TOÁN TIẾNG ANH CHO HỌC SINH LỚP 10

Toán nâng cao dành cho học sinh lớp 10 bằng tiếng Anh. Lưu ý: Trong tài liệu có ghi Grade 10 nhưng thực ra là kiến thức cơ bản THPT, bạn có thể xem và kiểm chứng. Lưu ý: Đây là toán tiếng anh không phải toán tiếng Việt

Everything Maths Grade 10 Mathematics Version – CAPS by Siyavula and volunteers Copyright notice Your freedom to legally copy this book You are allowed and encouraged to freely copy this book You can photocopy, print and distribute it as often as you like You can download it onto your mobile phone, iPad, PC or flash drive You can burn it to CD, e-mail it around or upload it to your website The only restriction is that you have to keep this book, its cover and short-codes unchanged For more information about the Creative Commons Attribution-NoDerivs 3.0 Unported (CC BY-ND 3.0) license see http://creativecommons.org/licenses/by-nd/3.0/ Authors List This book is based upon the original Free High School Science Text which was entirely written by volunteer academics, educators and industry professionals Their vision was to see a curriculum aligned set of mathematics and physical science textbooks which are freely available to anybody and exist under an open copyright license Siyavula core team Neels van der Westhuizen; Alison Jenkin; Leonard Gumani Mudau; Marina van Zyl; Helen Robertson; Carl Scheffler; Nicola du Toit; Josephine Mamaroke Phatlane; William Buthane Chauke; Thomas Masango Original Free High School Science Texts core team Mark Horner; Samuel Halliday; Sarah Blyth; Rory Adams; Spencer Wheaton Original Free High School Science Texts editors Jaynie Padayachee; Joanne Boulle; Diana Mulcahy; Annette Nell; René Toerien; Donovan Whitfield Siyavula and Free High School Science Texts contributors Sarah Abel; Dr Rory Adams; Andrea Africa; Matthew Amundsen; Ben Anhalt; Prashant Arora; Amos Baloyi; Bongani Baloyi; Raymond Barbour; Caro-Joy Barendse; Richard Baxter; Tara Beckerling; Tim van Beek; Jennifer de Beyer; Dr Sarah Blyth; Sebastian Bodenstein; Martin Bongers; Gareth Boxall; Stephan Brandt; Hannes Breytenbach; Alex Briell; Wilbur Britz; Graeme Broster; Craig Brown; Deanne de Bude; Richard Burge; Bianca Böhmer; George Calder-Potts; Eleanor Cameron; Richard Case; Sithembile Cele; Alice Chang; Richard Cheng; Fanny Cherblanc; Dr Christine Chung; Brett Cocks; Stefaan Conradie; Rocco Coppejans; Tim Craib; Andrew Craig; Tim Crombie; Dan Crytser; Dr Anne Dabrowski; Laura Daniels; Gareth Davies; Sean Dobbs; Buhle Donga; William Donkin; Esmi Dreyer; Matthew Duddy; Fernando Durrell; Dr Dan Dwyer; Frans van Eeden; Alex Ellis; Tom Ellis; Andrew Fisher; Giovanni Franzoni; Ingrid von Glehn; Tamara von Glehn; Lindsay Glesener; Kevin Godby; Dr Vanessa Godfrey; Terence Goldberg; Dr Johan Gonzalez; Saaligha Gool; Hemant Gopal; Dr Stephanie Gould; Umeshree Govender; Heather Gray; Lynn Greeff; Carine Grobbelaar; Dr Tom Gutierrez; Brooke Haag; Kate Hadley; Alex Hall; Dr Sam Halliday; Asheena Hanuman; Dr Melanie Dymond Harper; Dr Nicholas Harrison; Neil Hart; Nicholas Hatcher; Jason Hayden; Laura Hayward; Dr William P Heal; Pierre van Heerden; Dr Fritha Hennessy; Shaun Hewitson; Millie Hilgart; Grant Hillebrand; Nick Hobbs; Chris Holdsworth; Dr Benne Holwerda; Dr Mark Horner; Robert Hovden; Mfandaidza Hove; Jennifer Hsieh; Laura Huss; Rowan Jelley; Grant Jelley; Clare Johnson; Luke Jordan; Tana Joseph; Dr Fabian Jutz; Brian Kamanzi; Dr Lutz Kampmann; Simon Katende; Natalia Kavalenia; Nothando Khumalo; Paul Kim; Dr Jennifer Klay; Lara Kruger; Sihle Kubheka; Andrew Kubik; Dr Jannie Leach; Nkoana Lebaka; Dr Marco van Leeuwen; Dr Tom Leinster; Henry Liu; Christopher Loetscher; Mike Loseby; Amandla Mabona; Malothe Mabutho; Stuart Macdonald; Dr Anton Machacek; Tshepo Madisha; Batsirai Magunje; Dr Komal Maheshwari; Michael Malahe; Masoabi Malunga; Kosma von Maltitz; Masilo Mapaila; Bryony Martin; Nicole Masureik; John Mathew; Dr Will Matthews; Chiedza Matuso; JoEllen McBride; Nikolai Meures; Riana Meyer; Filippo Miatto; Jenny Miller; Abdul Mirza; Mapholo Modise; Carla Moerdyk; Tshwarelo Mohlala; Relebohile Molaoa; Marasi Monyau; Asogan Moodaly; Jothi Mood- ley; Robert Moon; Calvin Moore; Bhavani Morarjee; Kholofelo Moyaba; Nina Gitau Muchunu; Kate Murphy; Emmanuel Musonza; Tom Mutabazi; David Myburgh; Kamie Naidu; Nolene Naidu; Gokul Nair; Vafa Naraghi; Bridget Nash; Tyrone Negus; Huw Newton-Hill; Buntu Ngcebetsha; Dr Markus Oldenburg; Thomas O’Donnell; Dr Jaynie Padayachee; Poveshen Padayachee; Masimba Paradza; Dave Pawson; Justin Pead; Nicolette Pekeur; Sirika Pillay; Jacques Plaut; Barry Povey; Andrea Prinsloo; Joseph Raimondo; Sanya Rajani; Prof Sergey Rakityansky; Alastair Ramlakan; Dr Matina J Rassias; Dr Jocelyn Read; Jonathan Reader; Jane Reddick; Dr Matthew Reece; Razvan Remsing; Laura Richter; Max Richter; Sean Riddle; Dr David Roberts; Christopher Roberts; Helen Robertson; Evan Robinson; Raoul Rontsch; Dr Andrew Rose; Katie Ross; Jeanne-Marié Roux; Mark Roux; Bianca Ruddy; Nitin Rughoonauth; Katie Russell; Steven Sam; Dr Carl Scheffler; Cho Hee Shrader; Nathaniel Schwartz; Duncan Scott; Helen Seals; Relebohile Sefako; Sandra Serumaga-Zake; Paul Shangase; Cameron Sharp; Ian Sherratt; Dr James Short; Roger Sieloff; Brandon Sim; Bonga Skozana; Clare Slotow; Bradley Smith; Greg Solomon; Nicholas Spaull; Dr Andrew Stacey; Dr Jim Stasheff; Mike Stay; Mike Stringer; Masixole Swartbooi; Tshenolo Tau; Tim Teatro; Ben Thompson; Shen Tian; Xolani Timbile; Nicola du Toit; Robert Torregrosa; Jimmy Tseng; Pieter Vergeer; Rizmari Versfeld; Mfundo Vezi; Mpilonhle Vilakazi; Mia de Vos; Helen Waugh; Leandra Webb; Dr Dawn Webber; Michelle Wen; Neels van der Westhuizen; Dr Alexander Wetzler; Dr Spencer Wheaton; Vivian White; Dr Gerald Wigger; Harry Wiggins; Heather Williams; Wendy Williams; Julie Wilson; Timothy Wilson; Andrew Wood; Emma Wormauld; Dr Sahal Yacoob; Jean Youssef; Ewald Zietsman; Marina van Zyl iv Everything Maths Mathematics is commonly thought of as being about numbers but mathematics is actually a language! Mathematics is the language that nature speaks to us in As we learn to understand and speak this language, we can discover many of nature’s secrets Just as understanding someone’s language is necessary to learn more about them, mathematics is required to learn about all aspects of the world – whether it is physical sciences, life sciences or even finance and economics The great writers and poets of the world have the ability to draw on words and put them together in ways that can tell beautiful or inspiring stories In a similar way, one can draw on mathematics to explain and create new things Many of the modern technologies that have enriched our lives are greatly dependent on mathematics DVDs, Google searches, bank cards with PIN numbers are just some examples And just as words were not created specifically to tell a story but their existence enabled stories to be told, so the mathematics used to create these technologies was not developed for its own sake, but was available to be drawn on when the time for its application was right There is in fact not an area of life that is not affected by mathematics Many of the most sought after careers depend on the use of mathematics Civil engineers use mathematics to determine how to best design new structures; economists use mathematics to describe and predict how the economy will react to certain changes; investors use mathematics to price certain types of shares or calculate how risky particular investments are; software developers use mathematics for many of the algorithms (such as Google searches and data security) that make programmes useful But, even in our daily lives mathematics is everywhere – in our use of distance, time and money Mathematics is even present in art, design and music as it informs proportions and musical tones The greater our ability to understand mathematics, the greater our ability to appreciate beauty and everything in nature Far from being just a cold and abstract discipline, mathematics embodies logic, symmetry, harmony and technological progress More than any other language, mathematics is everywhere and universal in its application See introductory video by Dr Mark Horner: VMiwd at www.everythingmaths.co.za More than a regular textbook Everything Maths is not just a Mathematics textbook It has everything you expect from your regular printed school textbook, but comes with a whole lot more For a start, you can download or read it on-line on your mobile phone, computer or iPad, which means you have the convenience of accessing it wherever you are We know that some things are hard to explain in words That is why every chapter comes with video lessons and explanations which help bring the ideas and concepts to life Summary presentations at the end of every chapter offer an overview of the content covered, with key points highlighted for easy revision All the exercises inside the book link to a service where you can get more practice, see the full solution or test your skills level on mobile and PC We are interested in what you think, wonder about or struggle with as you read through the book and attempt the exercises That is why we made it possible for you to use your mobile phone or computer to digitally pin your question to a page and see what questions and answers other readers pinned up Everything Maths on your mobile or PC You can have this textbook at hand wherever you are – whether at home, on the the train or at school Just browse to the on-line version of Everything Maths on your mobile phone, tablet or computer To read it off-line you can download a PDF or e-book version To read or download it, go to www.everythingmaths.co.za on your phone or computer Using the icons and short-codes Inside the book you will find these icons to help you spot where videos, presentations, practice tools and more help exist The short-codes next to the icons allow you to navigate directly to the resources on-line without having to search for them (A123) Go directly to a section (V123) Video, simulation or presentation (P123) Practice and test your skills (Q123) Ask for help or find an answer To watch the videos on-line, practise your skills or post a question, go to the Everything Maths website at www.everythingmaths.co.za on your mobile or PC and enter the short-code in the navigation box Video lessons Look out for the video icons inside the book These will take you to video lessons that help bring the ideas and concepts on the page to life Get extra insight, detailed explanations and worked examples See the concepts in action and hear real people talk about how they use maths and science in their work See video explanation (Video: V123) Video exercises Wherever there are exercises in the book you will see icons and short-codes for video solutions, practice and help These short-codes will take you to video solutions of select exercises to show you step-by-step how to solve such problems See video exercise (Video: V123) You can get these videos by: • viewing them on-line on your mobile or computer • downloading the videos for off-line viewing on your phone or computer • ordering a DVD to play on your TV or computer • downloading them off-line over Bluetooth or Wi-Fi from select outlets To view, download, or for more information, visit the Everything Maths website on your phone or computer at www.everythingmaths.co.za Practice and test your skills One of the best ways to prepare for your tests and exams is to practice answering the same kind of questions you will be tested on At every set of exercises you will see a practice icon and short-code This on-line practice for mobile and PC will keep track of your performance and progress, give you feedback on areas which require more attention and suggest which sections or videos to look at See more practice (QM123) To practice and test your skills: Go to www.everythingmaths.co.za on your mobile phone or PC and enter the short-code CHAPTER 13 EXERCISE SOLUTIONS (5 − a)(x + 2y) (a − x)(a − 2) (y + 2)(5x − 3) (−a + b)(a + 1) Exercise 1-7 (a) (b) (c) (d) (e) (f) (a) (b) (c) (d) (e) (f) (a) (b) (c) (d) (a) (b) (c) (d) (x + 5)(x + 3) (x + 6)(x + 4) (x + 8)(x + 1) (x + 7)(x + 2) (x + 12)(x + 3) (x + 6)(x + 6) (x + 5)(x − 3) (x + 3)(x − 1) (x + 4)(x − 2) (x + 5)(x − 4) (x − 5)(x + 4) 2(x + 10)(x + 1) (3x + 1)(x + 6) (6x + 1)(x + 1) (6x + 1)(2x + 1) (2x + 1)(4x + 1) (3x + 1)(x + 6) (7x + 1)(x − 1) (4x − 1)(2x − 1) (6x + 3)(x − 3) Exercise 1-9 a5 a+5 a 3a a+3 a+2 a(3b+1) b 4xy p(y−2) 3y 10 3(a+3) 98 (a−5) 11 4a 6(a+5)2 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 Exercise 1-8 10 (x + 2)(x2 − 2x + 4) (3 − m)(9 + 3m + m2 ) 2(x − y)(x2 + xy + y ) 3(k + 3q)(k2 − 3kq + 9q ) (4t − 1)(16t2 + 4t + 1) (8x − 1)(8x + 1) (5x + 1)(25x2 − 5x + 1) No solution z(1 − 5z)(1 + 5z + 25z ) (2m2 + n3 )(4m4 − 2m2 n3 + n6 ) 11 (p5 − 12 y )(p1 + 12 p5 y + y 12 [1 − x + y][1 + x − y + x2 − 2xy + y ] 440 30 10 (3x+4)2 96p2 2a 3q 5(a+b) 24b af 2z+4y+3x xyz 11 12 13 2(k−1) (k2 +2)(k+2) 5t+7 6q (5p−2) (p−2)2 (p+2) −xy x2 −y m+1 m2 −mn+n2 f h3 −f 2x−1 x2 −3x−1 − (x−1) (x2 +x+1) 3(p2 −pq+q ) p2 a2 +4b−4b2 (a−2b2 )(a+2b) End of chapter exercises (a) (b) (c) (d) (a) (b) (c) (d) Rational Irrational Irrational Irrational 500 59 100 221 18 78 90 (a) 0,5 (b) 1,00 (c) 0,11 14 (d) 1,00 (a) 3,142 (b) 1,618 (c) 1,414 (d) 2,718 571 (a) 3,142 1500 809 (b) 1,618 500 (c) 1,414 707 500 359 (d) 2,718 1500 (a) and (b) and (c) and (d) and (e) and (f) and (g) and (h) and and 3 and (a) (a − 3)(a + 3) (b) (m + 6)(m − 6) (c) (3b − 9)(3b + 9) (d) (4b + 5a)(4b − 5a) (e) (m + 13 )(m − 13 ) (f) 5(1 − ab3 )(1 + ab3 ) (g) b(4a2 − 9)(2a + 3)(2a − 3) (h) (a − 5)(a − 5) (i) (4b + 7)(4b + 7) (j) (2a − 6b)(a − 3b) (k) −4b2 (6b3 − 1)(6b3 − 1) (l) (2−x)(2+x)(4+x2 ) (m) 7(x − 2)(x + y) (n) (y − 10)(y + 3) (o) (1 − x)2 (1 + x) (p) (p − 1)(−3p − 2) (q) (x+y)(1−x2 +xy − y2 ) (r) x(x−2)+(1+y)(1− y)(1 + y ) (s) (x − 1)(x2 + x + 1)(4b − x) (t) 3(p − 31 )(p2 + p3 + 19 ) (u) (2x2 − 5y )(4x4 + 10x2 y + 25y ) (v) (−p)(12+18p+7p2 ) (a) −8a + (b) 125a3 − 64b3 Focus Area: Mathematics CHAPTER 13 EXERCISE SOLUTIONS (c) 16m4 − 81 (d) a2 + 4ab + 4b2 − c2 (e) p2 − 2pq + q 2 (f) 12−x 6x −14 (g) (a+7)(a−7) 32x3 +x+2 2x3 4a−1 (2a+1)(2a−1)(a−1) x(x+1) x2 +x+6 (h) (i) (j) 15 (3x − 4)(x + 2) 16 17 18 19 5x 8,85 a2 + 2ab + 4b2 9x2 − 3x + Equations and inequalities EMAEW Exercise 2-1 −7 Exercise 2-3 29 − 16 30 −3 12 11 Exercise 2-2 117 20 15 −2 10 9 100 26 −3 10 11 12 13 17 12 14 20 15 26 16 or or −6 or − 12 or or −3 8 or or 11 17 18 − 36 11 11 or 13 √ √ 14 18 or − 18 24 −7 25 26 − 17 13 27 −12,5 28 − 17 Focus Area: Mathematics y = − 23 x + 2 15 −1 or 7 10 −3 −4 (b) x = � and y = y y = −x + −2 −1 −1 19 or −5 38 20 or x y =x−2 −2 −3 −4 (c) x = and y = y 10 y = 32 x or − 12 18 −2 or � 16 or − 11 17 −2 20 No solution 23 −6 y = − 21 x + 12 or 22 24 y 19 21 y = −3 and x = −7 y = and x = x = −1 and y = −1 a = 13 and b = 13 x = 17 and y = − 14 x = and y = −4 x −4 10 2 (a) (b) (c) (d) (e) (a) −4 −1 −2 or y = 14 x + � −2 −1 −1 x −2 (d) x = and y = 25 441 CHAPTER 13 EXERCISE SOLUTIONS y y = 12 x + � x −4 −3 −2 −1 −1 −3 −4 (e) x = and y = y y= −2 � x −4 −3 −2 −1 −1 4 r = √ h = h = A−2πr 2πr λ = D ft m = vw w−v 12 r = ± 13 C = � 160 R2 − A π − 59 F 14 r = sqrt[3] 3V 4π Exercise 2-4 t = hrs L = 18 and S = w = 28, l = 45 and d = 53 −34 30◦ and 60◦ l = 16 cm and b = cm x = or x = −3 l = cm and w = cm 19 50 and 11 years old Murunwa is Tshamano is 35 E gh+ V2 x = −a or x = −b √ 10 b = ± c2 − a2 y = −2x − −4 10 11 12 13 V πh Eλ C 11 u = −2 −3 x = 2b2 2b−1 −2 pV RT y = −2x + End of chapter exercises n = and Exercise 2-6 x < −2 x < −5 4 x > a = 1 442 No solution −2 10 x = or x = y = or y = −3 x = or x = −9 x≥2 x < −1 x > 588 10 a ≤ − 75 −3 ≤ k < x = or x = I = VP E c2 (c) t = v−u a (d) t = uv u+v (e) C = 59 (F − 32) −1 ≤ x < (f) (a) (b) (c) (d) (a) (b) (c) (d) (e) −1 < x ≤ x > x ≥ 11 y < 13 −20 < x < Exponentials Exercise 3-1 12 −4 ≤ y ≤ 2(s−ut) t2 −8 (b) m = x ≥ −3 10 −12 ≤ x ≤ Exercise 2-5 (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n) (o) (p) (q) (r) (a) y = mx + c x = and y = x = and y = x = and y = −2 No solution x = 120 13 Ruler: R 5; pen: R km km and 12 km R 20,00 EMAEX 16 36 x3t+3 40 32a+3 27 Focus Area: Mathematics CHAPTER 13 EXERCISE SOLUTIONS (f) − 12 (i) 10 8t12 (h) (k) 4x+3 11 32n+6 (i) 12 (j) a2x (g) p6 27 (j) 8x6a y 3b (l) 22 or (m) −27 (k) (l) Exercise 3-2 (n) −3 8x −3 12 (m) −3 3t2 (n) (o) (p) (q) 14 (o) −1 or (r) 22p − 2p + (a) −3 (b) 3p End of chapter exercises Exercise 3-3 (d) 864p (b) (e) 27m (c) −7 (f) 3x5 (g) 625 (e) (h) 27 (d) (d) (g) +k (a) − 52 (f) 16 (b) 55x+y+3z (c) (e) −1 (a) 2t3 (c) bk 1 27 16 (h) −1 (i) or 16 t (j) (k) 81 (l) 81 Number patterns EMAEY Exercise 4-1 (a) (b) (c) (a) (b) (c) (a) 35; 45; 55 7; 12; 17 21; 18; 15 −1 −9 and −5 Tn = 3n − 1; T10 = 29; T50 = 149; T100 = 299 (b) Tn = 4n − 4; T10 = Focus Area: Mathematics 36; T50 = 196; T100 = 396 (c) Tn = − 3n; T10 = −25; T50 = −145; T100 = −295 End of chapter exercises (a) 49 (b) −10 (c) 18,9 (a) (b) (c) (d) 111 (a) (b) (c) (d) 10 77 (c) Tn Tn Tn Tn = 4n − = 3n − = 4n + = n3 Tn = 3n + 76 + (10x + y) = 10(x + 1) + (y − 1) 443 CHAPTER 13 EXERCISE SOLUTIONS Functions EMAEZ Exercise 5-1 y (0; 25 ) (a) {x : x ∈ R, x ≤ 7} (b) {y : y ∈ R, −13 ≤ y < 4} (c) {z : z ∈ R, z > 35} (d) {t : t ∈ R, 34 ≤ t < 21} (e) {p : p ∈ R, − 12 ≤ p ≤ 12 } (f) {m : m ∈ R, m > √ − 3} (a) (−∞; 6] (b) (−5; 5) (c) ( 15 ; ∞) (d) [21; 41) � y P (1; 2) ( ; 0) (5; 0) � � � x (−4; 1) (0; −1) � y x h(x) (1; 6) (0; 3) � g(x) � (3; 2) (1; 0) x � � � (−3; 0) Exercise 5-5 f (x) y y= 1x y = 2x Exercise 5-3 � (a) Asymptote x (b) y = 12 (c) (0; 1) (0; 4) (a) (b) (c) (d) (2; 0) x � Exercise 5-2 x (−2; 0) 444 +3 y = −x2 − y= x � y (a) (0; 1) and (−1; 0); increasing (b) (0; −1) and (1; 0); increasing (c) (0; −1) and ( 12 ; 0); increasing (d) (0; 1) and ( 13 ; 0); decreasing (e) (0; 2) and (−3; 0); increasing (f) (0; 3); horizontal line (g) (0; 0); increasing (h) (0; 3) and (2; 0); decreasing (a) a(x) = − 34 x + (b) b(x) = 32 x − (c) p(x) = (d) d(x) = − 34 x x � � y= a = and p = −9 b = −1 and q = 23 x ≤ −4 or x ≥ x≥0 Exercise 5-4 (a) (b) (c) (d) (e) (f) f (x) = 3x h(x) = −3x Range: (−∞; 0) g(x) = 3−x j(x) = 2.3x k(x) = 3x − Exercise 5-6 y y � � x � y = − x6 (a) (a) (b) (c) (d) (e) Yes y = −24 Decrease y = and x = (−3; 2) � y 360◦ x 180◦ � (b) 180◦ 360◦ x y � 180◦ 360◦ x (c) Focus Area: Mathematics CHAPTER 13 EXERCISE SOLUTIONS y y y 360◦ x 180◦ −2 −8 −6 −4 −2 −2 � −4 y −2 (c) 180◦ 360◦ � � x y 360◦ x 180◦ −3 � (d) (e) (a) (b) (c) (d) (e) (f) (g) (a) (b) (c) (d) (e) 10 y (f) (a) y = cos θ (b) y = sin θ + 14 12 10 f (x) 11 −2 (c) (b) x −3 −2 −1 −1 (a) 1 (a) (b) (−3; 12) and (2; 2) (c) i x ∈ √ (−∞; 3) ∪ √ ( 3; ∞) ii x ∈ (3; ∞) iii x ∈ [−3; 2] (d) y = −2x2 + 1,6 units (a) x+y = 15; y = x+3 y x −2 −4 End of chapter exercises g(x) 12 18 15 −3 y =x+3 (a) −8−6−4−2 −3 2 −2 y (a) (a) 13 k(x) (b) y = x2 + (c) Domain h: (−∞; ∞) Range h: [−4; ∞) 180◦ 360◦ x 180◦ 360◦ x y x (b) (c) Focus Area: Mathematics (c) � −2 � y −4 −5 � −3 y −2 � (b) −4 −4 −3 −2 −1 −1 −3 −4 −2 y (c) x −3 −2 −1 −1 −9 (b) (c) x = and y = x 1 10 12 14 16 18 −6 h(x) x 2 y = 3x y = −2x2 + y = −3 x y =x+2 y = sin x + y = 2.2x + y = −tan x − M (0; 1), N (0; −1) M N = units P Q = unit y = 2−x Range y = 2x : (0; ∞); Range y = −2x : (−∞; 0) q=1 units y = −4x y = 4x + x=0 Range f (x): (0; ∞) Range g(x): (−∞; 1] y = −x + 15 −3 360◦ x � y (b) 180◦ y 360◦ 12 y −3 −2 −1 −1 (a) (b) (c) (d) (e) (f) y −2 180◦ F F T F F T −4 � −8 (a) (b) (c) (d) (e) (f) � x x −6 (e) (b) � (d) (a) � � 180◦ 14 360◦ x −2 360◦ x 180◦ � � (a) f (180◦ ) = (b) g(180◦ ) = −2 445 CHAPTER 13 EXERCISE SOLUTIONS (c) g(270◦ )−f (270◦ ) = (d) Domain: [0◦ ; 360◦ ] Range: [−2; 0] (e) Amplitude = 2; Period = 360◦ 15 √ √ (a) A( 8; 8) and √ √ B(− 8; − 8) √ (b) CD = units (c) AB = units 16 17 (d) EF = units (a) A(−1; 0), B(1; 0) and C(0; 3) 90◦ ; 270◦ } (b) Amplitude = (c) i {0◦ ; 180◦ ; 360◦ } ∪ ii (0◦ ; 90◦ ) ◦ ◦ (270 ; 360 ) iii {θ : 90◦ < θ < 270◦ , θ �= 180◦ } ∪ iv (0◦ ; 90◦ ) (270◦ ; 360◦ ) (b) The two graphs intersect in the fourth quadrant (c) D(2; −9) (d) y = −6x + (a) Domain: {θ : 0◦ ≤ θ ≤ 360◦ , θ �= Finance and growth Exercise 6-1 R 025 (a) R 324 (b) R 937,50 19 years 1,25% p.a EMAFA Exercise 6-3 R 044,69 8,45% p.a R 59 345,13 (a) (b) (c) (d) R 825 R 743 R 197,63 R 418 (a) R 12 962,50 (b) R 462,50 (c) R 360,07 (a) R 400 (b) R 251,97 Trigonometry 446 R 840 (a) R 534,25 (b) R 520 Bank B Exercise 6-4 Exercise 6-2 End of chapter exercises R 200 R 174,77 R 200 R 38,64 (a) Simple interest (b) Compound interest (a) R 205 (b) R 286,52 (c) R 128 553 babies Exercise 6-5 (a) (b) (c) (a) (b) R 400 R 200 R 100 USA Sollie AUD = 82,03 Yen 8,5% p.a 10 (a) 62,3 million people (b) ≈ 1,7% EMAFB Focus Area: Mathematics CHAPTER 13 EXERCISE SOLUTIONS Exercise 7-1 (a) a =adj; b c =opp (b) a =opp; b c =hyp (c) a =hyp; b c =adj (d) a =opp; b c =adj (e) a =adj; b c =opp (f) a =adj; b c =hyp (a) 2,14 (b) 0,62 (c) 0,28 (d) 0,21 (e) 0,90 (f) 1,15 (g) 0,23 (h) 2,52 (i) 0,67 (a) T (b) F (c) T (d) F (a) sin Aˆ = CB AC (b) cos Aˆ = AB AC (c) tan Aˆ = CB AB ˆ= (d) sin C AB AC ˆ= (e) cos C CB AC ˆ= (f) tan C √ (a) AB CB (b) (c) (d) (e) (f) (a) (b) (c) =hyp; =opp; =hyp; =opp; AD BD 53,13◦ 35,30◦ 26 m 15,05 m Exercise 7-6 (a) OB = (b) AD BD ˆ = AC = (c) tan B BC M N = 12,86 units; N P = 15,32 units AD AB √ 10 units √1 10 √ 10 −3 (c) CD AD (d) −3 √ (a) − 21 (b) −2 √ t2 +4 (a) (b) 2t (c) t24+4 (d) −1 End of chapter exercises Exercise 7-4 1 12 y (−3; 5) √1 √1 √1 Focus Area: Mathematics a = 37,31 units b = 8,91 units c = 10,90 units d = 21,65 units e = 41,04 units f = 33,43 units g = 29,46 units h = 10,00 units ˆ = AC = sin B AB = √ (a) (b) (c) (d) (e) (f) (g) (h) (a) ˆ = (b) cos D Exercise 7-2 (a) Exercise 7-3 =hyp; (c) 1 √ 3−1 =adj; √ Exercise 7-5 (b) 12 (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n) (o) (a) (b) (c) (d) (e) (f) 59,5◦ 53,1◦ 71,3◦ 80,1◦ 41,8◦ No solution 41,4◦ 18,1◦ 40,5◦ 53,1◦ 18,4◦ No solution 109,8◦ 26,6◦ 17,7◦ 24,0◦ 35,2◦ 50,6◦ 26,3◦ 37,0◦ 45,0◦ α x (a) −3 √ 34 (b) −1 (a) (b) (c) (d) (a) (b) (c) (d) (e) (f) (g) (a) (b) (c) 42,07◦ 63,43◦ 25◦ 45◦ a = 13,86 cm b = 12,56 cm c = 4,30 cm d = 7,78 cm e = 5,34 cm f = 9,20 cm g = 1,65 cm 17,32 cm 10 cm 25,08◦ 447 CHAPTER 13 EXERCISE SOLUTIONS 19,47◦ 53,54◦ 11,88 units 33,69◦ 10 23,96◦ 11 8,35 mm; 9,96 mm 12 (a) cm (b) 4,83 cm; 1,29 cm 13 (a) 18◦ (b) 23◦ 14 97,12◦ 15 5,65 cm; 8,70 cm Analytical geometry Exercise 8-1 EMAFC Exercise 8-4 y B(1; 7) � √ (a) 29 √ (b) 52 √ (c) 17 (a) x = or x = (b) y = or y = −5 C(4; 5) (a) (−1; 6) (b) (14; 32) ; 2y−5 ) (c) ( 2x−3 2 P (8; 13) S(4; −5) End of chapter exercises (c) (a) 11 −10 y O N (c) (b) H (a) E D P F M x −4 −3 −2 −1 −1 Exercise 8-3 I −2 X −3 −4 (a) (b) (c) (a) (b) (c) AB � CD Neither Neither On same line On same line Not on same line Statistics 448 (b) G W J � A(3; 2) (a) (b) � D(1; 3) Exercise 8-2 � 4 y C(6; 4) Z (d) � � A(1; 3) Y √ (a) F G = 26; IH = √ √ 41; GH = 8; √ F I = 29 (b) No (c) No (d) Ordinary quadrilateral x (a) √ (b) AB = 10;BC = √ 13; CD = √ 4;DA = √ (b) i 10 ii (c) Trapezium H(3; 3) √ (a) 34 (b) 13 (c) ( 32 ; − 12 ) a = 0; b = 92 10 � B(4; 1) (a) (c) ( 72 ; 72 ) (d) −2 (a) y = 13 x + √ (b) 40 x EMAFD Focus Area: Mathematics CHAPTER 13 EXERCISE SOLUTIONS Exercise 9-3 53; 50 < m ≤ 55; 50 < m ≤ 55 71,66; 65 < t ≤ 75; 65 < t ≤ 75 (a) 700 < x ≤ 800 (b) 33 600 (c) 700 (d) 750 (e) R 588 000 (a) Mean = 13,2; Median = 11; Mode = (b) Mean = 26; Median = 25; Mode = 24 (c) Mean = 11,2; Median = 11; Mode = 11 (d) Mean = 34,29; Median = 32; No mode Mean = 38,3; Median = 38; Mode = 33 and 42 (a) (b) Exercise 9-4 Q1 = 6,2; Q2 = 18; Q3 = 29 R = 70; Q1 = 41,5; Q2 = 49,5; Q3 = 66,5; IQR = 25 (a) 15; 9; 16 (b) 12; 7,5; 15,5 (c) 10; 8; 7,5 (d) 5; 4; 3,75 (e) 14; 12; 19 (f) 22; 15,5; 23 Exercise 9-2 Group (a) Freq 11 − 20 21 − 30 13 31 − 40 15 41 − 50 51 − 60 3; 5; 7; 13; 16 (a) 15; 22; 25; 28; 35 (b) 88; 92; 98; 100; 101 End of chapter exercises 44 (a) Mean and Mode (b) Bike 1= 1,02 s and Bike = 1,0 s (c) Bike (a) 19,9 (b) i 38% ii 14% iii 50% 11 10 Count Exercise 9-1 −1 Count 10 15 20 25 30 35 | | | | | | | | | | | | | | | | | | | | | | 118 125 129 132 137 Untrained ees: 10 employ- | | | | 17.5 | | 27 | | | 44 | | | 1; 12; 28,5; 46,5; 60 10 20 30 40 50 Range of guesses 60 | | | 12 | | | | 28.5 | | | 46.5 | 65 |||||||||||||||||||||||||||||||||| 126 139 144 149 157 | 60 (a) R 182 222,22 (b) R 100 000 (c) R 100 000 10 Probability Exercise 10-1 (a) Distance (km) (c) (a) 129; 144 (b) 7; 10 (c) Trained employees: 3; 17,5; 27; 44; 65 6 14 (b) Exercise 9-5 12 Focus Area: Mathematics EMAFE (b) (c) (d) (a) 52 (b) (c) 13 449 40 45 CHAPTER 13 EXERCISE SOLUTIONS (d) 13 (e) 13 Exercise 10-2 Y S X 11 10 13 12 14 15 16 M : 30 0,18 (a) 16 (b) H : 36 6 29 {1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12} (a) (b) {1; 2; 3; 4; 6; 12} (c) {2; 3; 5; 7; 11} S B 12 A 5 (a) 11 (b) 12 10 (a) (d) (b) (e) i 12 (c) ii iii 450 (a) (b) (c) (d) (a) (b) (c) (d) (e) (f) 0,21 0,64 0,72 0,5 0,23 0,67 0,67 0,67 11 14 S (b) (a) {deck of cards without clubs} (b) P = {J; Q; K of hearts, diamonds or spades} (c) N = {A; 2; 3; 4; 5; 6; 7; 8; 9; 10 of hearts, diamonds or spades} N P 7♦ A♦ 6♦ J♠ Q♠ 7♥ A♥ 7♠ A♠ 6♠ 6♥ 5♦ 10♦ 5♥ 9♦ 10♥ 5♠ 8♦ 9♥ 8♥ 10♠ 9♠ 4♦ 8♠ 2♥ 4♥ 2♦ 4♠ 3♦ 2♠ 3♥ 3♠ Q♥ Q♦ K♠ K♦ K♥ (d) (e) Mutually exclusive and complementary 19 30 11 30 50 V 300 100 200 S (e) (f) (a) 11 21 (b) (a) 11 14 56 14 (b) 53 56 (a) 103 (b) i ii (c) (a) S J♦ J♥ (d) 10 S (a) 9 13 (a) (b) (c) (d) (a) (b) (c) Exercise 10-3 (b) 10 (c) 0,11 0,33 0,16 0,01 0,80 0,55 0,16 0,01 End of chapter exercises G : 41 16 (a) (b) (c) (d) (e) (f) (g) (h) ii iii i ii iii iv v 30,8% 46,2% 7,7 15,4% 92,3% 10 D/S 40 19 30 58 103 14 19 i (a) (b) 5 25 D/B 15 (a) 50% (b) 31,25% (c) 6,25% Focus Area: Mathematics CHAPTER 13 EXERCISE SOLUTIONS 11 Euclidean Geometry Exercise 11-1 Exercise 11-6 1 a = 138◦ ; b = 42◦ ; c = 138◦ ; d = 138◦ ; e = 42◦ ; f = 138◦ ; g = 42◦ ˆ1 = 110◦ ; C ˆ1 = 80◦ ; B ◦ ˆ3 = 70◦ ; ˆ2 = 30 ; C C ˆ = 100◦ ; Fˆ1 = 70◦ ; D ˆ1 = ˆ F2 = 30◦ ; Fˆ3 = 80◦ ; G ◦ ˆ ◦ ˆ 70 ; G2 = 30 ; G3 = 80◦ 70◦ (a) (b) (c) (d) x = 14 units x = 3,5 units x = units x = 28 units; y = 80◦ (e) x = 24◦ ;y = 12 units End of chapter exercises Exercise 11-2 EMAFF (a) (b) (c) (d) (e) (f) x = y = 72◦ x = 98◦ x = 44◦ ; y = 112◦ x = 19 units x = 25 units x = 18 units; y = units (g) x = 12 units; y = 13 units (a) (b) (c) (d) (e) (f) (g) (h) (a) (b) (c) (d) Straight angle Obtuse angle Acute angle Right angle Reflex angle Obtuse angle Straight angle Reflex angle F T T F 10 (e) (f) (g) (h) (i) (a) (b) (c) T T T F F x = 25◦ x = 145◦ x = 10 units; y = 12,5 units (d) x = 60◦ (e) x = 36◦ (f) x = units; y = 10 units (a) a = 107◦ ; b = 73◦ ; c = 107◦ ; d = 73◦ (b) a = 80◦ ; b = 80◦ ; c = 80◦ ; d = 80◦ (c) a = 50◦ ; b = 45◦ ; c = 95◦ ; d = 85◦ (a) x = 4,24 cm (b) x = 12 cm (c) x = 7,28 cm (d) x = 40 mm x = 2,75 units; y = 30◦ a = units; b = 12 units 12 Measurements Exercise 12-1 25 cm2 50 cm2 79 cm2 40 cm2 128 cm2 17,5 cm2 60 cm Focus Area: Mathematics EMAFG 525 cm2 Exercise 12-2 (a) (b) (c) (a) 344 cm2 471 cm2 533 cm2 24 � (b) 22 � Exercise 12-3 420 cm3 500 cm3 785,4 cm3 451 CHAPTER 13 EXERCISE SOLUTIONS Exercise 12-4 a 282,7 cm2 b 45,6 cm2 c 180 cm2 d 256,6 cm2 a 108,33 cm3 b 270 cm3 c 144 cm3 d 188,8 cm3 175 cm3 ; 190 cm2 31 552 cm3 ; 96 112 cm2 Exercise 12-5 Volume doubles 452 End of chapter exercises (a) Cylinder= 352 cm2 Tri prism= 384 cm2 Rect prism= 72 cm2 (b) Cylinder= 502 cm3 Tri prism = 240 cm3 Rect prism= 40 cm3 (c) Cyl= 3166,7 cm2 Tri prism = 456 cm2 Rect prism= 684 cm2 (d) Cy= 13571,9 cm3 Tri prism = 480 cm3 Rect prism = 080 cm3 (a) Cone= 225 cm2 Sq pyramid= 585 cm2 Half sphere= 100 cm2 (b) Cone= 94 cm3 Sq pyramid= 900 cm3 Half sphere= 134 cm3 Volume= 335 103,2 cm3 S.A= 35 185,9 cm3 Focus Area: Mathematics CHAPTER 13 EXERCISE SOLUTIONS Focus Area: Mathematics 453 ... 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