#### Miscellaneous Mathematics

You may use your notes, textbook, homework, and any materials from Canvas. Be sure to write clearly and show all work. For all hypothesis tests: verify the requirements are met, state your null and alternative hypotheses, show your work for finding the test statistic, and state your conclusion. Round your test statistic and critical value to three decimals when appropriate. 1. Abstergo Industries keeps records on all its users: age (in years), ethnicity, weight, political affiliation, and concentration of First Civilization DNA (as a percent). Classify each of those variables as either qualitative or quantitative. If a variable is quantitative, then also state if it is discrete or continuous.

4. Using the digits 0, 2, 3, 5, 6, 7, 8 without repetitions, how many four-digit codes can be formed:

MHF What is the difference between the number of solutions for a trigonometric linear equation and a non-trigonometric linear equation? What type of solution would you expect for each and what techniques would you use to solve each equation? Worksheet Questions MCV Q- Let y = x/x+1 Find the first, second and third derivative (1:3)

Question 1 c) Describe a tool or technique which can be used to analyse and improve the performance of manufacturing systems e.g. Rank-order clustering, Ishikawa diagrams, Value Stream Mapping Root cause analysis, Pareto Analysis etc. Explain the basic principle behind the technique, how it would be applied and the potential benefits/inhibitors to implementation.

3. Leo is a self-employed plumber. One month, Leo's plumbing business had three contracts for \$2500.00, \$7000.00, and \$275.00. The cost of Leo's expenses and materials to complete these contracts was \$7200.00. a) How much did Leo earn an hour based on a 40-hour work week?

1. Convert 723.8 mm to cm.

2. Calculate the volume of a rectangular prism with a width of 5 cm, a length of 7 cm, and a height of 2 cm.

3. Calculate the surface area of a box with a width of 6 mm, a length of 5 mm, and a height of 7 mm.

4. Based on the following chart, convert 84.3 km to mi.

5. In a triangle labelled ∆ABC, ∠C = 90", A = 44", and AC = 27.3 cm. Calculate the length of BC.

6. In a triangle labelled ∆ABC,∠C = 90", AB = 17 mm, and BC = 15 mm. Calculate ∠A.

9. Given the following data: 35 67 56 78 56 67 82 73 67 72 82 67 85, find the: a) mean b) median c) mode

7. In a non-right-angled triangle labelled ∆PQR, ∠P = 124°, ∠R = 32", and p= 37.5 m. Calculate the length of r.

8. In a non-right-angled triangle labelled ∆PQR, p = 52.9 m, q = 10.4 m, and r = 47.6 m. Calculate the size of ∠R it to the nearest degree.

10.Using the formula C = Pi d, calculate the diameter of a circle with a circumference of 12.4 cm.

11.A cereal box is designed to hold a volume of 4096 cubic centimetres of cereal. What dimensions will minimize the cost of producing the box?

12. The average age of a university student was found to be 24 with a standard deviation of 2. What age group is within 2 standard deviations of the mean?

13. Given the following set of numbers, use the measures of central tendency to identify the type of distribution. Defend your answer. 70 67 75 60 67 65

14. Toss a coin 10 times and record how many times you tossed heads. a) What is your experimental probability of tossing heads? b) What is the theoretical probability of tossing heads? c) Compare the two probabilities and determine if you are considered to be "lucky" at tossing heads. d) If you were to toss the coin 100 times, what would you expect the experimental probability of heads to be? Defend your answer.

Q5 (i) Let f: N→ N² be the function given by f(n) = (1, n²). Prove that f is injective. What can you deduce about the relationship between the cardinalities of N and N²? (ii) Let p and q be integers. Prove that if 2^P = 3^q, then p = q = 0. [Hint: consider separately all the possible cases of p and/or q being positive/negative/zero.] (iii) Let g: N² → N be the function given by g(m, n) = 2m3n. Assuming the result from part (ii), or otherwise, prove that g is injective. What can you deduce about the relationship between the cardinalities of N and N²?

Q4 (a) Simplify each of the following complex numbers, expressing your answers in Cartesian form: (i) (7-i)(2+3i) (ii) i / 2+i (b) Using de Moivre's Theorem, and without using a calculator, show that (1 - i)^1066 = -(2^k)i, where k is a positive integer to be stated. (c) Find the cube roots of 4i, expressing your answers in the form z = 2^p e^qi, where p and q are rational numbers to be stated, with -1 <q≤1.

Q3 Let S = R² with addition and multiplication in S defined by (a, b) + (x, y) = (a + x, b+y), (a, b) x (x, y) = (ax, by). Throughout this question you may assume that R is a field. (i) Show that (x, y) is an additive identity element in S if and only if (x, y) = (0,0). (ii) Show that (x, y) is a multiplicative identity element in S if and only if (x, y) = (1, 1). (iii) Verify that the distributive law holds in S. (iv) By considering the element (1,0), or otherwise, show that S is not a field.

Q2 (a) Let U = {-5,5,7) and H: P(U)→→P(U) be the function defined by H(X)=XnN. Determine the image of H. (b) Consider the function g: P ({0,2,3}) → {0,2,3} defined by g(X) = |X|. Prove that g is surjective. (c) Let F: R² → R³ and G: R³ → R² be given by F(x, y) = (x, y,2y). G(x, y, z) = (x, y+z/3). (i) Determine the compositions F. G and G. F. (ii) Is G the inverse of F? Justify your answer. (iii) Show that G is not injective.

Q1 (a) Let X = {cat, } and Y= {0, {1,2},2}. Determine each of the following sets: (i) (Nn(-1,1)) UX (ii) (Y\Q)UX (iii) Y x X