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Determinants — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDeterminants1 Mark
MCQ
Using the factor theorem it is found that $a + b, b + c$ and $c + a$ are three factors of the determinant $\begin{vmatrix}-2\text{a}&\text{a}+\text{b}&\text{a}+\text{c}\\\text{b}+\text{a}&-2\text{b}&\text{b}+\text{c}\\\text{c}+\text{a}&\text{c}+\text{b}&-2\text{c}\end{vmatrix}$ the other factor in the value of the determinant is:
  • ✓
    $4$
  • B
    $2$
  • C
    $a + b + c$
  • D
    None of these.
✓

Answer

Correct option: A.
$4$
$\triangle=\begin{vmatrix}-2\text{a}&\text{a}+\text{b}&\text{a}+\text{c}\\\text{b}+\text{a}&-2\text{b}&\text{b}+\text{c}\\\text{c}+\text{a}&\text{c}+\text{b}&-2\text{c}\end{vmatrix}$
Let $a + b = 2C, b + c = 2A$ and $c + a = 2B$
$\Rightarrow a + b + b + c + c + a = 2A + 2B + 2C$
$\Rightarrow 2(a + b + c) = (A + B + C)$
Also, $a = (a + b + c) - (b + c) = (A + B + C) - 2A = B + C - A$
Similarly, $b = C + A - B, c = A + B - C$
Hence, $4$ is the order factor of the determinant.

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