GATE 2025 Data Science & AI

Let $\quad f: \mathbb{R} \rightarrow \mathbb{R} \quad$ be such that $|f(x)-f(y)| \leq(x-y)^2$ for all $x, y \in \mathbb{R}$. Then $\quad f(1)-f(0)=$ ____________

Consider the function $$ f(\mathrm{x})=\frac{x^3}{3}+\frac{7}{2} x^2+10 x+\frac{133}{2}, x \in[-8,0] . $$ Which of the following statements is/are correct?

Which of the following statements is/are correct in a Bayesian network?

Consider a hash table of size 10 with indices $\{0,1, \ldots, 9\}$, with the hash function $$ h(x)=3 x(\bmod 10) $$ where linear probing is used to handle collisions. The hash tabl...

Let $D=\left\{x^{(1)}, \ldots ., x^{(n)}\right\}$ be a dataset of $n$ observations where each $x^i \in \mathbb{R}^{100}$. It is given that $\sum_{i=1}^n x^{(\mathrm{i})}=0$ The cov...

Let $f(x)=\frac{e^x-e^{-x}}{2}, x \in R$. Let $f^{(k)}(a)$ denote the $k^{\text {th }}$ derivative of $f$ evaluated at $a$. What is the value of $f^{(10)}(0)$ ?(Note: ! denotes fac...

$$\mathop {\lim }\limits_{t \to + \infty } \sqrt{t^2+t}-t= $$ (Round off to one decimal place)

Consider two functions $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow(1, \infty)$. Both functions are differentiable at a point c . Which of the following fu...

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a twice-differentiable function and suppose its second derivative satisfies $f^{\prime \prime}(x)>0$ for all $x \in \mathbb{R}$. Which...

Consider a directed graph $G=(V, E)$, where $V=\{0,1,2, \ldots, 100\}$ and $E=\{(i$, $j): 0

Let $A \in \mathbb{R}^{n \times n}$ be such that $A^3=A$. Which one of the following statements is ALWAYS correct?

Which of the following statements is/are correct?

The sum of the elements in each row of $A \in \mathbb{R}^{n \times n}$ is 1 . If $B=A^3-2 A^2+A$, which one of the following statements is correct (for $x \in \mathbb{R}^n$ )?

Let $A=I_n+x x^T$, where $I_n$ is the $n \times n$ identity matrix and $x \in \mathbb{R}^n, x^T x=1$. Which of the following option is/are correct?

An $n \times n$ matrix $A$ with real entries satisfies the property: $\|A x\|^2=\|x\|^2$ for all $x \in R^n$ where $\|\cdot\|$ denotes the Euclidean norm. Which of the following st...

Let $p$ and $q$ be any two propositions. Consider the following propositional statements. $$ \begin{aligned} & S_1: p \rightarrow q, \quad S_2: \neg p \wedge q, \quad S_3: \neg p \...

Consider a coin-toss experiment where the probability of head showing up is $p$. In the $i^{\text {th }}$ coin toss, let $X_i=1$ if head appears, and $X_i=0$ if tail appears. Consi...

Suppose $X$ and $Y$ are random variables. The conditional expectation of $X$ given $Y$ is denoted by $E[X \mid Y]$. Then $E[E[X \mid Y]]$ equals

There are three boxes containing white balls and black balls. Box-1 contains 2 black and 1 white balls. Box-2 contains 1 black and 2 white balls. Box-3 contains 3 black and 3 white...

Consider the cumulative distribution function (CDF) of a random variable X : $$ F_X(x)=\left\{\begin{array}{cc} 0 & x \leq-1 \\ \frac{1}{4}(x+1)^2 & -1 \leq x \leq 1 \\ 1 & x \geq...

A random variable X is said to be distributed as $\operatorname{Bernoulli}(\theta)$, denoted by $X \sim \operatorname{Bernoulli}(\theta)$, if $$ P(X=1)=\theta, P(X=0)=1-\theta $$ f...

Let X be a continuous random variable whose cumulative distribution function (CDF) $F_X(x)$, for some $t$, is given as follows: $$ F_X(x)=\left\{\begin{array}{cc} 0 & x \leq t \\ \...

A bag contains 5 white balls and 10 black balls. In a random experiment, $n$ balls are drawn from the bag one at a time with replacement. Let $S_n$ denote the total number of black...

A random experiment consists of throwing 100 fair dice, each die having six faces numbered 1 to 6 . An event $A$ represents the set of all outcomes where at least one of the dice s...

For $x \in \mathbb{R}$, the floor function is denoted by $f(x)=\lfloor x\rfloor$ and defined as follows $\lfloor x\rfloor=k, k \leq x where $k$ is an integer. Let $Y=\lfloor X\rflo...

Let $X=a Z+b$, where Z is a standard normal random variable, and $a, b$ are two unknown constants. It is given that $$ \begin{aligned} E[X] & =1, E[(X-E[X]) Z] \\ & =-2, E\left[(X-...

Let $Y=Z^2, Z=\frac{X-\mu}{\sigma}$, where $X$ is a normal random variable with mean $\mu$ and variance $\sigma^2$. The variance of $Y$ is

It is given that $P(X \geq 2)=0.25$ for an exponentially distributed random variable $X$ with $E[X]=\frac{1}{\lambda}$, where $E[X]$ denotes the expectation of $X$. What is the val...

Weight of a person can be expressed as a function of their age. The function usually varies from person to person. Suppose this function is identical for two brothers, and it monot...

A rectangle has a length L and a width W . where $\mathrm{L}>\mathrm{W}$. If the width, W , is increased by $10 \%$, which one of the following statements is correct for all values...

A $4 \times 4$ digital image has pixel intensities $(U)$ as shown in the figure. The number of pixels with $U \leq 4$ is: 0 1 0 2 4 7 3 3 5 5 4 4 6 7 3 2

If a real variable $x$ satisfies $3^{x^2}=27 \times 9^x$, then the value of $\frac{2^{x^2}}{\left(2^x\right)^2}$ is :

Column-I has statements made by Shanthala; and, Column-II has responses given by Kanishk. Column - I Column-II P. This house is in a mess. 1. Alright, I won't bring it up during ou...

Courage : Bravery :: Yearning : ___________ Select the most appropriate option to complete the analogy.

We ___________ tennis in the lawn when it suddenly started to rain. Select the most appropriate option to complete the above sentence.

Consider designing a linear classifier $$ y=\operatorname{sign}(f(x ; w ; b)), f(x ; w, b)=w^{\mathrm{T}} x+b $$ on a dataset $$ \begin{aligned} & D=\left\{\left(x_1, y_1\right),\l...

(Round off to three decimal places) The naive Bayes classifier is used to solve a two-class classification problem with class labels $y_1, y_2$. Suppose the prior probabilities are...

Given data $\{(-1,1),(2,-5),(3,5)\}$ of the form $(x, y)$, we fit a model $y=w x$ using linear least-squares regression. The optimal value of $w$ is _________

Let $\left\{x_1, x_2, \ldots ., x_n\right\}$ be a set of linearly independent vectors in $\mathbb{R}^n$. Let the $(\mathrm{i}, \mathrm{j})$ - th element of matrix $A \in \mathbb{R}...

Let $C_1$ and $C_2$ be two sets of objects. Let $D(x, y)$ be a measure of dissimilarity between two objects $x$ and $y$. Consider the following definitions of dissimilarity between...

Let $x_1, x_2, x_3, x_4, x_5$ be a system of orthonormal vectors in $\mathbb{R}^{10}$. Consider the matrix $A=x_1 x_1^T+\ldots . .+x_5 x_5^T$. Which of the following statements is/...

Consider a two-class problem in $R^d$ with class labels red and green. Let $\mu_{\text {red }}$ and $\mu_{\text {green }}$ be the means of the two classes. Given test sample $x \in...

Which of the following statements is/are correct about the rectified linear unit (ReLU) activation function defined as $\operatorname{ReLU}(x)=\max (x, 0)$, where $\mathrm{x} \in \...

Consider designing a linear binary classifier $f(x)=\operatorname{sign} g\left(w^T x+b\right), x \in \mathbb{R}^2$ on the following training data: Class -1: $\left\{\binom{2}{0},\b...

Consider the following Python code snippet. $\operatorname{def} f(a, b)$ : if ( $a==0$ ): return b $$ \begin{aligned} &\begin{aligned} & \text { if }(\mathrm{a} \% 2==1) \text { :...

Consider the following Python declarations of two lists. $$ \begin{aligned} & A=[1,2,3] \\ & B=[4,5,6] \end{aligned} $$ Which one of the following statements results in $A=[1,2,3,4...

Consider the following pseudocode. Create empty stack S set $x=0$, flag $=0$, sum $=0$ Push $x$ onto $S$ while ( $S$ is not empty) $\{$ if (flag equals 0){ Set $x=x+1$ Push $x$ ont...

Consider the following Python code snippet. $\mathrm{A}=\{$ "this","that" $\}$ $B=\{$ "that","other" $\}$ $\mathrm{C}=\{$ "other","this"} while "other" in C : if "this" in A : $\ma...

Suppose that insertion sort is applied to the array $[1,3,5,7,9,11, x, 15,13]$ and it takes exactly two swaps to sort the array. Select all possible values of $x$.

Consider the following two relations, named Customer and Person, in a database: Person ( aadhaar CHAR(12) PRIMARY KEY, name VARCHAR(32)); Customer ( name VARCHAR(32), email VARCHAR...

Consider a database relation $R$ with attributes ABCDEFG , and having the following functional dependencies: $$ \mathrm{A} \rightarrow \mathrm{BCEF} \quad \mathrm{E} \rightarrow \m...

If a relational decomposition is not dependency-preserving, which one of the following relational operators will be executed more frequently in order to maintain the dependencies?

Consider the following three relations: Car (model, year, serial, color) Make (maker, model) Own (owner, serial) A tuple in Car represents a specific car of a given model, made in...

Consider the following tables, Loan and Borrower, of a bank. Loan loan_num branch_name amount L11 Banjara Hills 90000 L14 Kondapur 50000 L15 SR Nagar 40000 L22 SR Nagar 25000 L23 B...

Consider a fact table in an OLAP application: Facts (D1, D2, val), where D1 and D2 are its dimension attributes and val is a dependent attribute. Suppose attribute D1 takes 3 value...

$$ \text { On a relation named Loan of a bank: } $$ Loan Loan_number Branch_name Amount L11 Banjra Hills 90000 L14 Kondapur 50000 L15 SR Nagar 4000 L22 SR Nagar 25000 L23 Balanagar...

The number of additions and multiplications involved in performing Gaussian elimination on any $n \times n$ upper triangular matrix is of the order

For which of the following inputs does binary search take time $O(\log n)$ in the worst case?