Question

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

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

$\text{a.}D=\underset{}{\underset{\text{odd}}{\underset{}{(2m+1{)}^2}}-\underset{\text{even}}{\underset{}{4(2n+1)}}}$

For rational root, $D$ must be a perfect square. As $D$ is odd, let $D$ be perfect square of $2l+1,$ where $l\in Z$

$(2m+1{)}^2-4(2n+1)=(2l+1{)}^2$

$\Rightarrow (2m+1{)}^2-(2l+1{)}^2=4(2n+1)$

$\Rightarrow \left[\right(2m+1)+(2l+1\left)\right]\left[2\right(m-l\left)\right]=4(2n+1)$

$\Rightarrow (m+l+1)(m-l)=(2n+1)$

R.H.S. of (1) is always odd but L.H.S. is always even.

Hence, $D$ cannot be a perfect square. So, the roots cannot be rational.

Hence, statement 1 is true, statement 2 is true and statement 2 is correct explanation for statement 1.