Question

$\Delta \left(x\right)=\left|\begin{array}{ccc}\sin x& \cos x& \sin 2x+\cos 2x\\ 0& 1& 1\\ 1& 0& -1\end{array}\right|$

${\Delta }^{\prime }\left(x\right)$ vanishes at least once in

• A. $(0,\frac{\pi }{2})$
• B. $(\frac{\pi }{2},\pi )$
• C. $(0,\frac{\pi }{4})$
• D. $(-\frac{\pi }{2},0)$

Answer 1

Answer: (A)

$(0,\frac{\pi }{2})$

${\Delta }^{\prime }\left(x\right)=\left|\begin{array}{ccc}\cos x& -\sin x& 2\cos 2x-2\sin 2x\\ 0& 1& 1\\ 1& 0& -1\end{array}\right|$

$\Rightarrow {\Delta }^{\prime }\left(x\right)=-\cos x-\sin x-1(2\cos 2x-2\sin 2x)$

(Expanded along first column)

$\Rightarrow {\Delta }^{\prime }\left(x\right)=2\sin 2x-\cos x-\sin x-2\cos 2x$

Clearly,

${\Delta }^{\prime }\left(x\right)$ is continuous and ${\Delta }^{\prime }\left(0\right)=-3<0$ and ${\Delta }^{\prime }\left(\frac{\pi }{2}\right)=1>0$
Hence,

${\Delta }^{\prime }\left(x\right)$ will vanish at least once in $(0,\frac{\pi }{2})$

Hence, option A.