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  • Provide Solution For R. D. Sharma Maths Class 12 Chapter determinants Exercise 5.2 Question 31 Maths Textbook Solution.

Provide Solution For  R. D. Sharma Maths Class 12 Chapter  determinants Exercise 5.2 Question 31 Maths Textbook Solution.

Answers (1)

Answer: 2(a+b)(b+c)(c+a)

Hint Use determinant formula

Given:\left|\begin{array}{ccc} a+b+c & -c & -b \\ -c & a+b+c & -a \\ -b & -a & a+b+c \end{array}\right|=2(a+b)(b+c)(c+a)

Solution:

\begin{aligned} &\text { L.H.S }\left|\begin{array}{ccc} a+b+c & -c & -b \\ -c & a+b+c & -a \\ -b & -a & a+b+c \end{array}\right| \\ &\text { Use } \mathrm{C}_{1} \rightarrow C_{1}+\mathrm{C}_{2} \text { and } \mathrm{C}_{2} \rightarrow C_{2}+\mathrm{C}_{3} \\ &|\Delta|=\left|\begin{array}{ccc} a+b & -b-c & -b \\ a+b & b+c & -a \\ -a-b & b+c & a+b+c \end{array}\right| \end{aligned}

\begin{aligned} &\text { Common }(a+b)(b+c) \text { from } \mathrm{C}_{1} \& C_{2} \\ &|\Delta|=(a+b)(b+c)\left|\begin{array}{ccc} 1 & -1 & -b \\ 1 & 1 & -a \\ -1 & 1 & a+b+c \end{array}\right| \\ &\text { Use } \mathrm{R}_{1} \rightarrow R_{1}+\mathrm{R}_{2} \text { and } \mathrm{R}_{2} \rightarrow R_{2}+\mathrm{R}_{3} \\ &|\Delta|=(a+b)(b+c)\left|\begin{array}{ccc} 2 & 0 & -b+a \\ 0 & 2 & b+c \\ -1 & 1 & a+b+c \end{array}\right| \end{aligned}

\begin{aligned} &\text { Expanding w.r.t } \mathrm{C}_{1}\\ &|\Delta|=(a+b)(b+c)[2(2 a+2 b+2 c-b-c)+(-1)(2 b+2 a)]\\ &|\Delta|=(a+b)(b+c)[4 a+4 b+4 c-2 b-2 c-2 b-2 a]\\ &|\Delta|=2(a+b)(b+c)(c+a) \end{aligned}

 

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