All 65 questions of GATE 2024 DA

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

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?

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

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?

The sum of the following infinite series is

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?

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?

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?

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

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.

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?

Euclidean distance based $k$-means clustering algorithm was run on a dataset of 100 points with $k=3$. If the points

are both part of cluster 3, then which ONE of the following points is necessarily also part of cluster 3?

Consider the matrix

Which ONE of the following statements is TRUE?

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 $T \cap S$?

Match the items in Column 1 with the items in Column 2 in the following table:

For any twice differentiable function $f:\mathbb{R}\to\mathbb{R}$, if at some $x^* \in \mathbb{R}$,

then the function $f$ necessarily has a ______ at $x=x^*$.

Consider the dataset with six datapoints:

where

and the labels are given by

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?

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

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.

Match the items in Column 1 with the items in Column 2 in the following table:

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?

Consider the following statement:

In adversarial search, $\alpha$–$\beta$ pruning can be applied to game trees of any depth where $\alpha$ is the (m) value choice we have formed so far at any choice point along the path for the MAX player and $\beta$ 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?

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 $\phi$ denotes the empty set?

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?

Consider performing uniform hashing on an open address hash table with load factor

where $n$ elements are stored in the table with $m$ slots. The expected number of probes in an unsuccessful search is at most

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

Let $h_1$ and $h_2$ be two admissible heuristics used in A* search.

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

For any binary classification dataset, let $S_B \in \mathbb{R}^{d \times d}$ and $S_W \in \mathbb{R}^{d \times d}$ be the between-class and within-class scatter matrices, respectively. The Fisher linear discriminant is defined by $u^* \in \mathbb{R}^d$, that maximizes

If $\lambda = J(u^*)$, $S_W$ is non-singular and $S_B \ne 0$, then $(u^*, \lambda)$ must satisfy which ONE of the following equations?

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

Which ONE of the following statements is FALSE?

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?

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

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

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

Let $f:\mathbb{R} \to \mathbb{R}$ be the function

The value of the derivative of $f$ at $x$ where $f(x)=0.4$ is ______.

(Rounded off to two decimal places)

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 ______.

Consider the $3 \times 3$ matrix

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

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

Team

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 ______.

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 ______.

Let $f:\mathbb{R} \to \mathbb{R}$ be a function:

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

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?

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?

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?

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

Select all choices that are subspaces of $\mathbb{R}^3$.

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?

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

in increasing order after exactly two passes over the array?

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.

Which of the following statements is/are TRUE?

Note: $\mathbb{R}$ denotes the set of real numbers.

Let $\mathbb{R}$ be the set of real numbers, $U$ be a subspace of $\mathbb{R}^3$ and $M \in \mathbb{R}^{3\times 3}$ be the matrix corresponding to the projection on to the subspace $U$.

Which of the following statements is/are TRUE?

Consider the function

Which of the following statements is/are TRUE?

Given the relational schema

and the set of functional dependencies

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

Consider the table below, where the $(i,j)$-th element of the table is the distance between points $x_i$ and $x_j$. Single linkage clustering is performed on data points $x_1, x_2, x_3, x_4, x_5$.

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

Consider the two neural networks (NNs) shown in Figures 1 and 2, with ReLU activation

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

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?

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).

Evaluate the following limit:

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:

• $s_1: \forall ball\; \neg game(ball, rugby) \Rightarrow shape(ball, round)$
• $s_2: \forall ball\; \neg shape(ball, round) \Rightarrow game(ball, rugby)$
• $s_3: \forall ball\; game(ball, rugby) \Rightarrow \neg shape(ball, round)$
• $s_4: \forall ball\; shape(ball, round) \Rightarrow \neg game(ball, rugby)$

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

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

The value of

is ______ (rounded off to three decimal places).

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

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?

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).

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

If $5E(X) = \operatorname{Var}(X)$, where $E(X)$ and $\operatorname{Var}(X)$ indicate the expectation and variance of $X$, respectively, the value of $\lambda$ is ______ (rounded off to one decimal place).

Consider two events $T$ and $S$. Let $\bar T$ denote the complement of the event $T$. The probabilities associated with different events are given as follows:

Then,

is ______ (rounded off to two decimal places).

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?

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 relational 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?

Let

and let $\sigma_1, \sigma_2, \sigma_3, \sigma_4, \sigma_5$ be the singular values of the matrix

The value of

is ______.

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

is ______ (rounded off to three decimal places).

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

Then,

is ______.