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Determinants — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSDeterminants1 Mark
Question
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:
  1. 4
  2. 2
  3. a + b + c
  4. None of these.
✓

Answer

  1. 4
Solution:
$\triangle=\begin{vmatrix}-2\text{a}&\text{a}+\text{b}&\text{a}+\text{c}\\\text{b}+\text{a}&-2\text{b}&\text{b}+\text{c}\\\text{c}+\text{a}&\text{c}+\text{b}&-2\text{c}\end{vmatrix}$
Let a + b = 2C, b + c = 2A and c + a = 2B
⇒ a + b + b + c + c + a = 2A + 2B + 2C
⇒ 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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