P. D. James 
Unsuitable Job for a Woman 

Handsome Cambridge dropout Mark Callender died hanging by the neck with a faint trace of lipstick on his upper lip. When the official verdict is suicide, his wealthy father hires fledgling private investigator Cordelia Gray to find out what led him to self-destruction. What she discovers instead is a twisting trail of secrets and sins and the strong scent of murder.From the Paperback edition.