Question

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

• A. III only
• B. I and III only
• C. II and III only
• D. I, II and III

Answer 1

The correct option is B I and III only

Given equation,
$n{x}^2+7\sqrt{n}+n=0$
Discriminant, $D=49n-4n^2$
$\therefore D\ne 0\forall n\in I^{+}$
Statement I is correct.

For both roots to be real
$D\ge 0\Rightarrow n(49-4n)\ge 0\Rightarrow n\in (0,\frac{49}{4})$
So $n=1,2,3,.......12$ finite numbers.
Statement II is incorrect.

Product of roots $=\frac{n}{n}=1$
Statement III is correct.