Question

The number of solutions of equations
$\left|\begin{array}{ccc}\sin 3\theta & -1& 1\\ \cos 2\theta & 4& 3\\ 2& 7& 7\end{array}\right|=0$ in $[0,2\pi ]$ is

• A. $2$
• B. $3$
• C. $4$
• D. $5$

Answer 1

The correct option is A $2$

$\left[\begin{array}{ccc}\sin 3\theta & -1& 1\\ \cos 2\theta & 4& 3\\ 2& 7& 7\end{array}\right]=\sin 3\theta (28-21)+1(7\cos 2\theta -6)+1(7\cos 2\theta -8)=0$

$7\sin 3\theta +7\cos 2\theta +7\cos 2\theta =14$

$7\sin 3\theta +14\cos 2\theta =14$

$\sin 3\theta +2\cos 2\theta =2$

$\sin 3\theta =2(1-\cos 2\theta )$

$3\sin \theta -4{\sin }^3\theta =2\times 2{\sin }^2\theta$

$\sin \theta (3-4{\sin }^2\theta )=4{\sin }^2\theta$

$4{\sin }^2\theta +4\sin \theta -3=0$

$4{\sin }^2\theta +6\sin \theta -2\sin \theta -3=0$

$2\sin \theta (2\sin \theta +3)-1(2\sin \theta +3)=0$

$\sin \theta =\frac{1}{2}\sin \theta =-\frac{3}{2}$

$\therefore 2$ solutions i.e., $30°$ & $150°$