Question

Assertion :If $a$ and $b$ are integers and the roots of $x^2+ax+b=0$ are rational then they must be integers. Reason: If the coefficient of $x^2$ in a quadratic equation is unity then its roots must be integers.

• 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 C Assertion is correct but Reason is incorrect

Assertion:

Given a,b are integers

$x^2+ax+b=0$

Given that roots are rational

$x=-a\pm \sqrt{a^2-4b}$

As roots are rational $a^2-4b$ must be perfect square

Let $a^2-4b=m^2$

m is integer as a,b are integers

$\therefore x=-a\pm m$

$\therefore$Roots are also integers ($\because a,m$ are integers)

$\therefore$Assertion is true.

Reason:

In a quadratic equation if co efficient of $x^2$ is unity. Then it is not necessary that roots must be integer

For example; $x^2+2x-1$ has irrational roots

$\therefore$Reason is false.