If the mean (average) of the three numbers a, b and c is M and it is given that ab + bc + ca = 0, then what is the mean of the three numbers a², b² and c²?

Introduction:
This question tests algebraic manipulation of means and symmetric expressions in three variables. It connects the mean of the original numbers a, b, c with the mean of their squares using a condition on ab + bc + ca.

Given Data / Assumptions:
Mean of a, b, c is M, so (a + b + c) / 3 = M. Therefore, a + b + c = 3M. It is given that ab + bc + ca = 0. We need the mean of a², b², c², that is (a² + b² + c²) / 3.

Concept / Approach:
A key algebraic identity is: (a + b + c)² = a² + b² + c² + 2(ab + bc + ca). We already know a + b + c and ab + bc + ca. Using this identity, we can express a² + b² + c² in terms of M, then divide by 3 to get the required mean.

Step-by-Step Solution:
Step 1: Use the given mean. a + b + c = 3M. Step 2: Square both sides of a + b + c = 3M. (a + b + c)² = (3M)² = 9M². Step 3: Expand the left-hand side using the identity. (a + b + c)² = a² + b² + c² + 2(ab + bc + ca). Step 4: Substitute ab + bc + ca = 0. So, 9M² = a² + b² + c² + 2 * 0. This simplifies to a² + b² + c² = 9M². Step 5: Find the mean of a², b², c². Mean of squares = (a² + b² + c²) / 3 = 9M² / 3 = 3M².

Verification / Alternative Check:
You can test with a simple example. Suppose a, b, c satisfy a + b + c = 3 and ab + bc + ca = 0. One such triple is (1, 1, 1) but that gives ab + bc + ca = 3, not 0. Instead, consider more advanced algebraic triples. However, the identity used is standard and algebraically sound, so the result 3M² is reliable.

Why Other Options Are Wrong:
M and M²: These ignore the effect of the squared terms and the given condition ab + bc + ca = 0. 2(M/3): This is dimensionally inconsistent with the squared quantity and does not follow from the identity. 3M: Again, this ignores the squared nature of the expression and is not derived from any correct manipulation.

Common Pitfalls:
A common error is to think that the mean of the squares is simply the square of the mean, which would give M², but this is only true in special cases. Another pitfall is forgetting to use the condition ab + bc + ca = 0, which is crucial to simplifying the expression. Always use known algebraic identities systematically.

Final Answer:
The mean of a², b² and c² is 3M².