Question : If $a + b + c = 12$ and $a b + b c + c a = 22$ , then what is the value of $a^{3} + b^{3} + c^{3} - 3 a b c ?$
Option 1: 1052
Option 2: 936
Option 3: 924
Option 4: 876

Question

Question : If $a + b + c = 0$ and $a b + b c + c a = - 11$ , then what is the value of $a^{2} + b^{2} + c^{2}$ ?
Option 1: –11
Option 2: 22
Option 3: 0
Option 4: 11

9 Views

Answer 1

Correct Answer: 936

Solution : Using the given equation $a + b + c = 12$ , we can substitute this value into the formula:
$a^{3} + b^{3} + c^{3} - 3 a b c = (a + b + c) (a^{2} + b^{2} + c^{2} - a b - b c - c a)$
⇒ $a^{3} + b^{3} + c^{3} - 3 a b c = (12) (a^{2} + b^{2} + c^{2} - a b - b c - c a)$
To find the value of $a^{2} + b^{2} + c^{2} - a b - b c - c a$ , we'll square the equation $a + b + c = 12$
⇒ $(a + b + c)^{2} = 12^{2}$
⇒ $a^{2} + b^{2} + c^{2} + 2 (a b + b c + c a) = 144$
Substituting the given equation $a b + b c + c a = 22$ :
$a^{2} + b^{2} + c^{2} + 2 (22) = 144$
⇒ $a^{2} + b^{2} + c^{2} = 100$
Now we have the value of $a^{2} + b^{2} + c^{2} - a b - b c - c a$ :
$a^{2} + b^{2} + c^{2} - a b - b c - c a = 100 - 22 = 78$
Substituting this value back into the formula, we get:
$a^{3} + b^{3} + c^{3} - 3 a b c = (12) (78)$
⇒ $a^{3} + b^{3} + c^{3} - 3 a b c = 936$
Hence, the correct answer is 936.

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Question : If a+b+c=12 and ab+bc+ca=22, then what is the value of a3+b3+c3−3abc?Option 1: 1052Option 2: 936Option 3: 924Option 4: 876

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Question : If $a + b + c = 12$ and $a b + b c + c a = 22$ , then what is the value of $a^{3} + b^{3} + c^{3} - 3 a b c ?$
Option 1: 1052
Option 2: 936
Option 3: 924
Option 4: 876