Recent questions tagged tifr2023

Consider the following two statements:$\text{(P)}$ The current population of Bhutan is greater than the current population of India.$\text{(Q)}$ The Moon is smaller than ...

Which of the following is true about the set of regular languages and the set of recursively enumerable languages over a finite alphabet $\Sigma?$ The set of regular lang...

Given a set $\mathcal{F}$ of intervals $\left(s_{i}, t_{i}\right)_{i=1}^{n}$ on the integer line (assume all $s_{i}, t_{i}$ are distinct), a subset $S$ of $\mathcal{F}$ i...

Amar, Balu, and Chhaya are three friends and they play the following game. Chhaya first chooses a number $k \in U$ where $U=\{1,2, \ldots, 127\}$. She either gives $k$ to...

Consider unit vectors $\mathbf{a}$ and $\mathbf{b}$ in $\mathbb{R}^{n}$. Let $\mathbf{w}$ be an arbitrary vector in $\mathbb{R}^{n}$ and $\eta$ be a positive real number ...

What is the solution to the following recurrence?\[T(n)=\left\{\begin{array}{ll}1 & \text { if } n \leq 10, \\\sqrt{n} \cdot T(\sqrt{n})+n & \text { if } n>10.\end{array}...

Consider the random variables $N, H$, and $T$ sampled as follows. First, $N$ is sampled from the Poisson$(10)$ distribution. This means that for each integer $n \geq 0$,\...

Let $U$ be a finite set and let $h$ be a function mapping $U \times U$ to $U$. Consider the following process that assigns values to all nodes of a complete binary tree w...

Fix a positive integer $n$, and let $n=p_{1}^{e_{1}} p_{2}^{e_{2}} \cdots p_{k}^{e_{k}}$ be the prime factorization of $n$. Here, $p_{1}, \ldots, p_{k}$ are prime numbers...

A $d$-regular graph is one in which every vertex has degree $d$. Also, a minimum cut in a graph is a smallest set of edges which, upon removal, disconnects the graph, so ...

Let $m=2877426671$. It is known that $p=5754853343=2 m+1$ is a $10$ -digit prime number. What is $16^{m}(\bmod p)$ ?$1$$4$$16$$2877426671$$5754853342 \;($ which is actua...

A graph $G=(V, E)$ is said to be $k$-colourable if the set $V$ of vertices can be coloured with $k$ colours such that no edge has both its endpoints of the same colour. I...

You have a regular tetrahedron and $4$ distinct colours. You wish to paint the faces of the tetrahedron such that each face gets a different colour. How many ways can you...

Consider the language $L=\left\{a^{i} \$ a^{j} \$ b^{k} \$ \mid k \leqslant \max (i, j), i, j, k \geq 0\right\}$ over the alphabet $\Sigma=\{a, b, \$\}$. The complement o...

How many distinct rectangles can be formed using the vertices in the grid shown below? Squares are also counted as rectangles, and two rectangles are distinct if either t...

$A$ is an $n \times n$ matrix with real-valued entries. Further, there exists a vector $x \neq 0$ such that $A x=0$. Now consider a given vector $b$ in $\mathbb{R}^{n}$. ...

Let $\mathbb{R}$ be the set of all real numbers. Consider the relation $T$ defined as\[T=\{(x, y) \in \mathbb{R} \times \mathbb{R} \text { such that } 4+x y>0\}\]$($Recal...

Suppose $A$ is a $2 \times 2$ matrix such that the sum of the principal diagonal entries of $A$ is $10$ and the sum of the principal diagonal entries of $A^{2}$ is $20$. ...

For a function $f: \mathbb{R} \rightarrow \mathbb{R},$ consider the following conditions.$\text{(C1)}$ $|f(x)| \leq|x|$ for all $x \in \mathbb{R}$.$\text{(C2)}$ $|f(x)| \...

Suppose $f(x)$ is a polynomial of the form $a x^{2}+b x+c,$ with $a, b, c$ unknown real numbers. Suppose you are additionally told that $f(1)=2$ and $f(-1)=3$. Consider t...

Let $\mu$ be a probability distribution on the interval $[0,1]$ with probability density function $p(x)=c \cdot x^{2}$ where $c$ is an undetermined constant. Consider an ...

Let $Q$ be a set with $n$ elements. Consider a system with states of the form $(S, R)$ where $S \subseteq Q$ and $R \subseteq Q$, with the further restriction that $R \su...

We are given a set $S=\left\{x_{1}, \ldots, x_{n}\right\}$ of distinct positive integers such that $\operatorname{gcd}\left(x_{i}, x_{j}\right)= 1$ for any $i, j \in\{1, ...

Suppose is a piece on a chess board that attacks squares that are exactly two steps in the vertical direction, and squares that are adjacent horizontally (as marked with...

We have a coin that is equally likely to land heads (denoted $\text{‘H’})$ or tails (denoted $\text{‘T’})$ when tossed. Suppose we keep tossing this coin and stop the gam...

Let $f(x)=a x^{3}+b x^{2}+c x+d$ be a polynomial, where $a, b, c, d$ are unknown real numbers. It is further given that $f(1)=1, f(2)=2, f(3)=9$, and $f^{\prime}(1)=0$. T...

Consider the $n \times n$ matrix $M$ defined as follows:$$M=\left(\begin{array}{cccc}1 & 2 & \ldots & n \\n+1 & n+2 & \ldots & 2 n \\2 n+1 & 2 n+2 & \ldots & 3 n \\\vdots...