GATE CS Engineering Mathematics questions

The minimum number of colours that is sufficient to vertex-colour any planar graph is ________.

Suppose that the eigenvalues of matrix $A$ are $1, 2, 4$. The determinant of $\left(A^{-1}\right)^{T}$ is _________.

Consider the systems, each consisting of $m$ linear equations in $n$ variables. If $m < n$, then all such systems have a solution. If $m > n$, then none of these systems has a solution. If $m = n$, then there exists a system which has a solution. Which one of the following is CORRECT?

Suppose that a shop has an equal number of LED bulbs of two different types. The probability of an LED bulb lasting more than $100$ hours given that it is of Type $1$ is $0.7$, and given that it is of Type $2$ is $0.4$. The probability that an LED bulb chosen uniformly at random lasts more than $100$ hours is _________.

Let G be a connected planar graph with 10 vertices. If the number of edges on each face is three, then the number of edges in G is_______________.

Compute the value of: $$\large \int \limits_{\frac{1}{\pi}}^{\frac{2}{\pi}}\frac{\cos(1/x)}{x^{2}}dx$$

Consider the following $2 \times 2$ matrix $A$ where two elements are unknown and are marked by $a$ and $b$. The eigenvalues of this matrix are $-1$ and $7.$ What are the values of $a$ and $b$? $\qquad A = \begin{pmatrix}1 & 4\\ b&a \end{pmatrix}$

Suppose $L = \left\{ p, q, r, s, t\right\}$ is a lattice represented by the following Hasse diagram: For any $x, y \in L$, not necessarily distinct , $x \vee y$ and $x \wedge y$ are join and meet of $x, y$, respectively. Let $L^3 = \left\{\left(x, y, z\right): x, y, z \in L\right\}$ be the set of all ordered triplets of the elements of $L$. Let …

$\sum\limits_{x=1}^{99}\frac{1}{x(x+1)}$ = ______.

In the LU decomposition of the matrix $\begin{bmatrix}2 & 2 \\ 4 & 9\end{bmatrix}$, if the diagonal elements of $U$ are both $1$, then the lower diagonal entry $l_{22}$ of $L$ is_________________.

For a set $A$, the power set of $A$ is denoted by $2^{A}$. If $A = \left\{5,\left\{6\right\}, \left\{7\right\}\right\}$, which of the following options are TRUE? $\varnothing \in 2^{A}$ $\varnothing \subseteq 2^{A}$ $\left\{5,\left\{6\right\}\right\} \in 2^{A}$ $\left\{5,\left\{6\right\}\right\} \subseteq 2^{A}$

The binary operator $\neq$ is defined by the following truth table. …

Which one of the following is NOT equivalent to $p ↔ q$?

If $g(x) = 1 - x$ and $h(x) = \frac{x}{x-1}$, then $\frac{g(h(x))}{h(g(x))}$ is:

$\displaystyle \lim_{x\rightarrow \infty } x^{ \tfrac{1}{x}}$ is

Which one of the following well-formed formulae is a tautology?

Let $X$ and $Y$ denote the sets containing $2$ and $20$ distinct objects respectively and $F$ denote the set of all possible functions defined from $X$ to $Y$. Let $f$ be randomly chosen from $F$. The probability of $f$ being one-to-one is ______.

In a connected graph, a bridge is an edge whose removal disconnects the graph. Which one of the following statements is true?

The number of onto functions (surjective functions) from set $X = \{1, 2, 3, 4\}$ to set $Y=\{a,b,c\}$ is ______.

A graph is self-complementary if it is isomorphic to its complement. For all self-complementary graphs on $n$ vertices, $n$ is