A probability density function is given by $$p(x) = \,K\,\,\exp \,\,( - \,{x^2}/2),\,\, - \,\infty \, < \,x < \,\infty $$.

The value of K should be

A certain linear time invariant system has the state and the output equations given below $$$\left[ {\matrix{ {\mathop {{x_1}}\limits^ \bullet } \cr {\mathop {{x_2}}\limits^ \bullet } \cr } } \right] = \left[ {\matrix{ 1 & { - 1} \cr 0 & 1 \cr } } \right]\left[ {\matrix{ {{x_1}} \cr {{x_2}} \cr } } \right] + \left[ {\matrix{ 0 \cr 1 \cr } } \right]u$$$ $$$y = \left[ {\matrix{ 1 & 1 \cr } } \right]\left[ {\matrix{ {{x_1}} \cr {{x_2}} \cr } } \right], if$$$ $${x_1}\left( 0 \right) =1 ,{x_2}\left( 0 \right) = - 1,$$ $$u\left( 0 \right) = 0,$$ then $${{dy} \over {dt}}{|_{t = 0}}$$ is

In the signal flow graph of Fig. y/x equals

Following fig. shows the block diagram representation of control system. The system in block A has an impulse response h(t ) = e^−t u(t ). The system in block B has an impulse response h(t ) = e^−2t u(t ). The block 'K' amplifies its inputs by a factor k. For the overall system with input x(t) and output y(t). (a) Find the transfer function $$\frac{y\left(s\right)}{x\left(s\right)}$$ when k =1.
(b) Find the impulse response when k = 0.
(c) Find the values of k for which the system becomes unstable.