The exact time is 1 1: 30. According to the time of getting up in the morning, it can be inferred that his alarm clock is ahead of the exact time, and the time of walking to the library should be the same as that of walking home from the library. Then, it is set that the time for her to get to the library is x minutes, which means that 2X should be 1 hour. It can be concluded that his alarm clock was half an hour fast, so she pushed it forward for half an hour and got home.