Most answered questions in Engineering Mathematics / GATE Overflow for GATE CSE

54 votes

7 answers

151

GATE CSE 2009 | Question: 3
Which one of the following is TRUE for any simple connected undirected graph with more than $2$ vertices? No two vertices have the same degree. At least two vertices have the same degree. At least three vertices have the same degree. All vertices have the same degree.

Which one of the following is TRUE for any simple connected undirected graph with more than $2$ vertices? No two vertices have the same degree. At least two vertices ...

gatecse

11.4k views

gatecse asked Sep 15, 2014

28 votes

7 answers

152

GATE CSE 2009 | Question: 26
Consider the following well-formed formulae: $\neg \forall x(P(x))$ $\neg \exists x(P(x))$ $\neg \exists x(\neg P(x))$ $\exists x(\neg P(x))$ Which of the above are equivalent? $\text{I}$ and $\text{III}$ $\text{I}$ and $\text{IV}$ $\text{II}$ and $\text{III}$ $\text{II}$ and $\text{IV}$

Consider the following well-formed formulae:$\neg \forall x(P(x))$$\neg \exists x(P(x))$$\neg \exists x(\neg P(x))$$\exists x(\neg P(x))$Which of the above are equivalent...

gatecse

5.6k views

gatecse asked Sep 15, 2014

47 votes

7 answers

153

GATE CSE 2000 | Question: 2.7
Let $a, b, c, d$ be propositions. Assume that the equivalence $a ⇔ ( b \vee \neg b)$ and $b ⇔c$ hold. Then the truth-value of the formula $(a ∧ b) → (a ∧ c) ∨ d$ is always True False Same as the truth-value of $b$ Same as the truth-value of $d$

Let $a, b, c, d$ be propositions. Assume that the equivalence $a ⇔ ( b \vee \neg b)$ and $b ⇔c$ hold. Then the truth-value of the formula $(a ∧ b) → (a ∧ c) ∨...

Kathleen

12.2k views

Kathleen asked Sep 14, 2014

26 votes

7 answers

154

GATE CSE 2000 | Question: 1.3
The determinant of the matrix $\begin{bmatrix}2 &0 &0 &0 \\ 8& 1& 7& 2\\ 2& 0&2 &0 \\ 9&0 & 6 & 1 \end{bmatrix}$ $4$ $0$ $15$ $20$

The determinant of the matrix $$\begin{bmatrix}2 &0 &0 &0 \\ 8& 1& 7& 2\\ 2& 0&2 &0 \\ 9&0 & 6 & 1 \end{bmatrix}$$$4$$0$$15$$20$

Kathleen

6.6k views

Kathleen asked Sep 14, 2014

49 votes

7 answers

155

GATE CSE 1993 | Question: 01.1
The eigen vector $(s)$ of the matrix $\begin{bmatrix} 0 &0 &\alpha\\ 0 &0 &0\\ 0 &0 &0 \end{bmatrix},\alpha \neq 0$ is (are) $(0,0,\alpha)$ $(\alpha,0,0)$ $(0,0,1)$ $(0,\alpha,0)$

The eigen vector $(s)$ of the matrix $$\begin{bmatrix} 0 &0 &\alpha\\ 0 &0 &0\\ 0 &0 &0 \end{bmatrix},\alpha \neq 0$$ is (are)$(0,0,\alpha)$$(\alpha,0,0)$$(0,0,1)$$(0,\al...

Kathleen

11.6k views

Kathleen asked Sep 13, 2014

31 votes

7 answers

156

GATE CSE 2008 | Question: 31
$P$ and $Q$ are two propositions. Which of the following logical expressions are equivalent? $P ∨ \neg Q$ $\neg(\neg P ∧ Q)$ $(P ∧ Q) ∨ (P ∧ \neg Q) ∨ (\neg P ∧ \neg Q)$ $(P ∧ Q) ∨ (P ∧ \neg Q) ∨ (\neg P ∧ Q)$ Only I and II Only I, II and III Only I, II and IV All of I, II, III and IV

$P$ and $Q$ are two propositions. Which of the following logical expressions are equivalent?$P ∨ \neg Q$$\neg(\neg P ∧ Q)$$(P ∧ Q) ∨ (P ∧ \neg Q) ∨ (\neg P �...

Kathleen

8.4k views

Kathleen asked Sep 12, 2014

39 votes

7 answers

157

GATE CSE 2008 | Question: 23
Which of the following statements is true for every planar graph on $n$ vertices? The graph is connected The graph is Eulerian The graph has a vertex-cover of size at most $\frac{3n}{4}$ The graph has an independent set of size at least $\frac{n}{3}$

Which of the following statements is true for every planar graph on $n$ vertices?The graph is connectedThe graph is EulerianThe graph has a vertex-cover of size at most $...

Kathleen

65.1k views

Kathleen asked Sep 11, 2014

10 votes

6 answers

158

GO Classes CS 2025 | Weekly Quiz 3 | Propositional Logic | Question: 3
Consider the following atomic propositions: $\text{R}$: It is Raining $\text{S}$ ... , and vice versa It is raining is equivalent to sonu is sick It is raining or sonu is sick but not both

Consider the following atomic propositions:$\text{R}$: It is Raining$\text{S}$: Sonu is SickWhich of the following is/are correct English Translation of the following log...

GO Classes

677 views

GO Classes asked Apr 5, 2023

21 votes

6 answers

159

GO Classes CS 2025 | Weekly Quiz 3 | Propositional Logic | Question: 9
Which of the following statements is/are true? The argument form with premises $p_1, p_2, \ldots, p_n$ and conclusion $q \rightarrow r$ is valid iff the argument form with premises $p_1, p_2, \ldots, p_n, r$ ... the argument form with premises $p_1, p_2, \ldots, p_n, \sim r$, and conclusion $\sim q$ is valid.

Which of the following statements is/are true?The argument form with premises $p_1, p_2, \ldots, p_n$ and conclusion $q \rightarrow r$ is valid iff the argument form with...

GO Classes

897 views

GO Classes asked Apr 5, 2023

57 votes

6 answers

160

GO Classes CS 2025 | Weekly Quiz 2 | Propositional Logic | Question: 16
If $\text{F1, F2}$ and $\text{F3}$ are propositional formulae/expressions, over some set of propositional variables, such that $\mathrm{F} 1 \vee F 2 \rightarrow \mathrm{F} 3$ is a contradiction, then which of the following is/ ... is a tautology. $\text{F3}$ is a contradiction. $\text{F1} \mathrm{v} \text{F2}$ is a tautology.

If $\text{F1, F2}$ and $\text{F3}$ are propositional formulae/expressions, over some set of propositional variables, such that $\mathrm{F} 1 \vee F 2 \rightarrow \mathrm{...

GO Classes

1.7k views

GO Classes asked Mar 26, 2023

20 votes

6 answers

161

GO Classes CS/DA 2025 | Weekly Quiz 3 | Fundamental Course and Linear Algebra | Question: 10
Which of the following is(are) sufficient argument(s) to show that the vectors of set $\text{S}$ ... $7 u+(-1) v+1 w=\left[\begin{array}{l}0 \\ 0 \\ 0\end{array}\right]$

Which of the following is(are) sufficient argument(s) to show that the vectors of set $\text{S}$ are linearly dependent?$$\text{S}=\left\{u=\left[\begin{array}{c}1 \\-2 \...

GO Classes

1.4k views

GO Classes asked Mar 14, 2023

26 votes

6 answers

162

GO Classes CS 2025 | Weekly Quiz 1 | Propositional Logic | Question: 15
Consider the following popular puzzle. A boy and a girl are talking. One of them has black hair, another has white hair. I am a boy said the child with black hair. I am a girl said the child with white hair ... Which of them is lying? The boy only The girl only Both of them Information is not sufficient to find out the liar

Consider the following popular puzzle.A boy and a girl are talking. One of them has black hair, another has white hair.“I am a boy” said the child with black hair.“...

GO Classes

1.6k views

GO Classes asked Mar 30, 2022

26 votes

6 answers

163

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.$ ... $\frac{2x}{(1-x^{2})^{2}} + \frac{1}{1-x}$ $\frac{x}{(1-x^{2})^{2}} + \frac{1}{1-x}$

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, &...

Arjun

9.6k views

Arjun asked Feb 15, 2022

15 votes

6 answers

164

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 $\text{TRUE}?$ ... $f(2^{n}) = 1$ $f(5 \cdot 2^{n}) = 2^{n+1} + 1$ $f(2^{n} + 1) = 2^{n} + 1$

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...

Arjun

7.8k views

Arjun asked Feb 15, 2022

26 votes

6 answers

165

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 ___________

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

12.0k views

Arjun asked Feb 18, 2021

35 votes

6 answers

166

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 $G_2$ ... $\text{diam}(G_2) = \text{diam}(G)$ $\text{diam}(G)< \text{diam}(G_2)\leq 2\; \text{diam}(G)$

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 shortes...

Arjun

10.0k views

Arjun asked Feb 18, 2021

2 votes

6 answers

167

UGC NET CSE | January 2017 | Part 3 | Question: 60
The first order logic (FOL) statement $((R\vee Q)\wedge(P\vee \neg Q))$ is equivalent to which of the following? $((R\vee \neg Q)\wedge(P\vee \neg Q)\wedge (R\vee P))$ $((R\vee Q)\wedge(P\vee \neg Q)\wedge (R\vee P))$ $((R\vee Q)\wedge(P\vee \neg Q)\wedge(R\vee \neg P))$ $((R\vee Q)\wedge(P\vee \neg Q)\wedge (\neg R\vee P))$

The first order logic (FOL) statement $((R\vee Q)\wedge(P\vee \neg Q))$ is equivalent to which of the following?$((R\vee \neg Q)\wedge(P\vee \neg Q)\wedge (R\vee P))$$((R...

go_editor

2.9k views

go_editor asked Mar 24, 2020

3 votes

6 answers

168

UGC NET CSE | January 2017 | Part 2 | Question: 6
In propositional logic if $\left ( P \rightarrow Q \right )\wedge \left ( R \rightarrow S \right )$ and $\left ( P \vee R \right )$ are two premises such that $\begin{array}{c} (P \to Q) \wedge (R \to S) \\ P \vee R \\ \hline Y \\ \hline \end{array}$ $Y$ is the premise : $P \vee R$ $P \vee S$ $Q \vee R$ $Q \vee S$

In propositional logic if $\left ( P \rightarrow Q \right )\wedge \left ( R \rightarrow S \right )$ and $\left ( P \vee R \right )$ are two premises such that$$\begin{arr...

go_editor

3.0k views

go_editor asked Mar 24, 2020

28 votes

6 answers

169

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 _______

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

13.7k views

Arjun asked Feb 12, 2020

40 votes

6 answers

170

GATE CSE 2019 | Question: 38
Let $G$ be any connected, weighted, undirected graph. $G$ has a unique minimum spanning tree, if no two edges of $G$ have the same weight. $G$ has a unique minimum spanning tree, if, for every cut of $G$, there is a unique minimum-weight edge crossing the cut. Which of the following statements is/are TRUE? I only II only Both I and II Neither I nor II

Let $G$ be any connected, weighted, undirected graph.$G$ has a unique minimum spanning tree, if no two edges of $G$ have the same weight.$G$ has a unique minimum spanning...

Arjun

20.6k views

Arjun asked Feb 7, 2019

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