If $\left|\overrightarrow{\mathrm{a}}\right|=2,\left|\overrightarrow{\mathrm{b}}\right|=3\text{ and }|2\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}|=5\text{, then }|2\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}|$ equals :

 $$\begin{gathered} Given that \left|\overrightarrow{\mathrm{a}}\right|=2,\left|\overrightarrow{\mathrm{b}}\right|=3 \\ Now |2\overline{\mathrm{a}}-\overline{\mathrm{b}}{|}^2=4|\overline{\mathrm{a}}{|}^2+|\overline{\mathrm{b}}{|}^2-4\overline{\mathrm{a}}\cdot \overline{\mathrm{b}}=25 \end{gathered}$$
$$\begin{gathered} \Rightarrow 4\left(4\right)+9-4\overline{\mathrm{a}}\cdot \overline{\mathrm{b}}=25\Rightarrow \overline{\mathrm{a}}\cdot \overline{\mathrm{b}}=0 \\ \mathrm{Now} |2\overline{\mathrm{a}}+\overline{\mathrm{b}}{|}^2=4|\overline{\mathrm{a}}{|}^2+|\overline{\mathrm{b}}{|}^2+4\overline{\mathrm{a}}\cdot \overline{\mathrm{b}} \end{gathered}$$
$$\begin{gathered} =4\left(4\right)+9+0=25 \\ \Rightarrow |2\overline{\mathrm{a}}+\overline{\mathrm{b}}|=5 \end{gathered}$$