QJEEM19Ma7125: JEE Main 2019 Mathematics - Class 12 JEEM19Ma7125 Solution: The number of values of (0, ) for which the system of linear…

$$\left|\begin{array}{ccc}1& 3& 7\\ -1& 4& 7\\ \sin 3\theta & \cos 2\theta & 2\end{array}\right|=0$$

(8 $-$ 7 cos 2 $\theta$ ) $-$ 3( $-$ 2 $-$ 7 sin 3 $\theta$ )

      +7 ( $-$ cos 2 $\theta$ $-$ 4 sin 3 $\theta$ ) = 0

14 $-$ 7 cos 2 $\theta$ + 21 sin 3 $\theta$ $-$ 7 cos 2 $\theta$

       $-$ 28 sin 3 $\theta$ = 0

14 $-$ 7 sin 3 $\theta$ $-$ 14 cos 2 $\theta$ = 0

14 $-$ 7 (3 sin $\theta$ $-$ 4 sin³ $\theta$ ) $-$ 14 (1 $-$ 2 sin² $\theta$ ) = 0

$-$ 21 sin $\theta$ + 28 sin³ $\theta$ + 28 sin² $\theta$ = 0

7 sin $\theta$ [ $-$ 3 + 4 sin² $\theta$ + 4 sin $\theta$ ] = 0 sin $\theta$ ,

4 sin² $\theta$ + 6 sin $\theta$ $-$ 2 sin $\theta$ $-$ 3 = 0

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

sin $\theta$ = ${{ - 3} \over 2}$ ; sin $\theta$ = ${1 \over 2}$

Hence, 2 solutions in (0, $\pi$ )