$ABC$ is a triangular park with $AB = AC = 100 \text{ metres}$. $A$ vertical tower of height $h$ is situated at the mid-point $P$ of $BC$. If the angles of elevation of the top of the tower $Q$ at $A$ and $B$ are $\cot^{-1}(3\sqrt{2})$ and $\csc^{-1}(2\sqrt{2})$ respectively,then the height of the tower (in metres) is

An aeroplane flying with uniform speed horizontally $1 \ km$ above the ground is observed at an elevation of $60^{\circ}$. After $10 \ s$, if the elevation is observed to be $30^{\circ}$, then the speed of the plane (in $km/h$) is

$AB$ is a vertical pole with $B$ at the ground level and $A$ at the top. $A$ man finds that the angle of elevation of the point $A$ from a certain point $C$ on the ground is $60^{\circ}$. He moves away from the pole along the line $BC$ to a point $D$ such that $CD = 7 \ m$. From $D$,the angle of elevation of the point $A$ is $45^{\circ}$. Then the height of the pole is

An aeroplane flying at a constant speed,parallel to the horizontal ground,$\sqrt{3} \ km$ above it,is observed at an elevation of $60^\circ$ from a point on the ground. If,after five seconds,its elevation from the same point is $30^\circ$,then the speed (in $km/hr$) of the aeroplane is

$A$ tower of height $b$ subtends an angle $\alpha$ at a point $O$ on the level of the foot of the tower and at a distance $a$ from the foot of the tower. If a pole of height $p$ mounted on the tower also subtends an equal angle $\alpha$ at $O$,the height of the pole $p$ is:

Two vertical poles of equal heights are $120 \, m$ apart. On the line joining their bottoms,$A$ and $B$ are two points. The angle of elevation of the top of one pole from $A$ is $45^\circ$ and that of the other pole from $B$ is also $45^\circ$. If $AB = 30 \, m$,then the height of each pole is.....$m$.