Question : If $x=\frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}+\sqrt{2}}$ and $y=\frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}}$, then the value of $x^{3} + y^{3}$ is:
Option 1: 950
Option 2: 730
Option 3: 650
Option 4: 970
Correct Answer: 970
Solution :
$x=\frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}+\sqrt{2}}=\frac{(\sqrt{3}-\sqrt{2})×(\sqrt{3}-\sqrt{2})}{(\sqrt{3}+\sqrt{2})×(\sqrt{3}-\sqrt{2})}=(\sqrt{3}-\sqrt{2})^2$ = $5-2\sqrt6$
$y=\frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}}=\frac{(\sqrt{3}+\sqrt{2})×(\sqrt{3}+\sqrt{2})}{(\sqrt{3}-\sqrt{2})×(\sqrt{3}+\sqrt{2})}=(\sqrt{3}+\sqrt{2})^2$ = $5+2\sqrt6$
$⇒x+y=5-2\sqrt6+5+2\sqrt6 =10$
Also, $xy=(5-2\sqrt6)(5+2\sqrt6)=25-24=1$
$\therefore x^3+y^3=(x+y)^{3} -3xy(x+y)$
$=10^3-3\times 1 \times 10$
$=1000-30$
$= 970$
Hence, the correct answer is 970.
Related Questions
Know More about
Staff Selection Commission Sub Inspector ...
Result | Eligibility | Application | Selection Process | Cutoff | Admit Card | Preparation Tips
Get Updates Brochure
Your Staff Selection Commission Sub Inspector Exam brochure has been successfully mailed to your registered email id “”.
