Recent questions tagged tifr2025

What is the smallest positive integer that when multiplied with $2^{3} 3^{2} 4^{3} 5^{7} 6^{5}$ results in a perfect square?$6$$10$$15$$60$$210$ 

Assume that in a room there are $k$ persons. We know that $k=3$ with probability $\frac{1}{4}, k=5$ with probability $\frac{1}{4}$, and $k=10$ with probability $\frac{1}{...

How many distinct factors does $2^{2} 3^{3} 4^{4} 5^{5}$ have? For example, $12$ has $6$ distinct fact ors: $1,2,3,4,6$, and $12$.$240$$264$$324$$360$None of the above 

There are two bags, one with $10$ red balls and $90$ blue balls, and another with $90$ red balls and $10$ blue balls. You pick one of the two bags uniformly at random and...

There are three boxes. One box contains ten red socks, another contains ten blue socks, and the third contains a mix five red and five blue socks. The three boxes are lab...

Consider the following matrix:\[A=\left[\begin{array}{lll}3 & 0 & 1 \\0 & 2 & 0 \\x & 0 & 1\end{array}\right]\]If all eigenvalues of the matrix are the same, what is the ...

Suppose we had a conical container with some liquid filled in that reaches until half its height when placed with the circular face downwards. If we were to turn the cone...

Consider the collection $\mathcal{C}$ of all possible $n \times n$ matrices with entries in $\{1, \ldots, n\}$ that obey the following property: every row of $M$ is a per...

What is the largest number of knights that can be placed on a $60 \times 60$ chessboard such that no two knights attack each other (two knights attack each other if they ...

A clock shows $3:00$. After $10^{1024}$ minutes, what minute does the minute hand point to?$0$$20$$40$$50$$10$ 

How many distinct ways are there to paint the $6$ faces of a cube using six colors $( 1, 2,3,4,5,6)$, with each face painted a different color, such that no two painted c...

Consider a coin which has probability $p$ of coming up heads when tossed. Assume that $0<p<10 < p < 10<p<1$. A trial is performed by repeatedly tossing the coin until the...

Let $(X, Y, Z) \in \mathbb{R}^{3}$ be a point chosen uniformly at random from the unit sphere $S^{2}:=\left\{(a, b, c) \in \mathbb{R}^{3} \mid a^{2}+b^{2}+c^{2}=1\right\}...

How many subsets of $\{1,2, \ldots, 2 n\}$ do not contain two numbers with sum $2 n+1?$$3^{n}+\binom{2 n}{2}$$3^{n}$$4^{n}-\binom{2 n}{2}$$2^{n}$None of the above 

What is the solution to the following recursion?$$\begin{array}{l}T(n)=T\left(\dfrac{n}{2}\right)+T\left(\dfrac{n}{3}\right)+T\left(\dfrac{n}{6}\right)+O(n), \\T(n)=5 \qu...

Let $\varphi$ be a propositional formula on $n$ variables, with $n \geq 1$. Consider the following statements.$\varphi$ is satisfiable$\neg \varphi$ is unsatisfiable$\var...

Consider the following sets.The set of all natural numbers $\mathbb{N}$.The set of all rational numbers $\mathbb{Q}$.The set of all functions from $\mathbb{N}$ to $\{1,2,...

Consider the following languages:$L_{1}$ is the set of languages recognised by a deterministic pushdown automaton.$L_{2}$ is the set of languages recognised by a nondeter...

Given a directed graph $G$ and an initial vertex $s$, we would like to $explore$ the graph from $s$, that is, starting from $s$ see all vertices along a path.For example,...

The complexity class $\textsf{NP}$ corresponds to the class of languages which can be accepted by some nondeterministic Turing machine in polynomial time.The complexity c...

Consider the following non-deterministic automata for a unary language $L \subseteq\{a\}^{*}$ consisting of 11 states with the start state as the unique final state. All ...

Consider the following pseudocode. The program should take as input $n$, and when it ends, the element $A[i]$ of the vector $A$ should contain the binomial coefficient $\...

Suppose $3$ elements are hashed independently and uniformly at random one by one to slots in a hash table of size $6$ (assume that in case of a collision, the element is ...

Recall the insertion sort algorithm: procedure INSERTIONSORT(A) for i ← 1 to A.length - 1 do key ← A[i] j ← i - 1 while j ≥ 0 and A[j] key do A[j + ...

Let $m, n$ be positive integers such that $10 \leq m \leq n$. Let $A$ be an $m \times n$ matrix with real entries. Let $v$ be a unit vector in $\mathbb{R}^{n}$ with the m...