Question : If $x=(\sqrt{6}-1)^{\frac{1}{3}}$, then the value of $\left(x-\frac{1}{x}\right)^3+3\left(x-\frac{1}{x}\right)$ is:
Option 1: $\frac{2 \sqrt{6}-6}{5}$
Option 2: $\frac{4 \sqrt{6}-6}{5}$
Option 3: $\frac{4 \sqrt{6}-6}{3}$
Option 4: $\frac{4 \sqrt{3}-6}{5}$
New: SSC CHSL tier 1 answer key 2024 out | SSC CHSL 2024 Notification PDF
Recommended: How to crack SSC CHSL | SSC CHSL exam guide
Don't Miss: Month-wise Current Affairs | Upcoming government exams
Correct Answer: $\frac{4 \sqrt{6}-6}{5}$
Solution :
Given: $x=(\sqrt{6}-1)^{\frac{1}{3}}$
Cubing both sides, we get,
$x^3=\sqrt6-1$
$\therefore \frac{1}{x^3}=\frac{1}{\sqrt6-1}=\frac{\sqrt6+1}{5}$
Now, $\left(x-\frac{1}{x}\right)^3+3\left(x-\frac{1}{x}\right)$
$=x^3-\frac{1}{x^3}-3×x×\frac{1}{x^3}(x-\frac{1}{x})-3(x-\frac{1}{x})$
$=x^3-\frac{1}{x^3}$
Putting values, we get,
$=(\sqrt6-1)-\frac{\sqrt6+1}{5}$
$=\frac{4 \sqrt{6}-6}{5}$
Hence, the correct answer is $\frac{4 \sqrt{6}-6}{5}$.
Related Questions
Know More about
Staff Selection Commission Combined High ...
Result | Eligibility | Application | Admit Card | Answer Key | Preparation Tips | Cutoff
Get Updates Brochure
Your Staff Selection Commission Combined Higher Secondary Level Exam brochure has been successfully mailed to your registered email id “”.
