In the figure shown a₃ = 6m/s² (downwards) and a₂ = 4m/s² (upwards). Find acceleration of 1. 

Since the points 1,2,3 and are movable, so let their displacements are  x₁ , x₂ , x₃ and x₄ We observe that the length of the strings between 1 and 4 and 2 and 3 are constants.
${\mathrm{x}}_1+{\mathrm{x}}_4={\mathrm{\ell }}_1;\left({\mathrm{x}}_2-{\mathrm{x}}_4\right)+\left({\mathrm{x}}_3-{\mathrm{x}}_4\right)={\mathrm{\ell }}_2$
Differentiating twice w.r.t time, we get
$\begin{array}{l}{\mathrm{a}}_1+{\mathrm{a}}_4=0\Rightarrow {\mathrm{a}}_1=-{\mathrm{a}}_4\to \left(1\right)\\ {\mathrm{a}}_2-{\mathrm{a}}_4+{\mathrm{a}}_3-{\mathrm{a}}_4=0\\ \Rightarrow {\mathrm{a}}_2+{\mathrm{a}}_3-2{\mathrm{a}}_4=0\to \left(2\right)\\ {\mathrm{a}}_2=-4{\mathrm{ms}}^{-2},{\mathrm{a}}_3=6{\mathrm{ms}}^{-2};-4+6-2{\mathrm{a}}_4=0\\ 2{\mathrm{a}}_4=2\Rightarrow {\mathrm{a}}_4=1{\mathrm{ms}}^{-2}; \mathrm{From}(1), \\ {\mathrm{a}}_1=-{\mathrm{a}}_4;{\mathrm{a}}_1=-1{\mathrm{ms}}^{-2};{\mathrm{a}}_1=1{\mathrm{ms}}^{-2} \mathrm{upwards} \end{array}$