Q1: If the long chord and tangent length of a circular curve of radius R are equal the angle of deflection, is

ANS:D - 120°

Long chord = 2R*sinθ/2, and,
For Tangent length= R* tanθ/2.
A/c to question,
Long chord = tangent length.

2R * sinθ/2 = R * tanθ/2.
=> 2 sinθ/2 = (sinθ/2 ÷cosθ/2)
=> 2 = 1÷ cosθ/2.
=> cosθ/2 = 1/2 = cos60°

=> θ/2 = 60.
=> θ =120. Long chord = 2*Rsinθ/2,
Tangent = Rtanθ/2.
Equating both, we get,
θ/2 = 60.
θ =120. If LC = T, then you have an equilateral triangle: 60-60-60.

The deflection angle at the PI is simply 180 - interior angle 60 = 120.