Question

Assertion :If $f\left(x\right)$ is a quadratic expression such that $f\left(1\right)+f\left(2\right)=0$. If $-1$ is a root of $f\left(x\right)=0$, then the other root is $\frac{8}{5}$ Reason: If $f\left(x\right)=a{x}^2+bx+c$ then $\alpha +\beta =-\frac{b}{a}$ & $\alpha \beta =\frac{c}{a}$.

• 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. Assertion is incorrect but Reason is correct

Answer 1

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

Given $f\left(1\right)+f\left(2\right)=0$For $f\left(x\right)=a{x}^2+bx+c$, we have $a+b+c+4a+2b+c=0$
Therefore, $5a+3b+2c=0$ ......(1)
Since $-1$ is a root of the equation $f(-1)=a-b+c=0$ ......(2)
By eliminating $c$ and $b$ in turn by solving equations (1) and (2), we have $\frac{-b}{a}=\frac{3}{5},\frac{c}{a}=\frac{-8}{5}$.
Since $-1$ is one root, the other root is $\frac{8}{5}$.
Therefore, both assertion and reason are correct but reason is not a correct explanation.