GATE

There are around 440 colleges for Production Engineering in India, out of which 110 colleges accept the GATE exam scores for giving admission to the M.Tech Production Engineering course. Since the GATE exam is the top national-level entrance exam for PG courses, there is intense competition for seats at top colleges, and hence you need to work extra hard to get excellent marks and rank to grab a seat for the course.

You can get one of the top 10 IITs for 747 marks.

As cutoffs change every year you have a high chance of making to top 10 IITs for 747 marks for CS branch.

You should put 

IIT Bombay, IIT Delhi, IIT Madras, IIT Kanpur, IIT Roorkee as your top targets or preference.

Also you can get IIT Kharagpur.

No, you can't take admission in M.Tech at IIIT Delhi without appearing for the GATE exam. The institute offers this two-year full-time programme at the postgraduate level. Aspirants must meet the basic eligibility of passing graduation in the relevant stream from a recognised university in addition to clearing the GATE exam. For seat allotment, candidates have to register for the CCMT counselling. 

If your son qualifies in GATE, he's eligible for the TIST M.Tech GATE scholarship—this provides a INR 15,000 waiver each semester on tuition fees. This offer applies across all branches and is available to female students and the general category alike. Let me know if you need more details!

There is no such necessity to take coaching to clear the GATE exam for admission to MTech courses; you can get excellent marks and rank through dedicated effort and self-study. Basically, it depends on your learning style, dedication towards clearing the exam, and passion to pursue MTech.

Having said that, coaching can definitely add value because of the structured learning approach followed there, along with expert knowledge and guidance and access to study material that can turn out to be very helpful.