GATE EC 2014 SET-1

Question 1
For matrices of same dimension M, N and scalar c, which one of these properties DOES NOT ALWAYS hold ?
A
(M^{T})^{T}=M
B
(cM)^{T}=c(M)^{T}
C
(M+N)^{T}=M^{T}+N^{T}
D
MN=NM
Engineering Mathematics   Linear Algebra
Question 1 Explanation: 
Matrix multiplication is not commutative.
Question 2
In a housing society, half of the families have a single child per family, while the remaining half have two children per family. The probability that a child picked at random, has a sibling is _____
A
1.5
B
0.67
C
2
D
3
Engineering Mathematics   Probability and Statistics
Question 2 Explanation: 
Required probability
=\frac{\frac{1}{2} \times \frac{2}{3}}{\frac{1}{2} \times \frac{1}{3}+\frac{1}{2} \times \frac{2}{3}}=\frac{2}{3}=0.666
Question 3
C is a closed path in the z -plane by |z| = 3. The value of the integral \oint_{c}(\frac{z^{2}-z+4j}{z+2j})dz is
A
-4\pi (1+ j2 )
B
4\pi (3- j2 )
C
-4\pi (3+ j2 )
D
4\pi (1- j2)
Engineering Mathematics   Complex Analysis
Question 3 Explanation: 


\oint_{c} \frac{z^{2}-z+4 j}{z+2 j}
Pole =z=-2 j
which is inside of |z|=3
From Cauchy integral formula
\begin{aligned} \oint \frac{z^{2}-z+4 j}{z+2 j} &=2 \pi i\left[\lim _{z \rightarrow-2 j} z^{2}-z+4 j\right] \\ &=2 \pi j[-4+2 j+4 j] \\ &=2 \pi j[-4+6 j] \\ &=-4 \pi[2 j+3] \end{aligned}
Question 4
A real (4x4) matrix A satisfies the equation A_{2} = I, where I is the (4x4) identity matrix. The positive eigen value of A is _____.
A
1
B
2
C
3
D
4
Engineering Mathematics   Linear Algebra
Question 4 Explanation: 
\begin{aligned} \text{since, }A^{2}=l, \text{eig}\left(A^{2}\right)&=\text{eig}(I)=1 \\ \Rightarrow \quad \text{eig}(A)^{2}&=1\\ \Rightarrow \quad \text{eig}(A)&=\pm 1 \end{aligned}
Therefore, the positive eigen value of A is +1.
Question 5
Let X_{1} , X_{2}, \; and \; X_{3} be independent and identically distributed random variables with the uniform distribution on [0,1]. The probability P{X_{1} is the largest} is
A
0.5
B
0.33
C
0.25
D
0.75
Communication Systems   Random Processes
Question 5 Explanation: 
If multiple independent random variables are uniformly distributed in the same interval then each random variable will have equal chances to be largest and to be lowest.
P\left(X_{1} \text { is the largest) }=\frac{1}{3}\right.
Question 6
For maximum power transfer between two cascaded sections of an electrical network, the relationship between the output impedance Z_{1} of the first section to the input impedance Z_{2} of the second section is
A
Z_{2}=Z_{1}
B
Z_{2}=-Z_{1}
C
Z_{2}=Z_{1}^{*}
D
Z_{2}=-Z_{1}^{*}
Network Theory   Transient Analysis
Question 7
Consider the configuration shown in the figure which is a portion of a larger electrical network

For R = 1\Omega and currents i_{1} = 2 A , i_{4} =- 1A , i_{5}=- 4 A , which one of the following is TRUE ?
A
i_{6}=5A
B
i_{3}=-4A
C
Data is sufficient to conclude that the supposed currents are impossible
D
Data is insufficient to identify the currents i_{2},i_{3} \; and \; i_{6}
Network Theory   Basics of Network Analysis
Question 7 Explanation: 
Given data:
\begin{array}{l} i_{1}=2 \mathrm{A}, i_{4}=-1 \mathrm{A}, i_{5}=-4 \mathrm{A} \\ R=1 \Omega \end{array}
To calculate:
i_{6}=?


Using KVL at all the three nodes we get,
At node A
i_{5}-i_{3}+i_{2}=0\qquad \ldots(i)
At node B
i_{4}+i_{1}-i_{2}=0\qquad \ldots(ii)
At node C
i_{6}+i_{3}-i_{1}=0 \qquad \ldots(iii)
By putting the value of i_{3} and i_{2} from equation (i)
and (ii) in equation (iii) we get,
\begin{aligned} i_{6}+\left(i_{2}+i_{5}\right)-i_{1}&=0 \\ i_{6}+\left(i_{1}+i_{4}+i_{5}\right)-i_{1}&=0 \\ \therefore \quad i_{6}+(2-1-4)-2&=0 \\ i_{6}&=5 \mathrm{A} \end{aligned}
Question 8
When the optical power incident on a photodiode is 10\muW and the responsivity is 0.8 A/W, the photocurrent generated (in \mu A) is _____.
A
2
B
4
C
8
D
10
Electronic Devices   PN-Junction Diodes and Special Diodes
Question 8 Explanation: 
\begin{array}{l} \text { Responsivity }(R)=\frac{I_{p}}{P_{o}} \\ \text { where } I_{p}=\text { Photo current } \\ \qquad P_{0}=\text { Incident power } \\ \therefore \quad I_{p}=R \times P_{0}=8 \mu \mathrm{A} \end{array}
Question 9
In the figure, assume that the forward voltage drops of the PN diode D_{1} and Schottky diode D_{2} are 0.7 V and 0.3 V, respectively. If ON denotes conducting state of the diode and OFF denotes non-conducting state of the diode, then in the circuit,
A
both D_{1} \; and \; D_{2} are ON
B
D_{1} is ON and D_{2} are OFF
C
both D_{1} \; and \; D_{2} are OFF
D
D_{1} is OFF and D_{2} are ON
Analog Circuits   Diodes Applications
Question 9 Explanation: 
Consider D_{1} \rightarrow \text{OFF } and D_{2} \rightarrow \text{ON} then


Apply KVL
\begin{aligned} 10 &=1000 I+20 I+0.3 \\ I &=\frac{9.7}{1020} \\ I &=9.5 \mathrm{mA} \end{aligned}
Now, we calculate V_{D_{1}} of
\begin{aligned} & 10=9.5+V_{D_{1}} \\ \therefore V_{D_{1}} &=0.5 \mathrm{V} \end{aligned}
since v_{D_{1}} \lt 0.7 \mathrm{V}, \quad \mathrm{D}_{1} is in OFF state i.e. our assumption is correct and hence (D) is the correct option.
Question 10
If fixed positive charges are present in the gate oxide of an n-channel enhancement type MOSFET, it will lead to
A
a decrease in the threshold voltage
B
channel length modulation
C
an increase in substrate leakage current
D
an increase in accumulation capacitance
Electronic Devices   Ic Fabrication
Question 10 Explanation: 
Fixed charges reduces threshold voltage.
There are 10 questions to complete.
Like this FREE website? Please share it among all your friends and join the campaign of FREE Education to ALL.