Question : If $a+b+c=1, ab+bc+ca=-1$ and $abc=-1$, then the value of $a^{3}+b^{3}+c^{3}$ is:
Option 1: 1
Option 2: – 1
Option 3: 2
Option 4: – 2
Correct Answer: 1
Solution :
Given: $a+b+c=1, ab+bc+ca=-1$ and $abc=-1$
$a^2+b^2+c^2= (a+b+c)^2-2(ab+bc+ca)=3$
Consider, $a+b+ c = 1$ ......(1)
⇒ $a^3 +b^3+c^3 -3abc=(a+b+c)(a^2+b^2+c^2−ab−bc−ca)$
⇒ $a^3 +b^3+c^3 =(a+b+c)(a^2+b^2+c^2−ab−bc−ca)+3abc$
⇒ $a^3 +b^3+c^3 = (1)[3-(-1)]+3\times(-1)$
⇒ $a^3 +b^3+c^3 = 4-3$
⇒ $a^3 +b^3+c^3 = 1$
Hence, the correct answer is 1.
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