$A$ ladder leans on a wall to reach a height of $3 \, m$ on the wall. If the ladder makes an angle of $30^{\circ}$ with the ground,the length of the ladder is ........... $m$.

The angle of elevation of the top of a vertical pole as observed from a point $20\, m$ away from the base of the pole is $60^\circ$. Find the height of the pole. (in $m$)

From a balloon vertically above a straight road,the angles of depression of two cars at an instant are found to be $45^{\circ}$ and $60^{\circ}$. If the cars are $100 \, m$ apart,find the height of the balloon.

$A$ ladder $15 \, m$ long just reaches the top of a vertical wall. If the ladder makes an angle of $60^{\circ}$ with the wall,find the height of the wall in $m$.

Write 'True' or 'False' and justify your answer.
If the height of a tower and the distance of the point of observation from its foot,both,are increased by $10 \%$,then the angle of elevation of its top remains unchanged.

As observed from the top of a $510 \ m$ high tower,the angles of depression of two houses situated in the east and west of the tower are found to be $30^{\circ}$ and $60^{\circ}$ respectively. Find the distance between the two houses (in $m$).