Couple of questions Im having trouble with here folks.

1. Kates preference over two goods can be represented by the utility function:

She faces prices P1 and P2 for the two goods and has an income level of M.

(a) Using the lagrange method, solves Kate's optimal choice problem and determine the demand functions for goods 1 and 2.
(b) How much of the goods 1 and 2 will Kate consume if P1 = 2, P2 = 3 and M = 350?
(c) For any income M, what fraction of income is spent on good 1?
(d) Illustrate the Engel curve for good 1 when P1 = 2.
2. Simons utility function is given by:

Initially, P1 = 1, P2 = 4 and Simons income is M = 2400. Then, one day, the price of good one rises and P1 becomes = 4.
(a) Determine Simons optimally chosen consumption bundles for goods 1 and 2 before AND after the price of good 1 changed.
(b) Find the Slutsky substitution effect and income effect of this price change on good 1 and good 2 seperately. Explain your answer using a labelled diagram.
If anyone could show me step by step solutions to these, I would be grateful.