1.

If the arithmetic mean of a, b, c is \(\rm \frac M 3\) and \(\rm \frac{1}{a} + \frac{1}{b} = -\frac{1}{c} \), then the arithmetic mean of a2, b2, c2 is1. M2/32. 3M23. 6M24. 9M2

Answer» Correct Answer - Option 1 : M2/3

Concept:

The arithmetic mean is the sum of all the numbers in a data set divided by the quantity of numbers in that set.

 

Calculation:

Given: Arithmetic mean of a, b, c is \(\rm \frac M 3\) 

So, \(\rm \frac M 3 = \frac {a+b+c}{3}\)

⇒ M = a + b + c           .... (1)

Given: \(\rm \frac{1}{a} + \frac{1}{b} = -\frac{1}{c} \)

⇒ \(\rm \frac{1}{a} + \frac{1}{b} +\frac{1}{c} =0\)

⇒ \(\rm \frac {ab + bc+ac}{abc} = 0\)

⇒ ab + bc  + ac = 0      .... (ii)

As we know, (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc  + ac)

⇒ M2 = a2 + b2 + c2 + 2 × 0             [From equation (i) and (ii)]

∴ a2 + b2 + c2 = M2

Now, the arithmetic mean of a2, b2, c2 = \(\rm \frac {a^2+b^2+c^2} 3 = \frac {M^2}3\)


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