Consider the system of linear equations given below.

$$ \begin{aligned} a x+y & =b \\ 16 x+a y & =24 \end{aligned} $$

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 sets. Which one of the following options gives the lowest possible value for the Jaccard coefficient between any two spanning trees of $K_n$ ?

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 conditional probability $P\left(E_1 \mid E_2\right)$ is equal to $\_\_\_\_$ . (rounded off to one decimal place)

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)