1
GATE ECE 2015 Set 1
+2
-0.6
The solution of the differential equation $${{h\left( {t + 1} \right)} \over {h\left( t \right)}}\,\,\,\,\,{{{d^2}y} \over {d{t^{ \to 2}}}} + {{2\,dy} \over {dt}} + y\, = \,0$$ with $$\,y\left( 0 \right)\, = \,y'\left( 0 \right)\, = \,1$$ is
A
$$\left( {2 - t} \right){e^t}$$
B
$$\left( {1 + 2t} \right){e^{ - t}}$$
C
$$\left( {2 + t} \right){e^{ - t}}$$
D
$$\left( {1 - 2t} \right){e^t}$$
2
GATE ECE 2014 Set 4
+2
-0.6
The unilateral Laplace transform of F(t) is $${1 \over {{s^2} + s + 1}}$$. Which one of the following is the unilateral Laplace transform of g(t) = $$t \cdot f\left( t \right)$$
A
$${{ - s} \over {{{\left( {{s^2} + s + 1} \right)}^2}}}$$
B
$${{ - \left( {2s + 1} \right)} \over {{{\left( {{s^2} + s + 1} \right)}^2}}}$$
C
$${s \over {{{\left( {{s^2} + s + 1} \right)}^2}}}$$
D
$${{2s + 1} \over {{{\left( {{s^2} + s + 1} \right)}^2}}}$$
3
GATE ECE 2014 Set 4
+2
-0.6
A stable linear time invariant (LTI) system has a transfer function H(s) = $${1 \over {{s^2} + s - 6}}$$. To make this system casual it needs to be cascaded with another LTI system having a transfer function H1(s). A correct choice for H1(s) among the following options is
A
s + 3
B
s - 2
C
s - 6
D
s + 1
4
GATE ECE 2014 Set 4
Numerical
+2
-0
A casual LTI system has zero initial conditions and impulse response h(t). Its input x(t) and output y(t) are related through the linear constant - coefficient differential equation $${{{d^2}y\left( t \right)} \over {d{t^2}}} + \alpha {{dy\left( t \right)} \over {dt}} + {\alpha ^2}y\left( t \right) = x\left( t \right).$$\$

Let another signal g(t) be defined as $$\left( t \right) = {\alpha ^2}\int_0^t {h\left( \tau \right)d\tau + {{dh\left( t \right)} \over {dt}} + \alpha h\left( t \right)}$$.

If G(s) is the Laplace transform of g(t), then the number of poles of G(s) is ______.