1Mediummains10 marks
Mathematics
Let H and K be two subgroups of a group G such that o(H) > √o(G) and o(K) > √o(G). Show that H ∩ K ≠ {e}, where e is the identity element. Here o(H), o(K) and o(G) denote the order of H, K and G respectively.
Let G = {e, x, x², y, yx, yx²} be a non-Abelian group with o(x) = 3 and o(y) = 2. Show that xy = yx² (where e is the identity element of G and o(x), o(y) denote the order of the elements x, y respectively).
Examine whether the series Σ_{n=1}^{∞} (–1)^{n-1}/n is absolutely or conditionally convergent.
Expand f(z) = 1/[(z+1)(z+3)] in a Laurent series valid for 1 < |z| < 3.
How many basic solutions are there for the following system of equations?
2x₁ – x₂ + 3x₃ + x₄ = 6
4x₁ – 2x₂ – x₃ + 2x₄ = 10
Find all of them. Furthermore, find the number of basic solutions, which are feasible/non-feasible/non-degenerate.