Highest voted questions in Engineering Mathematics / GATE Overflow for GATE CSE

46 votes

4 answers

121

GATE CSE 1996 | Question: 2.4
Which one of the following is false? The set of all bijective functions on a finite set forms a group under function composition The set $\{1, 2, \dots p-1\}$ forms a group under multiplication mod $p$, where $p$ is a prime number The set of all strings over a finite ... $\langle G, * \rangle$ if and only if for any pair of elements $a, b \in S, a * b^{-1} \in S$

Which one of the following is false?The set of all bijective functions on a finite set forms a group under function compositionThe set $\{1, 2, \dots p-1\}$ forms a group...

Kathleen

9.6k views

Kathleen asked Oct 9, 2014

46 votes

6 answers

122

GATE CSE 2003 | Question: 39
Let $\Sigma = \left\{a, b, c, d, e\right\}$ be an alphabet. We define an encoding scheme as follows: $g(a) = 3, g(b) = 5, g(c) = 7, g(d) = 9, g(e) = 11$. Let $p_i$ denote the i-th prime number $\left(p_1 = 2\right)$ ... numbers is the encoding, $h$, of a non-empty sequence of strings? $2^73^75^7$ $2^83^85^8$ $2^93^95^9$ $2^{10}3^{10}5^{10}$

Let $\Sigma = \left\{a, b, c, d, e\right\}$ be an alphabet. We define an encoding scheme as follows:$g(a) = 3, g(b) = 5, g(c) = 7, g(d) = 9, g(e) = 11$.Let $p_i$ denote t...

Kathleen

7.6k views

Kathleen asked Sep 17, 2014

46 votes

6 answers

123

GATE CSE 2002 | Question: 1.4
The minimum number of colours required to colour the vertices of a cycle with $n$ nodes in such a way that no two adjacent nodes have the same colour is $2$ $3$ $4$ $n-2 \left \lfloor \frac{n}{2} \right \rfloor+2$

The minimum number of colours required to colour the vertices of a cycle with $n$ nodes in such a way that no two adjacent nodes have the same colour is$2$$3$$4$$n-2 \lef...

Kathleen

11.3k views

Kathleen asked Sep 15, 2014

46 votes

3 answers

124

GATE CSE 2013 | Question: 47
Which one of the following is NOT logically equivalent to $¬∃x(∀ y (α)∧∀z(β ))$ ? $∀ x(∃ z(¬β )→∀ y(α))$ $∀x(∀ z(β )→∃ y(¬α))$ $∀x(∀ y(α)→∃z(¬β ))$ $∀x(∃ y(¬α)→∃z(¬β ))$

Which one of the following is NOT logically equivalent to $¬∃x(∀ y (α)∧∀z(β ))$ ?$∀ x(∃ z(¬β )→∀ y(α))$$∀x(∀ z(β )→∃ y(¬α))$$∀x(∀ y(�...

gatecse

12.0k views

gatecse asked Aug 21, 2014

45 votes

4 answers

125

GATE CSE 2016 Set 1 | Question: 2
Let $a_n$ be the number of $n$-bit strings that do NOT contain two consecutive $1's$. Which one of the following is the recurrence relation for $a_n$? $a_n = a_{n-1}+ 2a_{n-2}$ $a_n = a_{n-1}+ a_{n-2}$ $a_n = 2a_{n-1}+ a_{n-2}$ $a_n = 2a_{n-1}+ 2a_{n-2}$

Let $a_n$ be the number of $n$-bit strings that do NOT contain two consecutive $1's$. Which one of the following is the recurrence relation for $a_n$?$a_n = a_{n-1}+ 2a_{...

Sandeep Singh

9.6k views

Sandeep Singh asked Feb 12, 2016

45 votes

5 answers

126

GATE CSE 2015 Set 1 | Question: 26
$\sum\limits_{x=1}^{99}\frac{1}{x(x+1)}$ = ______.

$\sum\limits_{x=1}^{99}\frac{1}{x(x+1)}$ = ______.

makhdoom ghaya

8.2k views

makhdoom ghaya asked Feb 13, 2015

45 votes

10 answers

127

GATE IT 2005 | Question: 33
Let $A$ be a set with $n$ elements. Let $C$ be a collection of distinct subsets of $A$ such that for any two subsets $S_1$ and $S_2$ in $C$, either $S_1 \subset S_2$ or $S_2\subset S_1$. What is the maximum cardinality of $C?$ $n$ $n+1$ $2^{n-1} + 1$ $n!$

Let $A$ be a set with $n$ elements. Let $C$ be a collection of distinct subsets of $A$ such that for any two subsets $S_1$ and $S_2$ in $C$, either $S_1 \subset S_2$ or $...

Ishrat Jahan

11.9k views

Ishrat Jahan asked Nov 3, 2014

45 votes

3 answers

128

GATE IT 2004 | Question: 32
Let $A$ be an $n \times n$ ...

Let $A$ be an $n \times n$ matrix of the following form.$$A = \begin{bmatrix}3&1&0&0&0&\ldots&0&0&0\\1&3&1&0&0&\ldots&0&0&0\\0&1&3&1&0&\ldots&0&0&0\\0&0&1&3&1&\ldots&0&0&...

Ishrat Jahan

8.1k views

Ishrat Jahan asked Nov 2, 2014

45 votes

4 answers

129

GATE IT 2006 | Question: 2
For the set $N$ of natural numbers and a binary operation $f : N \times N \to N,$ an element $z \in N$ is called an identity for $f,$ if $f (a, z) = a = f(z, a),$ for all $a \in N.$ Which of the following binary operations have an identity? $f (x, y) = x + y - 3$ $f (x, y) = \max(x, y)$ $f (x, y) = x^y$ I and II only II and III only I and III only None of these

For the set $N$ of natural numbers and a binary operation $f : N \times N \to N,$ an element $z \in N$ is called an identity for $f,$ if $f (a, z) = a = f(z, a),$ for all...

Ishrat Jahan

9.7k views

Ishrat Jahan asked Oct 30, 2014

45 votes

3 answers

130

GATE CSE 1997 | Question: 13
Let $F$ be the set of one-to-one functions from the set $\{1, 2, \dots, n\}$ to the set $\{1, 2,\dots, m\}$ where $m\geq n\geq1$. How many functions are members of $F$? How many functions $f$ in $F$ satisfy the property $f(i)=1$ for some $i, 1\leq i \leq n$? How many functions $f$ in $F$ satisfy the property $f(i)<f(j)$ for all $i,j \ \ 1\leq i \leq j \leq n$?

Let $F$ be the set of one-to-one functions from the set $\{1, 2, \dots, n\}$ to the set $\{1, 2,\dots, m\}$ where $m\geq n\geq1$.How many functions are members of $F$?How...

Kathleen

6.5k views

Kathleen asked Sep 29, 2014

45 votes

3 answers

131

GATE CSE 2006 | Question: 27
Consider the following propositional statements: $P_1: ((A ∧ B) → C)) ≡ ((A → C) ∧ (B → C))$ $P_2: ((A ∨ B) → C)) ≡ ((A → C) ∨ (B → C))$ Which one of the following is true? $P_1$ is a tautology, but not $P_2$ $P_2$ is a tautology, but not $P_1$ $P_1$ and $P_2$ are both tautologies Both $P_1$ and $P_2$ are not tautologies

Consider the following propositional statements:$P_1: ((A ∧ B) → C)) ≡ ((A → C) ∧ (B → C))$$P_2: ((A ∨ B) → C)) ≡ ((A → C) ∨ (B → C))$Which one of...

Rucha Shelke

8.6k views

Rucha Shelke asked Sep 18, 2014

44 votes

9 answers

132

GATE CSE 2017 Set 2 | Question: 23
$G$ is an undirected graph with $n$ vertices and $25$ edges such that each vertex of $G$ has degree at least $3$. Then the maximum possible value of $n$ is _________ .

$G$ is an undirected graph with $n$ vertices and $25$ edges such that each vertex of $G$ has degree at least $3$. Then the maximum possible value of $n$ is _________ .

Madhav

17.6k views

Madhav asked Feb 14, 2017

44 votes

11 answers

133

GATE CSE 2016 Set 2 | Question: 29
The value of the expression $13^{99}\pmod{17}$ in the range $0$ to $16$, is ________.

The value of the expression $13^{99}\pmod{17}$ in the range $0$ to $16$, is ________.

Akash Kanase

18.0k views

Akash Kanase asked Feb 12, 2016

44 votes

4 answers

134

GATE IT 2006 | Question: 1
In a certain town, the probability that it will rain in the afternoon is known to be $0.6$. Moreover, meteorological data indicates that if the temperature at noon is less than or equal to $25°C$, the probability that it will rain in the afternoon is $0.4$. The temperature ... in the afternoon on a day when the temperature at noon is above $25°C$? $0.4$ $0.6$ $0.8$ $0.9$

In a certain town, the probability that it will rain in the afternoon is known to be $0.6$. Moreover, meteorological data indicates that if the temperature at noon is les...

Ishrat Jahan

7.4k views

Ishrat Jahan asked Oct 30, 2014

44 votes

5 answers

135

GATE CSE 2006 | Question: 23
$F$ is an $n\times n$ real matrix. $b$ is an $n\times 1$ real vector. Suppose there are two $n\times 1$ vectors, $u$ and $v$ such that, $u ≠ v$ and $Fu = b, Fv = b$. Which one of the following statements is false? Determinant of $F$ is zero. There are an infinite number of solutions to $Fx = b$ There is an $x≠0$ such that $Fx = 0$ $F$ must have two identical rows

$F$ is an $n\times n$ real matrix. $b$ is an $n\times 1$ real vector. Suppose there are two $n\times 1$ vectors, $u$ and $v$ such that, $u ≠ v$ and $Fu = b, Fv = b$. Wh...

Rucha Shelke

10.1k views

Rucha Shelke asked Sep 17, 2014

44 votes

4 answers

136

GATE CSE 2003 | Question: 4
Let $A$ be a sequence of $8$ distinct integers sorted in ascending order. How many distinct pairs of sequences, $B$ and $C$ are there such that each is sorted in ascending order, $B$ has $5$ and $C$ has $3$ elements, and the result of merging $B$ and $C$ gives $A$ $2$ $30$ $56$ $256$

Let $A$ be a sequence of $8$ distinct integers sorted in ascending order. How many distinct pairs of sequences, $B$ and $C$ are there such thateach is sorted in ascending...

Kathleen

13.5k views

Kathleen asked Sep 16, 2014

43 votes

2 answers

137

GATE CSE 1989 | Question: 1-v
The number of possible commutative binary operations that can be defined on a set of $n$ elements (for a given $n$) is ___________.

The number of possible commutative binary operations that can be defined on a set of $n$ elements (for a given $n$) is ___________.

makhdoom ghaya

6.6k views

makhdoom ghaya asked Nov 27, 2016

43 votes

8 answers

138

GATE CSE 2006 | Question: 73
The $2^n$ vertices of a graph $G$ corresponds to all subsets of a set of size $n$, for $n \geq 6$. Two vertices of $G$ are adjacent if and only if the corresponding sets intersect in exactly two elements. The number of connected components in $G$ is: $n$ $n + 2$ $2^{\frac{n}{2}}$ $\frac{2^{n}}{n}$

The $2^n$ vertices of a graph $G$ corresponds to all subsets of a set of size $n$, for $n \geq 6$. Two vertices of $G$ are adjacent if and only if the corresponding set...

go_editor

8.8k views

go_editor asked Apr 24, 2016

43 votes

4 answers

139

GATE CSE 2016 Set 2 | Question: 03
The minimum number of colours that is sufficient to vertex-colour any planar graph is ________.

The minimum number of colours that is sufficient to vertex-colour any planar graph is ________.

Akash Kanase

15.5k views

Akash Kanase asked Feb 12, 2016

43 votes

7 answers

140

GATE CSE 2015 Set 3 | Question: 41
Let $R$ be a relation on the set of ordered pairs of positive integers such that $((p,q),(r,s)) \in R$ if and only if $p-s=q-r$. Which one of the following is true about $R$? Both reflexive and symmetric Reflexive but not symmetric Not reflexive but symmetric Neither reflexive nor symmetric

Let $R$ be a relation on the set of ordered pairs of positive integers such that $((p,q),(r,s)) \in R$ if and only if $p-s=q-r$. Which one of the following is true about ...

go_editor

13.0k views

go_editor asked Feb 15, 2015

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