$1.$Analyse the circuit given below and find the expression of Vout as a function of input voltages Vin and V$2.$

$2.$Obtain the voltage transfer function E$2/$E$1$ for the following circuit. Then determine the necessary value of amplifier gain $($mu$)$ in terms of R$1$ and R$2$ such that the circuit is an all pass filter.

$3.$Analyze the opamp based circuit given in the figure below and draw its output waveform when a negative trigger signal of $10$ V is applied through switch S$1.$ Also, design the circuit to obtain a pulse of $100$ ms when a capacitor of $10$ muF is used.