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UPSC CDS 2 2022 Math Paper

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

A chord of length

l

of a circle makes an angle

90^{\circ}

at the centre of the circle.

Question : 24 of 100

Marks: +1, -0

Solution: 👈: Video Solution

Area of circle

= \pi (\text{ radius })^2

Pythagoras theorem: In a right-angled triangle

Base

^2+

Perpendicular

^2=

Hypotenuse

^2

Now in

\triangle OAB

, by Pythagoras theorem,

1^2 = r^2 + r^2

\Rightarrow 1^2 = 2 r^2

\Rightarrow r = \frac{1}{\sqrt{2}}

We know that, the area of segment

A_1 = \frac{1}{2} \left(\frac{\pi}{2} - \sin\theta\right) r^2

\Rightarrow A_1 = \frac{1}{2} \left(\frac{\pi}{2} - \sin 90^{\circ}\right) \left(\frac{1}{\sqrt{2}}\right)^2

\Rightarrow A_1 = \frac{1^2}{4} \left(\frac{\pi}{2} - 1\right)

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