A tennis pro shop ordered racquets and shirts for a local tournament. The total cost of a racquet and shirt for each player was $250. Not all players wanted to order both items. The final order was placed for 15 racquets and 40 shirts and came to a total of $4750.
TPP7182 TPP7182 Alternative Assessment Semester 1 2021 Total Marks : 100 Examiner: Clare Robinson Moderator : Sue Worsley Special instructions: This is an open book examination. Write your answers on separate paper as there is not sufficient space to show logic on a printout of these questions. It is expected that you use a standard scientific calculator. You must show working to receive full marks for a question. TPP7182 – Mathematics Tertiary Preparation 2 Page 1 of 7 Examination Period - Semester 1, 2021 Please write all answers on provided answer booklet. QUESTION 1 (23 marks) (a) Make y the subject of the equation 5a y x b (4 marks) (b) Explain why do we not need to use the quadratic formula to solve (2 1)( 3) 0x x (3 marks) (c) What are the domain and range of the function 2y x (3 marks) (d) Given 2 4 3 ( 2) ( 2) a x x x x x , what is the value of a? (4 marks) (e) Find the range of values for x for which 3 2 4x and list two possible integer solutions for x. (5 marks) (f) Use indices to solve for n in the following equation 2 108 2 2n n (4 marks) Start the next question on a new page!! QUESTION 2 (16 marks) Consider the functions ( 3)(2 5)y x x and 3y x (a) Solve these two equations simultaneously using algebra. Give your answers to 2 decimal places. (5 marks) (b) For the function ( 3)(2 5)y x x calculate: i. the x intercepts (2 marks) ii. the y intercept (1 mark) iii. the co-ordinates of the turning point. (2 marks) iv. Sketch the graph of the function. Use the grid on the last page of this examination or your own graph paper. (2 marks) (c) On the same axes, plot the line 3y x (2 marks) (d) Use the graph to find approximate solutions to the equation ( 3)(2 5) 3x x x (2 marks) Start the next question on a new page!! TPP7182 – Mathematics Tertiary Preparation 2 Page 2 of 7 Examination Period - Semester 1, 2021 QUESTION 3 (14 marks) (a) Given 2 2 A 4 3 1 1 2 B 2 1 1 , find, if possible (i) 2A (3 marks) (ii) A+B (1 mark) (iii) B 2 (1 mark) (b) A tennis pro shop ordered racquets and shirts for a local tournament. The total cost of a racquet and shirt for each player was $250. Not all players wanted to order both items. The final order was placed for 15 racquets and 40 shirts and came to a total of $4750. (i) Use this information to form two equations that could be used to determine the cost of a racquet (R) and the cost of a shirt (S). (2 marks) (ii) Convert the equations from part (i) to a matrix equation. (1 mark) (iii) Show that 40 11 15 125 is the inverse of 1 1 15 40 (2 marks) (iv) Use matrix methods to solve the two equations formed in (i). (3 marks) (v) Verify that the solutions found in (iv) are correct. (1 mark) Start the next question on a new page!! TPP7182 – Mathematics Tertiary Preparation 2 Page 3 of 7 Examination Period - Semester 1, 2021 QUESTION 4 (8 marks) (a) Given the following equation: 2log log18 log2x , a student’s solution is shown below. 2 2 2log log18 log 2 log log16 16 4 x x x x Explain what error(s) has the student made. (3 marks) (b) Solve for x in the following equations. (i) 125log 3x (2 marks) (ii) 2 500x (3 marks) Start the next question on a new page!! QUESTION 5 (10 marks) An epidemic occurs in a city. On the day that authorities realize that this is an epidemic, hospitals record having 18 patients with this disease. By the following day, they record a total of 24 patients. The number of affected people is predicted to increase according to the equation 0 ktN N e where ?0 is the initial number affected at the time the authorities were alerted, N is the number affected after t days, and k is a constant. (a) Find the value of k (to 2 decimal places). What is the meaning of this value of k in the context of this question? (4 marks) (b) Find the predicted number of days that it will take for the number of affected people to reach 1 million if the disease is not controlled. (3 marks) (c) Find the predicted average rate of increase per day of those affected for the 30 days following the initial alarm. (3 marks) Start the next question on a new page!! TPP7182 – Mathematics Tertiary Preparation 2 Page 4 of 7 Examination Period - Semester 1, 2021 QUESTION 6 (11 marks) A university student using a residential flat is required to pay $600 per month for lodging plus $5 for every meal eaten in the University dining hall. Assume that the maximum number of meals that a student can eat in the dining hall in a month is 93. (a) From the above information, express the monthly account (C) as a function of the number of meals eaten (M). (2 marks) (b) Explain why this relationship is a function. (1 mark) (c) What are the domain and range of this function? (3 marks) (d) What is the inverse of the above function? Give your answer in function notation. (3 marks) (e) What are the domain and range of the inverse of the function? (1 mark) (f) What information would this inverse calculate? (1 mark) Start the next question on a new page!! TPP7182 – Mathematics Tertiary Preparation 2 Page 5 of 7 Examination Period - Semester 1, 2021 QUESTION 7 (7 marks) Consider the cosine function shown below, in the domain0 360x (a) What is the period and the amplitude of the function? (2 marks) (b) What is the frequency of cycles over this domain? (1 mark) (c) Given that the above graph is of the function 3cos2y x , link the answers to (a) and (b) to this equation. (2 marks) (d) If another cosine graph had 3 cycles over the same domain, and an amplitude of 4, what would the equation be? (2 marks) Start the next question on a new page!! TPP7182 – Mathematics Tertiary Preparation 2 Page 6 of 7 Examination Period - Semester 1, 2021 QUESTION 8 (11 marks) (a) Given tan x = –2, find the value(s) of x in the domain 0 360x (3 marks) (b) Given 5 sin 13 and 12 cos 13 , find tan as a fraction, without first finding θ. (2 marks) (c) Using the sine rule or cosine rule (or both), find the length of the side BC and the angles B and C in the triangle below. (6 marks) END OF EXAMINATION TPP7182 – Mathematics Tertiary Preparation 2 Page 7 of 7 Examination Period - Semester 1, 2021 Graph paper for Question 2