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.