# 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$\mu$W 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.