The value of $$\int_0^{\pi /4} {x\cos \left( {{x^2}} \right)dx} $$ correct to three decimal places (assuming that $$\pi = 3.14$$ ) is ________.

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?