Question

Let $\Delta \left(x\right)$ = $\left|\begin{array}{ccc}\sin x& \cos x& \sin 2x+\cos 2x\\ 0& 1& 1\\ 1& 0& -1\end{array}\right|$ then ${\Delta }^{{}^{\prime }}\left(x\right)$ vanishes at least once in

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

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A $(0,\pi /2)$
B $(\pi /2,\pi )$
C $(-\pi /2,0)$
D $(0,\pi /4)$

$\Delta \left(x\right)$ = $\left|\begin{array}{ccc}\sin x& \cos x& \sin 2x+\cos 2x\\ 0& 1& 1\\ 1& 0& -1\end{array}\right|$
$\Rightarrow {\Delta }^{\prime }\left(x\right)=\left|\begin{array}{ccc}\cos x& -\sin x& 2\cos 2x-2\sin 2x\\ 0& 1& 1\\ 1& 0& -1\end{array}\right|$
Now expanding along first column,
${\Delta }^{\prime }\left(x\right)=\cos x(-1-0)+1(-\sin x-2\cos 2x+2\sin 2x)=-(\sin x+\cos x)+2(\sin 2x-\cos 2x)$
Now ${\Delta }^{\prime }\left(0\right)=-3,{\Delta }^{\prime }\left(\pi /2\right)=1,{\Delta }^{\prime }\left(\pi \right)=-1,{\Delta }^{\prime }(-\pi /2)=3,{\Delta }^{\prime }\left(\pi /4\right)=2-\sqrt{2},$
Hence ${\Delta }^{\prime }\left(x\right)$ will vanish at least once in $(0,\pi /2),(\pi /2,\pi ),(0,\pi /4),(-\pi /2,0)$