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 -

Two parallel towers $A$ and $B$ of different heights are at some distance $d$ on the same level ground. If the angle of elevation of a point $P$ at $20\,m$ height on tower $B$ from a point $Q$ at $10\,m$ height on tower $A$ is $\theta$,and this angle is equal to half the angle of elevation of point $R$ at $50\,m$ height on tower $A$ from point $P$ on tower $B$,then $\theta$ is equal to....$^o$

$A$ man is observing,from the top of a tower,a boat speeding towards the tower from a certain point $A$,with uniform speed. At that point,the angle of depression of the boat from the man's eye is $30^{\circ}$ (ignore the man's height). After sailing for $20 \text{ seconds}$ towards the base of the tower (which is at the level of water),the boat reaches a point $B$,where the angle of depression is $45^{\circ}$. Then,the time taken (in seconds) by the boat from $B$ to reach the base of the tower is:

$P$ is a point on the segment joining the feet of two vertical poles of heights $a$ and $b$. The angles of elevation of the tops of the poles from $P$ are $45^{\circ}$ each. Then,the square of the distance between the tops of the poles is

Two vertical poles are $150 \ m$ apart and the height of one is three times that of the other. If from the middle point of the line joining their feet,an observer finds the angles of elevation of their tops to be complementary,then the height of the shorter pole (in meters) is

The angle of elevation of a jet plane from a point $A$ on the ground is $60^{\circ}$. After a flight of $20 \, s$ at the speed of $432 \, km/h$, the angle of elevation changes to $30^{\circ}$. If the jet plane is flying at a constant height, then its height is ..... $m$. (in $\sqrt{3}$)