If the sum of solutions of the system of equations $2 \sin ^{2} \theta-\cos 2 \theta=0$ and $2 \cos ^{2} \theta+3 \sin \theta=0$ in the interval $[0,2 \pi]$ is $k \pi$, then $k$ is equal to __________.

<p>Equation (1) $~~~2{\sin ^2}\theta = 1 - 2{\sin ^2}\theta$</p> <p>$\Rightarrow {\sin ^2}\theta = {1 \over 4}$</p> <p>$\Rightarrow \sin \theta = \, \pm \,{1 \over 2}$</p> <p>$$ \Rightarrow \theta = {\pi \over 6},{{5\pi } \over 6},{{7\pi } \over 6},{{11\pi } \over 6}$$</p> <p>Equation (2) $~~~2{\cos ^2}\theta + 3\sin \theta = 0$</p> <p>$\Rightarrow 2{\sin ^2}\theta - 3\sin \theta - 2 = 0$</p> <p>$\Rightarrow 2{\sin ^2}\theta - 4\sin \theta + \sin \theta - 2 = 0$</p> <p>$\Rightarrow (\sin \theta - 2)(2\sin \theta + 1) = 0$</p> <p>$\Rightarrow \sin \theta = {{ - 1} \over 2}$</p> <p>$\Rightarrow \theta = {{7\pi } \over 6},{{11\pi } \over 6}$</p> <p>$\therefore$ Common solutions $= {{7\pi } \over 6};\,{{11\pi } \over 6}$</p> <p>Sum of solutions $= {{7\pi + 11\pi } \over 6} = {{18\pi } \over 6} = 3\pi$</p> <p>$\therefore$ $k = 3$</p>