Being an IITK alumnus, I would say that core mechanical (especially Solid Mechanics and Design) in IITK is very good from both placememt and research perspective. So I think you can prefer it over IITB. The only thing which you may not like is even grading system of IITK. It is unlike any other IIT. They have an even grading system which means A=10 , B=8, C=6, D =4 and so on. They don’t have 9, 7, 5, 3 in calculation of credits for any course work which you do. This results in a low CPI in comparison to someone from any other IIT and the consequences of which yiu might have to suffer later when you go for further studies abroad. What I mean is, a person having same knowledge and marks may have a low CPI than the one who is from anybother IIT. So, if you are fully determined for IITK, then go for it because it is one of the best institutes for both placement and reseach in india.
Also in IITK, you have to register for summer semester (for 2 months) separately and the credits you get, will be mentioned in your gradereport. This system has started from last 3 years. This system is unlike any other IIT.