Question

Assertion :If one zero of polynomial $p\left(x\right)=(k^2+4)x^2+13x+4k$ is reciprocal of other, then $k=2$. Reason: If $x-\alpha$ is a factor of p(x), then $p\left(\alpha \right)$, then $p\left(\alpha \right)=0$ i.e. $\alpha$ is a zero of p(x).

• 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

$p\left(x\right)=(k^2+4)x^2+13x+4k$
Let one root of $p\left(x\right)$ be $\alpha$, then other root will be$\frac{1}{\alpha }$
Sum of roots $=\alpha +\frac{1}{\alpha }=\frac{-13}{k^2+4}$
Product of roots $=\alpha \times \frac{1}{\alpha }=\frac{4k}{k^2+4}$
i.e. $\frac{4k}{k^2+4}=1$
$\Rightarrow$ $4k=k^2+4$
$\Rightarrow$ $k^2+4-4k=0$
$\Rightarrow$ ${(k-2)}^2=0$
$\Rightarrow$ $k-2=0$
$\Rightarrow$ $k=2$
$\therefore$ Assertion is correct.
Also, if $(x-\alpha )$ is a factor of $p\left(x\right)$, it means $\alpha$ is a root of $p\left(x\right)$.
$\therefore p\left(\alpha \right)=0$ or $\alpha$ is a zero of $p\left(x\right)$.
Thus, Reason is also correct but it is not the correct explanation for assertion.