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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 \([-1, 1]^3 \subset \mathbb{R} \) . How many rotations are there in \(\mathbb{R}^3 \) which take C to itself?

A. 6; B. 12 C. 18. **D. 24**

**Discussion:**

Theorem: A finite subgroup of \(SO_3 \) is one of the following groups:

- \(C_k \) : the cyclic group of rotations by multiples of \(\frac {2 \pi } {k} \) about a line, with k arbitrary
- \(D_k \) : the Dihedral group of symmetries of a regular k-gon , with k arbitrary
- \(T \) the tetrahedral group of 12 rotational symmetries of a tetrahedron;
- \(O \) : the octahedral group of 24 rotational symmetries of a cube or an octahedron
- \(I \) : 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/tifr-2013-math-paper-answers-key/