Complex numbers sin x + i cos 2x and cos x - i sin 2x are conjugate to each other, for

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4-1.Complex numbers

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The complex numbers $sin\ x + i\ cos\ 2x$ and $cos\ x\ -\ i\ sin\ 2x$ are conjugate to each other, for

A

$x = n\pi ,\,n \in Z$

B

$x=0$

C

$x = \frac{{n\pi }}{2},\,n \in Z$

D

No value of $x$

Solution

$\sin x+i \cos 2 x=\cos x+i \sin 2 x$

$\Rightarrow \cos 2 x=\sin 2 x$ and $\sin x=\cos x$

$\Rightarrow \tan x=1$ and $\tan 2 x=1$

$x=\frac{\pi}{4}, \frac{5 \pi}{4}, \frac{9 \pi}{4} \quad x=\frac{\pi}{8}, \frac{5 \pi}{8}, \frac{9 \pi}{8}$

$\because$ both equation will not have solution simultaneously, hence answer is $( 4)$

Standard 11

Mathematics

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