GATE 2024 Question Paper for Data Science and Artificial Intelligence (DA)

GATE
2024
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.

General Aptitude (GA)
Q.1 – Q.5 Carry ONE mark Each

Q.1 If ‘→’ denotes increasing order of intensity, then the meaning of the words

[sick → infirm → moribund] is analogous to [silly → _______ → daft].

Which one of the given options is appropriate to fill the blank?

(A) frown

(B) fawn

(C) vein

(D) vain

Page 1 of 41

Q.2 The 15 parts of the given figure are to be painted such that no two adjacent parts
with shared boundaries (excluding corners) have the same color. The minimum
number of colors required is

(A) 4

(B) 3

(C) 5

(D) 6

Page 2 of 41

Q.3 How many 4-digit positive integers divisible by 3 can be formed using only the
digits {1, 3, 4, 6, 7}, such that no digit appears more than once in a number?

(A) 24

(B) 48

(C) 72

(D) 12

Q.4 The sum of the following infinite series is

1 1 1 1 1 1 1
2+ + + + + + + +⋯
2 3 4 8 9 16 27

(A) 11/3

(B) 7/2

(C) 13/4

(D) 9/2

Page 3 of 41

Q.5 In an election, the share of valid votes received by the four candidates A, B, C, and
D is represented by the pie chart shown. The total number of votes cast in the
election were 1,15,000, out of which 5,000 were invalid.

Share of valid votes

D
15%
A
C 40%
20%

B
25%

Based on the data provided, the total number of valid votes received by the
candidates B and C is

(A) 45,000

(B) 49,500

(C) 51,750

(D) 54,000

Page 4 of 41

Q.6 – Q.10 Carry TWO marks Each

Q.6 Thousands of years ago, some people began dairy farming. This coincided with a
number of mutations in a particular gene that resulted in these people developing
the ability to digest dairy milk.

Based on the given passage, which of the following can be inferred?

(A) All human beings can digest dairy milk.

(B) No human being can digest dairy milk.

(C) Digestion of dairy milk is essential for human beings.

(D) In human beings, digestion of dairy milk resulted from a mutated gene.

Q.7 The probability of a boy or a girl being born is 1/2. For a family having only
three children, what is the probability of having two girls and one boy?

(A) 3/8

(B) 1/8

(C) 1/4

(D) 1/2

Page 5 of 41

Q.8 Person 1 and Person 2 invest in three mutual funds A, B, and C. The amounts they
invest in each of these mutual funds are given in the table.

Mutual fund A Mutual fund B Mutual fund C

Person 1 ₹10,000 ₹20,000 ₹20,000

Person 2 ₹20,000 ₹15,000 ₹15,000

At the end of one year, the total amount that Person 1 gets is ₹500 more than
Person 2. The annual rate of return for the mutual funds B and C is 15% each. What
is the annual rate of return for the mutual fund A?

(A) 7.5%

(B) 10%

(C) 15%

(D) 20%

Page 6 of 41

Q.9 Three different views of a dice are shown in the figure below.

The piece of paper that can be folded to make this dice is

(A)

(B)

(C)

(D)

Page 7 of 41

Q.10 Visualize two identical right circular cones such that one is inverted over the other
and they share a common circular base. If a cutting plane passes through the vertices
of the assembled cones, what shape does the outer boundary of the
resulting cross-section make?

(A) A rhombus

(B) A triangle

(C) An ellipse

(D) A hexagon

Page 8 of 41

Q.11 – Q.35 Carry ONE mark Each

Q.11 Consider the following statements:

(i) The mean and variance of a Poisson random variable are equal.
(ii) For a standard normal random variable, the mean is zero and the
variance is one.

Which ONE of the following options is correct?

(A) Both (i) and (ii) are true

(B) (i) is true and (ii) is false

(C) (ii) is true and (i) is false

(D) Both (i) and (ii) are false

Q.12 Three fair coins are tossed independently. T is the event that two or more tosses
result in heads. S is the event that two or more tosses result in tails.

What is the probability of the event 𝑇 ∩ 𝑆 ?

(A) 0

(B) 0.5

(C) 0.25

(D) 1

Page 9 of 41

Q.13 2 −1
Consider the matrix 𝑴 = [ ].
3 1
Which ONE of the following statements is TRUE?

(A) The eigenvalues of 𝑴 are non-negative and real.

(B) The eigenvalues of 𝑴 are complex conjugate pairs.

(C) One eigenvalue of 𝑴 is positive and real, and another eigenvalue of 𝑴 is zero.

(D) One eigenvalue of 𝑴 is non-negative and real, and another eigenvalue of 𝑴 is
negative and real.

Q.14 Consider performing depth-first search (DFS) on an undirected and unweighted
graph G starting at vertex 𝑠. For any vertex 𝑢 in G, 𝑑[𝑢] is the length of the shortest
path from 𝑠 to 𝑢. Let (𝑢, 𝑣) be an edge in G such that 𝑑 [𝑢] < 𝑑[𝑣]. If the edge
(𝑢, 𝑣) is explored first in the direction from 𝑢 to 𝑣 during the above DFS, then (𝑢, 𝑣)
becomes a ______ edge.

(A) tree

(B) cross

(C) back

(D) gray

Page 10 of 41

Q.15 For any twice differentiable function 𝑓: ℝ → ℝ, if at some 𝑥 ∗ ∈ ℝ, 𝑓 ′ (𝑥 ∗ ) = 0
and 𝑓 ′′ (𝑥 ∗ ) > 0, then the function 𝑓 necessarily has a ______ at 𝑥 = 𝑥 ∗.

Note: ℝ denotes the set of real numbers.

(A) local minimum

(B) global minimum

(C) local maximum

(D) global maximum

Q.16 Match the items in Column 1 with the items in Column 2 in the following table:

Column 1 Column 2

(p) First In First Out (i) Stacks
(q) Lookup Operation (ii) Queues
(r) Last In First Out (iii) Hash Tables

(A) (p) − (ii), (q) − (iii), (r) − (i)

(B) (p) − (ii), (q) − (i), (r) − (iii)

(C) (p) − (i), (q) − (ii), (r) − (iii)

(D) (p) − (i), (q) − (iii), (r) − (ii)

Page 11 of 41

Q.17 Consider the dataset with six datapoints: {(𝒙𝟏 , 𝒚𝟏 ), (𝒙𝟐 , 𝒚𝟐 ), … , (𝒙𝟔 , 𝒚𝟔 )},
1 0 0 −1 2 −2
where 𝒙𝟏 = [ ] , 𝒙𝟐 = [ ] , 𝒙𝟑 = [ ] , 𝒙𝟒 = [ ] , 𝒙𝟓 = [ ] , 𝒙𝟔 = [ ]
0 1 −1 0 2 −2
and the labels are given by 𝒚𝟏 = 𝒚𝟐 = 𝒚𝟓 = 1, and 𝒚𝟑 = 𝒚𝟒 = 𝒚𝟔 = −1. A hard
margin linear support vector machine is trained on the above dataset.

Which ONE of the following sets is a possible set of support vectors?

(A) {𝒙𝟏 , 𝒙𝟐 , 𝒙𝟓 }

(B) {𝒙𝟑 , 𝒙𝟒 , 𝒙𝟓 }

(C) {𝒙𝟒 , 𝒙𝟓 }

(D) {𝒙𝟏 , 𝒙𝟐 , 𝒙𝟑 , 𝒙𝟒 }

Q.18 Match the items in Column 1 with the items in Column 2 in the following table:

Column 1 Column 2
(p) Principal Component Analysis (i) Discriminative Model
(q) Naïve Bayes Classification (ii) Dimensionality Reduction
(r) Logistic Regression (iii) Generative Model

(A) (p) − (iii), (q) − (i), (r) − (ii)

(B) (p) − (ii), (q) − (i), (r) − (iii)

(C) (p) − (ii), (q) − (iii), (r) − (i)

(D) (p) − (iii), (q) − (ii), (r) − (i)

Page 12 of 41

Q.19 Euclidean distance based 𝑘-means clustering algorithm was run on a dataset of 100
1 −1
points with 𝑘 = 3. If the points [ ] and [ ] are both part of cluster 3, then which
1 1
ONE of the following points is necessarily also part of cluster 3?

(A) 0
[ ]
0

(B) 0
[ ]
2

(C) 2
[ ]
0

(D) 0
[ ]
1

Q.20 Given a dataset with 𝐾 binary-valued attributes (where 𝐾 > 2) for a two-class
classification task, the number of parameters to be estimated for learning a naïve
Bayes classifier is

(A) 2𝐾 + 1

(B) 2𝐾 + 1

(C) 2𝐾+1 + 1

(D) 𝐾2 + 1

Page 13 of 41

Q.21 Consider performing uniform hashing on an open address hash table with load
𝑛
factor 𝛼 = 𝑚 < 1, where 𝑛 elements are stored in the table with 𝑚 slots. The
1
expected number of probes in an unsuccessful search is at most .
1−𝛼

Inserting an element in this hash table requires at most ______ probes, on average.

(A) 1
ln ( )
1−𝛼

(B) 1
1− 𝛼

(C) 𝛼
1+
2

(D) 1
1+ 𝛼

Q.22 For any binary classification dataset, let 𝑆𝐵 ∈ ℝ𝑑×𝑑 and 𝑆𝑊 ∈ ℝ𝑑×𝑑 be the
between-class and within-class scatter (covariance) matrices, respectively. The
Fisher linear discriminant is defined by 𝑢 ∗ ∈ ℝ𝑑 , that maximizes

𝑢𝑇 𝑆𝐵 𝑢
𝐽(𝑢) =
𝑢𝑇 𝑆𝑊 𝑢

If 𝜆 = 𝐽(𝑢 ∗ ), 𝑆𝑊 is non-singular and 𝑆𝐵 ≠ 0, then (𝑢 ∗ , 𝜆) must satisfy which ONE
of the following equations?

Note: ℝ denotes the set of real numbers.

−1
(A) 𝑆𝑊 𝑆𝐵 𝑢 ∗ = 𝜆𝑢 ∗

(B) 𝑆𝑊 𝑢∗ = 𝜆𝑆𝐵 𝑢 ∗

(C) 𝑆𝐵 𝑆𝑊 𝑢 ∗ = 𝜆𝑢 ∗

(D) 𝑢 ∗ 𝑇 𝑢∗ = 𝜆2

Page 14 of 41

Q.23 Let ℎ1 and ℎ2 be two admissible heuristics used in 𝐴∗ search.

Which ONE of the following expressions is always an admissible heuristic?

(A) ℎ1 + ℎ 2

(B) ℎ1 × ℎ2

(C) ℎ1 /ℎ2 , (ℎ2 ≠ 0)

(D) |ℎ1 − ℎ2 |

Q.24 Consider five random variables 𝑈, 𝑉, 𝑊, 𝑋, and 𝑌 whose joint distribution
satisfies:

𝑃 (𝑈, 𝑉, 𝑊, 𝑋, 𝑌) = 𝑃(𝑈)𝑃(𝑉)𝑃(𝑊|𝑈, 𝑉)𝑃(𝑋|𝑊)𝑃(𝑌|𝑊)

Which ONE of the following statements is FALSE?

(A) 𝑌 is conditionally independent of 𝑉 given 𝑊

(B) 𝑋 is conditionally independent of 𝑈 given 𝑊

(C) 𝑈 and 𝑉 are conditionally independent given 𝑊

(D) 𝑌 and 𝑋 are conditionally independent given 𝑊

Page 15 of 41

Q.25 Consider the following statement:

In adversarial search, 𝛼– 𝛽 pruning can be applied to game trees of any depth where
𝛼 is the (m) value choice we have formed so far at any choice point along the
path for the MAX player and 𝛽 is the (n) value choice we have formed so far
at any choice point along the path for the MIN player.

Which ONE of the following choices of (m) and (n) makes the above statement
valid?

(A) (m) = highest, (n) = highest

(B) (m) = lowest, (n) = highest

(C) (m) = highest, (n) = lowest

(D) (m) = lowest, (n) = lowest

Q.26 Consider a database that includes the following relations:

Defender(name, rating, side, goals)

Forward(name, rating, assists, goals)

Team(name, club, price)

Which ONE of the following relational algebra expressions checks that every name
occurring in Team appears in either Defender or Forward, where 𝜙 denotes the
empty set?

(A) Π𝑛𝑎𝑚𝑒 (Team) ∖ (Π𝑛𝑎𝑚𝑒 (Defender ) ∩ Π𝑛𝑎𝑚𝑒 (Forward)) = 𝜙

(B) (Π𝑛𝑎𝑚𝑒 (Defender) ∩ Π𝑛𝑎𝑚𝑒 (Forward)) ∖ Π𝑛𝑎𝑚𝑒 (Team) = 𝜙

(C) Π𝑛𝑎𝑚𝑒 (Team) ∖ (Π𝑛𝑎𝑚𝑒 (Defender ) ∪ Π𝑛𝑎𝑚𝑒 (Forward)) = 𝜙

(D) (Π𝑛𝑎𝑚𝑒 (Defender ) ∪ Π𝑛𝑎𝑚𝑒 (Forward)) ∖ Π𝑛𝑎𝑚𝑒 (Team) = 𝜙

Page 16 of 41

Q.27 Let the minimum, maximum, mean and standard deviation values for the attribute
income of data scientists be ₹46000, ₹170000, ₹96000, and ₹21000, respective ly.
The z-score normalized income value of ₹106000 is closest to which ONE of the
following options?

(A) 0.217

(B) 0.476

(C) 0.623

(D) 2.304

Q.28 Consider the following tree traversals on a full binary tree:

(i) Preorder
(ii) Inorder
(iii) Postorder

Which of the following traversal options is/are sufficient to uniquely reconstruct
the full binary tree?

(A) (i) and (ii)

(B) (ii) and (iii)

(C) (i) and (iii)

(D) (ii) only

Page 17 of 41

Q.29 Let 𝑥 and 𝑦 be two propositions. Which of the following statements is a tautology
/are tautologies?

(A) (¬𝑥 ∧ 𝑦 ) ⟹ (𝑦 ⟹ 𝑥)

(B) (𝑥 ∧ ¬𝑦 ) ⟹ (¬𝑥 ⟹ 𝑦)

(C) (¬𝑥 ∧ 𝑦 ) ⟹ (¬𝑥 ⟹ 𝑦)

(D) (𝑥 ∧ ¬𝑦 ) ⟹ (𝑦 ⟹ 𝑥)

Q.30 Consider sorting the following array of integers in ascending order using an in-place
Quicksort algorithm that uses the last element as the pivot.

60 70 80 90 100

The minimum number of swaps performed during this Quicksort is ______.

Page 18 of 41

Q.31 Consider the following two tables named Raider and Team in a relational database
maintained by a Kabaddi league. The attribute ID in table Team references the
primary key of the Raider table, ID.

Raider
ID Name Raids RaidPoints
1 Arjun 200 250
2 Ankush 190 219
3 Sunil 150 200
4 Reza 150 190
5 Pratham 175 220
6 Gopal 193 215

Team
City ID BidPoints
Jaipur 2 200
Patna 3 195
Hyderabad 5 175
Jaipur 1 250
Patna 4 200
Jaipur 6 200

The SQL query described below is executed on this database:

SELECT *
FROM Raider, Team
WHERE Raider.ID=Team.ID AND City=“Jaipur” AND
RaidPoints > 200;

The number of rows returned by this query is ______.

Page 19 of 41

Q.32 The fundamental operations in a double-ended queue D are:

insertFirst(e) – Insert a new element e at the beginning of D.
insertLast(e) – Insert a new element e at the end of D.
removeFirst() – Remove and return the first element of D.
removeLast() – Remove and return the last element of D.

In an empty double-ended queue, the following operations are performed:

insertFirst(10)

insertLast(32)

a ←removeFirst()

insertLast(28)

insertLast(17)

a ←removeFirst()

a ← removeLast()

The value of a is ______.

Q.33 1
Let 𝑓: ℝ → ℝ be the function 𝑓(𝑥 ) = .
1+𝑒 −𝑥

The value of the derivative of 𝑓 at 𝑥 where 𝑓(𝑥 ) = 0.4 is ______
(rounded off to two decimal places).

Note: ℝ denotes the set of real numbers.

Page 20 of 41

Q.34 The sample average of 50 data points is 40. The updated sample average after
including a new data point taking the value of 142 is ______.

Q.35 1 2 3
Consider the 3 × 3 matrix 𝑴 = [3 1 3].
4 3 6

The determinant of (𝑴𝟐 + 12𝑴) is ______.

Q.36
A fair six-sided die (with faces numbered 1, 2, 3, 4, 5, 6) is repeatedly thrown
independently.

What is the expected number of times the die is thrown until two consecutive throws
of even numbers are seen?

(A) 2

(B) 4

(C) 6

(D) 8

Page 21 of 41

Q.37 Let 𝑓: ℝ → ℝ be a function. Note: ℝ denotes the set of real numbers.

−𝑥, if 𝑥 < −2
𝑓(𝑥 ) = {𝑎𝑥 2 + 𝑏𝑥 + 𝑐, if 𝑥 ∈ [−2, 2]
𝑥, if 𝑥 > 2

Which ONE of the following choices gives the values of 𝑎, 𝑏, 𝑐 that make the
function 𝑓 continuous and differentiable?

(A) 1
𝑎= , 𝑏 = 0, 𝑐 = 1
4

(B) 1
𝑎= , 𝑏 = 0, 𝑐 = 0
2

(C) 𝑎 = 0, 𝑏 = 0, 𝑐 = 0

(D) 𝑎 = 1, 𝑏 = 1, 𝑐 = −4

Page 22 of 41

Q.38 Consider the following Python code:

def count(child_dict, i):
if i not in child_dict.keys():
return 1
ans = 1
for j in child_dict[i]:
ans += count(child_dict, j)
return ans

child_dict = dict()
child_dict[0] = [1,2]
child_dict[1] = [3,4,5]
child_dict[2] = [6,7,8]
print(count(child_dict,0))

Which ONE of the following is the output of this code?

(A) 6

(B) 1

(C) 8

(D) 9

Page 23 of 41

Q.39 Consider the function computeS(X) whose pseudocode is given below:

computeS(X)
𝑆[1] ← 1
for 𝑖 ←2 to 𝑙𝑒𝑛𝑔𝑡ℎ(𝑋)
𝑆[𝑖] ← 1
if 𝑋 [𝑖 − 1] ≤ 𝑋[𝑖]
𝑆[𝑖] ← 𝑆[𝑖] + 𝑆[𝑖 − 1]
end if
end for
return S
Which ONE of the following values is returned by the function computeS(X)
for X = [6, 3, 5, 4, 10]?

(A) [1, 1, 2, 3, 4]

(B) [1, 1, 2, 3, 3]

(C) [1, 1, 2, 1, 2]

(D) [1, 1, 2, 1, 5]

Page 24 of 41

Q.40 Let 𝐹 (𝑛) denote the maximum number of comparisons made while searching for
an entry in a sorted array of size 𝑛 using binary search.

Which ONE of the following options is TRUE?

(A) 𝐹(𝑛) = 𝐹(⌊𝑛/2⌋) + 1

(B) 𝐹(𝑛) = 𝐹(⌊𝑛/2⌋) + 𝐹(⌈𝑛/2⌉)

(C) 𝐹(𝑛) = 𝐹(⌊𝑛/2⌋)

(D) 𝐹(𝑛) = 𝐹(𝑛 − 1) + 1

Q.41 Consider the following Python function:

def fun(D, s1, s2):
if s1 < s2:
D[s1], D[s2] = D[s2], D[s1]
fun(D, s1+1, s2-1)

What does this Python function fun() do? Select the ONE appropriate option
below.

(A) It finds the smallest element in D from index s1 to s2, both inclusive.

(B) It performs a merge sort in-place on this list D between indices s1 and s2, both
inclusive.

(C) It reverses the list D between indices s1 and s2, both inclusive.

(D) It swaps the elements in D at indices s1 and s2, and leaves the remaining elements
unchanged.

Page 25 of 41

Q.42 Consider the table below, where the (𝑖, 𝑗)𝑡ℎ element of the table is the distance
between points 𝑥 𝑖 and 𝑥𝑗. Single linkage clustering is performed on data points,
𝑥 1 , 𝑥 2 , 𝑥 3 , 𝑥 4 , 𝑥 5.

𝑥1 𝑥2 𝑥3 𝑥4 𝑥5
𝑥1 0 1 4 3 6
𝑥2 1 0 3 5 3
𝑥3 4 3 0 2 5
𝑥4 3 5 2 0 1
𝑥5 6 3 5 1 0

Which ONE of the following is the correct representation of the clusters produced?

(A)

(B)

(C)

(D)

Page 26 of 41

Q.43 Consider the two neural networks (NNs) shown in Figures 1 and 2, with 𝑅𝑒𝐿𝑈
activation (𝑅𝑒𝐿𝑈(𝑧) = max{0, 𝑧} , ∀𝑧 ∈ ℝ). ℝ denotes the set of real numbers. The
connections and their corresponding weights are shown in the Figures. The biases
at every neuron are set to 0. For what values of 𝑝, 𝑞, 𝑟 in Figure 2 are the two NNs
equivalent, when 𝑥 1 , 𝑥 2 , 𝑥 3 are positive?

Figure 1

Figure 2

(A) 𝑝 = 36, 𝑞 = 24, 𝑟 = 24

(B) 𝑝 = 24, 𝑞 = 24, 𝑟 = 36

(C) 𝑝 = 18, 𝑞 = 36, 𝑟 = 24

(D) 𝑝 = 36, 𝑞 = 36, 𝑟 = 36

Page 27 of 41

Q.44 Consider a state space where the start state is number 1. The successor function for
the state numbered n returns two states numbered n+1 and n+2. Assume that the
states in the unexpanded state list are expanded in the ascending order of numbers
and the previously expanded states are not added to the unexpanded state list.

Which ONE of the following statements about breadth-first search (BFS) and
depth-first search (DFS) is true, when reaching the goal state number 6?

(A) BFS expands more states than DFS.

(B) DFS expands more states than BFS.

(C) Both BFS and DFS expand equal number of states.

(D) Both BFS and DFS do not reach the goal state number 6.

Q.45 Consider the following sorting algorithms:

(i) Bubble sort
(ii) Insertion sort
(iii) Selection sort

Which ONE among the following choices of sorting algorithms sorts the numbers
in the array [4, 3, 2, 1, 5] in increasing order after exactly two passes over the array?

(A) (i) only

(B) (iii) only

(C) (i) and (iii) only

(D) (ii) and (iii) only

Page 28 of 41

Q.46 Given the relational schema 𝑅 = (𝑈, 𝑉, 𝑊, 𝑋, 𝑌, 𝑍) and the set of functional
dependencies:

{𝑈 → 𝑉, 𝑈 → 𝑊, 𝑊𝑋 → 𝑌, 𝑊𝑋 → 𝑍, 𝑉 → 𝑋}

Which of the following functional dependencies can be derived from the above
set?

(A) 𝑉𝑊 → 𝑌𝑍

(B) 𝑊𝑋 → 𝑌𝑍

(C) 𝑉𝑊 → 𝑈

(D) 𝑉𝑊 → 𝑌

Page 29 of 41

Q.47 Select all choices that are subspaces of ℝ3 .

Note: ℝ denotes the set of real numbers.

(A) 𝑥1 1 1
{𝐱 = [𝑥 2 ] ∈ ℝ3 : 𝐱 = 𝛼 [1] + 𝛽 [ 0] , 𝛼, 𝛽 ∈ ℝ}
𝑥3 0 0

(B) 𝑥1 1 1
{𝐱 = [𝑥 2 ] ∈ ℝ3 : 𝐱 = 𝛼2 [ 2] + 𝛽2 [0] , 𝛼, 𝛽 ∈ ℝ}
𝑥3 0 1

(C) 𝑥1
{𝐱 = [𝑥 2 ] ∈ ℝ3 : 5𝑥1 + 2𝑥 3 = 0, 4𝑥 1 − 2𝑥 2 + 3𝑥 3 = 0}
𝑥3

(D) 𝑥1
{𝐱 = [𝑥 2 ] ∈ ℝ3 : 5𝑥1 + 2𝑥 3 + 4 = 0}
𝑥3

Page 30 of 41

Q.48 Which of the following statements is/are TRUE?

Note: ℝ denotes the set of real numbers.

(A) There exist 𝑴 ∈ ℝ3× 3 , 𝒑 ∈ ℝ3 , and 𝒒 ∈ ℝ3 such that 𝑴𝐱 = 𝒑 has a unique
solution and M𝐱 = 𝒒 has infinite solutions.

(B) There exist 𝑴 ∈ ℝ3× 3 , 𝒑 ∈ ℝ3 , and 𝒒 ∈ ℝ3 such that 𝑴𝐱 = 𝒑 has no solutions
and M𝐱 = 𝒒 has infinite solutions.

(C) There exist 𝑴 ∈ ℝ2× 3 , 𝒑 ∈ ℝ2 , and 𝒒 ∈ ℝ2 such that 𝑴𝐱 = 𝒑 has a unique
solution and M𝐱 = 𝒒 has infinite solutions.

(D) There exist 𝑴 ∈ ℝ3× 2 , 𝒑 ∈ ℝ3 , and 𝒒 ∈ ℝ3 such that 𝑴𝐱 = 𝒑 has a unique
solution and M𝐱 = 𝒒 has no solutions.

Page 31 of 41

Q.49 Let ℝ be the set of real numbers, 𝑈 be a subspace of ℝ3 and 𝑴 ∈ ℝ3×3 be the
matrix corresponding to the projection on to the subspace 𝑈.

Which of the following statements is/are TRUE?

(A)
If 𝑈 is a 1-dimensional subspace of ℝ3 , then the null space of 𝑴 is a
1-dimensional subspace.

(B) If 𝑈 is a 2-dimensional subspace of ℝ3 , then the null space of 𝑴 is a
1-dimensional subspace.

(C) 𝑴2 = 𝑴

(D) 𝑴3 = 𝑴

Q.50 Consider the function 𝑓: ℝ → ℝ where ℝ is the set of all real numbers.

𝑥 4 2𝑥 3 3𝑥 2
𝑓( 𝑥 ) = − − +1
4 3 2
Which of the following statements is/are TRUE?

(A) 𝑥 = 0 is a local maximum of f

(B) 𝑥 = 3 is a local minimum of f

(C) 𝑥 = −1 is a local maximum of f

(D) 𝑥 = 0 is a local minimum of f

Page 32 of 41

Q.51 Consider the directed acyclic graph (DAG) below:

Which of the following is/are valid vertex orderings that can be obtained from a
topological sort of the DAG?

(A) PQRSTUV

(B) PRQVSUT

(C) PQRSVUT

(D) PR QSV TU

Page 33 of 41

Q.52 Let H, 𝐼, 𝐿, and 𝑁 represent height, number of internal nodes, number of leaf nodes,
and the total number of nodes respectively in a rooted binary tree.

Which of the following statements is/are always TRUE?

(A) 𝐿 ≤ 𝐼 +1

(B) 𝐻 + 1 ≤ 𝑁 ≤ 2𝐻+1 − 1

(C) 𝐻 ≤ 𝐼 ≤ 2𝐻 − 1

(D) 𝐻 ≤ 𝐿 ≤ 2𝐻−1

Page 34 of 41

Q.53 Consider the following figures representing datasets consisting of
two-dimensional features with two classes denoted by circles and squares.

(i) (ii)

(iii) (iv)

Which of the following is/are TRUE?

(A) (i) is linearly separable.

(B) (ii) is linearly separable.

(C) (iii) is linearly separable.

(D) (iv) is linearly separable.

Page 35 of 41

Q.54 Let game(ball, rugby) be true if the ball is used in rugby and false otherwise.

Let shape(ball, round) be true if the ball is round and false otherwise.

Consider the following logical sentences:

s1: ∀ball ¬ game(ball, rugby) ⟹shape(ball, round)

s2: ∀ball ¬ shape(ball, round) ⟹game(ball, rugby)

s3: ∀ball game(ball, rugby) ⟹ ¬ shape(ball, round)

s4: ∀ball shape(ball, round) ⟹ ¬ game(ball, rugby)

Which of the following choices is/are logical representations of the assertion,
“All balls are round except balls used in rugby”?

(A) 𝑠1 ∧ 𝑠3

(B) 𝑠1 ∧ 𝑠2

(C) 𝑠2 ∧ 𝑠3

(D) 𝑠3 ∧ 𝑠4

Page 36 of 41

Q.55 An OTT company is maintaining a large disk-based relational database of differe nt
movies with the following schema:

Movie(ID, CustomerRating)
Genre(ID, Name)
Movie_Genre(MovieID, GenreID)

Consider the following SQL query on the relation database above:

SELECT *
FROM Movie, Genre, Movie_Genre
WHERE
Movie.CustomerRating > 3.4 AND
Genre.Name = “Comedy” AND
Movie_Genre.MovieID = Movie.ID AND
Movie_Genre.GenreID = Genre.ID;

This SQL query can be sped up using which of the following indexing options?

A B + tree on all the attributes.

B Hash index on Genre.Name and B + tree on the remaining attributes.

C Hash index on Movie.CustomerRating and B + tree on the remaining
attributes.

D Hash index on all the attributes.

Q.56 Let 𝑋 be a random variable uniformly distributed in the interval [1, 3] and 𝑌 be a
random variable uniformly distributed in the interval [2, 4]. If X and Y are
independent of each other, the probability P(𝑋 ≥ 𝑌) is ______ (rounded off to
three decimal places).

Page 37 of 41

Q.57 Let 𝑋 be a random variable exponentially distributed with parameter 𝜆 > 0. The
probability density function of X is given by:
−𝜆𝑥
𝑓𝑋 (𝑥 ) = { 𝜆 𝑒 , 𝑥≥0
0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

If 5𝐸(𝑋) = 𝑉𝑎𝑟(𝑋), where 𝐸 (𝑋) and 𝑉𝑎𝑟(𝑋) indicate the expectation and variance
of 𝑋, respectively, the value of 𝜆 is ______ (rounded off to one decimal place).

Q.58 Consider two events T and S. Let 𝑇̅ denote the complement of the event T. The
probability associated with different events are given as follows:

𝑃(𝑇̅ ) = 0.6, 𝑃(𝑆|𝑇 ) = 0.3, 𝑃(𝑆 |𝑇̅ ) = 0.6

Then, 𝑃(𝑇 |𝑆 ) is ______ (rounded off to two decimal places).

Q.59 Consider a joint probability density function of two random variables X and Y

2𝑥𝑦, 0 < 𝑥 < 2, 0<𝑦<𝑥
𝑓𝑋,𝑌 (𝑥, 𝑦) = {
0, otherwise

Then, 𝐸[𝑌|𝑋 = 1.5] is ______.

Q.60 Evaluate the following limit:

ln((𝑥2 +1) cos 𝑥)
lim = ______.
𝑥→0 𝑥2

Page 38 of 41

Q.61 1
2
Let 𝒖 = 3 , and let 𝜎1 , 𝜎2 , 𝜎3 , 𝜎4 , 𝜎5 be the singular values of the matrix
4
[5]

𝑴= 𝒖𝒖𝑻 (where 𝒖𝑻 is the transpose of 𝒖). The value of ∑5𝑖=1 𝜎𝑖 is ______.

Q.62 Details of ten international cricket games between two teams “Green” and “Blue ”
are given in Table C. This table consists of matches played on different pitches,
across formats along with their winners. The attribute Pitch can take one of two
values: spin-friendly (represented as 𝑆) or pace-friendly (represented as 𝐹). The
attribute Format can take one of two values: one-day match (represented as 𝑂) or
test match (represented as 𝑇).

A cricket organization would like to use the information given in Table C to develop
a decision-tree model to predict outcomes of future games between these two teams.

To develop such a model, the computed InformationGain(C, Pitch) with respect to
the Target is ______ (rounded off to two decimal places).

Table C

Match Pitch Format Winner (Target)
Number
1 𝑆 𝑇 Green
2 𝑆 𝑇 Blue
3 𝐹 𝑂 Blue
4 𝑆 𝑂 Blue
5 𝐹 𝑇 Green
6 𝐹 𝑂 Blue
7 𝑆 𝑂 Green
8 𝐹 𝑇 Blue
9 𝐹 𝑂 Blue
10 𝑆 𝑂 Green

Page 39 of 41

Q.63 Given the two-dimensional dataset consisting of 5 data points from two classes
(circles and squares) and assume that the Euclidean distance is used to measure the
distance between two points. The minimum odd value of 𝑘 in 𝑘-nearest neighbor
algorithm for which the diamond (⋄) shaped data point is assigned the label square
is ______.

Page 40 of 41

Q.64 Given the following Bayesian Network consisting of four Bernoulli random
variables and the associated conditional probability tables:

𝑃(⋅)
𝑈=0 0.5
𝑈=1 0.5

𝑃(𝑉 = 0| ⋅) 𝑃 (𝑉 = 1| ⋅ )
𝑈= 0 0.5 0.5
𝑈=1 0.5 0.5
𝑃(𝑊 = 0| ⋅) 𝑃(𝑊 = 1| ⋅ )
𝑈=0 1 0
𝑈=1 0 1

𝑃(𝑍 = 0| ⋅) 𝑃(𝑍 = 1| ⋅ )
𝑉=0 𝑊=0 0.5 0.5
𝑉=0 𝑊=1 1 0
𝑉=1 𝑊=0 1 0
𝑉=1 𝑊=1 0.5 0.5

The value of 𝑃(𝑈 = 1, 𝑉 = 1, 𝑊 = 1, 𝑍 = 1) = ______ (rounded off to three
decimal places).

Q.65 Two fair coins are tossed independently. X is a random variable that takes a value
of 1 if both tosses are heads and 0 otherwise. Y is a random variable that takes a
value of 1 if at least one of the tosses is heads and 0 otherwise.

The value of the covariance of X and Y is ______ (rounded off to three decimal
places).

Page 41 of 41

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