Question

Assertion :Consider the determinant $f\left(x\right)=\left|\begin{array}{ccc}0& x^2-a& x^3-b\\ x^2+a& 0& x^2+c\\ x^4+b& x-c& 0\end{array}\right|$
$f\left(x\right)=0$ has one root $x=0$. Reason: The value of skew-symmetric determinant of odd-order is always 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 A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

$f\left(x\right)=\left|\begin{array}{ccc}0& x^2-a& x^3-b\\ x^2+a& 0& x^2+c\\ x^4+b& x-c& 0\end{array}\right|$
Now, for assertion, we put $x=0$
$f\left(x\right)=\left|\begin{array}{ccc}0& -a& -b\\ a& 0& c\\ b& -c& 0\end{array}\right|$
i.e. determinant of skew-symmmetric matrix of odd order.
$\Rightarrow f\left(x\right)=0$
Reason is also correct. We know that determinant of skew-symmetric matrix of odd order is 0.
Hence, reason is correct explanation for the assertion