As at $31$st December, $2022$

Total LP$$s committed capital $=$ $$5,000,000$

The first capital call was made on $1$st January, $2017$ of $$1,000,000,$ and it exited with a return of $$3,000,000$ on $31$st December, $2020$

The second capital call was made on $1$st January, $2019$ of $$3,000,000,$ and it exits with a return of $$6,000,000$ on $31$st December, $2022$

a$.$ Write an equation to solve this investment's internal rate of return $($IRR$).$

b$.$ Calculate the distribution to paid$-$in capital $($DPI$)$ for the investment on $31$st December $2022.$

c$.$ Calculate the total value to paid$-$in capital $($TVPI$)$ for this investment on $31$st December $2022$ if the unrealised value was $$2,500,000$ at the current market valuation.