MCQ
If $\frac{\text{a}}{\text{b}}+\frac{\text{b}}{\text{a}}=1,$ then $a^3 + b^3 =$
- A$1$
- B$-1$
- ✓$0$
- D$\frac{1}{2}$
✓
Answer
Correct option: C.
$0$
Here, $\frac{\text{a}}{\text{b}}+\frac{\text{b}}{\text{a}}=1$
$\Rightarrow\frac{\text{a}^2+\text{b}^2}{\text{ab}}=1$
$\Rightarrow\text{a}^2+\text{b}^2=\text{ab}$
$\Rightarrow\text{a}^2+\text{b}^2-\text{ab}=0$
Using, $\text{a}^2+\text{b}^2=(\text{a}+\text{b})(\text{a}^2+\text{b}^2-\text{ab})$
$=(\text{a}+\text{b})(0)$
$=0$
$\Rightarrow\frac{\text{a}^2+\text{b}^2}{\text{ab}}=1$
$\Rightarrow\text{a}^2+\text{b}^2=\text{ab}$
$\Rightarrow\text{a}^2+\text{b}^2-\text{ab}=0$
Using, $\text{a}^2+\text{b}^2=(\text{a}+\text{b})(\text{a}^2+\text{b}^2-\text{ab})$
$=(\text{a}+\text{b})(0)$
$=0$
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