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Recent questions in Engineering Mathematics
3
votes
1
answer
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
by
admin
314
views
tifr2022
graph-theory
graph-matching
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
by
admin
119
views
tifr2022
combinatory
binomial-theorem
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
by
admin
117
views
tifr2022
graph-theory
graph-coloring
p-np-npc-nph
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
by
admin
122
views
tifr2022
graph-theory
graph-connectivity
rank-of-matrix
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
by
admin
137
views
tifr2022
linear-algebra
vector-space
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
tifr2022
combinatory
counting
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
tifr2022
probability
expectation
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
tifr2022
linear-algebra
matrix
determinant
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
tifr2022
probability
uniform-distribution
functions
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
tifr2022
calculus
maxima-minima
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
tifr2022
probability
uniform-distribution
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
tifr2022
linear-algebra
rank-of-matrix
eigen-value
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
tifr2022
set-theory&algebra
set-theory
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
tifr2022
probability
conditional-probability
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
tifr2022
set-theory&algebra
set-theory
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
tifr2022
probability
conditional-probability
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
tifr2022
calculus
differentiation
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
tifr2022
probability
uniform-distribution
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
by
ryandany07
121
views
engineering-mathematics
linear-algebra
matrix
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
by
Overflow04
165
views
engineering-mathematics
generating-functions
test-series
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