Assertion(A)=|a2+x2ab−cxac+bxab+cxb2+x2bc−axac−bxbc+axc2+x2|=|xc−b−cxab−ax|2Reason(R): Δc=Δn−1 where n is order of determinant and Δc is the determinant of cofactors of Δ

LetΔ=xc−b−cxab−axΔc=a2+x2cx+abac−bxab−cxx2+b2ax+bcac+bxbc−axc2+x2=a2+x2ab−cxac+bxcx+abx2+b2bc−axac−bxax+bcc2+x2=Δ2(∵n=3)=xc−b−cxab−ax2

Assertion(A)=|a2+x2ab−cxac+bxab+cxb2+x2bc−axac−bxbc+axc2+x2|=|xc−b−cxab−ax|2Reason(R): Δc=Δn−1 where n is order of determinant and Δc is the determinant of cofactors of Δ

LetΔ=xc−b−cxab−axΔc=a2+x2cx+abac−bxab−cxx2+b2ax+bcac+bxbc−axc2+x2=a2+x2ab−cxac+bxcx+abx2+b2bc−axac−bxax+bcc2+x2=Δ2(∵n=3)=xc−b−cxab−ax2

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$Assertion (A) = | \begin{matrix} a^{2} + x^{2} & a b - c x & a c + b x \\ a b + c x & b^{2} + x^{2} & b c - a x \\ a c - b x & b c + a x & c^{2} + x^{2} \end{matrix} | = {| \begin{matrix} x & c & - b \\ - c & x & a \\ b & - a & x \end{matrix} |}^{2}$
Reason(R): $Δ^{c} = Δ^{n - 1}$ where n is order of determinant and $Δ^{c}$ is the determinant of cofactors of $Δ$

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Assertion (A) is true; reason (R) is false

Assertion (A) is false, reason R is true

Assertion (A) is true, reason (R) is true and it is correct explanation for assertion A

Assertion (A) is true, reason (R) is true but it is not correct explanation for assertion A

answer is A.

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Detailed Solution

$Let Δ = |\begin{matrix} x & c & - b \\ - c & x & a \\ b & - a & x \end{matrix}|$

$Δ^{c} = |\begin{matrix} a^{2} + x^{2} & c x + a b & a c - b x \\ a b - c x & x^{2} + b^{2} & a x + b c \\ a c + b x & b c - a x & c^{2} + x^{2} \end{matrix}|$

$= |\begin{matrix} a^{2} + x^{2} & a b - c x & a c + b x \\ c x + a b & x^{2} + b^{2} & b c - a x \\ a c - b x & a x + b c & c^{2} + x^{2} \end{matrix}| = Δ^{2} (∵ n = 3)$