###
**If the mean of a, b, c, is M and ab + bc + ca = 0, then the mean of $\mathbf{a}^{2}, \mathbf{b}^{2}, \mathbf{c}^{2}$ is.
**

A. $\mathrm{M}^{2}$
B. $3 \mathrm{M}^{2}$
C. $6 \mathrm{M}^{2}$
D. $9 \mathrm{M}^{2}$
**Answer: Option B**

## Show Answer

Solution(By Apex Team)

$\begin{aligned}&\text{ We have: }\\
&\frac{a+b+c}{3}=\mathrm{M}\\
&\text{ or }(a+b+c)=3M\\
&\text{ Now, }(a+b+c)^2=(3M)^2=9M^2\\
&\Leftrightarrow a^2+b^2+c^2+2(ab+bc+ca)=9M^2\\
&\Leftrightarrow a^2+b^2+c^2=9M^2\\
&[\because(ab+bc+ca)=0]\\
&\text { : Required mean }\\
&=\left(\frac{a^{2}+b^{2}+c^{2}}{3}\right)\\
&=\frac{9 M^{2}}{3}\\
&=3 M^{2}
\end{aligned}$

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