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121

TIFR CSE 2022 | Part B | Question: 6
We are given a graph $G$ along with a matching $M$ and a vertex cover $C$ in it such that $|M|=|C|$. Consider the following statements: $M$ is a maximum matching in $G$. $C$ is a minimum vertex cover in $G$. $G$ is a bipartite graph. Which of ... $(1)$ and $(2)$ are correct All the three statements $(1), (2),$ and $(3)$ are correct

admin asked in Graph Theory Sep 1, 2022

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2 votes

1 answer

122

TIFR CSE 2022 | Part B | Question: 9
Let $n \geq 2$ be any integer. Which of the following statements is $\text{FALSE}?$ $n!$ divides the product of any $n$ consecutive integers $\displaystyle{}\sum_{i=0}^n\left(\begin{array}{c}n \\ i\end{array}\right)=2^n$ ... an odd prime, then $n$ divides $2^{n-1}-1$ $n$ divides $\left(\begin{array}{c}2 n \\ n\end{array}\right)$

admin asked in Combinatory Sep 1, 2022

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1 vote

0 answers

123

TIFR CSE 2022 | Part B | Question: 12
Given an undirected graph $G$, an ordering $\sigma$ of its vertices is called a perfect ordering if for every vertex $v$, the neighbours of $v$ which precede $v$ in $\sigma$ form a clique in $G$. Recall that given an undirected ... SPECIAL-COLOURING are in $\mathrm{P}$ Neither of SPECIAL-CLiQUE and SPECIAL-COLOURING is in $\mathrm{P},$ but both are decidable

admin asked in Graph Theory Sep 1, 2022

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117 views

1 vote

1 answer

124

TIFR CSE 2022 | Part B | Question: 14
Let $G$ be a directed graph (with no self-loops or parallel edges) with $n \geq 2$ vertices and $m$ edges. Consider the $n \times m$ incidence matrix $M$ of $G$, whose rows are indexed by the vertices of $G$ and the columns by the edges of $G$ ... . Then, what is the rank of $M?$ $m-1$ $m-n+1$ $\lceil m / 2\rceil$ $n-1$ $\lceil n / 2\rceil$

admin asked in Graph Theory Sep 1, 2022

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122 views

1 vote

0 answers

125

TIFR CSE 2022 | Part B | Question: 15
Let $\mathbb{R}$ denote the set of real numbers. Let $d \geq 4$ and $\alpha \in \mathbb{R}$ ... $\left(a_0, a_1, \ldots, a_d\right) \in S$, the function $ x \mapsto \sum_{i=0}^d a_i x^i $ has a local optimum at $\alpha$

admin asked in Linear Algebra Sep 1, 2022

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137 views

0 votes

1 answer

126

TIFR CSE 2022 | Part A | Question: 1
A snail crawls up a vertical pole $75$ feet high, starting from the ground. Each day it crawls up $5$ feet, and each night it slides down $4$ feet. When will it first reach the top of the pole? $75^{\text {th}}$ day $74^{\text {th}}$ day $73^{ \text{rd}}$ day $72^{\text {nd }}$ day $71^{\text {st }}$ day

Lakshman Patel RJIT asked in Combinatory Sep 1, 2022

by Lakshman Patel RJIT

226 views

0 votes

2 answers

127

TIFR CSE 2022 | Part A | Question: 2
We would like to invite a minimum number $n$ of people (their birthdays are independent of each other) to a party such that the expected number of pairs of people that share the same birthday is at least $1.$ What should $n$ be? (Ignore leap years, so ... birthdays fall with equal probability on each of the $365$ days of the year.) $23$ $28$ $92$ $183$ $366$

Lakshman Patel RJIT asked in Probability Sep 1, 2022

by Lakshman Patel RJIT

253 views

1 vote

1 answer

128

TIFR CSE 2022 | Part A | Question: 4
Consider the polynomial $p(x)=x^3-x^2+x-1$. How many symmetric matrices with integer entries are there whose characteristic polynomial is $p$? (Recall that the characteristic polynomial of a square matrix $A$ in the variable $x$ is defined to be the determinant of the matrix $(A-x I)$ where $I$ is the identity matrix.) $0$ $1$ $2$ $4$ Infinitely many

Lakshman Patel RJIT asked in Linear Algebra Sep 1, 2022

by Lakshman Patel RJIT

186 views

2 votes

1 answer

129

TIFR CSE 2022 | Part A | Question: 5
Let $\mathcal{F}$ be the set of all functions mapping $\{1, \ldots, n\}$ to $\{1, \ldots, m\}$. Let $f$ be a function that is chosen uniformly at random from $\mathcal{F}$. Let $x, y$ be distinct elements from the set $\{1, \ldots, n\}$. Let $p$ denote the probability ... Then, $p=0$ $p=\frac{1}{n^m}$ $0<p \leq \frac{1}{m^n}$ $p=\frac{1}{m}$ $p=\frac{1}{n}$

Lakshman Patel RJIT asked in Probability Sep 1, 2022

by Lakshman Patel RJIT

115 views

0 votes

0 answers

130

TIFR CSE 2022 | Part A | Question: 6
Let $f$ be a polynomial of degree $n \geq 3$ all of whose roots are non-positive real numbers. Suppose that $f(1)=1$. What is the maximum possible value of $f^{\prime}(1)?$ $1$ $n$ $n+1$ $\frac{n(n+1)}{2}$ $f^{\prime}(1)$ can be arbitrarily large given only the constraints in the question

Lakshman Patel RJIT asked in Calculus Sep 1, 2022

by Lakshman Patel RJIT

117 views

0 votes

1 answer

131

TIFR CSE 2022 | Part A | Question: 7
Initially, $N$ white beads are arranged in a circle. A number $k$ is chosen uniformly at random from $\{1, \ldots, N-1\}$. Then a set of $k$ beads is chosen uniformly from the white beads, and these $k$ beads are coloured black. The position of the beads remains ... None of the above

Lakshman Patel RJIT asked in Probability Sep 1, 2022

by Lakshman Patel RJIT

162 views

13 votes

1 answer

132

TIFR CSE 2022 | Part A | Question: 8
Let $A$ be the $(n+1) \times(n+1)$ matrix given below, where $n \geq 1$. For $i \leq n$, the $i$-th row of $A$ has every entry equal to $2i-1$ and the last row, i.e., the $(n+1)$-th row of $A$ has every entry equal to $-n^2$ ... $A$ has rank $n$ $A^2$ has rank $1$ All the eigenvalues of $A$ are distinct All the eigenvalues of $A$ are $0$ None of the above

Lakshman Patel RJIT asked in Linear Algebra Sep 1, 2022

by Lakshman Patel RJIT

294 views

0 votes

1 answer

133

TIFR CSE 2022 | Part A | Question: 9
You are given the following properties of sets $A, B, X$, and $Y$. For notation, $|A|$ denotes the cardinality of set $A$ (i.e., the number of elements in $A$ ), and $A \backslash B$ denotes the set of elements that are in $A$ but not in $B$. $A \cup B=X \cup Y$ ... $|X|=5$ $|Y|=5$ $|A \cup X|=|B \cup Y|$ $|A \cap X|=|B \cap Y|$ $|A|=|B|$

Lakshman Patel RJIT asked in Set Theory & Algebra Sep 1, 2022

by Lakshman Patel RJIT

123 views

0 votes

1 answer

134

TIFR CSE 2022 | Part A | Question: 10
Consider a bag containing colored marbles. There are $n$ marbles in the bag such that there is exactly one pair of marbles of color $i$ for each $i \in\{1, \ldots, m\}$ and the rest of the marbles are of distinct colors (different from colors $\{1, \ldots, m\}$ ). You draw ... $\frac{2m}{n}$ $\frac{2m}{n(n-1)}$ $\frac{2m}{n^2}$ $\frac{m}{n(n-1)}$

Lakshman Patel RJIT asked in Probability Sep 1, 2022

by Lakshman Patel RJIT

108 views

0 votes

0 answers

135

TIFR CSE 2022 | Part A | Question: 11
Let $X$ be a finite set. A family $\mathcal{F}$ of subsets of $X$ is said to be upward closed if the following holds for all sets $A, B \subseteq X$ ... $\mathcal{F} \sqcup \mathcal{G}=\mathcal{G} \backslash \mathcal{F}$ None of the above

Lakshman Patel RJIT asked in Set Theory & Algebra Sep 1, 2022

by Lakshman Patel RJIT

76 views

0 votes

1 answer

136

TIFR CSE 2022 | Part A | Question: 12
Alice plays the following game on a math show. There are $7$ boxes and identical prizes are hidden inside $3$ of the boxes. Alice is asked to choose a box where a prize might be. She chooses a box uniformly at random. From the unchosen boxes which do not have a prize, ... ). Her probability of winning the prize is $3 / 7$ $1 / 2$ $17 / 30$ $18 / 35$ $9 / 19$

Lakshman Patel RJIT asked in Probability Sep 1, 2022

by Lakshman Patel RJIT

128 views

0 votes

0 answers

137

TIFR CSE 2022 | Part A | Question: 14
Suppose $w(t)=4 e^{i t}, x(t)=3 e^{i(t+\pi / 3)}, y(t)=3 e^{i(t-\pi / 3)}$ and $z(t)=3 e^{i(t+\pi)}$ are points that move in the complex plane as the time $t$ varies in $(-\infty, \infty)$. Let $c(t)$ ... $\frac{1}{2 \pi}$ $2 \pi$ $\sqrt{3} \pi$ $\frac{1}{\sqrt{3} \pi}$ $1$

Lakshman Patel RJIT asked in Calculus Sep 1, 2022

by Lakshman Patel RJIT

97 views

0 votes

1 answer

138

TIFR CSE 2022 | Part A | Question: 15
Fix $n \geq 4$. Suppose there is a particle that moves randomly on the number line, but never leaves the set $\{1,2, \ldots, n\}$. The initial probability distribution of the particle is $\pi$ i.e., the probability that particle is in location $i$ is given by $\pi(i)$. In the ... $\pi(1)=1$ and $\pi(i)=0$ for $i \neq 1$ $\pi(n)=1$ and $\pi(i)=0$ for $i \neq n$

Lakshman Patel RJIT asked in Probability Sep 1, 2022

by Lakshman Patel RJIT

130 views

0 votes

0 answers

139

Linear Algebra
MX = O is a homogeneous equation and such an equation when |M| = 0 has non trivial solution. M: Square Matrix O: Null Matrix Kindly help me with the above statement.

ryandany07 asked in Mathematical Logic Aug 30, 2022

121 views

0 votes

0 answers

140

Engineering Mathematics
How the value of a1 = 3, a2 = 2 is calculated.

Overflow04 asked in Mathematical Logic Aug 25, 2022

165 views

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