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Previous GATE Questions in Engineering Mathematics

9 votes
2 answers
1
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 Arjun 1.2k views
4 votes
1 answer
2
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 srestha 443 views
3 votes
4 answers
3
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 srestha 861 views
17 votes
4 answers
4
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 Arjun 5k views
14 votes
6 answers
5
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 Arjun 4.3k views
20 votes
5 answers
6
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 Arjun 6.8k views
17 votes
9 answers
7
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 Arjun 9.3k views
8 votes
6 answers
8
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 Arjun 3.1k views
11 votes
13 answers
9
The value of $3^{51} \text{ mod } 5$ is _____
asked Feb 7, 2019 in Combinatory Arjun 9.4k views
28 votes
10 answers
10
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 $a \mid b$ ... $S3:$ 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 Arjun 9.5k views
17 votes
6 answers
11
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 Arjun 9k views
11 votes
4 answers
12
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 Arjun 7.3k views
18 votes
3 answers
13
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 Arjun 7.3k views
6 votes
3 answers
14
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 Ram Swaroop 2.3k views
0 votes
2 answers
15
What is the total number of different Hamiltonian cycles for the complete graph of n vertices?
asked Feb 3, 2019 in Graph Theory Atul Sharma 1 1.1k views
2 votes
1 answer
16
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 Mk Utkarsh 482 views
0 votes
1 answer
17
Can the answer to this be "∀x ∃y (teacher (x) ∧ student (y) ∧ likes (y,x))" ?
asked Sep 30, 2018 in Mathematical Logic rambo1987 299 views
5 votes
2 answers
18
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 Arjun 1.3k views
21 votes
4 answers
19
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 probability of ... , 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 gatecse 7k views
34 votes
5 answers
20
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 gatecse 9.4k views
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