$AB$ is a vertical pole resting at the end $A$ on the level ground. $P$ is a point on the level ground such that $AP = 3 \, AB$. If $C$ is the mid-point of $AB$ and $CB$ subtends an angle $\beta$ at $P$,the value of $\tan \beta$ is

$A$ 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:

Two vertical poles of heights $20 \ m$ and $80 \ m$ stand apart on a horizontal plane. The height (in meters) of the point of intersection of the lines joining the top of each pole to the foot of the other,from this horizontal plane is:

$ABCD$ is a rectangular field. $A$ vertical lamp post of height $12 \ m$ stands at the corner $A$. If the angle of elevation of its top from $B$ is $60^{\circ}$ and from $C$ is $45^{\circ}$,then the area of the field is

$A$ man from the top of a $100 \ m$ high tower sees a car moving towards the tower at an angle of depression of $30^\circ$. After some time,the angle of depression becomes $60^\circ$. The distance (in meters) travelled by the car during the time is

Two vertical poles are $150 \ m$ apart and the height of one is three times that of the other. If from the middle point of the line joining their feet,an observer finds the angles of elevation of their tops to be complementary,then the height of the shorter pole (in meters) is