GATE CS Engineering Mathematics questions

Let $\text{S}$ be the set of all ternary strings defined over the alphabet $\{a, b, c\}$. Consider all strings in $\text{S}$ that contain at least one occurrence of two consecutive symbols, that is, $\text{"aa"}$, $\text{"bb"}$ or $\text{"cc"}$. The number of such strings of length $5$ that are possible is __________. (Answer in integer).

Consider the given function $f(x)$. $$f(x)=\left\{\begin{array}{ll} a x+b & \text{ for } x<1 \\ x^{3}+x^{2}+1 & \text{ for } x \geq 1 \end{array}\right.$$ If the function is differentiable everywhere, the value of $b$ must be ________. (rounded off to one decimal place)

A box contains $5$ coins: $4$ regular coins and $1$ fake coin. When a regular coin is tossed, the probability $P($ head $)=0.5$ and for a fake coin, $P($ head $)=1$. You pick a coin at random and toss it twice, and get two heads. The probability that the coin you have chosen is the fake coin is __________. (rounded off to two decimal places)

Let $A$ be a $2 \times 2$ matrix as given. $$A=\left[\begin{array}{rr} 1 & 1 \\ 1 & -1 \end{array}\right]$$ What are the eigenvalues of the matrix $A^{13}$ ?

Which of the following predicate logic formulae/formula is/are CORRECT representation(s) of the statement: "Everyone has exactly one mother"? The meanings of the predicates used are: mother $(y, x): y$ is the mother of $x$ noteq $(x, y): x$ and $y$ are not equal

$A=\{0,1,2,3, \ldots\}$ is the set of non-negative integers. Let $F$ be the set of functions from $A$ to itself. For any two functions, $f_{1}, f_{2} \in \mathrm{~F}$, we define $$\left(f_{1} \odot f_{2}\right)(n)=f_{1}(n)+f_{2}(n)$$ for every number $n$ in $A$. Which of the following is/are CORRECT about the mathematical structure …

Consider a probability distribution given by the density function $P(x)$. $$P(x)=\left\{\begin{aligned} C x^{2}, & \text{ for } 1 \leq x \leq 4 \\ 0, & \text{ for } x<1 \text{ or } x>4 \end{aligned}\right.$$ The probability that $x$ lies between $2$ and $3$, i.e., $P(2 \leq x \leq 3)$ is _________. (rounded off to three decimal …

A quadratic polynomial $(x-\alpha)(x-\beta)$ over complex numbers is said to be square invariant if $(x-\alpha)(x-\beta)=\left(x-\alpha^{2}\right)\left(x-\beta^{2}\right)$. Suppose from the set of all square invariant quadratic polynomials we choose one at random. The probability that the roots of the chosen polynomial are equal is __________. (rounded …

The unit interval $(0,1)$ is divided at a point chosen uniformly distributed over $(0,1)$ in $\mathbb{R}$ into two disjoint subintervals. The expected length of the subinterval that contains $0.4$ is __________. (rounded off to two decimal places)

​​​​If $A=\left(\begin{array}{cc}1 & 2 \\ 2 & -1\end{array}\right)$, then which ONE of the following is $A^{8}$ ?

​​​​​The value of $x$ such that $x>1$, satisfying the equation $\int_1^x t \ln t \: dt = \frac{1}{4}$ is

Let $L$, $M$, and $N$ be non-singular matrices of order $3$ satisfying the equations $L^2 = L^{-1}$, $M=L^8$ and $N=L^2$. Which ONE of the following is the value of the deteminant of $(M-N)$?

​​​Let $P(x)$ be an arbitrary predicate over the domain of natural numbers. Which ONE of the following statements is TRUE?

​​Let $\mathcal{F}$ be the set of all functions from $\{1, \ldots, n\}$ to $\{0,1\}$. Define the binary relation $\leqslant$ on $\mathcal{F}$ as follows: $\forall f, g \in \mathcal{F}, f \preccurlyeq g$ if and only if $\forall x \in\{1, \ldots, n\}, f(x) \leq g(x)$, where $0 \leq 1$. Which of the following statement(s) is/are TRUE?

Consider a system of linear equations $P X=Q$ where $P \in \mathbb{R}^{3 \times 3}$ and $\mathrm{Q} \in \mathbb{R}^{3 \times 1}$. Suppose $P$ has an LU decomposition, $P=L U$, where …

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function such that $f(x)=\max \left\{x, x^3\right\}, x \in \mathbb{R}$, where $\mathbb{R}$ is the set of all real numbers. The set of all points where $f(x)$ is NOT differentiable is

The product of all eigenvalues of the matrix $\left[\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{array}\right]$ is

Consider a permutation sampled uniformly at random from the set of all permutations of $\{1,2,3, \cdots, n\}$ for some $n \geq 4$. Let $X$ be the event that $1$ occurs before $2$ in the permutation, and $Y$ the event that $3$ occurs before $4$. Which one of the following statements is TRUE?

Let $A$ and $B$ be two events in a probability space with $P(A)=0.3, P(B)=0.5$, and $P(A \cap B)=0.1$. Which of the following statements is/are TRUE?

Let $A$ and $B$ be non-empty finite sets such that there exist one-to-one and onto functions $\text{(i)}$ from $A$ to $B$ and $\text{(ii)}$ from $A \times A$ to $A \cup B$. The number of possible values of $\text{|A|}$ is ___________.