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.
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