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  • Please Solve R.D.Sharma class 12 Chapter determinants Exercise 5.2 Question 35 Maths textbook Solution.

Please Solve R.D.Sharma class 12 Chapter determinants Exercise  5.2 Question 35 Maths textbook Solution.

Answers (1)

Answer:4 a b c

Hint Use determinant formula

Given : \left|\begin{array}{ccc} \frac{a^{2}+b^{2}}{c} & c & c \\ a & \frac{b^{2}+c^{2}}{a} & a \\ b & b & \frac{c^{2}+a^{2}}{b} \end{array}\right|=4 a b c

Solution:

\text { L.H.S }\left|\begin{array}{ccc} \frac{a^{2}+b^{2}}{c} & c & c \\ a & \frac{b^{2}+c^{2}}{a} & a \\ b & b & \frac{c^{2}+a^{2}}{b} \end{array}\right|

\begin{aligned} &R_{1} \rightarrow c R_{1}, R_{2} \rightarrow c R_{2}, R_{3} \rightarrow b R_{3} \\ &=\frac{1}{a b c}\left|\begin{array}{ccc} a^{2}+b^{2} & c^{2} & c^{2} \\ a^{2} & b^{2}+c^{2} & a^{2} \\ b^{2} & b^{2} & c^{2}+a^{2} \end{array}\right| \end{aligned}

\begin{aligned} &\text { Use } R_{1} \rightarrow R_{1}-R_{2}-R_{3} \\ &=\frac{1}{a b c}\left|\begin{array}{ccc} 0 & -2 b^{2} & -2 a^{2} \\ a^{2} & b^{2}+c^{2} & a^{2} \\ b^{2} & b^{2} & c^{2}+a^{2} \end{array}\right| \\ &\text { Now } \mathrm{C}_{2} \rightarrow C_{2}-C_{1} \text { and } \mathrm{C}_{3} \rightarrow C_{3}-C_{1} \\ &=\frac{1}{a b c} \mid \begin{array}{ccc} 0 & -2 b^{2} & -2 a^{2} \\ a^{2} & b^{2}+c^{2}-a^{2} & a^{2} \\ b^{2} & b^{2} & c^{2}+a^{2}-b^{2} \end{array} \end{aligned}

\begin{aligned} &\text { Expanding w.r.t } \mathrm{C}_{1}\\ &=\frac{1}{a b c}\left[c-a^{2}\left(-2 b^{2}\left(c^{2}+a^{2}-b^{2}\right)-a\right)+b^{2}\left(0-\left(2 a^{2}\right)\left(b^{2}+c^{2}-a^{2}\right)\right)\right]\\ &=\frac{1}{a b c}\left[2 a^{2} b^{2}\left(c^{2}+a^{2}-b^{2}\right)+b^{2}\left(\left(2 a^{2}\right)\left(b^{2}+c^{2}-a^{2}\right)\right)\right]\\ &=\frac{2 a^{2} b^{2}}{a b c}\left(c^{2}+a^{2}-b^{2}+b^{2}+c^{2}-a^{2}\right)\\ &=\frac{2 a b}{c}\left(2 c^{2}\right)=4 a b c\\ &=R \cdot H \cdot S \end{aligned}

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