An aeroplane flying horizontally $1 \ km$ above the ground is observed at an elevation of $60^\circ$ and after $10$ seconds the elevation is observed to be $30^\circ$. The uniform speed of the aeroplane in $km/h$ is

$A$ 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?

$A$ balloon is descending at a rate of $4 \, m/min$. Its angle of elevation from a point on the ground is $45^\circ$,which reduces to $30^\circ$ after $10 \, minutes$. At what distance from the observer will the balloon touch the ground?

$A$ man on the top of a vertical tower observes a car moving at a uniform speed towards the tower on a horizontal road. If it takes $18 \text{ min}$ for the angle of depression of the car to change from $30^\circ$ to $45^\circ$,then after this,the time taken (in min) by the car to reach the foot of the tower is:

The angle of elevation of the sun,when the shadow of the pole is $\sqrt{3}$ times the height of the pole,is ....$^o$

$A$ flag-post $20 \ m$ high standing on the top of a house subtends an angle whose tangent is $\frac{1}{6}$ at a distance $70 \ m$ from the foot of the house. The height of the house is......$m$