Question

If $a$ is 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. $8$

Answer 1

The correct option is D

$8$

Explanation for the correct option:

Find the value of $a$:

AM $\mathbf{=}\mathbf{}\frac{\left(a+2\right)}{\mathbf{2}}$

GM $\mathbf{=}\mathbf{}\sqrt{\left(2a\right)}$

Given that AM exceeds GM by $1$

$\Rightarrow$$\frac{\left(a+2\right)}{2}-\sqrt{\left(2a\right)}=1$

$\Rightarrow$$\left(a+2\right)-2\sqrt{\left(2a\right)}=2$

$\Rightarrow$ $a=2\sqrt{\left(2a\right)}$

$\Rightarrow$ $\sqrt{a}=2\sqrt{2}$

$\mathbf{\therefore }\mathit{a}\mathbf{}\mathbf{=}\mathbf{}\mathbf{8}$

Hence, Option ‘D’ is Correct.