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CGPET 2017 Solved Paper

Section: Mathematics

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Question : 124 of 150

Marks: +1, -0

The number of solution of

\tan (5 \pi \cos \theta)=\cot (5 \pi \sin \theta)

\theta

(0,2\pi)

will be

28
14
4
2

Solution:

We have,

\tan (5 \pi \cos \theta)=\cot (5 \pi \sin \theta)

\Rightarrow \tan (5\pi\cos\theta)=\tan \left(\frac{\pi}{2}-5\pi\sin\theta\right)

\Rightarrow 5\pi\cos\theta+5\pi\sin\theta=\frac{\pi}{2}

\Rightarrow \cos\theta+\sin\theta=\frac{1}{10}

On multiplying both sides by

\frac{1}{\sqrt{2}}

, we get

\frac{\cos\theta}{\sqrt{2}}+\frac{\sin\theta}{\sqrt{2}}=\frac{1}{10\sqrt{2}}

\Rightarrow \cos\theta \sin \frac{\pi}{4}+\sin\theta \cos \frac{\pi}{4}=\frac{1}{10\sqrt{2}}

\Rightarrow \sin \left(\frac{\pi}{4}+\theta\right)=\frac{1}{10\sqrt{2}}

(\because \sin (A+B)=\sin A \cos B+\cos A \sin B)

\therefore

only 2 solutions are there in

(0,2\pi)

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