$A$ pole stands vertically on the ground. The length of the shadow of the pole is $\frac{1}{\sqrt{3}}$ times the height of the pole. Then,the angle of elevation of the sun is $\ldots \ldots \ldots \ldots . . .$ (in $^\circ$)

The approximate value of $\sqrt{3}$ up to $2$ decimal places is $\ldots \ldots \ldots$

Write 'True' or 'False' and justify your answer.
The value of $\tan \theta$ (where $\theta < 90^{\circ}$) increases as $\theta$ increases.

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

From a point on the ground,the angle of elevation of the top of a tower is found to be $30^{\circ}$. Walking $30\, m$ towards the tower from that point,the angle of elevation of the top of the tower is found to be $60^{\circ}$. Find the height of the tower (in $m$).

The height of a tower is $50 \, m$ and the height of a building is $30 \, m$. Watching from the midpoint of the line segment joining their bases,the angle of elevation of the top of the tower is $\alpha$ and that of the building is $\beta$. Then,which of the following holds true?