Assertion :The equation $x^{2} + (2 m + 1) x + (2 n + 1) = 0$ , where $m$ and $n$ are integers, cannot have any rational roots. Reason: The quantity $(2 m + 1)^{2} - 4 (2 n + 1)$ , where $m, n \in I$ , can never be a perfect square.

Both Assertion and Reason are correct and Reason is the correct explanation for Assertion.

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Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion.

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Assertion is correct but Reason is incorrect.

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Assertion is incorrect but Reason is correct.

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Solution

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion.
$x^{2} + (2 m + 1) x + (2 n + 1) = 0$
$D = (2 m + 1)^{2} - 4 (2 n + 1)$
For $D$ to be a perfect square the expression $4 m^{2} + 4 m + (1 - 8 n - 4)$ which is a quadratic in $m$ must have discriminant zero.
$\Rightarrow 4^{2} - (4) (4) (1 - 8 n - 4) = 0$
But $n$ is an integer.
Therefore, when $m$ & $n$ are integers $D$ cannot be perfect square.
Therefore, roots are irrational.
Ans: A

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Assertion :The equation x2+(2m+1)x+(2n+1)=0, where m and n are integers, cannot have any rational roots. Reason: The quantity (2m+1)2−4(2n+1), where m,n∈I, can never be a perfect square.

Assertion :The equation $x^{2} + (2 m + 1) x + (2 n + 1) = 0$ , where $m$ and $n$ are integers, cannot have any rational roots. Reason: The quantity $(2 m + 1)^{2} - 4 (2 n + 1)$ , where $m, n \in I$ , can never be a perfect square.