If $m_1=10kg$, $m_2=4kg$ and $m_3=2kg$ the tension in the string connecting $m_1$and $m_2$is,

For $m_1$

${\mathrm{m}}_1\mathrm{g}-{\mathrm{T}}_1={\mathrm{m}}_1\mathrm{a}$

$T_1=m_{1}g-m_{1}a$

For $m_2$

${\mathrm{T}}_1+{\mathrm{m}}_2\mathrm{gsin}\theta -{\mathrm{T}}_2={\mathrm{m}}_2\mathrm{a}$

${\mathrm{T}}_1-{\mathrm{T}}_2={\mathrm{m}}_2\mathrm{a}-{\mathrm{m}}_2\mathrm{gsin}\theta$

For $m_3$

${\mathrm{T}}_2={\mathrm{m}}_3\mathrm{g}+{\mathrm{m}}_3\mathrm{a}$

$\therefore m_{1}g-m_{1}a-m_{3}g-m_{3}a=m_{2}a-m_{2}g\sin \theta$

$10g-10a-2g-2a=4a$

$=4g\left(\frac{1}{2}\right)$

$10\mathrm{g}=16\mathrm{a}$

$a=\frac{10g}{16}=\frac{5g}{8}$

${\mathrm{T}}_1={\mathrm{m}}_1(\mathrm{g}-\mathrm{a})$

${\mathrm{T}}_1=\frac{15\mathrm{g}}{4}$