$A$ ladder leans on a wall to reach a height of $3 \, m$ on the wall. If the ladder makes an angle of $30^{\circ}$ with the ground,the length of the ladder is ........... $m$.

Watching from the top of a building and from a window $10 \, m$ below the top,the angles of depression of a car on the ground are found to be $60^{\circ}$ and $45^{\circ}$ respectively. Find the distance of the car from the building. (in $m$)

Watching from a point $A$ on the ground $x \ m$ away from the base of the building. The height of the building is $y$ and the angle of elevation of the top of the building is found to be $25^\circ$. Then,$\ldots \ldots \ldots \ldots .$

The lower window of a house is at a height of $2\, m$ above the ground and its upper window is $4\, m$ vertically above the lower window. At a certain instant,the angles of elevation of a balloon from these windows are observed to be $60^{\circ}$ and $30^{\circ}$,respectively. Find the height of the balloon above the ground (in $m$).

Write 'True' or 'False' and justify your answer.
$\tan \theta$ increases faster than $\sin \theta$ as $\theta$ increases for $0^\circ < \theta < 90^\circ$.

Watching from the top of a $30 \ m$ high tower,the angle of depression of a stone on the ground is found to be $45^{\circ}$. Then the distance of the stone from the base of the tower is $\ldots \ldots \ldots \ m$.