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Part A
1. D
2. B
3. C
4. C
5. D
6. C
7. D
8. A
9. C
10. B
11. D
12. C
13. B
14. B
15. D
16. D
17. B
18. C
19. B
20. D
Let C denote the cube . How many rotations are there in which take C to itself?
A. 6; B. 12 C. 18. D. 24
Discussion:
Theorem: A finite subgroup of is one of the following groups:
 : the cyclic group of rotations by multiples of about a line, with k arbitrary
 : the Dihedral group of symmetries of a regular kgon , with k arbitrary
 the tetrahedral group of 12 rotational symmetries of a tetrahedron;
 : the octahedral group of 24 rotational symmetries of a cube or an octahedron
 : the icosahedral group of 60 rotational symmetries of a dodecahedron or an icosahedron
Part B
21. C
22. A
23.
24. D
25. A
26. B
27. D
28. A
29. A
30. B
(courtesy: Tattwamasi Amrutam)
https://cheenta.com/2013/12/08/tifr2013mathpaperanswerskey/