$A$ pole stands vertically inside a triangular park $\Delta ABC$. Let the angle of elevation of the top of the pole from each corner of the park be $\frac{\pi}{3}$. If the radius of the circumcircle of $\Delta ABC$ is $2$,then the height of the pole is equal to:

The angle of elevation of the top of a mobile tower from three points $P, Q$ and $R$ (on a straight line through the foot of the tower) are $\alpha, \beta$ and $\gamma$ respectively. If all three points lie on the same side of the foot of the tower and $\alpha : \beta : \gamma = 1 : 2 : 3$ and $PQ = l$,then the height of the tower is -

The sum of angles of elevation of the top of a tower from two points distant $a$ and $b$ from the base and in the same straight line with it is $90^{\circ}$. Then, the height of the tower is

$A$ vertical pole subtends an angle $\tan ^{-1}\left(\frac{1}{2}\right)$ at a point $P$ on the ground. If the angles subtended by the upper half and the lower half of the pole at $P$ are respectively $\alpha$ and $\beta$, then $(\tan \alpha, \tan \beta)$ is equal to

An aeroplane is flying at a constant speed,parallel to the horizontal ground at a height of $5 \text{ km}$. $A$ person on the ground observed that the angle of elevation of the plane is changed from $15^{\circ}$ to $30^{\circ}$ in the duration of $50 \text{ seconds}$. The speed of the plane (in $\text{kmph}$) is:

$A$ vertical pole fixed to the horizontal ground is divided in the ratio $3:7$ by a mark on it,with the lower part shorter than the upper part. If the two parts subtend equal angles at a point on the ground $18 \ m$ away from the base of the pole,then the height of the pole (in $meters$) is: