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

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

Answers (1)

Answer:

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.

Posted by

Gurleen Kaur

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