IPMAT Indore 2020 (MCQ) PYQs

For $0\lt\theta\lt\frac{\pi}{4}$, let $a=\left((\sin \theta)^{\sin \theta}\right)\left(\log _{2} \cos \theta\right), b=\left((\cos \theta)^{\sin \theta}\right)\left(\log _{2} \sin \theta\right), c=\left((\sin \theta)^{\cos \theta}\right)\left(\log _{2} \cos \theta\right)$ and $d=\left((\sin \theta)^{\sin \theta}\right)\left(\log _{2} \sin \theta\right)$. Then, the median value in the sequence $a, b, c, d$ is

If $\sin \alpha+\sin \beta=\frac{\sqrt{2}}{\sqrt{3}}$ and $\cos \alpha+\cos \beta=\frac{1}{\sqrt{3}}$, then the value of $\left(20 \cos \left(\frac{\alpha-\beta}{2}\right)\right)^{2}$ is _________.

If $cos \alpha$ + $cos \beta$ = 1 then the maximum value of $sin \alpha - sin \beta$ is