$A$ ladder $5 \ m$ long leans against a vertical wall. The bottom of the ladder is $3 \ m$ from the wall. If the bottom of the ladder is pulled $1 \ m$ farther from the wall,how much does the top of the ladder slide down the wall?

$A$ person is standing on a tower of height $15(\sqrt{3} + 1) \ m$ and observing a car coming towards the tower. He observes that the angle of depression changes from $30^\circ$ to $45^\circ$ in $3 \ s$. What is the speed of the car in $km/hr$?

$A$ vertical pole subtends an angle $\tan ^{-1}\left(\frac{1}{2}\right)$ at a point $P$ on the ground. If the angles subtended by the upper half and the lower half of the pole at $P$ are respectively $\alpha$ and $\beta$, then $(\tan \alpha, \tan \beta)$ is equal to

If a flagstaff 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 angle of elevation of the top of a tower from the top of a house is $60^\circ$ and the angle of depression of its base is $30^\circ$. If the horizontal distance between the house and the tower is $12 \ m$,then the height of the tower is

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 $..........$.