$A$ man from the top of a $100 \ m$ high tower sees a car moving towards the tower. The angle of depression is $30^{\circ}$. After some time,the angle of depression becomes $60^{\circ}$. The distance travelled by the car during this time is:

The angle of elevation of the top $P$ of a vertical tower $PQ$ of height $10$ from a point $A$ on the horizontal ground is $45^{\circ}$. Let $R$ be a point on $AQ$ and from a point $B$,vertically above $R$,the angle of elevation of $P$ is $60^{\circ}$. If $\angle BAQ = 30^{\circ}$,$AB = d$ and the area of the trapezium $PQRB$ is $\alpha$,then the ordered pair $(d, \alpha)$ is.

$A$ 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.

The shadow of a tower standing on a level ground is found to be $60 \ m$ longer when the Sun's altitude is $30^{\circ}$ than when it is $45^{\circ}$. The height of the tower is

The upper part of a tree broken by the wind makes an angle of $30^\circ$ with the ground. The distance from the root to the point where the top of the tree touches the ground is $10 \, m$. What was the original height of the tree in meters?

Two poles standing on a horizontal ground are of heights $5 \, m$ and $10 \, m$ respectively. The line joining their tops makes an angle of $15^o$ with the ground. Then the distance (in $m$) between the poles is: