# GATE EE 2006

 Question 1
Which of the following is true?
 A A finite signal is always time bounded B A time bounded signal always possesses finite energy C A bounded signal is always zero outside the interval [$-t_0,t_0$ ] for some $t_0$ D A time bounded signal is always finite
Signals and Systems   Introduction of C.T. and D.T. Signals
 Question 2
x(t) is a real valued function of a real variable with period T. Its trigonometric Fourier Series expansion contains no terms of frequency $\omega = 2\pi$(2k)/T; k=1,2... Also, no sine terms are present. Then x(t) satisfies the equation
 A $x(t)=-x(t-T)$ B $x(t)=x(T-t)=-x(-t)$ C $x(t)=x(T-t)=-x(t-T/2)$ D $x(t)=x(t-T)=x(t-T/2)$
Signals and Systems   Fourier Series
Question 2 Explanation:
Since trigonometric fourier series has no sine terms and has only cosine terms therefore this will be an even signal i.e. it will satisfy.
$x(t)=x(-t)$
or, we can write,
$x(t-T)=x(-t+T)$
but signal is periodic with period T.
therefore $x(t-T)=x(t)$
therefore, $x(t)=x(T-t)\;\;...(i)$
Now, since signal contains only odd harmonics i.e. no terms of frequency
$\omega =\frac{2 \pi \times 2k}{T},\;\;\;k=1,2,3,4$
i.e. no even harmonics.
This means signal contains half wave symmetry
this implies that,
$x(t)=-x\left ( t-\frac{T}{2} \right )\;\;...(ii)$
From eq. (i) and (ii),
$x(t)=x(T-t)=-x\left ( t-\frac{T}{2} \right )$

 Question 3
In the figure the current source is $1\angle 0$ A, R=1$\Omega$, the impedances are $Z_{C} =-j \Omega$ and $Z_{L}= 2j \Omega$. The Thevenin equivalent looking into the circuit across X-Y is
 A $\sqrt{2}\angle 0V , (1+2j)\Omega$ B $2\angle 45^{\circ}V , (1-2j)\Omega$ C $2\angle 45^{\circ}V , (1+j)\Omega$ D $\sqrt{2}\angle 45^{\circ}V , (1+j)\Omega$
Electric Circuits   Network Theorems
Question 3 Explanation:
To calculate Thevenin's impedance, current-souce is open-circuited

$Z_{th}=R+Z_L+Z_c$
$\;\;=1+2j-j$
$\;\;=1+j \Omega$
Open-circuit voltage at terminals X-Y
$=I \times Z_{th}$
$\;\;=1\angle 0 \times (1+j)$
$\;\;=\sqrt{2}\angle 15^{\circ} volts$
 Question 4
The three limbed non ideal core shown in the figure has three windings with nominal inductances L each when measured individually with a single phase AC source. The inductance of the windings as connected will be
 A very low B L/3 C 3L D very high
Electrical Machines   Transformers
 Question 5
Which of the following statement holds for the divergence of electric and magnetic flux densities ?
 A Both are zero B These are zero for static densities but non zero for time varying densities. C It is zero for the electric flux density D It is zero for the magnetic flux density
Electromagnetic Fields   Magnetostatic Fields
Question 5 Explanation:
The divergence of magnetic field is always zzero because magnetic flux makes always a closed path.
So, $\bigtriangledown \cdot B=0$ (Maxweel's equation)
while divergence of electric field,
$\bigtriangledown \cdot \vec{E}=\frac{\rho _v}{\in}$

There are 5 questions to complete.