Question 39 on communication from GATE (Graduate Aptitude Test in Engineering) 2012 Electronics and Communication Engineering paper.
Q39. The signal as shown is applied both to a phase modulator (with as the phase constant) and a frequency modulator (with as the frequency constant) having the same carrier frequency.
The ratio for the same maximum phase deviation is,
To answer this question, let us understand basics of Frequency modulation and Phase modulation. Am referring to the discussion in Section 10 of Electronic Communication, 4th Edition, Dennis Roddy, John Coolen (buy from Amazon.com, buy from Flipkart.com).
In frequency modulation, the modulation signal will cause a change in the instantaneous carrier frequency and is related as
is the frequency deviation constant expressed in hertz/volt (Hz/V) and is the carrier frequency in Hz.
The angular velocity radians per second.
Since the rate of change of phase is the angular velocity,
and so, the instantaneous phase is
The frequency modulated carrier signal is
In phase modulation, the modulation signal will cause a change in phase and the instantaneous phase is
is the phase deviation constant expressed in radians/volt and is the carrier frequency in radians per second.
The phase modulated carrier signal is
Now, applying the above equations to solve the problem :
In frequency modulation case, the maximum phase deviation happens when the term hits the maximum.
In phase modulation case, the maximum phase deviation happens when the term hits the maximum.
The ratio is,
Based on the above, the right choice is (B)
 GATE Examination Question Papers [Previous Years] from Indian Institute of Technology, Madras http://gate.iitm.ac.in/gateqps/2012/ec.pdf