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
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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