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જો $\sin 5x + \sin 3x + \sin x = 0$, તો $x$ ની શૂન્ય સિવાયની $0 \le x \le \frac{\pi }{2}$ ની વચ્ચેની કિમત મેળવો.
A
$\frac{\pi }{6}$
B
$\frac{\pi }{{12}}$
C
$\frac{\pi }{3}$
D
$\frac{\pi }{4}$
Solution
(c) $ \sin 5x + \sin 3x + \sin x = 0$
$ \Rightarrow $ $ – \sin 3x = \sin 5x + \sin x = 2\sin 3x\cos 2x$
$ \Rightarrow $ $\sin 3x = 0$
$ \Rightarrow $ $x = 0$
or $\cos 2x = – \frac{1}{2} = – \cos \,\left( {\frac{\pi }{3}} \right) = \cos \,\left( {\pi – \frac{\pi }{3}} \right)$
$ \Rightarrow $ $2x = 2n\pi \pm \left( {\pi – \frac{\pi }{3}} \right)\, $
$\Rightarrow x = n\pi \pm \left( {\frac{\pi }{3}} \right)$
For $x$ lying between $0$ and $\frac{\pi }{2}$, we get $x = \frac{\pi }{3}$.
Trick : Check with options.
Standard 11
Mathematics
