A modulating signal given By $$x\left(t\right)=5\sin\left(4\mathrm\pi10^3\mathrm t\;-\;10\mathrm{πcos}2\mathrm\pi10^3\mathrm t\right)$$ V is fed to a phase modulator with phase deviation constant k_p=0.5rad/V. If the carrier frequency is 20 kHz, the instantaneous frequency (in kHz) at t = 0.5 ms is __________

In the system shown in figure (a), m(t) is a low-pass signal with bandwidth W Hz. The frequency response of the band-pass filter H(f) is shown in figure (b). If it is desired that the output signal z(t) = 10x(t), the maximum value of W (in Hz) should be strictly less than ____________________.

The signal m(t) as shown is applied both to a phase modulator (with k_p as the phase constant) and a frequency modulator (with k_f as the frequency constant) having the same carrier frequency.

The ratio k_p/k_f (in rad/Hz) for the same maximum phase deviation is

The signal $$\cos\;\omega_ct\;+\;0.5\;\cos\;\omega_mt\;s\mathrm{in}\;\omega_ct$$ is