You are using a net present value profile to compare Projects A and $B,$ which are mutually exclusive. The crossover rate is $12\%.$ At a required return of $9\%,$ Project A has NPV of $$10,000$ and Project B has NPV of $$7,500.$ Which of the following statements is FALSE?

Project $A$ is preferred at a discount rate of $5\%.$

Project B is preferred at a discount rate of $13\%$

Both projects has IRR$=12\%$

Project B must have a higher IRR than Project A