I don;t understand what you mean by how do you know what to do. I mean it's kind of clear if they give you the CoF and the height in which the particle reached before start.
For me if I see they give you the NEW HEIGHT REACHED, and also if they are on about KE, then I will consider energy.
If energy doesnt work then you shuold use force. That's how I am thinking.
Solution to this question:
GPE = EPE - Workdone against friction
you need to workout EPE by workout the difference between the EPE at the start, becasue the natural length is .9m but it is 1.5m down. i.e lambda .6^2 / 1.8 - lambda .1^2/1.8
then you need to workout workdone against friction which is Reaction force (worked out by resolving force) X CoF X distance moved.
Then sub everything into the energy formula and get the answer.