When a body of mass M1 is hanging freely and another of mass M2 lying on a smooth inclined plane ($$\alpha $$) are connected by a light index tensile string passing over a smooth pulley, the acceleration of the body of mass M1, will be given by A. $$\frac{{{\text{g}}\left( {{{\text{M}}_1} + {{\text{M}}_2}\sin \alpha } \right)}}{{{{\text{M}}_1} + {{\text{M}}_2}}}{\text{m/sec}}$$ B. $$\frac{{{\text{g}}\left( {{{\text{M}}_1} – {{\text{M}}_2}\sin \alpha } \right)}}{{{{\text{M}}_1} + {{\text{M}}_2}}}{\text{m/se}}{{\text{c}}^2}$$ C. $$\frac{{{\text{g}}\left( {{{\text{M}}_2} + {{\text{M}}_1}\sin \alpha } \right)}}{{{{\text{M}}_1} + {{\text{M}}_2}}}{\text{m/se}}{{\text{c}}^2}$$ D. $$\frac{{{\text{g}}\left( {{{\text{M}}_2} \times {{\text{M}}_1}\sin \alpha } \right)}}{{{{\text{M}}_2} – {{\text{M}}_1}}}{\text{m/se}}{{\text{c}}^2}$$

The correct answer is $\boxed{\frac{{{\text{g}}\left( {{{\text{M}}_1} – {{\text{M}}_2}\sin \alpha } \right)}}{{{{\text{M}}_1} + {{\text{M}}_2}}}{\text{m/se}}{{\text{c}}^2}}$.

Let’s analyze each option to see why it is or is not correct.

Option A: This option is incorrect because it does not take into account the force of friction between the mass $M_2$ and the inclined plane. The force of friction will cause the mass $M_2$ to accelerate down the inclined plane at a rate less than $g$.

Option B: This option is incorrect because it does not take into account the tension in the string. The tension in the string will cause the mass $M_1$ to accelerate upwards at a rate less than $g$.

Option C: This option is correct because it takes into account both the force of friction and the tension in the string. The force of friction will cause the mass $M_2$ to accelerate down the inclined plane at a rate less than $g$, and the tension in the string will cause the mass $M_1$ to accelerate upwards at a rate less than $g$. The net acceleration of the system will be the difference between the acceleration of the mass $M_2$ and the acceleration of the mass $M_1$.

Option D: This option is incorrect because it does not take into account the force of friction. The force of friction will cause the mass $M_2$ to accelerate down the inclined plane at a rate less than $g$.