# GATE EE 2011

 Question 1
Roots of the algebraic equation $x^{3}+x^{2}+x+1=0$ are
 A (+1, +j, -j) B (+1, -1, +1) C (0, 0, 0) D (-1, +j, -j)
Engineering Mathematics   Calculus
Question 1 Explanation:
$-1$ is one of the root since
$(-1)^3+(-1)^2+(-1)+1=0$
By polynomial division
$\frac{x^3+x^2+x+1}{(x-(-1))}=x^2+1 \Rightarrow \;\; x^3+x^2+x+1=(x^2+1)(x+1)$
So root are (-1,+j,-j)
 Question 2
With K as a constant, the possible solution for the first order differential equation $\frac{dy}{dx}=e^{-3x}$ is
 A $-\frac{1}{3}e^{-3x}+K$ B $-\frac{1}{3}e^{3x}+K$ C $-\frac{1}{3}e^{-3x}+K$ D $-3e^{-x}+K$
Engineering Mathematics   Differential Equations
Question 2 Explanation:
\begin{aligned} \frac{dy}{dx}&=e^{-3x}\\ \int dy&=\int e^{-3x}dx\\ y&=\frac{e^{-3x}}{-3}+K\\ y&=-\frac{1}{3}e^{-3x}+K \end{aligned}

 Question 3
The r.m.s value of the current i(t) in the circuit shown below is A $\frac{1}{2}A$ B $\frac{1}{\sqrt{2}}A$ C 1A D $\sqrt{2}A$
Electric Circuits   Steady State AC Analysis
Question 3 Explanation:
$V_s=1 \sin t\equiv V_m \sin \omega t$
$V_m=1 V$ and $\omega =1$ rad/sec
Impedance of the branch containing inductor and capacitor
$Z=j(X_L-X_C)$
$\;\;=j\left ( \omega L-\frac{1}{\omega C} \right )$
$\;\;=j\left ( 1 times 1 -\frac{1}{1 times 1} \right )=0$
So, this branch is short circuit and the whole current flow through it
$i(t)=\frac{1.0 \sin t}{1}=1.0 \sin t$
rms value of the current $=\frac{1}{\sqrt{2}}A$
 Question 4
The fourier series expansion $f(t)=a_{0}+\sum_{n=1}^{\infty }a_{n}cosn\omega t+b_{n}sin n\omega t$ of the periodic signal shown below will contain the following nonzero terms A $a_{0} \; and \; b_{n},n=1,3,5,...\infty$ B $a_{0} \; and \; a_{n},n=1,2,3,...\infty$ C $a_{0},a_{n} \; and \; b_{n},n=1,3,5,...\infty$ D $a_{0} \; and \; a_{n},n=1,3,5,...\infty$
Signals and Systems   Fourier Series
Question 4 Explanation:
Let, $x(t)$= Even and Hws Fourier series expansion of $x(t)$ contains cos terms with odd harmonics. NOw, $f(t)=1+x(t)$
Fourier series of $f(t)$ contains dc and cos terms with odd harmonics.
 Question 5
A 4-point starter is used to start and control the speed of a
 A dc shunt motor with armature resistance control B dc shunt motor with field weakening control C dc series motor D dc compound motor
Electrical Machines   DC Machines
Question 5 Explanation:
4-point starter is used to control the speed of shunt motor in field weakening region, i.e. above rates speeds.
In field weakening region field current will reduce in 3-point starter holding coil unable to hold the plunger in ON position.

There are 5 questions to complete.