An ideal band-pass channel 500Hz-2000 Hz is deployed for communication. A modem is designed to transmit bits at the rate of 4800 bits/s using 16-QAM. The roll-off factor of a pulse with a raised cosine spectrum that utilizes the entire frequency band is____________________

The response of the system $$G\left(s\right)\;=\;\frac{s\;-\;2}{\left(s\;+\;1\right)\left(s\;+\;3\right)}$$ to the unit step input u(t) is y(t). The value of $$\frac{\mathrm{dy}}{\mathrm{dt}}\;\mathrm{at}\;\mathrm t\;=\;0^+\;\mathrm{is}$$ ___________.

In the feedback system shown below $$$G\left(S\right)\;=\frac1{(s^2\;+\;2s)}$$$ The step response of the closed-loop system should have minimum settling time and have no overshoot. The required value of gain k to achieve this is __________.

In the feedback system shown below $${\rm{G(s) = }}{1 \over {\left( {s + 1} \right)\left( {s + 2} \right)\left( {s + 3} \right)}}$$ The positive value of 𝑘 for which the gain margin of the loop is exactly 0 dB and the phase margin of the loop is exactly zero degree is ____