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Previous GATE Questions in Engineering Mathematics
23
votes
5
answers
21
GATE CSE 2023 | Question: 16
Geetha has a conjecture about integers, which is of the form \[ \forall x(P(x) \Longrightarrow \exists y Q(x, y)), \] where $P$ is a statement about integers, and $Q$ is a statement about pairs of integers. Which of the following (one or more) option(s) would imply ... $\exists y \forall x(P(x) \Longrightarrow Q(x, y))$ $\exists x(P(x) \wedge \exists y Q(x, y))$
admin
asked
in
Mathematical Logic
Feb 15, 2023
by
admin
11.0k
views
gatecse-2023
mathematical-logic
first-order-logic
multiple-selects
1-mark
8
votes
2
answers
22
GATE CSE 2023 | Question: 18
Let $\qquad f(x)=x^{3}+15 x^{2}-33 x-36$ be a real-valued function. Which of the following statements is/are $\text{TRUE}?$ $f(x)$ does not have a local maximum. $f(x)$ has a local maximum. $f(x)$ does not have a local minimum. $f(x)$ has a local minimum.
admin
asked
in
Calculus
Feb 15, 2023
by
admin
5.1k
views
gatecse-2023
calculus
maxima-minima
multiple-selects
1-mark
8
votes
4
answers
23
GATE CSE 2023 | Question: 20
Let $A$ be the adjacency matrix of the graph with vertices $\{1,2,3,4,5\}.$ Let $\lambda_{1}, \lambda_{2}, \lambda_{3}, \lambda_{4}$, and $\lambda_{5}$ be the five eigenvalues of $A$. Note that these eigenvalues need not be distinct. The value of $\lambda_{1}+\lambda_{2}+\lambda_{3}+\lambda_{4}+\lambda_{5}=$____________
admin
asked
in
Linear Algebra
Feb 15, 2023
by
admin
12.8k
views
gatecse-2023
linear-algebra
eigen-value
numerical-answers
1-mark
5
votes
2
answers
24
GATE CSE 2023 | Question: 21
The value of the definite integral \[ \int_{-3}^{3} \int_{-2}^{2} \int_{-1}^{1}\left(4 x^{2} y-z^{3}\right) \mathrm{d} z \mathrm{~d} y \mathrm{~d} x \] is _________. (Rounded off to the nearest integer)
admin
asked
in
Calculus
Feb 15, 2023
by
admin
8.5k
views
gatecse-2023
calculus
definite-integral
numerical-answers
1-mark
14
votes
3
answers
25
GATE CSE 2023 | Question: 38
Let $U=\{1,2, \ldots, n\},$ where $n$ is a large positive integer greater than $1000.$ Let $k$ be a positive integer less than $n$. Let $A, B$ be subsets of $U$ with $|A|=|B|=k$ and $A \cap B=\emptyset$. We say that a permutation of $U$ separates $A$ from $B$ if ... $2\left(\begin{array}{c}n \\ 2 k\end{array}\right)(n-2 k) !(k !)^{2}$
admin
asked
in
Combinatory
Feb 15, 2023
by
admin
6.3k
views
gatecse-2023
combinatory
counting
2-marks
11
votes
2
answers
26
GATE CSE 2023 | Question: 39
Let $f: A \rightarrow B$ be an onto (or surjective) function, where $A$ and $B$ are nonempty sets. Define an equivalence relation $\sim$ on the set $A$ as \[ a_{1} \sim a_{2} \text { if } f\left(a_{1}\right)=f\left(a_{2}\right), \] ... is NOT well-defined. $F$ is an onto (or surjective) function. $F$ is a one-to-one (or injective) function. $F$ is a bijective function.
admin
asked
in
Set Theory & Algebra
Feb 15, 2023
by
admin
5.7k
views
gatecse-2023
set-theory&algebra
equivalence-class
multiple-selects
2-marks
10
votes
1
answer
27
GATE CSE 2023 | Question: 41
Let $X$ be a set and $2^{X}$ denote the powerset of $X$. Define a binary operation $\Delta$ on $2^{X}$ as follows: \[ A \Delta B=(A-B) \cup(B-A) \text {. } \] Let $H=\left(2^{X}, \Delta\right)$. Which of the following statements about $H$ is/are correct? ... $A \in 2^{X},$ the inverse of $A$ is the complement of $A$. For every $A \in 2^{X},$ the inverse of $A$ is $A$.
admin
asked
in
Set Theory & Algebra
Feb 15, 2023
by
admin
5.5k
views
gatecse-2023
set-theory&algebra
group-theory
multiple-selects
2-marks
8
votes
4
answers
28
GATE CSE 2023 | Question: 43
Consider a random experiment where two fair coins are tossed. Let $A$ be the event that denotes $\text{HEAD}$ on both the throws, $B$ be the event that denotes $\text{HEAD}$ on the first throw, and $C$ be the event that denotes $\text{HEAD}$ on the ... . $A$ and $C$ are independent. $B$ and $C$ are independent. $\operatorname{Prob}(B \mid C)=\operatorname{Prob}(B)$
admin
asked
in
Probability
Feb 15, 2023
by
admin
6.8k
views
gatecse-2023
probability
independent-events
multiple-selects
2-marks
7
votes
3
answers
29
GATE CSE 2023 | Question: 45
Let $G$ be a simple, finite, undirected graph with vertex set $\left\{v_{1}, \ldots, v_{n}\right\}$. Let $\Delta(G)$ denote the maximum degree of $G$ and let $\mathbb{N}=\{1,2, \ldots\}$ denote the set of all possible colors. Color the vertices ... $\Delta(G)$. The number of colors used is equal to the chromatic number of $G$.
admin
asked
in
Graph Theory
Feb 15, 2023
by
admin
8.1k
views
gatecse-2023
graph-theory
graph-coloring
multiple-selects
2-marks
0
votes
2
answers
30
GATE CSE 2023 | Memory Based Question: 11
Let $f(x)=x^3+15 x^2-33 x-36$ be a real valued function. Which statement is/are TRUE? $f(x)$ has a local maximum. $f(x)$ does NOT have a local maximum. $f(x)$ has a local minimum. $f(x)$ does NOT have a local minimum.
closed
GO Classes
asked
in
Calculus
Feb 5, 2023
by
GO Classes
3.9k
views
memorybased-gatecse2023
goclasses
calculus
maxima-minima
3
votes
1
answer
31
GATE CSE 2023 | Memory Based Question: 13
Let $ A=\left[\begin{array}{llll} 1 & 2 & 3 & 4 \\ 4 & 1 & 2 & 3 \\ 3 & 4 & 1 & 2 \\ 2 & 3 & 4 & 1 \end{array}\right] $ ... $\operatorname{det} \mathrm{B}=-\operatorname{det} \mathrm{A}$ $\operatorname{det} \mathrm{A}=\operatorname{det} \mathrm{B}$
GO Classes
asked
in
Linear Algebra
Feb 5, 2023
by
GO Classes
2.3k
views
memorybased-gatecse2023
goclasses
linear-algebra
determinant
0
votes
1
answer
32
GATE CSE 2023 | Memory Based Question: 15
The Lucas sequence $L_n$ is defined by the recurrence relation: $L_n=L_{n-1}+L_{n-2}$, for $n \geq 3$ with $L_1=1$ and $L_2=3$. Which one of the options given is TRUE? $L_n=\left(\frac{1+\sqrt{5}}{2}\right)^n+\left(\frac{1-\sqrt{5}}{3}\right)^n$ ... $L_n=\left(\frac{1+\sqrt{5}}{2}\right)^n+\left(\frac{1-\sqrt{5}}{2}\right)^n$
closed
GO Classes
asked
in
Combinatory
Feb 5, 2023
by
GO Classes
1.5k
views
memorybased-gatecse2023
goclasses
combinatory
recurrence-relation
1
vote
1
answer
33
GATE CSE 2023 | Memory Based Question: 16
How many permutations of $U$ separate $A$ from $B?$ $2\left(\begin{array}{c}n \\ 2 k\end{array}\right)(n-2 k) !(k!)^2$ $\left(\begin{array}{c}n \\ 2 k\end{array}\right)(n-2 k)!(n!)$ $n!$ $\left(\begin{array}{c}n \\ 2 k\end{array}\right)(n-2 k) !(k !)^2$
closed
GO Classes
asked
in
Combinatory
Feb 5, 2023
by
GO Classes
1.0k
views
memorybased-gatecse2023
goclasses
combinatory
counting
1
vote
2
answers
34
GATE CSE 2023 | Memory Based Question: 17
Let $x$ be a set, $2^x=$ power $2 \mathrm{k}$ set of $\mathrm{X}$. define A binary operation $\Delta$ on $2^x$ as $A \Delta B=(A-B) \cup(B-A)$. Let $H=\left(2^x, \Delta\right)$, then for every $A \in 2^x$; inverse of $A$ ... $\mathrm{H}$ is a group. $\mathrm{H}$ satisfies inverse prop, but not a group for every $A \in 2^x$; the inverse of $A$ is $A$.
closed
GO Classes
asked
in
Set Theory & Algebra
Feb 5, 2023
by
GO Classes
868
views
memorybased-gatecse2023
goclasses
set-theory&algebra
group-theory
multiple-selects
18
votes
4
answers
35
GATE CSE 2022 | Question: 10
Consider the following two statements with respect to the matrices $\textit{A}_{m \times n}, \textit{B}_{n \times m}, \textit{C}_{n \times n}$ and $ \textit{D}_{n \times n}.$ Statement $1: tr \text{(AB)} = tr \text{(BA)}$ ... $2$ is correct. Both Statement $1$ and Statement $2$ are correct. Both Statement $1$ and Statement $2$ are wrong.
Arjun
asked
in
Linear Algebra
Feb 15, 2022
by
Arjun
10.4k
views
gatecse-2022
linear-algebra
matrix
1-mark
14
votes
2
answers
36
GATE CSE 2022 | Question: 17
Which of the following statements is/are $\text{TRUE}$ for a group $\textit{G}?$ If for all $x,y \in \textit{G}, \; (xy)^{2} = x^{2} y^{2},$ then $\textit{G}$ is commutative. If for all $x \in \textit{G}, \; x^{2} = 1,$ then ... $2,$ then $\textit{G}$ is commutative. If $\textit{G}$ is commutative, then a subgroup of $\textit{G}$ need not be commutative.
Arjun
asked
in
Set Theory & Algebra
Feb 15, 2022
by
Arjun
7.4k
views
gatecse-2022
set-theory&algebra
group-theory
multiple-selects
1-mark
19
votes
5
answers
37
GATE CSE 2022 | Question: 20
Consider a simple undirected graph of $10$ vertices. If the graph is disconnected, then the maximum number of edges it can have is _______________ .
Arjun
asked
in
Graph Theory
Feb 15, 2022
by
Arjun
8.6k
views
gatecse-2022
numerical-answers
graph-theory
graph-connectivity
1-mark
17
votes
3
answers
38
GATE CSE 2022 | Question: 22
The number of arrangements of six identical balls in three identical bins is _____________ .
Arjun
asked
in
Combinatory
Feb 15, 2022
by
Arjun
10.2k
views
gatecse-2022
numerical-answers
combinatory
balls-in-bins
1-mark
9
votes
4
answers
39
GATE CSE 2022 | Question: 24
The value of the following limit is ________________. $\lim_{x \rightarrow 0^{+}} \frac{\sqrt{x}}{1-e^{2\sqrt{x}}}$
Arjun
asked
in
Calculus
Feb 15, 2022
by
Arjun
6.3k
views
gatecse-2022
numerical-answers
calculus
limits
1-mark
26
votes
6
answers
40
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
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