Question : If $x - \frac{3}{x} = 6, x \neq 0$ , then the value of $\frac{x^{4} - \frac{27}{x^{2}}}{x^{2} - 3 x - 3}$ is:
Option 1: 80
Option 2: 270
Option 3: 54
Option 4: 90

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Question : If $x - \frac{1}{x} = 5, x \neq 0$ , then what is the value of $\frac{x^{6} + 3 x^{3} - 1}{x^{6} - 8 x^{3} - 1} ?$

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Answer 1

Correct Answer: 90

Solution : $x - \frac{3}{x} = 6$
⇒ $x^{2} - 3 = 6 x$
⇒ $x^{2} - 6 x - 3 = 0$
⇒ $x^{2} - 3 x - 3 = 3 x$ ---(1)
( $x - \frac{3}{x})^{3} = 6^{3}$
⇒ $x^{3} - (\frac{3}{x})^{3} - 3 (x - \frac{3}{x}) = 216$
⇒ $x^{3} - (\frac{3}{x})^{3} - 9 (6) = 216$
⇒ $x^{3} - (\frac{3}{x})^{3} - 54 = 216$
⇒ $x^{3} - (\frac{3}{x})^{3} = 270$ --- (2)
Now, $\frac{x^{4} - \frac{27}{x^{2}}}{x^{2} - 3 x - 3}$
= $\frac{x^{4} - \frac{27}{x^{2}}}{3 x}$ [from equation (1)]
= $\frac{x^{3} - \frac{27}{x^{3}}}{3}$
= $\frac{270}{3}$ [From the above equation (2)]
= 90
Hence the correct answer is 90.

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Question : If x−3x=6,x≠0, then the value of x4−27x2x2−3x−3 is:Option 1: 80Option 2: 270Option 3: 54Option 4: 90

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Question : If $x - \frac{3}{x} = 6, x \neq 0$ , then the value of $\frac{x^{4} - \frac{27}{x^{2}}}{x^{2} - 3 x - 3}$ is:
Option 1: 80
Option 2: 270
Option 3: 54
Option 4: 90