Let f:(-1,1) arrow I R be such that f( 4 θ)= 2 2- ^ 2 θ for θ ∈(0, (π)/(4)) ((π)/(4), (π)/(2)) . Then the value (s) of f((1)/(3)) is (are)

Correct answer: A. 1- (3)/(2) Given : f( 4 θ)= 2 2- ^ 2 θ = 2 ^ 2 θ 2 ^ 2 θ-1 =(1+ 2 θ)/( 2 θ)=1+(1)/( 2 θ) Let 4 θ=(1)/(3) 2 ^ 2 2 θ-1=(1)/(3) 2 θ=± (2)/(3) f( 4 θ)=1+(1)/( 2 θ)=1 ± (3)/(2) or f((1)/(3))=1 ± (3)/(2)

JEE Advanced 2012Paper 2

Exam: JEE Advanced
Year: 2012
Paper: Paper 2
Subject: Mathematics
Chapter: Trigonometric Ratios & Identities

JEE Advanced 2012 · Paper 2 · Mathematics · Trigonometric Ratios & Identities

Let f:(-1,1) arrow I R be such that f( 4 θ)= 2 2- ^ 2 θ for θ ∈(0,...

Let

f:(-1,1) \rightarrow I R

be such that

f(\cos 4 \theta)=\frac{2}{2-\sec ^{2} \theta}

for

\theta \in\left(0, \frac{\pi}{4}\right) \cup\left(\frac{\pi}{4}, \frac{\pi}{2}\right)

. Then the value (s)

of f\left(\frac{1}{3}\right)

is (are)

Options

A $1-\sqrt{\frac{3}{2}}$
B $1+\sqrt{\frac{3}{2}}$
C $1-\sqrt{\frac{2}{3}}$
D $1+\sqrt{\frac{2}{3}}$

Correct answer

A. $1-\sqrt{\frac{3}{2}}$

Step-by-step solution

Given :

f(\cos 4 \theta)=\frac{2}{2-\sec ^{2} \theta}=\frac{2 \cos ^{2} \theta}{2 \cos ^{2} \theta-1}

=\frac{1+\cos 2 \theta}{\cos 2 \theta}=1+\frac{1}{\cos 2 \theta}

Let

\cos 4 \theta=\frac{1}{3} \Rightarrow 2 \cos ^{2} 2 \theta-1=\frac{1}{3} \Rightarrow \cos 2 \theta=\pm \sqrt{\frac{2}{3}}

\therefore f(\cos 4 \theta)=1+\frac{1}{\cos 2 \theta}=1 \pm \sqrt{\frac{3}{2}}

f\left(\frac{1}{3}\right)=1 \pm \sqrt{\frac{3}{2}}

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