GATE 2025 Question Paper Electronics and Communication Engineering (EC)
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Page 1
GATE
2025
Question Paper | Answer Key
Graduate Aptitude Test in Engineering
(GATE) is a prestigious national-level exam
that assesses candidates for
comprehensive understanding in various
undergraduate-level subjects in
Engineering, Technology, Science,
Architecture, and Humanities.
2025
Question Paper | Answer Key
Graduate Aptitude Test in Engineering
(GATE) is a prestigious national-level exam
that assesses candidates for
comprehensive understanding in various
undergraduate-level subjects in
Engineering, Technology, Science,
Architecture, and Humanities.
Page 2
General Aptitude
Q.1 – Q.5 Carry ONE mark Each
Q.1 Here are two analogous groups, Group-I and Group-II, that list words in their
decreasing order of intensity. Identify the missing word in Group-II.
Group-I: Abuse → Insult → Ridicule
Group-II: ________ → Praise → Appreciate
(A) Extol
(B) Prize
(C) Appropriate
(D) Espouse
Q.2 Had I learnt acting as a child, I ________ a famous film star.
Select the most appropriate option to complete the above sentence.
(A) will be
(B) can be
(C) am going to be
(D) could have been
Organizing Institute: IIT Roorkee Page 1 of 52
Q.1 – Q.5 Carry ONE mark Each
Q.1 Here are two analogous groups, Group-I and Group-II, that list words in their
decreasing order of intensity. Identify the missing word in Group-II.
Group-I: Abuse → Insult → Ridicule
Group-II: ________ → Praise → Appreciate
(A) Extol
(B) Prize
(C) Appropriate
(D) Espouse
Q.2 Had I learnt acting as a child, I ________ a famous film star.
Select the most appropriate option to complete the above sentence.
(A) will be
(B) can be
(C) am going to be
(D) could have been
Organizing Institute: IIT Roorkee Page 1 of 52
Page 3
Q.3 The 12 musical notes are given as 𝐶, 𝐶 # , 𝐷, 𝐷# , 𝐸, 𝐹, 𝐹 # , 𝐺, 𝐺 # , 𝐴, 𝐴# , and 𝐵.
12
Frequency of each note is √2 times the frequency of the previous note. If the
frequency of the note C is 130.8 Hz, then the ratio of frequencies of notes 𝐹 # and 𝐶
is:
(A) 6
√2
(B)
√2
(C) 4
√2
(D) 2
Organizing Institute: IIT Roorkee Page 2 of 52
12
Frequency of each note is √2 times the frequency of the previous note. If the
frequency of the note C is 130.8 Hz, then the ratio of frequencies of notes 𝐹 # and 𝐶
is:
(A) 6
√2
(B)
√2
(C) 4
√2
(D) 2
Organizing Institute: IIT Roorkee Page 2 of 52
Page 4
Q.4 The following figures show three curves generated using an iterative algorithm. The
total length of the curve generated after ‘Iteration 𝑛’ is:
Note: The figures shown are representative.
1
Iteration 0:
1 1
Iteration 1: Length of each segment:
3 3
1
1
Iteration 2: 9 Length of each segment:
9
𝑛
(A) 5 2
( )
3
(B) 5 𝑛
( )
3
(C) 5 2𝑛
( )
3
(D) 5 𝑛(2𝑛−1)
( )
3
Organizing Institute: IIT Roorkee Page 3 of 52
total length of the curve generated after ‘Iteration 𝑛’ is:
Note: The figures shown are representative.
1
Iteration 0:
1 1
Iteration 1: Length of each segment:
3 3
1
1
Iteration 2: 9 Length of each segment:
9
𝑛
(A) 5 2
( )
3
(B) 5 𝑛
( )
3
(C) 5 2𝑛
( )
3
(D) 5 𝑛(2𝑛−1)
( )
3
Organizing Institute: IIT Roorkee Page 3 of 52
Page 5
Q.5 |𝑥|
Which one of the following plots represents 𝑓 (𝑥 ) = − , where 𝑥 is a non-
𝑥
zero real number?
Note: The figures shown are representative.
f(x)
(A)
(0,1)
x
(0, −1)
f(x)
(B)
(0,1)
x
(0, −1)
f(x)
(C)
(1,0)
x
(−1,0)
f(x)
(D)
(0,1)
x
(0, −1)
Organizing Institute: IIT Roorkee Page 4 of 52
Which one of the following plots represents 𝑓 (𝑥 ) = − , where 𝑥 is a non-
𝑥
zero real number?
Note: The figures shown are representative.
f(x)
(A)
(0,1)
x
(0, −1)
f(x)
(B)
(0,1)
x
(0, −1)
f(x)
(C)
(1,0)
x
(−1,0)
f(x)
(D)
(0,1)
x
(0, −1)
Organizing Institute: IIT Roorkee Page 4 of 52
Page 6
Q.6 – Q.10 Carry TWO marks Each
Q.6 Identify the option that has the most appropriate sequence such that a coherent
paragraph is formed:
P. Over time, such adaptations lead to significant evolutionary changes with
the potential to shape the development of new species.
Q. In natural world, organisms constantly adapt to their environments in
response to challenges and opportunities.
R. This process of adaptation is driven by the principle of natural selection,
where favorable traits increase an organism’s chances of survival and
reproduction.
S. As environments change, organisms that can adapt their behavior, structure
and physiology to such changes are more likely to survive.
(A) P→Q→R→S
(B) Q→S→R→P
(C) R→S→Q→P
(D) S→P→R→Q
Organizing Institute: IIT Roorkee Page 5 of 52
Q.6 Identify the option that has the most appropriate sequence such that a coherent
paragraph is formed:
P. Over time, such adaptations lead to significant evolutionary changes with
the potential to shape the development of new species.
Q. In natural world, organisms constantly adapt to their environments in
response to challenges and opportunities.
R. This process of adaptation is driven by the principle of natural selection,
where favorable traits increase an organism’s chances of survival and
reproduction.
S. As environments change, organisms that can adapt their behavior, structure
and physiology to such changes are more likely to survive.
(A) P→Q→R→S
(B) Q→S→R→P
(C) R→S→Q→P
(D) S→P→R→Q
Organizing Institute: IIT Roorkee Page 5 of 52
Page 7
Q.7 A stick of length one meter is broken at two locations at distances of 𝑏1 and 𝑏2 from
the origin (0), as shown in the figure. Note that 0 < 𝑏1 < 𝑏2 < 1. Which one of the
following is NOT a necessary condition for forming a triangle using the three
pieces?
Note: All lengths are in meter. The figure shown is representative.
(A) 𝑏1 < 0.5
(B) 𝑏2 > 0.5
(C) 𝑏2 < 𝑏1 + 0.5
(D) 𝑏1 + 𝑏2 < 1
Organizing Institute: IIT Roorkee Page 6 of 52
the origin (0), as shown in the figure. Note that 0 < 𝑏1 < 𝑏2 < 1. Which one of the
following is NOT a necessary condition for forming a triangle using the three
pieces?
Note: All lengths are in meter. The figure shown is representative.
(A) 𝑏1 < 0.5
(B) 𝑏2 > 0.5
(C) 𝑏2 < 𝑏1 + 0.5
(D) 𝑏1 + 𝑏2 < 1
Organizing Institute: IIT Roorkee Page 6 of 52
Page 8
Q.8 Eight students (P, Q, R, S, T, U, V, and W) are playing musical chairs. The figure
indicates their order of position at the start of the game. They play the game by
moving forward in a circle in the clockwise direction.
After the 1st round, 4th student behind P leaves the game. After 2nd round, 5th student
behind Q leaves the game. After 3rd round, 3rd student behind V leaves the game.
After 4th round, 4th student behind U leaves the game. Who all are left in the game
after the 4th round?
Note: The figure shown is representative.
P
W R
U T
Q
V
S
(A) P; T; Q; S
(B) V; P; T; Q
(C) W; R; Q; V
(D) Q; T; V; W
Organizing Institute: IIT Roorkee Page 7 of 52
indicates their order of position at the start of the game. They play the game by
moving forward in a circle in the clockwise direction.
After the 1st round, 4th student behind P leaves the game. After 2nd round, 5th student
behind Q leaves the game. After 3rd round, 3rd student behind V leaves the game.
After 4th round, 4th student behind U leaves the game. Who all are left in the game
after the 4th round?
Note: The figure shown is representative.
P
W R
U T
Q
V
S
(A) P; T; Q; S
(B) V; P; T; Q
(C) W; R; Q; V
(D) Q; T; V; W
Organizing Institute: IIT Roorkee Page 7 of 52
Page 9
Q.9 The table lists the top 5 nations according to the number of gold medals won in a
tournament; also included are the number of silver and the bronze medals won by
them. Based only on the data provided in the table, which one of the following
statements is INCORRECT?
Nation Gold Silver Bronze
USA 40 44 41
Canada 39 27 24
Japan 20 12 13
Australia 17 19 16
France 16 26 22
(A) France will occupy the third place if the list were made on the basis of the total
number of medals won.
(B) The order of the top two nations will not change even if the list is made on the basis
of the total number of medals won.
(C) USA and Canada together have less than 50% of the medals awarded to the nations
in the above table.
(D) Canada has won twice as many total medals as Japan.
Organizing Institute: IIT Roorkee Page 8 of 52
tournament; also included are the number of silver and the bronze medals won by
them. Based only on the data provided in the table, which one of the following
statements is INCORRECT?
Nation Gold Silver Bronze
USA 40 44 41
Canada 39 27 24
Japan 20 12 13
Australia 17 19 16
France 16 26 22
(A) France will occupy the third place if the list were made on the basis of the total
number of medals won.
(B) The order of the top two nations will not change even if the list is made on the basis
of the total number of medals won.
(C) USA and Canada together have less than 50% of the medals awarded to the nations
in the above table.
(D) Canada has won twice as many total medals as Japan.
Organizing Institute: IIT Roorkee Page 8 of 52
Page 10
Q.10 An organization allows its employees to work independently on consultancy
projects but charges an overhead on the consulting fee. The overhead is 20% of the
consulting fee, if the fee is up to ₹ 5,00,000. For higher fees, the overhead is
₹ 1,00,000 plus 10% of the amount by which the fee exceeds ₹ 5,00,000. The
government charges a Goods and Services Tax of 18% on the total amount (the
consulting fee plus the overhead). An employee of the organization charges this
entire amount, i.e., the consulting fee, overhead, and tax, to the client. If the client
cannot pay more than ₹ 10,00,000, what is the maximum consulting fee that the
employee can charge?
(A) ₹ 7,01,438
(B) ₹ 7,24,961
(C) ₹ 7,51,232
(D) ₹ 7,75,784
Organizing Institute: IIT Roorkee Page 9 of 52
projects but charges an overhead on the consulting fee. The overhead is 20% of the
consulting fee, if the fee is up to ₹ 5,00,000. For higher fees, the overhead is
₹ 1,00,000 plus 10% of the amount by which the fee exceeds ₹ 5,00,000. The
government charges a Goods and Services Tax of 18% on the total amount (the
consulting fee plus the overhead). An employee of the organization charges this
entire amount, i.e., the consulting fee, overhead, and tax, to the client. If the client
cannot pay more than ₹ 10,00,000, what is the maximum consulting fee that the
employee can charge?
(A) ₹ 7,01,438
(B) ₹ 7,24,961
(C) ₹ 7,51,232
(D) ₹ 7,75,784
Organizing Institute: IIT Roorkee Page 9 of 52
Page 11
Q.11 – Q.35 Carry ONE mark Each
Q.11 Consider the matrix 𝐴 below:
2 3 4 5
0 6 7 8
𝐴= [ ]
0 0 𝛼 𝛽
0 0 0 𝛾
For which of the following combinations of 𝛼, 𝛽, and 𝛾, is the rank of 𝐴 at least
three?
(i) 𝛼 = 0 and 𝛽 = 𝛾 ≠ 0.
(ii) 𝛼 = 𝛽 = 𝛾 = 0.
(iii) 𝛽 = 𝛾 = 0 and 𝛼 ≠ 0.
(iv) 𝛼 = 𝛽 = 𝛾 ≠ 0.
(A) Only (i), (iii), and (iv)
(B) Only (iv)
(C) Only (ii)
(D) Only (i) and (iii)
Organizing Institute: IIT Roorkee Page 10 of 52
Q.11 Consider the matrix 𝐴 below:
2 3 4 5
0 6 7 8
𝐴= [ ]
0 0 𝛼 𝛽
0 0 0 𝛾
For which of the following combinations of 𝛼, 𝛽, and 𝛾, is the rank of 𝐴 at least
three?
(i) 𝛼 = 0 and 𝛽 = 𝛾 ≠ 0.
(ii) 𝛼 = 𝛽 = 𝛾 = 0.
(iii) 𝛽 = 𝛾 = 0 and 𝛼 ≠ 0.
(iv) 𝛼 = 𝛽 = 𝛾 ≠ 0.
(A) Only (i), (iii), and (iv)
(B) Only (iv)
(C) Only (ii)
(D) Only (i) and (iii)
Organizing Institute: IIT Roorkee Page 10 of 52
Page 12
Q.12 Consider the following series:
1
(i) ∑∞
𝑛=1 √𝑛
1
(ii) ∑∞
𝑛=1 𝑛(𝑛+1)
1
(iii) ∑∞
𝑛=1 𝑛!
Choose the correct option.
(A) Only (ii) converges
(B) Only (ii) and (iii) converge
(C) Only (iii) converges
(D) All three converge
Organizing Institute: IIT Roorkee Page 11 of 52
1
(i) ∑∞
𝑛=1 √𝑛
1
(ii) ∑∞
𝑛=1 𝑛(𝑛+1)
1
(iii) ∑∞
𝑛=1 𝑛!
Choose the correct option.
(A) Only (ii) converges
(B) Only (ii) and (iii) converge
(C) Only (iii) converges
(D) All three converge
Organizing Institute: IIT Roorkee Page 11 of 52
Page 13
Q.13 A pot contains two red balls and two blue balls. Two balls are drawn from this pot
randomly without replacement.
What is the probability that the two balls drawn have different colours?
(A) 2⁄
3
(B) 1⁄
3
(C) 1⁄
2
(D) 1
Organizing Institute: IIT Roorkee Page 12 of 52
randomly without replacement.
What is the probability that the two balls drawn have different colours?
(A) 2⁄
3
(B) 1⁄
3
(C) 1⁄
2
(D) 1
Organizing Institute: IIT Roorkee Page 12 of 52
Page 14
Q.14 Consider a frequency-modulated (FM) signal
𝑓(𝑡) = 𝐴𝑐 cos(2𝜋𝑓𝑐 𝑡 + 3 sin(2𝜋𝑓1 𝑡) + 4 sin(6𝜋𝑓1 𝑡)),
where 𝐴𝑐 and 𝑓𝑐 are, respectively, the amplitude and frequency (in Hz) of the
carrier waveform. The frequency 𝑓1 is in Hz, and assume that 𝑓𝑐 > 100𝑓1 .
The peak frequency deviation of the FM signal in Hz is _______.
(A) 15𝑓1
(B) 12𝑓1
(C) 4𝑓1
(D) 2𝑓1
Organizing Institute: IIT Roorkee Page 13 of 52
𝑓(𝑡) = 𝐴𝑐 cos(2𝜋𝑓𝑐 𝑡 + 3 sin(2𝜋𝑓1 𝑡) + 4 sin(6𝜋𝑓1 𝑡)),
where 𝐴𝑐 and 𝑓𝑐 are, respectively, the amplitude and frequency (in Hz) of the
carrier waveform. The frequency 𝑓1 is in Hz, and assume that 𝑓𝑐 > 100𝑓1 .
The peak frequency deviation of the FM signal in Hz is _______.
(A) 15𝑓1
(B) 12𝑓1
(C) 4𝑓1
(D) 2𝑓1
Organizing Institute: IIT Roorkee Page 13 of 52
Page 15
Q.15 Consider an additive white Gaussian noise (AWGN) channel with bandwidth 𝑊
𝑁0
and noise power spectral density . Let 𝑃𝑎𝑣 denote the average transmit power
2
constraint.
Which one of the following plots illustrates the dependence of the channel capacity
𝐶 on the bandwidth 𝑊 (keeping 𝑃𝑎𝑣 and 𝑁0 fixed)?
(A)
(B)
(C)
(D)
Organizing Institute: IIT Roorkee Page 14 of 52
𝑁0
and noise power spectral density . Let 𝑃𝑎𝑣 denote the average transmit power
2
constraint.
Which one of the following plots illustrates the dependence of the channel capacity
𝐶 on the bandwidth 𝑊 (keeping 𝑃𝑎𝑣 and 𝑁0 fixed)?
(A)
(B)
(C)
(D)
Organizing Institute: IIT Roorkee Page 14 of 52
Page 16
Q.16 The Nyquist plot of a system is given in the figure below. Let 𝜔𝑃 , 𝜔𝑄 , 𝜔𝑅 , and 𝜔𝑆
be the positive frequencies at the points 𝑷, 𝑸, 𝑹, and 𝑺, respectively.
Which one of the following statements is TRUE?
(A) 𝜔𝑆 is the gain crossover frequency and 𝜔𝑃 is the phase crossover frequency
(B) 𝜔𝑄 is the gain crossover frequency and 𝜔𝑅 is the phase crossover frequency
(C) 𝜔𝑄 is the gain crossover frequency and 𝜔𝑆 is the phase crossover frequency
(D) 𝜔𝑆 is the gain crossover frequency and 𝜔𝑄 is the phase crossover frequency
Organizing Institute: IIT Roorkee Page 15 of 52
be the positive frequencies at the points 𝑷, 𝑸, 𝑹, and 𝑺, respectively.
Which one of the following statements is TRUE?
(A) 𝜔𝑆 is the gain crossover frequency and 𝜔𝑃 is the phase crossover frequency
(B) 𝜔𝑄 is the gain crossover frequency and 𝜔𝑅 is the phase crossover frequency
(C) 𝜔𝑄 is the gain crossover frequency and 𝜔𝑆 is the phase crossover frequency
(D) 𝜔𝑆 is the gain crossover frequency and 𝜔𝑄 is the phase crossover frequency
Organizing Institute: IIT Roorkee Page 15 of 52
Page 17
Q.17 Consider the discrete-time system below with input 𝑥[𝑛] and output 𝑦[𝑛]. In the
figure, ℎ1 [𝑛] and ℎ2 [𝑛] denote the impulse responses of LTI Subsystems 1 and 2,
respectively. Also, 𝛿[𝑛] is the unit impulse, and 𝑏 > 0.
Assuming ℎ2 [𝑛] ≠ 𝛿[𝑛], the overall system (denoted by the dashed box) is _______.
(A) linear and time invariant
(B) linear and time variant
(C) nonlinear and time invariant
(D) nonlinear and time variant
Organizing Institute: IIT Roorkee Page 16 of 52
figure, ℎ1 [𝑛] and ℎ2 [𝑛] denote the impulse responses of LTI Subsystems 1 and 2,
respectively. Also, 𝛿[𝑛] is the unit impulse, and 𝑏 > 0.
Assuming ℎ2 [𝑛] ≠ 𝛿[𝑛], the overall system (denoted by the dashed box) is _______.
(A) linear and time invariant
(B) linear and time variant
(C) nonlinear and time invariant
(D) nonlinear and time variant
Organizing Institute: IIT Roorkee Page 16 of 52
Page 18
Q.18 Consider a continuous-time, real-valued signal 𝑓(𝑡) whose Fourier transform
∞
𝐹(𝜔) = ∫−∞ 𝑓(𝑡) exp(−𝑗 𝜔𝑡) 𝑑𝑡 exists.
Which one of the following statements is always TRUE?
(A) ∞
|𝐹(𝜔)| ≤ ∫−∞ |𝑓(𝑡)| 𝑑𝑡
(B) ∞
|𝐹(𝜔)| > ∫−∞ |𝑓(𝑡)| 𝑑𝑡
(C) ∞
|𝐹(𝜔)| ≤ ∫−∞ 𝑓(𝑡) 𝑑𝑡
(D) ∞
|𝐹(𝜔)| ≥ ∫−∞ 𝑓(𝑡) 𝑑𝑡
Organizing Institute: IIT Roorkee Page 17 of 52
∞
𝐹(𝜔) = ∫−∞ 𝑓(𝑡) exp(−𝑗 𝜔𝑡) 𝑑𝑡 exists.
Which one of the following statements is always TRUE?
(A) ∞
|𝐹(𝜔)| ≤ ∫−∞ |𝑓(𝑡)| 𝑑𝑡
(B) ∞
|𝐹(𝜔)| > ∫−∞ |𝑓(𝑡)| 𝑑𝑡
(C) ∞
|𝐹(𝜔)| ≤ ∫−∞ 𝑓(𝑡) 𝑑𝑡
(D) ∞
|𝐹(𝜔)| ≥ ∫−∞ 𝑓(𝑡) 𝑑𝑡
Organizing Institute: IIT Roorkee Page 17 of 52
Page 19
Q.19 Consider a part of an electrical network as shown below. Some node voltages, and
the current flowing through the 3Ω resistor are as indicated.
The voltage (in Volts) at node 𝑋 is ______.
(A) 20⁄
3
(B) 32⁄
3
(C) 22⁄
3
(D) 2⁄
3
Organizing Institute: IIT Roorkee Page 18 of 52
the current flowing through the 3Ω resistor are as indicated.
The voltage (in Volts) at node 𝑋 is ______.
(A) 20⁄
3
(B) 32⁄
3
(C) 22⁄
3
(D) 2⁄
3
Organizing Institute: IIT Roorkee Page 18 of 52
Page 20
Q.20 Let 𝑖𝐶 , 𝑖𝐿 , and 𝑖𝑅 be the currents flowing through the capacitor, inductor, and
resistor, respectively, in the circuit given below. The AC admittances are given in
Siemens (S).
Which one of the following is TRUE?
(A) 𝑖𝐶 = 0.25∠180° A, 𝑖𝐿 = 0.1∠0° A, 𝑖𝑅 = 0.2∠90° A
(B) 𝑖𝐶 = 4∠180° A, 𝑖𝐿 = 10∠0° A, 𝑖𝑅 = 5∠90° A
(C) 𝑖𝐶 = 0.25∠270° A, 𝑖𝐿 = 0.1∠90° A, 𝑖𝑅 = 0.2∠90° A
(D) 𝑖𝐶 = 4∠90° A, 𝑖𝐿 = 10∠270° A, 𝑖𝑅 = 5∠0° A
Organizing Institute: IIT Roorkee Page 19 of 52
resistor, respectively, in the circuit given below. The AC admittances are given in
Siemens (S).
Which one of the following is TRUE?
(A) 𝑖𝐶 = 0.25∠180° A, 𝑖𝐿 = 0.1∠0° A, 𝑖𝑅 = 0.2∠90° A
(B) 𝑖𝐶 = 4∠180° A, 𝑖𝐿 = 10∠0° A, 𝑖𝑅 = 5∠90° A
(C) 𝑖𝐶 = 0.25∠270° A, 𝑖𝐿 = 0.1∠90° A, 𝑖𝑅 = 0.2∠90° A
(D) 𝑖𝐶 = 4∠90° A, 𝑖𝐿 = 10∠270° A, 𝑖𝑅 = 5∠0° A
Organizing Institute: IIT Roorkee Page 19 of 52
Page 21
Q.21 A simplified small-signal equivalent circuit of a BJT-based amplifier is given
below.
The small-signal voltage gain vo/vs (in V/V) is ___________.
(A) −𝛽R L
R S + rπ
(B) +𝛽R L
RS
(C) −𝛽R L
RS
(D) +𝛽R L
R S + rπ
Organizing Institute: IIT Roorkee Page 20 of 52
below.
The small-signal voltage gain vo/vs (in V/V) is ___________.
(A) −𝛽R L
R S + rπ
(B) +𝛽R L
RS
(C) −𝛽R L
RS
(D) +𝛽R L
R S + rπ
Organizing Institute: IIT Roorkee Page 20 of 52
Page 22
Q.22 The ideal BJT in the circuit given below is biased in the active region with a 𝛽 of
100.
If 𝐼𝐵 is 10 µA, then 𝑉𝐶𝐸 (in Volts, rounded off to two decimal places) is _______ .
(A) 4.95
(B) 3.03
(C) 1.92
(D) 3.73
Organizing Institute: IIT Roorkee Page 21 of 52
100.
If 𝐼𝐵 is 10 µA, then 𝑉𝐶𝐸 (in Volts, rounded off to two decimal places) is _______ .
(A) 4.95
(B) 3.03
(C) 1.92
(D) 3.73
Organizing Institute: IIT Roorkee Page 21 of 52
Page 23
Q.23 A 3-input majority logic gate has inputs 𝑋, 𝑌, and 𝑍. The output 𝐹 of the gate is
logic ‘1’ if two or more of the inputs are logic ‘1’. The output 𝐹 is logic ‘0’ if two
or more of the inputs are logic ‘0’.
Which one of the following options is a Boolean expression of the output 𝐹?
(A) 𝑋𝑌 + 𝑌𝑍 + 𝑍𝑋
(B) 𝑋⊕𝑌⊕𝑍
(C) 𝑋+𝑌+𝑍
(D) 𝑋𝑌𝑍
Organizing Institute: IIT Roorkee Page 22 of 52
logic ‘1’ if two or more of the inputs are logic ‘1’. The output 𝐹 is logic ‘0’ if two
or more of the inputs are logic ‘0’.
Which one of the following options is a Boolean expression of the output 𝐹?
(A) 𝑋𝑌 + 𝑌𝑍 + 𝑍𝑋
(B) 𝑋⊕𝑌⊕𝑍
(C) 𝑋+𝑌+𝑍
(D) 𝑋𝑌𝑍
Organizing Institute: IIT Roorkee Page 22 of 52
Page 24
Q.24 A full adder and an XOR gate are used to design a digital circuit with inputs X, Y,
and Z, and output F, as shown below. The input Z is connected to the carry-in input
of the full adder.
If the input Z is set to logic ‘1’, then the circuit functions as ________ with X and
Y as inputs.
(A) an adder
(B) a subtractor
(C) a multiplier
(D) a binary to Gray code converter
Organizing Institute: IIT Roorkee Page 23 of 52
and Z, and output F, as shown below. The input Z is connected to the carry-in input
of the full adder.
If the input Z is set to logic ‘1’, then the circuit functions as ________ with X and
Y as inputs.
(A) an adder
(B) a subtractor
(C) a multiplier
(D) a binary to Gray code converter
Organizing Institute: IIT Roorkee Page 23 of 52
Page 25
Q.25 Consider the function 𝑓: ℝ → ℝ, defined as
𝑓(𝑥) = 2𝑥 3 − 3𝑥 2 − 12𝑥 + 1.
Which of the following statements is/are correct?
(Here, ℝ is the set of real numbers.)
(A) 𝑓 has no global maximizer
(B) 𝑓 has no global minimizer
(C) 𝑥 = −1 is a local minimizer of 𝑓
(D) 𝑥 = 2 is a local maximizer of 𝑓
Organizing Institute: IIT Roorkee Page 24 of 52
𝑓(𝑥) = 2𝑥 3 − 3𝑥 2 − 12𝑥 + 1.
Which of the following statements is/are correct?
(Here, ℝ is the set of real numbers.)
(A) 𝑓 has no global maximizer
(B) 𝑓 has no global minimizer
(C) 𝑥 = −1 is a local minimizer of 𝑓
(D) 𝑥 = 2 is a local maximizer of 𝑓
Organizing Institute: IIT Roorkee Page 24 of 52
Page 26
Q.26 Consider the unity-negative-feedback system shown in Figure (i) below, where
gain 𝐾 ≥ 0. The root locus of this system is shown in Figure (ii) below.
For what value(s) of 𝐾 will the system in Figure (i) have a pole at −1 + 𝑗1?
(A) 𝐾= 5
(B) 𝐾 = 1⁄5
(C) For no positive value of 𝐾
(D) For all positive values of 𝐾
Organizing Institute: IIT Roorkee Page 25 of 52
gain 𝐾 ≥ 0. The root locus of this system is shown in Figure (ii) below.
For what value(s) of 𝐾 will the system in Figure (i) have a pole at −1 + 𝑗1?
(A) 𝐾= 5
(B) 𝐾 = 1⁄5
(C) For no positive value of 𝐾
(D) For all positive values of 𝐾
Organizing Institute: IIT Roorkee Page 25 of 52
Page 27
Q.27 Let 𝑥[𝑛] be a discrete-time signal whose 𝑧-transform is 𝑋(𝑧).
Which of the following statements is/are TRUE?
(A) The discrete-time Fourier transform (DTFT) of 𝑥[𝑛] always exists
(B) The region of convergence (RoC) of 𝑋(𝑧) contains neither poles nor zeros
(C) The discrete-time Fourier transform (DTFT) exists if the region of convergence
(RoC) contains the unit circle
(D) If 𝑥[𝑛] = 𝛼𝛿[𝑛], where 𝛿[𝑛] is the unit impulse and 𝛼 is a scalar, then the
region of convergence (RoC) is the entire 𝑧-plane
Organizing Institute: IIT Roorkee Page 26 of 52
Which of the following statements is/are TRUE?
(A) The discrete-time Fourier transform (DTFT) of 𝑥[𝑛] always exists
(B) The region of convergence (RoC) of 𝑋(𝑧) contains neither poles nor zeros
(C) The discrete-time Fourier transform (DTFT) exists if the region of convergence
(RoC) contains the unit circle
(D) If 𝑥[𝑛] = 𝛼𝛿[𝑛], where 𝛿[𝑛] is the unit impulse and 𝛼 is a scalar, then the
region of convergence (RoC) is the entire 𝑧-plane
Organizing Institute: IIT Roorkee Page 26 of 52
Page 28
Q.28 Consider a message signal 𝑚(𝑡) which is bandlimited to [−𝑊, 𝑊], where 𝑊 is in
Hz. Consider the following two modulation schemes for the message signal:
• Double sideband-suppressed carrier (DSB-SC):
𝑓𝐷𝑆𝐵 (𝑡) = 𝐴𝑐 𝑚(𝑡) cos(2𝜋𝑓𝑐 𝑡)
• Amplitude modulation (AM):
𝑓𝐴𝑀 (𝑡) = 𝐴𝑐 (1 + 𝜇 𝑚(𝑡)) cos(2𝜋𝑓𝑐 𝑡)
Here, 𝐴𝑐 and 𝑓𝑐 are the amplitude and frequency (in Hz) of the carrier, respectively.
In the case of AM, 𝜇 denotes the modulation index.
Consider the following statements:
(i) An envelope detector can be used for demodulation in the DSB-SC
scheme if 𝑚(𝑡) > 0 for all 𝑡.
(ii) An envelope detector can be used for demodulation in the AM scheme
only if 𝑚(𝑡) > 0 for all 𝑡.
Which of the following options is/are correct?
(A) (i) is TRUE
(B) (i) is FALSE
(C) (ii) is TRUE
(D) (ii) is FALSE
Organizing Institute: IIT Roorkee Page 27 of 52
Hz. Consider the following two modulation schemes for the message signal:
• Double sideband-suppressed carrier (DSB-SC):
𝑓𝐷𝑆𝐵 (𝑡) = 𝐴𝑐 𝑚(𝑡) cos(2𝜋𝑓𝑐 𝑡)
• Amplitude modulation (AM):
𝑓𝐴𝑀 (𝑡) = 𝐴𝑐 (1 + 𝜇 𝑚(𝑡)) cos(2𝜋𝑓𝑐 𝑡)
Here, 𝐴𝑐 and 𝑓𝑐 are the amplitude and frequency (in Hz) of the carrier, respectively.
In the case of AM, 𝜇 denotes the modulation index.
Consider the following statements:
(i) An envelope detector can be used for demodulation in the DSB-SC
scheme if 𝑚(𝑡) > 0 for all 𝑡.
(ii) An envelope detector can be used for demodulation in the AM scheme
only if 𝑚(𝑡) > 0 for all 𝑡.
Which of the following options is/are correct?
(A) (i) is TRUE
(B) (i) is FALSE
(C) (ii) is TRUE
(D) (ii) is FALSE
Organizing Institute: IIT Roorkee Page 27 of 52
Page 29
Q.29 Which of the following statements is/are TRUE with respect to an ideal opamp?
(A) It has an infinite input resistance
(B) It has an infinite output resistance
(C) It has an infinite open-loop differential gain
(D) It has an infinite open-loop common-mode gain
Q.30 Which of the following statements is/are TRUE with respect to ideal MOSFET-
based DC-coupled single-stage amplifiers having finite load resistors?
(A) The common-gate amplifier has an infinite input resistance
(B) The common-source amplifier has an infinite input resistance
(C) The input and output voltages of the common-source amplifier are in phase
(D) The input and output voltages of the common-drain amplifier are in phase
Organizing Institute: IIT Roorkee Page 28 of 52
(A) It has an infinite input resistance
(B) It has an infinite output resistance
(C) It has an infinite open-loop differential gain
(D) It has an infinite open-loop common-mode gain
Q.30 Which of the following statements is/are TRUE with respect to ideal MOSFET-
based DC-coupled single-stage amplifiers having finite load resistors?
(A) The common-gate amplifier has an infinite input resistance
(B) The common-source amplifier has an infinite input resistance
(C) The input and output voltages of the common-source amplifier are in phase
(D) The input and output voltages of the common-drain amplifier are in phase
Organizing Institute: IIT Roorkee Page 28 of 52
Page 30
Q.31 Which of the following can be used as an n-type dopant for silicon?
Select the correct option(s).
(A) Arsenic
(B) Boron
(C) Gallium
(D) Phosphorous
Q.32 The function 𝑦(𝑡) satisfies
𝑡 2 𝑦 ′′ (𝑡) − 2𝑡𝑦 ′ (𝑡) + 2𝑦(𝑡) = 0,
where 𝑦 ′ (𝑡) and 𝑦 ′′ (𝑡) denote the first and second derivatives of 𝑦(𝑡),
respectively.
Given 𝑦 ′ (0) = 1 and 𝑦 ′ (1) = −1, the maximum value of 𝑦(𝑡) over [0,1] is
_____ (rounded off to two decimal places).
Q.33 The generator matrix of a (6,3) binary linear block code is given by
1 0 0 1 0 1
𝐺 = [0 1 0 0 1 1].
0 0 1 1 1 0
The minimum Hamming distance 𝑑𝑚𝑖𝑛 between codewords equals _____ (answer
in integer).
Organizing Institute: IIT Roorkee Page 29 of 52
Select the correct option(s).
(A) Arsenic
(B) Boron
(C) Gallium
(D) Phosphorous
Q.32 The function 𝑦(𝑡) satisfies
𝑡 2 𝑦 ′′ (𝑡) − 2𝑡𝑦 ′ (𝑡) + 2𝑦(𝑡) = 0,
where 𝑦 ′ (𝑡) and 𝑦 ′′ (𝑡) denote the first and second derivatives of 𝑦(𝑡),
respectively.
Given 𝑦 ′ (0) = 1 and 𝑦 ′ (1) = −1, the maximum value of 𝑦(𝑡) over [0,1] is
_____ (rounded off to two decimal places).
Q.33 The generator matrix of a (6,3) binary linear block code is given by
1 0 0 1 0 1
𝐺 = [0 1 0 0 1 1].
0 0 1 1 1 0
The minimum Hamming distance 𝑑𝑚𝑖𝑛 between codewords equals _____ (answer
in integer).
Organizing Institute: IIT Roorkee Page 29 of 52
Page 31
Q.34 All the components in the bandpass filter given below are ideal. The lower −3 dB
frequency of the filter is 1 MHz.
The upper −3 dB frequency (in MHz, rounded off to the nearest integer) is
__________.
Q.35 A 4-bit weighted-resistor DAC with inputs b3, b2, b1, and b0 (MSB to LSB) is
designed using an ideal opamp, as shown below. The switches are closed when the
corresponding input bits are logic ‘1’ and open otherwise.
When the input b3b2b1b0 changes from 1110 to 1101, the magnitude of the change
in the output voltage VO (in mV, rounded off to the nearest integer) is ________.
Organizing Institute: IIT Roorkee Page 30 of 52
frequency of the filter is 1 MHz.
The upper −3 dB frequency (in MHz, rounded off to the nearest integer) is
__________.
Q.35 A 4-bit weighted-resistor DAC with inputs b3, b2, b1, and b0 (MSB to LSB) is
designed using an ideal opamp, as shown below. The switches are closed when the
corresponding input bits are logic ‘1’ and open otherwise.
When the input b3b2b1b0 changes from 1110 to 1101, the magnitude of the change
in the output voltage VO (in mV, rounded off to the nearest integer) is ________.
Organizing Institute: IIT Roorkee Page 30 of 52
Page 32
Q.36 – Q.65 Carry TWO marks Each
Q.36 1
Let 𝐺(𝑠) = be the transfer function of a second-order system. A controller
10𝑠 2
𝑀(𝑠) is connected to the system 𝐺(𝑠) in the configuration shown below.
Consider the following statements.
𝐾𝐼
(i) There exists no controller of the form 𝑀(𝑠) = , where 𝐾𝐼 is a positive real
𝑠
number, such that the closed loop system is stable.
(ii) There exists at least one controller of the form 𝑀(𝑠) = 𝐾𝑃 + 𝑠𝐾𝐷 , where 𝐾𝑃
and 𝐾𝐷 are positive real numbers, such that the closed loop system is stable.
Which one of the following options is correct?
(A) (i) is TRUE and (ii) is FALSE
(B) (i) is FALSE and (ii) is TRUE
(C) Both (i) and (ii) are FALSE
(D) Both (i) and (ii) are TRUE
Organizing Institute: IIT Roorkee Page 31 of 52
Q.36 1
Let 𝐺(𝑠) = be the transfer function of a second-order system. A controller
10𝑠 2
𝑀(𝑠) is connected to the system 𝐺(𝑠) in the configuration shown below.
Consider the following statements.
𝐾𝐼
(i) There exists no controller of the form 𝑀(𝑠) = , where 𝐾𝐼 is a positive real
𝑠
number, such that the closed loop system is stable.
(ii) There exists at least one controller of the form 𝑀(𝑠) = 𝐾𝑃 + 𝑠𝐾𝐷 , where 𝐾𝑃
and 𝐾𝐷 are positive real numbers, such that the closed loop system is stable.
Which one of the following options is correct?
(A) (i) is TRUE and (ii) is FALSE
(B) (i) is FALSE and (ii) is TRUE
(C) Both (i) and (ii) are FALSE
(D) Both (i) and (ii) are TRUE
Organizing Institute: IIT Roorkee Page 31 of 52
Page 33
Q.37 Consider the polynomial 𝑝(𝑠) = 𝑠 5 + 7𝑠 4 + 3𝑠 3 − 33𝑠 2 + 2𝑠 − 40. Let
(𝐿, 𝐼, 𝑅) be defined as follows.
𝐿 is the number of roots of 𝑝(𝑠) with negative real parts.
𝐼 is the number of roots of 𝑝(𝑠) that are purely imaginary.
𝑅 is the number of roots of 𝑝(𝑠) with positive real parts.
Which one of the following options is correct?
(A) 𝐿 = 2, 𝐼 = 2, and 𝑅 = 1
(B) 𝐿 = 3, 𝐼 = 2, and 𝑅 = 0
(C) 𝐿 = 1, 𝐼 = 2, and 𝑅 = 2
(D) 𝐿 = 0, 𝐼 = 4, and 𝑅 = 1
Organizing Institute: IIT Roorkee Page 32 of 52
(𝐿, 𝐼, 𝑅) be defined as follows.
𝐿 is the number of roots of 𝑝(𝑠) with negative real parts.
𝐼 is the number of roots of 𝑝(𝑠) that are purely imaginary.
𝑅 is the number of roots of 𝑝(𝑠) with positive real parts.
Which one of the following options is correct?
(A) 𝐿 = 2, 𝐼 = 2, and 𝑅 = 1
(B) 𝐿 = 3, 𝐼 = 2, and 𝑅 = 0
(C) 𝐿 = 1, 𝐼 = 2, and 𝑅 = 2
(D) 𝐿 = 0, 𝐼 = 4, and 𝑅 = 1
Organizing Institute: IIT Roorkee Page 32 of 52
Page 34
Q.38 Consider a continuous-time finite-energy signal 𝑓(𝑡) whose Fourier transform
vanishes outside the frequency interval [−𝜔 𝑐 , 𝜔𝑐 ], where 𝜔𝑐 is in rad/sec.
The signal 𝑓(𝑡) is uniformly sampled to obtain 𝑦(𝑡) = 𝑓(𝑡) 𝑝(𝑡). Here,
∞
𝑝(𝑡) = ∑ 𝛿(𝑡 − 𝜏 − 𝑛𝑇𝑠 ),
𝑛=−∞
with 𝛿(𝑡) being the Dirac impulse, 𝑇𝑠 > 0, and 𝜏 > 0. The sampled signal 𝑦(𝑡)
sin (𝜔𝑐 𝑡)
is passed through an ideal lowpass filter ℎ(𝑡) = 𝜔𝑐 𝑇𝑠 with cutoff
𝜋𝜔𝑐 𝑡
frequency 𝜔𝑐 and passband gain 𝑇𝑠 .
The output of the filter is given by ________________.
(A) 𝑓(𝑡) if 𝑇𝑠 < 𝜋/𝜔𝑐
(B) 𝑓(𝑡 − 𝜏) if 𝑇𝑠 < 𝜋/𝜔𝑐
(C) 𝑓(𝑡 − 𝜏) if 𝑇𝑠 < 2𝜋/𝜔𝑐
(D) 𝑇𝑠 𝑓(𝑡) if 𝑇𝑠 < 2𝜋/𝜔𝑐
Organizing Institute: IIT Roorkee Page 33 of 52
vanishes outside the frequency interval [−𝜔 𝑐 , 𝜔𝑐 ], where 𝜔𝑐 is in rad/sec.
The signal 𝑓(𝑡) is uniformly sampled to obtain 𝑦(𝑡) = 𝑓(𝑡) 𝑝(𝑡). Here,
∞
𝑝(𝑡) = ∑ 𝛿(𝑡 − 𝜏 − 𝑛𝑇𝑠 ),
𝑛=−∞
with 𝛿(𝑡) being the Dirac impulse, 𝑇𝑠 > 0, and 𝜏 > 0. The sampled signal 𝑦(𝑡)
sin (𝜔𝑐 𝑡)
is passed through an ideal lowpass filter ℎ(𝑡) = 𝜔𝑐 𝑇𝑠 with cutoff
𝜋𝜔𝑐 𝑡
frequency 𝜔𝑐 and passband gain 𝑇𝑠 .
The output of the filter is given by ________________.
(A) 𝑓(𝑡) if 𝑇𝑠 < 𝜋/𝜔𝑐
(B) 𝑓(𝑡 − 𝜏) if 𝑇𝑠 < 𝜋/𝜔𝑐
(C) 𝑓(𝑡 − 𝜏) if 𝑇𝑠 < 2𝜋/𝜔𝑐
(D) 𝑇𝑠 𝑓(𝑡) if 𝑇𝑠 < 2𝜋/𝜔𝑐
Organizing Institute: IIT Roorkee Page 33 of 52
Page 35
Q.39 In the circuit below, 𝑀1 is an ideal AC voltmeter and 𝑀2 is an ideal AC ammeter.
The source voltage (in Volts) is 𝑣𝑠 (𝑡) = 100 cos(200𝑡).
What should be the value of the variable capacitor 𝐶 such that the RMS readings
on 𝑀1 and 𝑀2 are 25 V and 5 A, respectively?
(A) 25 μF
(B) 4 μF
(C) 0.25 μF
(D) Insufficient information to find 𝐶
Organizing Institute: IIT Roorkee Page 34 of 52
The source voltage (in Volts) is 𝑣𝑠 (𝑡) = 100 cos(200𝑡).
What should be the value of the variable capacitor 𝐶 such that the RMS readings
on 𝑀1 and 𝑀2 are 25 V and 5 A, respectively?
(A) 25 μF
(B) 4 μF
(C) 0.25 μF
(D) Insufficient information to find 𝐶
Organizing Institute: IIT Roorkee Page 34 of 52
Page 36
Q.40 The 𝑍-parameter matrix of a two port network relates the port voltages and port
currents as follows:
𝑉 𝐼
[ 1] = 𝑍 [ 1]
𝑉2 𝐼2
The 𝑍-parameter matrix (with each entry in Ohms) of the network shown below is
__________.
(A) 10⁄ 2⁄
3 3
[ ]
2⁄ 10⁄
3 3
(B) 2⁄ 10⁄
3 3
[ ]
10⁄ 2⁄
3 3
(C) 10 2
[ ]
2 10
(D) 10⁄ 1⁄
3 3
[ ]
1⁄ 10⁄
3 3
Organizing Institute: IIT Roorkee Page 35 of 52
currents as follows:
𝑉 𝐼
[ 1] = 𝑍 [ 1]
𝑉2 𝐼2
The 𝑍-parameter matrix (with each entry in Ohms) of the network shown below is
__________.
(A) 10⁄ 2⁄
3 3
[ ]
2⁄ 10⁄
3 3
(B) 2⁄ 10⁄
3 3
[ ]
10⁄ 2⁄
3 3
(C) 10 2
[ ]
2 10
(D) 10⁄ 1⁄
3 3
[ ]
1⁄ 10⁄
3 3
Organizing Institute: IIT Roorkee Page 35 of 52
Page 37
Q.41 A source transmits symbol 𝑆 that takes values uniformly at random from the set
{−2,0,2}. The receiver obtains 𝑌 = 𝑆 + 𝑁, where 𝑁 is a zero-mean Gaussian
random variable independent of 𝑆. The receiver uses the maximum likelihood
decoder to estimate the transmitted symbol 𝑆.
Suppose the probability of symbol estimation error 𝑃𝑒 is expressed as follows:
𝑃𝑒 = 𝛼 𝑃(𝑁 > 1),
where 𝑃(𝑁 > 1) denotes the probability that 𝑁 exceeds 1.
What is the value of 𝛼 ?
(A) 1⁄
3
(B) 1
(C) 2⁄
3
(D) 4⁄
3
Organizing Institute: IIT Roorkee Page 36 of 52
{−2,0,2}. The receiver obtains 𝑌 = 𝑆 + 𝑁, where 𝑁 is a zero-mean Gaussian
random variable independent of 𝑆. The receiver uses the maximum likelihood
decoder to estimate the transmitted symbol 𝑆.
Suppose the probability of symbol estimation error 𝑃𝑒 is expressed as follows:
𝑃𝑒 = 𝛼 𝑃(𝑁 > 1),
where 𝑃(𝑁 > 1) denotes the probability that 𝑁 exceeds 1.
What is the value of 𝛼 ?
(A) 1⁄
3
(B) 1
(C) 2⁄
3
(D) 4⁄
3
Organizing Institute: IIT Roorkee Page 36 of 52
Page 38
Q.42 Consider a real-valued random process
𝑁
𝑓(𝑡) = ∑ 𝑎𝑛 𝑝(𝑡 − 𝑛𝑇),
𝑛=1
where 𝑇 > 0 and 𝑁 is a positive integer. Here, 𝑝(𝑡) = 1 for 𝑡 ∈ [0, 0.5𝑇] and
0 otherwise. The coefficients 𝑎𝑛 are pairwise independent, zero-mean unit-
variance random variables.
Read the following statements about the random process and choose the correct
option.
(i) The mean of the process 𝑓(𝑡) is independent of time 𝑡.
(ii) The autocorrelation function 𝐸[ 𝑓(𝑡)𝑓(𝑡 + 𝜏)] is independent of
time 𝑡 for all 𝜏.
(Here, 𝐸[⋅] is the expectation operation.)
(A) (i) is TRUE and (ii) is FALSE
(B) Both (i) and (ii) are TRUE
(C) Both (i) and (ii) are FALSE
(D) (i) is FALSE and (ii) is TRUE
Organizing Institute: IIT Roorkee Page 37 of 52
𝑁
𝑓(𝑡) = ∑ 𝑎𝑛 𝑝(𝑡 − 𝑛𝑇),
𝑛=1
where 𝑇 > 0 and 𝑁 is a positive integer. Here, 𝑝(𝑡) = 1 for 𝑡 ∈ [0, 0.5𝑇] and
0 otherwise. The coefficients 𝑎𝑛 are pairwise independent, zero-mean unit-
variance random variables.
Read the following statements about the random process and choose the correct
option.
(i) The mean of the process 𝑓(𝑡) is independent of time 𝑡.
(ii) The autocorrelation function 𝐸[ 𝑓(𝑡)𝑓(𝑡 + 𝜏)] is independent of
time 𝑡 for all 𝜏.
(Here, 𝐸[⋅] is the expectation operation.)
(A) (i) is TRUE and (ii) is FALSE
(B) Both (i) and (ii) are TRUE
(C) Both (i) and (ii) are FALSE
(D) (i) is FALSE and (ii) is TRUE
Organizing Institute: IIT Roorkee Page 37 of 52
Page 39
Q.43 The identical MOSFETs M1 and M2 in the circuit given below are ideal and biased
in the saturation region. M1 and M2 have a transconductance 𝑔𝑚 of 5 mS.
The input signals (in Volts) are:
𝑉1 = 2.5 + 0.01 sin 𝜔𝑡
𝑉2 = 2.5 − 0.01 sin 𝜔𝑡
The output signal 𝑉3 (in Volts) is __________.
(A) 3 + 0.05 sin 𝜔𝑡
(B) 3 − 0.1 sin 𝜔𝑡
(C) 4 + 0.1 sin 𝜔𝑡
(D) 4 − 0.05 sin 𝜔𝑡
Organizing Institute: IIT Roorkee Page 38 of 52
in the saturation region. M1 and M2 have a transconductance 𝑔𝑚 of 5 mS.
The input signals (in Volts) are:
𝑉1 = 2.5 + 0.01 sin 𝜔𝑡
𝑉2 = 2.5 − 0.01 sin 𝜔𝑡
The output signal 𝑉3 (in Volts) is __________.
(A) 3 + 0.05 sin 𝜔𝑡
(B) 3 − 0.1 sin 𝜔𝑡
(C) 4 + 0.1 sin 𝜔𝑡
(D) 4 − 0.05 sin 𝜔𝑡
Organizing Institute: IIT Roorkee Page 38 of 52
Page 40
Q.44 A 10-bit analog-to-digital converter (ADC) has a sampling frequency of 1 MHz and
a full scale voltage of 3.3 V.
For an input sinusoidal signal with frequency 500 kHz, the maximum SNR (in dB,
rounded off to two decimal places) and the data rate (in Mbps) at the output of the
ADC are ________, respectively.
(A) 61.96 and 10
(B) 61.96 and 5
(C) 33.36 and 10
(D) 33.36 and 5
Organizing Institute: IIT Roorkee Page 39 of 52
a full scale voltage of 3.3 V.
For an input sinusoidal signal with frequency 500 kHz, the maximum SNR (in dB,
rounded off to two decimal places) and the data rate (in Mbps) at the output of the
ADC are ________, respectively.
(A) 61.96 and 10
(B) 61.96 and 5
(C) 33.36 and 10
(D) 33.36 and 5
Organizing Institute: IIT Roorkee Page 39 of 52
Page 41
Q.45 A positive-edge-triggered sequential circuit is shown below. There are no timing
violations in the circuit. Input P0 is set to logic ‘0’ and P1 is set to logic ‘1’ at all
times. The timing diagram of the inputs SEL and S are also shown below.
The sequence of output Y from time T0 to T3 is ________.
(A) 1011
(B) 0100
(C) 0010
(D) 1101
Organizing Institute: IIT Roorkee Page 40 of 52
violations in the circuit. Input P0 is set to logic ‘0’ and P1 is set to logic ‘1’ at all
times. The timing diagram of the inputs SEL and S are also shown below.
The sequence of output Y from time T0 to T3 is ________.
(A) 1011
(B) 0100
(C) 0010
(D) 1101
Organizing Institute: IIT Roorkee Page 40 of 52
Page 42
Q.46 The intrinsic carrier concentration of a semiconductor is 2.5 × 1016 /m3 at 300 K.
If the electron and hole mobilities are 0.15 m2/Vs and 0.05 m2/Vs, respectively, then
the intrinsic resistivity of the semiconductor (in kΩ.m) at 300 K is ________.
(Charge of an electron 𝑒 = 1.6 × 10−19 C.)
(A) 1.65
(B) 1.25
(C) 0.85
(D) 1.95
Organizing Institute: IIT Roorkee Page 41 of 52
If the electron and hole mobilities are 0.15 m2/Vs and 0.05 m2/Vs, respectively, then
the intrinsic resistivity of the semiconductor (in kΩ.m) at 300 K is ________.
(Charge of an electron 𝑒 = 1.6 × 10−19 C.)
(A) 1.65
(B) 1.25
(C) 0.85
(D) 1.95
Organizing Institute: IIT Roorkee Page 41 of 52
Page 43
Q.47 In the circuit shown, the identical transistors Q1 and Q2 are biased in the active
region with 𝛽 = 120. The Zener diode is in the breakdown region with VZ = 5 V
and IZ = 25 mA.
If IL = 12 mA and VEB1 = VEB2 = 0.7 V, then the values of R1 and R2 (in kΩ,
rounded off to one decimal place) are ______, respectively.
(A) 0.6 and 0.4
(B) 1.4 and 2.5
(C) 14.0 and 25.0
(D) 6.0 and 4.0
Organizing Institute: IIT Roorkee Page 42 of 52
region with 𝛽 = 120. The Zener diode is in the breakdown region with VZ = 5 V
and IZ = 25 mA.
If IL = 12 mA and VEB1 = VEB2 = 0.7 V, then the values of R1 and R2 (in kΩ,
rounded off to one decimal place) are ______, respectively.
(A) 0.6 and 0.4
(B) 1.4 and 2.5
(C) 14.0 and 25.0
(D) 6.0 and 4.0
Organizing Institute: IIT Roorkee Page 42 of 52
Page 44
Q.48 The electron mobility µn in a non-degenerate germanium semiconductor at 300 K
is 0.38 m2/Vs.
The electron diffusivity 𝐷𝑛 at 300 K (in cm2/s, rounded off to the nearest integer)
is _______.
(Consider the Boltzmann constant 𝑘B = 1.38 × 10−23 J/K and the charge of an
electron 𝑒 = 1.6 × 10−19 C.)
(A) 26
(B) 98
(C) 38
(D) 10
Organizing Institute: IIT Roorkee Page 43 of 52
is 0.38 m2/Vs.
The electron diffusivity 𝐷𝑛 at 300 K (in cm2/s, rounded off to the nearest integer)
is _______.
(Consider the Boltzmann constant 𝑘B = 1.38 × 10−23 J/K and the charge of an
electron 𝑒 = 1.6 × 10−19 C.)
(A) 26
(B) 98
(C) 38
(D) 10
Organizing Institute: IIT Roorkee Page 43 of 52
Page 45
Q.49 A square metal sheet of 4 m × 4 m is placed on the x-y plane as shown in the figure
below.
If the surface charge density (in µC/m2) on the sheet is 𝜌𝑠 (𝑥, 𝑦) = 4|𝑦|, then the
total charge (in µC, rounded off to the nearest integer) on the sheet is ________.
(A) 16
(B) 85
(C) 64
(D) 256
Organizing Institute: IIT Roorkee Page 44 of 52
below.
If the surface charge density (in µC/m2) on the sheet is 𝜌𝑠 (𝑥, 𝑦) = 4|𝑦|, then the
total charge (in µC, rounded off to the nearest integer) on the sheet is ________.
(A) 16
(B) 85
(C) 64
(D) 256
Organizing Institute: IIT Roorkee Page 44 of 52
Page 46
Q.50 An electric field of 0.01 V/m is applied along the length of a copper wire of circular
cross-section with diameter 1 mm. Copper has a conductivity of 5.8 × 107 S/m.
The current (in Amperes, rounded off to two decimal places) flowing through the
wire is _________.
(A) 0.46
(B) 1.82
(C) 0.58
(D) 1.12
Q.51 Consider a non-negative function 𝑓(𝑥) which is continuous and bounded over the
interval [2, 8]. Let 𝑀 and 𝑚 denote, respectively, the maximum and the minimum
values of 𝑓(𝑥) over the interval.
Among the combinations of 𝛼 and 𝛽 given below, choose the one(s) for which the
inequality
8
𝛽 ≤ ∫2 𝑓(𝑥) 𝑑𝑥 ≤ 𝛼
is guaranteed to hold.
(A) 𝛽 = 5𝑚, 𝛼 = 7𝑀
(B) 𝛽 = 6𝑚, 𝛼 = 5𝑀
(C) 𝛽 = 7𝑚, 𝛼 = 6𝑀
(D) 𝛽 = 7𝑚, 𝛼 = 5𝑀
Organizing Institute: IIT Roorkee Page 45 of 52
cross-section with diameter 1 mm. Copper has a conductivity of 5.8 × 107 S/m.
The current (in Amperes, rounded off to two decimal places) flowing through the
wire is _________.
(A) 0.46
(B) 1.82
(C) 0.58
(D) 1.12
Q.51 Consider a non-negative function 𝑓(𝑥) which is continuous and bounded over the
interval [2, 8]. Let 𝑀 and 𝑚 denote, respectively, the maximum and the minimum
values of 𝑓(𝑥) over the interval.
Among the combinations of 𝛼 and 𝛽 given below, choose the one(s) for which the
inequality
8
𝛽 ≤ ∫2 𝑓(𝑥) 𝑑𝑥 ≤ 𝛼
is guaranteed to hold.
(A) 𝛽 = 5𝑚, 𝛼 = 7𝑀
(B) 𝛽 = 6𝑚, 𝛼 = 5𝑀
(C) 𝛽 = 7𝑚, 𝛼 = 6𝑀
(D) 𝛽 = 7𝑚, 𝛼 = 5𝑀
Organizing Institute: IIT Roorkee Page 45 of 52
Page 47
Q.52 Which of the following statements involving contour integrals (evaluated counter-
clockwise) on the unit circle 𝐶 in the complex plane is/are TRUE?
(A) ∮𝐶 𝑒 𝑧 𝑑𝑧 = 0
(B) ∮𝐶 𝑧 𝑛 𝑑𝑧 = 0, where 𝑛 is an even integer
(C) ∮𝐶 cos 𝑧 𝑑𝑧 ≠ 0
(D) ∮𝐶 sec 𝑧 𝑑𝑧 ≠ 0
Q.53 Consider a system where 𝑥1 (𝑡), 𝑥2 (𝑡), and 𝑥3 (𝑡) are three internal state signals
and 𝑢(𝑡) is the input signal. The differential equations governing the system are
given by
𝑥 (𝑡) 2 0 0 𝑥1 (𝑡) 1
𝑑 1
𝑥 (𝑡)
[ 2 ] = [0 𝑥 (𝑡)
−2 0] [ 2 ] + [1] 𝑢(𝑡).
𝑑𝑡
𝑥3 (𝑡) 0 0 0 𝑥3 (𝑡) 1
Which of the following statements is/are TRUE?
(A) The signals 𝑥1 (𝑡), 𝑥2 (𝑡), and 𝑥3 (𝑡) are bounded for all bounded inputs
(B) There exists a bounded input such that at least one of the signals 𝑥1 (𝑡),
𝑥2 (𝑡), and 𝑥3 (𝑡) is unbounded
(C) There exists a bounded input such that the signals 𝑥1 (𝑡), 𝑥2 (𝑡), and 𝑥3 (𝑡) are
unbounded
(D) The signals 𝑥1 (𝑡), 𝑥2 (𝑡), and 𝑥3 (𝑡) are unbounded for all bounded inputs
Organizing Institute: IIT Roorkee Page 46 of 52
clockwise) on the unit circle 𝐶 in the complex plane is/are TRUE?
(A) ∮𝐶 𝑒 𝑧 𝑑𝑧 = 0
(B) ∮𝐶 𝑧 𝑛 𝑑𝑧 = 0, where 𝑛 is an even integer
(C) ∮𝐶 cos 𝑧 𝑑𝑧 ≠ 0
(D) ∮𝐶 sec 𝑧 𝑑𝑧 ≠ 0
Q.53 Consider a system where 𝑥1 (𝑡), 𝑥2 (𝑡), and 𝑥3 (𝑡) are three internal state signals
and 𝑢(𝑡) is the input signal. The differential equations governing the system are
given by
𝑥 (𝑡) 2 0 0 𝑥1 (𝑡) 1
𝑑 1
𝑥 (𝑡)
[ 2 ] = [0 𝑥 (𝑡)
−2 0] [ 2 ] + [1] 𝑢(𝑡).
𝑑𝑡
𝑥3 (𝑡) 0 0 0 𝑥3 (𝑡) 1
Which of the following statements is/are TRUE?
(A) The signals 𝑥1 (𝑡), 𝑥2 (𝑡), and 𝑥3 (𝑡) are bounded for all bounded inputs
(B) There exists a bounded input such that at least one of the signals 𝑥1 (𝑡),
𝑥2 (𝑡), and 𝑥3 (𝑡) is unbounded
(C) There exists a bounded input such that the signals 𝑥1 (𝑡), 𝑥2 (𝑡), and 𝑥3 (𝑡) are
unbounded
(D) The signals 𝑥1 (𝑡), 𝑥2 (𝑡), and 𝑥3 (𝑡) are unbounded for all bounded inputs
Organizing Institute: IIT Roorkee Page 46 of 52
Page 48
Q.54 The random variable 𝑋 takes values in {−1,0,1} with probabilities
𝑃(𝑋 = −1) = 𝑃(𝑋 = 1) = 𝛼 and 𝑃(𝑋 = 0) = 1 − 2𝛼, where 0 < 𝛼 < 1⁄2.
Let 𝑔(𝛼) denote the entropy of 𝑋 (in bits), parameterized by 𝛼.
Which of the following statements is/are TRUE?
(A) 𝑔(0.4) > 𝑔(0.3)
(B) 𝑔(0.3) > 𝑔(0.4)
(C) 𝑔(0.3) > 𝑔(0.25)
(D) 𝑔(0.25) > 𝑔(0.3)
Q.55 Let 𝑓(𝑡) be a periodic signal with fundamental period 𝑇0 > 0. Consider the signal
𝑦(𝑡) = 𝑓(𝛼𝑡), where 𝛼 > 1.
The Fourier series expansions of 𝑓(𝑡) and 𝑦(𝑡) are given by
∞ ∞
2𝜋 2𝜋
𝑗 𝑘𝑡 𝑗𝛼𝑘𝑡
𝑓(𝑡) = ∑ 𝑐𝑘 𝑒 𝑇0 and 𝑦(𝑡) = ∑ 𝑑𝑘 𝑒 𝑇0 .
𝑘=−∞ 𝑘=−∞
Which of the following statements is/are TRUE?
(A) 𝑐𝑘 = 𝑑𝑘 for all 𝑘
(B) 𝑦(𝑡) is periodic with a fundamental period 𝛼𝑇0
(C) 𝑐𝑘 = 𝑑𝑘 /𝛼 for all 𝑘
(D) 𝑦(𝑡) is periodic with a fundamental period 𝑇0 /𝛼
Organizing Institute: IIT Roorkee Page 47 of 52
𝑃(𝑋 = −1) = 𝑃(𝑋 = 1) = 𝛼 and 𝑃(𝑋 = 0) = 1 − 2𝛼, where 0 < 𝛼 < 1⁄2.
Let 𝑔(𝛼) denote the entropy of 𝑋 (in bits), parameterized by 𝛼.
Which of the following statements is/are TRUE?
(A) 𝑔(0.4) > 𝑔(0.3)
(B) 𝑔(0.3) > 𝑔(0.4)
(C) 𝑔(0.3) > 𝑔(0.25)
(D) 𝑔(0.25) > 𝑔(0.3)
Q.55 Let 𝑓(𝑡) be a periodic signal with fundamental period 𝑇0 > 0. Consider the signal
𝑦(𝑡) = 𝑓(𝛼𝑡), where 𝛼 > 1.
The Fourier series expansions of 𝑓(𝑡) and 𝑦(𝑡) are given by
∞ ∞
2𝜋 2𝜋
𝑗 𝑘𝑡 𝑗𝛼𝑘𝑡
𝑓(𝑡) = ∑ 𝑐𝑘 𝑒 𝑇0 and 𝑦(𝑡) = ∑ 𝑑𝑘 𝑒 𝑇0 .
𝑘=−∞ 𝑘=−∞
Which of the following statements is/are TRUE?
(A) 𝑐𝑘 = 𝑑𝑘 for all 𝑘
(B) 𝑦(𝑡) is periodic with a fundamental period 𝛼𝑇0
(C) 𝑐𝑘 = 𝑑𝑘 /𝛼 for all 𝑘
(D) 𝑦(𝑡) is periodic with a fundamental period 𝑇0 /𝛼
Organizing Institute: IIT Roorkee Page 47 of 52
Page 49
Q.56 Consider a system represented by the block diagram shown below. Which of the
following signal flow graphs represent(s) this system? Choose the correct option(s).
(A)
(B)
(C)
(D)
Organizing Institute: IIT Roorkee Page 48 of 52
following signal flow graphs represent(s) this system? Choose the correct option(s).
(A)
(B)
(C)
(D)
Organizing Institute: IIT Roorkee Page 48 of 52
Page 50
Q.57 All the diodes in the circuit given below are ideal.
Which of the following plots is/are correct when VI (in Volts) is swept from –M to
M?
(A)
(B)
(C)
(D)
Organizing Institute: IIT Roorkee Page 49 of 52
Which of the following plots is/are correct when VI (in Volts) is swept from –M to
M?
(A)
(B)
(C)
(D)
Organizing Institute: IIT Roorkee Page 49 of 52
Page 51
Q.58 Two fair dice (with faces labeled 1, 2, 3, 4, 5, and 6) are rolled. Let the random
variable 𝑋 denote the sum of the outcomes obtained.
The expectation of 𝑋 is______ (rounded off to two decimal places).
Q.59 Consider the vectors
1 0
𝒂 = [ ], 𝒃=[ ].
1 3√2
For real-valued scalar variable 𝑥, the value of
min ‖𝒂𝑥 − 𝒃‖2
𝑥
is ____________ (rounded off to two decimal places).
𝑦1
‖⋅‖2 denotes the Euclidean norm, i.e., for 𝒚 = [ ], ‖𝒚‖2 = √𝑦12 + 𝑦22 .
𝑦2
Q.60 𝑋 and 𝑌 are Bernoulli random variables taking values in {0,1}. The joint
probability mass function of the random variables is given by:
𝑃(𝑋 = 0, 𝑌 = 0) = 0.06
𝑃(𝑋 = 0, 𝑌 = 1) = 0.14
𝑃(𝑋 = 1, 𝑌 = 0) = 0.24
𝑃(𝑋 = 1, 𝑌 = 1) = 0.56
The mutual information 𝐼(𝑋; 𝑌) is _________ (rounded off to two decimal places).
Organizing Institute: IIT Roorkee Page 50 of 52
variable 𝑋 denote the sum of the outcomes obtained.
The expectation of 𝑋 is______ (rounded off to two decimal places).
Q.59 Consider the vectors
1 0
𝒂 = [ ], 𝒃=[ ].
1 3√2
For real-valued scalar variable 𝑥, the value of
min ‖𝒂𝑥 − 𝒃‖2
𝑥
is ____________ (rounded off to two decimal places).
𝑦1
‖⋅‖2 denotes the Euclidean norm, i.e., for 𝒚 = [ ], ‖𝒚‖2 = √𝑦12 + 𝑦22 .
𝑦2
Q.60 𝑋 and 𝑌 are Bernoulli random variables taking values in {0,1}. The joint
probability mass function of the random variables is given by:
𝑃(𝑋 = 0, 𝑌 = 0) = 0.06
𝑃(𝑋 = 0, 𝑌 = 1) = 0.14
𝑃(𝑋 = 1, 𝑌 = 0) = 0.24
𝑃(𝑋 = 1, 𝑌 = 1) = 0.56
The mutual information 𝐼(𝑋; 𝑌) is _________ (rounded off to two decimal places).
Organizing Institute: IIT Roorkee Page 50 of 52
Page 52
Q.61 The diode in the circuit shown below is ideal. The input voltage (in Volts) is given
by 𝑉𝐼 = 10 sin 100𝜋𝑡, where time 𝑡 is in seconds.
The time duration (in ms, rounded off to two decimal places) for which the diode is
forward biased during one period of the input is ___________.
Q.62 In the circuit shown below, the AND gate has a propagation delay of 1 ns. The edge-
triggered flip-flops have a set-up time of 2 ns, a hold-time of 0 ns, and a clock-to-Q
delay of 2 ns.
The maximum clock frequency (in MHz, rounded off to the nearest integer) such
that there are no setup violations is ________.
Organizing Institute: IIT Roorkee Page 51 of 52
by 𝑉𝐼 = 10 sin 100𝜋𝑡, where time 𝑡 is in seconds.
The time duration (in ms, rounded off to two decimal places) for which the diode is
forward biased during one period of the input is ___________.
Q.62 In the circuit shown below, the AND gate has a propagation delay of 1 ns. The edge-
triggered flip-flops have a set-up time of 2 ns, a hold-time of 0 ns, and a clock-to-Q
delay of 2 ns.
The maximum clock frequency (in MHz, rounded off to the nearest integer) such
that there are no setup violations is ________.
Organizing Institute: IIT Roorkee Page 51 of 52
Page 53
Q.63 An ideal p-n junction germanium diode has a reverse saturation current of 10 µA at
300 K.
The voltage (in Volts, rounded off to two decimal places) to be applied across the
junction to get a forward bias current of 100 mA at 300 K is _____.
(Consider the Boltzmann constant 𝑘B = 1.38 × 10−23 J/K and the charge of an
electron 𝑒 = 1.6 × 10−19 C.)
Q.64 A 50 Ω lossless transmission line is terminated with a load ZL of (50 – j75) Ω.
If the average incident power on the line is 10 mW, then the average power
delivered to the load (in mW, rounded off to one decimal place) is __________.
Q.65 Two resistors are connected in a circuit loop of area 5 m2, as shown in the figure
below. The circuit loop is placed on the x-y plane.
When a time-varying magnetic flux, with flux-density 𝐵(𝑡) = 0.5𝑡 (in Tesla), is
applied along the positive z-axis, the magnitude of current I (in Amperes, rounded
off to two decimal places) in the loop is ________.
Organizing Institute: IIT Roorkee Page 52 of 52
300 K.
The voltage (in Volts, rounded off to two decimal places) to be applied across the
junction to get a forward bias current of 100 mA at 300 K is _____.
(Consider the Boltzmann constant 𝑘B = 1.38 × 10−23 J/K and the charge of an
electron 𝑒 = 1.6 × 10−19 C.)
Q.64 A 50 Ω lossless transmission line is terminated with a load ZL of (50 – j75) Ω.
If the average incident power on the line is 10 mW, then the average power
delivered to the load (in mW, rounded off to one decimal place) is __________.
Q.65 Two resistors are connected in a circuit loop of area 5 m2, as shown in the figure
below. The circuit loop is placed on the x-y plane.
When a time-varying magnetic flux, with flux-density 𝐵(𝑡) = 0.5𝑡 (in Tesla), is
applied along the positive z-axis, the magnitude of current I (in Amperes, rounded
off to two decimal places) in the loop is ________.
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Document Details
| Board / Org | IIT |
| Exam | Graduate Aptitude Test in Engineering |
| Type | Question Paper |
| Pages | 54 |
| Updated | 30 Apr 2026 |