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  • Please Solve R.D. Sharma class 12 Chapter determinants Exercise 5.2 Question 32 Maths textbook Solution.

Please Solve R.D. Sharma class 12 Chapter determinants Exercise  5.2 Question 32 Maths textbook Solution.

Answers (1)

Answer: 4\: abc

Hint: Use determinant formula

Given: \left|\begin{array}{ccc} b+c & a & a \\ b & a+c & b \\ c & c & a+b \end{array}\right|=4 a b c

Solution:

\text { L.H.S }\left|\begin{array}{ccc} b+c & a & a \\ b & a+c & b \\ c & c & a+b \end{array}\right|

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

\begin{aligned} &\text { Use } R_{1} \rightarrow R_{1}+R_{2}+R_{3}\\ &|\Delta|=2\left|\begin{array}{ccc} 0 & c & b \\ -c & 0 & -a \\ -b & -a & 0 \end{array}\right|\\ &\text { Expanding w.r.t } \mathrm{C}_{1}\\ &|\Delta|=2[-c(0-a b)+b(a c-0)]\\ &|\Delta|=2(a b c+a b c)\\ &|\Delta|=2(2 a b c)=4 a b c\\ &=R \cdot H . S \end{aligned}

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