how much weightage will be given to modern physics and mechanics?

Mechanics is probably the most important topic any engineer should know. This invariably makes it the most important part of the paper. Indeed, the bulk of questions in JEE are either directly mechanics questions or require you to have considerable skill in mechanics, which will help you solve a part of the question.
JEE mechanics talks about some of the most fundamental laws –Newton’s law of motion, universal law of gravitation, the law of conservation of linear momentum and so on. These laws form the framework of physics, and every student is expected to know them properly. Students *must* be able to apply these laws to a wide variety of situations.

Any JEE level mechanics problem can be solved by a combination of one or more of these:

• Newton’s laws
• Work energy theorem
• Law of conservation of energy
• Law of conservation of momentum (linear/angular)
• Equation of continuity
• Bernoulli’s theorem

These are probably the most important chapters for JEE and have applications in other chapters for JEE Physics too.

Also, do not ignore the chapter on elasticity.