Question

Assertion :The equation $f\left(x\right)=x^3+x^2-8x-12=0$ has $x=-2$ as repeated root. Reason: If the equation $f\left(x\right)=0$ & $f{}^{\prime }\left(x\right)=0$ have a common root then roots of $f\left(x\right)=0$ are repeated.

• 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

Given, $f\left(x\right)=x^3+x^2-8x+12$
Now $f\left(x\right)=0$
Implies $(x+2{)}^2(x-3)=0$
$x=-2$ is the repeated root and $x=3$
$f^{\prime }\left(x\right)$ $=3x^2+2x-8$
$=(3x-4)(x+2)$
$=0$ implies
$x=\frac{4}{3}$ and $x=-2$
Hence, assertion and reason are both correct and the reason is the correct explanation of assertion.