GATE IN
Engineering Mathematics
Transform Theory
Previous Years Questions

## Marks 1

If $$x\left[ N \right] = {\left( {1/3} \right)^{\left| n \right|}} - {\left( {1/2} \right)^n}\,u\left[ n \right],$$ then the region of convergence $$(... The unilateral Laplace transform of$$f(t)$$is$$\,{1 \over {{s^2} + s + 1}}.$$The unilateral Laplace transform of$$tf(t)$$is$$u(t)$$represents the unit step function. The Laplace transform of$$u\left( {t - \tau } \right)$$is The laplace transform of a function$$f(t)$$is defined by$$F\left( s \right) = L\left\{ {f\left( t \right)} \right\} = \int\limits_0^\infty {{e^{ ...
Find $$L\left\{ {{e^{at}}\,\cos \,\omega t} \right\}$$ when $$L\left\{ {\cos \,\,\omega t} \right\} = {s \over {{s^2} + {\omega ^2}}}$$

## Marks 2

Consider the differential equation $${{{d^2}y\left( t \right)} \over {d{t^2}}} + 2{{dy\left( t \right)} \over {dt}} + y\left( t \right) = \delta \lef... The integral$$\int\limits_{ - \alpha }^\alpha \delta \left( {t - {\pi \over 6}} \right)6\,\sin \,\left( t \right)dt evaluates to
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