$A$ person standing at a distance of $10 \ m$ from a pole observes that the angle subtended by the lower $\frac{1}{3}$rd part and the upper $\frac{2}{3}$rd part of the pole are the same. Find the height of the pole.

$A$ flag-post $20 \ m$ high standing on the top of a house subtends an angle whose tangent is $\frac{1}{6}$ at a distance $70 \ m$ from the foot of the house. The height of the house is......$m$

Two vertical poles $AB = 15 \ m$ and $CD = 10 \ m$ are standing apart on a horizontal ground with points $A$ and $C$ on the ground. If $P$ is the point of intersection of $BC$ and $AD$,then the height of $P$ (in $m$) above the line $AC$ 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

If the angles of elevation of the top of a tower from three collinear points $A, B$ and $C$ on a line leading to the foot of the tower are $30^o, 45^o$ and $60^o$ respectively,then the ratio $AB : BC$ is:

$A$ sphere with centre $O$ sits on the top of a pole as shown in the figure. An observer on the ground is at a distance $50 \ m$ from the foot of the pole. She notes that the angles of elevation from the observer to points $P$ and $Q$ on the sphere are $30^{\circ}$ and $60^{\circ}$,respectively. Then,the radius of the sphere in metres is