Four forces are acting at a point $P$ in equilibrium as shown in figure. The ratio of force $F_1$ to $F_2$ is $1:x$, where $x=$_____ .

For a free body diagram shown in the figure, the four forces are applied in the '$x$' and '$y$' directions. What additional force must be applied and at what angle with positive $x$-axis so that the net acceleration of body is zero?

Three masses $M=100 \mathrm{kg},{ m}_1=10 \mathrm{kg}$ and $m_2=20 \mathrm{kg}$ are arranged in a system as shown in figure. All the surfaces are frictionless and strings are inextensible and weightless. The pulleys are also weightless and frictionless. A force $F$ is applied on the system so that the mass $m_2$ moves upward with an acceleration of $2{ \mathrm{ms}}^{-2}$. The value of $F$ is

(Take $g=10{ \mathrm{ms}}^{-2}$)

Two masses $M_1$ and $M_2$ are tied together at the two ends of a light inextensible string that passes over a frictionless pulley. When the mass $M_2$ is twice that of $M_1$, the acceleration of the system is $a_1$. When the mass $M_2$ is thrice that of $M_1$, the acceleration of the system is $a_2$. The ratio $\frac{a_1}{a_2}$ will be