Graph Theory and State Equations

Question 1
In the following graph, the number of trees (P) and the number of cut-set (Q) are
A
P = 2,Q = 2
B
P = 2,Q = 6
C
P = 4,Q = 6
D
P = 4,Q = 10
GATE EC 2008   Network Theory
Question 1 Explanation: 
Different trees (P) are shown below

Different cut-sets (Q) are shown below:

Question 2
Consider the network graph shown in the figure. Which one of the following is NOT a 'tree' of this graph ?
A
a
B
b
C
c
D
d
GATE EC 2004   Network Theory
Question 2 Explanation: 
It is forming a closed loop. So it can't be a tree.
Question 3
The differential equation for the current i(t) in the circuit of the figure is
A
2\frac{d^{2}i}{dt^{2}}+2\frac{di}{dt}+i(t)=\sin t
B
\frac{d^{2}i}{dt^{2}}+2\frac{di}{dt}+2i(t)=\cos t
C
2\frac{d^{2}i}{dt^{2}}+2\frac{di}{dt}+i(t)=\cos t
D
\frac{d^{2}i}{dt^{2}}+2\frac{di}{dt}+2i(t)=\sin t
GATE EC 2003   Network Theory
Question 3 Explanation: 
Applying KVL,
\sin t=i(t) \times 2+L \frac{d i(t)}{d t}+\frac{1}{c} \int i(t) d t
\sin t=2 i(t)+2 \frac{d i(t)}{d t}+\int i(t) d t
Differentiating with respect to t
\cos (t)=\frac{2 d i(t)}{d t}+\frac{2 d^{2} i(t)}{d t^{2}}+i(t)
There are 3 questions to complete.
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