$A$ $4 \, m$ long ladder leans on a wall such that its lower end remains $2 \, m$ away from the base of the wall. Then,the ladder makes an angle of measure $\ldots \ldots$ with the ground. (in $^\circ$)

From the top of a $50 \, m$ high tower,the angle of depression of a parked car is found to be $30^\circ$. Find the distance between the car and the base of the tower. (in $m$)

Watching from the top of an $80 \, m$ high hill,the angle of elevation of the top of a tower is $30^\circ$ and the angle of depression of the base of the tower is $45^\circ$. Find the height of the tower 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.

The height of a lighthouse is $100\, m$. Two boats are anchored in the opposite directions from the lighthouse. The angle of depression of each boat from the top of the lighthouse is found to be $30^\circ$. Find the distance between these two boats (in $m$).

$A$ ladder rests against a vertical wall at an inclination $\alpha$ to the horizontal. Its foot is pulled away from the wall through a distance $p$ so that its upper end slides a distance $q$ down the wall and then the ladder makes an angle $\beta$ to the horizontal.
Show that $\frac{p}{q}=\frac{\cos \beta-\cos \alpha}{\sin \alpha-\sin \beta}$