Question : If $\frac{a}{b}+\frac{b}{a}=1$, then the value of $a^{3}+b^{3}$ will be:
Option 1: 1
Option 2: 0
Option 3: –1
Option 4: 2
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Correct Answer: 0
Solution : Given: $\frac{a}{b}+\frac{b}{a}=1$ ⇒ $\frac{a^2 +b^2}{ab}=1$ ⇒ ${(a^2 +b^2)}={ab}$__________(equation 1) Now, $a^{3}+b^{3}={(a+b)(a^2+b^2–ab)}$ Putting the value of ${a^2+b^2}$ from equation 1, we get: ${(a+b)(a^2+b^2–ab)}$ = ${(a+b)(ab–ab)}$ = ${(a+b)×0}=0$ Thus, $a^{3}+b^{3}=0$ Hence, the correct answer is $0$.
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