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  • Explain Solution R.D.Sharma Class 12 Chapter deteminants Exercise 5.2 Question 34 maths Textbook Solution.

Explain Solution R.D.Sharma Class 12 Chapter  deteminants Exercise 5.2 Question 34 maths Textbook Solution.

Answers (1)

Answer:2 a^{3} b^{3} c^{3}

Hint Use determinant formula

Given: \left|\begin{array}{ccc} 0 & a b^{2} & a c^{2} \\ a^{2} b & 0 & b c^{2} \\ a^{2} c & b^{2} c & 0 \end{array}\right|=2 a^{3} b^{3} c^{3}

Solution:

\text { L.H.S }\left|\begin{array}{ccc} 0 & a b^{2} & a c^{2} \\ a^{2} b & 0 & b c^{2} \\ a^{2} c & b^{2} c & 0 \end{array}\right|

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

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

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