Two forces $$\overline{\mathrm{F}}_1$$ and $$\overline{\mathrm{F}}_2$$ are acting on a body. One force has magnitude thrice that of the other force and the resultant of the two forces is equal to the force of larger magnitude. The angle between $$\vec{F}_1$$ and $$\vec{F}_2$$ is $$\cos ^{-1}\left(\frac{1}{n}\right)$$. The value of $$|n|$$ is _______.

Four forces are acting at a point $$\mathrm{P}$$ in equilibrium as shown in figure. The ratio of force $$\mathrm{F}_{1}$$ to $$\mathrm{F}_{2}$$ is $$1: x$$ where $$x=$$ _____________.

A hanging mass M is connected to a four times bigger mass by using a string-pulley arrangement, as shown in the figure. The bigger mass is placed on a horizontal ice-slab and being pulled by 2 Mg force. In this situation, tension in the string is $${x \over 5}$$ Mg for x = ______________. Neglect mass of the string and friction of the block (bigger mass) with ice slab.

(Given g = acceleration due to gravity)

A mass of 10 kg is suspended vertically by a rope of length 5 m from the roof. A force of 30 N is applied at the middle point of rope in horizontal direction. The angle made by upper half of the rope with vertical is $$\theta$$ = tan^$$-$$1 (x $$\times$$ 10^$$-$$1). The value of x is ____________.

(Given, g = 10 m/s²)