Question : If $(a + b + c) = 14$ and $(a^{3} + b^{3} + c^{3} - 3 a b c) = 98$ , find the value of $(a b + b c + c a)$ .
Option 1: 60
Option 2: 64
Option 3: 65
Option 4: 63

Question

Question : If $(a + b + c) = 13$ and $(a b + b c + c a) = 54$ , find the value of $(a^{2} + b^{2} + c^{2})$ .

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Answer 1

Correct Answer: 63

Solution : Given that:
$a + b + c = 14$ and $a^{3} + b^{3} + c^{3} - 3 a b c = 98$
We have to find the value of $a b + b c + c a$
We know that,
$a^{3} + b^{3} + c^{3} - 3 a b c = 98$
⇒ $(a^{2} + b^{2} + c^{2} - a b - b c - c a) (a + b + c) = 98$
⇒ $(a^{2} + b^{2} + c^{2} - a b - b c - c a) (14) = 98$
⇒ $(a^{2} + b^{2} + c^{2} - a b - b c - c a)$ = $\frac{98}{14}$
⇒ $(a^{2} + b^{2} + c^{2}) = 7 + a b + b c + c a$
Now,
$(a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2 (a b + b c + c a)$
⇒ $14^{2} = 7 + a b + b c + c a + 2 (a b + b c + c a)$
⇒ $196 = 7 + 3 (a b + b c + c a)$
⇒ $196 - 7 = 3 (a b + b c + c a)$
⇒ $189 = 3 (a b + b c + c a)$
$∴$ $a b + b c + c a = 63$
Hence, the correct answer is 63.

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Question : If (a+b+c)=14 and (a3+b3+c3−3abc)=98, find the value of (ab+bc+ca).Option 1: 60Option 2: 64Option 3: 65Option 4: 63

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Question : If $(a + b + c) = 14$ and $(a^{3} + b^{3} + c^{3} - 3 a b c) = 98$ , find the value of $(a b + b c + c a)$ .
Option 1: 60
Option 2: 64
Option 3: 65
Option 4: 63