Maximum value of the given expression will be equal to the reciprocal of the minimum value of

 sin²$\mathrm{\theta }$ + 3 sin $\mathrm{\theta }$ cos $\mathrm{\theta }$ + 5 cos²$\mathrm{\theta }$

 $\therefore \mathrm{E}=1+\frac{3}{2}\sin 2\mathrm{\theta }+2(1+\cos 2\mathrm{\theta })=3+\frac{3}{2}\sin 2\mathrm{\theta }+2\cos 2\mathrm{\theta }=3\pm \sqrt{\frac{9}{4}+4}=3\pm \frac{5}{2}$

 $\therefore {\mathrm{E}}_{\min }=\frac{1}{2}$

Hence maximum value of given expression = 2