The length of AB is:

1. 20 cm

2. 10 ($\sqrt{3}$ - 1) cm

3. $10\sqrt{3}$ cm

4. $10(\sqrt{3} + 1)$ cm

Since CD is perpendicular on AB.

By formula,

tan θ = $\dfrac{\text{Perpendicular}}{\text{Base}}$

From figure,

$\Rightarrow \text{tan B} = \dfrac{CD}{DB}\\[1em] \Rightarrow \text{tan 30°} = \dfrac{10}{DB}\\[1em] \Rightarrow \dfrac{1}{\sqrt{3}} = \dfrac{10}{DB}\\[1em] \Rightarrow DB = 10\sqrt{3}.$

As ΔACD is a right angled triangle,

$\Rightarrow \text{tan A} = \dfrac{CD}{AD} \\[1em] \Rightarrow \text{tan 45°} = \dfrac{10}{AD} \\[1em] \Rightarrow 1 = \dfrac{10}{AD} \\[1em] \Rightarrow AD = 10.$

From figure,

AB = AD + DB = 10 + $10\sqrt{3} = 10(1 + \sqrt{3})$ cm.

Hence, option 4 is the correct option.