Cubic $left| {begin{array}{*{20}{c}} 0&{a - x}&{b - x} { - a - x}&0

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3 and 4 .Determinants and Matrices

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The cubic $\left| {\begin{array}{*{20}{c}}
0&{a - x}&{b - x} \\
{ - a - x}&0&{c - x} \\
{ - b - x}&{ - c - x}&0
\end{array}} \right| = 0$ has a reperated root in $x$ then,

A

$2ac = ab + bc$

B

$ac = ab + bc$

C

$ac = 2ab + 2bc$

D

$a^2c^2 = a^2b^ 2 + b^2c^2$

Solution

Determinant $=-2 x^{3}+2(b c-a c+a b) x$

$=-2 x\left(x^{2}-(b c-a c+a b)\right),$ has repeated roots

$\Rightarrow \mathrm{b} \mathrm{c}-\mathrm{ac}+\mathrm{ab}=0$

Standard 12

Mathematics

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(JEE MAIN-2019)