# GATE EC 2003

 Question 1
The minimum number of equations required to analyze the circuit shown in the figure is
 A $3$ B $4$ C $6$ D $7$
Network Theory   Basics of Network Analysis
Question 1 Explanation:
As voltage at 1 node is known
$\therefore$ using nodal analysis only 3 equations required.
 Question 2
A source of angular frequency 1 rad/sec has a source impedance consisting of 1 $\Omega$ resistance in series with 1 H inductance. The load that will obtain the maximum power transfer is
 A 1 $\Omega$ resistance B 1 $\Omega$ resistance in parallel with 1 H inductance C 1 $\Omega$ resistance in series with 1 F capacitor D 1 $\Omega$ resistance in parallel with 1 F capacitor
Network Theory   Network Theorems
Question 2 Explanation:
\begin{aligned} Z_{L}&=R_{S}-j X_{s} \\ \therefore \quad Z_{L}&=1-1 j \end{aligned}

 Question 3
A series RLC circuit has a resonance frequency of 1 kHz and a quality factor Q=100. If each of R, L and C is doubled from its original value, the new Q of the circuit is
 A $25$ B $50$ C $100$ D $200$
Network Theory   Sinusoidal Steady State Analysis
Question 3 Explanation:
\begin{aligned} Q=& \frac{f_{o}}{B W} \\ f_{0} &=\frac{1}{2 \pi \sqrt{L C}} \\ B W &=\frac{R}{L} \\ \text { (Characteristic equation } &\left.=s^{2}+\frac{R s}{L}+\frac{1}{L C}\right)\\ \text{or}\qquad&=\frac{1}{R} \sqrt{\frac{L}{C}} \\ \text{When R, L, C are doubled,}\\ Q&=\frac{1}{2} Q=50 \end{aligned}
 Question 4
The Laplace transform of i(t) is given by
$I(s)=\frac{2}{s\left ( 1+s \right )}$
At $t\rightarrow \infty$
The value of i(t) tends to
 A $0$ B $1$ C $2$ D $\infty$
Signals and Systems   Laplace Transform
Question 4 Explanation:
\begin{aligned} \lim _{t \rightarrow \infty} i(t) &=\lim _{s \rightarrow 0} s I(s) \\ &=\operatorname{lims}_{s \rightarrow 0} \frac{2}{s(1+s)}=2 \end{aligned}
 Question 5
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 $\frac{d^{2}i}{dt^{2}}+2\frac{di}{dt}+2i(t)=\sin t$
Network Theory   Graph Theory and State Equations
Question 5 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 5 questions to complete.