Sahar is a lecturer of a large class of economics students. Sahar can’t stand being ill and so decides to get a flu vaccination. On the other hand, some of Sahar’s colleagues decide to skip their vaccinations because they are too busy. Do the lecturers’ decisions on whether or not to get vaccinated affect their students and if so how? Describe any market failures that arise from this situation. Describe how the government can address any of these market failures. What are some of the potential downsides to government intervention in this case?

Sahar is a lecturer of a large class of economics students. Sahar can’t stand being ill and so decides to get a flu vaccination. On the other hand, some of Sahar’s colleagues decide to skip their vaccinations because they are too busy. Do the lecturers’ decisions on whether or not to get vaccinated affect their students and if so how? Describe any market failures that arise from this situation. Describe how the government can address any of these market failures. What are some of the potential downsides to government intervention in this case?

The students in a Math class decide to divide themselves into groups of equal size to study multiple congruencesWhen they try groups of 4, 3 are left over; groups of 5, 2 are left; finally they try groups of 7 and discover that if the professor joins in then they can use 7. The professor, reasoning that he never really understood modular arithmetic anyway, thinks it will probably do him some good and agrees to join in. What is the number of students in the class if you are told it is between 150 and 250

1. Which of the following maps define an isomorphism of binary structures? Explain. (a) The function (R, +) → (R, +) given by x → x 2 (b) The function (R>0, ·) → (R>0, ·) given by x → x 2 (c) The function (R>0, +) → (R>0, +) given by x → 2x 2. For each of the following, prove or disprove that it is a group. If it is a group, prove or disprove that it is abelian. (a) (Q +, ·) (b) (R \ {0}, ∗), where a ∗ b = ab 2 3. Assume that φ : (S, ∗) → (S 0 , ∗ 0 ) is an isomorphism of binary structures, and that ∗ is associative. Prove that ∗ 0 is associative. 4. Consider the set H = a −b b a ∈ M2(R) : a 6= 0 or b 6= 0 , with matrix multiplication as the operation. (a) Show that the operation is closed. (b) Define the function φ : H → C ∗ (where C ∗ is the group C \ {0} with multiplication as the operation) by φ : a −b b a 7→ a + bi. Show that φ is an isomorphism of binary structures. (c) Explain briefly why we can now conclude that H is a group. 5. Consider the set V = a 0 0 b ∈ M2(R) : a, b ∈ {1, −1} , with the operation · (matrix multiplication). Note that (V, ·) is a group (you do not need to prove this). Prove that (V, ·) is not isomorphic to U4.

Q 12

LCC has total budgeted production overheads for next year of GHS816,000 and has traditionally absorbed overheads on machine hour basis. It makes two products Product V and Product W

Product V Product W

Direct materials cost per unit GHS20 GHS60

Direct labour cost per unit GHS50 GHS40

Machine time per unit 3 hours 4 hours

Annual production 6000 units 4,000 units

Required:

a) Calculate the product cost for each of the two products on the assumption the firm continues to absorb overhead costs on a machine hour basis

b) The company is considering changing to an activity based costing (ABC) system and has identified the following information:

Product V Product W

Number of setups 18 32

Number of purchase orders 48 112

Overhead cost analysis GHS

Machine related overhead costs 240,000

Setup related overhead costs 280,000

Purchasing related overhead costs 332,000

Total production overheads 816,000

Calculate the unit cost for each of the two products on the assumption that the firm changes to an ABC system, using whatever assumption you consider appropriate.

c) Suggest how ABC analysis could be useful for measuring performance and improving profibability

Tests were carried out on a three-phase 220 V, 50 Hz, 4-pole delta-connected induction motor. The following results were obtained from open circuit (no-load) and short circuit tests.

            Open Circuit (no-load) Test

            Short Circuit (locked rotor) Test

            A dc resistance test indicates that the stator resistance per phase is 1.165 W.

a)        Estimate the machine’s friction and windage loss and determine the parameters of the full equivalent circuit of the machine.

Using an appropriate form of the machine equivalent circuit:

b)         Calculate the losses in the machine and draw a power flow diagram which illustrates the flow of power through the machine when it runs on-load with a slip of 0.06

c)         Determine the starting torque, and also the torque when running with a slip of 0.06.

3a. If the molar absorptivity constant for the red dye solution is 5.56×104 M-1 cm-1 , calculate the molarity of the red dye solution with absorption = 0.900. You may assume L = 1.22cm. Hint: A=εLC

b. Convert the molarity in part a to w/v%. Show your work. Molar mass of FD&C red #3 = 879.86g/mol w/v% is defined as weight (in grams) of solute/100 mL of solution You should remember that w/v% always refers to the grams of materials in 100mL of solution. ppm is defined as mg of solute/ L of solution (1mg = 10-3 g) ppb is defined as µg of solute / L of solution (1µg = 10-6 g)

c. If you want to dilute the red dye solution in part a by 5 times in a single dilution step, explain in two sentences on how one should proceed with the dilution. In addition, what is the final concentration of the diluted solution? Show your work for the numerical part of this question. You are only allowed to use a 5-mL volumetric pipet, a 10-mL volumetric pipet, a 50-mL volumetric flask and/or a 100-mL volumetric flask during the dilution. You may NOT reuse the same pipet during the dilution.

Consider the following three projects:

The cost of capital of the firm is 10%.

Based on the IRR decision rule:

1. Should you invest in Project A?

Below are multiple choice for the IRR calculation

i) 16.55%

ii) 17.61%

iii) 18.31%

iv) 20.22%

v) Cannot calculate

Answer: I should/should not (choose one) invest in Project A.

2. Should you invest in Project B?

Below are multiple choice for the IRR calculation

i) 16.55%

ii) 17.61%

iii) 18.31%

iv) 20.22%

v) Cannot calculate

Answer: I should/should not (choose one) invest in Project B.

3. Should you invest in Project C?

Below are multiple choice for the IRR calculation

i) 16.55%

ii) 17.61%

iii) 18.31%

iv) 20.22%

v) Cannot calculate

Answer: I should/should not (circle one) invest in Project C.

4. Assume that we can only take one of the projects. Which one would we choose?

Answer: You would choose project ____________

An amateur astronomer is researching statistical properties of known stars using a variety of databases. They collect the color index, or B−V index, and distance (in light years) from Earth for 30 stars. The color index of a star is the difference in the light absorption measured from the star using two different light filters (a B and a V filter). This then allows the scientist to know the star's temperature and a negative value means a hot blue star. A light year is the distance light can travel in 1 year, which is approximately 5.9 trillion miles. The data is provided below. Use Excel to calculate the correlation coefficient r between the two data sets, rounding to two decimal places.

B-V index   Distance (ly)
1.1   1380
0.4   556
1.0   771
0.5   304
1.4   532
1.0   751
0.5   267
0.8   229
0.5   552
0.2   896
1.5   1819
0.5   381
0.5   257
1.1   541
0.7   133
0.5   300
0.0   985
0.4   525
1.0   408
1.1   1367
1.07   2848
1.1   128.9
1.12   1766.2
0.64   186.5
0.87   8269.2
0.19   828.9
1.03   153
0.55   223.6
1.39   963.9
0.89   91.7