ABC26GN1549 · Average
Subject: General Aptitude · Chapter: Average · Exam: · Marks: · Difficulty:
If the mean of $a, b, c$ is M and $a b+b c+c a=0$, then the mean of $a^{2}, b^{2}, c^{2}$ is
(a)$\mathrm{M}^{2}$
(b)$3 \mathrm{M}^{2}$
(c)$6 \mathrm{M}^{2}$
(d)$9 \mathrm{M}^{2}$
Answer
Explanation
We have : $\left(\frac{a+b+c}{3}\right)=\mathrm{M} \quad$ or $\quad(a+b+c)=3 \mathrm{M}$. Now, $\quad(a+b+c)^{2}=(3 \mathrm{M})^{2}=9 \mathrm{M}^{2}$ $\Leftrightarrow \quad a^{2}+b^{2}+c^{2}+2(a b+b c+c a)=9 \mathrm{M}^{2}$ $\Leftrightarrow \quad a^{2}+b^{2}+c^{2}=9 \mathrm{M}^{2} . \quad[\because(a b+b c+c a)=0]$ $$\therefore \quad \text { Required mean }=\left(\frac{a^{2}+b^{2}+c^{2}}{3}\right)=\frac{9 \mathrm{M}^{2}}{3}=3 \mathrm{M}^{2} .$$
Explanation as extracted from the printed page; notation may be imperfect.
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