IPMAT Indore 2019 (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 _________.

The angle of elevation of the top of a pole from a point A on the ground is 30°. The angle of elevation changes to 45°, after moving 20 meters towards the base of the pole. Then the height of the pole, in meters, is

The set of all real value of $p$ for which the equation $3 \sin^2x + 12 \cos x - 3 = p$ has at least one solution is