VERSION 1 CAPS VERSION 1 CAPS EVERYTHING MATHS BY GRAdE 12 GRAdE 12 MATHEMATICS MATHEMATICS WRITTEN BY VOLUNTEERS WRITTEN BY VOLUNTEERS S R E E T N 6 21 179 THIS TEXTBOOK IS AVAILABLE U L 89 O 4 2.5 ON YOUR MOBILE V 27 000 Y B N 39 4 100 29 39 333 61 E T T Trigonometry exercises in Litres of ink used in the Breadth of this book (cm) Number of words used in Hours spent being taught I R this book production of all the grade 10, this book this book W 11 and 12 textbooks Depth of this book (cm) Geometry exercises in this book Number of pages Hours spent doing homework Litres of glue used in the Height of this book (cm) from this book Algebra exercises in this book production of all the grade 10, 11 and 12 textbooks 2 1 3.75 E 3 759 87 52 Everything d Maths a 913 R 5 191 249 30 5 690 3 800 81 19 3.75 G Length of pages side by side How many times student Number of females who Masters students who Hours spent getting book to (cm) scratches head while reading helped write this book contributed to this book school per week This book is available on web, mobi and Mxit. Length of pages top to bottom this book Number of males who helped Honours students who Hours spent getting book (cm) How many times student picks write this book contributed to this book home per week Read, check solutions and practise intelligently at www.everythingmaths.co.za nose while reading this book Undergraduate students who Hours spent with book in class How many times student clicks contributed to this book per week S pen while reading this book C I 179 46 7 50 T 40 206 A 2 5 M 342 0.083 3 290 1 E 63 36 6 H Average size of class being Number of pages in Grade 12 Number of Afrikaans Number of hours spent Weekly UCT hackathons that taught from this book Maths textbook volunteers who helped write conceptualising this cover contributed to this book this book Average age of a maths teacher Number of pages in Grade 11 Number of hours it takes to Small office hackathons that T teaching from this book Maths textbook Number of English volunteers manufacture this book contributed to this book who helped write this book Number of pages in Grade 10 Number of hours spent Afrikaans hackathons that A Maths textbook designing this cover contributed to this book Virtual hackathons that M contributed to this book EVERYTHING MATHS GRADE 12 MATHEMATICS VERSION 1 CAPS WRITTEN BY VOLUNTEERS COPYRIGHT NOTICE Your freedom to legally copy this book You are allowed and encouraged to copy any of the Everything Maths and Everything Science textbooks. You can legally photocopy any page or even the entire book. You can download it from www.everythingmaths.co.za and www.everythingscience.co.za, read it on your phone, tablet, iPad, or computer. You can burn it to CD, put on your flash drive, e-mail it around or upload it to your website. The only restriction is that you have to keep this book, its cover, title, contents and short-codes unchanged. This book was derived from the original Free High School Science Texts written by volunteer academics, educators and industry professionals. Everything Maths and Everything Science are trademarks of Siyavula Education. 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 AND CONTRIBUTORS Siyavula Education Siyavula Education is a social enterprise launched in 2012 with capital and support from the PSG Group Limited and the Shuttleworth Foundation. The Everything Maths and Science series is one of the titles developed and openly released by Siyavula. For more information about the writing and distribution of these or other openly licensed titles: www.siyavula.com [email protected] 021 469 4771 Siyavula Authors Alison Jenkin; Marina van Zyl; Luke Kannemeyer Siyavula and DBE team Dr Carl Scheffler; Bridget Nash; Ewald Zietsman; William Buthane Chauke; Leonard Gumani Mudau; Sthe Khanyile; Josephine Mamaroke Phatlane Siyavula and Free High School Science Text contributors Dr Mark Horner; Dr Samuel Halliday; Dr Sarah Blyth; Dr Rory Adams; Dr Spencer Wheaton Iesrafeel Abbas; Sarah Abel; Taskeen Adam; Ross Adams; Tracey Adams; Dr Rory Adams; Andrea Africa; Wiehan Agenbag; Ismail Akhalwaya; Matthew Amundsen; Ben Anhalt; Prashant Arora; Bianca Bôhmer; Amos Baloyi; Bongani Baloyi; Raymond Barbour; Caro-Joy Barendse; Katie Barry; Dr Ilsa Basson; Richard Baxter; Tara Beckerling; Tim van Beek; Lisette de Beer; Jessie Bester; Mariaan Bester; Jennifer de Beyer; Dr Sarah Blyth; Sebastian Bodenstein; Martin Bongers; Dr Thinus Booysen; Ena Bosman; Janita Botha; Pieter Botha; Gareth Boxall; Stephan Brandt; Hannes Breytenbach; Alexander Briell; Wilbur Britz; Graeme Broster; Craig Brown; Michail Brynard; Richard Burge; Jan Buys; George Calder-Potts; Biddy Cameron; Eleanor Cameron; Mark Carolissen; Shane Carollisson; Richard Case; Sithembile Cele; Alice Chang; Faith Chaza; Richard Cheng; Fanny Cherblanc; Lizzy Chivaka; Dr Christine Chung; Dr Mareli Claasens; Brett Cocks; Zelmari Coetzee; Roché Compaan; Willem Conradie; Stefaan Conradie; Deanne Coppejans; Rocco Coppejans; Tim Craib; Dr Andrew Craig; Tim Crombie; Dan Crytser; Jock Currie; Dr Anne Dabrowski; Laura Daniels; Gareth Davies; Mia de; Tariq Desai; Sandra Dickson; Sean Dobbs; Buhle Donga; William Donkin; Esmi Dreyer; Matthew Duddy; Christel Durie; Fernando Durrell; Dr Dan Dwyer; Frans van Eeden; Kobus Ehlers; Alexander Ellis; Tom Ellis; Charl Esterhuysen; Andrew Fisher; Dr Philip Fourie; Giovanni Franzoni; Sanette Gildenhuys; Olivia Gillett; Ingrid von Glehn; Tamara von Glehn; Nicola Glenday; Lindsay Glesener; Kevin Godby; Dr Vanessa Godfrey; Terence Goldberg; Dr Johan Gonzalez; Saaligha Gool; Hemant Gopal; Dr Stephanie Gould; Umeshree Govender; Dr Ilse le Grange; Heather Gray; Lynn Greeff; Jaco Greyling; Martli Greyvenstein; Carine Grobbelaar; Suzanne Grové; Dr Tom Gutierrez; Brooke Haag; Kate Hadley; Alex Hall; Dr Sam Halliday; Asheena Hanuman; Dr Melanie Dymond Harper; Ebrahim Harris; Dr Nicholas Harrison; Neil Hart; Nicholas Hatcher; Jason Hayden; Laura Hayward; Dr William P. 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Rassias; Ona Rautenbach; Dr Jocelyn Read; Jonathan Reader; Jane Reddick; Robert Reddick; Dr Matthew Reece; Chris Reeders; Brice Reignier; Razvan Remsing; Dr Liezel Retief; Adam Reynolds; Laura Richter; Max Richter; Sean Riddle; Dr David Roberts; Christopher Roberts; Helen Robertson; William Robinson; Evan Robinson; Christian Roelofse; Raoul Rontsch; Dr Andrew Rose; Katie Ross; Karen Roux; Dr Maritha le Roux; Jeanne-Mariè Roux; Karen Roux; Mark Roux; Bianca Ruddy; Heinrich Rudman; Nitin Rughoonauth; Katie Russell; Steven Sam; Jason Avron Samuels; Rhoda van Schalkwyk; Christo van Schalkwyk; Dr Carl Scheffler; Nathaniel Schwartz; Duncan Scott; Helen Seals; Relebohile Sefako; Sandra Serumaga-Zake; Paul Shangase; Cameron Sharp; Ian Sherratt; Ryman Shoko; Dr James Short; Cho Hee Shrader; Roger Sieloff; Brandon Sim; Bonga Skozana; Bradley Smith; Greg Solomon; Zamekile Sondzaba; Nicholas Spaull; Margaret Spicer; Hester Spies; Dr Andrew Stacey; Dr Jim Stasheff; Mike Stay; Nicol Steenkamp; Nicky Stocks; Dr Fred Strassberger; Mike Stringer; Stephanie Strydom; Abdulhuck Suliman; Bianca Swart; Masixole Swartbooi; Ketan Tailor; Tshenolo Tau; Tim Teatro; Ben Thompson; Shen Tian; Xolani Timbile; Dr Francois Toerien; René Toerien; Liezel du Toit; Nicola du Toit; Dr Johan du Toit; Robert Torregrosa; Jimmy Tseng; Theresa Valente; Alida Venter; Pieter Vergeer; Rizmari Versfeld; Nina Verwey; Mfundo Vezi; Mpilonhle Vilakazi; Katie Viljoen; Adele de Villiers; Daan Visage; Wetsie Visser; Alexander Volkwyn; Kosma von Maltitz; Dr Karen Wallace; John Walmsley; Duncan Watson; Helen Waugh; Leandra Webb; Dr Dawn Webber; Michelle Wen; Dr Rufus Wesi; Francois Wessels; Wessel Wessels; Leandi van der Westhuizen; Neels van der Westhuizen; Sabet van der Westhuizen; Dr Alexander Wetzler; Dr Spencer Wheaton; Vivian White; Mark Whitehead; Dr Gerald Wigger; Harry Wiggins; Heather Williams; Wendy Williams; Julie Wilson; Timothy Wilson; Andrew Wood; Emma Wormauld; Dr Sahal Yacoob; Jean Youssef; Ewald Zietsman; Johan Zietsman; Marina van Zyl EVERYTHING MATHS Mathematics is commonly thought of as being about numbers but mathematics is actu- ally 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, mathemat- ics 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 mathemat- ics used to create these technologies was not developed for its own sake, but was avail- able 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 ab- stract discipline, mathematics embodies logic, symmetry, harmony and technological progress. More than any other language, mathematics is everywhere and universal in its application. SPONSOR This textbook was developed with corporate social investment funding from MMI Holdings. Well structured, impactful Corporate Social Investment (CSI) has the ability to contribute positively to nation building and drive positive change in the communities. MMI’s commitment to social investment means that we are constantly looking for ways in which we can assist some of South Africa’s most vulnerable citizens to expand their horizons and gain greater access to life’s opportunities. This means that we do not view social investment as a nice to have or as an exercise in marketing or sponsorship but rather as a critical part of our contribution to society. The merger between Metropolitan and Momentum was lauded for the complementary fit between two companies. This complementary fit is also evident in the focus areas of CSI programmes where Metropolitan and Momentum together cover and support the most important sectors and where the greatest need is in terms of social participation. HIV/AIDS is becoming a manageable disease in many developed countries but in a country such as ours, it remains a disease where people are still dying of this scourge unnecessarily. Metropolitan continues to make a difference in making sure that HIV AIDS moves away from being a death sentence to a manageable disease. Metropolitan’s other focus area is education which remains the key to economic prosperity for our country. Momentum’s focus on persons with disabilities ensures that this community is included and allowed to make their contribution to society. Orphaned and vulnerable children are another focus area for Momentum and projects supported ensure that children are allowed to grow up safely, to assume their role along with other children in inheriting a prosperous future. EVERYTHING MATHS & SCIENCE The Everything Mathematics and Science series covers Mathematics, Physical Sciences, Life Sciences and Mathematical Literacy. The Siyavula Everything Science textbooks The Siyavula Everything Maths textbooks DIGITAL TEXTBOOKS READ ONLINE Watch this textbook come alive on the web. In addition to all the content in this printed copy, the online version is also full of videos, presentations and simulations to give you a more comprehensive learning experience. www.everythingmaths.co.za and www.everythingscience.co.za CHECK YOUR ANSWERS ONLINE OR ON YOUR PHONE Want the answers? View the fully worked solutions to any question in this textbook by entering its shortcode (4 digit combination of letters and numbers) into the search box on the web or mobi sites. www.everythingmaths.co.za and www.everythingscience.co.za or m.everythingmaths.co.za and m.everythingscience.co.za from your cellphone. MOBILE & TABLET MOBI You can access this whole textbook on your mobile phone. Yes, the whole thing, anytime, anywhere. Visit the mobi sites at: m.everythingmaths.co.za and m.everythingscience.co.za MXIT Don’t stress if you haven’t got a smart phone. All Mxit users can read their Everything Series textbooks on Mxit Reach too. 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