Question

If the mean of a, b, c is M and ab + bc + ca = 0, then the mean of a², b², c² is
1. M²
2. 3M²
3. 6M²
4. 9M²

Answer 1

Correct Answer - Option 2 : 3M²

Given:

The mean of a, b, c is M and ab + bc + ca = 0

Formula used:

Mean = (Sum of all the terms/Total number of terms)

(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

Calculation:

Mean = (a + b + c)/3 = M

⇒ (a + b + c) = 3M      ----(1)

Now, (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)

⇒ (3M)² =  a2 + b2 + c2 + 2 × 0      ----(from eq (1))

⇒ 9M² =  a2 + b2 + c²      ----(2)

Now, Mean = (a2 + b2 + c2)/3

⇒ (9M²)/3      ----(from eq 2)

⇒ 3M2     

∴ The mean of the given expression is 3M2