Question

If the number of distinct real roots of $\left|\begin{array}{ccc}\sin x& \cos x& \cos x\\ \cos x& \sin x& \cos x\\ \cos x& \cos x& \sin x\end{array}\right|$ = 0 in the interval $-\frac{\pi }{4}\le x\le \frac{\pi }{4}$ is

• A. $0$
• B. $2$
• C. $1$
• D. $3$

Answer 1

Answer: (C)

$1$

opening determinant

$\sin x({\sin }^{2}x-{\cos }^{2}x)-\cos x(\sin x\cos x-{\cos }^{2}x)+\cos x({\cos }^{2}x-\sin x\cos )$

$={\sin }^{3}x-\sin x{\cos }^{2}x-\sin x.{\cos }^{2}x+{\cos }^{3}x+{\cos }^{3}x-\sin x{\cos }^{2}x$

$\Rightarrow {\sin }^{3}x+2{\cos }^{3}x-3\sin x{\cos }^{2}x=0$

$\Rightarrow$ Dividing overall by ${\cos }^{3}x$

$\frac{{\sin }^{3}x}{{\cos }^{3}x}+2-\frac{3\sin x{\cos }^{2}x}{{\cos }^{3}x}=0$

$\Rightarrow {\tan }^{3}x+2-3{\tan }^{2}x=0$

net turn $=t$

$t^3-3t^2+2=0$

$t-1)\overline{t^3-3t^2+2}(t^2-2t-2t^3-t^2-+\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_-2t^2+2-2t^2+2t+-\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_-2t+2-2t+2\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_x$

$\therefore (t-1)(t^2-x-2)=0$

$\Rightarrow t=1$ in internal of $-\frac{\pi }{4}$ to $\frac{\pi }{4}$

$\Rightarrow \tan x=1$

$\Rightarrow x=\pi /4$

Hence only $1$ mole exist.