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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)