Let S be the set of all alpha in | vedclass

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Question

Given, $\cos 2 x+2 \sin x=2 \alpha-7$

$\Rightarrow 1-2 \sin ^{2} x+\alpha \sin x=2 \alpha-7$

$\Rightarrow 2 \sin ^{2} x-\alpha \sin x+2 \alpha-8

Vedclass · Accepted Answer

$[2, 6]$

Vedclass · Answer

Given, $\cos 2 x+2 \sin x=2 \alpha-7$

$\Rightarrow 1-2 \sin ^{2} x+\alpha \sin x=2 \alpha-7$

$\Rightarrow 2 \sin ^{2} x-\alpha \sin x+2 \alpha-8=0$

$\Rightarrow \sin x=\frac{\alpha \pm \sqrt{\alpha^{2}-8(2 \alpha-8)}}{4}$

$\Rightarrow \sin x=\frac{\alpha \pm(\alpha-8)}{4}$

$\Rightarrow \sin x=\frac{\alpha+\alpha-8}{4}, \frac{\alpha-\alpha+8}{4}$

$\sin x=2$ (Not possible)

For solution $-1 \leq \frac{2 \alpha-8}{4} \leq 1$

$-4 \leq 2 \alpha-8 \leq 4$

$\Rightarrow 4 \leq 2 \alpha \leq 12$

$\Rightarrow \alpha \in[2,6]$

Let $S$ be the set of all $\alpha \in R$ such that the equation, $cos\,2 x + \alpha \,sin\, x = 2\alpha -7$ has a solution. Then $S$ is equal to

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