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Trigonometrical Equations
hard
अन्तराल $\left(\frac{\pi}{4}, \frac{7 \pi}{4}\right)$ में $x$ के मानों की संख्या, जिसके लिए $14 \operatorname{cosec}^2 x-2 \sin ^2 x=21-4$ $\cos ^2 x$ सत्य हो, होगी
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
