Turing Prep | Previous Year Questions

Question : 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?

frown

fawn

vein

vain

Correct Answer : D

Question Type : MCQ

Max Marks : 1

Negative Marks : 0.33

Question : 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

4

3

5

6

Correct Answer : A

Question Type : MCQ

Max Marks : 1

Negative Marks : 0.33

Question : 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?

24

48

72

12

Correct Answer : B

Question Type : MCQ

Max Marks : 1

Negative Marks : 0.33

Question : 4

The sum of the following infinite series is

\(2 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{8} + \frac{1}{9} + \frac{1}{16} + \frac{1}{27} + \ldots \)

11/3

7/2

13/4

9/2

Correct Answer : B

Question Type : MCQ

Max Marks : 1

Negative Marks : 0.33

Question : 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.

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

45,000

49,500

51,750

54,000

Correct Answer : B

Question Type : MCQ

Max Marks : 1

Negative Marks : 0.33

Question : 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?

All human beings can digest dairy milk.

No human being can digest dairy milk.

Digestion of dairy milk is essential for human beings.

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

Correct Answer : D

Question Type : MCQ

Max Marks : 2

Negative Marks : 0.67

Question : 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?

3/8

1/8

1/4

1/2

Correct Answer : A

Question Type : MCQ

Max Marks : 2

Negative Marks : 0.67

Question : 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?

7.5%

10%

15%

20%

Correct Answer : B

Question Type : MCQ

Max Marks : 2

Negative Marks : 0.67

Question : 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

Correct Answer : A

Question Type : MCQ

Max Marks : 2

Negative Marks : 0.67

Question : 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 rhombus

A triangle

An ellipse

A hexagon

Correct Answer : A

Question Type : MCQ

Max Marks : 2

Negative Marks : 0.67

Question : 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?

Both (i) and (ii) are true

(i) is true and (ii) is false

(ii) is true and (i) is false

Both (i) and (ii) are false

Correct Answer : A

Question Type : MCQ

Max Marks : 1

Negative Marks : 0.33

Question : 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 ?

0

0.5

0.25

1

Correct Answer : A

Question Type : MCQ

Max Marks : 1

Negative Marks : 0.33

Question : 13

Consider the matrix \(M = \begin{bmatrix} 2 & -1 \\ 3 & 1 \\ \end{bmatrix} \).

Which ONE of the following statements is TRUE?

The eigenvalues of 𝑴 are non-negative and real.

The eigenvalues of 𝑴 are complex conjugate pairs.

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

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

Correct Answer : B

Question Type : MCQ

Max Marks : 1

Negative Marks : 0.33

Question : 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.

tree

cross

back

gray

Correct Answer : A

Question Type : MCQ

Max Marks : 1

Negative Marks : 0.33

Question : 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.

local minimum

global minimum

local maximum

global maximum

Correct Answer : A

Question Type : MCQ

Max Marks : 1

Negative Marks : 0.33

Question : 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

(q) Lookup Operation

(r) Last In First Ou

(i) Stacks

(ii) Queues

(iii) Hash Tables

(p) − (ii), (q) − (iii), (r) − (i)

(p) − (ii), (q) − (i), (r) − (iii)

(p) − (i), (q) − (ii), (r) − (iii)

(p) − (i), (q) − (iii), (r) − (ii)

Correct Answer : A

Question Type : MCQ

Max Marks : 1

Negative Marks : 0.33

Question : 17

Consider the dataset with six datapoints: {(𝒙_𝟏 , 𝒚_𝟏 ), (𝒙_𝟐 ,𝒚_𝟐 ),… , (𝒙_𝟔 ,𝒚_𝟔 )}, where 𝒙_𝟏 = \(\begin{bmatrix} 1 \\ 0 \end{bmatrix} \), 𝒙_𝟐 = \(\begin{bmatrix} 0 \\ 1 \end{bmatrix} \), 𝒙_𝟑 = \(\begin{bmatrix} 0 \\ -1 \end{bmatrix} \), 𝒙_𝟒 = \(\begin{bmatrix} -1 \\ 0 \end{bmatrix} \), 𝒙_𝟓 = \(\begin{bmatrix} 2 \\ 2 \end{bmatrix} \) , 𝒙_𝟔 = \(\begin{bmatrix} -2 \\ -2 \end{bmatrix} \) and the labels are given by 𝒚_𝟏 = 𝒚_𝟐 = 𝒚_𝟓 = 1, and 𝒚_𝟑 = 𝒚_𝟒 = 𝒚_𝟔 = −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?

{𝒙_𝟏 , 𝒙_𝟐 ,𝒙_𝟓 }

{𝒙_𝟑 , 𝒙_𝟒 ,𝒙_𝟓 }

{𝒙_𝟒 , 𝒙_𝟓 }

{𝒙_𝟏 , 𝒙_𝟐 ,𝒙_𝟑 , 𝒙_𝟒 }

Correct Answer : D

Question Type : MCQ

Max Marks : 1

Negative Marks : 0.33

Question : 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

(q) Naïve Bayes Classification (r)

Logistic Regression

(i) Discriminative Model

(ii) Dimensionality Reduction

(iii) Generative Model

(p) − (iii), (q) − (i), (r) − (ii)

(p) − (ii), (q) − (i), (r) − (iii)

(p) − (ii), (q) − (iii), (r) − (i)

(p) − (iii), (q) − (ii), (r) − (i)

Correct Answer : C

Question Type : MCQ

Max Marks : 1

Negative Marks : 0.33

Question : 19

Euclidean distance based 𝑘-means clustering algorithm was run on a dataset of 100 points with 𝑘 = 3. If the points \(\begin{bmatrix} 1 \\ 1 \end{bmatrix} \) and \(\begin{bmatrix} -1 \\ 1 \end{bmatrix} \) are both part of cluster 3, then which ONE of the following points is necessarily also part of cluster 3?

\(\begin{bmatrix} 0 \\ 0 \end{bmatrix} \)

\(\begin{bmatrix} 0 \\ 2 \end{bmatrix} \)

\(\begin{bmatrix} 2 \\ 0 \end{bmatrix} \)

\(\begin{bmatrix} 0 \\ 1 \end{bmatrix} \)

Correct Answer : D

Question Type : MCQ

Max Marks : 1

Negative Marks : 0.33

Question : 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

2^𝐾 + 1

2𝐾 + 1

2^𝐾+1 + 1

𝐾² + 1

Correct Answer : B

Question Type : MCQ

Max Marks : 1

Negative Marks : 0.33

Question : 21

Consider performing uniform hashing on an open address hash table with load factor \(\alpha = \frac{n}{m} < 1 \), where 𝑛 elements are stored in the table with 𝑚 slots. The expected number of probes in an unsuccessful search is at most \(\frac{1}{1 - \alpha} \) . Inserting an element in this hash table requires at most ______ probes, on average.

\(\ln\left(\frac{1}{1 - \alpha}\right) \)

\(\frac{1}{1 - \alpha} \)

\(1 + \frac{\alpha}{2} \)

\(\frac{1}{1 + \alpha} \)

Correct Answer : B

Question Type : MCQ

Max Marks : 1

Negative Marks : 0.33

Question : 22

For any binary classification dataset, let 𝑆_𝐵 ∈ ℝ^𝑑×𝑑 and 𝑆_𝑊 ∈ ℝ^𝑑×𝑑 be the between-class and within-class scatter (covariance) matrices, respectively. The Fisher linear discriminant is defined by 𝑢^∗ ∈ ℝ^𝑑 , that maximizes

\(J(u) = \frac{u^T S_B u}{u^T S_W u} \)

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

Note: ℝ denotes the set of real numbers.

\(S_W^{-1} S_B u^* = \lambda u^* \)

\(S_W u^* = \lambda S_B u^* \)

\(S_B S_W u^* = \lambda u^* \)

\(u^{*T} u^* = \lambda^2 \)

Correct Answer : A

Question Type : MCQ

Max Marks : 1

Negative Marks : 0.33

Question : 23

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

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

ℎ₁ + ℎ₂

ℎ₁ × ℎ₂

ℎ₁ /ℎ₂ , (ℎ₂ ≠ 0)

|ℎ₁ − ℎ₂ |

Correct Answer : D

Question Type : MCQ

Max Marks : 1

Negative Marks : 0.33

Question : 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?

𝑌 is conditionally independent of 𝑉 given 𝑊

𝑋 is conditionally independent of 𝑈 given 𝑊

𝑈 and 𝑉 are conditionally independent given 𝑊

𝑌 and 𝑋 are conditionally independent given 𝑊

Correct Answer : C

Question Type : MCQ

Max Marks : 1

Negative Marks : 0.33

Question : 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?

(m) = highest, (n) = highest

(m) = lowest, (n) = highest

(m) = highest, (n) = lowest

(m) = lowest, (n) = lowest

Correct Answer : C

Question Type : MCQ

Max Marks : 1

Negative Marks : 0.33

Question : 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?

Π_{𝑛𝑎𝑚𝑒} (Team) ∖ (Π_{𝑛𝑎𝑚𝑒} (Defender)∩ Π_{𝑛𝑎𝑚𝑒} (Forward)) = 𝜙

(Π_{𝑛𝑎𝑚𝑒}(Defender)∩ Π_{𝑛𝑎𝑚𝑒} (Forward)) ∖ Π_{𝑛𝑎𝑚𝑒} (Team) = 𝜙

Π_{𝑛𝑎𝑚𝑒} (Team) ∖ (Π_{𝑛𝑎𝑚𝑒} (Defender)∪ Π_{𝑛𝑎𝑚𝑒} (Forward)) = 𝜙

(Π_{𝑛𝑎𝑚𝑒} (Defender)∪ Π_{𝑛𝑎𝑚𝑒} (Forward)) ∖ Π_{𝑛𝑎𝑚𝑒} (Team) = 𝜙

Correct Answer : C

Question Type : MCQ

Max Marks : 1

Negative Marks : 0.33

Question : 27

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

0.217

0.476

0.623

2.304

Correct Answer : B

Question Type : MCQ

Max Marks : 1

Negative Marks : 0.33

Question : 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?

(i) and (ii)

(ii) and (iii)

(i) and (iii)

(ii) only

Correct Answer : ABC

Question Type : MSQ

Max Marks : 1

Negative Marks : 0

Question : 29

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

(¬𝑥 ∧ 𝑦 ) ⟹ (𝑦 ⟹ 𝑥)

(𝑥 ∧ ¬𝑦 ) ⟹ (¬𝑥 ⟹ 𝑦)

(¬𝑥 ∧ 𝑦 ) ⟹ (¬𝑥 ⟹ 𝑦)

(𝑥 ∧ ¬𝑦 ) ⟹ (𝑦 ⟹ 𝑥)

Correct Answer : BCD

Question Type : MSQ

Max Marks : 1

Negative Marks : 0

Question : 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 ______.

Correct Answer : 0

Question Type : NAT

Max Marks : 1

Negative Marks : 0

Question : 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 ______.

Correct Answer : 3

Question Type : NAT

Max Marks : 1

Negative Marks : 0

Question : 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 ______.

Correct Answer : 17

Question Type : NAT

Max Marks : 1

Negative Marks : 0

Question : 33

Let 𝑓:ℝ → ℝ be the function \(f(x) = \frac{1}{1 + e^{-x}} \) .

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.

Correct Answer : 0.24 to 0.24

Question Type : NAT

Max Marks : 1

Negative Marks : 0

Question : 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 ______.

Correct Answer : 42

Question Type : NAT

Max Marks : 1

Negative Marks : 0

Question : 35

Consider the 3 × 3 matrix \(M = \begin{bmatrix} 1 & 2 & 3 \\ 3 & 1 & 3 \\ 4 & 3 & 6 \end{bmatrix} \).

The determinant of (𝑴^𝟐 + 12𝑴) is ______.

Correct Answer : 0

Question Type : NAT

Max Marks : 1

Negative Marks : 0

Question : 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?

2

4

6

8

Correct Answer : C

Question Type : MCQ

Max Marks : 2

Negative Marks : 0.67

Question : 37

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

\(f(x) = \begin{cases} -x, & \text{if } x < -2 \\ ax^2 + bx + c, & \text{if } x ∈ [−2,2] \\ x, & \text{if } x > 2 \end{cases} \)

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

𝑎 = \(\frac{1}{4} \), 𝑏 = 0, 𝑐 = 1

𝑎 = \(\frac{1}{2} \), 𝑏 = 0, 𝑐 = 0

𝑎 = 0, 𝑏 = 0, 𝑐 = 0

𝑎 = 1, 𝑏 = 1, 𝑐 = −4

Correct Answer : A

Question Type : MCQ

Max Marks : 2

Negative Marks : 0.67

Question : 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?

6

1

8

9

Correct Answer : D

Question Type : MCQ

Max Marks : 2

Negative Marks : 0.67

Question : 39

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

Which ONE of the following values is returned by the function computeS(X) for X = [6, 3, 5, 4, 10]?

[1, 1, 2, 3, 4]

[1, 1, 2, 3, 3]

[1, 1, 2, 1, 2]

[1, 1, 2, 1, 5]

Correct Answer : C

Question Type : MCQ

Max Marks : 2

Negative Marks : 0.67

Question : 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?

𝐹(𝑛) = 𝐹(⌊𝑛/2⌋) + 1

𝐹(𝑛) = 𝐹(⌊𝑛/2⌋) + 𝐹(⌈𝑛/2⌉)

𝐹(𝑛) = 𝐹(⌊𝑛/2⌋)

𝐹(𝑛) = 𝐹(𝑛 − 1) + 1

Correct Answer : A

Question Type : MCQ

Max Marks : 2

Negative Marks : 0.67

Question : 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.

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

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

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

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

Correct Answer : C

Question Type : MCQ

Max Marks : 2

Negative Marks : 0.67

Question : 42

Consider the table below, where the (𝑖,𝑗)^𝑡ℎ element of the table is the distance between points 𝑥𝑖 and 𝑥𝑗 . Single linkage clustering is performed on data points, 𝑥₁ ,𝑥₂ , 𝑥₃ ,𝑥₄ , 𝑥₅ .

	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?

Correct Answer : A

Question Type : MCQ

Max Marks : 2

Negative Marks : 0.67

Question : 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 𝑥₁ ,𝑥₂ , 𝑥₃ are positive?

𝑝 = 36, 𝑞 = 24,𝑟 = 24

𝑝 = 24, 𝑞 = 24,𝑟 = 36

𝑝 = 18, 𝑞 = 36,𝑟 = 24

𝑝 = 36, 𝑞 = 36,𝑟 = 36

Correct Answer : A

Question Type : MCQ

Max Marks : 2

Negative Marks : 0.67

Question : 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?

BFS expands more states than DFS.

DFS expands more states than BFS.

Both BFS and DFS expand equal number of states.

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

Correct Answer : C

Question Type : MCQ

Max Marks : 2

Negative Marks : 0.67

Question : 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?

(i) only

(iii) only

(i) and (iii) only

(ii) and (iii) only

Correct Answer : B

Question Type : MCQ

Max Marks : 2

Negative Marks : 0.67

Question : 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?

𝑉𝑊 → 𝑌𝑍

𝑊𝑋 → 𝑌𝑍

𝑉𝑊 → 𝑈

𝑉𝑊 → 𝑌

Correct Answer : ABD

Question Type : MSQ

Max Marks : 2

Negative Marks : 0

Question : 47

Select all choices that are subspaces of ℝ³ .

Note: ℝ denotes the set of real numbers.

\(\{ \mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \in \mathbb{R}^3 : \mathbf{x} = \alpha \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} + \beta \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \, \alpha, \beta \in \mathbb{R} \} \)

\(\{ \mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \in \mathbb{R}^3 : \mathbf{x} = \alpha^2 \begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix} + \beta^2 \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}, \, \alpha, \beta \in \mathbb{R} \} \)

\(\{ \mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \in \mathbb{R}^3 : 5x_1 + 2x_3 = 0, \, 4x_1 - 2x_2 + 3x_3 = 0 \} \)

\(\{ \mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \in \mathbb{R}^3 : 5x_1 + 2x_3 + 4 = 0 \} \)

Correct Answer : AC

Question Type : MSQ

Max Marks : 2

Negative Marks : 0

Question : 48

Which of the following statements is/are TRUE?

Note: ℝ denotes the set of real numbers.

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

There exist 𝑴 ∈ ℝ^3×3 , 𝒑 ∈ ℝ³ , and 𝒒 ∈ ℝ³ such that 𝑴𝐱 = 𝒑 has no solutions and M𝐱 = 𝒒 has infinite solutions

There exist 𝑴 ∈ ℝ^2×3 , 𝒑 ∈ ℝ² , and 𝒒 ∈ ℝ² such that 𝑴𝐱 = 𝒑 has a unique solution and M𝐱 = 𝒒 has infinite solutions.

There exist 𝑴 ∈ ℝ^3×2 , 𝒑 ∈ ℝ³ , and 𝒒 ∈ ℝ³ such that 𝑴𝐱 = 𝒑 has a unique solution and M𝐱 = 𝒒 has no solutions.

Correct Answer : BD

Question Type : MSQ

Max Marks : 2

Negative Marks : 0

Question : 49

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

Which of the following statements is/are TRUE?

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

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

𝑴² = 𝑴

𝑴³ = 𝑴

Correct Answer : BCD

Question Type : MSQ

Max Marks : 2

Negative Marks : 0

Question : 50

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

\(f(x) = \frac{x^4}{4} - \frac{2x^3}{3} - \frac{3x^2}{2} + 1 \)

Which of the following statements is/are TRUE?

𝑥 = 0 is a local maximum of f

𝑥 = 3 is a local minimum of f

𝑥 = −1 is a local maximum of f

𝑥 = 0 is a local minimum of f

Correct Answer : AB

Question Type : MSQ

Max Marks : 2

Negative Marks : 0

Question : 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?

P Q R S T U V

P R Q V S U T

P Q R S V U T

P R Q S V T U

Correct Answer : BD

Question Type : MSQ

Max Marks : 2

Negative Marks : 0

Question : 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?

𝐿 ≤ 𝐼 + 1

𝐻 + 1 ≤ 𝑁 ≤ 2^𝐻+1 − 1

𝐻 ≤ 𝐼 ≤ 2^𝐻 − 1

𝐻 ≤ 𝐿 ≤ 2^𝐻−1

Correct Answer : ABC

Question Type : MSQ

Max Marks : 2

Negative Marks : 0

Question : 53

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

Which of the following is/are TRUE?

(i) is linearly separable.

(ii) is linearly separable.

(iii) is linearly separable.

(iv) is linearly separable.

Correct Answer : AD

Question Type : MSQ

Max Marks : 2

Negative Marks : 0

Question : 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”?

𝑠1 ∧ 𝑠3

𝑠1 ∧ 𝑠2

𝑠2 ∧ 𝑠3

𝑠3 ∧ 𝑠4

Correct Answer : AC

Question Type : MSQ

Max Marks : 2

Negative Marks : 0

Question : 55

An OTT company is maintaining a large disk-based relational database of different
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?

B⁺ tree on all the attributes.

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

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

Hash index on all the attributes

Correct Answer : AB

Question Type : MSQ

Max Marks : 2

Negative Marks : 0

Question : 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).

Correct Answer : 0.125 to 0.125

Question Type : NAT

Max Marks : 2

Negative Marks : 0

Question : 57

Let 𝑋 be a random variable exponentially distributed with parameter 𝜆 > 0. The probability density function of X is given by:

\(f_X(x) = \begin{cases} \lambda e^{-\lambda x}, & \text{if } x \geq 0 \\ 0, & \text{otherwise} \end{cases} \)

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).

Correct Answer : 0.2 to 0.2

Question Type : NAT

Max Marks : 2

Negative Marks : 0

Question : 58

Consider two events T and S. Let \(\overline T\) denote the complement of the event T. The probability associated with different events are given as follows:

\(𝑃(\overline T) = 0.6, 𝑃(𝑆|𝑇) = 0.3, 𝑃(𝑆|\overline T) = 0.6\)

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

Correct Answer : 0.25 to 0.25

Question Type : NAT

Max Marks : 2

Negative Marks : 0

Question : 59

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

\(f_{X,Y}(x, y) = \begin{cases} 2xy, & \text{if } 0 < x < 2 \text{ and } 0 < y < x \\ 0, & \text{otherwise} \end{cases} \)

Then, 𝐸[𝑌|𝑋 = 1.5] is ______.

Correct Answer : MTA

Question Type : NAT

Max Marks : 2

Negative Marks : 0

Question : 60

Evaluate the following limit:

\(\lim_{{x \to 0}} \frac{{\ln((x^2 + 1) \cos x)}}{{x^2}} \)

Correct Answer : 0.5

Question Type : NAT

Max Marks : 2

Negative Marks : 0

Question : 61

Let 𝒖 = \(\begin{bmatrix} 1 \\ 2 \\ 3 \\ 4 \\ 5 \\ \end{bmatrix} \) , and let 𝜎₁ , 𝜎₂ ,𝜎₃ , 𝜎₄ , 𝜎₅ be the singular values of the matrix

𝑴= 𝒖𝒖^𝑻 (where 𝒖^𝑻 is the transpose of 𝒖). The value of \(\sum_{i=1}^{5} \sigma_i \) is ______.

Correct Answer : 55

Question Type : NAT

Max Marks : 2

Negative Marks : 0

Question : 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 Number	Pitch	Format	Winner (Target)
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

Correct Answer : 0.12 to 0.13

Question Type : NAT

Max Marks : 2

Negative Marks : 0

Question : 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 ______.

Correct Answer : 5

Question Type : NAT

Max Marks : 2

Negative Marks : 0

Question : 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).

Correct Answer : 0.125 to 0.125

Question Type : NAT

Max Marks : 2

Negative Marks : 0

Question : 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).

Correct Answer : 0.062 to 0.063

Question Type : NAT

Max Marks : 2

Negative Marks : 0

GATE DA - 2024