1
GATE EE 2013
+2
-0.6
The state variable formulation of a system is given as
$$\left[ {\matrix{ {\mathop {{x_1}}\limits^ \bullet } \cr {\mathop {{x_2}}\limits^ \bullet } \cr } } \right] = \left[ {\matrix{ { - 2} & 0 \cr 0 & { - 1} \cr } } \right]\left[ {\matrix{ {{x_1}} \cr {{x_2}} \cr } } \right] + \left[ {\matrix{ 1 \cr 1 \cr } } \right]u,\,\,{x_1}\left( 0 \right) = 0,$$
$${x_2}\left( 0 \right) = 0$$ and $$y = \left[ {\matrix{ 1 & 0 \cr } } \right]\left[ {\matrix{ {{x_1}} \cr {{x_2}} \cr } } \right]$$

The response $$y(t)$$ to a unit step input is

A
$${1 \over 2} - {1 \over 2}{e^{ - 2t}}$$
B
$$1 - {1 \over 2}{e^{ - 2t}} - {1 \over 2}{e^{ - t}}$$
C
$${e^{ - 2t}} - {e^{ - t}}$$
D
$$1 - {e^{ - t}}$$
2
GATE EE 2013
+2
-0.6
The state variable formulation of a system is given as
$$\left[ {\matrix{ {\mathop {{x_1}}\limits^ \bullet } \cr {\mathop {{x_2}}\limits^ \bullet } \cr } } \right] = \left[ {\matrix{ { - 2} & 0 \cr 0 & { - 1} \cr } } \right]\left[ {\matrix{ {{x_1}} \cr {{x_2}} \cr } } \right] + \left[ {\matrix{ 1 \cr 1 \cr } } \right]u,\,\,{x_1}\left( 0 \right) = 0,$$
$${x_2}\left( 0 \right) = 0$$ and $$y = \left[ {\matrix{ 1 & 0 \cr } } \right]\left[ {\matrix{ {{x_1}} \cr {{x_2}} \cr } } \right]$$

The system is

A
controllable but not observable
B
not controllable but obserable
C
both controllable and observable
D
both not controllable and not Observable
3
GATE EE 2013
+1
-0.3
A bulb in staircase has two switches, one switch being at the ground floor and the other one at the first floor. The bulb can be turned $$ON$$ and also can be turned $$OFF$$ by any one of the switches irrespective of the state of the other switch. The logic of switching of the bulb resembles
A
an $$AND$$ gate
B
an $$OR$$ gate
C
an $$XOR$$ gate
D
a $$NAND$$ gate
4
GATE EE 2013
+2
-0.6
The clock frequency applied to the digital circuit shown in the figure below is $$1$$ $$kHz.$$ If the initial state of the output $$Q$$ of the flip-flop is $$‘0’,$$ then the frequency of the output waveform $$Q$$ in $$kHz$$ is
A
$$0.25$$
B
$$0.5$$
C
$$1$$
D
$$2$$
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