GATE EC 2014 SET-1 - PracticePaper

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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

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}}

\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.