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. \N
• B. 2
• C. 1
• D. 3

Answer 1

The correct option is C 1

Given, $\left|\begin{array}{ccc}sinx& cosx& cosx\\ cosx& sinx& cosx\\ cosx& cosx& sinx\end{array}\right|=0$
Applying $C_1\to C_1+C_2+C_3$
$=\left|\begin{array}{ccc}sinx+2cosx& cosx& cosx\\ sinx+2cosx& sinx& cosx\\ sinx+2cosx& cosx& sinx\end{array}\right|$
$=(2cosx+sinx)\left|\begin{array}{ccc}1& cosx& cosx\\ 1& sinx& cosx\\ 1& cosx& sinx\end{array}\right|=0$
Applying $R_2\to R_2-R_1,R_3\to R_3-R_1$
$\Rightarrow (2cosx+sinx)\left|\begin{array}{ccc}1& cosx& cosx\\ 0& sinx-cosx& 0\\ 0& 0& sinx-cosx\end{array}\right|=0\Rightarrow (2cosx+sinx)(sinx-cosx{)}^2=0$
$\Rightarrow$ 2 cos x + sin x = 0 or sin x - cos x = 0
$\Rightarrow$ 2 cos x =-sin x or sin x = cos x
$\Rightarrow cotx=-\frac{1}{2}$ gives no solution in $-\frac{\pi }{4}\le x\le \frac{\pi }{4}$
and sin x = cos x $\Rightarrow$ tan x = 1 $\Rightarrow$ $x=\frac{\pi }{4}$