# GATE EE 2008

 Question 1
The number of chords in the graph of the given circuit will be A 3 B 4 C 5 D 6
Electric Circuits   Magnetically Coupled Circuits, Network Topology and Filters
Question 1 Explanation: Number of branches =b =6
No. of nodes = n= 4
No. of chords =b-(n-1) =6-(4-1)=3
 Question 2
The Thevenin's equivalent of a circuit operation at $\omega$=5 rads/s, has $V_{oc}=3.71\angle -15.9^{\circ}$ V and $Z_0=2.38-j0.667\Omega$. At this frequency, the minimal realization of the Thevenin's impedance will have a
 A resistor and a capacitor and an inductor B resistor and a capacitor C resistor and an inductor D capacitor and an inductor
Electric Circuits   Network Theorems
Question 2 Explanation:
Thevenin's Impedance:
$Z_0=2.38-j0.667\Omega$
as real part is not zero, so $Z_0$ has resistor
$Img[Z_0]=-j0.667$
CASE-I:
$Z_0$ has capacitor (as $Img[Z_0]$ is negative)
CASE-II:
$Z_0$ has both capacitor and inductor, but inductive reactance $\lt$ capacitive reactance.
At, $\omega$=5 rad/sec
For minimal realization case-I is considered. Therefore, $Z_0$ will have a resistor and a capacitor.

 Question 3
A signal $e^{-\alpha t}sin(\omega t)$ is the input to a real Linear Time Invariant system. Given K and $\phi$ are constants, the output of the system will be of the form $Ke^{-\beta t}sin(vt+\phi )$ where
 A $\beta$ need not be equal to $\alpha$ but v equal to $\omega$ B v need not be equal to $\omega$ but $\beta$ equal to $\alpha$ C $\beta$ equal to $\alpha$ and v equal to $\omega$ D $\beta$ need not be equal to $\alpha$ and v need not be equal to $\omega$
Signals and Systems   Linear Time Invariant Systems
 Question 4
X is a uniformly distributed random variable that takes values between 0 and 1. The value of E{$X^{3}$} will be
 A 0 B $1/8$ C $1/4$ D $1/2$
Engineering Mathematics   Probability and Statistics
Question 4 Explanation:
x is uniformly distributes in [0,1]
Therefore, probability density function
\begin{aligned} f(x)&=\frac{1}{b-a} =\frac{1}{1-0}=1\\ \therefore \; f(x) &=1\;\;\;0 \lt x \lt 1 \\ &=0\;\;\;\text{elsewhere} \\ \text{Now, } E(x^3)&=\int_{0}^{1}x^3f(x)\; dx \\ &= \int_{0}^{1}x^3 \times 1 \times dx\\ &= \left [ \frac{x^4}{4} \right ]_0^1=\frac{1}{4} \end{aligned}
 Question 5
The characteristic equation of a (3x3) matrix P is defined as
$a(\lambda )|\lambda I-P|= \lambda ^{3}+ \lambda ^{2}+ 2\lambda +1=0$
If $I$ denotes identity matrix, then the inverse of matrix $P$ will be
 A $(P^{2}+P+2I)$ B $(P^{2}+P+I)$ C $-(P^{2}+P+I)$ D $-(P^{2}+P+2I)$
Engineering Mathematics   Linear Algebra
Question 5 Explanation:
If characteristic equation is
$\lambda ^3+\lambda ^2+2\lambda +1=0$
Then by cayley- hamilton theorem,
$P^3+P^2+2P+I=0$
$I=-P^3-P^2-2P$
Multiplying by $P^{-1}$ on both sides,
$P^{-1}=-P^2-P-2I=-(P^2+P+2I)$

There are 5 questions to complete.