Question

Assertion (A): If $$\alpha$$ and $$\beta$$ be the zeros of polynomial $$(x^{2}-6x+k)$$ such that $$(\alpha-\beta)=2$$ then $$k=8$$.

Reason (R): If $$\alpha$$ is a zero of the polynomial $$p(x)$$ then $$p(-\alpha)=0$$.

Options

• 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

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

Explanation:

Step 1 – Assertion: A is true. By the sum of zeros, $$\alpha+\beta=6$$.

Combining this with $$\alpha-\beta=2$$ gives $$\alpha=4$$ and $$\beta=2$$. 

Substituting $$\alpha=4$$ into the polynomial gives:

$$4^{2}-6(4)+k=0$$

$$16-24+k=0\Rightarrow k=8$$

Step 2 – Reason: R is false. A zero $$\alpha$$ of $$p(x)$$ implies $$p(\alpha)=0$$, not necessarily $$p(-\alpha)=0$$.