GATE 2026 Question Paper Data Science and Artificial Intelligence (DA)

FOR GATE EXAM PREPARATION

GATE 2026
Question Paper · Data
Science Artificial
Intelligence (DA)
EXAM YEAR TYPE SUBJECT

GATE 2026 Question Paper Data Science Artificial Intelligence (DA)

Notes · Sample Papers · Previous Year Papers · Mock Tests

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Data Science & Artificial Intelligence (DA)

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

Q. 1 Verbosity : Brevity :: Insolence :___________

Choose the word that best fills the blank.
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(A)
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(B) Respect

(C) Solace

(D) Wealth
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Q.2
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The product of the digits ofaagthree-digit number is 70. The sum of the digits of
this three-digit number is _____

(A) 12

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Data Science & Artificial Intelligence (DA)

Q.3 The four pieces of a puzzle are shown in the figure below.

Which one of the figures labelled as P, Q, R, and S can be constructed by using
each of the four pieces only once without overlaps?

P Q

R S

(A) P

(B) Q

(C) R

(D) S

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Q.4 Consider two distinct positive real numbers 𝑚, 𝑛, with 𝑚 > 𝑛.
Let 𝑥 = 𝑛log10(𝑚) and 𝑦 = 𝑚log10(𝑛) . The relation between 𝑥 and 𝑦 is _______.

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(C) a 𝑥 = 𝑦

(D) 𝑥 = log10 (𝑦)

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Q.5
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‘If his latest movie had been a commercial success, the actor would have made
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enough money to sponsor his next movie.’
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Based only on the above sentence,

(A) The actor will certainly sponsor his next movie.

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(B) His latest movie was a commercial success.

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Data Science & Artificial Intelligence (DA)

Q.6 – Q.10 Carry TWO marks Each

Q.6 ‘My friend and I parted __ the door __ the cabin that I had rented __ the night.’

Choose the option with the correct sequence of words to fill the blanks.

(A) at; of; for

(B) for; at; of

(C) of; for; in

(D) in; of; for

Q.7 Five integers are picked from 0 to 20, with possible repetitions, such that their
mean is 12, median is 18, and they have a single mode of 20.

Ignoring permutations, the number of ways to pick these five integers is _____

(A) 0

(B) 1

(C) 2

(D) 3

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Q.8 Rishi and Swathi are students of Class 5. Pavan and Tanvi are students of Class 4.
Rishi and Pavan are boys. Swathi and Tanvi are girls. The four students played a
total of three games of chess. The games were played one after another. A player
who lost a game did not participate in any more games. It was observed that:

(i) the first game was the only game where two students of the same class
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(ii)
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(A) Pavan

(B) Rishi
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Data Science & Artificial Intelligence (DA)

Q.9 𝑃, 𝑄, 𝑅, 𝑆, 𝑋, and 𝑌 are distinct single-digit whole numbers taking values from
0 to 9.

𝑃𝑄 is a two-digit number with 𝑄 being in the units place and 𝑃 in the tens place.
Similarly, 𝑅𝑆 is a two-digit number.

It is known that 𝑃𝑄 and 𝑅𝑆 are consecutive numbers and
(𝑃𝑄)2 + (𝑅𝑆)2 = 𝑋𝑌𝑃, with 𝑋𝑌𝑃 being a three-digit number.

The value of 𝑌 is __________

(A) 4

(B) 5

(C) 6

(D) 7

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Q.10 In the given figure, 𝑃, 𝑄, and 𝑅 are three points on a circle of radius 10 cm
with 𝑂 as its center, ̅̅̅̅ ̅̅̅̅, and ∠𝑃𝑄𝑅 = 45°. The figure is representative.
𝑃𝑄 = 𝑅𝑄
The area of the shaded region 𝑃𝑄𝑅𝑂 is ______________ cm2 .

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(B) 25√2

(C) 50√2

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(D) 100
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Q.11 – Q.35 Carry ONE mark Each

Q.11 For a classification problem, Principal Component Analysis (PCA) has been used
to reduce the dimensionality of a feature space from 100 to 10.

Which of the following options is true about the angle 𝜃 between the first and the
tenth principal components?

(A) 𝜃 = 0°

(B) 𝜃 = 90°

(C) 90° < 𝜃 ≤ 180°

(D) 0 < 𝜃 < 90°

Consider that you are training a classifier for a 10-class classification problem.
Q.12
Each input is represented as a 512-dimensional vector. There are 1000 samples,
out of which first 100 will be used for testing. Let Leave-One-Out-Cross-
Validation (LOOCV) be used for selection of the classifier model before testing.
Which of the following options is the correct number of validation splits that will
be generated?

10
(A)

512
(B)

900
(C)

1000
(D)

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Q.13 Which of the following algorithms is NOT an example of uninformed search?

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(A) Breadth First Search

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Depth First. Search
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(D) Depth-limited Search

Which of the following statements is NOT true?
Q.14
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(The names of the predicates are intuitive.)

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(A) ∀𝑥 ∀𝑦 𝐶𝑙𝑎𝑠𝑠𝑚𝑎𝑡𝑒(𝑥, 𝑦) ⇒ 𝐶𝑙𝑎𝑠𝑠𝑚𝑎𝑡𝑒(𝑦, 𝑥)

(B) ∀𝑥 𝐿𝑖𝑘𝑒𝑠(𝑥, 𝐼𝑐𝑒𝑐𝑟𝑒𝑎𝑚) ⇒ ¬∃𝑥 ¬𝐿𝑖𝑘𝑒𝑠(𝑥, 𝐼𝑐𝑒𝑐𝑟𝑒𝑎𝑚)

“Each king is a person” is equivalent to ∀𝑥 𝐼𝑠𝐾𝑖𝑛𝑔(𝑥) ⋀ 𝐼𝑠𝑃𝑒𝑟𝑠𝑜𝑛(𝑥)
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“All humans are mortal” is equivalent to ∀𝑥 𝐼𝑠𝐻𝑢𝑚𝑎𝑛(𝑥) ⇒ 𝐼𝑠𝑀𝑜𝑟𝑡𝑎𝑙(𝑥)
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Q.15 Consider that the quick sort algorithm is used to sort an array of n distinct
randomly ordered elements. In every call, the pivot is chosen as the first element
of the current subarray.

Let 𝑇(𝑛) denote the expected time to sort the array. Assume that the time to
partition is linear in the size of the current subarray.

Which of the following recurrence relations correctly represents 𝑇(𝑛) in this
scenario?

(A) 𝑇(𝑛) = 𝑇(1) + 𝑇(𝑛 − 1) + 𝑂(𝑛)

(B) 𝑛 3𝑛
𝑇(𝑛) = 𝑇 ( ) + 𝑇 ( ) + 𝑂(𝑛)
4 4

(C) 𝑛
𝑇(𝑛) = 2𝑇 ( ) + 𝑂(𝑛)
2

1
(D) 𝑇(𝑛) = ∑𝑛−1
𝑘=0[𝑇(𝑘) + 𝑇(𝑛 − 𝑘 − 1)] + 𝑂(𝑛)
𝑛

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Q.16 Consider the given Python program.

def append_to_lst(val, lst=[]):
lst.append(val)
return lst
print(append_to_lst(1))
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print(append_to_lst(2))
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print(append_to_lst(3, []))
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Which ofmthe following is the correct output of this program?
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(A)
[1]
[2]
[3]

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

(D)
[1]
[1, 2]
[1, 3]
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Q.17 Let R(A,B,C,D,E) be a relational schema with functional dependency set
F={ABC, CDE, EA}.

Which of the following statements is correct?

(A) AD, ED and CD are the only candidate keys of R.

(B) AD and ED are the only candidate keys of R.

(C) A, E and CD are the only candidate keys of R.

(D) A and CD are the only candidate keys of R.

Q.18 Consider that the visualization of a 3-dimensional data cube is showing Sales
Quantity for each combination of the attributes Product Type, Month and Country.

From this, if we want to further visualize the Sales Quantity for each combination
of Product Type, Month and State, which of the following OLAP operations
should be performed?

(A) Slicing

(B) Dicing

(C) Roll-up

(D) Drill-down

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Q.19 Let 𝑀 be a randomly chosen non-empty subset of 𝑆 = {1,2,3, … ,2026}.

Which of the following is the probability that the product of all the elements of 𝑀
is even?

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1013 (21013 2026
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(A)
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(B)
a 2 1013 2026

(C) 21013 (21013 − 1)/(22026 − 1)

(D) 1/(22026 − 1)

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Q.20
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Suppose that a computer program provides a non-negative and integer-valued
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random solution to the equation 𝑛1 + 𝑛2 + 𝑛3 + 𝑛4 = 20.
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a probability that all of 𝑛 , 𝑛 , 𝑛 , 𝑛 in the provided
Which of the following is the 1 2 3 4
solution are positive?

(A) (19)/(23)
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Q.21 𝑐𝑜𝑠 𝜃 − 𝑠𝑖𝑛 𝜃 2π
Let 𝑀 = ( ) be a 2 × 2 matrix, where 𝜃 = , and
𝑠𝑖𝑛 𝜃 𝑐𝑜𝑠 𝜃 5

1 0
𝐼2 = ( ).
0 1
Which of the following options is equal to 𝑀2026 ?

(A) 𝑀2

(B) 𝑀

(C) 𝑀−1

(D) 𝐼2

Q.22 Consider a set 𝑆1 = {𝑥 = (𝑥1 , 𝑥2 , 𝑥3 )𝑇 ∈ ℝ3 | 𝑥 𝑇 𝑥 ≤ 16}. Let 𝑆2 be another set
which is a subspace of ℝ3 with dimension two.

Which of the following gives the area of 𝑆1 ∩ 𝑆2 ?

(A) 16𝜋

(B) 4𝜋

(C) 4𝜋 2

(D) 16𝜋 2

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Q.23 In the following table, the Task column lists a few tasks related to machine
learning. The Algorithm column lists a few algorithms.

Each entry “t” from the Task column is to be matched with an appropriate entry
“a” from the Algorithm column such that the task “t” can be solved using the
algorithm “a”. Denote such a match as t:a
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Task Algorithm
T1 – Clustering
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A1 – Markov Chain Monte Carlo
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a T4 – Feature Extraction A4 – Naive Bayes

Which of the following options is/are the correct matching(s)?

(A) T1:A4, T2:A3, T3:A1, T4:A2
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(B) T1:A2, T2:A4, T3:A1, T4:A3
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(C)
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(D) T1:A4, T2:A2, T3:A1, T4:A3

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Q.24 Sentence 𝑋 is said to entail Sentence 𝑌 if whenever 𝑋 is 𝑇𝑅𝑈𝐸, 𝑌 also must hold
𝑇𝑅𝑈𝐸.

Which of the following statements is/are correct if 𝑋 entails 𝑌?

(A) 𝑋 ⇒𝑌

(B) 𝑋 ⋀¬𝑌 is 𝐹𝐴𝐿𝑆𝐸

(C) if 𝑋 then 𝑌

(D) if 𝑌 then 𝑋

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Q.25 You are given the following Pre-order and In-order traversals of a Binary Tree T
with nodes E, F, G, P, Q, R, S.

Pre-order: P Q S E R F G

In-order: S Q E P F R G
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Which of the following statements is/are true about the Binary Tree T?
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(A) a Node P is the root of T

(B) The Post-order traversal of T is:

S E Q F G R P

Node Q has only one child
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The left subtree of node R contains the node G

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Q.26 Consider two relations r and s defined on the relational schemas R(A,B) and
S(E,C), respectively. A is the primary key of R and E is a foreign key of S
referencing A in R.

Which of the following operations will NEVER violate the foreign key
constraint?

(A) Inserting records into relation r

(B) Deleting records from relation s

(C) Deleting records from relation r

(D) Inserting records into relation s

Q.27 Let 𝑓(𝑥) = 𝑥 3 − 3𝑥 2 + 2 be a function defined on (−1, 3].

Which of the following statements is/are correct?

(A) 𝑓(𝑥) has exactly two roots in [−0.9, 0].

(B) 𝑓(𝑥) has a minimum at 2 only.

(C) 𝑓(𝑥) has maximum at 0 only.

(D) 𝑓(𝑥) has a root at 1.

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Q.28 Suppose a random variable 𝑍 follows 𝑁𝑜𝑟𝑚𝑎𝑙(𝜇 = 0, 𝜎 2 = 1) distribution with
probability density function 𝑔(𝑧) and cumulative distribution function 𝐺(𝑧).
Another random variable 𝑌 follows 𝑡1 distribution with probability density
function ℎ(𝑦) and cumulative distribution function 𝐻(𝑦). Let 𝑐 be the positive
real number for which 𝑔(𝑐) = ℎ(𝑐).

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Which of the following statements is/are correct?
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(A) a 𝐺(0) = 𝐻(0)
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(B) 𝐺(𝑐) < 𝐻(𝑐)

(C) 𝐺(−𝑐) < 𝐻(−𝑐)

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(D) 𝑔(0) = ℎ(0)

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Q.29 Consider that for a supervised learning task, the objective function being
minimized is 𝑓𝑤 (𝑥) = 𝑤𝑥, where 𝑥 ∈ ℝ is the input and 𝑤 ∈ ℝ is the parameter.
Stochastic Gradient Descent with learning rate of 0.10 is used for parameter
updates.

Suppose that at the end of iteration 𝑖, the value of 𝑤 becomes 10.00.

Let 𝑥 = 10.00 be the input for iteration (𝑖 + 1).

The value of 𝑤 at the end of iteration (𝑖 + 1) is __________ . (Rounded off to
two decimal places)

Q.30 Consider the game tree for a two-player turn-taking minimax game as shown in
the figure. The value of a terminal node represents the utility of the game state if
the game ends there. The numbers written next to the edges denote the strategies.

There are two players MAX and MIN. At any particular state of the game, MAX
prefers to move to a state of maximum value. On the other hand, MIN prefers to
move to a state of minimum value.

Suppose MAX starts the game at the root and has three strategies: 1, 2 and 3.
Next, MIN plays and also has three strategies: 1, 2 and 3. The game ends there.
Both players always take optimal strategies throughout the game.

At the root, the best strategy for MAX is ___________ . (Answer in integer)

Q.31 Let A be a sorted array containing 1000 distinct integers. You perform a recursive
binary search on A to find an element y. Suppose each comparison checks whether
the middle element computed during the current recursive step is equal to, less
than, or greater than y.

The maximum number of comparisons that may have to be performed if y is not
an element of A is _______ . (Answer in integer)

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Q.32 In a relational database, a B+ Tree Index is to be constructed for a relation on a
key field. In a B+ Tree, a Node Pointer points to a sub-tree and a Data Record
Pointer points to a block of database records.

Let, Node size = 4096 bytes, Node Pointer size = 10 bytes, Search Key Field size
= 11 bytes and Data Record Pointer size = 12 bytes.
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The maximum number of Node Pointers that can be present in a non-leaf node of
the B+ Tree is ________ . (Answer in integer)
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l asnumber of bijections 𝑓(⋅) from the set 𝑆 = {1, 2, 3, 4} to itself such that ag
ag𝑓(𝑓(𝑛)) = 𝑛, for all 𝑛 ∈ 𝑆, is __________ . (Answer in integer)
Q.33 The

Q.34 Let 𝑋 be an exponentially distributed random variable with mean 𝜆(> 0). If
𝑃(𝑋 > 5) = 0.35, then the conditional probability 𝑃(𝑋 > 10|𝑋 > 5) is
___________ . (Rounded off to two decimal places)

Q.35 The value of ∑∞ ∞ −𝑖 −𝑗
𝑖=0 ∑𝑗=1 2 3 is ______________ . (Answer in integer)
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Q.36 – Q.65 Carry TWO marks Each

Q.36 Let four points in three-dimensional space be:

P1: [2, 3, −1], P2: [3, 1, 1], P3: [5, −2, 3] and P4: [3, 3, 3].

Hierarchical Agglomerative Clustering is used to cluster the above points. If
Manhattan Distance is used as the distance metric during clustering, which of the
following options indicates the two points that will be merged first?

(A) 𝑃1, 𝑃2

(B) 𝑃2, 𝑃3

(C) 𝑃3, 𝑃4

(D) 𝑃2, 𝑃4

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Q.37 Which of the following statements is true for Ridge Regression?

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(A) The regularizer in the objective function of Ridge Regression is used to guard
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against scenarios where the model works well for the test data, but poorly for the
training data.
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(B)
g la regularizer of Ridge Regression uses 𝐿 norm.
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(C) Ridge Regression aims to reduce the number of parameters that have negative
values.

(D) The regularizer of Ridge Regression may increase the bias of the model, but it
helps in reducing the variance in predictions.

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s e
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Assume that a Creative (𝐶) person will Succeed (𝑆) if the person is also
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Q.38
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Disciplined (𝐷), but will not succeed otherwise. Now, consider the following
statements:
(i) 𝐶 ⋀ 𝑆 ⇔𝐷
(ii) 𝐶 ⇒ (𝑆 ⟺ 𝐷)
(iii) 𝐶 ⇔ ((𝐷 ⇒ 𝑆) ∨ ¬𝑆)
Which of the following options is correct?

m
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c. o(A) Both (i) and (ii) are TRUE
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m s e
s e g l a
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a
a
Both (ii) and (iii) are TRUE
(C)

Only (iii) is TRUE
(D)

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Q.39 A recursive function in Python is given.

def mystery(n):
if n <= 0:
return 1
else:
return mystery(n-1) + mystery(n-2)

Now, consider the following function call:

mystery(4)

Assume that a typical runtime stack is used to manage function calls. Each
function call is pushed onto the stack and removed only after it finishes execution.

Which of the following options denotes the total number of function calls (i.e., the
total number of stack activations), including the initial call, to compute
mystery(4)?

(A) 5

(B) 9

(C) 15

(D) 17

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Q.40 Consider a directed graph 𝐺 = (𝑉, 𝐸), where 𝑉 is the finite set of vertices and 𝐸 is
the set of directed edges between the vertices. 𝐺 may contain cycles but there is no
self-loop. Further, 𝐺 may not be strongly connected.

Let 𝐺 𝑅 be the graph obtained by reversing the directions of all the edges in 𝐺

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without changing the set of vertices.

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Assume that Breadth First Search (BFS) or Depth First Search (DFS) from any
given vertex 𝑣 of a graph visits only the reachable vertices from 𝑣 in that graph.
e m
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Which

ag 𝐺?

(A) If 𝑢 is a reachable vertex in the BFS of 𝐺 𝑅 from 𝑣, then 𝑢 is also a reachable
vertex in the DFS of 𝐺 from 𝑣.

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(B) In 𝐺 𝑅 , the BFS traversal from 𝑣 will visit exactly the same set of vertices as the
DFS from 𝑣 in 𝐺.

s em
g a BFS of 𝐺 from 𝑣 is the reverse of the order of
The order of vertices visited inlthe 𝑅

vertices visited in the DFS a
(C)
of 𝐺 from 𝑣.

(D) If 𝑢 is a reachable vertex in the DFS of 𝐺 from 𝑣, then 𝑣 is also a reachable vertex
in the BFS of 𝐺 𝑅 from 𝑢.

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Q.41 Consider a B+ Tree where the maximum number of key values in each leaf node
is 2 and the maximum number of pointers in each non-leaf node is 3. Let the
content of the B+ Tree be as shown in the figure.

Which of the following options denotes the key value(s) stored in the root node
after inserting a key value 3 in the given B+ Tree?

(A) 5

(B) 8

(C) 3 and 5

(D) 3, 5 and 8

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Q.42 Consider the given relations 𝑋, 𝑌 and 𝑍. The relation 𝑋 has three columns 𝑃, 𝑄
and 𝑅. The relation 𝑌 has three columns 𝑃, 𝑄 and 𝑆. The relation 𝑍 has two
columns 𝑃 and 𝑇.

𝑋 𝑌 𝑍
𝑃 𝑄 𝑅 𝑃 𝑄 𝑆 𝑃 𝑇 m
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P1 Q1 R1 P1 Q1 2 P1 T1
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s em Q3
P2 R2 P1 Q2 5 P3 T2
T3 gl
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P3 R2 P2 Q1 6 P4

ag P3 Q3 1 P4 NULL

Consider the relational algebra expression

∏𝑃𝑅𝑆 [(𝜎(𝑄 = 𝑄3 ∨ 𝑅 = 𝑅2) [𝑋 ⨝ 𝑌])⨝(𝜎(𝑆 > 1) [𝑌 ⨝ 𝑍])]

where ⨝ denotes natural join operation.

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Which of the following options is the correct output for the given expression?

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s em
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(A) a R1, 5)
Two rows (P1, R1, 2) and (P1,

(B) Three rows (P1, R1, 2), (P1, R1, 5) and (P2, R2, 6)

(C) One row (P1, R1, 2)

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(D) Zero rows
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Q.43 Consider the concept hierarchies as shown in the figure.

Which of the following options denotes the total number of possible data cuboids
from these concept hierarchies?

(A) 43

(B) 23

(C) 2!

(D) 4!

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Q.44 Let 𝑋 and 𝑌 be two independent random variables. 𝑋 follows 𝐵𝑒𝑟𝑛𝑜𝑢𝑙𝑙𝑖(𝑝 = 0.3)
distribution and 𝑌 follows 𝑁𝑜𝑟𝑚𝑎𝑙(𝜇 = 0, 𝜎 2 = 100) distribution.

Which of the following options is the variance of (2𝑋 − 1)𝑌 ?

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(A)
s e
100
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(B)
a 90

(C) 49

(D) 21

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Let
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Q.45 𝑛
𝑒 −𝑛 𝑛𝑘 s e
𝐿 = lim ∑
𝑘!
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a
𝑛→∞
𝑘=0

Which of the following is the value of 𝐿?

(A) 0.5
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c. o(B) m.co
e
1.0

e m l as
las (C) 0
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(D) 𝑒 −1

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Q.46 1 0 0
Let 𝛾1 , 𝛾2 , 𝛾3 be the eigenvalues of the matrix [0 cos 𝑡 sin 𝑡 ],
0 − sin 𝑡 cos 𝑡
where 𝑡 ∈ [−𝜋, 𝜋] is in radians.

Which one of the following options lists all the possible values of 𝑡 satisfying
𝛾1 + 𝛾2 + 𝛾3 = 1 + √2 ?

(A) 𝜋 𝜋
{ ,− }
3 4

(B) 𝜋 𝜋
{ ,− }
4 3

(C) 𝜋 𝜋
{ ,− }
4 4

(D) 𝜋 𝜋
{ ,− }
3 3

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Q.47 Consider that 20 stories of Author X and 10 stories of Author Y were kept together
without mentioning the names of the authors. A classifier was then asked to
predict the author (X or Y) of each of these stories. Let, out of X’s stories, 6 were
classified as that of Y. On the other hand, out of Y’s stories, 2 were classified as
that of X.

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Considering X and Y as two classes, which of the following statements is/are true?
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(A) a Accuracy of the classifier is 11/15.

(B) Precision of Class X is higher than the Precision of Class Y.

(C) Recall of Class X is higher than the Recall of Class Y.

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(D) Accuracy of the classifier is 14/15.

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Q.48 Let 𝑃(𝑥) be a predicate.

Which of the following statements is/are NOT valid in first-order logic?

∀𝑥 𝑃(𝑥) ⇒ ∃𝑥 𝑃(𝑥) m
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(A)
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c. o(B) s e m
e m ∃𝑥 𝑃(𝑥) ⇒ ∀𝑥 𝑃(𝑥)
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ag (C) ∃𝑥 𝑃(𝑥) ⇔ ∀𝑥 𝑃(𝑥)

(D) ∀𝑥 𝑃(𝑥) ⇒ ∃𝑥 ¬𝑃(𝑥)

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Q.49 Consider the problem of sorting the given array in ascending order:

P = [1, 2, 3, 5, 4]

Consider two sorting algorithms Bubble Sort (BS) and Insertion Sort (IS).

Let N1 be the total number of comparisons done by BS on the elements of P and
N2 be the total number of comparisons done by IS on the elements of P.

Which of the following options is/are correct?

(A) N1 = 10, N2 = 4

(B) N1 > N2

(C) IS on P will perform only one swap

(D) Both BS and IS on P will make at least one unnecessary comparison (i.e.,
comparing elements that are already in correct order)

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Q.50 Consider the given Python program.

def outer():
x = []
def inner(val):
x.append(val)
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return x
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return inner
f1 = outer()
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s e
f2 = outer()
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print(f1(10)) # Line P
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print(f2(30))
# Line Q
# Line R
print(f1(40)) # Line S

Which of the following options is/are correct?

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(A) f1 and f2 share the same list x

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Output of Line Q is [10, 20]as
(B)
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(C) Output of Line R is [10, 20, 30]

(D) Output of Line S is [10, 20, 40]

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Q.51 Consider a table Employee(EmpID, TeamID), where the column EmpID (ID of
an employee) is the primary key. The column TeamID denotes the team ID of the
team of which the employee is a member. TeamID is a NOT NULL column.

We want to display the size of the team (denoted as TeamSize) in which each
employee is a member by using SQL. As an example, the desired output for the
given Employee table is also shown in tabular form.

Which of the following is/are correct?

Employee Output
EmpID TeamID EmpID TeamSize
1 8 1 3
2 8 2 3
3 8 3 3
4 7 4 2
5 7 5 2
6 9 6 1

(A) SELECT E.EmpID, B.TeamSize
FROM Employee AS E, (SELECT TeamID, COUNT(TeamID) AS
TeamSize FROM Employee GROUP BY TeamID) AS B
WHERE E.TeamID = B.TeamID
(B) SELECT A.EmpID, COUNT(B.TeamID) AS TeamSize
FROM Employee AS A, Employee AS B
WHERE A.TeamID = B.TeamID AND A.EmpID = B.EmpID
GROUP BY A.EmpID
(C) SELECT B.EmpID, B.TeamSize
FROM (SELECT EmpID, COUNT(TeamID) AS TeamSize
FROM Employee GROUP BY EmpID) AS B
(D) SELECT A.EmpID, B.TeamSize
FROM Employee AS A, (SELECT COUNT(TeamID) AS TeamSize FROM
Employee GROUP BY TeamID) AS B
WHERE A.TeamID = B.TeamID

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Q.52 Let 𝑀 = (𝐼𝑛 − 𝟏𝟏𝑇 ) be a matrix, where 𝟏 = (1,1,1, … ,1)𝑇 ∈ ℝ𝑛 and 𝐼𝑛 is the
𝑛
identity matrix of order 𝑛.

Which of the following options is/are correct?

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(A)
s em 𝑇
𝑀 =𝑀
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(B) a 𝑀 = 𝐼 2
𝑛

(C) 𝑇𝑟𝑎𝑐𝑒(𝑀) = 𝑛

(D) 𝑀 is a projection matrix

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Q.53 Let 𝑋1 , 𝑋2 , … . , 𝑋𝑛 be 𝑛 independent random variables. Each of the random
g 1
variables follows 𝑁𝑜𝑟𝑚𝑎𝑙(𝜇 = 0, 𝜎 2 = 1) distribution. Define 𝑋̅ = ∑𝑛𝑖=1 𝑋𝑖 .
a 𝑛

Which of the following statements is/are correct?

(A) ∑𝑛𝑖=1 𝑋𝑖 2 follows Chi-square distribution with 𝑛 degrees of freedom.

o m
m c
. freedom.
c. o(B) m
∑ (𝑋 − 𝑋̅) follows Chi-square distribution with (𝑛 − 1) degrees of
𝑛 2

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𝑖=1 𝑖

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𝑋 + 𝑋 follows exponential distribution with mean 2. g
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(C) 2 2

g
1 𝑛

a
2
(D) (√𝑛 𝑋̅) follows Chi-square distribution with 2 degrees of freedom.

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Q.54 Let 𝑋 be a discrete valued random variable with cumulative distribution function
𝐹(𝑥).

Which of the following statements is/are correct?

(A) 𝐹(𝑥) is always a positive function.

(B) 𝐹(𝑥) is a non-decreasing function.

(C) 𝐹(𝑥) has jump discontinuity.

(D) 𝐹(𝑥) is a left continuous function.

Q.55 Consider that Linear Ridge Regression is being used to learn a prediction function
𝑦𝑝𝑟𝑒𝑑 = 𝑤 𝑇 𝑥, where 𝑤, 𝑥 ∈ ℝ2 and Mean Absolute Error (MAE) is used to
measure the prediction error. A weight of 0.20 is associated with the regularizer.

At an intermediate step of the training process, assume that the parameter 𝑤 =
[−3.00, 4.00]𝑇 . In the next step, for the input 𝑥 = [1.00, 2.00]𝑇 , the predicted
value of 𝑦 is noted. Let the relation between 𝑥 = [𝑥1 , 𝑥2 ]𝑇 and the true value of 𝑦
be 𝑦𝑡𝑟𝑢𝑒 = 𝑥1 + 𝑥2 .

The value of the overall regularized loss function for this instance is _______ .
(Rounded off to two decimal places)

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Q.56 Consider a fully-connected feed-forward multi-layer perceptron. It has 30 neurons
in the input layer, followed by two hidden layers and an output layer. The first
hidden layer has 4 neurons and the second hidden layer has 3 neurons. The output
layer has only one neuron. Assume that no bias parameters are used.

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The number of learnable parameters in the multi-layer perceptron is __________ .
(Answer in integer) m
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s e
A clinic specializes in testing for a disease D. The result of the test can be either g l a
la or negative.
Q.57

g a
a positive

A study revealed that if a person suffers from the disease D, the test result in that
clinic comes out positive 80% of the time, and negative 20% of the time. If a
person is not suffering from the disease D, the test comes out positive 10% of the
time and negative 90% of the time. It is also known that among the general
population, the disease D occurs in 30% of the individuals.

If a person tests positive for D in that clinic, the probability that he/she actually
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suffers from the disease D is __________ . (Rounded off to two decimal places)

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Q.58
la s
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Consider the given Python program.

def fun(L, i=0):
a
if i >= len(L)-1:
return 0
if L[i] > L[i+1]:
L[i+1], L[i] = L[i], L[i+1]
return 1+fun(L, i+1)
else:
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return fun(L, i+1)
m data = [5, 3, 4, 1, 2]

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s e
for _ in range(len(data)):
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count += fun(data)
g print(count) a
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The output of the program is __________ . (Answer in integer)

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Q.59 Let there be two relations 𝑋 and 𝑌 as shown. 𝑋 has three columns 𝑃, 𝑄 and 𝑅. 𝑌
has two columns 𝑃 and 𝑆.

𝑋 𝑌
𝑃 𝑄 𝑅 𝑃 𝑆
P1 Q1 R1 P1 10
P2 Q2 R2 P1 15
P3 Q3 R2 P2 20
P3 1

Consider that the following tuple relational calculus expression is evaluated.

{𝑡|𝑡 ∈ 𝑋 ⋀ ∃𝑧 ∈ 𝑋(𝑡[𝑃] = 𝑧[𝑃]) ⋀ ∃𝑚 ∈ 𝑌 (𝑚[𝑃] = 𝑡[𝑃] ⋀ 𝑚[𝑆] > 1)}

The number of tuples that will be returned is __________ . (Answer in integer)

Q.60 Let Account be a relation as shown.

Account
AccNo Balance
A1 5000
A2 5000
A3 10000
A4 15000
A5 18000

Consider the given SQL query.

SELECT AccNo FROM Account AS A
WHERE (SELECT COUNT(*) FROM Account AS B
WHERE A.Balance < B.Balance) >= (SELECT COUNT(*)
FROM Account AS C WHERE A.Balance > C.Balance)

The number of rows returned by the SQL query is __________ . (Answer in
integer)

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Q.61 Consider an ER model with the entities E1(A11,A12,A13) and E2(A21,A22,A23),
where, A11, A12, A13 are the attributes of E1, and A21, A22, A23 are the attributes of
E2. Let A22 be a multi-valued attribute. A11 and A21 are the primary keys of E1 and
E2, respectively.

Let R12 be a many-to-many relationship between E1 and E2. Participation of both
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E1 and E2 in R12 is total.
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The minimum number of relations required to convert the ER model into
s e m
s em
relational model (assuming there is no other functional dependency) where each
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relation is in third normal form (3NF) is __________ . (Answer in integer)
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Q.62
a For a given data set {𝑥 , 𝑥 , … . , 𝑥 }, where 𝑛 = 100, it is known that
1 2 𝑛

𝑛 𝑛
1
∑ ∑(𝑥𝑖 − 𝑥𝑗 )2 = 99
2000
𝑖=1 𝑗=1

1
Let us denote 𝑥̅ = ∑𝑛𝑖=1 𝑥𝑖
𝑛
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1
The value of ∑𝑛𝑖=1(𝑥𝑖 − 𝑥̅ )2 is __________ . (Answer in integer)
em
99

s
Let 𝑋 be a random variable thatla
agrandom variable 𝑌 given 𝑋 = 𝑥 is the
Q.63 follows 𝑈𝑛𝑖𝑓𝑜𝑟𝑚(−1,1) distribution. The
conditional distribution of the
𝑈𝑛𝑖𝑓𝑜𝑟𝑚(𝑥 2 − 0.1, 𝑥 2 + 0.1) distribution.

The value of 𝑐𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛(𝑋, 𝑌) is __________ . (Answer in integer)

Q.64 Let 𝐴5×5 be a matrix such that each of its elements follows 𝐵𝑒𝑟𝑛𝑜𝑢𝑙𝑙𝑖(𝑝 = 0.50)
distribution independently.
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The probability that the row-sum of the second row and the column-sum of the

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third column are both equal to 3 is ________ . (Rounded off to two decimal

m
places)
s em
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Q.65 𝑇 𝑇 𝑛
Let 𝐴 = (𝐼 − 𝟏𝟏 ) be a matrix, where 𝟏 = (1,1,1, … ,1) ∈ ℝ and 𝐼 is the
𝑛 𝑛
𝑛
identity matrix of order 𝑛.

The value of max 𝑥 𝑇 𝐴𝑥 , where 𝑆 = {𝑥 ∈ ℝ𝑛 | 𝑥 𝑇 𝑥 = 1}, is __________ .
𝑆
(Answer in integer)

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