WJEC Maths for A2 – Pure

7 How to use this book • Step by steps are included to help you answer questions that do not guide you bit by bit towards the ϐinal answer (called unstructured questions). In the past you would be guided to the ϐinal answer by the question being structured. For example, there may have been parts (a), (b), (c) and (d). Now you can get questions which ask you to go to the answer to part (d) on your own. You have to work out for yourself the steps (a), (b) and (c) you would need to take to arrive at the ϐinal answer. The ‘step by steps’ help teach you to look carefully at the question to analyse what steps need to be completed in order to arrive at the answer. • Active learning – are short tasks which you carry out on your own which aid understanding of a topic or help with revision. • Summaries – are provided for each topic and present the formulae and the main points in a topic. They can be used for quick reference or help with your revision. Exam Practice and Technique Helping you understand how to answer examination questions lies at the heart of this book. This means that we have included questions throughout the book that will build up your skills and knowledge until you are at a stage to answer full exam questions on your own. Examples are included, some of which are full examination style questions. These are annotated with pointers and general advice about the knowledge, skills and techniques needed to answer them. Test yourself answers 196 ➎ ȋƒȌ ‡– y α e ͷԜΫԜ x ʹ Ϊ͸ y Ϋ͸α e ͷԜΫԜ x ʹ ƒ‹‰Ž‘ˆ„‘–Š•‹†‡•ǡ™‡‘„–ƒ‹ŽԜȋԜ y Ϋ͸ȌαͷΫ x ʹ ‡ƒ””ƒ‰‹‰ˆ‘” x ǡ™‡‘„–ƒ‹ x αʹȑͷێԜȍԜ y Ϋ͸ȎȒ ‡…‡ f ౶ Ϋͳ ౷ ȋ x ȌαʹȑͷێԜȍ x Ϋ͸ȎȒ ȋ„Ȍ ‘ƒ‹‘ˆ f ౶ Ϋͳ αȋ͸ǡλȌ ➏ ȋ ‘–‡–Šƒ–‹–Š‡‡šƒ–Š‹•‰”ƒ’Š™‘—Ž†„‡Šƒ†Ǧ†”ƒ™ǤȌ y x ȋͲǡͳȌ y α e x y αΫʹ ȋͲǡͲȌ y αʹ e x Ϋʹ ‘” y α f ౶ ȋ x ȌǤ —”˜‡…—–•–Š‡ y Ǧƒš‹•ƒ–ȋͲǡͳȌǤ ➐ ȋƒȌ ‘ƒ‹‘ˆ fg ‹•–Š‡†‘ƒ‹‘ˆ g αȋͲǡλȌǤ ƒ‰‡‘ˆ fg ‹•ȋͲǡλȌǤ ȋ„Ȍ fg ȋ x Ȍαʹ e ͵ g ȋ x Ȍ fg ȋ x Ȍαʹ e ͵ԜŽԜʹ x fg ȋ x Ȍαʹ e ŽԜȋʹ x Ȍ ͵ fg ȋ x Ȍαʹ e ŽԜͺ x ͵ fg ȋ x Ȍαͳ͸ x ͵ ‹…‡ fg ȋ x Ȍαͳʹͺ ͳ͸ x ͵ αͳʹͺ x ͵ αͺ x αʹ Topic 3 ➊ ͳΪ x √ ͳΫͶ x αȋͳΪ x ȌȋͳΫͶ x Ȍ ΫԜ ͳ ʹ αȋͳΪ x Ȍ ȏ ͳΪ ȋ Ϋԝ ͳ ʹ Ȍ ȋΫͶ x ȌΪ ȋ Ϋ ͳ ʹ Ȍȋ Ϋ ͵ ʹ Ȍ ȍΫͶ x Ȏ ʹ ʹǨ Ϊǥ Ȑ αͳΪʹ x Ϊ͸ x ʹ Ϊ x Ϊʹ x ʹ Ϊ͸ x ͵ Ϊǥ αͳΪ͵ x Ϊͺ x ʹ Ϊǥ ȁͶ x ȁδͳ•‘–Š‹•‡š’ƒ•‹‘‹•…‘˜‡”‰‡–ˆ‘”ȁ x ȁδ ͳ Ͷ Ԙ‘”ԘΫԝ ͳ Ͷ δ x δ ͳ Ͷ ŽԜȋ x Ϋ͸Ȍ…ƒ‘Ž›„‡ˆ‘—† ‹ˆ x ‹•‰”‡ƒ–‡”–Šƒ͸ƒ•›‘— …ƒ‘–ϐ‹†Ž‘ˆͲ‘”ƒ ‡‰ƒ–‹˜‡—„‡”Ǥ Š‹•‡ƒ• x Šƒ•–‘„‡ ‰”‡ƒ–‡”–Šƒ͸ƒ†…ƒ–ƒ‡ ƒ›˜ƒŽ—‡ˆ”‘͸–‘‹ϐ‹‹–›Ǥ Š‡‰”ƒ’Š‘ˆ y α f Ԝȋ x Ȍ ”‡’”‡•‡–•ƒ•–”‡–…Š‘ˆ–Š‡ ‘”‹‰‹ƒŽ‰”ƒ’Šȋ y α e x Ȍ„› •…ƒŽ‡ˆƒ…–‘”ʹ’ƒ”ƒŽŽ‡Ž–‘ –Š‡ y Ǧƒš‹•ˆ‘ŽŽ‘™‡†„›ƒ –”ƒ•Žƒ–‹‘‘ˆ ȋ Ͳ Ϋʹ Ȍ Ǥ ‘–‡–Š‡•‡–”ƒ•ˆ‘”ƒ–‹‘• …ƒ„‡ƒ’’Ž‹‡†–Š‡‘–Š‡”™ƒ› ƒ”‘—†Ǥ Š‡†‘ƒ‹‘ˆƒ…‘’‘•‹–‡ ˆ—…–‹‘ fg ‹•–Š‡•‡–‘ˆƒŽŽ x ‹–Š‡†‘ƒ‹‘ˆ g ˆ‘”™Š‹…Š g ȋ x Ȍ‹•‹–Š‡†‘ƒ‹‘ˆ f Ǥ R ȋ g ȌαȋΫλǡλȌƒ† D ȋ f ȌαȋΫλǡλȌŠ‡…‡ D ȋ fg Ȍα D ȋ g ȌαȋͲǡλȌ Š‡”ƒ‰‡‘ˆ fg ‹•ˆ‘—†„› •—„•–‹–—–‹‰˜ƒŽ—‡•ˆ”‘–Š‡ †‘ƒ‹‹–‘–Š‡…‘’‘•‹–‡ ˆ—…–‹‘ȋ‹Ǥ‡Ǥ fg ȋšȌαͳ͸ x ͵ ȌǤ ͵ԜŽԜʹ x ᎐Ԝȋʹ x Ȍ ͵ e ŽԜ a α a ‘–‡–Šƒ–ͳΪ x ‹•‘– ‹˜‘Ž˜‡†‹–Š‡…‘†‹–‹‘ˆ‘” …‘˜‡”‰‡…‡„‡…ƒ—•‡ͳΪ x ‹• ‘–ƒ‡š’ƒ•‹‘Ǥ [1608]TYQAnswers.indd 196 2.12 Exponential and logarithmic functions 49 Test yourself ➊ Š‡ˆ—…–‹‘ f Šƒ•†‘ƒ‹ x ζΫʹƒ†‹•†‡ϐ‹‡†„› f ౶ ȋ x Ȍαȋ x ΪʹȌ ʹ ΫͳǤ ȋƒȌ ‹†–Š‡”ƒ‰‡‘ˆ f Ǥ ȏʹȐ ȋ„Ȍ ‹†ƒ‡š’”‡••‹‘ˆ‘” f ౶ Ϋͳ ౷ ȋ x ȌǤ –ƒ–‡–Š‡†‘ƒ‹ƒ†”ƒ‰‡‘ˆ f ౶ Ϋͳ Ǥ ȏ͵Ȑ ➋ Š‡†‹ƒ‰”ƒ•Š‘™•ƒ•‡–…Š‘ˆ–Š‡‰”ƒ’Š‘ˆ y α f ౶ ȋ x ȌǤ Š‡‰”ƒ’ŠŠƒ•‹–• Š‹‰Š‡•–’‘‹–ƒ–ȋ͵ǡͶȌƒ†‹–‡”•‡…–•–Š‡ x Ǧƒš‹•ƒ––Š‡’‘‹–•ȋͳǡͲȌƒ†ȋͷǡͲȌǤ y x ȋͳǡͲȌ Ͳ ȋ͵ǡͶȌ ȋͷǡͲȌ ‡–…Š–Š‡‰”ƒ’Š‘ˆ y αʹ f ౶ ȋ x ΫͳȌǡ‹†‹…ƒ–‹‰–Š‡…‘‘”†‹ƒ–‡•‘ˆ–Š”‡‡’‘‹–• ‘–Š‡‰”ƒ’ŠǤ ȏͶȐ ➌ ‘Ž˜‡–Š‡ˆ‘ŽŽ‘™‹‰ ȋƒȌ ͵ȁ x ΫͳȁΪ͹αͳͻ ȏʹȐ ȋ„Ȍ ͸ȁ x ȁΫ͵αʹȁ x ȁΪͷ ȏʹȐ ➍ ȋƒȌ ‡–…Š–Š‡‰”ƒ’Š‘ˆ x ʹ Ϊ͸ x Ϊͳ͵ǡ‹†‡–‹ˆ›‹‰–Š‡•–ƒ–‹‘ƒ”›’‘‹–Ǥ ȏʹȐ ȋ„Ȍ Š‡ˆ—…–‹‘ f ‹•†‡ϐ‹‡†„› f ౶ ȋ x Ȍα x ʹ Ϊ͸ x Ϊͳ͵™‹–Š†‘ƒ‹ȋ a ǡ b ȌǤ ȋ‹Ȍ š’Žƒ‹™Š› f ౶ Ϋͳ †‘‡•‘–‡š‹•–™Š‡ a αΫͳͲƒ† b αͳͲǤ ȏͳȐ ȋ‹‹Ȍ ”‹–‡†‘™ƒ˜ƒŽ—‡‘ˆ a ƒ†ƒ˜ƒŽ—‡‘ˆ b ˆ‘”™Š‹…Š–Š‡‹˜‡”•‡‘ˆ f †‘‡•‡š‹•–ƒ††‡”‹˜‡ƒ‡š’”‡••‹‘ˆ‘” f ౶ Ϋͳ ౷ ȋ x ȌǤ ȏͷȐ ➎ Š‡ˆ—…–‹‘ f Šƒ•†‘ƒ‹ȋΫλǡͳͲȌƒ†‹•†‡ϐ‹‡†„›ǣ f ౶ ȋ x Ȍα e ͷԜΫԜ x ʹ Ϊ͸ ȋƒȌ ‹†ƒ‡š’”‡••‹‘ˆ‘” f ౶ Ϋͳ ౷ ȋ x ȌǤ ȏͶȐ ȋ„Ȍ ”‹–‡†‘™–Š‡†‘ƒ‹‘ˆ f ౶ Ϋͳ Ǥ ȏʹȐ ➏ ‹˜‡–Šƒ– f ౶ ȋ x Ȍα e x ǡ•‡–…Š‘–Š‡•ƒ‡†‹ƒ‰”ƒǡ–Š‡‰”ƒ’Š•‘ˆ y α f ౶ ȋ x Ȍƒ† y αʹ f ౶ ȋ x ȌΫʹǤ ƒ„‡Ž–Š‡…‘‘”†‹ƒ–‡•‘ˆ–Š‡’‘‹–•‘ˆ‹–‡”•‡…–‹‘‘ˆ‡ƒ…Š‘ˆ –Š‡‰”ƒ’Š•™‹–Š–Š‡ x Ǧƒš‹•Ǥ †‹…ƒ–‡–Š‡„‡Šƒ˜‹‘—”‘ˆ‡ƒ…Š‘ˆ–Š‡‰”ƒ’Š•ˆ‘” Žƒ”‰‡’‘•‹–‹˜‡ƒ†‡‰ƒ–‹˜‡˜ƒŽ—‡•‘ˆ x Ǥ ȏ͸Ȑ ➐ Š‡ˆ—…–‹‘ f Šƒ•†‘ƒ‹ȋΫλǡλȌƒ†‹•†‡ϐ‹‡†„› f ౶ ȋ x Ȍαʹ e ͵ x Š‡ˆ—…–‹‘ g Šƒ•†‘ƒ‹ȋͲǡλȌƒ†‹•†‡ϐ‹‡†„› g ȋ x Ȍ᎐ ౶ ʹ x ȋƒȌ ”‹–‡†‘™–Š‡†‘ƒ‹ƒ†”ƒ‰‡‘ˆ fg. ȏʹȐ ȋ„Ȍ ‘Ž˜‡–Š‡‡“—ƒ–‹‘Ԙ fg ȋ x ȌαͳʹͺǤ ȏ͵Ȑ 11/09/2018 12:16