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 $(A): a^2+b^2+c^2-a b-b c-c a=0$ if and only if $a=b=c$.
Statement-2 (R):$(a+b+c)^2=a^2+b^2+c^2+2 a b+2 b c+2 c a$

A
Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
✓
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

Correct option: B.

Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1.

(b)
Statement-2, being a standard result, is true.
Now, $\quad a^2+b^2+c^2-a b-b c-c a=0$
$ \Rightarrow \quad 2 a^2+2 b^2+2 c^2-2 a b-2 b c-2 c a=0$[Multiplying both sides by 2]
$\Rightarrow \quad\left(a^2+b^2-2 a b\right)+\left(b^2+c^2-2 b c\right)+\left(c^2+a^2-2 c a\right)=0$
$\Rightarrow \quad(a-b)^2+(b-c)^2+(c-a)^2=0$
$\Rightarrow\quad a-b=0$ and $b-c=0$ and $c-a=0 \Rightarrow a=b=c$.
So, statement-1 is also true. Hence, option (b) is correct.

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