Complex Numbers 2 Question 17

17.

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

(a) $x=n \pi$

(b) $x=0$

(c) $x=(n+1 / 2) \pi$

(d) no value of $x$

(1988, 2M)

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Answer:

Correct Answer: 17. ( d)

Solution:

  1. Since, $(\overline{\sin x+i \cos 2 x})=\cos x-i \sin 2 x$

$\Rightarrow \quad \sin x-i \cos 2 x=\cos x-i \sin 2 x$

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

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

$\Rightarrow \quad x=\pi / 4$ and $x=\pi / 8$ which is not possible at same time.

Hence, no solution exists.



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