Q27 / 58 · KVPY 2016 (SA) · Mathematics
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 $\mathrm{n}$, the roots are distinct.
II. There are infinitely many values of $\mathrm{n}$ for which both roots are real.
III. The product of the roots is necessarily an integer.
- AIII only
- Correct answerBI and III only
- CII and III only
- DI, II and III
Solution
$\begin{aligned}
\mathrm{D} &=49 \mathrm{n}-4 \mathrm{n}^{2} \\
&=\mathrm{n}(49-4 \mathrm{n})
\end{aligned}$
$\mathrm{D} \neq 0$ for any $\mathrm{n} \in \mathrm{I}^{+}$. So roots are distinct For roots to be real $\mathrm{D} \geq 0$
So $\mathrm{n} \leq \frac{49}{4}$
So n can be $\{1,2,3, \ldots \ldots 12\}$
Clearly product of the roots is 1
Hence, the correct answer is option: 'B'.