Shown below is a submarine scouting an enemy ship in the ocean using a sonar device. Sonar devices send out a sound pulse from a transducer, and then precisely measure the time it takes for the sound pulses to be reflected back to the transducer.

A sonar wave sent by the submarine hits the ship and returns back in $2 \mathrm{seconds}$. The speed of a sonar wave underwater is $1500 \mathrm{m}/\mathrm{s}$ and the submarine is diving at a depth of $750 \mathrm{m}$ below sea level.

Find the angle of elevation '$a$' (without units) of the ship from the submarine (without symbol). Show your steps.

The position of an eagle and two identical geese are shown in the figure below. All the birds are at the same height from the ground. Assume that the eagle can fly at the same speed in all directions and that the geese are unaware of the Eagle's intention and will not move from their positions.

(Note: The figure is not to scale)

If the eagle wants to attack the goose that is nearer to it, which one should it attack? Show your steps.

Two trees are $2d \mathrm{meters}$ apart. Ajay stood at a point midway between them and started walking in a direction perpendicular to the line connecting the two trees. After walking $d \mathrm{metres}$, he observed the angle of elevations to the tops of the two trees and found them to be complementary.

(Note: The figure is not to scale)

If one of the trees is thrice as tall as the other, find the height of the shorter tree, in terms of $d$. Show your work.

If the angle of elevation of a cloud from a point $h$ metres above a lake is $\alpha$ and the angle of depression of its reflection in the lake is $\beta$, prove that the height of the cloud is :

$\frac{h\left( \tan \beta +\tan \alpha \right)}{\tan \beta -\tan \alpha }$

The interior angles of the triangular swimming pool below are $80°, 60°,$ and $40°$. A place for pedestal $\left(P\right)$ marked in the middle of the pool, $5 \mathrm{m}$ away from each of its vertices.

Find the perimeter of the pool.

An electrician has to repair an electric fault on a pole of height $5 \mathrm{m}$. She needs to reach a point $1.3 \mathrm{m}$ below the top of the pole to undertake the repair work. What should be the length of the ladder that she should use which, when inclined at an angle of $60°$ to the horizontal, would enable her to reach the required position? Also, how far from the foot of the pole should she place the foot of the ladder? (You may take $\sqrt{3}=1.73$)

The distance between two points $\mathrm{A}$ and $\mathrm{B}$on the same side of a river bank is  $40 \mathrm{m}$. There is a point $\mathrm{C}$ on the opposite, parallel river bank such that $\angle \mathrm{BAC}=60°$ and $\angle \mathrm{ABC}=30°$.Find the width of the river.