Trivia quiz – three music folks seemingly unconnected, and a two times British PM with no mandate

One politics, one music question. Will Howells will get both straight away.

1. Politics

Which British Prime Minister was in Number Ten (as PM) for two full terms but never received the majority mandate of the British electorate?

2. Music

What do the rock band C.C.S, Phil Lynott of Thin Lizzy and tune meister Paul Hardcastle have in common?

Please try answers in the comments. Then I’ll confirm the answers at the same time tomorrow. Thanks for your involvement!