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$-\frac{\pi}{4} \leq x \leq \frac{\pi}{4}$ અંતરાલમાં $\left|\begin{array}{lll}\sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x\end{array}\right|=0$ ના વાસ્તવિક ભિન્ન બીજની સંખ્યા મેળવો.
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$2$
$3$
$4$
Solution
$\left|\begin{array}{lll} \sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x \end{array}\right|=0$, $\frac{\pi}{4} \leq x \leq \frac{\pi}{4}$
Apply: $R_{1} \rightarrow R_{1}-R_{2}$ and $R_{2} \rightarrow R_{2}-R_{3}$
$\left|\begin{array}{ccc} \sin x-\cos x & \cos x-\sin x & 0 \\ 0 & \sin x-\cos x & \cos x-\sin x \\ \cos x & \cos x & \sin x \end{array}\right|=0$
$(\sin x-\cos x)^{2}\left|\begin{array}{ccc}1 & -1 & 0 \\ 0 & 1 & -1 \\ \cos x & \cos x & \sin x\end{array}\right|=0$
$(\sin x-\cos x)^{2}(\sin x+2 \cos x)=0$
$\sin x=\cos x \quad$ or $\quad \sin x=-2 \cos x$
$\tan x=1$ or $\quad \tan x=-2$ (Not valid)
$\therefore x=\frac{\pi}{4}$
