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Previous GATE Questions in Discrete Mathematics
26
votes
6
answers
21
GATE CSE 2022 | Question: 26
Which one of the following is the closed form for the generating function of the sequence $\{ a_{n} \}_{n \geq 0}$ defined below? $ a_{n} = \left\{\begin{matrix} n + 1, & \text{n is odd} & \\ 1, & \text{otherwise} & \end{matrix}\right.$ ... $\frac{2x}{(1-x^{2})^{2}} + \frac{1}{1-x}$ $\frac{x}{(1-x^{2})^{2}} + \frac{1}{1-x}$
Arjun
asked
in
Combinatory
Feb 15, 2022
by
Arjun
9.4k
views
gatecse-2022
combinatory
generating-functions
2-marks
14
votes
3
answers
22
GATE CSE 2022 | Question: 27
Consider a simple undirected unweighted graph with at least three vertices. If $\textit{A}$ is the adjacency matrix of the graph, then the number of $3–$cycles in the graph is given by the trace of $\textit{A}^{3}$ $\textit{A}^{3}$ divided by $2$ $\textit{A}^{3}$ divided by $3$ $\textit{A}^{3}$ divided by $6$
Arjun
asked
in
Graph Theory
Feb 15, 2022
by
Arjun
7.4k
views
gatecse-2022
graph-theory
graph-connectivity
2-marks
14
votes
2
answers
23
GATE CSE 2022 | Question: 40
The following simple undirected graph is referred to as the Peterson graph. Which of the following statements is/are $\text{TRUE}?$ The chromatic number of the graph is $3.$ The graph has a Hamiltonian path. The following graph is isomorphic to the Peterson ... $3.$ (A subset of vertices of a graph form an independent set if no two vertices of the subset are adjacent.)
Arjun
asked
in
Graph Theory
Feb 15, 2022
by
Arjun
7.4k
views
gatecse-2022
graph-theory
graph-isomorphism
multiple-selects
2-marks
hard
15
votes
6
answers
24
GATE CSE 2022 | Question: 41
Consider the following recurrence: $\begin{array}{} f(1) & = & 1; \\ f(2n) & = & 2f(n) - 1, & \; \text{for}\; n \geq 1; \\ f(2n+1) & = & 2f(n) + 1, & \; \text{for}\; n \geq 1. \end{array}$ Then, which of the following statements is/are $\text{TRUE}?$ ... $f(2^{n}) = 1$ $f(5 \cdot 2^{n}) = 2^{n+1} + 1$ $f(2^{n} + 1) = 2^{n} + 1$
Arjun
asked
in
Combinatory
Feb 15, 2022
by
Arjun
7.7k
views
gatecse-2022
combinatory
recurrence-relation
multiple-selects
2-marks
16
votes
4
answers
25
GATE CSE 2022 | Question: 42
Which of the properties hold for the adjacency matrix $A$ of a simple undirected unweighted graph having $n$ vertices? The diagonal entries of $A^{2}$ ... . If there is at least a $1$ in each of $A\text{'s}$ rows and columns, then the graph must be connected.
Arjun
asked
in
Graph Theory
Feb 15, 2022
by
Arjun
7.3k
views
gatecse-2022
graph-theory
graph-connectivity
multiple-selects
2-marks
24
votes
3
answers
26
GATE CSE 2021 Set 2 | Question: 11
Consider the following sets, where $n \geq 2$: $S_1$: Set of all $n \times n$ matrices with entries from the set $\{ a, b, c\}$ $S_2$: Set of all functions from the set $\{0,1,2, \dots, n^2-1\}$ ... There exists a surjection from $S_1$ to $S_2$ There exists a bijection from $S_1$ to $S_2$ There does not exist an injection from $S_1$ to $S_2$
Arjun
asked
in
Set Theory & Algebra
Feb 18, 2021
by
Arjun
6.3k
views
gatecse-2021-set2
multiple-selects
set-theory&algebra
functions
1-mark
27
votes
9
answers
27
GATE CSE 2021 Set 2 | Question: 15
Choose the correct choice(s) regarding the following proportional logic assertion $S$: $S: (( P \wedge Q) \rightarrow R) \rightarrow (( P \wedge Q) \rightarrow (Q \rightarrow R))$ $S$ is neither a tautology nor a contradiction $S$ is a tautology $S$ is a contradiction The antecedent of $S$ is logically equivalent to the consequent of $S$
Arjun
asked
in
Mathematical Logic
Feb 18, 2021
by
Arjun
8.7k
views
gatecse-2021-set2
multiple-selects
mathematical-logic
propositional-logic
1-mark
21
votes
1
answer
28
GATE CSE 2021 Set 2 | Question: 37
For two $n$-dimensional real vectors $P$ and $Q$, the operation $s(P,Q)$ is defined as follows: $s(P,Q) = \displaystyle \sum_{i=1}^n (P[i] \cdot Q[i])$ Let $\mathcal{L}$ be a set of $10$-dimensional non-zero real vectors such that for every pair ... $s(P,Q)=0$. What is the maximum cardinality possible for the set $\mathcal{L}$? $9$ $10$ $11$ $100$
Arjun
asked
in
Set Theory & Algebra
Feb 18, 2021
by
Arjun
7.0k
views
gatecse-2021-set2
set-theory&algebra
set-theory
2-marks
25
votes
6
answers
29
GATE CSE 2021 Set 2 | Question: 50
Let $S$ be a set of consisting of $10$ elements. The number of tuples of the form $(A,B)$ such that $A$ and $B$ are subsets of $S$, and $A \subseteq B$ is ___________
Arjun
asked
in
Combinatory
Feb 18, 2021
by
Arjun
11.8k
views
gatecse-2021-set2
combinatory
counting
numerical-answers
2-marks
14
votes
8
answers
30
GATE CSE 2021 Set 1 | Question: 7
Let $p$ and $q$ be two propositions. Consider the following two formulae in propositional logic. $S_1: (\neg p\wedge(p\vee q))\rightarrow q$ $S_2: q\rightarrow(\neg p\wedge(p\vee q))$ Which one of the following choices is correct? Both $S_1$ and ... but $S_2$ is not a tautology $S_1$ is not a tautology but $S_2$ is a tautology Neither $S_1$ nor $S_2$ is a tautology
Arjun
asked
in
Mathematical Logic
Feb 18, 2021
by
Arjun
8.1k
views
gatecse-2021-set1
mathematical-logic
propositional-logic
1-mark
13
votes
5
answers
31
GATE CSE 2021 Set 1 | Question: 16
In an undirected connected planar graph $G$, there are eight vertices and five faces. The number of edges in $G$ is _________.
Arjun
asked
in
Graph Theory
Feb 18, 2021
by
Arjun
8.0k
views
gatecse-2021-set1
graph-theory
graph-planarity
numerical-answers
easy
1-mark
38
votes
3
answers
32
GATE CSE 2021 Set 1 | Question: 19
There are $6$ jobs with distinct difficulty levels, and $3$ computers with distinct processing speeds. Each job is assigned to a computer such that: The fastest computer gets the toughest job and the slowest computer gets the easiest job. Every computer gets at least one job. The number of ways in which this can be done is ___________.
Arjun
asked
in
Combinatory
Feb 18, 2021
by
Arjun
11.7k
views
gatecse-2021-set1
combinatory
counting
numerical-answers
1-mark
23
votes
4
answers
33
GATE CSE 2021 Set 1 | Question: 34
Let $G$ be a group of order $6$, and $H$ be a subgroup of $G$ such that $1<|H|<6$. Which one of the following options is correct? Both $G$ and $H$ are always cyclic $G$ may not be cyclic, but $H$ is always cyclic $G$ is always cyclic, but $H$ may not be cyclic Both $G$ and $H$ may not be cyclic
Arjun
asked
in
Set Theory & Algebra
Feb 18, 2021
by
Arjun
8.2k
views
gatecse-2021-set1
set-theory&algebra
group-theory
2-marks
35
votes
6
answers
34
GATE CSE 2021 Set 1 | Question: 36
Let $G=(V, E)$ be an undirected unweighted connected graph. The diameter of $G$ is defined as: $\text{diam}(G)=\displaystyle \max_{u,v\in V} \{\text{the length of shortest path between $u$ and $v$}\}$ Let $M$ be the adjacency matrix of $G$. Define graph $G_2$ ... $\text{diam}(G_2) = \text{diam}(G)$ $\text{diam}(G)< \text{diam}(G_2)\leq 2\; \text{diam}(G)$
Arjun
asked
in
Graph Theory
Feb 18, 2021
by
Arjun
9.9k
views
gatecse-2021-set1
graph-theory
graph-connectivity
2-marks
27
votes
5
answers
35
GATE CSE 2021 Set 1 | Question: 43
A relation $R$ is said to be circular if $a\text{R}b$ and $b\text{R}c$ together imply $c\text{R}a$. Which of the following options is/are correct? If a relation $S$ is reflexive and symmetric, then $S$ is an equivalence relation ... and circular, then $S$ is an equivalence relation. If a relation $S$ is transitive and circular, then $S$ is an equivalence relation.
Arjun
asked
in
Set Theory & Algebra
Feb 18, 2021
by
Arjun
8.1k
views
gatecse-2021-set1
multiple-selects
set-theory&algebra
relations
2-marks
15
votes
5
answers
36
GATE CSE 2020 | Question: 17
Let $\mathcal{R}$ be the set of all binary relations on the set $\{1,2,3\}$. Suppose a relation is chosen from $\mathcal{R}$ at random. The probability that the chosen relation is reflexive (round off to $3$ decimal places) is ______.
Arjun
asked
in
Set Theory & Algebra
Feb 12, 2020
by
Arjun
9.1k
views
gatecse-2020
numerical-answers
probability
relations
1-mark
15
votes
4
answers
37
GATE CSE 2020 | Question: 18
Let $G$ be a group of $35$ elements. Then the largest possible size of a subgroup of $G$ other than $G$ itself is _______.
Arjun
asked
in
Set Theory & Algebra
Feb 12, 2020
by
Arjun
9.1k
views
gatecse-2020
numerical-answers
group-theory
easy
1-mark
42
votes
8
answers
38
GATE CSE 2020 | Question: 39
Which one of the following predicate formulae is NOT logically valid? Note that $W$ is a predicate formula without any free occurrence of $x$. $\forall x (p(x) \vee W) \equiv \forall x \: ( px) \vee W$ ... $\exists x(p(x) \rightarrow W) \equiv \forall x \: p(x) \rightarrow W$
Arjun
asked
in
Mathematical Logic
Feb 12, 2020
by
Arjun
17.0k
views
gatecse-2020
first-order-logic
mathematical-logic
2-marks
28
votes
8
answers
39
GATE CSE 2020 | Question: 42
The number of permutations of the characters in LILAC so that no character appears in its original position, if the two L’s are indistinguishable, is ______.
Arjun
asked
in
Combinatory
Feb 12, 2020
by
Arjun
16.4k
views
gatecse-2020
numerical-answers
combinatory
2-marks
28
votes
6
answers
40
GATE CSE 2020 | Question: 52
Graph $G$ is obtained by adding vertex $s$ to $K_{3,4}$ and making $s$ adjacent to every vertex of $K_{3,4}$. The minimum number of colours required to edge-colour $G$ is _______
Arjun
asked
in
Graph Theory
Feb 12, 2020
by
Arjun
13.5k
views
gatecse-2020
numerical-answers
graph-theory
graph-coloring
2-marks
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