Question : If $x^{2} - 5 \sqrt{5} x + 1 = 0$ and $x > 0$ , what is the value of $(x^{3} - \frac{1}{x^{3}})$ ?
Option 1: 1331
Option 2: 1364
Option 3: 1296
Option 4: 1244

Question

Question : If $x = \frac{1}{x - 5} (x > 0)$ , then the value of $x + \frac{1}{x}$ is:
Option 1: $\sqrt{41}$
Option 2: $\sqrt{29}$
Option 3: $\sqrt{23}$
Option 4: $\sqrt{43}$

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Answer 1

Correct Answer: 1364

Solution : Given:
$x^{2} - 5 \sqrt{5} x + 1 = 0$ , and $x > 0$
Dividing by $x$ on both sides, we get
$x - 5 \sqrt{5} + \frac{1}{x} = 0$
⇒ $x + \frac{1}{x} = 5 \sqrt{5}$
Squaring both sides, we get
⇒ $(x + \frac{1}{x})^{2} = (5 \sqrt{5})^{2} = 125$
We know that $(x - \frac{1}{x})^{2} = (x + \frac{1}{x})^{2} - 4$
⇒ $(x - \frac{1}{x})^{2} = 125 - 4 = 121$
⇒ $(x - \frac{1}{x}) = 11$
Now, $(x - \frac{1}{x})^{3} = x^{3} - \frac{1}{x^{3}} - 3 \times x \times \frac{1}{x} (x - \frac{1}{x})$
⇒ $11^{3} = x^{3} - \frac{1}{x^{3}} - 3 \times 11$
⇒ $x^{3} - \frac{1}{x^{3}} = 1331 + 33$
$∴ x^{3} - \frac{1}{x^{3}} = 1364$
Hence, the correct answer is 1364.

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Question : If x2−55x+1=0 and x>0, what is the value of (x3−1x3)?Option 1: 1331Option 2: 1364Option 3: 1296Option 4: 1244

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Question : If $x^{2} - 5 \sqrt{5} x + 1 = 0$ and $x > 0$ , what is the value of $(x^{3} - \frac{1}{x^{3}})$ ?
Option 1: 1331
Option 2: 1364
Option 3: 1296
Option 4: 1244