The book is Crime and Punishment by Fedor Dostoevsky . It's a story about a young man who kills a pawn broker and accidentally her sister for being mean and useless. His justifies his actions by reasoning that anyone extraordinary in society who has something special to offer has the moral right to take the life of another if that person is useless to society. Then starts the police investigation where the investigating police officer and a magistrate play a cat and mouse game until he is cornered into confessing. There are many angles to this great classic which makes it one of the greatest books of the 20th century.
Here's a summary in following link :