Introduction:
This question deals with the force acting on a particle of mass 1 kg and how it depends on its X and Y coordinates. We are given that the particle is at rest at point O and we need to find out the force when X=2m and Y=3m.

Given:
Mass of the particle, m = 1 kg
X-coordinate, X = 2 m
Y-coordinate, Y = 3 m

Methodology:
To find out the force acting on the particle, we need to use the equation of force, F = ma, where m is the mass of the particle and a is the acceleration. Since the particle is at rest at point O, its initial velocity is zero. Therefore, the acceleration of the particle is given by the gradient of the force-distance graph.

Calculating the force:
To find out the force acting on the particle when X=2m and Y=3m, we need to plot the force-distance graph using the given information. Let's assume that the force is given by F = kXY, where k is a constant.

When X = 0, Y = 2N (given)
When Y = 0, X = 0 (as the particle is at rest at point O)

Using these two points, we can plot the graph of force against X and Y coordinates. The graph is shown below:

https://edurev.in/api/image/11785970" />

The gradient of the graph at point (2,3) gives us the acceleration of the particle, which is given by:

a = dF/ds = dF/dX * dX/ds + dF/dY * dY/ds

where s is the distance traveled by the particle.

Differentiating the equation of force with respect to X and Y, we get:

dF/dX = kY
dF/dY = kX

Substituting the values of X and Y, we get:

dF/dX = 3k
dF/dY = 2k

Substituting these values in the equation for acceleration, we get:

a = (3k)(2) + (2k)(3) = 12k

Using the equation of force, F = ma, we get:

F = ma = (1 kg)(12k) = 12kN

Therefore, the force acting on the particle when X=2m and Y=3m is 12kN.

Conclusion:
In this question, we used the equation of force, F = ma, to find out the force acting on a particle of mass 1 kg. The force depended on the X and Y coordinates of the particle, and we used the given information to plot the force-distance graph. From the graph, we found out the acceleration of the particle and then used the equation of force to find out the force acting on the particle.