QJEEM22Mad1e9: JEE Main 2022 Mathematics - Class 11 JEEM22Mad1e9 Solution: Let . Then, the sum of roots of all the equations , is…

$7{\cos ^2}\theta - 3{\sin ^2}\theta - 2{\cos ^2}2\theta = 2$

$\Rightarrow 4\left( {{{1 + \cos 2\theta } \over 2}} \right) + 3\cos 2\theta - 2{\cos ^2}2\theta = 2$

$\Rightarrow 2 + 5{\cos ^2}\theta - 2{\cos ^2}2\theta = 2$

$\Rightarrow \cos 2\theta = 0$ or ${5 \over 2}$ (rejected)

$\Rightarrow \cos 2\theta = 0 = {{1 - {{\tan }^2}\theta } \over {1 + {{\tan }^2}\theta }} \Rightarrow {\tan ^2}\theta = 1$

$\therefore$ Sum of roots $= 2({\tan ^2}\theta + {\cot ^2}\theta ) = 2 \times 2 = 4$

But as $\tan \theta = \, \pm \,1$ for ${\pi \over 4},{{3\pi } \over 4},\,{{5\pi } \over 4},\,{{7\pi } \over 4}$ in the interval $(0,\,2\pi )$

$\therefore$ Four equations will be formed

Hence sum of roots of all the equations $= 4 \times 4 = 16.$