1
GATE ECE 2004
+2
-0.6
The state variable equations of a system are: $${\mathop {{x_1} = - 3{x_1} - x}\limits^ \bullet _2} + u$$$$${\mathop x\limits^ \bullet _2} = 2{x_1}$$$ $$y = {x_1} + u.$$$The system is A controllable but not observable. B observable but not controllable. C neither controllable nor observable. D controllable and observable. 2 GATE ECE 2004 MCQ (Single Correct Answer) +2 -0.6 Given A $$= \left[ {\matrix{ 1 & 0 \cr 0 & 1 \cr } } \right],$$ the state transition matrix eAt is given by A $$\left[ {\matrix{ 0 & {{e^{ - t}}} \cr {{0^{ - t}}} & 0 \cr } } \right]$$ B $$\left[ {\matrix{ {{e^t}} & 0 \cr 0 & {{e^t}} \cr } } \right]$$ C $$\left[ {\matrix{ {{e^{ - t}}} & 0 \cr 0 & {{e^{ - t}}} \cr } } \right]$$ D $$\left[ {\matrix{ 0 & {{e^t}} \cr {{e^t}} & 0 \cr } } \right]$$ 3 GATE ECE 2004 MCQ (Single Correct Answer) +2 -0.6 If A = $$\left[ {\matrix{ { - 2} & 2 \cr 1 & { - 3} \cr } } \right],$$ then sin At is A $${1 \over 3}\left[ {\matrix{ {\sin \left( { - 4t} \right) + 2\sin \left( { - t} \right)} & { - 2\sin \left( { - 4t} \right) + 2\sin \left( { - t} \right)} \cr { - \sin \left( { - 4t} \right) + \sin \left( { - t} \right)} & {2\sin \left( { - 4t} \right) + \sin \left( { - t} \right)} \cr } } \right]$$ B $$\left[ {\matrix{ {\sin \left( { - 2t} \right)} & {\sin \left( {2t} \right)} \cr {\sin \left( t \right)} & {\sin \left( { - 3t} \right)} \cr } } \right]$$ C $${1 \over 3}\left[ {\matrix{ {\sin \left( {4t} \right) + 2\sin \left( t \right)} & {2\sin \left( { - 4t} \right) - 2\sin \left( { - t} \right)} \cr { - \sin \left( { - 4t} \right) + \sin \left( t \right)} & {2\sin \left( {4t} \right) + \sin \left( t \right)} \cr } } \right]$$ D $${1 \over 3}\left[ {\matrix{ {\cos \left( { - t} \right) + 2\cos \left( t \right)} & {2\cos \left( { - 4t} \right) + 2\sin \left( { - t} \right)} \cr { - \cos \left( { - 4t} \right) + \sin \left( { - t} \right)} & { - 2\cos \left( { - 4t} \right) + \cos \left( { - t} \right)} \cr } } \right]$$ 4 GATE ECE 2003 MCQ (Single Correct Answer) +2 -0.6 The zero, input response of a system given by the state space equation $$\left[ {{{\mathop {{x_1}}\limits^ \bullet } \over {\mathop {{x_2}}\limits^ \bullet }}} \right] = \left[ {\matrix{ 1 & 0 \cr 1 & 1 \cr } } \right]\left[ {\matrix{ {{x_1}} \cr {{x_2}} \cr } } \right]and\left[ {\matrix{ {{x_1}} & {\left( 0 \right)} \cr {{x_2}} & {\left( 0 \right)} \cr } } \right] = \left[ {\matrix{ 1 \cr 0 \cr } } \right]is$$$
A
$$\left[ {\matrix{ {t{e^t}} \cr t \cr } } \right]$$
B
$$\left[ {\matrix{ {{e^t}} \cr t \cr } } \right]$$
C
$$\left[ {\matrix{ {{e^t}} \cr {t{e^t}} \cr } } \right]$$
D
$$\left[ {\matrix{ t \cr {t{e^t}} \cr } } \right]$$
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