Recent questions and answers in Engineering Mathematics - GATE Overflow

+54 votes

9 answers

1

GATE2015-1-39
Consider the operations $\textit{f (X, Y, Z) = X'YZ + XY' + Y'Z'}$ and $\textit{g (X, Y, Z) = X'YZ + X'YZ' + XY}$ Which one of the following is correct? Both $\left\{\textit{f} \right\}$ and $\left\{ \textit{g}\right\}$ are ... Only $\left\{ \textit{g}\right\}$ is functionally complete Neither $\left\{ \textit{f}\right\}$ nor $\left\{\textit{g}\right\}$ is functionally complete

answered 13 hours ago in Set Theory & Algebra by Kuldeep Pal Active (1.5k points) | 7.6k views

+23 votes

7 answers

2

GATE2006-28
A logical binary relation $\odot$ ... $(\sim A\odot B)$ $\sim(A \odot \sim B)$ $\sim(\sim A\odot\sim B)$ $\sim(\sim A\odot B)$

answered 14 hours ago in Set Theory & Algebra by haralk10 (11 points) | 1.5k views

+2 votes

5 answers

3

TIFR2019-A-15
Consider the matrix $A = \begin{bmatrix} \frac{1}{2} &\frac{1}{2} & 0\\ 0& \frac{3}{4} & \frac{1}{4}\\ 0& \frac{1}{4} & \frac{3}{4} \end{bmatrix}$ What is $\lim_{n→\infty}$A^n$ ? $\begin{bmatrix} \ 0 & 0 & 0\\ 0& 0 ... $\text{The limit exists, but it is none of the above}$

answered 18 hours ago in Calculus by severustux (121 points) | 383 views

+6 votes

3 answers

4

ISRO2007-29
The set of all Equivalence Classes of a set A of Cardinality C is of cardinality $2^c$ have the same cardinality as A forms a partition of A is of cardinality $C^2$

answered 1 day ago in Set Theory & Algebra by JashanArora Active (1.7k points) | 2.4k views

+8 votes

5 answers

5

TIFR2018-A-6
What is the minimum number of students needed in a class to guarantee that there are at least $6$ students whose birthdays fall in the same month ? $6$ $23$ $61$ $72$ $91$

answered 2 days ago in Combinatory by `JEET Boss (12.9k points) | 417 views

+4 votes

5 answers

6

TIFR2012-A-17
A spider is at the bottom of a cliff, and is $n$ inches from the top. Every step it takes brings it one inch closer to the top with probability $1/3$, and one inch away from the top with probability $2/3$, unless it is at the bottom in which case, it ... function of $n$? It will never reach the top. Linear in $n$. Polynomial in $n$. Exponential in $n$. Double exponential in $n$.

answered 2 days ago in Probability by pritishc Junior (657 points) | 594 views

+1 vote

2 answers

7

#set theory #groups
Consider the set H of all 3 × 3 matrices of the type: $\begin{bmatrix} a&f&e\\ 0&b&d\\ 0&0&c\\ \end{bmatrix}$ where a, b, c, d, e and f are real numbers and $abc ≠ 0$. Under the matrix multiplication operation, the set H is: (a) a group (b) a monoid but not a group (c) a semigroup but not a monoid (d) neither a group nor a semigroup

answered 2 days ago in Set Theory & Algebra by smsubham Loyal (9.6k points) | 75 views

+3 votes

4 answers

8

ISRO2008-34
If a square matrix A satisfies $A^TA=I$, then the matrix $A$ is Idempotent Symmetric Orthogonal Hermitian

answered 2 days ago in Linear Algebra by JashanArora Active (1.7k points) | 1.4k views

+34 votes

9 answers

9

GATE1994-1.6, ISRO2008-29
The number of distinct simple graphs with up to three nodes is $15$ $10$ $7$ $9$

answered 2 days ago in Graph Theory by JashanArora Active (1.7k points) | 9.7k views

+24 votes

13 answers

10

GATE2018-46
The number of possible min-heaps containing each value from $\{1,2,3,4,5,6,7\}$ exactly once is _______

answered 3 days ago in Combinatory by Praveenk99 (49 points) | 8.4k views

+37 votes

11 answers

11

GATE2016-1-26
The coefficient of $x^{12}$ in $\left(x^{3}+x^{4}+x^{5}+x^{6}+\dots \right)^{3}$ is ___________.

answered 3 days ago in Combinatory by pritishc Junior (657 points) | 9k views

+28 votes

5 answers

12

GATE2000-6
Let $S$ be a set of $n$ elements $\left\{1, 2,....., n\right\}$ and $G$ a graph with 2$^{n}$ vertices, each vertex corresponding to a distinct subset of $S$. Two vertices are adjacent iff the symmetric difference of the corresponding sets has exactly $2$ ... $G$ has the same degree. What is the degree of a vertex in $G$? How many connected components does $G$ have?

answered 3 days ago in Set Theory & Algebra by Vimal Patel (137 points) | 1.8k views

+1 vote

3 answers

13

GATE2001-4
Consider the function $h: N \times N \rightarrow N$ so that $h(a,b) = (2a +1)2^b - 1$, where $N=\{0,1,2,3,\dots\}$ is the set of natural numbers. Prove that the function $h$ is an injection (one-one). Prove that it is also a Surjection (onto)

answered 3 days ago in Set Theory & Algebra by Verma Ashish Boss (11.8k points) | 453 views

+1 vote

1 answer

14

TIFR2019-A-6
A function $f: \mathbb{R} \rightarrow \mathbb{R}$ is said to be $\textit{convex}$ if for all $x,y \in \mathbb{R}$ and $\lambda$ such that $0 \leq \lambda \leq1,$ $f(\lambda x+ (1-\lambda)y) \leq \lambda f (x) + (1-\lambda) f(y)$. Let $f:$\mathbb{R}$ $→$ $\mathbb ... $p,q$ and $r$ must be convex? Only $p$ Only $q$ Only $r$ Only $p$ and $r$ Only $q$ and $r$

answered 4 days ago in Set Theory & Algebra by Satbir Boss (21.5k points) | 339 views

+4 votes

3 answers

15

GATE2019-44
Consider the following matrix: $R = \begin{bmatrix} 1 & 2 & 4 & 8 \\ 1 & 3 & 9 & 27 \\ 1 & 4 & 16 & 64 \\ 1 & 5 & 25 & 125 \end{bmatrix}$ The absolute value of the product of Eigen values of $R$ is _______

answered 4 days ago in Linear Algebra by JashanArora Active (1.7k points) | 2.7k views

+7 votes

7 answers

16

GATE2019-35
Consider the first order predicate formula $\varphi$: $\forall x [ ( \forall z \: z \mid x \Rightarrow (( z=x) \vee (z=1))) \rightarrow \exists w ( w > x) \wedge (\forall z \: z \mid w \Rightarrow ((w=z) \vee (z=1)))]$ Here $a \mid b$ ... Set of all positive integers $S3:$ Set of all integers Which of the above sets satisfy $\varphi$? S1 and S2 S1 and S3 S2 and S3 S1, S2 and S3

answered 5 days ago in Mathematical Logic by JashanArora Active (1.7k points) | 5.1k views

+5 votes

4 answers

17

GATE2019-38
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

answered 5 days ago in Graph Theory by JashanArora Active (1.7k points) | 3k views

+5 votes

9 answers

18

GATE2019-12
Let $G$ be an undirected complete graph on $n$ vertices, where $n > 2$. Then, the number of different Hamiltonian cycles in $G$ is equal to $n!$ $(n-1)!$ $1$ $\frac{(n-1)!}{2}$

answered 5 days ago in Graph Theory by JashanArora Active (1.7k points) | 3.3k views

+3 votes

5 answers

19

GATE2019-9
Let $X$ be a square matrix. Consider the following two statements on $X$. $X$ is invertible Determinant of $X$ is non-zero Which one of the following is TRUE? I implies II; II does not imply I II implies I; I does not imply II I does not imply II; II does not imply I I and II are equivalent statements

answered 5 days ago in Linear Algebra by JashanArora Active (1.7k points) | 2k views

+19 votes

5 answers

20

GATE2014-3-6
If $\int \limits_0^{2 \pi} |x \: \sin x| dx=k\pi$, then the value of $k$ is equal to ______.

answered 5 days ago in Calculus by Lakshman Patel RJIT Veteran (54.8k points) | 2.8k views

+17 votes

4 answers

21

GATE2014-3-47
The value of the integral given below is $\int \limits_0^{\pi} \: x^2 \: \cos x\:dx$ $-2\pi$ $\pi$ $-\pi$ $2\pi$

answered 5 days ago in Calculus by Lakshman Patel RJIT Veteran (54.8k points) | 1.9k views

0 votes

2 answers

22

self doubt
What is the general formula for number of simple graph having n unlabelled vertices ??

answered 6 days ago in Graph Theory by Muneendra1337 (11 points) | 60 views

0 votes

1 answer

23

group
if (G,*) is a cyclic group of order 97 , then number of generator of G is equal to ___

answered 6 days ago in Set Theory & Algebra by GoalSet1 (203 points) | 122 views

+7 votes

2 answers

24

GATE1995-2.13
A unit vector perpendicular to both the vectors $a=2i-3j+k$ and $b=i+j-2k$ is: $\frac{1}{\sqrt{3}} (i+j+k)$ $\frac{1}{3} (i+j-k)$ $\frac{1}{3} (i-j-k)$ $\frac{1}{\sqrt{3}} (i+j-k)$

answered Nov 28 in Linear Algebra by Satbir Boss (21.5k points) | 1.1k views

0 votes

1 answer

25

Probability- Gravner- 79.c
A random variable $X$ has the density function $f(x)= \begin{Bmatrix} c(x+\sqrt{x}) & x\epsilon [0,1]\\ 0& otherwise \end{Bmatrix}.$ (c) Determine the probability density function of $Y$ $=$ $X^2$

answered Nov 28 in Probability by Mk Utkarsh Boss (35.7k points) | 37 views

+4 votes

3 answers

26

ISRO-2013-49
What is the least value of the function $f(x) = 2x^{2}-8x-3$ in the interval $[0, 5]$? $-15$ $7$ $-11$ $-3$

answered Nov 28 in Calculus by Lakshman Patel RJIT Veteran (54.8k points) | 1.9k views

+1 vote

2 answers

27

ISI2014-DCG-71
Five letters $A, B, C, D$ and $E$ are arranged so that $A$ and $C$ are always adjacent to each other and $B$ and $E$ are never adjacent to each other. The total number of such arrangements is $24$ $16$ $12$ $32$

answered Nov 28 in Combinatory by noob_coder Junior (657 points) | 26 views

0 votes

1 answer

28

Probability - Gravner-69.b
$f(x) = \begin{Bmatrix} cx & if (0<x<4) \\ 0 & otherwise \end{Bmatrix}$ (c) Determine EX and Var(X).

answered Nov 27 in Probability by Mk Utkarsh Boss (35.7k points) | 22 views

0 votes

1 answer

29

Probability - Gravner-69.b
$f(x) = \begin{Bmatrix} cx & if (0<x<4) \\ 0 & otherwise \end{Bmatrix}$ (b) Compute $P(1\leqslant X\leqslant 2)$

answered Nov 27 in Probability by Mk Utkarsh Boss (35.7k points) | 22 views

0 votes

1 answer

30

Probability - Gravner-69.a
$f(x) = \begin{Bmatrix} cx & if (0<x<4) \\ 0 & otherwise \end{Bmatrix}$ (a) Determine $c$.

answered Nov 27 in Probability by Mk Utkarsh Boss (35.7k points) | 16 views

+20 votes

4 answers

31

GATE2006-IT-22
When a coin is tossed, the probability of getting a Head is $p, 0 < p < 1$. Let $N$ be the random variable denoting the number of tosses till the first Head appears, including the toss where the Head appears. Assuming that successive tosses are independent, the expected value of $N$ is $\dfrac{1}{p}$ $\dfrac{1}{(1 - p)}$ $\dfrac{1}{p^{2}}$ $\dfrac{1}{(1 - p^{2})}$

answered Nov 26 in Probability by Mk Utkarsh Boss (35.7k points) | 2.2k views

+2 votes

1 answer

32

Kenneth Rosen Edition 6th Exercise 6.4 Question 13 (Page No. 440)
Use Generating function to determine,the number of different ways $10$ identical balloons can be given to four children if each child receives atleast $2$ ballons? Ans given $(x^{2}+x^{3}+.........................)^{4}$ But as there is a upper ... Which one is correct? plz confirm

answered Nov 26 in Combinatory by Kushagra गुप्ता Active (1.8k points) | 198 views

+36 votes

7 answers

33

GATE2008-IT-21
Which of the following first order formulae is logically valid? Here $\alpha(x)$ is a first order formula with $x$ as a free variable, and $\beta$ ... $[(\forall x, \alpha(x)) \rightarrow \beta] \rightarrow [\forall x, \alpha(x) \rightarrow \beta]$

answered Nov 26 in Mathematical Logic by JashanArora Active (1.7k points) | 4.2k views

+1 vote

1 answer

34

TIFR-2011-Maths-A-19
The derivative of the function $\int_{0}^{\sqrt{x}} e^{-t^{2}}dt$ at $x = 1$ is $e^{-1}$ .

answered Nov 25 in Calculus by seetal samal (37 points) | 151 views

+1 vote

1 answer

35

TIFR-2011-Maths-A-21
Any continuous function from the open unit interval $(0, 1)$ to itself has a fixed point.

answered Nov 25 in Calculus by seetal samal (37 points) | 105 views

0 votes

1 answer

36

ISI2015-MMA-44
Let $P_1$, $P_2$ and $P_3$ denote, respectively, the planes defined by $\begin{array} {} a_1x +b_1y+c_1z=\alpha _1 \\ a_2x +b_2y+c_2z=\alpha _2 \\ a_3x +b_3y+c_3z=\alpha _3 \end{array}$ It is given that $P_1$, $P_2$ and $P_3$ ... then the planes do not have any common point of intersection intersect at a unique point intersect along a straight line intersect along a plane

answered Nov 25 in Linear Algebra by Lakshman Patel RJIT Veteran (54.8k points) | 14 views

+2 votes

2 answers

37

ISI2015-MMA-39
The eigenvalues of the matrix $X = \begin{pmatrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{pmatrix}$ are $1,1,4$ $1,4,4$ $0,1,4$ $0,4,4$

answered Nov 25 in Linear Algebra by Lakshman Patel RJIT Veteran (54.8k points) | 31 views

0 votes

1 answer

38

ISI2014-DCG-64
The value of $\lambda$ such that the system of equation $\begin{array}{} 2x & – & y & + & 2z & = & 2 \\ x & – & 2y & + & z & = & -4 \\ x & + & y & + & \lambda z & = & 4 \end{array}$ has no solution is $3$ $1$ $0$ $-3$

answered Nov 24 in Linear Algebra by Lakshman Patel RJIT Veteran (54.8k points) | 31 views

+28 votes

3 answers

39

GATE2002-1.25, ISRO2008-30, ISRO2016-6
The maximum number of edges in a n-node undirected graph without self loops is $n^2$ $\frac{n(n-1)}{2}$ $n-1$ $\frac{(n+1)(n)}{2}$

answered Nov 23 in Graph Theory by Shailendra_ Junior (555 points) | 5.5k views

+2 votes

2 answers

40

Integration
Solve the following $\int_{0}^{\infty}e^{-x^2}x^4dx$

answered Nov 22 in Calculus by Lakshman Patel RJIT Veteran (54.8k points) | 185 views

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