1
GATE ECE 2012
+1
-0.3
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
A
$${1 \over 3} < \left| z \right| < 3$$
B
$${1 \over 3} < \left| z \right| < {1 \over 2}$$
C
$${1 \over 2} < \left| z \right| < 3$$
D
$${1 \over 3} < \left| z \right|$$
2
GATE ECE 2012
+1
-0.3
The unilateral Laplace transform of $$f(t)$$ is
$$\,{1 \over {{s^2} + s + 1}}.$$ The unilateral Laplace transform of $$t$$ $$f(t)$$ is
A
$$- {s \over {{{\left( {{s^2} + s + 1} \right)}^2}}}$$
B
$$- {{2s + 1} \over {{{\left( {{s^2} + s + 1} \right)}^2}}}$$
C
$${s \over {{{\left( {{s^2} + s + 1} \right)}^2}}}$$
D
$${{2s + 1} \over {{{\left( {{s^2} + s + 1} \right)}^2}}}$$
3
GATE ECE 2009
+1
-0.3
Given that $$F(s)$$ is the one-sided Laplace transform of $$f(t),$$ the Laplace transform of $$\int\limits_0^t {f\left( \tau \right)} d\tau$$ is
A
$$s\,\,F\left( s \right) - f\left( 0 \right)$$
B
$${1 \over s}F\left( s \right)$$
C
$$\int\limits_0^s {f\left( \tau \right)} d\tau$$
D
$${1 \over s}\left[ {F\left( s \right) - f\left( 0 \right)} \right]$$
4
GATE ECE 2006
+1
-0.3
Consider the function $$f(t)$$ having laplace transform
$$F\left( s \right) = {{{\omega _0}} \over {{s^2} + \omega _0^2}},\,\,{\mathop{\rm Re}\nolimits} \left( s \right) > 0.$$ The final value of $$f(t)$$ would be ____________.
A
$$0$$
B
$$1$$
C
$$- 1 - f\left( \infty \right) \le 1$$
D
$$\infty$$
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