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#### Need Solution for R.D.Sharma Maths Class 12 Chapter determinants  Exercise 5.2 Question 40  Maths Textbook Solution.

Answer: $a b c\left(a^{2}+b^{2}+c^{2}\right)^{3}$

Hint Use determinant formula

Given: $\left|\begin{array}{ccc} -a\left(b^{2}+c^{2}-a^{2}\right) & 2 b^{3} & 2 c^{3} \\ 2 a^{3} & -b\left(c^{2}+a^{2}-b^{2}\right) & 2 c^{3} \\ 2 a^{3} & 2 b^{3} & -c\left(a^{2}+b^{2}-c^{2}\right) \end{array}\right|=a b c\left(a^{2}+b^{2}+c^{2}\right)^{3}$

Solution:

\begin{aligned} &\text { L.H.S }\left|\begin{array}{ccc} -a\left(b^{2}+c^{2}-a^{2}\right) & 2 b^{3} & 2 c^{3} \\ 2 a^{3} & -b\left(c^{2}+a^{2}-b^{2}\right) & 2 c^{3} \\ 2 a^{3} & 2 b^{3} & -c\left(a^{2}+b^{2}-c^{2}\right) \end{array}\right|\\ &\text { Common a from } \mathrm{C}_{1}, b \text { from } \mathrm{C}_{2} \text { and c from } \mathrm{C}_{3}\\ &=a b c\left|\begin{array}{ccc} -\left(b^{2}+c^{2}-a^{2}\right) & 2 b^{2} & 2 c^{2} \\ 2 a^{2} & -\left(c^{2}+a^{2}-b^{2}\right) & 2 c^{2} \\ 2 a^{2} & 2 b^{2} & -\left(a^{2}+b^{2}-c^{2}\right) \end{array}\right| \end{aligned}

\begin{aligned} &\text { Apply } \mathrm{C}_{1} \rightarrow C_{1}+C_{2}+C_{3} \\ &=a b c\left|\begin{array}{ccc} -b^{2}-c^{2}+a^{2}+2 b^{2}+2 c^{2} & 2 b^{2} & 2 c^{2} \\ 2 a^{2}+2 c^{2}-c^{2}-a^{2}+b^{2} & -c^{2}+a^{2}-b^{2} & 2 c^{2} \\ 2 a^{2}+2 b^{2}-a^{2}-b^{2}+c^{2} & 2 b^{2} & -a^{2}+b^{2}-c^{2} \end{array}\right| \\ &=a b c\left|\begin{array}{ccc} a^{2}+b^{2}+c^{2} & 2 b^{2} & 2 c^{2} \\ a^{2}+b^{2}+c^{2} & -c^{2}+a^{2}-b^{2} & 2 c^{2} \\ a^{2}+b^{2}+c^{2} & 2 b^{2} & -a^{2}+b^{2}-c^{2} \end{array}\right| \\ &=a b c\left(a^{2}+b^{2}+c^{2}\right)\left|\begin{array}{ccc} 1 & 2 b^{2} & 2 c^{2} \\ 1 & -c^{2}+a^{2}-b^{2} & 2 c^{2} \\ 1 & 2 b^{2} & -a^{2}+b^{2}-c^{2} \end{array}\right| \end{aligned}

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

Expanding w.r.t $C_{1}$

\begin{aligned} &=a b c\left(a^{2}+b^{2}+c^{2}\right)\left(a^{2}+b^{2}+c^{2}\right)^{2} \\ &=a b c\left(a^{2}+b^{2}+c^{2}\right)^{3} \\ &=R \cdot H \cdot S \end{aligned}