JEE Main 2024 — 27 January, Shift 1 — Question 25

$\cos 2 \mathrm{x}+\mathrm{a} \cdot \sin \mathrm{x}=2 \mathrm{a}-7$

$a(\sin x-2)=2(\sin x-2)(\sin x+2)$

$\sin x=2, a=2(\sin x+2)$

$\Rightarrow a \in[2,6]$

$p=2 , q=6$

$r=\tan 9^{\circ}+\cot 9^{\circ}-\tan 27-\cot 27$

$r=\frac{1}{\sin 9 \cdot \cos 9}-\frac{1}{\sin 27 \cdot \cos 27}$

$=2\left[\frac{4}{\sqrt{5}-1}-\frac{4}{\sqrt{5}+1}\right]$

$r=4$

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