How does SPICE calculate Drain current of a MOSFET model?
how do we know that SPICE/ADS uses the particular equations for that >particular model?


You invoke a particular model on Spice (I am using Ngapice but the idea is similar in the Spice you use) by passing to it a .model parameter, if you did not do that a default model will be assumed by the Spice simulator (usually level 1). 

The models are the equations you are refereeing to, for example if you passed level=1 to the simulator you will be using the Shichman-Hodges model if you passed level=8 you will be using the BSIM model. off-course The models will vary in complexity, the more sophisticated circuits you build the more complex models you may wish to use to account for parasitics and other circuit behaviors.   

Ngspice manual have a list of models available (see the figure) you may want to read section 11.2 form the manual.