In a rectangle,if the angle between a diagonal and a side is $30^{\circ}$ and the length of the diagonal is $6 \text{ cm}$,the area (in $\text{cm}^2$) of the rectangle is:

$A$ person standing on the bank of a river observes that the angle of elevation of the top of a tree on the opposite bank is $60^{\circ}$. When he moves $40 \ m$ away from the bank,he finds the angle of elevation to be $30^{\circ}$. The breadth of the river is (in $m$):

If $x = a \cos \theta + b \sin \theta$ and $y = b \cos \theta - a \sin \theta,$ then $x^{2} + y^{2}$ is equal to

The angle of elevation of a tower from a distance of $100 \ m$ from its foot is $30^{\circ}$. Then the height of the tower is (in $m$):

$A$ 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 $m$) travelled by the car during this time is

The angles of elevation of the top of a tower from the points $P$ and $Q,$ at distances of $a$ and $b$ respectively from the base of the tower and in the same straight line with it,are complementary. The height of the tower is