What is the extent to which projected coordinate systems conserve the shape, area and the distance?

[LawekushalsinghRamphul]

The extent to which the PCS conserve the original

 shape

(ii) area and

(iii)distance.

[JimCousins]

This is an extremely vague question. There are 100's of books devoted to the subject of different map projections and their strengths and weaknesses.

Start with this if you need the basics:

What are map projections?—Help | ArcGIS for Desktop 

there are links for more specific information in the help document, as well.

Regards,

Jim