UPSC 2014 CDS II Math Paper

In ΔADC,AC = CD (Radius of the circle)
∠ADC = ∠DAC
Remember: Equal sides have equal angles opposite to them.
Now,⇒2∠ADC=180°−60°=120°
⇒∠ADC=60°
So, ΔACD is an equilateral triangle.
Hence, statement I is incorrect
In equilateral ΔACD,CF ⊥ AD
∴ ∠DCF = ∠SCF = 30°
Remember:
Perpendicular from the vertex bisects the opposite side and the vertex angle.
In ΔCDF,sin‌30°=

DF

CD

=

1

2

=

DF

x

⇒DF=

x

2

Hence, statement II is correct