Let the set of all such that the equation has a solution be and then is equal to

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Solution

Q.
Let the set of all $a \in R$ such that the equation $\cos 2 x + a \sin x = 2 a - 7$ has a solution be $[p, q]$ and $r = \tan 9 ° - \tan 27 ° - \frac{1}{\cot 63 °} + \tan 81 °,$ then $p q r$ is equal to _____ . [2024]

Ans.
(48)

$\cos 2 x + a \sin x = 2 a - 7$

$1 - 2 \sin^{2} x + a \sin x = 2 a - 7$

Let $t = \sin x$    $∴$ $2 t^{2} - a t + 2 a - 8 = 0$

$\Rightarrow (t - 2) (2 t + 4 - a) = 0 \Rightarrow t = 2 or t = \frac{a - 4}{2}$

$∴$ $\sin x = 2 (not possible) or 2 \sin x = a - 4$

$\Rightarrow a = 2 \sin x + 4 \in [2, 6]$    $[∵$ $\sin x \in [- 1, 1]]$

$So, p = 2, q = 6$

Now, $r = \tan 9 ° - \tan 27 ° - \frac{1}{\cot 63 °} + \tan 81 °$

$= (\tan 9 ° + \cot 9 °) - (\tan 27 ° + \cot 27 °)$

$= 2 c o s e c 18 ° - 2 c o s e c 54 °$    $[∵$ $\tan θ + \cot θ = 2 c o s e c 2 θ]$

$= 2 [\frac{4}{\sqrt{5} - 1} - \frac{4}{\sqrt{5} + 1}] = 4 \Rightarrow r = 4$

$∴$ $p q r = 2 \times 6 \times 4 = 48$

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