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 3.5. Reason: If $g\left(x\right)=p{x}^2+qx+r=0$ has roots $\alpha ,\beta$, then $\alpha +\beta =-q/p\text{and}\alpha \beta =\left(r/p\right)$.

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

Since unity is the root of the quadratic equation

So the given quadratic equation can be written as $(x-1)(ax+b)=0$

$f\left(x\right)=$$(x-1)(ax+b)$

Given $f\left(2\right)+f\left(4\right)=0$

So$f\left(2\right)=2a+b$

$f\left(4\right)=12a+3b$

$f\left(2\right)+f\left(4\right)=2a+b+12a+3b=14a+4b$

$f\left(2\right)+f\left(4\right)=0$

$⟹14a+4b=0$

$b=-3.5a$

So equation becomes

$(x-1)(ax-3.5a)=0$

$f\left(x\right)=a(x-1)(x-3.5)$

So other root is $3.5$

So Assertion is correction but reason is not correct explanation for it.