$$X$$ and $$Y$$ are non-zero square matrices of size $$n \times n$$. If $$XY = {O_{n \times n}}$$ then

Consider the differential equation $${{dy} \over {dx}} + y = {e^x}$$ with $$y(0)=1.$$ Then the value of $$y(1)$$ is

The integral $$\int\limits_{ - \alpha }^\alpha \delta \left( {t - {\pi \over 6}} \right)6\,\sin \,\left( t \right)dt$$ evaluates to

$$u(t)$$ represents the unit step function. The Laplace transform of $$u\left( {t - \tau } \right)$$ is