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.