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4-1.Complex numbers
medium
The complex numbers $\sin x + i\cos 2x$ and $\cos x - i\sin 2x$ are conjugate to each other for
A
$x = n\pi $
B
$x = \left( {n + \frac{1}{2}} \right)\pi $
C
$x = 0$
D
No value of $x$
(IIT-1988)
Solution
(d) $\sin x + i\cos 2x$and $\cos x – i\sin 2x$are conjugate to each other if $sin x=cos x$ and $\cos 2x = \sin 2x$
or $\tan x = 1$==> $x = \frac{\pi }{4},\frac{{5\pi }}{4},\frac{{9\pi }}{4},……$ $(i)$
and $\tan 2x = 1$==> $2x = \frac{\pi }{4},\frac{{5\pi }}{4},\frac{{9\pi }}{4},……..$
or $x = \frac{\pi }{8},\frac{{5\pi }}{8},\frac{{9\pi }}{8}$……. $(ii)$
There exists no value of $x$common in $(i) $ and $ (ii)$. Therefore there is no value of $x$ for which the given complex numbers are conjugate.
Standard 11
Mathematics
