An observer on the top of a tree finds the angle of depression of a car moving towards the tree to be $30^\circ$. After $3 \text{ minutes}$,this angle becomes $60^\circ$. After how much more time will the car reach the tree? (in minutes)

If the angles of depression of the top and bottom of a short building from the top of a tall building are $30^{\circ}$ and $60^{\circ}$ respectively, then the ratio of the heights of the short and tall buildings is

The angle of elevation of the top $P$ of a vertical tower $PQ$ of height $10$ from a point $A$ on the horizontal ground is $45^{\circ}$. Let $R$ be a point on $AQ$ and from a point $B$,vertically above $R$,the angle of elevation of $P$ is $60^{\circ}$. If $\angle BAQ = 30^{\circ}$,$AB = d$ and the area of the trapezium $PQRB$ is $\alpha$,then the ordered pair $(d, \alpha)$ 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

$A$ $20 \ m$ high flag pole is fixed on an $80 \ m$ high pillar. At a point on the ground $50 \ m$ away from the base of the pillar,the flag pole subtends an angle $\alpha$. Find the value of $\tan \alpha$.

The shadow of a tower standing on a level ground is found to be $60 \ m$ longer when the Sun's altitude is $30^{\circ}$ than when it is $45^{\circ}$. The height of the tower is