Rationalise the denominator of the following:

$\frac{4 \sqrt{3}+5 \sqrt{2}}{\sqrt{48}+\sqrt{18}}$

Insert a rational number and an irrational number between the following:

$0$ and $0.1$

Find the value of $a$ :

$\frac{3-\sqrt{5}}{3+2 \sqrt{5}}=a \sqrt{5}-\frac{19}{11}$

Insert a rational number and an irrational number between the following:

$\frac{-2}{5}$ and $\frac{1}{2}$

Rationalise the denominator in each of the following and hence evaluate by taking $\sqrt{2}=1.414, \sqrt{3}=1.732$ and $\sqrt{5}=2.236,$ upto three places of decimal.

$\frac{\sqrt{10}-\sqrt{5}}{2}$

Insert a rational number and an irrational number between the following:

$2.357$ and $3.121$