QJEEM22Ma79a4: JEE Main 2022 Mathematics - Class 11 JEEM22Ma79a4 Solution: If the sum of solutions of the system of equations and in the…

Equation (1) $~~~2{\sin ^2}\theta = 1 - 2{\sin ^2}\theta$

$\Rightarrow {\sin ^2}\theta = {1 \over 4}$

$\Rightarrow \sin \theta = \, \pm \,{1 \over 2}$

$\Rightarrow \theta = {\pi \over 6},{{5\pi } \over 6},{{7\pi } \over 6},{{11\pi } \over 6}$

Equation (2) $~~~2{\cos ^2}\theta + 3\sin \theta = 0$

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

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

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

$\Rightarrow \sin \theta = {{ - 1} \over 2}$

$\Rightarrow \theta = {{7\pi } \over 6},{{11\pi } \over 6}$

$\therefore$ Common solutions $= {{7\pi } \over 6};\,{{11\pi } \over 6}$

Sum of solutions $= {{7\pi + 11\pi } \over 6} = {{18\pi } \over 6} = 3\pi$

$\therefore$ $k = 3$