If $r \sin \theta = 1$ and $r \cos \theta = \sqrt{3}$,then the value of $(\sqrt{3} \tan \theta + 1)$ is

From an aeroplane vertically over a straight horizontal road,the angles of depression of two consecutive milestones on the opposite sides of the aeroplane are observed to be $30^{\circ}$ and $60^{\circ}$. Find the height in miles of the aeroplane above the road.

$A$ tower standing on a horizontal plane subtends a certain angle at a point $160 \; m$ apart from the foot of the tower. On advancing $100 \; m$ towards it,the tower is found to subtend an angle twice as before. The height of the tower is (in $m$):

$A$ man standing at a point $P$ (or $L$ in the diagram) is watching the top of a tower $N$,which makes an angle of elevation of $30^{\circ}$ with the man's eye. The man walks some distance towards the tower to point $Q$ to watch its top and the angle of elevation becomes $60^{\circ}$. What is the distance (in units) between the base of the tower $M$ and the point $P$ (or $L$)?

The angle of elevation of the top of an unfinished tower at a point $120 \ m$ from its base is $45^{\circ}.$ How much higher (in $m$) must the tower be raised so that its angle of elevation is $60^{\circ}$ at the same point?

If a flagstaff of $6 \, m$ height placed on the top of a tower throws a shadow of $2 \sqrt{3} \, m$ along the ground,then the angle that the sun makes with the ground is