Question

The number of distinct real roots of $\left|\begin{array}{ccc}sinx& cosx& cosx\\ cosx& sinx& cosx\\ cosx& cosx& sinx\end{array}\right|=0$ 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)

$1$

$\left|\begin{array}{ccc}sinx& cosx& cosx\\ cosx& sinx& cosx\\ cosx& cosx& sinx\end{array}\right|=0$
Applying $C_1\to C_1+C_2+C_3$
$\left|\begin{array}{ccc}sinx+2cosx& cosx& cosx\\ sinx+2cosx& sinx& cosx\\ sinx+2cosx& cosx& sinx\end{array}\right|=0\Rightarrow (sinx+2cosx)\left|\begin{array}{ccc}1& cosx& cosx\\ 1& sinx& cosx\\ 1& cosx& sinx\end{array}\right|=0$
Applying $R_2\to R_2-R_1,R_3\to R_3-R_1$
$(sinx+2cosx)\left|\begin{array}{ccc}1& cosx& cosx\\ 0& sinx-cosx& 0\\ 0& 0& sinx-cosx\end{array}\right|=0\Rightarrow (sinx+2cosx){(sinx-cosx)}^2=0$
$\Rightarrow 2cosx+sinx=0$ or $sinx-cosx=0$
$\Rightarrow cotx=-\frac{1}{2}$ gives no solution in $-\frac{\pi }{4}\le x\le \frac{\pi }{4}$
or $tanx=1\Rightarrow x=\frac{\pi }{4}$
Hence, option 'C' is correct.