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  • Explain Solution R.D.Sharma Class 12 Chapter deteminants Exercise 5.2 Question 45 maths Textbook Solution.

Explain Solution R.D.Sharma Class 12 Chapter  deteminants Exercise 5.2 Question 45 maths Textbook Solution.

Answers (1)

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

Hint Use determinant formula

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

Solution:

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

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

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