What is the gain margin of this system with gain of K?

Consider the system shown. Draw bode diagram of the open loop transfer function, and determine the value of K such that the phase margin is 50 degrees. What is the gain margin of this system with the gain of K? Please see attachment.

Frequency response via Fourier series

Stretch receptors in lungs are modeled as second-order systems whose input is the applied stretch, x(t), and output is the instantaneous firing frequency of the receptor, y(t). Assume that one such receptor can be described by the differential equation: d^2y/dt^2 + 2*d/dt + 4y(t) = 10x(t) +dx/dt Determine the frequency response of this receptor to stretching. There are devices that can apply sinusoidal stretch to a small segment of tissue. Calculate the steady-state response, y(t), to stretching an airway segment according to x(t) = 2cos(pi*t)u(t). I am not sure how to go about this using the Fourier Series...

Fourier Transform / Sampling

Please provide a detailed solution for the attached question (.jpeg).

Pole-Zero Plot

Please solve all parts (a through c) of the attached (.jpeg) problem.

Signal Processing Problem

Signal Processing & Wavelength. Problems # 6-7. See attached file for full problem description.