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Hello Shree, to become a Game Designer, you must get a good knowledge of 2D/3D software as well as pre-production. Usually, in Games, the environment is created in 3DSMax and Character animations in Maya, Only the models are made with fewer polygons and more textures. So choose a programme that teaches you this software. It would not matter whether the programme is called game design programme or not. Rest, to become a level designer you must have a passion for game design. Also, you must have the passion for playing all kinds of games including card games, dice games, sports etc. To do programming of game design you can choose computer Science engineering from colleges such as-
Vishwakarma Institute of Technology
College of Engineering, Pune
MIT - World Peace University
D. Y. Patil College of Engineering
Good Luck.

Computer Science is traditionally more concerned with the theoretical knowledge of computation and (Algorithmic) programming. Thus one typically finds courses in programming, algorithms, numerical analysis and the theory of computation (what can and cannot in principle be computed) in Computer Science departments.
Computer engineering programs largely developed in engineering departments strong in electrical engineering. Computer Engineering focuses on the practical aspects of development and use of computers, and so courses in digital logic design and processor interfacing which build on an engineering student's knowledge of electronics and circuits are typically found in CEN programs.
You should opt CSE. You can do a graduation and can pursue a Game Design and Development Course or Diploma is VFX, 3D Arts, Game Design,etc. There are various institutes that offer diploma course in VFX, 3D, Game Design, etc and you can also find Online Courses on Game Design on Coursera, Edx, Udemy,etc.
You can find steps to become Game Designer on the link below. They have explained in a simple way. Please refer the link below.
https://www.gamedesigning.org/become-a-game-designer/
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Solve at-least last 10 years Previous Papers. It will make you get the idea about the type of Questions being asked in the Exam. Focus more on 2 mark questions. They boost your Score exponentially.
Practice as many Mock-tests as you can (that too as per the Standard pattern of the GATE Exam).
Keep your fundamentals and concepts clear. The Questions asked in the GATE Exam are mostly based on the fundamental concepts. By giving these you will actually know about question pattern and gain experience for giving objective kind of test and also you will be able to measure your standard as per other student so that you can change yourself better to crack the exam.