Question

Assertion :If $f\left(x\right)$ is a quadratic polynomial satisfying $f\left(2\right)+f\left(4\right)=0$. If unity is a root of $f\left(x\right)=0$, then the other root is $\frac{7}{2}$. Reason: If $g\left(x\right)=p{x}^2+qx+r=0$ has roots $a,\beta$,
then $a+\beta =-\frac{q}{p}$ and
$\alpha \beta =$ $\frac{r}{p}$.

• 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

Let 1 and $\frac{7}{2}$ are roots of f(x)=0
So, sum of roots $=1+\frac{7}{2}=\frac{9}{2}$
Product of roots $=\left(1\right)\frac{7}{2}=\frac{7}{2}$
So, the quadratic equation is
$x^2-\frac{9}{2}x+\frac{7}{2}=0$
Also, $f\left(2\right)=-\frac{3}{2}$
$f\left(4\right)=\frac{3}{2}$
Here, $f\left(2\right)+f\left(4\right)=0$
Hence, our assumption is correct.
$\frac{7}{2}$ is a root of $f\left(x\right)=0$
Now,if $f\left(x\right)=a{x}^2+bx+c$
Let $\alpha ,\beta$ are the roots of f(x)
Then, sum of roots =$\alpha +\beta =-\frac{b}{a}$
& product of roots =$\alpha \beta =\frac{c}{a}$.
Hence, assertion and reason are correct but reason is not the correct explanation for assertion