Question

If Y = $\sqrt{2}+1$, then the value of $Y+\frac{1}{Y}$ is:

• A. $\sqrt{\frac{3}{2}}$
• B. $\frac{\sqrt{3}}{2}$
• C. $\frac{\sqrt{2}}{2}$
• D. $\frac{4}{\sqrt{2}}$

Answer 1

Answer: (D)

Given $Y=\sqrt{2}+1$.
Now, $Y+\frac{1}{Y}=\sqrt{2}+1+\frac{1}{\sqrt{2}+1}$

$=\frac{(\sqrt{2}+1{)}^2+1}{\sqrt{2}+1}$

$=\frac{2+1+2\sqrt{2}+1}{\sqrt{2}+1}\times \frac{\sqrt{2}-1}{\sqrt{2}-1}$

$=\frac{(4+2\sqrt{2})(\sqrt{2}-1)}{2-1}$

$=\frac{4\sqrt{2}-4+4-2\sqrt{2}}{1}$

$=2\sqrt{2}\times \frac{\sqrt{2}}{\sqrt{2}}$

$=\frac{4}{\sqrt{2}}$.

Therefore, option $D$ is correct.