If a → = 2 ,   b → = 3 and 2 a → - b → = 5 , then 2 a → + b →…

$\left|\overrightarrow{a}\right|=2, \left|\overrightarrow{b}\right|=3, \left|2\overrightarrow{a}-\overrightarrow{b}\right|=5$

${\left|2 \overrightarrow{a}-\overrightarrow{b}\right|}^2=\left(2 \overrightarrow{a}-\overrightarrow{b}\right)·\left(2 \overrightarrow{a}-\overrightarrow{b}\right)$

$=4{\left|\overrightarrow{a}\right|}^2-4\left[\overrightarrow{a}·\overrightarrow{b}\right]+{\left|\overrightarrow{b}\right|}^2 ...\left(\mathrm{i}\right)$

${\left|2\overrightarrow{a}+\overrightarrow{b}\right|}^2=4{\left|\overrightarrow{a}\right|}^2+4\left[\overrightarrow{a}·\overrightarrow{b}\right]+{\left|\overrightarrow{b}\right|}^2 ...\left(\mathrm{ii}\right)$

Adding equation (i) and equation (ii), we get

${\left|2 \overrightarrow{a}+\overrightarrow{b}\right|}^2+{\left|2 \overrightarrow{a}-\overrightarrow{b}\right|}^2=8{\left|\overrightarrow{a}\right|}^2+{2\left|\overrightarrow{b}\right|}^2$

$\Rightarrow {\left|2 \overrightarrow{a}+\overrightarrow{b}\right|}^2+5^2=8\left(2^2\right)+2\left(3^2\right)$

$\Rightarrow {\left|2\overrightarrow{a}+\overrightarrow{b}\right|}^2=32+18-25$$=25$

$\Rightarrow \left|2\overrightarrow{a}+\overrightarrow{b}\right|=5$