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 $= {{7\pi + 11\pi } \over 6} = {{18\pi } \over 6} = 3\pi$ $\therefore$ $k = 3$