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 $A^{13}$?

$A = \{0, 1, 2, 3, \dots \}$ 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)

A pot contains two red balls and two blue balls. Two balls are drawn from this pot randomly without replacement. What is the probability that the two balls drawn have different colours?

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?