# 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,
$P^{-1}=-P^2-P-2I=-(P^2+P+2I)$
 Question 6
If the rank of a (5x6) matrix $Q$ is 4, then which one of the following statement is correct ?
 A $Q$ will have four linearly independent rows and four linearly independent columns B $Q$ will have four linearly independent rows and five linearly independent columns C $QQ^{T}$will be invertible D $Q^{T}Q$ will be invertible
Engineering Mathematics   Linear Algebra
Question 6 Explanation:
If rank of (5 x 6) matrix is 4, then surely it must have exactly 4 linearly independent rows as will as 4 linearly independent columns.
 Question 7
A function y(t) satisfies the following differential equation :
$\frac{dy(t)}{dt}+y(t)=\delta (t)$
where $\delta$(t) is the delta function. Assuming zero initial condition, and denoting the unit step function by u(t), y(t) can be of the form
 A $e^{t}$ B $e^{-t}$ C $e^{t}$ u(t) D $e^{-t}$ u(t)
Control Systems   Mathematical Models of Physical Systems
Question 7 Explanation:
Taking (L.T.) on both sides
$Y(s)(s+1)=1$
$\therefore \;\; Y(s)=\frac{1}{s+1}$
Taking inverse laplace transform
$y(t)=e^{-t}u(t)$
 Question 8
The equivalent circuits of a diode, during forward biased and reverse biased conditions, are shown in the figure.

If such a diode is used in clipper circuit of figure given above, the output voltage $v_0$ of the circuit will be
 A A B B C C D D
Analog Electronics   Diodes and their Applications
Question 8 Explanation:

$V_P=\frac{10}{10+10}\times 10sin\omega t$
$\;\; =5sin\omega t$
Since maximum voltage across at the point P may be 5 V, hence voltage across the diode always will be less than or equal to zero. So it will be reversed always.

$\therefore V_0=\frac{10}{10+10}\times 10sin\omega t=5sin\omega t$
 Question 9
Two 8-bit ADCs, one of single slope integrating type and other of successive approximate type, take $T_A \; and \; T_B$ times to convert 5 V analog input signal to equivalent digital output. If the input analog signal is reduced to 2.5 V, the approximate time taken by the two ADCs will respectively, be
 A $T_{A},T_{B}$ B $T_{A}/2,T_{B}$ C $T_{A},T_{B}/2$ D $T_{A}/2,T_{B}/2$
Electrical and Electronic Measurements   CRO and Electronic Measurements
Question 9 Explanation:
Single slope integrating type ADC utilize digital counter techniques to measure time required for a voltage ramp to rise from zero to the input voltage.
If conversion time for input voltage $5V=T_A$
So, conversion time for input voltage $2.5V=T_A/2$
Conversion time in successive type ADC does not depend on input voltage. So, conversion time for input voltage 2.5V is also $T_B$.
 Question 10
An input device is interfaced with Intel 8085A microprocessor as memory mapped I/O. The address of the device is 2500H. In order to input data from the device to accumulator, the sequence of instructions will be
 A LXI H, 2500H MOV A, M B LXI H, 2500H MOV M, A C LHLD 2500H MOV A, M D LHLD 2500H MOV M, A
Digital Electronics   Microprocessors
There are 10 questions to complete.