UPSC CDS 2 2018 Math Paper

Angle subtended by the arc at the centre is twice the angle subtended by the same chord in the remaining part of the circle.
Therefore,∠AOB = 2x
Applying sine rule in Δ OAB, we get:
⇒‌‌

r

sin(90−x)

=‌‌

√3r

sin‌2‌x

⇒‌‌

r

cosx

=‌‌

√3r

2‌sinx⋅cosx

⇒sinx=‌‌

√3

2

⇒x=60∘
Since the sum of opposite angles in a cyclic quadrilateral is 180°, therefore,kx+x=180.
60k+60=180
60k=120⇒k=2