Let $$G$$ be a finite group on $$84$$ elements. The size of a largest possible proper subgroup of $$G$$ is ________.

The chromatic number of the following graph is _______.

Consider a matrix P whose only eigenvectors are the multiples of $$\left[ {\matrix{ 1 \cr 4 \cr } } \right].$$

Consider the following statements.

$$\left( {\rm I} \right)$$ $$\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$P$$ does not have an inverse
$$\left( {\rm II} \right)$$ $$\,\,\,\,\,\,\,\,\,\,\,$$ $$P$$ has a repeated eigenvalue
$$\left( {\rm III} \right)$$ $$\,\,\,\,\,\,\,\,\,$$ $$P$$ cannot be diagonalized

Which one of the following options is correct?

Let N be the set of natural numbers. Consider the following sets.

$$\,\,\,\,\,\,\,\,$$ $$P:$$ Set of Rational numbers (positive and negative)
$$\,\,\,\,\,\,\,\,$$ $$Q:$$ Set of functions from $$\left\{ {0,1} \right\}$$ to $$N$$
$$\,\,\,\,\,\,\,\,$$ $$R:$$ Set of functions from $$N$$ to $$\left\{ {0,1} \right\}$$
$$\,\,\,\,\,\,\,\,$$ $$S:$$ Set of finite subsets of $$N.$$

Which of the sets above are countable?