MCQ
If $a + b + c = 9$ and $ab + bc + ca = 23,$ than $a^3+b^3+c^3-3 a b c=$
- A$729$
- B$207$
- C$669$
- ✓$108$
✓
Answer
Correct option: D.
$108$
$(a+b+c)^2=a^2+b^2+c^2+2 a b+2 b c+2 c a$
$\Rightarrow(9)^2=a^2+b^2+c^2+2(a b+b c+c a)$
$\Rightarrow(9)^2=a^2+b^2+c^2+2(23)$
$\Rightarrow a^2+b^2+c^2=81-46=35$
as we know that $a^3+b^3+c^3-3 a b c=(a+b+c)\left(a^2+b^2+c^2-a b-b c-c a\right)$
$\Rightarrow a^3+b^3+c^3-3 a b c=9 \times(35-23)$
$\Rightarrow a^3+b^3+c^3-3 a b c=108$
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