If the mean of $$ p, q, r $$ is $$ M $$ and $$ pq + qr + rp = 0 $$, then the mean of $$ p^2, q^2, r^2 $$ is:

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Accepted Answer

### Step-by-Step Solution:

1. **Given Information:**  
   - The mean of $$ p, q, r $$:  
     $$
     \frac{p + q + r}{3} = M.
     $$  
     Therefore,  
     $$
     p + q + r = 3M.
     $$

- It is also given that:  
     $$
     pq + qr + rp = 0.
     $$

2. **Square the Sum of $$ p, q, r $$:**  
   Using the identity for squaring a sum:  
   $$
   (p + q + r)^2 = p^2 + q^2 + r^2 + 2(pq + qr + rp).
   $$  
   Substituting $$ p + q + r = 3M $$:  
   $$
   (3M)^2 = p^2 + q^2 + r^2 + 2(0).
   $$  
   Simplify:  
   $$
   9M^2 = p^2 + q^2 + r^2.
   $$

3. **Calculate the Mean of $$ p^2, q^2, r^2 $$:**  
   The mean is:  
   $$
   	ext{Mean} = \frac{p^2 + q^2 + r^2}{3}.
   $$  
   Substituting $$ p^2 + q^2 + r^2 = 9M^2 $$:  
   $$
   	ext{Mean} = \frac{9M^2}{3} = 3M^2.
   $$

### Final Answer:  
The mean of $$ p^2, q^2, r^2 $$ is **$$ 3M^2 $$**. Hence, the correct answer is **(b)**.

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