GATE Overflow for GATE CSE - Recent questions in Engineering Mathematics
https://gateoverflow.in/questions/mathematics
Powered by Question2AnswerProvided by teacher, topic -arguments
https://gateoverflow.in/389396/provided-by-teacher-topic-arguments
Show that: (r → ~ q, r ∪ S, S → ~ q, p → q) ↔ ~ p<br />
<br />
are inconsistent.Mathematical Logichttps://gateoverflow.in/389396/provided-by-teacher-topic-argumentsSat, 26 Nov 2022 13:38:07 +0000Solve the simultaneous recurrence relations
https://gateoverflow.in/389383/solve-the-simultaneous-recurrence-relations
an = an−1 + bn−1<br />
bn = an−1 − bn−1<br />
with a0 = 1 and b0 = 2.Combinatoryhttps://gateoverflow.in/389383/solve-the-simultaneous-recurrence-relationsFri, 25 Nov 2022 17:44:20 +0000Probability Distribution
https://gateoverflow.in/389366/probability-distribution
Let X have a Poisson distribution with parameter λ = 1. What is the probability that X ≥ 2 given that X ≤ 4?Probabilityhttps://gateoverflow.in/389366/probability-distributionFri, 25 Nov 2022 13:01:40 +0000Conditional Probability
https://gateoverflow.in/389295/conditional-probability
English and American spelling are rigour and rigor, respectively. A man staying at Al Rashid hotel writes this word, and a letter taken at random from his spelling is found to be a vowel. If 40 percent of the English-speaking men at the hotel are English and 60 percent are American, what is the probability that the writer is an Englishman?Probabilityhttps://gateoverflow.in/389295/conditional-probabilityThu, 24 Nov 2022 15:42:45 +0000#LinearAlgebra
https://gateoverflow.in/389262/%23linearalgebra
<ul>
<li>Is this correct?</li>
<li><img alt="" src="https://gateoverflow.in/?qa=blob&qa_blobid=8216766089028809665"></li>
</ul>Linear Algebrahttps://gateoverflow.in/389262/%23linearalgebraThu, 24 Nov 2022 07:02:21 +0000TIFR system Science 2017 question 7
https://gateoverflow.in/389135/tifr-system-science-2017-question-7
A circulant matrix is a square matrix whose each row is the preceding row rotated to the right by one element, e.g., the following is a 3 × 3 circulant matrix \begin{pmatrix}<br />
1 & 2 & 3\\<br />
3 & 1 & 2\\<br />
2 & 3 & 1<br />
\end{pmatrix}<br />
<br />
For any n × n circulant matrix (n > 5), which of the following n-length vectors is always an eigenvector?<br />
<br />
(a) A vector whose k-th element is k<br />
<br />
(b) A vector whose k-th element is nk<br />
<br />
(c) A vector whose k-th element is $\exp \left(j\dfrac{2π(n − 5)k}{ n}\right)$ where $j = \sqrt{−1}$<br />
<br />
(d) A vector whose k-th element is $\sin h (2πk/n)$<br />
<br />
(e) None of the aboveLinear Algebrahttps://gateoverflow.in/389135/tifr-system-science-2017-question-7Tue, 22 Nov 2022 14:24:49 +0000TIFR CS & SS 2018 SS part Question 12
https://gateoverflow.in/389114/tifr-cs-%26-ss-2018-ss-part-question-12
Suppose that Amitabh Bachchan has ten coins in his pocket. 3 coins have tails on both sides. 4 coins have heads on both sides. 3 coins have heads on one side and tails on the other and both the outcomes are equally likely when that coin is flipped. In a bet with Dharmendra, Amitabh picks up a coin at random (each coin is equally likely to be picked) from these ten coins, flips it and finds that the outcome is tails. What is the probability that the other side of this coin is heads?Probabilityhttps://gateoverflow.in/389114/tifr-cs-%26-ss-2018-ss-part-question-12Tue, 22 Nov 2022 09:46:36 +0000KSET 2015 computer science
https://gateoverflow.in/388944/kset-2015-computer-science
Let X and Y be two sets having n elements<br />
each. Then the total number of bijective<br />
functions from X and Y is<br />
(A) n (B) 2n<br />
(C) n! (D) 2nSet Theory & Algebrahttps://gateoverflow.in/388944/kset-2015-computer-scienceSat, 19 Nov 2022 15:21:00 +0000gateforum
https://gateoverflow.in/388918/gateforum
<p><img alt="" src="https://gateoverflow.in/?qa=blob&qa_blobid=3503592563800676053"></p>Mathematical Logichttps://gateoverflow.in/388918/gateforumSat, 19 Nov 2022 12:42:34 +0000gateforum
https://gateoverflow.in/388917/gateforum
<p><img alt="" src="https://gateoverflow.in/?qa=blob&qa_blobid=3503592563800676053"></p>Mathematical Logichttps://gateoverflow.in/388917/gateforumSat, 19 Nov 2022 12:41:48 +0000graph theory ,discrete math
https://gateoverflow.in/388839/graph-theory-discrete-math
how many subgraph with atleast 1 vertex does k2 have? (graph theory question)Mathematical Logichttps://gateoverflow.in/388839/graph-theory-discrete-mathFri, 18 Nov 2022 05:49:47 +0000mathe
https://gateoverflow.in/388147/mathe
<p><img alt="" src="https://gateoverflow.in/?qa=blob&qa_blobid=6236960930499781645"></p>Mathematical Logichttps://gateoverflow.in/388147/matheFri, 11 Nov 2022 09:08:27 +0000Can a uniquely complemented lattice may not be distributive?
https://gateoverflow.in/387612/can-uniquely-complemented-lattice-may-not-be-distributive
Set Theory & Algebrahttps://gateoverflow.in/387612/can-uniquely-complemented-lattice-may-not-be-distributiveSun, 06 Nov 2022 07:02:19 +0000#Unacademy
https://gateoverflow.in/387450/%23unacademy
**Every group of prime order is always an abelian group<br />
<br />
on this fact please explain how can a 2 element group be an abelian group, please give an exampleSet Theory & Algebrahttps://gateoverflow.in/387450/%23unacademyFri, 04 Nov 2022 09:45:24 +0000#Unacademy
https://gateoverflow.in/387405/%23unacademy
Find inverse in a*b=a+b-ab for all a,b belongs to Q-{-1}. where Q is a rational number?<br />
<br />
please explain how the inverse will satisfy the equation of inverse(a*b=b*a=e)Set Theory & Algebrahttps://gateoverflow.in/387405/%23unacademyFri, 04 Nov 2022 09:45:19 +0000GATE 2008 | Question 21
https://gateoverflow.in/387302/gate-2008-question-21
<p>I have been trying to solve the question <a href="https://gateoverflow.in/3282/gate-it-2008-question-21" rel="nofollow">GATE CSE 2008 Question</a>.</p>
<p>Are the following two representations logically equivalent ?</p>
<ol>
<li>$\beta \rightarrow (\exists x, \alpha (x))$</li>
<li>$\exists x, \beta \rightarrow \alpha (x)$</li>
</ol>Mathematical Logichttps://gateoverflow.in/387302/gate-2008-question-21Thu, 03 Nov 2022 12:58:59 +0000Madeasy Test series 2023 Discrete Maths
https://gateoverflow.in/387285/madeasy-test-series-2023-discrete-maths
<p><img alt="" src="https://gateoverflow.in/?qa=blob&qa_blobid=14494547340703992786"></p>Mathematical Logichttps://gateoverflow.in/387285/madeasy-test-series-2023-discrete-mathsWed, 02 Nov 2022 18:27:35 +0000Engineering Mathematics
https://gateoverflow.in/387056/engineering-mathematics
<p>Can somebody brief description about this topic(means formula or link to study).</p>
<p>Actually this question is asked in TOC test, but looking more like mathematics part.<img alt="" src="https://gateoverflow.in/?qa=blob&qa_blobid=10428623918757308315"></p>Combinatoryhttps://gateoverflow.in/387056/engineering-mathematicsSun, 30 Oct 2022 09:59:52 +0000Let 𝐼(𝑥)be the statement “𝑥has an Internet connection” and 𝐶(𝑥,𝑦)be the statement “𝑥and𝑦have chatted over the Internet,” where the domain for the variables 𝑥and 𝑦consists of all students in your class. Use quantifiers to express the followingstatement:“Everyone in your class with an Internet connection has chatted over the Internet with at least one other student in your class.
https://gateoverflow.in/386896/connection-variables-quantifiers-followingstatement-connection
Mathematical Logichttps://gateoverflow.in/386896/connection-variables-quantifiers-followingstatement-connectionSat, 29 Oct 2022 10:02:58 +0000Translate the given statement into propositional logicexpressions:“A man qualifies for the marathon if his best previous time is less than 3 hours and a woman qualifies for the marathon if her best previous time is less than 3.5 hours.
https://gateoverflow.in/386894/translate-statement-propositional-logicexpressions-qualifies
Mathematical Logichttps://gateoverflow.in/386894/translate-statement-propositional-logicexpressions-qualifiesSat, 29 Oct 2022 10:02:08 +0000Set theory
https://gateoverflow.in/386463/set-theory
Let A = set of all real numbers. '*' be a binary operation <br />
<br />
a*b = a + b + a. b<br />
<br />
(A, *) is a group or not. <br />
<br />
<br />
<br />
Doubt : Till finding identity element (=0) every thing is okay. <br />
<br />
While finding inverse: <br />
<br />
a * $a^{-1}$ = e<br />
<br />
a + $a^{-1}$ + a$a^{-1}$ = 0<br />
<br />
a + $a^{-1}$ + 1 = 0<br />
<br />
<br />
<br />
Or <br />
<br />
a + $a^{-1}$(1+a) = 0<br />
<br />
<br />
<br />
Both will lead to different answer. <br />
<br />
What is correct.Set Theory & Algebrahttps://gateoverflow.in/386463/set-theoryTue, 25 Oct 2022 12:23:34 +0000Set Theory
https://gateoverflow.in/386458/set-theory
If A = {1, 3,5,7,.......}<br />
<br />
Then (A, +) is semi group or not. <br />
<br />
<br />
<br />
Doubt :<br />
<br />
a +(b+c) =( a + b) +c<br />
<br />
So associativity property is present <br />
<br />
But a + b doesn't belong to A<br />
<br />
So closure property is not present. <br />
<br />
<br />
<br />
<br />
<br />
So A is semi group or not.Set Theory & Algebrahttps://gateoverflow.in/386458/set-theoryTue, 25 Oct 2022 11:55:43 +0000Gate@Zeal 2022 Test Series
https://gateoverflow.in/385951/gate%40zeal-2022-test-series
<p><img alt="" height="70" src="https://cdn-test-assets.classplus.co/production_1631533749291/814ec1ea-8276-4931-b6fd-d4f02418a916.png" width="673"></p>
<p>Answer -3.7</p>Calculushttps://gateoverflow.in/385951/gate%40zeal-2022-test-seriesSat, 22 Oct 2022 02:29:44 +0000Matrix
https://gateoverflow.in/385569/matrix
<p><img alt="" src="https://gateoverflow.in/?qa=blob&qa_blobid=10166992849518532842"></p>
<p>While studying Linear algebra I got 2 perspectives. Which meaning out of these 2 is more accurate?</p>Linear Algebrahttps://gateoverflow.in/385569/matrixMon, 17 Oct 2022 15:18:34 +0000Poisson distribution
https://gateoverflow.in/384545/poisson-distribution
An 800 page book has 400 misprints. If the misprints are distributed uniformly throughout the book and the Poisson approximation to the binomial distribution is used to calculate the probability of exactly 2 misprints on page 16, which of the following represents the correct use of the Poisson approximation?Mathematical Logichttps://gateoverflow.in/384545/poisson-distributionWed, 05 Oct 2022 10:26:16 +0000is this argument valid: {P->Q , Q->R}-> R'
https://gateoverflow.in/384411/is-this-argument-valid-p-q-q-r-r
Mathematical Logichttps://gateoverflow.in/384411/is-this-argument-valid-p-q-q-r-rMon, 03 Oct 2022 09:39:22 +0000Topic: Combining relations (Composition of relations)
https://gateoverflow.in/383860/topic-combining-relations-composition-of-relations
Let R be a relation.<br />
<br />
Why $R^2 oR^2 !=R^4$ while $R^3 oR =R^4$?<br />
<br />
Please explain.Set Theory & Algebrahttps://gateoverflow.in/383860/topic-combining-relations-composition-of-relationsMon, 26 Sep 2022 12:11:40 +0000KENNITH ROSEN LATTICE
https://gateoverflow.in/383808/kennith-rosen-lattice
Find a compatible total order for the divisibility relation<br />
on the set {1, 2, 3, 6, 8, 12, 24, 36}.Set Theory & Algebrahttps://gateoverflow.in/383808/kennith-rosen-latticeMon, 26 Sep 2022 06:45:33 +0000Ace Book for Discreet Mathematics , Combinatorics.
https://gateoverflow.in/383442/ace-book-for-discreet-mathematics-combinatorics
Number of ways to assign 5 different people in 3 different rooms, so that each room contains at least one person?Mathematical Logichttps://gateoverflow.in/383442/ace-book-for-discreet-mathematics-combinatoricsThu, 22 Sep 2022 13:58:52 +0000Internet
https://gateoverflow.in/383405/internet
For f(x)=√x x€[0,b] the number c satisfying mean value therom is c=1Calculushttps://gateoverflow.in/383405/internetWed, 21 Sep 2022 17:31:12 +0000Engineering mathematics
https://gateoverflow.in/383330/engineering-mathematics
If A is a non-zero column matrix of order n×1 and B is a non-zero row matrix of order 1×n then rank of AB equals ?<br />
<br />
<br />
<br />
Rank(ab) can be zero???Linear Algebrahttps://gateoverflow.in/383330/engineering-mathematicsWed, 21 Sep 2022 05:47:59 +0000Made Easy: Linear Algebra (MSQ)
https://gateoverflow.in/383232/made-easy-linear-algebra-msq
<p>Let $A$ be a $3$ x $3$ matrix with rank $2$. Then, $AX=0$ has</p>
<ol style="list-style-type:lower-alpha" type="a">
<li>The trivial solution $X=0$.</li>
<li>One independent solution.</li>
<li>Two independent solution.</li>
<li>Three independent solution.</li>
</ol>Linear Algebrahttps://gateoverflow.in/383232/made-easy-linear-algebra-msqMon, 19 Sep 2022 16:57:07 +0000Gate@Zeal Test Series 2023
https://gateoverflow.in/383200/gate%40zeal-test-series-2023
<p><img alt="" src="https://cms-cdn.classplus.co/parserImages/13c0a68a-e0aa-472a-ac06-646780aa2b55.png"></p>
<p>Ans: 211</p>Combinatoryhttps://gateoverflow.in/383200/gate%40zeal-test-series-2023Mon, 19 Sep 2022 08:45:42 +0000Gate@Zeal Test Series 2023
https://gateoverflow.in/383199/gate%40zeal-test-series-2023
<p><img alt="" height="85" src="https://cms-cdn.classplus.co/parserImages/da00de4b-f7f6-48ce-8dc0-1b2d23a5f6e3.png" width="552"></p>
<p>Ans: 43333</p>Combinatoryhttps://gateoverflow.in/383199/gate%40zeal-test-series-2023Mon, 19 Sep 2022 08:43:58 +0000Self doubt.
https://gateoverflow.in/383198/self-doubt
If A, B & C are matrices & AB=AC then B=C?<br />
<br />
as we know it is not always true because when A is singular matrix then B=C not possible so what is the right ans to say B=C is (always not equal) or (may or may not be equal) or (simply not equal)?Linear Algebrahttps://gateoverflow.in/383198/self-doubtMon, 19 Sep 2022 08:36:04 +0000Gate At Zeal
https://gateoverflow.in/383087/gate-at-zeal
Question → If (G,*) is a group of order 960 and there exist a in G such that a^m=e for some integer m<=960 where e is identity element of G then total number of possible value of m is___________<br />
<br />
<br />
<br />
Answer==28Set Theory & Algebrahttps://gateoverflow.in/383087/gate-at-zealSat, 17 Sep 2022 10:58:31 +0000PhD Admissions Written Test (Basic)
https://gateoverflow.in/382697/phd-admissions-written-test-basic
Let x1, x2, ...x8 be 8 propositional variables. Let · represent AND connective ⊕ represent the Exclusive-or connective.<br />
<br />
The number of satisfying assignments of the formula x1 ⊕ x2 ⊕ ...x8 is _________________<br />
<br />
The number of satisfying assignments of the formula (x1·x2) ⊕ (x3·x4)... ⊕ (x7·x8) is __________________Mathematical Logichttps://gateoverflow.in/382697/phd-admissions-written-test-basicSun, 11 Sep 2022 05:17:18 +0000A First Course In Probability, 9th Edition, Indian Version, Sheldon Ross, Chapter 3, Problems, 33.
https://gateoverflow.in/382652/course-probability-edition-version-sheldon-chapter-problems
The meteorology department predicts the rain correctly with a probability of 4/5. When rain is predicted Amit travels by his car. When rain is not forecasted, he travels with his car with a probability of 0.5. Assuming that it rains in a day with a probability of 1/2. Find the probability that Amit does not travel by car given it rains.Probabilityhttps://gateoverflow.in/382652/course-probability-edition-version-sheldon-chapter-problemsSat, 10 Sep 2022 06:15:08 +0000igate test series
https://gateoverflow.in/382385/igate-test-series
Selection of how many integers from the first ten positive integers (1, 2, ...) guarantees that there must be a pair of these integers with a sum equal to 11 ?Combinatoryhttps://gateoverflow.in/382385/igate-test-seriesTue, 06 Sep 2022 19:15:35 +0000Predicate Translation
https://gateoverflow.in/382336/predicate-translation
<p><em><strong>S(x):</strong></em> <strong>x is a Student</strong><br>
<strong><em>P(x):</em></strong> <strong>x is a Professor</strong><br>
<em><strong>A(x, y):</strong></em> <strong>x has asked a question to y<br>
Domain not given, so we have to think about default domain</strong><br>
<strong>Q1)</strong> Translate <strong>“There is a student who has asked every professor a question”</strong><br>
<strong>Q2)</strong> Translate <strong>“There is a professor who has asked every student a question”</strong><br>
<strong>Q3)</strong> Translate <strong>“There is a professor who has been asked a question by every student”</strong><br>
<strong>Q4)</strong> Translate <strong>“There is a student who has been asked a question by every professor”</strong></p>Mathematical Logichttps://gateoverflow.in/382336/predicate-translationTue, 06 Sep 2022 06:03:30 +0000Eigen Value Of A Matrix
https://gateoverflow.in/382175/eigen-value-of-a-matrix
For given Matrix:<br />
<br />
[ 1 2 3<br />
<br />
1 5 1<br />
<br />
3 1 1 ]<br />
<br />
Why does the sum of the eigen values of above matrix is the sum of diagonal elements of that matrix?Linear Algebrahttps://gateoverflow.in/382175/eigen-value-of-a-matrixSun, 04 Sep 2022 08:46:39 +0000TIFR CSE 2022 | Part B | Question: 2
https://gateoverflow.in/382007/tifr-cse-2022-part-b-question-2
<p>Let $G=(V, E)$ be an undirected simple graph. A subset $M \subseteq E$ is a <em>matching</em> in $G$ if distinct edges in $M$ do not share a vertex. A matching is <em>maximal </em>if no strict superset of $M$ is a matching. How many maximal matchings does the following graph have?</p>
<p><img alt="" src="https://gateoverflow.in/?qa=blob&qa_blobid=12788851749305281586"></p>
<ol start="1" style="list-style-type:upper-alpha">
<li>$1$</li>
<li>$2$</li>
<li>$3$</li>
<li>$4$</li>
<li>$5$</li>
</ol>Graph Theoryhttps://gateoverflow.in/382007/tifr-cse-2022-part-b-question-2Thu, 01 Sep 2022 17:42:50 +0000TIFR CSE 2022 | Part B | Question: 6
https://gateoverflow.in/382003/tifr-cse-2022-part-b-question-6
<p>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:</p>
<ol>
<li>$M$ is a maximum matching in $G$.</li>
<li>$C$ is a minimum vertex cover in $G$.</li>
<li>$G$ is a bipartite graph.</li>
</ol>
<p>Which of the following is $\text{TRUE}$?</p>
<ol start="1" style="list-style-type:upper-alpha">
<li>Only statement $(1)$ is correct</li>
<li>Only statement $(2)$ is correct</li>
<li>Only statement $(3)$ is correct</li>
<li>Only statements $(1)$ and $(2)$ are correct</li>
<li>All the three statements $(1), (2),$ and $(3)$ are correct</li>
</ol>Graph Theoryhttps://gateoverflow.in/382003/tifr-cse-2022-part-b-question-6Thu, 01 Sep 2022 17:42:49 +0000TIFR CSE 2022 | Part B | Question: 9
https://gateoverflow.in/382000/tifr-cse-2022-part-b-question-9
<p>Let $n \geq 2$ be any integer. Which of the following statements is $\text{FALSE}?$</p>
<ol start="1" style="list-style-type:upper-alpha">
<li>$n!$ divides the product of any $n$ consecutive integers</li>
<li>$\displaystyle{}\sum_{i=0}^n\left(\begin{array}{c}n \\ i\end{array}\right)=2^n$</li>
<li>$\left(\begin{array}{c}n \\ i\end{array}\right)=\left(\begin{array}{c}n-1 \\ i\end{array}\right)+\left(\begin{array}{c}n-1 \\ i-1\end{array}\right)$, where $1 \leq i \leq n-1$</li>
<li>If $n$ is an odd prime, then $n$ divides $2^{n-1}-1$</li>
<li>$n$ divides $\left(\begin{array}{c}2 n \\ n\end{array}\right)$</li>
</ol>Combinatoryhttps://gateoverflow.in/382000/tifr-cse-2022-part-b-question-9Thu, 01 Sep 2022 17:42:49 +0000TIFR CSE 2022 | Part B | Question: 12
https://gateoverflow.in/381997/tifr-cse-2022-part-b-question-12
<p>Given an undirected graph $G$, an ordering $\sigma$ of its vertices is called a <em>perfect ordering</em> if for every vertex $v$, the neighbours of $v$ which precede $v$ in $\sigma$ form a clique in $G$.</p>
<p>Recall that given an undirected graph $G$, a <em>clique</em> in $G$ is a subset of vertices every two of which are connected by an edge, while a <em>perfect colouring</em> of $G$ <em>with</em> $k$ colours is an assignment of labels from the set $\{1,2, \ldots, k\}$ to the vertices of $G$ such that no two vertices which are adjacent in $G$ receive the same label.</p>
<p>Consider the following problems.</p>
<p><strong>Problem SPECIAL-CLIQUE</strong></p>
<p><strong>INPUT:</strong> An undirected graph $G$, a positive integer $k$, and a perfect ordering $\sigma$ of the vertices of $G$.</p>
<p><strong>OUTPUT:</strong> Yes, if $G$ has a clique of size at least $k$, No otherwise.</p>
<p><strong>Problem SPECIAL-COLOURING</strong></p>
<p><strong>INPUT:</strong> An undirected graph $G$, a positive integer $k$, and a perfect ordering $\sigma$ of the vertices of $G$.</p>
<p><strong>OUTPUT:</strong> Yes, if $G$ has a proper colouring with at most $k$ colours, No otherwise.</p>
<p>Assume that $\mathrm{P} \neq N P$. Which of the following statements is true?</p>
<ol start="1" style="list-style-type:upper-alpha">
<li>Both SPECIAL-CLIQUE and SPECIAL-COLOURING are undecidable</li>
<li>Only SPECIAL-CLIQUE is in $\mathrm{P}$</li>
<li>Only SPECIAL-COLOURING is in $\mathrm{P}$</li>
<li>Both SPECIAL-CLIQUE and SPECIAL-COLOURING are in $\mathrm{P}$</li>
<li>Neither of SPECIAL-CLiQUE and SPECIAL-COLOURING is in $\mathrm{P},$ but both are decidable</li>
</ol>Graph Theoryhttps://gateoverflow.in/381997/tifr-cse-2022-part-b-question-12Thu, 01 Sep 2022 17:42:48 +0000TIFR CSE 2022 | Part B | Question: 14
https://gateoverflow.in/381995/tifr-cse-2022-part-b-question-14
<p>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 $G$. The entry $m_{v, e}$ is defined as follows.</p>
<p>$$ m_{v, e}= \begin{cases}-1 & \text { if } e=(v, w) \text { for some vertex } w, \\ +1 & \text { if } e=(u, v) \text { for some vertex } u, \\ 0 & \text { otherwise. }\end{cases} $$</p>
<p>Suppose every vertex of $G$ is reachable from a special source vertex of $G$. Then, what is the rank of $M?$ </p>
<ol start="1" style="list-style-type:upper-alpha">
<li>$m-1$</li>
<li>$m-n+1$</li>
<li>$\lceil m / 2\rceil$</li>
<li>$n-1$</li>
<li>$\lceil n / 2\rceil$</li>
</ol>Graph Theoryhttps://gateoverflow.in/381995/tifr-cse-2022-part-b-question-14Thu, 01 Sep 2022 17:42:48 +0000TIFR CSE 2022 | Part B | Question: 15
https://gateoverflow.in/381994/tifr-cse-2022-part-b-question-15
<p>Let $\mathbb{R}$ denote the set of real numbers. Let $d \geq 4$ and $\alpha \in \mathbb{R}$. Let</p>
<p>$$ S=\left\{\left(a_0, a_1, \ldots, a_d\right) \in \mathbb{R}^{d+1}: \sum_{i=0}^d a_i \alpha^i=0 \text { and } \sum_{i=0}^d i a_i \alpha^{i-1}=0\right\} \text {. } $$</p>
<p>Then,</p>
<ol start="1" style="list-style-type:upper-alpha">
<li>$S$ is finite or infinite depending on the value of $\alpha$</li>
<li>$S$ is a $2$-dimensional vector subspace of $\mathbb{R}^{d+1}$</li>
<li>$S$ is a $d$-dimensional vector subspace of $\mathbb{R}^{d+1}$</li>
<li>$S$ is a $(d-1)$-dimensional vector subspace of $\mathbb{R}^{d+1} $</li>
<li>For each $\left(a_0, a_1, \ldots, a_d\right) \in S$, the function</li>
</ol>
<p>$$ x \mapsto \sum_{i=0}^d a_i x^i $$</p>
<p>has a local optimum at $\alpha$</p>Linear Algebrahttps://gateoverflow.in/381994/tifr-cse-2022-part-b-question-15Thu, 01 Sep 2022 17:42:48 +0000TIFR CSE 2022 | Part A | Question: 1
https://gateoverflow.in/381951/tifr-cse-2022-part-a-question-1
<p>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?</p>
<ol start="1" style="list-style-type:upper-alpha">
<li>$75^{\text {th}}$ day</li>
<li>$74^{\text {th}}$ day</li>
<li>$73^{ \text{rd}}$ day</li>
<li>$72^{\text {nd }}$ day</li>
<li>$71^{\text {st }}$ day</li>
</ol>Combinatoryhttps://gateoverflow.in/381951/tifr-cse-2022-part-a-question-1Thu, 01 Sep 2022 01:29:14 +0000TIFR CSE 2022 | Part A | Question: 2
https://gateoverflow.in/381950/tifr-cse-2022-part-a-question-2
<p>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?</p>
<p>(Ignore leap years, so there are only $365$ possible birthdays. Assume that birthdays fall with equal probability on each of the $365$ days of the year.)</p>
<ol start="1" style="list-style-type:upper-alpha">
<li>$23$</li>
<li>$28$</li>
<li>$92$</li>
<li>$183$</li>
<li>$366$</li>
</ol>Probabilityhttps://gateoverflow.in/381950/tifr-cse-2022-part-a-question-2Thu, 01 Sep 2022 01:29:13 +0000TIFR CSE 2022 | Part A | Question: 4
https://gateoverflow.in/381948/tifr-cse-2022-part-a-question-4
<p>Consider the polynomial $p(x)=x^3-x^2+x-1$. How many <em>symmetric</em> matrices with integer entries are there whose characteristic polynomial is $p$? (Recall that the <em>characteristic polynomial</em> 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.)</p>
<ol start="1" style="list-style-type:upper-alpha">
<li>$0$</li>
<li>$1$</li>
<li>$2$</li>
<li>$4$</li>
<li>Infinitely many</li>
</ol>Linear Algebrahttps://gateoverflow.in/381948/tifr-cse-2022-part-a-question-4Thu, 01 Sep 2022 01:29:13 +0000