Question : If $x=2-2^{\frac{1}{3}}+2^{\frac{2}{3}}$, then find the value of $x^3-6 x^2+18 x$.
Option 1: 40
Option 2: 33
Option 3: 45
Option 4: 22
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Correct Answer: 22
Solution : $x=2-2^{\frac{1}{3}}+2^{\frac{2}{3}}$ ⇒ $(x-2) = (2^{\frac{2}{3}}-2^{\frac{1}{3}})$ Cubing both sides, ⇒ $(x-2)^3 = (2^{\frac{2}{3}}-2^{\frac{1}{3}})^3$ ⇒ $x^3 - 6x^2 +12x-8 = 2^2 -3 × 2^{\frac{2}{3}} × 2^{\frac{1}{3}}(2^{\frac{2}{3}}-2^{\frac{1}{3}}) - 2$ ⇒ $x^3 - 6x^2 +12x-8 = 4 - 3 × 2(x-2) - 2$ ⇒ $x^3 - 6x^2 + 12x = 10 - 6x +12$ ⇒ $x^3 - 6x^2 + 12x + 6x = 22$ Hence, the correct answer is 22.
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Question : If $x+\frac{1}{x}=6$, then find the value of $\frac{3 x}{2 x^2-5 x+2}$.
Option 1: $1$
Option 2: $\frac{3}{7}$
Option 3: $\frac{2}{3}$
Option 4: $0$
Question : If $(x+y):(x-y)=11:1$, find the value of $(\frac{5x+3y}{x-2y})$.
Option 1: $\frac{45}{4}$
Option 2: $\frac{4}{45}$
Option 3: $-\frac{45}{4}$
Option 4: $-\frac{4}{45}$
Question : If $x+\frac{1}{x}=5$, then the value of $\frac{x}{1+x+x^2}$ is:
Option 1: $\frac{1}{5}$
Option 2: $\frac{1}{6}$
Option 3: $5$
Option 4: $6$
Question : If $x=2+\sqrt3$, then the value of $\frac{x^{2}-x+1}{x^{2}+x+1}$ is:
Option 1: $\frac{2}{3}$
Option 2: $\frac{3}{4}$
Option 3: $\frac{4}{5}$
Option 4: $\frac{3}{5}$
Question : If $x^{3}+\frac{1}{x^{3}}=110$, then find the value of $x+\frac{1}{x}$:
Option 1: 2
Option 2: 3
Option 3: 4
Option 4: 5
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