Mathematics A-Level Induction

Transcription

1 Name Myton School Mathematics A-Level Induction Refresher Course to Help with the Transition from GCSE to AS-Level Maths Perhaps I could best describe my experience of doing mathematics in terms of entering a dark mansion. You go into the first room and it's dark, completely dark. You stumble around, bumping into the furniture. Gradually, you learn where each piece of furniture is. And finally, after six months or so, you find the light switch and turn it on. Suddenly, it's all illuminated and you can see exactly where you were. Then you enter the next dark room Sir Andrew Wiles Mathematics Department July 2014

2 Contents How to use this booklet... 2 Getting Help... 3 Recommended Resources for the AS Course Arithmetic of Fractions Rules and Manipulation of Indices Expanding Brackets and Factorising Algebraic Fractions Surds Linear Equations Changing the Subject of a Formulae Solving Quadratic Equations Factorising Solving Quadratic Equations Completing the Square and Using the Quadratic Formula Solving Simultaneous Linear Equations Answers Mock Assessment Mock Assessment Answers

3 How to use this booklet Thank you for choosing to study Mathematics in the sixth form at Myton School. At the end of year 12, you will sit two modules in Pure Mathematics (C1 and C2) as well as either Mechanics (M1) or Decision (D1), depending which option block you chose. If you have chosen to study Further Mathematics you will study those four modules and also Statistics (S1) and Further Pure 1. The Mathematics Department is committed to ensuring that you make good progress throughout your A-level course. In order that you make the best possible start to the course, we have prepared this booklet. The material in this refresher course has been designed to enable you to prepare for the demands of A-Level maths. When your course starts in September you will find that your ability to get the most from lessons, and to understand new material, depends crucially upon both having a good facility with algebraic manipulation and undertaking plenty of independent study. We think that this is so important that we are providing you with this workbook to complete before you start the course in September. It is vitally important that you spend some time working through the questions in this booklet over the summer - you will need to have a good knowledge of these topics before you commence the course in September. A-Level maths is a demanding course and good skills in the areas covered within this booklet will be paramount to your success. You will most-likely have met all the topics before at GCSE. At the start of each section the relevant MyMaths lessons have been indicated; use these to help you if you are stuck with anything or unsure of how to proceed. You should attempt at least the majority of questions in each exercise not necessarily every question, but enough to ensure you understand the topic thoroughly. The answers are given at the back of the booklet and you should mark your work. If you get questions wrong you should revise the material and try again until you are getting the majority of questions correct. You may find some questions towards the end of each section quite tricky this is ok, but still give them a try! In September, you will need to return your booklet to your A-Level teacher, who will not mark it but rather will consider if an appropriate amount of work has been completed. You will also be tested at the start of September to check how well you understand these topics, so it is important that you have worked through the booklet before then. If you do not achieve a good enough score in this test, a letter of concern will be sent home and you will then be re-tested in October. A mock test is provided at the back of this booklet, with answers. We hope that you will use this introduction to give you a good start to your AS work and that it will help you enjoy and benefit from the course more. Emma Atkins Head of Mathematics Matt O Reilly Key Stage 5 Coordinator 2

4 Getting Help You may also find the following books useful in helping you to prepare for your year 12 studies. Head Start to AS Maths Preparation for AS/A Level Mathematics Published by CGP Workbooks Published by Alpha Workbooks ISBN: Recommended Resources for the AS Course The following are resources you may find helpful alongside studying the course from September. Mathsnet A-Level Plus 12 for 3 modules of your choice, 1 year subscription 24 for all modules, 1 year subscription *Ensure you choose Edexcel as the syllabus Revise Edexcel: AS Mathematics Revision Guide Covers all AS modules (C1, C2, D1, M1, S1) ISBN: Revise Edexcel: AS Mathematics Revision Workbook Covers all AS modules (C1, C2, D1, M1, S1) ISBN: It is advised that you purchase both the Workbook and Revision Guide, rather than one or the other, both of which are available from the LRC at the discounted price of 4 each. 3

5 1. Arithmetic of Fractions Number > Fractions > Adding Subtracting, Multiplying, Dividing 4

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7 2. Rules and Manipulation of Indices Number > Powers and Roots > Indices 1, 2 and 3 6

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12 3. Expanding Brackets and Factorising Algebra > Algebraic Manipulation > Single Brackets, Brackets, Factorising Linear, Factorising Quadratics 1 and 2 11

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17 4. Algebraic Fractions Algebra > Algebraic Manipulation > Cancelling. Adding and Multiplying Algebraic Fractions 16

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19 5. Surds Number > Powers and Roots > Surds 1 and Surds 2 18

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21 6. Linear Equations Algebra > Equations Linear > Solving Equations, Equations with Fractions 20

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24 7. Changing the Subject of a Formulae Algebra > Expressions and Formulae > Rearranging 1 and Rearranging 2 23

25 8. Solving Quadratic Equations Factorising Algebra > Equations Quadratic > Quadratic Equations 24

26 9. Solving Quadratic Equations Completing the Square and Using the Quadratic Formula Algebra > Equations Quadratic > Completing the Square and Quadratic Formula 25

27 10. Solving Simultaneous Linear Equations Algebra > Equations Simultaneous > Simultaneous 1, 2, 3 and Negative Solve each of the pairs of simultaneous equations. Where the answer is not a whole number, give your answer as an improper fraction. 1. 4x + 3y = 6 5x 3y = x + 5y = 13 2x + 3y = 8 3. x + 4y = 5 4x 2y = a + b = 3 4a 5b = x + 3y = 4 3x + 4y = x 2y = 13 2x + 3y = a 2b = 14 4a + 3b = x + 4y = 5 2x + 7y = x 4y = 39 2x + y = 6 26

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30 Section 10. Solving simultaneous linear equations 1. x = 3, y = x = 1, y = 2 3. x = 3, y = ½ 4. a = 5/2, y = x = -1, y = 3 6. x = 3/2, y = a = 4, b = x = -3, y = 5 9. x = 9/2, y = -3 29

31 Mock Assessment The test you will sit in September will ask questions similar to this one. Be sure you are able to answer these questions well! You should also be able to complete this test within 1 hour. You may NOT use a calculator Useful Formulae If ax 2 + bx + c = 0 then x = b± b2 4ac 2a 1. Expand and simplify (a) (x + 3)(x 1) (b) (a + 3) 2 (c) (2x + 3)(x - 4) 2. Factorise (a) x 2 7x (b) y 2 64 (c) x 2 5x + 6 (d) 6t 2 13t Simplify (a) 4x3 y 3x + 2 8x 2 (b) + 4x 1 y Solve the following equations (a) h h 5 = 4 (b) x2 8x = 0 (c) p 2 + 4p - 12 = 0 5. Write each of the following as single powers of x and/or y (a) y 3 y 2 (b) 1 x 4 (c) (x2 y) 3 (d) x5 6. Work out the values of the following, giving your answers as fractions (a) 4-2 (b) 10 0 (c) ( 8 27 ) 1 3 x 2 7. Solve the simultaneous equations 3x 5y = -11 5x 2y = 7 8. Rearrange the following equations to make x the subject (a) v 2 = u 2 + 2ax (b) V = 1 3 πx2 h (c) y = x+2 x+1 9. Solve 5x 2 x 1 = 0 giving your solutions in surd form 30

32 Mock Assessment Answers 1) a) x 2 + 2x 3 b) a 2 + 6a + 9 c) 2x 2-5x -12 2) a) x(x 7) b) (y + 8)(y 8) c) (x - 2)(x - 3) d) (3t - 5)(2t 1) x 3) a) 2 2y b) 10 x 3 6 4) a) h = 5 b) x = 0 or x = 8 c) p = -6 or p = 2 5) a) y 5 b) x -4 c) x 6 y 3 d) x 7 6) a) 1 16 b) 1 c) 2 3 7) x = 3, y = 4 8) a) 2 2 v u x b) 2a 3V x c) h x 2 y y 1 9) 1 21 x 10 31