Two persons $$A$$ and $$B$$ perform same amount of work in moving a body through a certain distance $$d$$ with application of forces acting at angles 45° and 60° with the direction of displacement respectively. The ratio of force applied by person $$A$$ to the force applied by person $$B$$ is $$\frac{1}{\sqrt{x}}$$. The value of $$x$$ is _________.

We know that “work” is defined as the dot product of force and displacement. Mathematically, the magnitude of work done by a force $$\vec F$$ in producing a displacement $$\vec d$$ is given by the formula

$$W = F\,d \,\cos\theta,$$

where $$F$$ is the magnitude of the force, $$d$$ is the magnitude of the displacement and $$\theta$$ is the angle between the directions of $$\vec F$$ and $$\vec d$$.

Person $$A$$ applies a force $$F_A$$ at an angle $$45^\circ$$ with the displacement. Therefore the work done by $$A$$ is

$$W_A \;=\; F_A \, d \, \cos 45^\circ.$$

Person $$B$$ applies a force $$F_B$$ at an angle $$60^\circ$$ with the displacement. Hence the work done by $$B$$ is

$$W_B \;=\; F_B \, d \, \cos 60^\circ.$$

According to the statement of the problem, both persons perform the same amount of work in moving the body through the same distance $$d$$. So we equate the two expressions:

$$W_A \;=\; W_B.$$

Substituting the detailed expressions for $$W_A$$ and $$W_B$$, we get

$$F_A \, d \, \cos 45^\circ \;=\; F_B \, d \, \cos 60^\circ.$$

The distance $$d$$ is common on both sides, so we cancel it out:

$$F_A \,\cos 45^\circ \;=\; F_B \,\cos 60^\circ.$$

Now we isolate the required ratio $$\dfrac{F_A}{F_B}$$ by dividing both sides by $$F_B \cos 45^\circ$$:

$$\frac{F_A}{F_B} \;=\; \frac{\cos 60^\circ}{\cos 45^\circ}.$$

We recall the standard cosine values:

$$\cos 60^\circ = \frac{1}{2}, \qquad \cos 45^\circ = \frac{1}{\sqrt{2}}.$$

Substituting these numerical values, we obtain

$$\frac{F_A}{F_B} \;=\; \frac{\dfrac{1}{2}}{\dfrac{1}{\sqrt{2}}}.$$

To simplify the complex fraction, we multiply numerator and denominator appropriately:

$$\frac{F_A}{F_B} \;=\; \frac{1}{2} \times \frac{\sqrt{2}}{1} \;=\; \frac{\sqrt{2}}{2}.$$

Notice that $$\dfrac{\sqrt{2}}{2}$$ can also be written as $$\dfrac{1}{\sqrt{2}}$$, because

$$\frac{\sqrt{2}}{2} = \frac{1}{\sqrt{2}}.$$

The problem statement expresses the ratio in the form $$\dfrac{1}{\sqrt{x}}$$, so by direct comparison we identify

$$x = 2.$$

Hence, the correct answer is Option 2.