KVPY SA Exam 2016 Question Paper

Question : 62 of 80

Marks: +1, -0

Consider the quadratic equation

nx^2+7√{nx}+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 only
II and III only
I, II and III

Validate

Solution:

D = 49n - 4n^2

= n(49 - 4n)

D ≠ 0 for any

n ∈ I^{+}

.
So roots are distinct
For roots to be real D ≥ 0
So

n ≤ 49/4

So n can be {1, 2, 3, ..... 12}
Clearly product of the roots is 1

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