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#### Provide solution for RD Sharma maths class 12 chapter Determinants exercise multiple choise question 10

Correct option (a)

Hint:

Solve given determinant

Given:

$Here\; \begin{vmatrix} -2a &a+b &a+c \\ b+a &-2b &b+c \\ c+a &c+b &-2c \end{vmatrix} is \; given$

The three factor are given, we have to find another one factor.

Solution:

$\begin{vmatrix} -2a &a+b &a+c \\ b+a &-2b &b+c \\ c+a &c+b &-2c \end{vmatrix}$

Appling C1→C1+C2 , we get

$=\begin{vmatrix} b-a &a+b &a+c \\ a-b &-2b &b+c \\ a+b+2c &b+c &-2c \end{vmatrix}$

Appling C2→C2+C3 , we get

$=\left | b-a2a+b+ca+ca-bc-bb+ca+b+2cb-c-2c \right |$

Appling R3→R3+R2 , we get

$=\begin{vmatrix} b-a &2a+b+a &a+c \\ a-b &c-b &b+c \\ 2(c+a) &0 &b-c \end{vmatrix}$

Appling R2→R2+R1 , we get

$=\begin{vmatrix} b-a &2a+b+a &a+c \\ 0 &2(a+c) &a+b+2c \\ 2(c+a) &0 &b-c \end{vmatrix}$

Expanding along C1 , we get

$=(b-a)[2(a+c)(b-c)]+2(a+c)[(2a+b+c)(a+b+2c)-2(a+c)^{2}]$

$=2(a+c)[(b-a)(b-c)+(2a+b+c)(a+b+2c)-2(a+c)^{2}]$

$=2(a+c)[b^{2}-bc-ab+ac+2a^{2}+2ab+4ac+ab+b^{2}+2bc+ac+bc+2c^{2}-2a^{2}-2c^{2}-4ac]$

$=2(a+c)[2b^{2}+2ab+2bc+2ac]$

$=4(a+c)[b^{2}+ab+bc+ca]$

$=4(a+c)(a+b)(b+c)$

Hence another factor of the given determinant is 4.