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Question

Find the value of: $$\begin{vmatrix}
b+c & a+b & a\\
c+a& b+c &b \\
a+b& c+a & c
\end{vmatrix}$$ 


A
a3+b3+c3
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B
3abc
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C
a3+b3+c33abc
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D
a2+b2+c2abbcca
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Solution

The correct option is C $$\mathrm{a}^{3}+\mathrm{b}^{3}+\mathrm{c}^{3}-3\mathrm{a}\mathrm{b}\mathrm{c}$$
Let $$A=\begin{vmatrix}b+c & a+b & a\\ c+a& b+c &b \\ a+b& c+a & c\end{vmatrix}$$ 

Performing $$C_{2}\rightarrow C_{2}-C_{3}$$
$$A=\begin{vmatrix}b+c & b & a\\ c+a & c & b\\  a+b& a &c \end{vmatrix}$$
Performing $$C_{1}\rightarrow C_{1}-C_{2}$$
$$A=\begin{vmatrix}c & b & a\\ a & c & b\\ b& a &c \end{vmatrix}$$
$$A=c(c^{2}-ab)-b(ac-b^{2})+a(a^{2}-bc)$$
$$A=c^{3}-abc-bac+b^{3}+a^{3}-abc$$

$$A=a^{3}+b^{3}+c^{3}-3abc$$

Mathematics

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