Question

If the mean of a, b, c is M and $ab + bc + ca = 0$, then the mean of ${a^2},{b^2},{c^2}$ is:
A) ${M^2}$
B) $3{M^2}$
C) $6{M^2}$
D) $9{M^2}$

Answer 1

Hint: First find the value of the sum of the numbers in terms of the mean. After that square the sum of numbers and expand it by the formula ${\left( {x + y + z} \right)^2} = {x^2} + {y^2} + {z^2} + 2\left( {xy + yz + zx} \right)$. Then substitute the values and do simplification to get the value of the sum of ${a^2},{b^2},{c^2}$, After that divide by 3 to get the mean of ${a^2},{b^2},{c^2}$ which is the desired result.

Complete step-by-step answer:
Given: - The mean of a, b, c is M.
$ab + bc + ca = 0$
Mean is the arithmetic average of a data set. This is found by adding the numbers in a data set and dividing by the number of observations in the data set.
We know that the general formula to find the mean value is,
Mean $ = \dfrac{{\sum x }}{N}$
Where $x$ is observation and $N$ is the number of observations.
Here,
$ \Rightarrow \sum x = a + b + c$
$ \Rightarrow N = 3$
Substitute these values in the above formula,
$ \Rightarrow M = \dfrac{{a + b + c}}{3}$
Cross-multiply the terms,
$ \Rightarrow a + b + c = 3M$
Square both sides,
$ \Rightarrow {\left( {a + b + c} \right)^2} = {\left( {3M} \right)^2}$
We know that,
${\left( {x + y + z} \right)^2} = {x^2} + {y^2} + {z^2} + 2\left( {xy + yz + zx} \right)$
Use this formula to expand the terms,
$ \Rightarrow {a^2} + {b^2} + {c^2} + 2\left( {ab + bc + ca} \right) = 9{M^2}$
Since $ab + bc + ca = 0$. Substitute it in the equation,
$ \Rightarrow {a^2} + {b^2} + {c^2} + 2\left( 0 \right) = 9{M^2}$
The addition of 0 doesn’t make changes on any value. So remove 0 from it,
$ \Rightarrow {a^2} + {b^2} + {c^2} = 9{M^2}$
Divide both sides by 3,
$\therefore \dfrac{{{a^2} + {b^2} + {c^2}}}{3} = 3{M^2}$
So, the mean of ${a^2},{b^2},{c^2}$ is $3{M^2}$.

Hence, option (B) is the correct answer.

Note: Mean is given by the sum of all the observations divided by the number of observations.
Range = Highest Observation – Lowest Observation
The students must also note that if one entry would have been exactly equal to the mean which is 73, then that entry would not be counted as the student getting more than mean marks, because that is just equal to the mean marks, not greater than it.
The students should also note that the average and mean represent the same thing.