MacBook Pro screen goes black just after booting

My girlfriend's MacBook Pro has just developed the problem that just after (or sometimes during the boot process) the screen goes black. The power supply is connected.

I have tried a number of things, reset SMC, reset PRAM, hard reset and tried booting into Recovery Mode and Safe Mode. In Safe Mode, the user login screen comes up but only for about a second then the screen goes black, likewise, in Recovery Mode.

In all cases, holding the power key down for a couple of seconds brings the screen up (whether in 'normal' mode, Recovery Mode or Safe Mode) but it only lasts for about a second then goes black again.

It seems strange that the screen will stay on during most of the boot process (lasting about a minute) but after that, will only stay on for about a second at a time).