GATE 2021 Question Paper EC Electronics and Communication Engineering

General Aptitude (GA)

Q.1 – Q.5 Multiple Choice Question (MCQ), carry ONE mark each (for each wrong
answer: – 1/3).

Q.1 The current population of a city is 11,02,500. If it has been increasing at the
rate of 5% per annum, what was its population 2 years ago?

(A) 9,92,500

(B) 9,95,006

(C) 10,00,000

(D) 12,51,506

Q.2 p q
p and q are positive integers and + = 3,
q p

p2 q2
then, 2
+ =
q p2

(A) 3

(B) 7

(C) 9

(D) 11

EC - Copyright © GATE 2021 Page 1 of 36

Q.3

The least number of squares that must be added so that the line P-Q
becomes the line of symmetry is ________

(A) 4

(B) 3

(C) 6

(D) 7

Q.4 Nostalgia is to anticipation as _______ is to ________
Which one of the following options maintains a similar logical relation in the
above sentence?

(A) Present, past

(B) Future, past

(C) Past, future

(D) Future, present

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Q.5 Consider the following sentences:
(i) I woke up from sleep.
(ii) I woked up from sleep.
(iii) I was woken up from sleep.
(iv) I was wokened up from sleep.
Which of the above sentences are grammatically CORRECT?

(A) (i) and (ii)

(B) (i) and (iii)

(C) (ii) and (iii)

(D) (i) and (iv)

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Q. 6 – Q. 10 Multiple Choice Question (MCQ), carry TWO marks each (for each wrong
answer: – 2/3).

Q.6 Given below are two statements and two conclusions.
Statement 1: All purple are green.
Statement 2: All black are green.
Conclusion I: Some black are purple.
Conclusion II: No black is purple.
Based on the above statements and conclusions, which one of the following
options is logically CORRECT?

(A) Only conclusion I is correct.

(B) Only conclusion II is correct.

(C) Either conclusion I or II is correct.

(D) Both conclusion I and II are correct.

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Q.7 Computers are ubiquitous. They are used to improve efficiency in almost all
fields from agriculture to space exploration. Artificial intelligence (AI) is
currently a hot topic. AI enables computers to learn, given enough training
data. For humans, sitting in front of a computer for long hours can lead to
health issues.
Which of the following can be deduced from the above passage?
(i) Nowadays, computers are present in almost all places.
(ii) Computers cannot be used for solving problems in engineering.
(iii) For humans, there are both positive and negative effects of using
computers.
(iv) Artificial intelligence can be done without data.

(A) (ii) and (iii)

(B) (ii) and (iv)

(C) (i), (iii) and (iv)

(D) (i) and (iii)

Q.8 Consider a square sheet of side 1 unit. In the first step, it is cut along the
main diagonal to get two triangles. In the next step, one of the cut triangles
is revolved about its short edge to form a solid cone. The volume of the
resulting cone, in cubic units, is ________

(A) 𝜋
3
(B) 2𝜋
3
(C) 3𝜋
2
(D) 3π

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Q.9
65
Sunday 55

50
Saturday 60

35
Friday 20

55
Thursday 60

50
Wednesday 60

65
Tuesday 55

70
Monday 45

0 10 20 30 40 50 60 70 80
Y X

The number of minutes spent by two students, X and Y, exercising every day
in a given week are shown in the bar chart above.
The number of days in the given week in which one of the students spent a
minimum of 10% more than the other student, on a given day, is

(A) 4

(B) 5

(C) 6

(D) 7

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Q.10

Corners are cut from an equilateral triangle to produce a regular convex
hexagon as shown in the figure above.
The ratio of the area of the regular convex hexagon to the area of the original
equilateral triangle is

(A) 2:3

(B) 3:4

(C) 4:5

(D) 5:6

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Q.1 – Q.15 Multiple Choice Question (MCQ), carry ONE mark each (for each wrong
answer: – 1/3).

Q.1 The vector function 𝐅 (𝐫) = −𝒙 𝒊̂ + 𝒚 𝒋̂ is defined over a circular arc C
shown in the figure.

ĵ C
0
45
1
iˆ

The line integral of ∫𝑪 𝐅 (𝐫). 𝒅𝐫 is

(A) 1
2
(B) 1
4
(C) 1
6
(D) 1
3

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Q.2 Consider the differential equation given below.
𝒅𝒚 𝒙
+ 𝒚 = 𝒙√𝒚
𝒅𝒙 𝟏 − 𝒙𝟐
The integrating factor of the differential equation is

(A) 3
(1 − 𝑥 2 )− ⁄4

(B) 1
(1 − 𝑥 2 )− ⁄4

(C) 3
(1 − 𝑥 2 )− ⁄2

(D) 1
(1 − 𝑥 2 )− ⁄2

Q.3 Two continuous random variables 𝑿 and 𝒀 are related as
𝒀 = 𝟐𝑿 + 𝟑
Let 𝝈𝟐𝑿 and 𝝈𝟐𝒀 denote the variances of 𝑿 and 𝒀, respectively. The variances
are related as

(A) 𝜎𝑌2 = 2 𝜎𝑋2

(B) 𝜎𝑌2 = 4 𝜎𝑋2

(C) 𝜎𝑌2 = 5 𝜎𝑋2

(D) 𝜎𝑌2 = 25 𝜎𝑋2

Q.4 Consider a real-valued base-band signal 𝒙(𝒕), band limited to 𝟏𝟎 kHz. The
𝒕
Nyquist rate for the signal 𝒚(𝒕) = 𝒙(𝒕) 𝒙 ( 𝟏 + 𝟐 ) is

(A) 15 kHz

(B) 30 kHz

(C) 60 kHz

(D) 20 kHz

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Q.5 Consider two 𝟏𝟔-point sequences 𝒙[𝒏] and 𝒉[𝒏]. Let the linear convolution
of 𝒙[𝒏] and 𝒉[𝒏] be denoted by 𝒚[𝒏], while 𝒛[𝒏] denotes the 𝟏𝟔-point inverse
discrete Fourier transform (IDFT) of the product of the 𝟏𝟔-point DFTs of
𝒙[𝒏] and 𝒉[𝒏]. The value(s) of 𝒌 for which 𝒛[𝒌] = 𝒚[𝒌] is/are

(A) 𝑘 = 0, 1, 2, … , 15

(B) 𝑘=0

(C) 𝑘 = 15

(D) 𝑘 = 0 and 𝑘 = 15

Q.6 A bar of silicon is doped with boron concentration of 𝟏𝟎𝟏𝟔 𝐜𝐦−𝟑 and
assumed to be fully ionized. It is exposed to light such that electron-hole pairs
are generated throughout the volume of the bar at the rate of 𝟏𝟎𝟐𝟎 𝐜𝐦−𝟑 𝐬−𝟏 .
If the recombination lifetime is 𝟏𝟎𝟎 µs, intrinsic carrier concentration of
silicon is
𝟏𝟎 −𝟑
𝟏𝟎 𝐜𝐦 and assuming 𝟏𝟎𝟎% ionization of boron, then the approximate
product of steady-state electron and hole concentrations due to this light
exposure is

(A) 1020 cm−6

(B) 2 × 1020 cm−6

(C) 1032 cm−6

(D) 2 × 1032 cm−6

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Q.7 The energy band diagram of a p-type semiconductor bar of length 𝐋 under
equilibrium condition (i.e., the Fermi energy level EF is constant) is shown in
the figure. The valance band EV is sloped since doping is non-uniform along
the bar. The difference between the energy levels of the valence band at the
two edges of the bar is ∆.

p-type

EF
EV

Δ

z=0 z=L

If the charge of an electron is q, then the magnitude of the electric field
developed inside this semiconductor bar is

(A) ∆
qL

(B) 2∆
qL

(C) ∆
2qL

(D) 3∆
2qL

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Q.8 In the circuit shown in the figure, the transistors M1 and M2 are operating in
saturation. The channel length modulation coefficients of both the
transistors are non-zero. The transconductance of the MOSFETs M1 and M2
are 𝒈𝒎𝟏 and 𝒈𝒎𝟐 , respectively, and the internal resistance of the MOSFETs
M1 and M2 are 𝒓𝟎𝟏 and 𝒓𝟎𝟐 , respectively.

VDD

Vin M2

Vout

M1

Ignoring the body effect, the ac small signal voltage gain (𝝏𝑽𝒐𝒖𝒕 ⁄𝝏𝑽𝒊𝒏 ) of the
circuit is

(A) −𝑔𝑚2 ( 𝑟01 ||𝑟02 )

(B) 1
−𝑔𝑚2 ( ||𝑟 )
𝑔𝑚1 02

(C) 1
−𝑔𝑚1 ( ||𝑟 ||𝑟 )
𝑔𝑚2 01 02

(D) 1
−𝑔𝑚2 ( ||𝑟 ||𝑟 )
𝑔𝑚1 01 02

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Q.9 For the circuit with an ideal OPAMP shown in the figure, VREF is fixed.

RIN RF

+12 V

_
VIN
R1 VOUT
+

-12 V
RL = 100 Ω
VREF R2

If VOUT = 𝟏 volt for VIN = 𝟎. 𝟏 volt and VOUT = 𝟔 volt for VIN = 𝟏 volt, where
VOUT is measured across RL connected at the output of this OPAMP, the
value of 𝐑 𝐅 ⁄𝐑 𝐈𝐍 is

(A) 3.285

(B) 2.860

(C) 3.825

(D) 5.555

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Q.10 Consider the circuit with an ideal OPAMP shown in the figure.

R RF

+ VCC
R
_
VIN
VOUT
VREF
+

‒ VCC

Assuming |𝐕𝐈𝐍 | ≪ |𝐕𝐂𝐂 | and |𝐕𝐑𝐄𝐅 | ≪ |𝐕𝐂𝐂 |, the condition at which VOUT
equals to zero is

(A) VIN = VREF

(B) VIN = 0.5 VREF

(C) VIN = 2 VREF

(D) VIN = 2 + VREF

Q.11 If (𝟏𝟐𝟑𝟓)𝒙 = (𝟑𝟎𝟑𝟑)𝒚, where 𝒙 and 𝒚 indicate the bases of the
corresponding numbers, then

(A) 𝑥 = 7 and 𝑦 = 5

(B) 𝑥 = 8 and 𝑦 = 6

(C) 𝑥 = 6 and 𝑦 = 4

(D) 𝑥 = 9 and 𝑦 = 7

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Q.12 Addressing of a 𝟑𝟐K × 𝟏𝟔 memory is realized using a single decoder. The
minimum number of AND gates required for the decoder is

(A) 28

(B) 232

(C) 215

(D) 219

Q.13 The block diagram of a feedback control system is shown in the figure.

G2

+ + +
X (s) G1 Y (s)
−
H

𝒀(𝒔)
The transfer function 𝑿(𝒔) of the system is

(A) 𝐺1 + 𝐺2 + 𝐺1 𝐺2 𝐻
1 + 𝐺1 𝐻

(B) 𝐺1 + 𝐺2
1 + 𝐺1 𝐻 + 𝐺2 𝐻

(C) 𝐺1 + 𝐺2
1 + 𝐺1 𝐻

(D) 𝐺1 + 𝐺2 + 𝐺1 𝐺2 𝐻
1 + 𝐺1 𝐻 + 𝐺2 𝐻

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Q.14 The complete Nyquist plot of the open-loop transfer function 𝑮(𝒔)𝑯(𝒔) of a
feedback control system is shown in the figure.

j Im GH

G(s)H(s)-plane

Re GH
0

( −1, j 0 )

If 𝑮(𝒔)𝑯(𝒔) has one zero in the right-half of the 𝒔-plane, the number of poles
that the closed-loop system will have in the right-half of the 𝒔-plane is

(A) 0

(B) 1

(C) 4

(D) 3

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Q.15 Consider a rectangular coordinate system (x, y, z) with unit vectors ax, ay,
and az. A plane wave traveling in the region 𝒛 ≥ 𝟎 with electric field vector
𝐄 = 𝟏𝟎 𝐜𝐨𝐬(𝟐 × 𝟏𝟎𝟖 𝒕 + 𝜷𝒛)𝐚𝒚 is incident normally on the plane at 𝒛 = 𝟎,
where β is the phase constant. The region 𝒛 ≥ 𝟎 is in free space and the
region 𝒛 < 𝟎 is filled with a lossless medium (permittivity 𝜺 = 𝜺𝟎 ,
permeability 𝝁 = 𝟒𝝁𝟎 , where 𝜺𝟎 = 𝟖. 𝟖𝟓 × 𝟏𝟎−𝟏𝟐 F/m and 𝝁𝟎 = 𝟒𝝅 ×
𝟏𝟎−𝟕 H/m). The value of the reflection coefficient is

(A) 1
3
(B) 3
5
(C) 2
5
(D) 2
3

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Q.16 – Q.25 Numerical Answer Type (NAT), carry ONE mark each (no negative
marks).

Q.16 If the vectors (𝟏. 𝟎, −𝟏. 𝟎, 𝟐. 𝟎), (𝟕. 𝟎, 𝟑. 𝟎, 𝒙) and (𝟐. 𝟎, 𝟑. 𝟎, 𝟏. 𝟎) in ℝ𝟑
are linearly dependent, the value of 𝒙 is __________

Q.17 Consider the vector field 𝐅 = 𝐚𝒙 (𝟒𝒚 − 𝒄𝟏 𝒛) + 𝐚𝒚 (𝟒𝒙 + 𝟐𝒛) + 𝐚𝒛 (𝟐𝒚 + 𝒛) in
a rectangular coordinate system (x, y, z) with unit vectors ax, ay, and az. If the
field F is irrotational (conservative), then the constant 𝒄𝟏 (in integer) is
__________

Q.18 Consider the circuit shown in the figure.

3Ω 2Ω
5A

7Ω
P Q
I

6Ω 2Ω

The current I flowing through the 7 Ω resistor between P and Q (rounded off
to one decimal place) is __________ A.

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Q.19 Consider the circuit shown in the figure.

6 mA 1 kΩ

4V
v0 1 kΩ 2 mA

1 kΩ

The value of v0 (rounded off to one decimal place) is __________ V.

Q.20 An 𝟖-bit unipolar (all analog output values are positive) digital-to-analog
converter (DAC) has a full-scale voltage range from 𝟎 V to 𝟕. 𝟔𝟖 V. If the
digital input code is 10010110 (the leftmost bit is MSB), then the analog
output voltage of the DAC (rounded off to one decimal place) is __________
V.

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Q.21 The autocorrelation function 𝑹𝑿 (𝝉) of a wide-sense stationary random
process 𝑿(𝒕) is shown in the figure.

RX ( )
2

−2 0 2 

The average power of 𝑿(𝒕) is __________

Q.22 Consider a carrier signal which is amplitude modulated by a single-tone
sinusoidal message signal with a modulation index of 𝟓𝟎%. If the carrier and
one of the sidebands are suppressed in the modulated signal, the percentage
of power saved (rounded off to one decimal place) is __________

Q.23 A speech signal, band limited to 𝟒 kHz, is sampled at 𝟏. 𝟐𝟓 times the Nyquist
rate. The speech samples, assumed to be statistically independent and
uniformly distributed in the range −𝟓 V to +𝟓 V, are subsequently
quantized in an 𝟖-bit uniform quantizer and then transmitted over a voice-
grade AWGN telephone channel. If the ratio of transmitted signal power to
channel noise power is 𝟐𝟔 dB, the minimum channel bandwidth required to
ensure reliable transmission of the signal with arbitrarily small probability
of transmission error (rounded off to two decimal places) is __________ kHz.

Q.24 A 𝟒 kHz sinusoidal message signal having amplitude 𝟒 V is fed to a delta
modulator (DM) operating at a sampling rate of 𝟑𝟐 kHz. The minimum step
size required to avoid slope overload noise in the DM (rounded off to two
decimal places) is __________ V.

Q.25 The refractive indices of the core and cladding of an optical fiber are 𝟏. 𝟓𝟎
and 𝟏. 𝟒𝟖, respectively. The critical propagation angle, which is defined as
the maximum angle that the light beam makes with the axis of the optical
fiber to achieve the total internal reflection, (rounded off to two decimal
places) is __________ degree.

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Q. 26 – Q. 35 Multiple Choice Question (MCQ), carry TWO marks each (for each
wrong answer: – 2/3).

Q.26 Consider the integral
𝐬𝐢𝐧(𝒙)
∮ 𝒅𝒙
𝑪 𝒙𝟐 (𝒙𝟐 + 𝟒)
where 𝑪 is a counter-clockwise oriented circle defined as |𝒙 − 𝒊| = 𝟐. The
value of the integral is

(A) 𝜋
− sin(2𝑖)
8

(B) 𝜋
sin(2𝑖)
8

(C) 𝜋
− sin(2𝑖)
4

(D) 𝜋
sin(2𝑖)
4

Q.27 A box contains the following three coins.
I. A fair coin with head on one face and tail on the other face.
II. A coin with heads on both the faces.
III. A coin with tails on both the faces.
A coin is picked randomly from the box and tossed. Out of the two remaining
coins in the box, one coin is then picked randomly and tossed. If the first toss
results in a head, the probability of getting a head in the second toss is

(A) 2
5
(B) 1
3
(C) 1
2
(D) 2
3

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Q.28 The switch in the circuit in the figure is in position P for a long time and then
moved to position Q at time 𝒕 = 𝟎.

5 kΩ P t =0
5 kΩ

Q
10 kΩ
20VV
IN 20 kΩ v(t ) 1 mF
0.1 mH

𝒅𝒗(𝒕)
The value of at 𝒕 = 𝟎+ is
𝒅𝒕

(A) 0 V/s

(B) 3 V/s

(C) −3 V/s

(D) −5 V/s

EC - Copyright © GATE 2021 Page 22 of 36

Q.29 Consider the two-port network shown in the figure.

I1 1Ω I2
+ +

V1 1Ω 1Ω V2
3V2

_ _

The admittance parameters, in siemens, are

(A) 𝑦11 = 2, 𝑦12 = −4, 𝑦21 = −4, 𝑦22 = 2

(B) 𝑦11 = 1, 𝑦12 = −2, 𝑦21 = −1, 𝑦22 = 3

(C) 𝑦11 = 2, 𝑦12 = −4, 𝑦21 = −1, 𝑦22 = 2

(D) 𝑦11 = 2, 𝑦12 = −4, 𝑦21 = −4, 𝑦22 = 3

Q.30 For an n-channel silicon MOSFET with 𝟏𝟎 nm gate oxide thickness, the
substrate sensitivity (𝝏𝑽𝑻 ⁄𝝏|𝑽𝑩𝑺 |) is found to be 𝟓𝟎 mV/V at a substrate
voltage
|𝑽𝑩𝑺 | = 𝟐 𝑽, where 𝑽𝑻 is the threshold voltage of the MOSFET. Assume
that, |𝑽𝑩𝑺 | ≫ 𝟐𝚽𝑩, where 𝐪𝚽𝑩 is the separation between the Fermi energy
level EF and the intrinsic level Ei in the bulk. Parameters given are
Electron charge (q) = 𝟏. 𝟔 × 𝟏𝟎−𝟏𝟗 C
Vacuum permittivity (𝜺𝟎 ) = 𝟖. 𝟖𝟓 × 𝟏𝟎−𝟏𝟐 F/m
Relative permittivity of silicon (𝜺𝐒𝐢 ) = 𝟏𝟐
Relative permittivity of oxide (𝜺𝐨𝐱 ) = 𝟒
The doping concentration of the substrate is

(A) 7.37 × 1015 cm−3

(B) 4.37 × 1015 cm−3

(C) 2.37 × 1015 cm−3

(D) 9.37 × 1015 cm−3

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Q.31 The propagation delays of the XOR gate, AND gate and multiplexer (MUX)
in the circuit shown in the figure are 4 ns, 2 ns and 1 ns, respectively.

P 0
Q
MUX
0 Y
R MUX 1

S 1
S0 S0

T

If all the inputs P, Q, R, S and T are applied simultaneously and held
constant, the maximum propagation delay of the circuit is

(A) 3 ns

(B) 5 ns

(C) 6 ns

(D) 7 ns

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Q.32 The content of the registers are R1 = 𝟐𝟓H, R2 = 𝟑𝟎H and R3 = 𝟒𝟎H. The
following machine instructions are executed.
PUSH{R1}
PUSH{R2}
PUSH{R3}
POP{R1}
POP{R2}
POP{R3}
After execution, the content of registers R1, R2, R3 are

(A) R1 = 40H, R 2 = 30H, R 3 = 25H

(B) R1 = 25H, R 2 = 30H, R 3 = 40H

(C) R1 = 30H, R 2 = 40H, R 3 = 25H

(D) R1 = 40H, R 2 = 25H, R 3 = 30H

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Q.33 The electrical system shown in the figure converts input source current 𝒊𝒔 (𝒕)
to output voltage vo ( t ) .

iL ( t )

1H
vo ( t )

1Ω
+
is ( t )
1F
vC ( t ) 1Ω
−

Current iL ( t ) in the inductor and voltage vC ( t ) across the capacitor are
taken as the state variables, both assumed to be initially equal to zero, i.e.,
iL ( 0) = 0 and vC ( 0) = 0. The system is

(A) completely state controllable as well as completely observable

(B) completely state controllable but not observable

(C) completely observable but not state controllable

(D) neither state controllable nor observable

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Q.34 A digital transmission system uses a (𝟕, 𝟒) systematic linear Hamming code
for transmitting data over a noisy channel. If three of the message-codeword
pairs in this code ( 𝐦𝒊 ; 𝐜𝒊 ), where 𝐜𝒊 is the codeword corresponding to the
ith message 𝐦𝒊 , are known to be ( 𝟏 𝟏 𝟎 𝟎 ; 𝟎 𝟏 𝟎 𝟏 𝟏 𝟎 𝟎 ),
( 𝟏 𝟏 𝟏 𝟎 ; 𝟎 𝟎 𝟏 𝟏 𝟏 𝟏 𝟎 ) and (𝟎 𝟏 𝟏 𝟎 ; 𝟏 𝟎 𝟎 𝟎 𝟏 𝟏 𝟎 ), then which of the
following is a valid codeword in this code?

(A) 1101001

(B) 1011010

(C) 0001011

(D) 0110100

EC - Copyright © GATE 2021 Page 27 of 36

Q.35 The impedance matching network shown in the figure is to match a lossless
line having characteristic impedance 𝒁𝟎 = 𝟓𝟎 Ω with a load impedance 𝒁𝐋 .
A quarter-wave line having a characteristic impedance 𝒁𝟏 = 𝟕𝟓 Ω is
connected to 𝒁𝐋 . Two stubs having characteristic impedance of 𝟕𝟓 Ω each
are connected to this quarter-wave line. One is a short-circuited (S.C.) stub
of length 𝟎. 𝟐𝟓𝝀 connected across PQ and the other one is an open-circuited
(O.C.) stub of length 𝟎. 𝟓𝝀 connected across RS.

P 0.25 R

0.25
Z 0 = 50  Z1 = 75  ZL

Q S
5
=7

ub

O.C = 75 
. St

ub
Z1
S. C

. St
Z1

0.5

The impedance matching is achieved when the real part of 𝒁𝐋 is

(A) 112.5 Ω

(B) 75.0 Ω

(C) 50.0 Ω

(D) 33.3 Ω

EC - Copyright © GATE 2021 Page 28 of 36

Q.36 – Q.55 Numerical Answer Type (NAT), carry TWO mark each (no negative
marks).

Q.36 A real 𝟐 × 𝟐 non-singular matrix 𝐀 with repeated eigenvalue is given as
𝒙 −𝟑. 𝟎
𝐀= [ ]
𝟑. 𝟎 𝟒. 𝟎
where 𝒙 is a real positive number. The value of 𝒙 (rounded off to one decimal
place) is __________

Q.37 For a vector field 𝐃 = 𝝆 𝐜𝐨𝐬𝟐 𝝋 𝐚𝝆 + 𝒛𝟐 𝐬𝐢𝐧𝟐 𝝋 𝐚𝝋 in a cylindrical coordinate
system (𝝆, 𝝋, 𝒛) with unit vectors 𝐚𝝆 , 𝐚𝝋 𝐚𝐧𝐝 𝐚𝒛 , the net flux of 𝐃 leaving the
closed surface of the cylinder (𝝆 = 𝟑, 𝟎 ≤ 𝒛 ≤ 𝟐) (rounded off to two decimal
places) is __________

Q.38 In the circuit shown in the figure, the switch is closed at time 𝒕 = 𝟎, while
the capacitor is initially charged to −𝟓 V (i.e., vc(𝟎) = −𝟓 V).

250 Ω t=0

+ VR ‒
+
250 Ω vc(t) 0.6 µF
5V _
VR /500

The time after which the voltage across the capacitor becomes zero (rounded
off to three decimal places) is __________ ms.

EC - Copyright © GATE 2021 Page 29 of 36

Q.39 The exponential Fourier series representation of a continuous-time periodic
signal 𝒙(𝒕) is defined as
∞

𝒙(𝒕) = ∑ 𝒂𝒌 𝒆𝒋𝒌𝝎𝟎 𝒕
𝒌 = −∞

where 𝝎𝟎 is the fundamental angular frequency of 𝒙(𝒕) and the coefficients
of the series are 𝒂𝒌 . The following information is given about 𝒙(𝒕) and 𝒂𝒌 .
I. 𝒙(𝒕) is real and even, having a fundamental period of 𝟔
II. The average value of 𝒙(𝒕) is 𝟐
𝒌, 𝟏 ≤ 𝒌 ≤ 𝟑
III. 𝒂𝒌 = {
𝟎, 𝒌>𝟑
The average power of the signal 𝒙(𝒕) (rounded off to one decimal place)
is __________

Q.40 For a unit step input 𝒖[𝒏], a discrete-time LTI system produces an output
signal ( 𝟐𝜹[𝒏 + 𝟏] + 𝜹[𝒏] + 𝜹[𝒏 − 𝟏] ). Let 𝒚[𝒏] be the output of the system
𝟏 𝒏
for an input ((𝟐) 𝒖[𝒏]). The value of 𝒚[𝟎] is __________

Q.41 Consider the signals 𝒙[𝒏] = 𝟐𝒏−𝟏 𝒖[−𝒏 + 𝟐] and 𝒚[𝒏] = 𝟐−𝒏+𝟐 𝒖[𝒏 + 𝟏],
where 𝒖[𝒏] is the unit step sequence. Let 𝑿(𝒆𝒋𝝎 ) and 𝒀(𝒆𝒋𝝎 ) be the discrete-
time Fourier transform of 𝒙[𝒏] and 𝒚[𝒏], respectively. The value of the
integral
𝟐𝝅
𝟏
∫ 𝑿(𝒆𝒋𝝎 ) 𝒀(𝒆−𝒋𝝎 ) 𝒅𝝎
𝟐𝝅
𝟎

(rounded off to one decimal place) is __________

EC - Copyright © GATE 2021 Page 30 of 36

Q.42 A silicon P-N junction is shown in the figure. The doping in the P region is
𝟓 × 𝟏𝟎𝟏𝟔 cm−3 and doping in the N region is 𝟏𝟎 × 𝟏𝟎𝟏𝟔 cm−3. The parameters
given are
Built-in voltage (𝚽𝐛𝐢 ) = 0.8 V
Electron charge (𝐪) = 𝟏. 𝟔 × 𝟏𝟎−𝟏𝟗 C
Vacuum permittivity (𝜺𝟎 ) = 𝟖. 𝟖𝟓 × 𝟏𝟎−𝟏𝟐 F/m
Relative permittivity of silicon (𝜺𝐒𝐢 ) = 𝟏𝟐

P N

1.2 µm 0.2 µm

The magnitude of reverse bias voltage that would completely deplete one of
the two regions (P or N) prior to the other (rounded off to one decimal place)
is __________ V.

Q.43 An asymmetrical periodic pulse train vin of 𝟏𝟎 V amplitude with on-time
TON = 1 ms and off-time TOFF = 1 µs is applied to the circuit shown in the
figure. The diode D1 is ideal.

20 nF

+

500 kΩ D1 vo
vvinin

-
The difference between the maximum voltage and minimum voltage of the
output waveform vo (in integer) is __________ V.

EC - Copyright © GATE 2021 Page 31 of 36

Q.44 For the transistor M1 in the circuit shown in the figure, µ𝒏 𝑪𝐨𝐱 = 𝟏𝟎𝟎 µ𝐀/𝐕 𝟐
and (𝑾⁄𝑳) = 𝟏𝟎, where µ𝒏 is the mobility of electron, 𝑪𝐨𝐱 is the oxide
capacitance per unit area, 𝑾 is the width and 𝑳 is the length.

VDD = 3 V

RD = 20 kΩ

VOUT

VGS M1

The channel length modulation coefficient is ignored. If the gate-to-source
voltage 𝑽𝐆𝐒 is 1 V to keep the transistor at the edge of saturation, then the
threshold voltage of the transistor (rounded off to one decimal place)
is __________ V.

EC - Copyright © GATE 2021 Page 32 of 36

Q.45 A circuit with an ideal OPAMP is shown in the figure. A pulse 𝐕𝐈𝐍 of 𝟐𝟎 m𝐬
duration is applied to the input. The capacitors are initially uncharged.

5V
10 kΩ
0V

t=0 t = 20 ms 1 µF
+12 V

VIN ‒

VOUT
1 µF +

‒ 12 V

The output voltage 𝐕𝐎𝐔𝐓 of this circuit at 𝒕 = 𝟎+ (in integer) is __________
V.

Q.46 The propagation delay of the exclusive-OR (XOR) gate in the circuit in the
figure is 3 ns. The propagation delay of all the flip-flops is assumed to be
zero. The clock (Clk) frequency provided to the circuit is 500 MHz.

Q2 Q1 Q0
D2 D1 D0

Clk

Starting from the initial value of the flip-flop outputs 𝐐𝟐 𝐐𝟏 𝐐𝟎 = 𝟏 𝟏 𝟏 with
𝐃𝟐 = 𝟏, the minimum number of triggering clock edges after which the flip-
flop outputs 𝐐𝟐 𝐐𝟏 𝐐𝟎 becomes 1 0 0 (in integer) is __________

EC - Copyright © GATE 2021 Page 33 of 36

Q.47 The circuit in the figure contains a current source driving a load having an
inductor and a resistor in series, with a shunt capacitor across the load. The
ammeter is assumed to have zero resistance. The switch is closed at time
𝒕 = 𝟎.

Ammeter
Switch
Load
A
Current
Source 10 mH
Shunt
1A Capacitor 100 pF

5 kΩ

Initially, when the switch is open, the capacitor is discharged and the
ammeter reads zero ampere. After the switch is closed, the ammeter reading
keeps fluctuating for some time till it settles to a final steady value. The
maximum ammeter reading that one will observe after the switch is closed
(rounded off to two decimal places) is __________ A.

EC - Copyright © GATE 2021 Page 34 of 36

Q.48 A unity feedback system that uses proportional-integral (PI) control is
shown in the figure.

X (s)
+ KP +
KI 2
Y (s)
s s + 4 s + 5s + 2
3 2

−

The stability of the overall system is controlled by tuning the PI control
parameters 𝑲𝑷 and 𝑲𝑰 . The maximum value of 𝑲𝑰 that can be chosen so as
to keep the overall system stable or, in the worst case, marginally stable
(rounded off to three decimal places) is __________

Q.49 A sinusoidal message signal having root mean square value of 𝟒 V and
frequency of 𝟏 kHz is fed to a phase modulator with phase deviation constant
𝟐 rad/volt. If the carrier signal is 𝒄(𝒕) = 𝟐 𝐜𝐨𝐬(𝟐𝝅𝟏𝟎𝟔 𝒕), the maximum
instantaneous frequency of the phase modulated signal (rounded off to one
decimal place) is __________ Hz.

Q.50 Consider a superheterodyne receiver tuned to 𝟔𝟎𝟎 kHz. If the local oscillator
feeds a 𝟏𝟎𝟎𝟎 kHz signal to the mixer, the image frequency (in integer)
is __________ kHz.

Q.51 In a high school having equal number of boy students and girl students, 𝟕𝟓%
of the students study Science and the remaining 𝟐𝟓% students study
Commerce. Commerce students are two times more likely to be a boy than
are Science students. The amount of information gained in knowing that a
randomly selected girl student studies Commerce (rounded off to three
decimal places) is __________ bits.

EC - Copyright © GATE 2021 Page 35 of 36

Q.52 A message signal having peak-to-peak value of 2 V, root mean square value
of 0.1 V and bandwidth of 5 kHz is sampled and fed to a pulse code
modulation (PCM) system that uses a uniform quantizer. The PCM output
is transmitted over a channel that can support a maximum transmission rate
of 50 kbps. Assuming that the quantization error is uniformly distributed,
the maximum signal to quantization noise ratio that can be obtained by the
PCM system (rounded off to two decimal places) is __________

Q.53 Consider a polar non-return to zero (NRZ) waveform, using +𝟐 V and −𝟐
V for representing binary ‘1’ and ‘0’ respectively, is transmitted in the
presence of additive zero-mean white Gaussian noise with variance 𝟎. 𝟒 V2.
If the a priori probability of transmission of a binary ‘1’ is 𝟎. 𝟒, the optimum
threshold voltage for a maximum a posteriori (MAP) receiver (rounded off to
two decimal places) is __________ V.

Q.54 A standard air-filled rectangular waveguide with dimensions 𝒂 = 𝟖 cm,
𝒃 = 𝟒 cm, operates at 𝟑. 𝟒 GHz. For the dominant mode of wave
propagation, the phase velocity of the signal is 𝒗𝒑 . The value (rounded off to
two decimal places) of 𝒗𝒑 ⁄𝒄, where 𝒄 denotes the velocity of light, is
__________

Q.55 An antenna with a directive gain of 𝟔 dB is radiating a total power of 𝟏𝟔 kW.
The amplitude of the electric field in free space at a distance of 𝟖 km from
the antenna in the direction of 𝟔 dB gain (rounded off to three decimal places)
is __________ V/m.

END OF THE QUESTION PAPER

EC - Copyright © GATE 2021 Page 36 of 36