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Question

Consider the quadratic equation nx2+7nx+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
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B
I and III only
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C
II and III only
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D
I, II and III
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Solution

The correct option is B I and III only
Given equation,
nx2+7n+n=0
Discriminant, D=49n4n2
D0 nI+
Statement I is correct.

For both roots to be real
D0n(494n)0n(0,494)
So n=1,2,3,.......12 finite numbers.
Statement II is incorrect.

Product of roots =nn=1
Statement III is correct.

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