A system with input $$x(t)$$ and output $$y(t)$$ is defined by the input $$-$$ output relation:
$$y\left( t \right) = \int\limits_{ - \infty }^{ - 2t} {x\left( \tau \right)} d\tau .$$ The system will be

A signal $$x\left( t \right) = \sin c\left( {\alpha t} \right)$$ where $$\alpha $$ is a real constant $$\left( {\sin \,c\left( x \right) = {{\sin \left( {\pi x} \right)} \over {\pi x}}} \right)$$ is the input to a linear Time invariant system whose impulse response $$h\left( t \right) = \sin c\left( {\beta t} \right)$$ where $$\beta $$ is a real constant. If $$\min \left( {\alpha ,\,\,\beta } \right)$$ denotes the minimum of $$\alpha $$ and $$\beta $$, and similarly $$\max \left( {\alpha ,\,\,\beta } \right)$$ denotes the maximum of $$\alpha $$ and $$\beta $$, and $$K$$ is a constant, which one of the following statements is true about the output of the system?

A function y(t) satisfies the following differential equation:$$$\frac{\operatorname dy\left(t\right)}{\operatorname dt}+\;y\left(t\right)\;=\;\delta\left(t\right)$$$ where $$\delta\left(t\right)$$ is the delta function. Assuming zero initial condition, and denoting the unit step function by u(t), y(t) can be of the form

Given X(z)=$$\frac z{\left(z-a\right)^2}$$ with $$\left|z\right|$$ > a, the residue of X(z)z^n-1 at z = a for n $$\geq$$ 0 will be