List of top Questions asked in GATE AR- 2024

Consider a joint probability density function of two random variables X and Y
\(f_{x,y}(x,y) = \left\{ \begin{array}{ll} 2xy, & 0\lt x \lt2 , & 0\lt y\lt x\\ 0 & \text{otherwise } \end{array} \right.\)
Then, E[Y | X = 1.5] is ______.

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 ?

Let f: R → R be the function \(f(x) = \frac{1}{1+e^{-x}}\)
The value of the derivative of ƒ at x where f(x) = 0.4 is ______
(rounded off to two decimal places).
Note: R denotes the set of real numbers.

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 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 ______.

Evaluate the following limit :
\(\lim\limits_{x\rightarrow0}\frac{\text{ln}((x^2+1)\cos x)}{x^2}\) = ________.

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 _______.

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 \(u=\begin{bmatrix} 1 \\ 2 \\ 3 \\ 4 \\ 5 \end{bmatrix}\), 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 \(Σ^{5}_{i=1}σ_i\) is

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)

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 0) 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

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 ________.

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 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 three decimal places).

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 ?

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, Customer Rating)
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.Genre ID = Genre. ID;
This SQL query can be sped up using which of the following indexing options ?

Consider two events T and S. Let \(\bar{T}\) denote the complement of the event T. The probability associated with different events are given as follows :
 P(\(\bar{T}\)) = 0.6, P(S|T) = 0.3, P(SI\(\bar{T}\)) = 0.6
Then, P(TIS) is ______
(rounded off to two decimal places)

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”?

Let X be a random variable exponentially distributed with parameter λ > 0. The probability density function of X is given by :
\(f(x) = \left\{ \begin{array}{ll} \lambda e^{-\lambda x,} & x \ge0 \\ 0 & \text{otherwise} \end{array} \right.\)
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).

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 ?

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 ?

Let \(\R\) be the set of real numbers, U be a subspace of \(\R^3\) 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 ?

Which of the following statements is/are TRUE ?
Note: \(\R\) denotes the set of real numbers.

Select all choices that are subspaces of \(\R^3\).
Note : \(\R\) denotes the set of real numbers.