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3 and 4 .Determinants and Matrices
normal
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