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माना $S=\left\{\theta \in(0,2 \pi): 7 \cos ^2 \theta-3 \sin ^2 \theta-2\right.$ $\left.\cos ^2 2 \theta=2\right\}$ है। तब सभी समीकरणों $x ^2-2\left(\tan ^2 \theta+\cot ^2 \theta\right) x +6 \sin ^2 \theta=0, \theta \in S$ के मूलों का योग है $..............$
$15$
$14$
$13$
$16$
Solution
$7 \cos ^{2} \theta-3 \sin ^{2} \theta-2 \cos ^{2} 2 \theta=2$
$4 \cos ^{2} \theta+3 \cos 2 \theta-2 \cos ^{2} 2 \theta=2$
$2(1+\cos 2 \theta)+3 \cos 2 \theta-2 \cos ^{2} 2 \theta=2$
$2 \cos ^{2} 2 \theta-5 \cos 2 \theta=0$
$\cos 2 \theta(2 \cos 2 \theta-5)=0$
$\cos 2 \theta=0$
$2 \theta=(2 n+1) \frac{\pi}{2}$
$\theta=(2 n+1) \frac{\pi}{4}$
$S=\left\{\frac{\pi}{4}, \frac{3 \pi}{4}, \frac{5 \pi}{4}, \frac{7 \pi}{4}\right\}$
For all four values of $\theta$
$X^{2}-2\left(\tan ^{2} \theta+\cot ^{2} \theta\right) x+6 \sin ^{2} \theta=0$
$x^{2}-4 x+3=0$
Sum of roots of all four equations $=4 \times 4=16$.
