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

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

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

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

Given that $\cos x + \cos y = 1$, the range of $\sin x - \sin y$ 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 _________.