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What will be the value of $a 3 + b 3 + c 3 − 3 a b c$ , if $a + b + c = 9$ , if and $a b + b c + c a = 26$ and?

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answer is A.

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Detailed Solution

Given,
a+b+c=9
ab+bc+ca=26
We have to find the value of

a 3 + b 3 + c 3 − 3 a b c

.
We know that,

p + q + r 2 = p 2 + q 2 + r 2 + 2 p q + q r + p r

Squaring both sides in the equation a + b + c = 9 and substituting ab + bc + ac = 26,

⇒ (a + b + c) 2 = (9) 2 ⇒ a 2 + b 2 + c 2 + 2 (a b + b c + a c) = 81 ⇒ a 2 + b 2 + c 2 + 2 (26) = 81 ⇒ a 2 + b 2 + c 2 = 29 … … (1)

We know that,

(p + q + r) p 2 + q 2 + r 2 − p q − q r − p r = p 3 + q 3 + r 3 − 3 p q r

Thus,

a 3 + b 3 + c 3 − 3 a b c = (a + b + c) a 2 + b 2 + c 2 − (a b + b c + a c) ⇒ a 3 + b 3 + c 3 − 3 a b c = (9) (29 − 26) ⇒ a 3 + b 3 + c 3 − 3 a b c = 9 (3) ⇒ a 3 + b 3 + c 3 − 3 a b c = 27

So,

a 3 + b 3 + c 3 − 3 a b c = 27

.
Therefore, option 1 is correct.

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