If a person travels from a point $L$,towards east for $12 \ km$ and then travels $5 \ km$ towards north and reaches a point $M$,then the shortest distance from $L$ to $M$ is: (in $km$)

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

An aeroplane flying at a height of $5000 \; m$ from the ground passes vertically above another aeroplane at an instant. At that instant,the angles of elevation of the two aeroplanes from the same point on the ground are $60^{\circ}$ and $45^{\circ}$ respectively. The vertical distance between the aeroplanes at that instant is:

$A$ $10 \ m$ long ladder is placed against a wall. It is inclined at an angle of $30^{\circ}$ to the ground. The distance (in $m$) of the foot of the ladder from the wall is (Given $\sqrt{3} = 1.732$)

If $x = a \cos \theta + b \sin \theta$ and $y = b \cos \theta - a \sin \theta,$ then $x^{2} + y^{2}$ is equal to

From the top of a $25 \ m$ high cliff,the angle of elevation of a tower is found to be equal to the angle of depression of the foot of the tower. Find the height of the tower in $m$.