# GATE EC 2009

 Question 1
The order of the differential equation $\frac{d^{2}y}{dt^{2}}+(\frac{dy}{dt})^{3}+y^{4}=e^{-t}$ is
 A 1 B 2 C 3 D 4
Engineering Mathematics   Differential Equations
Question 1 Explanation:
Highest derivative of differential equation is 2.
 Question 2
The Fourier series of a real periodic function has only
(P) cosine terms if it is even
(Q) sine terms if it is even
(R) cosine terms if it is odd
(S) sine terms if it is odd
Which of the above statements are correct ?
 A P and S B P and R C Q and S D Q and R
Signals and Systems   Fourier Series
Question 2 Explanation:
The Fourier series of a real periodic function has only cosine terms if it is even and only sine terms if it is odd.
 Question 3
A function is given by $f(t)=sin^{2}t+cos2t$. Which of the following is true ?
 A f has frequency components at 0 and $\frac{1}{2\pi } Hz$ B f has frequency components at 0 and $\frac{1}{\pi } Hz$ C f has frequency components at $\frac{1}{2\pi } \; and \; \frac{1}{\pi}Hz$ D f has frequency components at 0, $\frac{1}{2\pi } \; and \; \frac{1}{\pi}Hz$
Signals and Systems   Fourier Transforms, Frequency Response and Correlation
Question 3 Explanation:
$f(t)=\frac{1}{2}(1-\cos 2 t)+\cos 2 t$
frequency components are
\begin{aligned} f_{1}&=0 \\ f_{2}&=\frac{\omega_{2}}{2 \pi}=\frac{2}{2 \pi}=\frac{1}{\pi} \mathrm{Hz} \end{aligned}
 Question 4
A fully charged mobile phone with a 12 V battery is good for a 10 minute talktime. Assume that, during the talk-time the battery delivers a constant current of 2 A and its voltage drops linearly from 12 V to 10 V as shown in the figure. How much energy does the battery deliver during this talk-time?
 A 220J B 12kJ C 13.2kJ D 14.4J
Network Theory   Basics of Network Analysis
Question 4 Explanation:
\begin{aligned} P &=V I \\ \text { Energy } &=P \cdot t=V \cdot|t=(V . t)| \\ I &=2 \mathrm{A} \quad(\text { given }) \\ V \cdot t &=\text { Area under } V-t \text { curve } \\ V \cdot t &=\left(\frac{1}{2} \times 2 \times 600\right)+(10 \times 600) \\ &=600+6000 \\ V \cdot t &=6600 \\ E &=(6600) \times 2=13200=13.2 \mathrm{kJ} \end{aligned}
 Question 5
In an n-type silicon crystal at room temperature, which of the following can have a concentration of $4 \times 10^{19}cm^{-3}$?
 A Silicon atoms B Holes C Dopant atoms D Valence electrons
Electronic Devices   Basic Semiconductor Physics
 Question 6
The full form of the abbreviations TTL and CMOS in reference to logic families are
 A Triple Transistor Logic and Chip Metal Oxide Semiconductor B Tristate Transistor Logic and Chip Metal Oxide Semiconductor C Transistor Transistor Logic and Complementary Metal Oxide Semiconductor D Tristate Transistor Logic and Complementary Metal Oxide Silicon
Digital Circuits   Logic Families
Question 6 Explanation:
TTL- Transistor-transistor logic
CMOS - Complementary Metal Oxide Semiconductor
 Question 7
The ROC of z -transform of the discrete time sequence $x(n)=(\frac{1}{3})^{n}u(n)-(\frac{1}{2})^{n}u(-n-1)$ is
 A $\frac{1}{3} \lt |z| \lt \frac{1}{2}$ B $|z|\gt \frac{1}{2}$ C $|z| \lt \frac{1}{3}$ D $2 \lt |z| \lt 3$
Signals and Systems   Z-Transform
Question 7 Explanation:
$x(n)=(1 / 3)^{n} u(n)-(1 / 2)^{n} u(-n-1)$
$(1 / 3)^{n} u(n)$ is right sided signal, so ROC will be
$|z| \gt 1 / 3$\;\; ...(i)
$-(1 / 2)^{n} u(-n-1)$ is left sided signal so ROC will be
$|z| \lt 1 / 2 \; \; ...(ii)$
from (i) and (ii) we see that ROC of the function will be
$1 / 3 \lt |z| \lt 1 / 2$
 Question 8
The magnitude plot of a rational transfer function G(s) with real coefficients is shown below. Which of the following compensators has such a magnitude plot ?
 A Lead compensator B Lag compensator C PID compensator D Lead-lag compensator
Control Systems   Compensators and Controllers
 Question 9
A white noise process X(t) with two-sided power spectral density $1 \times 10^{-10}W/Hz$ is input to a filter whose magnitude squared response is shown below. The power of the output process Y(t) is given by
 A $5 \times 10^{-7}W$ B $1 \times 10^{-6}W$ C $2 \times 10^{-6}W$ D $1 \times 10^{-5}W$
Communication Systems   Random Processes
Question 9 Explanation:
PSD of white noise $=1 \times 10^{-10} \mathrm{W} / \mathrm{Hz} (\equiv k)$
PSD of output
\begin{aligned} G_{0}(t) &=|H(t)|^{2} \cdot G_{i}(t) \\ &=k \cdot|H(t)|^{2} \end{aligned}
output noise power
\begin{aligned} &N_{0}=\int_{-f_{0}}^{+f_{0}} G_{0}(f) d f=k \times\left(\text { areaunder }|H(f)|^{2} \text { curve }\right) \\ &=k \times 2\left(\frac{1}{2} b h\right) \\ &=k f_{0} \times 1 \\ &=1 \times 10^{-10} \times 10 \times 10^{3} \\ &=10^{-6} \mathrm{W} \end{aligned}
 Question 10
Which of the following statements is true regarding the fundamental mode of the metallic waveguides shown ?
 A Only P has no cutoff-frequency B Only Q has no cutoff-frequency C Only R has no cutoff-frequency D All three have cutoff-frequencies
Electromagnetics   Waveguides
Question 10 Explanation:
P is coaxial line and support TEM wave
$\therefore$ P has no cutoff frequency
Q and R are wave-guides and cutoff frequency each depends upon their dimensions.
There are 10 questions to complete.