Let a causal $$LTI$$ system be characterized by the following differential equation, with initial rest condition
$${{{d^2}y} \over {d{t^2}}} + 7{{dy} \over {dt}} + 10y\left( t \right) = 4x\left( t \right) + 5{{dx\left( t \right)} \over {dt}}\,\,$$

Where, $$x(t)$$ and $$y(t)$$ are the input and output respectively. The impulse response of the system is ($$u(t)$$ is the unit step function)

In the system whose signal flow graph is shown in the figure, $${{U_1}\left( s \right)}$$ and $${{U_2}\left( s \right)}$$ are inputs. The transfer function $${{Y\left( s \right)} \over {{U_1}\left( s \right)}}\,$$ is

A closed loop system has the characteristic equation given by
$${s^3} + K{s^2} + \left( {K + 2} \right)s + 3 = 0.$$ For this system to be stable, which one of the following conditions should be satisfied?

The transfer function of a system is given by $${{{V_0}\left( s \right)} \over {{V_i}\left( s \right)}} = {{1 - s} \over {1 + s}}$$

Let the output of the system be $${v_0}\left( t \right) = {v_m}\sin \left( {\omega t + \phi } \right)$$ for the input $${v_i}\left( t \right) = {v_m}\sin \left( {\omega t} \right).$$ Then the minimum and maximum values of ϕ (in radians) are respectively