1.1
The number of solutions depends on parameters a,b,c, and is either infinite or zero; by Gaussian elimination it turns out that the multiplier matrix is not of full rank and a condition for parameters a,b,c is found such that there are either no solution or infinitely many of them.

Correctly done Gaussian elimination 3 points
The case of inf solutions 1p, the case of no solutions 1p.

1.2
Correct answer with anything else than the cramers rule 2p
Note: missing the condition that the determinant of the multiplier matrix non-zero -1p.

2.3
Solutions, i.e., partial derivatives of $z$ involving non-numeric value for z were accepted.

2.4. Note: byt the chain rule the solution is d z(x(1),y(1))/dt= dz/dx *dx(1)/dt+ dz/dy dy(1)/dt
where dz/dx and dz/dy are the partial derivates obtained in 2.3.

3.1
Definition of homogeneity +1
Correctly verifying it for the given function +1

3.2 If only some lements in the profit function were correct 1p

3.3 Note: all principal minors were needed, not only leading principal minors (-1p if only leading pm's were used)

4.1-4.2
Correct FOCs +2p. Only one critical point was asked. The function has two and both were ok: (x,y)=(0,1) and (x,y)=(e^{-2},e^2), and both are saddle points. 

4.3
Function is not bounded; log goes to minus infinity when x_2 goes to zero and infinity when x_2 goes to infinity. Hence, there are no global optima.