A force $$F=\alpha+\beta x^2$$ acts on an object in the $$x$$-direction. The work done by the force is  $$5\,J$$ when the object is displaced by $$1\,m.$$ If the constant $$\alpha=1\,N$$  then $$\beta$$ will be:

Given the force $$F = \alpha + \beta x^2$$, the work done is 5 J over a displacement of 1 m and α = 1 N, so we need to find β. Since the work is given by the integral of the force from 0 to 1, we have $$W = \int_0^1(\alpha+\beta x^2)dx = [\alpha x + \frac{\beta x^3}{3}]_0^1 = \alpha + \frac{\beta}{3}$$. Substituting the values gives $$5 = 1 + \frac{\beta}{3}$$, and therefore solving for β yields $$\beta = 12$$ N/m².

Thus the correct answer is Option 2: 12 N/m².