IPMAT Indore (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 _________.

Ayesha is standing atop a vertical tower $200 m$ high and observes a car moving away from the tower on a straight, horizontal road from the foot of the tower. At 11:00 AM, she observes the angle of depression of the car to be $45^{\circ}$. At 11:02 AM, she observes the angle of depression of the car to be $30^{\circ}$. The speed at which the car is moving is approximately

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