If a, b, c, d, e, f are in G.P.,then the valu | vedclass

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(D) Let the common ratio of the $G.P.$ be $r$. Then $b=ar, c=ar^2, d=ar^3, e=ar^4, f=ar^5$.Substituting these into the determinant:$\Delta = \left| \b

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(D) Let the common ratio of the $G.P.$ be $r$. Then $b=ar, c=ar^2, d=ar^3, e=ar^4, f=ar^5$.Substituting these into the determinant:$\Delta = \left| \begin{array}{ccc} a^2 & (ar^3)^2 & x \ (ar)^2 & (ar^4)^2 & y \ (ar^2)^2 & (ar^5)^2 & z \end{array} ight| = \left| \begin{array}{ccc} a^2 & a^2r^6 & x \ a^2r^2 & a^2r^8 & y \ a^2r^4 & a^2r^{10} & z \end{array} ight|$.Taking $a^2$ common from the first column and $a^2r^6$ common from the second column:$\Delta = a^4r^6 \left| \begin{array}{ccc} 1 & 1 & x \ r^2 & r^2 & y \ r^4 & r^4 & z \end{array} ight|$.Since the first and second columns are identical,the value of the determinant is $0$.Therefore,the value of the determinant is independent of $x, y,$ and $z$.

If $a, b, c, d, e, f$ are in $G.P.$,then the value of $\left| \begin{array}{ccc} a^2 & d^2 & x \\ b^2 & e^2 & y \\ c^2 & f^2 & z \end{array} \right|$ depends on

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