If 3x + 5y = 11 and xy = 2,

We have:

$(3 x+5 y)^{2}=(3 x)^{2}+2(3 x)(5 y)+(5 y)^{2}$

$\Rightarrow(3 x+5 y)^{2}=9 x^{2}+30 x y+25 y^{2}$

$\Rightarrow 9 x^{2}+25 y^{2}=(3 x+5 y)^{2}-30 x y$

$\Rightarrow 9 x^{2}+25 y^{2}=11^{2}-30 \times 2 \quad(\because 3 x+5 y=11$ and $x y=2)$

$\Rightarrow 9 x^{2}+25 y^{2}=121-60$

$\Rightarrow 9 x^{2}+25 y^{2}=61$