Previous GATE Questions in Engineering Mathematics

+5 votes

2 answers

1

GATE1995-25b
Determine the number of positive integers $(\leq 720)$ which are not divisible by any of $2,3$ or $5.$

asked Jun 6, 2019 in Set Theory & Algebra by Arjun Veteran (431k points) | 406 views

+2 votes

1 answer

2

Mathematics: GATE2017 EC-2-22
Consider the random process: $X\left ( t \right )=U+Vt$ where $U$ is zero-mean Gaussian random variable and $V$ is a random variable uniformly distributed between $0$ and $2.$ Assume $U$ and $V$ statistically independent. The mean value of random process at $t=2$ is ___________

asked Jun 3, 2019 in Probability by srestha Veteran (119k points) | 168 views

+2 votes

4 answers

3

GATE2017 EC
The rank of the matrix $\begin{bmatrix} 1 & -1 & 0 &0 & 0\\ 0 & 0 & 1 &-1 &0 \\ 0 &1 &-1 &0 &0 \\ -1 & 0 &0 & 0 &1 \\ 0&0 & 0 & 1 & -1 \end{bmatrix}$ is ________. Ans 5?

asked Jun 1, 2019 in Linear Algebra by srestha Veteran (119k points) | 297 views

+10 votes

3 answers

4

GATE2019-5
Let $U = \{1, 2, \dots , n\}$ Let $A=\{(x, X) \mid x \in X, X \subseteq U \}$. Consider the following two statements on $\mid A \mid$. $\mid A \mid = n2^{n-1}$ $\mid A \mid = \Sigma_{k=1}^{n} k \begin{pmatrix} n \\ k \end{pmatrix}$ Which of the above statements is/are TRUE? Only I Only II Both I and II Neither I nor II

asked Feb 7, 2019 in Combinatory by Arjun Veteran (431k points) | 3.1k views

+6 votes

5 answers

5

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

asked Feb 7, 2019 in Linear Algebra by Arjun Veteran (431k points) | 2.3k views

+11 votes

3 answers

6

GATE2019-10
Let $G$ be an arbitrary group. Consider the following relations on $G$: $R_1: \forall a , b \in G, \: a R_1 b \text{ if and only if } \exists g \in G \text{ such that } a = g^{-1}bg$ $R_2: \forall a , b \in G, \: a R_2 b \text{ if and only if } a= b^{-1}$ Which of the above is/are equivalence relation/relations? $R_1$ and $R_2$ $R_1$ only $R_2$ only Neither $R_1$ nor $R_2$

asked Feb 7, 2019 in Set Theory & Algebra by Arjun Veteran (431k points) | 3.5k views

+8 votes

9 answers

7

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 $n!$ $(n-1)!$ $1$ $\frac{(n-1)!}{2}$

asked Feb 7, 2019 in Graph Theory by Arjun Veteran (431k points) | 4k views

+4 votes

6 answers

8

GATE2019-13
Compute $\displaystyle \lim_{x \rightarrow 3} \frac{x^4-81}{2x^2-5x-3}$ $1$ $53/12$ $108/7$ Limit does not exist

asked Feb 7, 2019 in Calculus by Arjun Veteran (431k points) | 2k views

+7 votes

10 answers

9

GATE2019-21
The value of $3^{51} \text{ mod } 5$ is _____

asked Feb 7, 2019 in Combinatory by Arjun Veteran (431k points) | 4.1k views

+11 votes

8 answers

10

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

asked Feb 7, 2019 in Mathematical Logic by Arjun Veteran (431k points) | 5.9k views

+6 votes

5 answers

11

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

asked Feb 7, 2019 in Graph Theory by Arjun Veteran (431k points) | 3.8k views

+7 votes

4 answers

12

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 _______

asked Feb 7, 2019 in Linear Algebra by Arjun Veteran (431k points) | 3.5k views

+8 votes

3 answers

13

GATE2019-47
Suppose $Y$ is distributed uniformly in the open interval $(1,6)$. The probability that the polynomial $3x^2 +6xY+3Y+6$ has only real roots is (rounded off to $1$ decimal place) _______

asked Feb 7, 2019 in Probability by Arjun Veteran (431k points) | 3.9k views

+3 votes

4 answers

14

GATE2019-23
Two numbers are chosen independently and uniformly at random from the set $ [ 1, 2, \dots, 13]$. The probability (rounded off to $3$ decimal places ) that their $4-bit$ (unsigned) binary representations have the same most significant bit is

asked Feb 7, 2019 in Probability by Ram Swaroop Loyal (5.3k points) | 1.1k views

0 votes

2 answers

15

GATE2019
What is the total number of different Hamiltonian cycles for the complete graph of n vertices?

asked Feb 3, 2019 in Graph Theory by Atul Sharma 1 (81 points) | 860 views

+2 votes

1 answer

16

GATE2016 37 Mathematics
Let $M = \begin{bmatrix} a & b &c \\ b &d & e\\ c & e & f \end{bmatrix}$ be a real matrix with eigenvalues 1, 0 and 3. If the eigenvectors corresponding to 1 and 0 are $\left ( 1,1,1 \right )^T$ and $\left ( 1,-1, 0 \right )^T$ respectively, then the value of 3f is equal to _______.

asked Oct 4, 2018 in Linear Algebra by Mk Utkarsh Boss (36.5k points) | 308 views

0 votes

0 answers

17

First Order Logic: GATE2005-41 ( From gate Overflow volume 1)
Can the answer to this be "∀x ∃y (teacher (x) ∧ student (y) ∧ likes (y,x))" ?

asked Sep 30, 2018 in Mathematical Logic by rambo1987 (19 points) | 115 views

+3 votes

2 answers

18

GATE1998-10b
Let $R$ be a binary relation on $A = \{a, b, c, d, e, f, g, h\}$ represented by the following two component digraph. Find the smallest integers $m$ and $n$ such that $m < n$ and $R^m = R^n$.

asked Aug 12, 2018 in Set Theory & Algebra by Arjun Veteran (431k points) | 394 views

+29 votes

13 answers

19

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

asked Feb 14, 2018 in Combinatory by gatecse Boss (17.5k points) | 9.7k views

+14 votes

4 answers

20

GATE2018-44
Consider Guwahati, (G) and Delhi (D) whose temperatures can be classified as high $(H)$, medium $(M)$ and low $(L)$. Let $P(H_G)$ denote the probability that Guwahati has high temperature. Similarly, $P(M_G)$ and $P(L_G)$ denotes the ... , then the probability (correct to two decimal places) that Guwahati has high temperature given that Delhi has high temperature is _____

asked Feb 14, 2018 in Probability by gatecse Boss (17.5k points) | 4.6k views

+18 votes

9 answers

21

GATE2018-30
Let $G$ be a simple undirected graph. Let $T_D$ be a depth first search tree of $G$. Let $T_B$ be a breadth first search tree of $G$. Consider the following statements. No edge of $G$ is a cross edge with respect to $T_D$. (A cross edge in $G$ is between ... then $\mid i-j \mid =1$. Which of the statements above must necessarily be true? I only II only Both I and II Neither I nor II

asked Feb 14, 2018 in Graph Theory by gatecse Boss (17.5k points) | 6.3k views

+29 votes

3 answers

22

GATE2018-28
Consider the first-order logic sentence $\varphi \equiv \exists \: s \: \exists \: t \: \exists \: u \: \forall \: v \: \forall \: w \forall \: x \: \forall \: y \: \psi(s, t, u, v, w, x, y)$ ... than or equal to $3$ There exists no model of $\varphi$ with universe size of greater than $7$ Every model of $\varphi$ has a universe of size equal to $7$

asked Feb 14, 2018 in Mathematical Logic by gatecse Boss (17.5k points) | 7.8k views

+29 votes

4 answers

23

GATE2018-26
Consider a matrix P whose only eigenvectors are the multiples of $\begin{bmatrix} 1 \\ 4 \end{bmatrix}$. Consider the following statements. P does not have an inverse P has a repeated eigenvalue P cannot be diagonalized Which one of the following options ... and III are necessarily true Only II is necessarily true Only I and II are necessarily true Only II and III are necessarily true

asked Feb 14, 2018 in Linear Algebra by gatecse Boss (17.5k points) | 6.8k views

+27 votes

5 answers

24

GATE2018-27
Let $N$ be the set of natural numbers. Consider the following sets, $P:$ Set of Rational numbers (positive and negative) $Q:$ Set of functions from $\{0,1\}$ to $N$ $R:$ Set of functions from $N$ to $\{0, 1\}$ $S:$ Set of finite subsets of $N$ Which of the above sets are countable? $Q$ and $S$ only $P$ and $S$ only $P$ and $R$ only $P, Q$ and $S$ only

asked Feb 14, 2018 in Set Theory & Algebra by gatecse Boss (17.5k points) | 5.5k views

+11 votes

3 answers

25

GATE2018-17
Consider a matrix $A= uv^T$ where $u=\begin{pmatrix}1 \\ 2 \end{pmatrix} , v = \begin{pmatrix}1 \\1 \end{pmatrix}$. Note that $v^T$ denotes the transpose of $v$. The largest eigenvalue of $A$ is ____

asked Feb 14, 2018 in Linear Algebra by gatecse Boss (17.5k points) | 2.9k views

+18 votes

4 answers

26

GATE2018-18
The chromatic number of the following graph is _____

asked Feb 14, 2018 in Graph Theory by gatecse Boss (17.5k points) | 2.9k views

+12 votes

4 answers

27

GATE2018-19
Let $G$ be a finite group on $84$ elements. The size of a largest possible proper subgroup of $G$ is _____

asked Feb 14, 2018 in Set Theory & Algebra by gatecse Boss (17.5k points) | 3.9k views

+12 votes

6 answers

28

GATE2018-16
The value of $\int^{\pi/4} _0 x \cos(x^2) dx$ correct to three decimal places (assuming that $\pi = 3.14$) is ____

asked Feb 14, 2018 in Calculus by gatecse Boss (17.5k points) | 4.8k views

+10 votes

5 answers

29

GATE2018-15
Two people, $P$ and $Q$, decide to independently roll two identical dice, each with $6$ faces, numbered $1$ to $6$. The person with the lower number wins. In case of a tie, they roll the dice repeatedly until there is no tie. Define a trial ... probable and that all trials are independent. The probability (rounded to $3$ decimal places) that one of them wins on the third trial is ____

asked Feb 14, 2018 in Probability by gatecse Boss (17.5k points) | 3.5k views

+18 votes

10 answers

30

GATE2018-1
Which one of the following is a closed form expression for the generating function of the sequence $\{a_n\}$, where $a_n = 2n +3 \text{ for all } n=0, 1, 2, \dots$? $\frac{3}{(1-x)^2}$ $\frac{3x}{(1-x)^2}$ $\frac{2-x}{(1-x)^2}$ $\frac{3-x}{(1-x)^2}$

asked Feb 14, 2018 in Combinatory by gatecse Boss (17.5k points) | 7k views

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