If you don't want to spend MS point's as suggested in the the first solution....
I had tried this with a fully upgraded tier 3 Audi TT, but no joy it was just too slow.
I'm a cheapskate and didn't want to buy a car, so got thru the career and after doing the G-Mac car delivery (last car delivered is the Veyron) I got the Tier 2 Lambo Gallardo LP560-4 on a pink slip and added full PRO upgrades, retried it and did it first attempt.
I used the first shortcut, avoided the 2nd as it made the next bend awkward, then made sure it hit the nitro up the hill after that bend. I always took the 3rd shortcut at the top of the hill as this saves lots of time.