# GATE EC 2001

 Question 1
The voltage $e_o$ in figure is
 A 2V B $\frac{4}{3}$V C 4V D 8V
Network Theory   Basics of Network Analysis
Question 1 Explanation:
Applying KCL,
\begin{aligned} \frac{e_{0}-12}{4}+\frac{e_{0}}{4}+\frac{e_{0}}{4}&=0 \\ \Rightarrow \quad 3 e_{0}&=12 \\ \therefore \quad e_{0}&=4 \mathrm{V} \end{aligned}
 Question 2
If each branch of a Delta circuit has impedance $\sqrt{3}Z$, then each branch of the equivalent Wye circuit has impedance.
 A $\frac{Z}{\sqrt{3}}$ B 3Z C $3\sqrt{3}Z$ D $\frac{Z}{3}$
Network Theory   Basics of Network Analysis
Question 2 Explanation:
\begin{aligned} Z_{\Delta} &=3 Z_{Y} \\ \Rightarrow \quad \sqrt{3} Z_{\Delta} &=3 Z_{Y} \\ Z_{Y} &=\frac{Z_{\Delta}}{\sqrt{3}} \end{aligned}

 Question 3
The transfer function of a system is given by $H(s)=\frac{1}{s^2(s-2)}$. The impulse response of the system is: (* denotes convolution, and U(t) is unit step function)
 A $(t^2 * e^{-2t})U(t)$ B $(t * e^{2t})U(t)$ C $(te^{-2t})U(t)$ D $(te^{-2}t)U(t)$
Signals and Systems   Laplace Transform
Question 3 Explanation:
Impulse response of system is $L^{-1}[H(s)]$
$\frac{1}{s^{2}(s-2)}=\frac{1}{s^{2}} \times \frac{1}{s-2}=\left(t * e^{+2 t}\right) u(t)$
 Question 4
The admittance parameter $Y_{12}$ in the 2-port network in figure is
 A 0.2 mho B 0.1 mho C -0.05 mho D 0.05 mho
Network Theory   Two Port Networks
Question 4 Explanation:
$\left[\begin{array}{cc} y_{1}+y_{3} & -y_{3} \\ -y_{3} & y_{2}+y_{3} \end{array}\right]=\left[\begin{array}{cc} y_{11} & y_{12} \\ y_{21} & y_{22} \end{array}\right]$
$y_{12}=-y_{3}$

$y_{12}=-\frac{1}{20}=-0.05 \mathrm{mho}$
 Question 5
The region of convergence of the z-transform of a unit step function is
 A $\mid z\mid \gt 1$ B $\mid z\mid \lt 1$ C (real part of Z)$\gt$0 D (real part of Z)$\lt$0
Signals and Systems   Z-Transform
Question 5 Explanation:
\begin{aligned} h(n)&=u(n) \\ H(z)&=\sum_{n=0}^{\infty} 1 . z^{-n} \end{aligned}
For the convergence of H(z)
$\sum_{n=0}^{n}\left(z^{-1}\right)^{n}\lt \infty$
\therefore ROC is the range of values of z for which
$\left|z^{-1}\right| \lt 1\text{ or } |z|\gt |1|$

There are 5 questions to complete.