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Trigonometrical Equations
hard
समुच्चय $S=\left\{\theta \epsilon[-4 \pi, 4 \pi]: 3 \cos ^2 2 \theta+\right.$ $6 \cos 2 \theta-10 \cos ^2 \theta+5=0$ में अवयवों की संख्या है $........$
A
$32$
B
$33$
C
$34$
D
$35$
(JEE MAIN-2022)
Solution
$3 \cos ^{2} 2 \theta+6 \cos 2 \theta-10 \cos ^{2} \theta+5=0$
$3 \cos ^{2} 2 \theta+6 \cos 2 \theta-5(1+\cos 2 \theta)+5=0$
$3 \cos ^{2} 2 \theta+\cos 2 \theta=0$
$\operatorname{Cos} 2 \theta=0$ OR $\cos 2 \theta=-1 / 3$
$\theta \in[-4 \pi, 4 \pi]$
$2 \theta=(2 n +1) \cdot \frac{\pi}{2}$
$\therefore \quad \theta=\pm \pi / 4 . \pm 3 \pi / 4 \ldots \ldots \pm 15 \pi / 4$
Similarly $\cos 2 \theta=-1 / 3$ gives $16$ solution
Standard 11
Mathematics
