IPMAT Indore 2023 (MCQ) PYQs

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 _________.

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 the angles $A, B, C$ of a triangle are in arithmetic progression such that $\sin(2A + B) = 1/2$ then $\sin(B + 2C)$ is equal to