Question

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

• A. 0
• B. 2
• C. 1
• D. 3

Answer 1

Answer: (C)

1

Answer 2

1 & \cos x & \cos x \\ 1 & \sin x & \cos x \\ 1 & \cos x & \sin x \end{matrix} \right| = 0$$ Applying, $R_{2} \rightarrow R_{2} - R_{1}$ and $R_{3} \rightarrow R_{3} - R_{1}$ $$(2\cos x + \sin x)\left| \begin{matrix} 1 & \cos x & \cos x \\ 0 & \sin x - \cos x & 0 \\ 0 & 0 & \sin x - \cos x \end{matrix} \right| = 0$$ ⇒ $(2\cos x + \sin x)(\sin x - \cos x)^{2} = 0$ $\therefore\tan x = - 2,1$ But $\tan x \neq - 2$ in $\left\lbrack - \frac{\pi}{4},\frac{\pi}{4} \right\rbrack$. Hence $$\tan x = 1 \Rightarrow x = \frac{\pi}{4}$$