Consider the quadratic equation $n x^2+7 \sqrt{n x+n}=0$ 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.
Consider the quadratic equation $n x^2+7 \sqrt{n x+n}=0$ 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.
- [KVPY 2016]
- A
$III$ only
- B
$I$ and $III$
- C
$II$ and $III$
- D
$I, II$ and $III$
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