The internal rate of return of a project or IRR is the rate of return which when used to discount all cash-flows associated with the project and determine the net value gives as NPV of 0. If the IRR is a positive value, the project is a profit making one, else a loss is being on the project.
From the information provided about the three projects in the question the IRR can be calculated for each of them. It has to be kept in mind that depreciation does not play a role in calculating the IRR; only the cash inflows and outflows are relevant.
Consider project Alpha, here the initial cash outflow is $240000; there is a cash inflow of $13300 for the next 6 years which is the life of the project. Multiplying 13300 by 6 gives $79800 which is less than the initial investment, it is evident that a loss is being on the project.
If the loss as a percentage is R, the equation 240000 = 13300/(1 + R) + 13300/(1 + R)^2 + 13300/(1 + R)^3+...+13300/(1+R)^6 has to be solved for R. This can be done using a calculator or software like EXCEL. The value obtained for R is -0.2458 or -24.58%. The IRR of project Alpha is -24.58%.
Similarly, the IRR for project Beta is -0.082 or -8.2% and the IRR for project Gamma is -0.1290 or -12.9%
None of the projects is profitable.