Question : If $x(x-5)=-1$, then the value of $x^3\left(x^3-110\right)=$?
Option 1: 0
Option 2: –1
Option 3: 1
Option 4: 2
Correct Answer: –1
Solution : Given: $x(x-5)=-1$ ⇒ $x-5=-\frac{1}{x}$ ⇒ $x+\frac{1}{x}=5$ Cubing both sides, we get: ⇒ $(x+\frac{1}{x})^3=5^3$ ⇒ $x^3+\frac{1}{x^3}+3(x\times\frac{1}{x})(x+\frac{1}{x})=125$ ⇒ $x^3+\frac{1}{x^3}+3(x+\frac{1}{x})=125$ ⇒ $x^3+\frac{1}{x^3}+3\times5=125$ ⇒ $x^3+\frac{1}{x^3}=110$ ⇒ $x^3-110=-\frac{1}{x^3}$ $\therefore x^3(x^3-110)=-1$ Hence, the correct answer is –1.
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