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Further Mathematics Further Mathematics Solutions ©Ezy Math Tutoring All Rights Reserved www.ezymathtutoring.com.au Table of Contents CHAPTER 1: Data Analysis 3 Exercise 1: Data Types & Representation 4 Exercise 2: Summary Statistics 10 Exercise 3: Normal Distribution 14 Exercise 4: Box Plots 16 Exercise 5: Correlation 21 CHAPTER 2: Number Patterns 30 Exercise 1: Arithmetic Sequences 31 Exercise 2: Geometric Sequences 33 Exercise 3: Sum to Infinity 35 Exercise 4: Difference Equations 37 CHAPTER 3: Geometry & Trigonometry 40 Exercise 1: Pythagoras’ Theorem 41 Exercise 2: Similarity 43 Exercise 3: Volume & Surface Area 45 Exercise 4: Change of Scale 48 Exercise 5: Trigonometry (I) 50 Exercise 6: Trigonometry (II) 52 CHAPTER 4: Graphs & Relations 55 Exercise 1: Linear Relationships 56 Exercise 2: Simultaneous Equations 64 Exercise 3: Non-linear Relationships 69 Exercise 4: Proportional Relationships 87 Exercise 5: Linear Programming 93 CHAPTER 5: Networks 97 Exercise 1: Representation of Networks 98 Exercise 2: Trees 101 Exercise 3: Paths & Flow 102 ©Ezy Math Tutoring All Rights Reserved www.ezymathtutoring.com.au 1 Exercise 4:Optimisation 105 CHAPTER 6: Matrices 107 Exercise 1: Representation & Operations 108 Exercise 2: Simultaneous Equations 113 Exercise 3: Transition Matrices 115 ©Ezy Math Tutoring All Rights Reserved www.ezymathtutoring.com.au 2 Further Mathematics Data Analysis ©Ezy Math Tutoring All Rights Reserved www.ezymathtutoring.com.au 3 Further Mathematics: Solutions Chapter 1: Data Analysis Exercise 1: Data Types & Representation 1) a) Data b) Sample c) Data d) Sample e) Data 2) a) Quantitative, continuous b) Quantitative, discrete c) Categorical d) Categorical e) Categorical f) Quantitative, discrete 3) 7 6 5 4 3 2 1 0 7 8 9 10 11 12 14 15 16 17 21 22 23 67 ©Ezy Math Tutoring All Rights Reserved www.ezymathtutoring.com.au 4 4) 120 C 100 u m 80 u l % 60 a t 40 i v 20 e 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Weight 5) a) 12.0 to 12.49; 12.5 to 12.99, 13 to 13.49, 13.5 to 13.99, 14 to 14.49, 14.5 to 14.99, 15 to 15.49, 15.5 to 15.99, 16 to 16.49 b) 6 F r 5 e 4 q u 3 e 2 n c 1 y 0 12 - 12.49 12.5 - 13 - 13.49 13.5 - 14 - 14.49 14.5 - 15 - 15.49 15.5 - 16 - 16.49 12.99 13.9 14.99 15.99 Time groups c) 14.4 ©Ezy Math Tutoring All Rights Reserved www.ezymathtutoring.com.au 5 d) 13 – 13.49 6) a) Stem Leaf 2 0 3 3 5 5 6 6 7 3 0 0 1 2 3 4 7 4 1 1 2 2 3 5 3 b) Stem Leaf 6 2 6 6 9 7 0 1 3 3 4 5 6 8 8 0 1 3 8 9 2 9 9 c) Stem Leaf 1 0 2 2 0 1 1 1 2 2 2 4 6 7 3 0 1 2 2 2 5 7 4 0 3 3 5 6 7 9 5 0 6 6 0 ©Ezy Math Tutoring All Rights Reserved www.ezymathtutoring.com.au 6 7) a) Skewed left b) Skewed right c) Skewed right d) Normal 8) 100% 90% 80% 70% 60% Music 50% Clothes 40% Petrol 30% Lunch 20% 10% 0% Monday Tuesday Wednesday ©Ezy Math Tutoring All Rights Reserved www.ezymathtutoring.com.au 7 9) 10) Score range Cumulative frequency Cumulative % 26-30 2 6.25 31-35 2 6,25 36-40 3 9.4 41-45 5 15.6 46-50 8 25 51-55 8 25 56-60 12 37.5 61-65 18 56.25 66-70 23 71.9 71-75 26 81.25 76-80 26 81.25 81-85 28 87.5 86-90 29 90.6 91-95 31 96.9 96-100 32 100 32 students sat the test, 24 passed ©Ezy Math Tutoring All Rights Reserved www.ezymathtutoring.com.au 8 11) Stem Leaf 3 4 8 4 2 3 5 8 9 5 1 3 7 8 6 0 1 3 7 7 1 4 7 9 8 5 Stem Leaf 2 3 9 3 5 9 4 6 7 9 5 2 3 3 9 6 7 7 3 9 8 6 9 1 7 10 1 11 7 12 6 Team A median scores was 57.5 Team B median score was 56 Team A had the least range of scores and was more consistent ©Ezy Math Tutoring All Rights Reserved www.ezymathtutoring.com.au 9 Further Mathematics: Solutions Chapter 1: Data Analysis Exercise 2: Summary Statistics 1) a) Mean = 8 Mode = 7 Median = 7 b) Mean = 16 Mode = 8, 13, 15 and 20 Median = 13 c) Mean = 12.6 Mode = 4 and 16 Median = 15 d) Mean = 8.6 Mode = 1 Median = 8 e) Mean = 9.7 Mode = 1, 3, 5 and 17 Median = 11 2) a) Mean = 4.26 Mode = 7 Median = 5 b) Mean = 23.39 Mode = 25 Median = 23.5 c) Mean = 14 No mode Median = 14 ©Ezy Math Tutoring All Rights Reserved www.ezymathtutoring.com.au 10 d) Mean = 39.82 Mode = 7 Median = 5 3) An outlier biases the mean toward it, has no effect on the mode, and usually no effect on the median (depending on the distribution of scores around the original median) 4) a) Stem Leaf 4 6 7 7 8 5 2 6 7 8 6 0 1 2 7 1 2 7 8 8 0 1 3 6 9 Mean = 65.55 Mode = 47 Median = 61.5 Range = 43 b) Stem Leaf 4 7 8 8 8 9 9 5 0 1 4 6 7 6 2 5 9 7 5 9 8 2 4 8 9 Mean = 62.2 Mode = 48 Median = 56.5 Range = 42 ©Ezy Math Tutoring All Rights Reserved www.ezymathtutoring.com.au 11 c) Stem Leaf 4 7 8 5 0 1 2 3 6 6 6 1 1 9 7 4 6 7 9 8 1 4 8 9 9 0 Mean = 67.1 Mode = 56, 61 Median = 65 Range = 43 5) Mean = 23 Mode = 24 Median = 24 6) The mean will increase slightly, the mode will remain unchanged, and the median may change slightly depending on the distribution of scores around the median 7) 92 8) 75% 9) 76.67% 10) Due to the fact that in the second distribution there are more students contributing to the higher average, therefore the overall score for this group (girls) was higher than in the previous example 11) y = 16, x = 9 ©Ezy Math Tutoring All Rights Reserved www.ezymathtutoring.com.au 12 12) a) 33 b) 49 c) 88 d) 56 13) a) 17 b) 25 c) 67 d) 33 14) A No, since the IQR measures the difference between points within a data set, and the range measures the difference between points at either end of the data set 15) Yes but only in the case where all values of the data set are equal 16) a) 0 b) 1.58 c) 4.74 d) 37.35 ©Ezy Math Tutoring All Rights Reserved www.ezymathtutoring.com.au 13 Further Mathematics: Solutions Chapter 1: Data Analysis Exercise 3: Normal Distribution 1) a) The score is the mean b) The score is one standard deviation above the mean c) The score is two standard deviations below the mean d) The score is more than two standard deviations above the mean 2) The score is one standard deviation above the mean 3) 9.5 4) 10 5) 17 6) 68% 7) Mean = 90, s.d. = 6.5 8) Three 9) NAME SCORE James C Mark E Karen B Janine C Carol C June C Peter D Kevin D Brian C Alan C Bree C ©Ezy Math Tutoring All Rights Reserved www.ezymathtutoring.com.au 14 10) a) 96.4 to 103.6 kg b) 34% c) 8 ©Ezy Math Tutoring All Rights Reserved www.ezymathtutoring.com.au 15 Further Mathematics: Solutions Chapter 1: Data Analysis Exercise 4: Box Plots 1) a) b) The mean will be higher since it is influenced by the outlier c) Mean = 63.22, standard deviation = 12.97 d) The IQR, since the standard deviation is affected by the outlier 2) a) 40 b) 75% c) 86% d) 25% e) The test was easy, since the majority of the class scored well on it, the median was 86%, and three quarters of the class scored 70% or better ©Ezy Math Tutoring All Rights Reserved www.ezymathtutoring.com.au 16 3) a) b) 27 c) 103.5 d) There is a wide spread of scores, indicating that there are some evenly matched teams and some not so evenly matched. The spread may also be due to differing conditions in different parts of the country and over time 4) Hobart and Perth have similar distributions, but over a different set of data. Perth’s range of temperatures is higher and slightly more widespread. Darwin has a small range of temperatures which are higher at all times than at any time in Hobart, and for most times in Perth. ©Ezy Math Tutoring All Rights Reserved www.ezymathtutoring.com.au 17 5) 6) 7) A a) Water World (230) b) Water World ( 0 and 400) c) Water World ©Ezy Math Tutoring All Rights Reserved www.ezymathtutoring.com.au 18 d) Water World e) Great Lake; its distribution of rainfall is less random 8) a) b) IQR = 6, 6 x 1.5 = 9 14 + 9 = 23 8-9 = -1 ©Ezy Math Tutoring All Rights Reserved www.ezymathtutoring.com.au 19 9) a) b) The outliers are 16, 15, and 13 from the set of more education c) The outliers increase the range of the “educated” set d) With outliers, the range of the data for the “educated” group is higher.
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