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

Two inclined planes are placed as shown in figure.
A block is projected from the Point $A$ of inclined plane $AB$ along its surface with a velocity just sufficient to carry it to the top Point $B$ at a height $10 \mathrm{m}$. After reaching the Point $B$ the block slides down on inclined plane $BC$. Time it takes to reach to the point $C$ from point $A$ is $t\left(\sqrt{2}+1\right)\mathrm{s}$. The value of t is _____ (use $g=10 \mathrm{m} {\mathrm{s}}^{-2}$)