Question

The values of $\theta$ lying between $\theta =0$ and $\theta =\frac{\pi }{2}$ and satisfying the equation
$\left|\begin{array}{ccc}1+{\sin }^2\theta & {\cos }^2\theta & 4\sin 6\theta \\ {\sin }^2\theta & 1+{\cos }^2\theta & 4\sin 6\theta \\ {\sin }^2\theta & {\cos }^2\theta & 1+4\sin 6\theta \end{array}\right|$
are given by

• A. $\frac{\pi }{36},\frac{5\pi }{36}$
• B. $\frac{7\pi }{36},\frac{11\pi }{3}$
• C. $\frac{5\pi }{36},\frac{7\pi }{36}$
• D. $\frac{11\pi }{36},\frac{\pi }{36}$

Answer 1

The correct option is B $\frac{7\pi }{36},\frac{11\pi }{3}$

$\left|\begin{array}{ccc}1+{\sin }^2\theta & {\cos }^2\theta & 4\sin 6\theta \\ {\sin }^2\theta & 1+{\cos }^2\theta & 4\sin 6\theta \\ {\sin }^2\theta & {\cos }^2\theta & 1+4\sin 6\theta \end{array}\right|=0$

Applying $R_3\to R_3-R_1,R_2\to R_2-R_1$

$\Rightarrow \left|\begin{array}{ccc}1+{\sin }^2\theta & {\cos }^2\theta & 4\sin 6\theta \\ -1& 1& 0\\ -1& 0& 1\end{array}\right|=0$

Applying $C_1\to C_1+C_2$

$\Rightarrow \left|\begin{array}{ccc}2& {\cos }^2\theta & 4\sin 6\theta \\ 0& 1& 0\\ -1& 0& 1\end{array}\right|=0\Rightarrow 2+4\sin 6\theta =0\Rightarrow \sin 6\theta =-\frac{1}{2}\Rightarrow 6\theta =n\pi +{(-1)}^n(-\frac{\pi }{6})\Rightarrow \theta =\frac{n\pi }{6}+{(-1)}^{n+1}\left(\frac{\pi }{36}\right)\Rightarrow \theta =\frac{7\pi }{36},\frac{11\pi }{36}$