Problem on Tests on mean and variance

Looks pretty tuff. What do you have so far, what part are you stuck on? Maybe someone can nudge you in the right direction.
were you able to derive the LRT in i) ? Seems like low hanging fruit.

The independance thing in iii) sort of recognize, when you have orthogonal projections onto orthogonal subspaces, normal variables are independant. I forget how this is proved but iii) looks like it is in that mold. The Wp thing is asserting that the sums of squares are chis square. This is based on the sampling distribution of quadratic forms. They always have degrees of freedom relatedto the rank of A in xt A x, or something like that, you have to google 'sampling distribution of quadratic form.

Any help?