A) \[abc\left( 1+\frac{x}{a}+\frac{y}{b}+\frac{z}{c} \right)\]
B) \[abc\left( 1+\frac{a}{x}+\frac{b}{y}+\frac{c}{z} \right)\]
C) \[xyz\left( 1+\frac{a}{x}+\frac{b}{y}+\frac{c}{z} \right)\]
D) \[xyz\left( 1+\frac{x}{a}+\frac{y}{b}+\frac{z}{c} \right)\]
E) \[xyz(a+b+c+1)\]
Correct Answer: C
Solution :
\[\left| \begin{matrix} a+x & b & c \\ a & b+y & c \\ a & b & c+z \\ \end{matrix} \right|\] \[\left| \begin{matrix} a+x & b & c \\ -x & y & 0 \\ -x & 0 & z \\ \end{matrix} \right|\] \[=(a+x)(yz)-b(-xz)+c(xy)\] \[=ayz+xyz+bxz+cxy\] \[=xyz\left[ \frac{a}{x}+\frac{b}{y}+\frac{c}{z}+1 \right]\]
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