Question

Find the sum of the squares of the roots of the equation $\left|\begin{array}{ccc}3x^2& x^2+xcos\theta +co{s}^2\theta & x^2+xsin\theta +si{n}^2\theta \\ x^2+xcos\theta +co{s}^2\theta & 3co{s}^2\theta & 1+\frac{sin2\theta }{2}\\ x^2+xsin\theta +si{n}^2\theta & 1+\frac{sin2\theta }{2}& 3si{n}^2\theta \end{array}\right|=0$ are ............

Answer 1

Given equation is $\left|\begin{array}{ccc}3x^2& x^2+xcos\theta +co{s}^2\theta & x^2+xsin\theta +si{n}^2\theta \\ x^2+xcos\theta +co{s}^2\theta & 3co{s}^2\theta & 1+\frac{sin2\theta }{2}\\ x^2+xsin\theta +si{n}^2\theta & 1+\frac{sin2\theta }{2}& 3si{n}^2\theta \end{array}\right|=0$

if $x=\cos \theta$ equation becomes
$\left|\begin{array}{ccc}3{\cos }^2\theta & 3{\cos }^2\theta & 1+\sin \theta \cos \theta \\ 3{\cos }^2\theta & 3{\cos }^2\theta & 1+\sin \theta \cos \theta \\ 1+\sin \theta \cos \theta & 1+\sin \theta \cos \theta & 3{\sin }^2\theta \end{array}\right|=0$

as $C_1$ and $C_2$ are identical.
if $x=\sin \theta$ equation becomes

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

as $C_1$ and $C_3$ are identical.
$\therefore x=\sin \theta$ and $x=\cos \theta$ are roots of the given eqaution.
$\therefore$ Sum of squares of roots $={\sin }^2\theta +{\cos }^2\theta =1$