If $P$ is a point on the altitude $AD$ of the $\triangle ABC$,and $\angle ABP = \frac{2B}{3}$,then $AP$ is equal to

If the equation $2 \sin^2 x + \frac{\sin 2x}{2} = k$ has at least one real solution,then the sum of all integral values of $k$ is

In a $\triangle ABC$,if $\sin^2 B = \sin A$ and $2 \cos^2 A = 3 \cos^2 B$,then the triangle is

If $\alpha$ and $\beta$ are solutions of $\sin^2 x + a \sin x + b = 0$ as well as $\cos^2 x + c \cos x + d = 0$,then $\sin(\alpha + \beta)$ is equal to

In $\triangle ABC$,if $\sin^2 A + \sin^2 B = \sin^2 C$ and $l(AB) = 10$,then the maximum value of the area of $\triangle ABC$ is

$ABC$ is a triangular park with $AB = AC = 100 \, m$. $A$ clock tower is situated at the mid-point $D$ of $BC$. The angles of elevation of the top of the tower at $A$ and $B$ are $\cot^{-1} 3.2$ and $\csc^{-1} 2.6$ respectively. The height of the tower is .... $m$.