Consider the given function $f(x)$.

$$f(x)=\left\{\begin{array}{cc} a x+b & \text { for } x<1 \\ x^3+x^2+1 & \text { for } x \geq 1 \end{array}\right.$$

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.

$$A=\left[\begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array}\right]$$

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:

$\bullet$ mother $(y, x): y$ is the mother of $x$

$\bullet$ noteq $(x, y): x$ and $y$ are not equal