Indicate whether the following statement is TRUE/FALSE: Give reason for your answer.

If G(s) is a stable transfer function, then $$F\left( s \right) = {1 \over {G\left( s \right)}}$$ is always a stable transfer function.

Match each of the items A, B and C with an appropriate item from 1, 2, 3, 4 and 5.

List - 1
(A) $${a_1}{{{d^{2y}}} \over {d{x^2}}} + {a_2}y{{dy} \over {dx}} + {a_3}y = {a_4}$$
(B) $${a_1}{{{d^3}y} \over {d{x^3}}} + {a_2}y = {a_3}$$
(C) $$\eqalign{ & {a_1}{{{d_2}y} \over {d{x_2}}} + {a_2}x{{dy} \over {dx}} + {a_3}\,{x^2}y = 0 \cr & \cr} $$

List - 2
(1) Non linear differential equation.
(2) Linear differential equation with constant coefficients.
(3) Linear homogeneous differential equation.
(4) Non - Linear homogeneous differential equation.
(5) Non - linear first order differential equation.

Increased pulse-width in the flat-top sampling, leads to

A signal, f(t) = $${e^{ - at}}$$ u(t), where u(t) is the unit step function, is applied to the input of a low-pass filter having $$\left| {H(\omega )} \right| = {b \over {\sqrt {{\omega ^2} + {b^2}} }}$$.

Calculate the value of the ratio, $${a \over b}$$, for which 50% of the input signal energy is transferred to the output.