GATE 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?

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...