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 __________

For the system shown in the figure, Y (s) / X (s) = __________.

A unity feedback control system is characterized by the open-loop transfer function $$$G\left(s\right)\;=\;\frac{2\left(s+1\right)}{s^3+ks^2+2s+1}$$$ The value of k for which the system oscillates at 2 rad/s is ________.

A unity feedback control system is characterized by the open loop transfer function $$G(s) = {{10k\left( {s + 2} \right)} \over {\left( {{s^3} + 3{s^2} + 10} \right)}}$$ The Nyquist path and the corresponding Nyquist plot of g(s) are shown in the figures below. If 0 < K < 1, then number of poles of the closed loop transfer function that lie in the right half of the s-plane is