\[\frac{11}{2\sqrt{3} + 1} = \frac{11\left( 2\sqrt{3} - 1 \right)}{\left( 2\sqrt{3} + 1 \right)\left( 2\sqrt{3} - 1 \right)} =\]
\[= \frac{11\left( 2\sqrt{3} - 1 \right)}{4 \cdot 3 - 1} = \frac{11\left( 2\sqrt{3} - 1 \right)}{11} =\]
\[= 2\sqrt{3} - 1.\]