Consider the quadratic equation $nx^{2} + 7\sqrt {n}x + n = 0$, where $n$ is a positive integer. Which of the following statements are necessarily correct?

I. For any $n$, the roots are distinct.

II. There are infinitely many values of $n$ for which both roots are real.

III. The product of the roots is necessarily an integer.

Let the discriminant of the quadratic equation $ax^{2} + bx + c$, where $a, b$ and $c$ are real, be given by $D = b^{2} - 4ac$. Roots of the given equation are

i) real and distinct if $D > 0$

ii) real and equal if $D = 0$

iii) imaginary and distinct if $D < 0$

Product of the roots is $\dfrac{c}{a}$.

For the given problem, $D = 49n-4n^{2}$

I) roots are distinct if $D \neq 0$

$\Rightarrow 49n \neq 4n^{2}$

$\Rightarrow n \neq \dfrac{49}{4}$

$n$ is a positive integer, so for all n, the roots are distinct.

II) Both roots are real $\Rightarrow D \geq 0$

$\Rightarrow 49n\geq 4n^{2}$

$\Rightarrow n\leq \dfrac{49}{4}$

So, such values of $n$ are not infinite.

III) Product of roots $= \dfrac{n}{n} = 1$, which is an integer.