a+x&b&c &b+y&c &b&c+z =....... - Vidyadip

MCQ

$\begin{vmatrix}a+x&b&c\\a&b+y&c\\a&b&c+z\end{vmatrix}=.......$

A
$abc\left( {1 + \frac{x}{a} + \frac{y}{b} + \frac{z}{c}} \right)$
B
$abc\left( {1 + \frac{a}{x} + \frac{b}{y} + \frac{c}{z}} \right)$
✓
$xyz\left( {1 + \frac{a}{x} + \frac{b}{y} + \frac{c}{z}} \right)$
D
$xyz\left( {1 + \frac{x}{a} + \frac{y}{b} + \frac{z}{c}} \right)$

✓

Answer

Correct option: C.

$xyz\left( {1 + \frac{a}{x} + \frac{b}{y} + \frac{c}{z}} \right)$

C

$C_1(\frac{1}{x}),C_2(\frac{1}{y}),C_3(\frac{1}{z})$

$D=xyz\begin{vmatrix}\frac{a}{x}+1&\frac{b}{y}&\frac{c}{z}\\\frac{a}{x}&\frac{b}{y}+1&\frac{c}{z}\\\frac{a}{x}&\frac{b}{y}&\frac{c}{z}+1\end{vmatrix}$

$\frac{C_{21}(1)}{C_{31}(1)}$

$=xyz\begin{vmatrix}\frac{a}{x}+\frac{b}{y}+\frac{c}{z}+1&\frac{b}{y}&\frac{c}{z}\\\frac{a}{x}+\frac{b}{y}+\frac{c}{z}+1&\frac{b}{y}+1&\frac{c}{z}\\\frac{a}{x}+\frac{b}{y}+\frac{c}{z}+1&\frac{b}{y}&\frac{c}{z}+1\end{vmatrix}$

$=xyz(\frac{a}{x}+\frac{b}{y}+\frac{c}{z}+1)\begin{vmatrix}1&\frac{b}{y}&\frac{c}{z}\\1&\frac{b}{y}+1&\frac{c}{z}\\1&\frac{b}{y}&\frac{c}{z}+1\end{vmatrix}$

$ \frac{R_{21}(-1)}{R_{32}(-1)}$

$=xyz(\frac{a}{x}+\frac{b}{y}+\frac{c}{z}+1)\begin{vmatrix}1&-1&0\\1&1&-1\\1&\frac{b}{y}&\frac{c}{z}+1\end {vmatrix}$

$=xyz(\frac{a}{x}+\frac{b}{y}+\frac{c}{z}+1).1$

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