Test Index

UPSC CDS 2 2022 Math Paper

Question Numbers: 46-47

Consider the following for the next two (02) items that follow :

The angle of elevation of a cloud at C from a point (P), H metres above the surface of a lake is 30°. The height of the cloud from the surface of the lake is 2H metres. Let θ be the angle of depression of the reflection of the cloud in the lake from the point P.

Question : 46 of 100

Marks: +1, -0

What is the value of θ ?

30°
45°
60°
Cannot be determined due to insufficient data

Validate

Solution:

Concept:

Use the tangent ratio:

\tan \theta = \frac{\text{perpendicular}}{\text{base}}

, and the known value

\tan 30^\circ = \frac{1}{\sqrt{3}}

Explanation:

From the geometry, let the cloud be at height

H

above the lake.

\text{triangle PAC}

\tan 30^\circ = \frac{AC}{AP} = \frac{H}{AP} = \frac{1}{\sqrt{3}}

This gives

AP = H\sqrt{3}

The reflection of the cloud in the lake appears at a depth equal to its height above the lake.

Thus, the vertical distance from the observer’s point to the reflection is

H

(cloud)

+

2H

(reflection depth)

= 3H

\text{triangle PAB}

\tan \theta = \frac{AB}{AP} = \frac{3H}{H\sqrt{3}} = \sqrt{3}

Since

\sqrt{3} = \tan 60^\circ

, we get

\theta = 60^\circ

Answer:

\theta = 60^\circ

(Option C).

Go to Question:

Prev Question

Next Question