When the sun is $30^{\circ}$ above the horizontal,the length (in $m$) of the shadow cast by a building $50 \ m$ high is

Two towers are situated on opposite banks of a river. One tower is $60 \ m$ high. From the top of this tower,the angles of depression of the top and the bottom of the other tower are $30^{\circ}$ and $60^{\circ}$ respectively. Find the width of the river and the height of the other tower (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

If $r \sin \theta = 1$ and $r \cos \theta = \sqrt{3}$,then the value of $(\sqrt{3} \tan \theta + 1)$ is

$A$ $10 \ m$ long ladder is placed against a wall. It is inclined at an angle of $30^{\circ}$ to the ground. The distance (in $m$) of the foot of the ladder from the wall is (Given $\sqrt{3} = 1.732$)

$A$ man standing on the bank of a river observes that the angle subtended by a tree on the opposite bank is $60^{\circ}$. When he moves $36 \ m$ away from the bank,he finds the angle to be $30^{\circ}$. Find the breadth of the river (in $m$).