1
GATE EE 2003
+1
-0.3
A second order system starts with an initial condition of $$\left( {\matrix{ 2 \cr 3 \cr } } \right)$$ without any external input. The state transition matrix for the system is given by $$\left( {\matrix{ {{e^{ - 2t}}} & 0 \cr 0 & {{e^{ - t}}} \cr } } \right).$$ The state of the system at the end of $$1$$ second is given by.
A
$$\,\,\left( {\matrix{ {0.271} \cr {1.100} \cr } } \right)$$
B
$$\left( {\matrix{ {0.135} \cr {0.368} \cr } } \right)$$
C
$$\left( {\matrix{ {0.271} \cr {0.736} \cr } } \right)$$
D
$$\left( {\matrix{ {0.135} \cr {1.100} \cr } } \right)$$
2
GATE EE 2003
+2
-0.6
The following equation defines a separately exited $$dc$$ motor in the form of a differential equation $${{{d^2}\omega } \over {d{t^2}}} + {{B\,d\omega } \over {j\,\,dt}} + {{{K^2}} \over {LJ}}\omega = {K \over {LJ}}{V_a}$$

The above equation may be organized in the state space form as follows
$$\left( {\matrix{ {{{{d^2}\omega } \over {d{t^2}}}} \cr {{{d\omega } \over {dt}}} \cr } } \right) = P\left( {\matrix{ {{{d\omega } \over {dt}}} \cr \omega \cr } } \right) + Q{V_a}$$

where the $$P$$ matrix is given by

A
$$\left( {\matrix{ { - {B \over J}} & { - {{{K^2}} \over {LJ}}} \cr 1 & 0 \cr } } \right)$$
B
$$\left( {\matrix{ { - {{{K^2}} \over {LJ}}} & { - {B \over J}} \cr 0 & 1 \cr } } \right)$$
C
$$\left( {\matrix{ 0 & 1 \cr { - {{{K^2}} \over {LJ}}} & { - {B \over J}} \cr } } \right)$$
D
$$\left( {\matrix{ 1 & 0 \cr { - {B \over J}} & { - {{{K^2}} \over {LJ}}} \cr } } \right)$$
3
GATE EE 2003
+2
-0.6
The Boolean expression $$X\overline Y Z + XYZ + \overline X Y\overline Z + \overline X \overline Y Z + XY\overline Z$$ can be simplified to
A
$$X\overline Z + \overline X Z + YZ$$
B
$$XZ + \overline Y Z + Y\overline Z$$
C
$$\overline X Z + YZ + XZ$$
D
$$\overline X \overline Y + Y\overline Z + \overline X Z$$
4
GATE EE 2003
+1
-0.3
Figure shows a $$4$$ to $$1$$ $$MUX$$ to be used to implement the sum $$S$$ of a $$1$$-bit full adder with input bits $$P$$ and $$Q$$ and the carry input $${C_{in}}.$$ Which of the following combinations of inputs to $${{\rm I}_0},\,\,{{\rm I}_1},\,\,{{\rm I}_2}$$ and $$\,\,{{\rm I}_3}$$ of the $$MUX$$ will realize the sum $$S$$?
A
$${{\rm I}_0} = {{\rm I}_1} = {C_{in}};\,{{\rm I}_2} = {{\rm I}_3} = {\overline C _{in}}$$
B
$${{\rm I}_0} = {{\rm I}_1} = {\overline C _{in}};\,{{\rm I}_2} = {{\rm I}_3} = {C_{in}}$$
C
$${{\rm I}_0} = {{\rm I}_3} = {\overline C _{in}};\,{{\rm I}_1} = {{\rm I}_2} = {\overline C _{in}}$$
D
$${{\rm I}_0} = {{\rm I}_3} = {\overline C _{in}};\,{{\rm I}_1} = {{\rm I}_2} = {C_{in}}$$
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