If a flag-staff of $6 \ m$ high placed on the top of a tower throws a shadow of $2 \sqrt{3} \ m$ along the ground,then the angle (in degrees) that the sun makes with the ground is.....$^o$

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?

The angle of elevation of the top $P$ of a tower from the feet of one person standing due South of the tower is $45^{\circ}$ and from the feet of another person standing due West of the tower is $30^{\circ}$. If the height of the tower is $5 \text{ m}$,then the distance (in meters) between the two persons is equal to $..........$.

The angle of elevation of the top of a $TV$ tower from three points $A$,$B$ and $C$ in a straight line through the foot of the tower are $\alpha, 2 \alpha$ and $3 \alpha$ respectively. If $AB = a$,then the height of the tower is

From the top $A$ of a vertical wall $AB$ of height $30 \ m$,the angles of depression of the top $P$ and bottom $Q$ of a vertical tower $PQ$ are $15^{\circ}$ and $60^{\circ}$ respectively. $B$ and $Q$ are on the same horizontal level. If $C$ is a point on $AB$ such that $CB = PQ$,then the area (in $m^2$) of the quadrilateral $BCPQ$ is equal to

$A$ tower, of $x$ metres high, has a flagstaff at its top. The tower and the flagstaff subtend equal angles at a point distant $y$ metres from the foot of the tower. Then, the length of the flagstaff (in metres) is: