# GATE EC 2011

 Question 1
Consider the following statements regarding the complex Poynting vector $\vec{P}$ for the power radiated by a point source in an infinite homogeneous and lossless medium. Re($\vec{P}$) denotes the real part of $\vec{P}$, S denotes a spherical surface whose centre is at the point source, and $\hat{n}$ denotes the unit surface normal on S. Which of the following statements is TRUE?
 A Re($\vec{P}$) remains constant at any radial distance from the source B Re($\vec{P}$) increases with increasing radial distance from the source C $\oint \oint_{S}Re(\vec{P}).\hat{n} dS$ remains constant at any radial distance from the source D $\oint \oint_{S}Re(\vec{P}).\hat{n} dS$ decreases with increasing radial distance form the source
Electromagnetics   Antennas
Question 1 Explanation:
Power density of any point source decreases with distance i.e. the density decreases and area of cross-over increases with the product being constant.
 Question 2
A transmission line of characteristic impedance 50 W is terminated by a 50 $\Omega$ load. When excited by a sinusoidal voltage source at 10 GHz, the phase difference between two points spaced 2 mm apart on the line is found to $\frac{\pi }{4}$ radians. The phase velocity of the wave along the line is
 A $0.8 \times 10^{8} m/s$ B $1.2 \times 10^{8} m/s$ C $1.6 \times 10^{8} m/s$ D $3 \times 10^{8} m/s$
Electromagnetics   Transmission Lines
Question 2 Explanation:
\begin{aligned} \text { Phase difference }&=\frac{2 \pi}{\lambda} \text { (path difference) } \\ \Rightarrow \frac{\pi}{4}&=\frac{2 \pi}{\lambda}\left(2 \times 10^{-3}\right) \\ \therefore \quad &=8 \times 2 \times 10^{-3} \\ &=16 \times 10^{-3} \mathrm{m} \\ f&= 10 \mathrm{GHz}=10 \times 10^{9} \mathrm{Hz} \end{aligned}
Hence the phase velocity of wave along the line is,
\begin{aligned} & v=f \lambda=10 \times 10^{9} \times 16 \times 10^{-3} \mathrm{m} / \mathrm{sec} \\ \therefore \quad v &=1.6 \times 10^{8} \mathrm{m} / \mathrm{s} \end{aligned}

 Question 3
An analog signal is band-limited to 4 kHz, sampled at the Nyquist rate and the samples are quantized into 4 levels. The quantized levels are assumed to be independent and equally probable. If we transmit two quantized samples per second, the information rate is ________ bits / second.
 A 1 B 2 C 3 D 4
Communication Systems   Digital Communications
Question 3 Explanation:
Quantized levels are equiprobable; hence
\begin{aligned} H&=\log _{2} 4=2 \text { bits/sample }\\ r&=2 \text{samples/sec} \\ \end{aligned}
Hence information rate $R=r \cdot H=2 \text{ samples/sec}$
$\times 2\text{ bits/sample}$
$\Rightarrow R=4 \mathrm{bits} / \mathrm{sec}$
 Question 4
The root locus plot for a system is given below. The open loop transfer function corresponding to this plot is given by A $G(S)H(S)=k\frac{s(s+1)}{(s+2)(s+3)}$ B $G(S)H(S)=k\frac{(s+1)}{(s+2)(s+3)^{2}}$ C $G(S)H(S)=k\frac{1}{s(s-1)(s+2)(s+3)}$ D $G(S)H(S)=k\frac{(s+1)}{(s+2)(s+3)}$
Control Systems   Root Locus
Question 4 Explanation:
From plot we can observe that one pole terminates at one zero at position -1 and three poles terminates to $\infty$. It means there are four poles and 1 zero. Pole at -3 goes on both sides. It means there are two poles at -3.
 Question 5
A system is defined by its impulse response $h(n) =2^{n} u(n - 2)$. The system is
 A stable and causal B causal but not stable C stable but not causal D unstable and non-causal
Signals and Systems   LTI Systems Continuous and Discrete
Question 5 Explanation:
$h(n)=2^{n} u(n-2)$
For causal system $h(n)=0$ for $n \lt 0$
Hence given system is causal.
For stability:
$\sum_{n=2}^{\infty} 2^{n}=\infty,$ so given system is not stable.

There are 5 questions to complete.