Problem 1 - Algebra: This section contains six questions, each is worth one marks.

Prof Rogers put on a few pounds this fall, jeopardizing his long-shot chance at competing the pole vault at the 2028 Olympics. He has decided to lose a few pounds in hopes of increasing his chances, however slightly, of making the Canadian Olympic team.

Since the days of his Mc Diet he claims to have learned a lot about exercise and has decided to take his extreme training regime into account in dieting. In particular, he knows that to reach his rather arbitrary goals, his caloric needs for a day are 2400 + 200 for every hour he trains. He also needs to consume 130g of protein per day and 320g of carbohydrates. So he wants to know what to eat and how much to exercise.

His new and shockingly unhealthy diet is composed exclusively of fruit smoothies and protein bars, and he is taking exercise into consideration, so arguably this is a step up from last years McDiet. The nutritional information for these, and a few other items, are given in the table below.

Product	Fat	Protein	Carbohydrates	Calories
Smoothie	6	20	55	500
Protein Bar	11	25	50	400
Mc Double	17	21	34	370
Large Fries	27	6	74	560
Scotch, 1.5 oz	0	0	0	100

Structure Prof Rogers problem as system of linear equations in the form Ax = b and solve it using matrix-based elimination (i.e. Gaussian or Gauss-Jordan).

Note: Any solution is likely to involve non-integer solutions. That is not a problem since one can consume a partial serving, or think of it as averaging over multiple days per week.

Q5 - AQ1: Select the most true statement as it pertains to Prof Rogers diet problem as described above.

1. It requires two variables and two constraints.
2. It requires three variables and three constraints.
3. It requires three variables and two constraints.
4. It requires more variables than constraints.
5. It requires more constraints than variables.
6. One of the parameters is 100
7. None of the above are true.
8. Two or more of the above are true.

Q6 - AQ2: Select the most true statement as it pertains to Prof Rogers diet problem.

1. There is no way for Prof Rogers to meet his goals, since that would require a negative amount of one or more types of food.
2. Prof Rogers will have to eat 4 protein bars on average per day to meet his goals.
3. Prof Rogers has to train 2 hrs. per day to meet his goals if he eats 2 protein bars per day on average.
4. Prof Rogers can meet this goal in multiple ways.
5. Prof Rogers can only meet his goal in one way.
6. None of the above are true.
7. Two or more of the above are true.

Q7 - AQ3: Fill in the blank. Enter the solution for the Prof Rogers diet problem as described above, if there is one. If there is no solution, write N next to each variable, if there are multiple solutions parameterize them.

Q8 - AQ4: Why wouldnt having 4 smoothies and 4 protein bars be part of a practical solution?

1. Actually, it could be.
2. It would require a negative amount of exercise.
3. It would require more than 24 hrs. per day of exercise.
4. It would have too much protein
5. It would have too few calories
6. None of the above are true.
7. Two or more of the above are true.

Q9 - AQ5: If Prof Rogers decides, against all medical advice and common sense, to add two shots of scotch per day to his diet, how would this change the model.

1. We need to add a variable for scotch consumption.
2. We could simply change one of the parameters in the model but nothing else, though we would need to re-solve the model.
3. We would need a new A matrix that would have an additional row and column, the vectors x and b would also gain a row.
4. Two of the above are true.
5. The parametric solution could be used as is since, aside from calories, scotch has no nutritional value.
6. None of the above are true.

Q10 - AQ6: In the one good health-related decision he has made all year, Prof Rogers decides NOT to have any scotch in his diet and to train three hours per day. Select the most true statement as it pertains to Prof Rogers diet problem under these circumstances.

1. It can now be represented as a 2x2 matrix for A.
2. The solution must have 3 smoothies and 5 protein bars.
3. The problem could be solved with matrix inversion.
4. All of the above answers are correct.
5. Two of the answers of A through C are correct.
6. None of the answers above are correct.