CSS 2022 Pure Mathematics Paper

Let G be a group and H be a subgroup of index 2 in G. Show that H is normal in G.

Let G be any group, g a fixed element in G. Define f: G -> G by f(x) = gxg^-1, for all x in G. Prove that f is an automorphism of G onto G.

Prove that every finite integral domain is a field.

Let W be the subspace of R^5 spanned by u=(1,2,-1,3,4), v=(2,4,-2,6,8), w=(1,3,2,2,6), x=(1,4,5,1,8), and y=(2,7,3,3,9). Find a subset of the vectors that form a basis of W. Also extend the basis of W to a basis of R^5.

Let T: R^4 -> R^3 be defined by T(x,y,z,w) = (x-y+z+w, 2x-2y+3z+4w, 3x-3y+4z+5w). Find the rank and nullity of T.

Find all possible solutions of the following homogeneous system of equations: x+y+z-w=0, x+2y-2z+w=0, 2x+4y-3z+w=0, 4x+7y-4z+w=0.

Find the limit as x approaches infinity of (1 + 2/x)^x.

Evaluate the integral of x*cos(2x) dx.