$A$ ladder rests against a wall so that its top touches the roof of the house. If the ladder makes an angle of $60^{\circ}$ with the horizontal and the height of the house is $6\sqrt{3} \text{ m}$,then the length of the ladder is ..... $m$.

At a distance $2h$ from the foot of a tower of height $h$,the tower and a pole at the top of the tower subtend equal angles. 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

At a point on the ground,the angle of elevation of a tower is such that its cotangent is $3/5$. On walking $32 \, m$ towards the tower,the cotangent of the angle of elevation becomes $2/5$. The height of the tower is .... $m$

The angle of elevation of the summit of a mountain from a point on the ground is $45^{\circ}$. After climbing up $1 \ km$ towards the summit at an inclination of $30^{\circ}$ from the ground,the angle of elevation of the summit is found to be $60^{\circ}$. Then the height (in $km$) of the summit from the ground is

In a triangular park $PQR$,$PQ = PR = 200 \ m$. $A$ $TV$ tower stands at the mid-point $M$ of $QR$. If the angles of elevation of the top of the tower at $P, Q,$ and $R$ are $45^o, 30^o,$ and $30^o$ respectively,then the height of the tower (in $m$) is: