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3 and 4 . determinant and metrices — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths3 and 4 . determinant and metrices1 Mark
MCQ
If $\left| {\begin{array}{*{20}{c}}   {a - b}&{b - c}&{c - a} \\    {b - c}&{c - a}&{a - b} \\    {c - a + 1}&{a - b}&{b - c}  \end{array}} \right| = 0$ ,$\left( {a,b,c \in R - \left\{ 0 \right\}} \right),$ then
  • ✓
    $a$,$b$ and $c$ must be equal
  • B
    $a$,$b$ and $c$ may not be all equal
  • C
    $a$,$b$ and $c$ must be in $G.P$.
  • D
    $a$,$b$ and $c$ must be distinct
✓

Answer

Correct option: A.
$a$,$b$ and $c$ must be equal
a
$\left|\begin{array}{lll}{a-b} & {b-c} & {c-a} \\ {b-c} & {c-a} & {a-b} \\ {c-a} & {a-b} & {b-c}\end{array}\right|+\left|\begin{array}{ccc}{0} & {b-c} & {c-a} \\ {0} & {c-a} & {a-b} \\ {1} & {a-b} & {b-c}\end{array}\right|=0$

$\Rightarrow 0+(a-b)(b-c)=(c-a)^{2}$

$\Rightarrow a b-a c-b^{2}+b c=c^{2}+a^{2}-2 a c$

$\Rightarrow a^{2}+b^{2}+c^{2}=a b+b c+c a$

$\Rightarrow a=b=c$

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