JEE Main 2026 — 23 January, Morning Shift — Question 18

$f(\theta)=4\left(\sin ^{4}\left(\frac{7 \pi}{2}-\theta\right)+\sin ^{4}(11 \pi+\theta)\right)-2\left(\sin ^{6}\left(\frac{3 \pi}{2}-\theta\right)+\sin ^{6}(9 \pi-\theta)\right)$ $\begin{aligned} & f(\theta)=4\left(\cos ^{4}(\theta)+\sin ^{4}(\theta)\right)-2\left(\cos ^{6} \theta+\sin ^{6} \theta\right) & \end{aligned}$

$f(\theta)=4\left(1-2 \sin ^{2} \theta \cos ^{2} \theta\right)-2\left(1-3 \sin ^{2} \theta \cos ^{2} \theta\right)$

$f(\theta)=2-2 \sin ^{2} \theta \cos ^{2} \theta$

$f(\theta)=2-\frac{\sin ^{2}(2 \theta)}{2}$

$\alpha=f(\theta)_{\max }=2$

$\beta=f(\theta)_{\min }=\frac{3}{2}$

$\Rightarrow \alpha+2 \beta=5$

$\text { Ans. }=5$