Marks 1
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 $$(ROC)$$ of its $$Z$$-transform in the $$Z$$-plane will be
GATE IN 2012
2
The unilateral Laplace transform of $$f(t)$$ is
$$\,{1 \over {{s^2} + s + 1}}.$$ The unilateral Laplace transform of $$t$$ $$f(t)$$ is
$$\,{1 \over {{s^2} + s + 1}}.$$ The unilateral Laplace transform of $$t$$ $$f(t)$$ is
GATE IN 2012
3
$$u(t)$$ represents the unit step function. The Laplace transform of $$u\left( {t - \tau } \right)$$ is
GATE IN 2010
4
The laplace transform of a function $$f(t)$$ is defined by
$$F\left( s \right) = L\left\{ {f\left( t \right)} \right\} = \int\limits_0^\infty {{e^{ - st}}f\left( t \right)dt.} $$.
Find the inverse laplace transform of $$F(s-a)$$
$$F\left( s \right) = L\left\{ {f\left( t \right)} \right\} = \int\limits_0^\infty {{e^{ - st}}f\left( t \right)dt.} $$.
Find the inverse laplace transform of $$F(s-a)$$
GATE IN 1995
5
Find $$L\left\{ {{e^{at}}\,\cos \,\omega t} \right\}$$ when
$$L\left\{ {\cos \,\,\omega t} \right\} = {s \over {{s^2} + {\omega ^2}}}$$
$$L\left\{ {\cos \,\,\omega t} \right\} = {s \over {{s^2} + {\omega ^2}}}$$
GATE IN 1995
Marks 2
1
Consider the differential equation
$${{{d^2}y\left( t \right)} \over {d{t^2}}} + 2{{dy\left( t \right)} \over {dt}} + y\left( t \right) = \delta \left( t \right)$$
with $$y\left( t \right)\left| {_{t = 0} = - 2} \right.$$ and $${{dy} \over {dt}}\left| {_{t = 0}} \right. = 0.$$
$${{{d^2}y\left( t \right)} \over {d{t^2}}} + 2{{dy\left( t \right)} \over {dt}} + y\left( t \right) = \delta \left( t \right)$$
with $$y\left( t \right)\left| {_{t = 0} = - 2} \right.$$ and $${{dy} \over {dt}}\left| {_{t = 0}} \right. = 0.$$
The numerical value of $${{dy} \over {dt}}\left| {_{t = 0}.} \right.$$ is
GATE IN 2012
2
The integral $$\int\limits_{ - \alpha }^\alpha \delta \left( {t - {\pi \over 6}} \right)6\,\sin \,\left( t \right)dt$$ evaluates to
GATE IN 2010