。
How on earth did she manage to pass that exam, I wonder?
Literal
She [topic-は] on-earth how that exam [object-を] passed [explanatory-のか] [I wonder-かしら].
一体 ('on earth / in the world') is an intensifier specifically for question words — 一体何 ('what on earth'), 一体どうやって ('how on earth'), 一体なぜ ('why on earth'). It carries strong emotional flavor: surprise, irritation, or genuine bafflement. のかしら combines two pieces: のか (the explanatory の + question か, asking for an explanation) and the feminine sentence-final かしら ('I wonder'). The combination is a soft, musing question — half to oneself, half to the listener — and is strongly associated with feminine speech in modern Japanese. パスする is a verbed loanword from English 'pass.'