GATE CS Engineering Mathematics questions

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)

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 one of the following is true. All members of $A$ appear in the …

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 $$a_{1} \sim a_{2} \text{ if } f\left(a_{1}\right)=f\left(a_{2}\right),$$ where $a_{1}, a_{2} \in A$. Let $\mathcal{E}=\{[x]: x \in A\}$ be the set of all the equivalence classes under …

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?

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 second throw. Which of the following statements is/are $\text{TRUE}?$

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 of $G$ using the following greedy strategy: for $i=1, \ldots, n$ …

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 $1: tr \text{(AB)} = tr \text{(BA)}$ Statement $2: tr \text{(CD)} = tr \text{(DC)}$ where $tr()$ represents the trace of a matrix. Which one of the following holds?

Which of the following statements is/are $\text{TRUE}$ for a group $\textit{G}?$

Consider a simple undirected graph of $10$ vertices. If the graph is disconnected, then the maximum number of edges it can have is _______________ .

The number of arrangements of six identical balls in three identical bins is _____________ .

The value of the following limit is ________________. $$\lim_{x \rightarrow 0^{+}} \frac{\sqrt{x}}{1-e^{2\sqrt{x}}}$$

Which one of the following is the closed form for the generating function of the sequence $\{ a_{n} \}_{n \geq 0}$ defined below? $$a_{n} = \left\{\begin{matrix} n + 1, & \text{n is odd} & \\ 1, & \text{otherwise} & \end{matrix}\right.$$

Consider a simple undirected unweighted graph with at least three vertices. If $\textit{A}$ is the adjacency matrix of the graph, then the number of $3–$cycles in the graph is given by the trace of

Consider solving the following system of simultaneous equations using $\text{LU}$ decomposition. $$x_{1} + x_{2} – 2x_{3} = 4$$ $$x_{1} + 3x_{2} – x_{3} = 7$$ $$2x_{1} + x_{2} – 5x_{3} = 7$$ where $\textit{L}$ and $\textit{U}$ are denoted as …

The following simple undirected graph is referred to as the Peterson graph. Which of the following statements is/are $\text{TRUE}?$ The size of the largest independent set of the given graph is $3.$ (A subset of vertices of a graph form an independent set if no two vertices of the subset are adjacent.)

Consider the following recurrence: $$\begin{array}{} f(1) & = & 1; \\ f(2n) & = & 2f(n) – 1, & \; \text{for}\; n \geq 1; \\ f(2n+1) & = & 2f(n) + 1, & \; \text{for}\; n \geq 1. \end{array}$$ Then, which of the following statements is/are $\text{TRUE}?$

Which of the properties hold for the adjacency matrix $A$ of a simple undirected unweighted graph having $n$ vertices?

​​​​​Which of the following is/are the eigenvector(s) for the matrix given below? $$\begin{pmatrix} – 9 & – 6 & – 2 & – 4 \\ – 8 & – 6 & – 3 & – 1 \\ 20 & 15 & 8 & 5 \\ 32 & 21 & 7 & 12 \end{pmatrix}$$

Let $p$ and $q$ be two propositions. Consider the following two formulae in propositional logic. $S_1: (\neg p\wedge(p\vee q))\rightarrow q$ $S_2: q\rightarrow(\neg p\wedge(p\vee q))$ Which one of the following choices is correct?

In an undirected connected planar graph $G$, there are eight vertices and five faces. The number of edges in $G$ is _________.