Statement-1 (A): a^2+b^2+c^2-a b-b c-c a=0 if and only if a=b=c. … - Vidyadip

MCQ

Statement-1 (A): $\quad a^2+b^2+c^2-a b-b c-c a=0$ if and only if $a=b=c$.
Statement-2 $(R): \quad a^3+b^3+c^3-3 a b c=(a+b+c)\left(a^2+b^2+c^2-a b-b c-c a\right)$

A
Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-6
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. Statement-2 is true.$
\begin{array}{ll}
\text { Now, } & a^2+b^2+c^2-a b-b c-c a=0 \\
\Leftrightarrow & 2\left(a^2+b^2+c^2-a b-b c-c a\right)=0 \\
\Leftrightarrow & (a-b)^2+(b-c)^2+(c-a)^2=0 \Leftrightarrow a-b=0 \text { and } b-c=0 \text { and } c-a=0 \Leftrightarrow a=b=c
\end{array}
$
So, statement 1 is also true. Hence, options (b) is correct.

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