GATE CS Engineering Mathematics questions

An urn contains one red ball and one blue ball. At each step, a ball is picked uniformly at random from the urn, and this ball together with another ball of the same color is put back in the urn. The probability that there are equal number of red and blue balls after two steps is

For $n>1$, the maximum multiplicity of any eigenvalue of an $n \times n$ matrix with elements from $\mathbb{R}$ is

Let $n>1$. Consider an $n \times n$ matrix $M$ with its elements from $\mathbb{R}$. Let the vector $(0,1,0,0, \ldots, 0) \in \mathbb{R}^{n}$ be in the null space of $M$. Which of the following options is/are always correct?

Consider the function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined as follows: $$f(x)=\left\{\begin{array}{lr} c_{1} e^{x}-c_{2} \log _{\mathrm{e}}\left(\frac{1}{x}\right), & \text{ if } x>0 \\ 3 & \text{ otherwise } \end{array}\right.$$ where $c_{1}, c_{2} \in \mathbb{R}$. If $f$ is continuous at $x=0$, then $c_{1}+c_{2}=$ …

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be defined as follows: $$f(x)=\left(\frac{|x|}{2}-x\right)\left(x-\frac{|x|}{2}\right)$$ Which of the following statements is/are true?

Let $G(V, E)$ be a simple, undirected graph. A vertex cover of $G$ is a subset $V^{\prime} \subseteq V$ such that for every $(u, v) \in E, u \in V^{\prime}$ or $v \in V^{\prime}$. Let the size of the smallest vertex cover in $G$ be $k$. Let $S$ be any vertex cover of size $k$. For a vertex $v \in V$, which of the following constraints …

An undirected, unweighted, simple graph $G(V, E)$ is said to be $2$-colorable if there exists a function $c: V \rightarrow\{0,1\}$ such that for every $(u, v) \in E, c(u) \neq c(v)$. Which of the following statements about $2$-colorable graphs is/are true?

Let $G$ be an undirected graph, which is a path on $8$ vertices. The number of matchings in $G$ is $\_\_\_\_$. (answer in integer)

Let $X$ be a random variable which takes values in the set $\{1,2,3,4,5,6,7,8\}$. Further, $\operatorname{Pr}(X=1)=\operatorname{Pr}(X=2)=\operatorname{Pr}(X=5)=\operatorname{Pr}(X=7)=\frac{1}{6}$ and $\operatorname{Pr}(X=3)=\operatorname{Pr}(X=4)=\operatorname{Pr}(X=6)=\operatorname{Pr}(X=8)=\frac{1}{12}$. The expected value of $X$, denoted by …

For two different persons $x$ and $y$, the predicate $M(x, y)$ denotes that $x$ knows $y$. Consider the following statement. There is a person who does not know anyone else, but that person is known by everyone else. Which one of the following expressions represents the above statement?

The probability density function $f(x)$ of a random variable $X$ which takes real values is $$f(x)=\frac{1}{3 \sqrt{2 \pi}} \exp \left(-\frac{x^{2}}{18}\right), \quad x \in(-\infty,+\infty)$$ Which one of the following statements is correct about the random variable $X$ ?

Let $R$ be a binary relation on the set $\{1,2, \ldots, 10\}$, where $(x, y) \in R$ if the product of $x$ and $y$ is square of an integer. Which of the following properties is/are satisfied by $R$?

For a real number $a$, let $I(a)=\int_{-1}^{1}\left(3 x^{2}-a x+1\right) d x$. Which of the following statements is/are true?

Consider the system of linear equations given below. $$\begin{array}{c} a x+y=b \\ 16 x+a y=24 \end{array}$$ Suppose the values of $a$ and $b$ are chosen such that the system of linear equations produce multiple solutions. Then the product of $a$ and $b$ is $\_\_\_\_$. (answer in integer)

Consider a complete graph $K_{n}$ with $n$ vertices ( $n>4$ ). Note that multiple spanning trees can be constructed over $K_{n}$. Each of these spanning trees is represented as a set of edges. The Jaccard coefficient between any two sets is defined as the ratio of the size of the intersection of the two sets to the size of the union of the two …

The determinant of a $4 \times 4$ matrix $A$ is $3$. The value of the determinant of $2 A$ is $\_\_\_\_$. (answer in integer)

Suppose an unbiased coin is tossed $6$ times. Each coin toss is independent of all previous coin tosses. Let $E_{1}$ be the event that among the second, fourth, and sixth coin tosses, there are at least two heads. Let $E_{2}$ be the event that among the first, second, third, and fifth coin tosses, there are equal number of heads and tails. The …

Consider a function $f:(0,1) \rightarrow\{0,1\}$ defined as follows. For a real number $r \in(0,1), f(r)=1$ if the second digit after the decimal point in $r$ is one of the four digits $2,3,6$ and $7$. Otherwise, $f(r)$ is equal to $0$. The number of points in $(0,1)$ at which $f$ is discontinuous is $\_\_\_\_\_\_$. (answer in integer)

$g(\cdot)$ is a function from $A$ to $B$, $f(\cdot)$ is a function from $B$ to $C$, and their composition defined as $f(g(\cdot))$ is a mapping from $A$ to $C$. If $f(\cdot)$ and $f(g(\cdot))$ are onto (surjective) functions, which ONE of the following is TRUE about the function $g(\cdot)$ ?

Consider the given system of linear equations for variables $x$ and $y$, where $k$ is a real-valued constant. Which of the following option(s) is/are CORRECT? $$\begin{array}{l} x+k y=1 \\ k x+y=-1 \end{array}$$