^2θ=2nπ±{π\cosθ}/2

⇒

1/2-\cos^2θ=2n±{\cosθ}/2

⇒

1 -2\cos^2θ= 4n ± \cos θ

⇒

2\cos^2θ± \cosθ= 4k + 1

⇒

2\cos^2θ+ \cosθ= 1,

2\cos^2θ-\cos θ= 1

⇒

\cosθ= -1,1/2,1,-1/2

⇒ 7 solutions

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