Trigonometrical Equations
hard

અંતરાલ $\left(\frac{\pi}{4}, \frac{7 \pi}{4}\right)$ માં $x$ ની એવી કેટલી કિંમતો મળે કે જેથી  $14 \operatorname{cosec}^{2} x-2 \sin ^{2} x=21-4 \cos ^{2} x$ થાય?

A

$2$

B

$7$

C

$5$

D

$4$

(JEE MAIN-2022)

Solution

$x \in\left(\frac{\pi}{4}, \frac{7 \pi}{4}\right)$

$14 \operatorname{cosec}^{2} x-2 \sin ^{2} x=21-4 \cos ^{2} x$

$=21-4\left(1-\sin ^{2} x\right)$

$=17+4 \sin ^{2} x$

$14 \operatorname{cosec} x-6 \sin ^{2} x=17$

let $\sin ^{2} x=p$

$\frac{14}{p}-6 p=17 \Rightarrow 14-6 p^{2}=17 p$

$6 p^{2}+17 p-14=0$

$p=-3.5, \frac{2}{3} \Rightarrow \sin ^{2} x=\frac{2}{3}$

$\Rightarrow \sin x=\pm \sqrt{\frac{2}{3}}$

$\therefore$ Total $4$ solutions

Standard 11
Mathematics

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