Shown below is a rectangular tub of water of depth $34 \mathrm{cm}$. An object $\mathrm{O}$ is at the bottom of the tub. The image of the object is formed at $\mathrm{I}$ for an observer at $\mathrm{Q}$.

(Note: The figure is not to scale)

Find the distance by which the object seem to be moved for the observer. Show your work and give valid reasons.

(Note: Take $\sqrt{2}=1.4, \sqrt{3}=1.7$)

In the figure below, the height of the girl is $1.5 \mathrm{m}$ and the height of the tree is $13.5 \mathrm{m}$.

(Note: The figure is not to scale)

If $\mathrm{AB}=12\sqrt{3} \mathrm{m}$, what is the angle of elevation (without symbol) of the top of the tree from her eyes? Show your steps.

In the giant wheel shown below, Gagan is sitting in one of the cabins which is $12 \mathrm{m}$ high from the platform. Jyoti and Karan are sitting in the lowest and the highest cabins from the platform respectively.

(Note: The figure is not to scale)

From Gagan, the angle of depression of Jothi and the angle of elevation of Kiran is $30°$ and $60°$ respectively.

i) What will be the angle of elevation of Gagan from Jyoti?

ii) What will be the angle of depression of Gagan from Kiran?

iii) Find the diameter of the giant wheel.

Show your steps with a diagram.

A ship was moving towards the shore at a uniform speed of $36 \mathrm{km}/\mathrm{h}$. Initially, the ship was $1.3 \mathrm{km}$ away from the foot of a lighthouse which is $173.2 \mathrm{m}$ in height.

(Note: The figure is not to scale)

Find the angle of depression, $x$, (without symbol) of the top of the lighthouse from the ship after the ship had been moving for $2 \mathrm{minutes}$.

Show your steps and give reasons. (Take $\sqrt{3}$ as $1.732$ and $\sqrt{2}$ as $1.414$)

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 :

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