Question

Assertion (A): If one zero of the polynomial $$p(x)=(k^{2}+4)x^{2}+9x+4k$$ is the reciprocal of the other zero then $$k=2$$.

Reason (R): If $$(x-\alpha)$$ is a factor of the polynomial $$p(x)$$ then $$\alpha$$ is a zero of $$p(x)$$.

विकल्प

• Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).

• Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).

• Assertion (A) is true and Reason (R) is false.

• Assertion (A) is false and Reason (R) is true.

Answer 1

Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).

Explanation:

Both statements are true. If the zeros are $$\alpha$$ and $$\frac{1}{\alpha}$$, their product is 1.

For the given polynomial, the product of its zeros is $$\frac{4k}{k^{2}+4}$$.

Therefore:

$$\frac{4k}{k^{2}+4}=1$$

$$k^{2}-4k+4=0$$

$$(k-2)^{2}=0\Rightarrow k=2$$

The factor theorem in the Reason is true, but it does not explain the reciprocal-root condition or the value of $$k$$.