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 :

Three masses $$M=100 \mathrm{~kg}, \mathrm{~m}_{1}=10 \mathrm{~kg}$$ and $$\mathrm{m}_{2}=20 \mathrm{~kg}$$ 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 $$\mathrm{F}$$ is applied on the system so that the mass $$\mathrm{m}_{2}$$ moves upward with an acceleration of $$2 \mathrm{~ms}^{-2}$$. The value of $$\mathrm{F}$$ is :

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

A monkey of mass $$50 \mathrm{~kg}$$ climbs on a rope which can withstand the tension (T) of $$350 \mathrm{~N}$$. If monkey initially climbs down with an acceleration of $$4 \mathrm{~m} / \mathrm{s}^{2}$$ and then climbs up with an acceleration of $$5 \mathrm{~m} / \mathrm{s}^{2}$$. Choose the correct option $$\left(g=10 \mathrm{~m} / \mathrm{s}^{2}\right)$$.

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?