Question : If $x+\left [\frac{1}{(4x)} \right]=\frac{5}{2}$; then, what is the value of $\frac{64x^{6}+1}{8x^{3}} \;?$
Option 1: 110
Option 2: 115
Option 3: 125
Option 4: 140
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Correct Answer: 110
Solution :
The given expression $x+\left [ \frac{1}{(4x)} \right ]=\frac{5}{2}$ can be written as $4x^2+1 = 10x$.
We have, $\frac{64x^{6}+1}{8x^{3}}$ .......(1)
Here, we can write $64x^{6}+1$ as $(4x^2+1)(16x^4+1-4x^2)$.
⇒ $(4x^2+1)(16x^4+1-4x^2)$
⇒ $(4x^2+1)((4x^2+1)^2 - 8x^2-4x^2)$
⇒ $(4x^2+1)((4x^2+1)^2-12x^2)$
⇒ $(4x^2+1)((10x)^2-12x^2)$
⇒ $(10x)(88x^2) = 880x^3$
Putting this value in equation (1), we get,
⇒ $\frac{64x^{6}+1}{8x^{3}}$ = $\frac{880x^3}{8x^3}$ = 110
Hence, the correct answer is 110.
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