Question

The values of $\theta$ lying between $\theta =0$ and $\theta =\pi /2$ and satisfying the equation $\Delta =\left|\begin{array}{ccc}1+{\cos }^2\theta & {\sin }^2\theta & 4\sin 4\theta \\ {\cos }^2\theta & 1+{\sin }^2\theta & 4\sin 4\theta \\ {\cos }^2\theta & {\sin }^2\theta & 1+4\sin 4\theta \end{array}\right|=0$ are given by

• A. $\pi /24,5\pi /24$
• B. $7\pi /24,11\pi /24$
• C. $5\pi /24,7\pi /24$
• D. $11\pi /24,\pi /24$

Answer 1

Answer: (B)

$C_3\to C_3+C_1+C_2$, gives $\Delta =(2+4\sin 4\theta )\left|\begin{array}{ccc}1+{\cos }^2\theta & {\sin }^2\theta & 1\\ {\cos }^2\theta & 1+{\sin }^2\theta & 1\\ {\cos }^2\theta & {\sin }^2\theta & 1\end{array}\right|$
Applying $R_3\to R_3-R_2,R_2\to R_2-R_1$, we get $\Delta =2(1+2\sin 4\theta )\left|\begin{array}{ccc}1+{\cos }^2\theta & {\sin }^2\theta & 1\\ -1& 1& 0\\ 0& -1& 0\end{array}\right|$
$=2(1+2\sin 4\theta )$
$\Delta =0\Rightarrow \sin 4\theta =-1/2$
Now, $0\le \theta \le \frac{\pi }{2}\Rightarrow 0\le 4\theta \le 2\pi$
$\therefore 4\theta =\frac{7\pi }{6},\frac{11\pi }{6}\Rightarrow \theta =\frac{7\pi }{24},\frac{11\pi }{24}$