Question

Assertion (A): If $$(5 + \sqrt{7})$$ is a root of a quadratic equation with rational coefficients then the other root is $$(5 - \sqrt{7})$$.

Reason (R): The surd roots of a quadratic equation with rational coefficients occur in conjugate pairs.

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

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

Explanation:

Step 1 – Assertion: A is true. The conjugate of $$5+\sqrt{7}$$ is $$5-\sqrt{7},$$ so this is the other root.

Step 2 – Reason: R is true. For a quadratic equation with rational coefficients, surd roots occur in conjugate pairs.

Step 3 – Link: The conjugate-pair property directly explains why the other root must be $$5-\sqrt{7}.$$