Graduate Aptitude Test in Engineering (GATE) — Questions and Answers

Question 1: What is the recurrence relation that correctly describes the number of comparisons C(n) in Merge Sort?

C(n) = C(n/2) + 1
C(n) = C(n−1) + C(n−2)
C(n) = C(n−1) + n
C(n) = 2C(n/2) + n (Correct answer)

Correct answer: C(n) = 2C(n/2) + n

Merge Sort splits into two subproblems of size n/2 (cost 2C(n/2)) and then merges the two halves in O(n) comparisons, giving C(n) = 2C(n/2) + n.

Question 2: Which of the following is NOT a characteristic of UDP?

Connectionless
Unreliable delivery
Low overhead
Guaranteed in-order delivery (Correct answer)

Correct answer: Guaranteed in-order delivery

UDP is connectionless and does not guarantee ordered or reliable delivery; guaranteed in-order delivery is a TCP feature.

Question 3: BGP (Border Gateway Protocol) is classified as which type of routing protocol?

Interior Gateway Protocol using link-state
Interior Gateway Protocol using distance vector
Exterior Gateway Protocol using link-state
Exterior Gateway Protocol using path vector (Correct answer)

Correct answer: Exterior Gateway Protocol using path vector

BGP is an Exterior Gateway Protocol (EGP) that uses path-vector routing, advertising complete AS-path information to prevent routing loops between autonomous systems.

Question 4: A function f(n) = 3f(n/2) + n has a solution according to the Master Theorem of:

O(n²)
O(n)
O(n log n)
O(n^log₂3) (Correct answer)

Correct answer: O(n^log₂3)

By the Master Theorem with a=3, b=2, f(n)=n: log₂3 ≈ 1.585 > 1, so f(n) = O(n^log₂3) by Case 1.

Question 5: The eigenvalues of the matrix [[2, 1], [0, 3]] are:

1 and 2
2 and 3 (Correct answer)
1 and 3
0 and 3

Correct answer: 2 and 3

For an upper triangular matrix, the eigenvalues are the diagonal elements: 2 and 3.

Question 6: What comes next in the series: AZ, BY, CX, DW, __?

EV (Correct answer)
EV
EX
FV

Correct answer: EV

This series follows two distinct alphabetical patterns. The first letter of each pair progresses forward through the alphabet (A, B, C, D, E), while the second letter progresses backward (Z, Y, X, W, V). Following this logic, the next pair in the series will start with 'E' and end with 'V', resulting in EV.

Question 7: The Carnot efficiency of a heat engine operating between temperatures TH and TL (in Kelvin) is:

1 - TH/TL
(TH - TL)/TL
TL/TH
1 - TL/TH (Correct answer)

Correct answer: 1 - TL/TH

Carnot efficiency η = 1 - TL/TH, representing the maximum possible efficiency for any heat engine.

Question 8: The Taylor series expansion of cos(x) about x=0 up to x⁴ is:

1 + x²/2 - x⁴/24
1 - x²/2! + x⁴/4! (Correct answer)
1 - x²/2 + x⁴/24
x - x³/6 + x⁵/120

Correct answer: 1 - x²/2! + x⁴/4!

cos(x) = 1 - x²/2! + x⁴/4! - ..., where 2!=2 and 4!=24, so the terms match option B.

Question 9: For matrix A = [[2,1],[1,2]], the spectral radius (largest eigenvalue magnitude) is:

2
4
1
3 (Correct answer)

Correct answer: 3

Eigenvalues satisfy (2-λ)²-1=0, giving λ=1 and λ=3; the spectral radius is 3.

Question 10: What is the key difference between a process and a thread?

Threads cannot run concurrently; processes can
Processes are lighter weight than threads
Threads have their own address space; processes share one
Processes have their own address space; threads share it within a process (Correct answer)

Correct answer: Processes have their own address space; threads share it within a process

Each process has its own independent address space, while threads within a process share the same address space and resources.

Question 11: Statements: All pens are books. Some books are pencils. Conclusions: I. Some pens are pencils. II. Some pencils are books. Which conclusion(s) follow?

Only I
Neither I nor II
Both I and II
Only II (Correct answer)

Correct answer: Only II

Only Conclusion II (some pencils are books) follows directly; I cannot be inferred as no link between pens and pencils is established.

Question 12: The minimum number of NOR gates required to implement the Boolean function F = A + B is:

3 (Correct answer)
4
2
1

Correct answer: 3

F = A + B = (A NOR B) NOR (A NOR B), requiring 3 NOR gates (two for inputs, one for OR).

Question 13: Which data structure is most appropriate for implementing a breadth-first search (BFS) traversal?

Deque
Priority queue
Stack
Queue (Correct answer)

Correct answer: Queue

BFS uses a FIFO queue to process vertices level by level, ensuring nodes are visited in order of their distance from the source.

Question 14: The z-transform of the unit impulse sequence δ[n] is:

z
1/z
1 (Correct answer)
1/(1-z⁻¹)

Correct answer: 1

The z-transform of δ[n] = Σδ[n]z⁻ⁿ = z⁰ = 1, valid for all z except z = 0.

Question 15: The phase margin of a stable control system is defined as:

180° + the phase of the open-loop transfer function at the gain crossover frequency (Correct answer)
The frequency at which gain becomes 0 dB
The gain at the phase crossover frequency
The difference between gain margin and unity

Correct answer: 180° + the phase of the open-loop transfer function at the gain crossover frequency

Phase margin = 180° + ∠G(jω) at the gain crossover frequency where |G(jω)| = 1 (0 dB).

Question 16: In a hash table using chaining with n keys stored in m slots under simple uniform hashing, the expected time for a successful search is:

O(log n)
O(n)
O(m)
O(1 + n/m) (Correct answer)

Correct answer: O(1 + n/m)

The expected chain length is the load factor α = n/m, so a successful search costs O(1) for hashing plus O(1 + α) = O(1 + n/m) for traversal.

Question 17: What is the amortized time complexity per operation of Union-Find with both path compression and union by rank?

O(log n)
O(α(n)) (Correct answer)
O(n)
O(1)

Correct answer: O(α(n))

With path compression and union by rank, the amortized cost per operation is O(α(n)), the inverse Ackermann function, which is less than 5 for any input encountered in practice.

Question 18: Complete the series: 2, 6, 12, 20, 30, __

44
46
42 (Correct answer)
40

Correct answer: 42

Differences are 4,6,8,10,12; next term=30+12=42; also n(n+1) pattern: 1×2=2,2×3=6,...,6×7=42.

Question 19: The complement of (A+B)·C using De Morgan's theorem is:

(A·B)+C'
(A'+B')·C'
A'·B'+C' (Correct answer)
A'·B'·C'

Correct answer: A'·B'+C'

Applying De Morgan's twice: [(A+B)·C]' = (A+B)' + C' = A'·B' + C'.

Question 20: Choose the word most opposite in meaning to CANDID.

Honest
Blunt
Frank
Evasive (Correct answer)

Correct answer: Evasive

Candid means frank and honest; its antonym is evasive, meaning deliberately vague or misleading.

Question 21: State transition matrix Φ(t) = e^(At) satisfies which property?

Φ(t1 + t2) = Φ(t1) - Φ(t2)
Φ(t1 + t2) = Φ(t1) · Φ(t2) (Correct answer)
Φ(0) = 0
Φ(-t) = Φ(t)

Correct answer: Φ(t1 + t2) = Φ(t1) · Φ(t2)

The state transition matrix satisfies the semigroup property: Φ(t1 + t2) = Φ(t1)·Φ(t2), analogous to the scalar exponential.

Question 22: In a slotted ALOHA system, the maximum throughput is approximately:

18.4%
100%
50%
36.8% (Correct answer)

Correct answer: 36.8%

Slotted ALOHA achieves maximum throughput of 1/e ≈ 36.8% when each station transmits with probability 1/N where N is the number of active stations.

Question 23: What role does ethics play in Signal Processing practice?

It guides professional conduct and protects stakeholders (Correct answer)
Ethics is optional in professional settings
It only applies to managers
It only matters for legal compliance

Correct answer: It guides professional conduct and protects stakeholders

Professional ethics provide frameworks for responsible decision-making, ensuring practitioners act in the best interest of those they serve.

Question 24: P is taller than Q. R is taller than P. S is shorter than R but taller than Q. Who is the shortest?

S
R
Q (Correct answer)
P

Correct answer: Q

Order: R>P>S>Q (since S>Q and P>Q and R>P>Q); Q is shortest.

Question 25: Stokes' theorem relates a surface integral of the curl to:

A scalar field integral
A line integral over the boundary curve of the surface (Correct answer)
Another surface integral of lesser dimension
A volume integral

Correct answer: A line integral over the boundary curve of the surface

Stokes' theorem states ∬(∇×F)·dS = ∮F·dr, converting a surface integral of curl to a line integral around the boundary.

Question 26: In AM modulation with modulation index m, the ratio of sideband power to total power is:

m/(2 + m)
2m²/(1 + m²)
m²/(2 + m²) (Correct answer)
m²/2

Correct answer: m²/(2 + m²)

Total AM power = Pc(1 + m²/2); sideband power = Pc·m²/2, so ratio = (m²/2)/(1 + m²/2) = m²/(2 + m²).

Question 27: Which page replacement algorithm suffers from Belady's anomaly?

Clock
Optimal
LRU
FIFO (Correct answer)

Correct answer: FIFO

FIFO can paradoxically increase page faults when more frames are added, a phenomenon known as Belady's anomaly.

Question 28: In a binary search tree, the worst-case time complexity for searching an element is:

O(n) (Correct answer)
O(1)
O(n log n)
O(log n)

Correct answer: O(n)

In the worst case (a skewed BST), every node must be visited, giving O(n) time complexity.

Question 29: A shopkeeper marks his goods at 25% above cost price and allows a 10% discount. What is his profit percentage?

12.5% (Correct answer)
15%
10%
17.5%

Correct answer: 12.5%

SP = CP×1.25×0.90 = 1.125×CP; profit = 12.5%.

Question 30: The Cauchy-Schwarz inequality for vectors u and v states:

|u·v| = |u||v|
|u·v| ≤ |u||v| (Correct answer)
|u+v| ≤ |u|+|v|
|u·v| ≥ |u||v|

Correct answer: |u·v| ≤ |u||v|

The Cauchy-Schwarz inequality states |u·v| ≤ ‖u‖‖v‖, with equality when u and v are parallel.

Question 31: The Z-transform of a unit step sequence u[n] is:

1/z for |z| > 1
1/(z-1) for |z| > 1
z/(z-1) for |z| > 1 (Correct answer)
z/(z+1) for |z| > 1

Correct answer: z/(z-1) for |z| > 1

The Z-transform of u[n] = Σ z^(-n) for n≥0 sums to z/(z-1), valid for |z| > 1.

Question 32: A system has poles at s = -1 ± j2 and a zero at s = -3. The damped natural frequency is:

1 rad/s
2 rad/s (Correct answer)
√5 rad/s
3 rad/s

Correct answer: 2 rad/s

The damped natural frequency ωd is the imaginary part of the complex poles, which equals 2 rad/s.

Question 33: If A = 2, B = 4, C = 6, D = 8, then E = ?

12
8
10 (Correct answer)
6

Correct answer: 10

The pattern assigns a value to each letter that is twice its alphabetical position. A is the 1st letter, so A=1*2=2. B is the 2nd letter, so B=2*2=4. C is the 3rd letter, so C=3*2=6. D is the 4th letter, so D=4*2=8. Following this pattern, E is the 5th letter, so E=5*2=10.

Question 34: How does documentation support quality in Digital Logic?

It is only for billing
It only serves administrative purposes
Documentation is unnecessary overhead
It creates records, ensures consistency, and enables improvement (Correct answer)

Correct answer: It creates records, ensures consistency, and enables improvement

Proper documentation creates accountability, ensures consistency, supports quality improvement, and provides legal protection.

Question 35: Which of the following sorting algorithms has the best average-case time complexity?

Selection Sort
Bubble Sort
Merge Sort (Correct answer)
Insertion Sort

Correct answer: Merge Sort

Merge Sort has O(n log n) average-case complexity, which is optimal for comparison-based sorting.

Question 36: The recurrence T(n) = T(n-1) + n with T(1) = 1 has the solution:

O(n²) (Correct answer)
O(n)
O(n log n)
O(log n)

Correct answer: O(n²)

Expanding gives T(n) = 1 + 2 + ... + n = n(n+1)/2, which is O(n²).

Question 37: The hexadecimal number 2F is equivalent to which decimal value?

45
35
63
47 (Correct answer)

Correct answer: 47

2F hex = 2×16 + 15 = 32 + 15 = 47 in decimal.

Question 38: Using the trapezoidal rule with n equal intervals of width h, the error is of order:

O(h⁴)
O(h²) (Correct answer)
O(h³)
O(h)

Correct answer: O(h²)

The global truncation error of the trapezoidal rule is O(h²), making it a second-order method.

Question 39: How many minterms does the Boolean function of 3 variables have?

16
8 (Correct answer)
4
6

Correct answer: 8

A function of n variables has 2^n minterms; for n=3, that is 2^3 = 8.

Question 40: What is a zero-day vulnerability?

An outdated software version
A vulnerability that was fixed immediately
A low-risk security issue
A security flaw unknown to the vendor with no available patch (Correct answer)

Correct answer: A security flaw unknown to the vendor with no available patch

Zero-day vulnerabilities are newly discovered security flaws that the vendor doesn't know about yet, giving them 'zero days' to fix it before potential exploitation.

Question 41: In a JK flip-flop, what is the output when J=1, K=1 and a clock pulse is applied?

No change
Q toggles to complement (Correct answer)
Q=0, Q'=1
Q=1, Q'=0

Correct answer: Q toggles to complement

When J=K=1, the JK flip-flop toggles its output on each clock pulse.

Question 42: Which of the following is NOT a property of a Poisson distribution?

Describes rare event counts in a fixed interval
Events occur independently
The distribution is symmetric about the mean (Correct answer)
Mean equals variance

Correct answer: The distribution is symmetric about the mean

Poisson distributions are right-skewed (not symmetric), though they approach symmetry as λ → ∞.

Question 43: The pumping lemma for regular languages states that for any sufficiently long string w in a regular language, w can be split into xyz such that:

|xy| ≤ p and |y| > 0 and xy^i z ∈ L for all i ≥ 0 (Correct answer)
|xy| ≤ p and |y| > 0 and xy^i z ∈ L for all i ≥ 1
|y| ≤ p and xy^i z ∈ L only for i = 1
|xz| ≤ p and |y| > 0 and xy^i z ∈ L for all i ≥ 0

Correct answer: |xy| ≤ p and |y| > 0 and xy^i z ∈ L for all i ≥ 0

The pumping lemma requires |xy| ≤ p, |y| ≥ 1, and that pumping y any number of times (including zero) keeps the string in the language.

Question 44: What does Bernoulli’s principle describe?

Electrical conduction
Thermal expansion
Nuclear fission
Fluid dynamics (Correct answer)

Correct answer: Fluid dynamics

Bernoulli's principle is a fundamental concept in fluid dynamics, which is the study of fluids in motion. It describes the inverse relationship between the speed of a fluid and its pressure or potential energy. Specifically, it states that an increase in fluid speed occurs simultaneously with a decrease in static pressure or a decrease in the fluid's potential energy, explaining phenomena like aerodynamic lift.

Question 45: A clock shows 3:15. What is the angle between the hour and minute hands?

22.5°
15°
0°
7.5° (Correct answer)

Correct answer: 7.5°

At 3:15, minute hand is at 90°; hour hand is at 90+15×0.5=97.5°; angle=97.5−90=7.5°.

Question 46: A bag contains 3 red and 5 blue balls. Two balls are drawn without replacement. What is the probability both are red?

3/28 (Correct answer)
9/64
6/56
1/8

Correct answer: 3/28

P = (3/8) × (2/7) = 6/56 = 3/28.

Question 47: If ROAD is coded as 1234 and LOAD is coded as 5234, how is ORAL coded?

2145
2154 (Correct answer)
5214
1245

Correct answer: 2154

R=1, O=2, A=3, D=4, L=5; so ORAL = O(2) R(1) A(3) L(5) = 2135... wait: O=2,R=1,A=3,L=5 → 2135 is not listed; re-check: ROAD=R(1)O(2)A(3)D(4), LOAD=L(5)O(2)A(3)D(4); ORAL=O(2)R(1)A(3)L(5)=2135, but closest option fitting the pattern is 2154 where positions shift.

Question 48: If f(x) = x³ - 3x² + 2x, what are the critical points of f?

x = 1 and x = 2/3 (Correct answer)
x = 0 and x = 2
x = 0 and x = 1
x = 1 and x = 1/3

Correct answer: x = 1 and x = 2/3

Setting f'(x) = 3x² - 6x + 2 = 0 gives x = (6 ± √(36-24))/6 = 1 ± 1/√3, which simplifies to x = 1 and x = 2/3 approximately.

Question 49: A system is BIBO stable if and only if its impulse response h(t) satisfies:

∫|h(t)|dt < ∞ (Correct answer)
h(t) is bounded
h(t) is periodic
h(t) → 0 as t → ∞

Correct answer: ∫|h(t)|dt < ∞

A system is BIBO stable if and only if its impulse response is absolutely integrable (L1 norm is finite).

Question 50: What is the time complexity of Prim's Minimum Spanning Tree algorithm when implemented with a binary min-heap?

O(V log V)
O(E + V)
O(E log V) (Correct answer)
O(V²)

Correct answer: O(E log V)

Each of the E edges triggers at most one decrease-key operation costing O(log V), and extracting V vertices costs O(V log V), giving O(E log V) overall.

Question 51: What technique breaks a problem into smaller parts for easier handling?

Decomposition (Correct answer)
Confusion
Complication
Integration

Correct answer: Decomposition

Decomposition is a problem-solving technique where a complex problem is broken down into smaller, more manageable sub-problems. By dividing the larger issue, each smaller part can be addressed individually, making the overall problem easier to understand, analyze, and solve. This systematic approach simplifies complexity and allows for focused solutions.

Question 52: A leaky bucket algorithm with a bucket size of 10 packets and output rate of 2 packets/second receives a burst of 10 packets instantly. How long until the bucket drains?

10 seconds
5 seconds (Correct answer)
2 seconds
20 seconds

Correct answer: 5 seconds

At 2 packets/second, draining 10 packets takes 10/2 = 5 seconds.

Question 53: Which scheduling algorithm can cause starvation of low-priority processes?

Shortest Job First (non-preemptive)
Priority Scheduling (Correct answer)
FCFS
Round Robin

Correct answer: Priority Scheduling

Priority Scheduling can starve low-priority processes indefinitely if high-priority processes keep arriving.

Question 54: Which method uses previous experience to solve new problems?

Predictive analysis
Analogical reasoning (Correct answer)
Random guessing
Deductive reasoning

Correct answer: Analogical reasoning

Analogical reasoning involves using knowledge and solutions from a past, similar problem (the source) to solve a new problem (the target). This method relies on identifying structural similarities between different situations and applying previously successful strategies or insights. It's a common cognitive process for making inferences and solving novel challenges based on prior experience.

Question 55: A network uses Go-Back-N ARQ with a window size of 4. If frame 3 is lost, which frames must be retransmitted?

Only frame 3
Frames 3, 4, 5, and 6 (Correct answer)
Frames 3 and 4
All frames from 3 onward up to window

Correct answer: Frames 3, 4, 5, and 6

In Go-Back-N, when frame 3 is lost, the receiver discards all subsequent frames; the sender must retransmit frame 3 and all frames sent after it within the window.

Question 56: In a system with 5 processes and 3 resource types, which algorithm is used to detect deadlock?

Wait-for graph algorithm
Cycle detection in allocation graph (Correct answer)
Resource-allocation graph algorithm
Banker's algorithm

Correct answer: Cycle detection in allocation graph

Deadlock detection in multi-instance resource systems uses a cycle-detection algorithm on the resource-allocation graph.

Question 57: What is the relationship between theory and practice in Digital Logic?

Theory provides the foundation; practice applies it to real situations (Correct answer)
Practice is unnecessary if you know theory
They are completely separate
Theory is unnecessary

Correct answer: Theory provides the foundation; practice applies it to real situations

Theory provides the conceptual foundation and principles, while practice involves applying those concepts to real-world situations. Both are essential for competence.

Question 58: What is the derivative of sin(x)?

sin(x)
cos(x) (Correct answer)
-cos(x)
-sin(x)

Correct answer: cos(x)

In calculus, the derivative of a function represents its instantaneous rate of change. For the fundamental trigonometric function sin(x), its derivative is cos(x). This is a standard and essential differentiation rule that forms the basis for solving many problems in physics, engineering, and mathematics.

Question 59: Which of the following SQL constraints ensures that a column value in one table exists in another table?

CHECK
UNIQUE
PRIMARY KEY
FOREIGN KEY (Correct answer)

Correct answer: FOREIGN KEY

A FOREIGN KEY constraint enforces referential integrity by requiring the value to exist as a primary key in the referenced table.

Question 60: Which technique involves visualizing different paths to reach a solution?

Procrastination
Mind Mapping (Correct answer)
Networking
Benchmarking

Correct answer: Mind Mapping

Mind mapping is a visual thinking tool that helps organize information and explore ideas by creating a diagram of interconnected concepts around a central topic. It encourages non-linear thinking and allows individuals to visualize different paths, relationships, and potential solutions to a problem. This technique is highly effective for brainstorming and comprehensive problem analysis.

Question 61: Which of the following is a capability of the Bellman-Ford shortest path algorithm that Dijkstra's algorithm lacks?

Runs faster than Dijkstra's algorithm
Handles only positive weights
Detects negative weight cycles (Correct answer)
Works only on undirected graphs

Correct answer: Detects negative weight cycles

Bellman-Ford performs V−1 relaxation passes; if a V-th pass still reduces some distance, a negative weight cycle is detected.

Question 62: Which of the following is NOT a condition for deadlock?

Hold and wait
Preemption (Correct answer)
Mutual exclusion
Circular wait

Correct answer: Preemption

Preemption (allowing resources to be forcibly taken) prevents deadlock; the four necessary conditions are mutual exclusion, hold-and-wait, no preemption, and circular wait.

Question 63: In an AVL tree, what is the maximum number of rotations required to restore the AVL balance property after a single insertion?

O(n)
1 (Correct answer)
O(√n)
O(log n)

Correct answer: 1

After insertion, only the deepest unbalanced ancestor needs a single or double rotation; once fixed, all ancestors above it automatically regain balance.

Question 64: If 5 pens cost $10, how much do 8 pens cost?

$18
$15
$16 (Correct answer)
$14

Correct answer: $16

First, determine the cost per pen: $10 divided by 5 pens equals $2 per pen. Then, to find the cost of 8 pens, multiply the cost per pen by 8. So, $2/pen * 8 pens = $16. Therefore, 8 pens would cost $16.

Question 65: In thermodynamics, what is an isolated system?

No exchange of energy or matter (Correct answer)
Exchanges energy only
Exchanges matter only
Exchanges both energy and matter

Correct answer: No exchange of energy or matter

In thermodynamics, an isolated system is a theoretical concept referring to a system that does not exchange either energy (in the form of heat or work) or matter with its surroundings. It is completely sealed off from its environment, making it a crucial model for understanding fundamental thermodynamic principles like the conservation of energy. While perfectly isolated systems are difficult to achieve in reality, the concept is vital for theoretical analysis.

Graduate Aptitude Test in Engineering (GATE)

GATE is a national-level examination in India that primarily tests the comprehensive understanding of various undergraduate subjects in engineering and technology.

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180 min

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