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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.
$III$ only
$I$ and $III$
$II$ and $III$
$I, II$ and $III$
Solution
(b)
Given, $n x^2+7 \sqrt{n} x+n=0$ $D=49 n-4 n^2=n(49-4 n)$ $D \neq 0 ; \quad \therefore \forall n \in I^{+}$
$\therefore$ Roots are distinct.
For roots are real $D \geq 0$
$\therefore \quad n(49-4 n) \geq 0 \Rightarrow n \leq \frac{49}{4}$
So, $n \in\{1,2,3,4, \ldots, 12\}$
So, $x$ have finite value.
Product of roots is $\frac{n}{n}=1$
$\therefore$ Products of root is necessarily integer.
Hence, option $(b)$ is correct.
