Question

Let $a$ be a positive number such that the arithmetic mean of $a$ and $2$ exceeds their geometric mean by $1$. Then, the value of $a$ is

• A. 3
• B. 5
• C. 9
• D. 8
• E. 10

Answer 1

Answer: (D)

$ \frac{a+2}{2}=\sqrt{2a}+1 $
$ \Rightarrow $ $ \frac{a}{2}+1=\sqrt{2a}+1 $
$ \Rightarrow $ $ \frac{a}{2}=\sqrt{2a} $
$ \Rightarrow $ $ \frac{{{a}^{2}}}{4}=2a $
$ \Rightarrow $ $ a\left( \frac{a}{4}-2 \right)=0 $
$ \Rightarrow $ $ a=0,a=8 $