Question

Assertion :Let p < 0 and ${\alpha }_1,{\alpha }_2,...,{\alpha }_9$ be the nine roots of $x^9=p$, then
$\Delta =\left|\begin{array}{ccc}{\alpha }_1& {\alpha }_2& {\alpha }_3\\ {\alpha }_4& {\alpha }_5& {\alpha }_6\\ {\alpha }_4& {\alpha }_8& {\alpha }_9\end{array}\right|=0$ Reason: If two rows of a determinant are identical, then determinant equals zero.

• 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 D Assertion is incorrect but Reason is correct

${\alpha }_i$ is complex, and roots of the equation $x^9=p$ will occur in conjugate pairs,
such that $\sum {\alpha }_i=0$.
Hence $△\ne 0$.
Hence assertion is incorrect.
Now considering a determinant, it is the property of a determinant, that if any two rows are equal, then the determinant value is 0.
Hence reason is correct.