A car goes round a one mile circuit at 30mph. At what speed must it travel on a 2nd circuit to achieve?

Infinite.

To do two (one mile) laps at the 60mph average would take two minutes.

The single lap at 30mph has already taken that two minutes, so it's impossible to get the average up to 60mph.

Average speed = total distance / total time

first lap takes 1/30 hours

second lap takes 1/x hours

average takes 2x/60 hours

60 = 2 / ((1/30) + (1/x))

60((1/30) + (1/x)) = 2

2 + 60/x = 2

60/x = 0

no solution possible. ⬅

And that makes sense. Even at huge speeds, you can't make up for the lost time in the first lap.

other answer said 90 mph, but the average of that and 30 is not 60, but:

s = 2 / (1/30+1/90)

s(1/30+1/90) = 2

s(2+1) = 120

3s = 120

s = 40 mph average, not 60

90 mph.