Let $A$ be a $2 \times 2$ matrix as given: $$A = \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}$$ 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

$A = \{0, 1, 2, 3, \dots \}$ is the set of non-negative integers. Let $F$ be the set of functions from $A$ to itself. For any two functions $f1, f2 \in F$, we define $$(f1 \odot f2)(n) = f1(n) + f2(n)$$ for every number $n$ in $A$. Which of the following is/are CORRECT about the mathematical structure $(F, \odot)$?

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

$A = \{0, 1, 2, 3, \dots\}$ is the set of non-negative integers. Let $F$ be the set of functions from $A$ to itself. For any two functions $f1, f2 \in F$, we define $$(f1 \odot f2)(n) = f1(n) + f2(n)$$ for every number $n$ in $A$. Which of the following is/are CORRECT about the mathematical structure $(F, \odot)$?

Suppose a $5$-bit message is transmitted from a source to a destination through a noisy channel. The probability that a bit of the message gets flipped during transmission is $0.01$. Flipping of each bit is independent of one another. The probability that the message is delivered error-free to the destination is \\\\\\\. (rounded off to three decimal places)

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

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? $$x + ky = 1$$ $$kx + y = -1$$

Consider the given function $f(x)$. $$f(x) = \begin{cases} ax + b & \text{for } x < 1 \\ x^3 + x^2 + 1 & \text{for } x \ge 1 \end{cases}$$ 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)

The number of spanning trees in a complete graph of $4$ vertices labelled A, B, C, and D is \\\\\\\\\\\

Let $A$ be any $n \times m$ matrix, where $m > n$. Which of the following statements is/are TRUE about the system of linear equations $Ax = \mathbf{0}$?

The chromatic number of a graph is the minimum number of colours used in a proper colouring of the graph. Let $G$ be any graph with $n$ vertices and chromatic number $k$. Which of the following statements is/are always TRUE?

Consider the operators $\diamond$ and $\square$ defined by $a \diamond b = a + 2b$, $a \square b = ab$, for positive integers. Which of the following statements is/are TRUE?