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
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(3M^2\)
The squares of p, q and r are linked to their sum by the identity \((p+q+r)^2=p^2+q^2+r^2+2(pq+qr+rp)\). The mean gives the sum, and the given condition removes the last term.
Mean of p, q, r is M, so \(p+q+r=3M\)
Putting the sum and \(pq+qr+rp=0\) into the identity: \((3M)^2=p^2+q^2+r^2+2\times0\), so \(p^2+q^2+r^2=9M^2\)
Mean of \(p^2,q^2,r^2\) \(\displaystyle =\frac{9M^2}{3}=3M^2\)
Tip: \(9M^2\), Option (4), is the sum of the squares; the question asks for their mean, so divide by 3.