Question : If $a + b + c = 6$ and $a^{2} + b^{2} + c^{2} = 14$ , then what is the value of $(a - b)^{2} + (b - c)^{2} + (c -$ a) $^{2}$ ?
Option 1: –8
Option 2: 8
Option 3: 10
Option 4: 6

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Question : If $a + b + c = 7$ and $a^{3} + b^{3} + c^{3} - 3 a b c = 175$ , then what is the value of $(a b + b c + c a)$ ?

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Answer 1

Correct Answer: 6

Solution : $a^{2} + b^{2} + c^{2} = 14$
$a + b + c = 6$
Squaring on both sides,
$(a + b + c)^{2} = 6^{2}$
⇒ $a^{2} + b^{2} + c^{2} + 2 a b + 2 b c + 2 c a = 36$
⇒ $14 + 2 a b + 2 b c + 2 c a = 36$
⇒ $2 a b + 2 b c + 2 c a = 36 - 14 = 22$ --------------------(i)
$(a - b)^{2} + (b - c)^{2} + (c -$ a) $^{2}$
$= a^{2} + b^{2} - 2 a b + b^{2} + c^{2} - 2 b c + c^{2} + a^{2} - 2 c a$
$= 2 (a^{2} + b^{2} + c^{2}) - (2 a b + 2 b c + 2 c a)$
$= 2 \times 14 - 22$
$= 6$
Hence, the correct answer is 6.

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Question : If a+b+c=6 and a2+b2+c2=14, then what is the value of (a−b)2+(b−c)2+(c− a) 2?Option 1: –8Option 2: 8Option 3: 10Option 4: 6

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Question : If $a + b + c = 6$ and $a^{2} + b^{2} + c^{2} = 14$ , then what is the value of $(a - b)^{2} + (b - c)^{2} + (c -$ a) $^{2}$ ?
Option 1: –8
Option 2: 8
Option 3: 10
Option 4: 6