Statement-1 (A): (a+b+c) (a-b)^2+(b-c)^2+(c-a)^2 =2 (a^3+b^3+c^3-… - Vidyadip

MCQ

Statement-1 $(A): \quad(a+b+c)\left\{(a-b)^2+(b-c)^2+(c-a)^2\right\}=2\left(a^3+b^3+c^3-3 a b c\right)$
Statement-2 (R): If $a+b+c=0$ then $(a+b)^3+(b+c)^3+(c+a)^3=-3 a b c$

A
Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-4
B
Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1.
C
Statement-1 is True, Statement-2 is False.
D
Statement-1 is False, Statement-2 is True.

✓

Answer

B.
$
\begin{aligned}
\text {} & (a+b+c)\left\{(a-b)^2+(b-c)^2+(c-a)^2\right\} \\
& =2(a+b+c)\left(a^2+b^2+c^2-a b-b c-c a\right)=2\left(a^3+b^3+c^3-3 a b c\right)
\end{aligned}
$
So, statement- 1 is true.
If $a+b+c=0$, then.
$
\begin{array}{ll}
& 2(a+b+c)=0 \\
\Rightarrow & (a+b)+(b+c)+(c+a)=0 \\
\Rightarrow \quad & (a+b)^3+(b+c)^3+(c+a)^3=3(a+b)(b+c)(c+a) \\
\Rightarrow \quad & (a+b)^3+(b+c)^3+(c+a)^3=3(-c)(-a)(-b)=-3 a b c
\end{array}
$
So, statement-2 is also true. Hence, option (b) is true.

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