CONTENTS. IBDP Mathematics HL Page 1
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1 CONTENTS ABOUT THIS BOOK... 3 THE NON-CALCULATOR PAPER... 4 ALGEBRA... 5 Sequences and Series... 5 Sequences and Series Applications... 7 Exponents and Logarithms... 8 Permutations and Combinations... 1 Binomial Expansion Proof by Induction Complex Numbers The Complex Plane De Moivre's Theorem... 0 Systems of Equations... FUNCTIONS AND EQUATIONS... 4 Basics of Functions... 4 Graphs of Functions... 8 Reciprocal Functions... 3 Quadratic Functions Solving Quadratic Equations Sum and Product of Roots Inequalities Polynomial Functions Exponential and Logarithmic Functions CIRCULAR FUNCTIONS AND TRIGONOMETRY... 4 Definitions and Formulae... 4 Trigonometric Formulae Solving Trigonometric Equations The Solution of Triangles Graphing Periodic Functions VECTORS... 5 Basics of Vectors... 5 Scalar (Dot) Product Vector (Cross) Product Equations of Lines Equations of Planes Equations of Lines and Planes Summary Intersections Angles in Three Dimensions Miscellaneous Vector Questions... 6 STATISTICS AND PROBABILITY Statistics Probability Notation and Formulae Lists and Tables of Outcomes Venn Diagrams Tree Diagrams Bayes Theorem... 7 Discrete Probability Distributions Binomial Distribution Poisson Distribution Continuous Distributions The Normal Distribution CALCULUS Differentiation The Basics Differentiation from First Principles The Chain Rule Product and Quotient Rules IBDP Mathematics HL Page 1
2 Second Derivative...88 Applications of Differentiation...89 Implicit Differentiation...91 Graphical behaviour of functions...93 Sketching Graphs Examples...94 Indefinite Integration...95 Definite Integration...97 Integration By Substitution...98 Integration by Parts...99 General Methods for Integration Further Kinematics Volumes of Revolution Calculus Non-Calculator Techniques MAXIMISING YOUR MARKS ASSESSMENT DETAILS PRACTICE QUESTIONS Answers to Practice Questions Page IBDP Mathematics HL
3 ABOUT THIS BOOK This is a revision book, not a text book. It will show you everything you need to know in the Math Higher syllabus, but it assumes that you have already covered the work, and that you are now going through it for the second (or third, or fourth.) time. I would expect you to use your other resources (text book, class notes) to fill in much of the detail. The exam is not so much a test of your knowledge and understanding (you will not get a question which begins "What do you know about.?"); but a test of how you use your understanding to solve mathematical problems. So the emphasis in this revision book is on how to answer questions. In particular you will find plenty of worked exam style questions, as well as further ones for you to solve. All the questions in boxes are of a standard and of a type that could occur in your exams. Do not skim over these much useful revision material is contained in the working which is not contained in the text. The option topics are all quite substantial. So, rather than increase the size (and cost) of this book by including all of them here, each option topic has its own revision guide. You are expected to be able to understand and use your graphic display calculator (GDC) in all relevant areas of the syllabus. Indeed, some questions require you to use, for example, the graphing or the equation solving features. Since different people use different calculators, it is not possible for this book to explain the detail of their use; but I have indicated (using the calculator symbol ) where the GDC can be particularly useful. If you have a calculator from the TI family, you might like to know that another book in the OSC Revision Guide series, "Using the TI calculator in IB Maths", will guide you through all the techniques you need. This is your revision book. Every page has a wide column for you to make notes and scribblings and write down questions to ask your teacher; and there are plenty of questions for you to work through. You Solve questions appear in boxes like this with a grey indicator bar at the side. And towards the end of the book there are some important points about how to maximise your exam mark. Do follow the suggestions there, and perhaps add some more of your own. At the very end there are some practice questions testing you on the basic work contained in each area of the syllabus. I am enormously grateful to Peter Gray, from Malvern College, who has proof read this book and, in the process, made some eminently sensible suggestions for numerous improvements; he has also tactfully pointed to a number of errors in both the text and the calculations which I have gratefully corrected! Any remaining errors are entirely my responsibility, and I would be grateful for any corrections from readers. Through Oxford Study Courses I have been privileged to help many students revise towards their IB Mathematics exams, and much of what I have learnt from teaching them has been distilled into this book. I would value any feedback so that later editions can continue to help students around the world. Please feel free to me on inlucas@greentrees.fsnet.co.uk. All correspondence will be answered personally. Ian Lucas IBDP Mathematics HL Page 3
4 x = 5 x = ± 5 Example: Solve the equation x 4x = x 3 x 4x + 3 = 0 (x 3)(x 1) = 0 x = 3 or 1 Example: Solve the equation x 4x = x + x x = 5 0 ± x = ( 5) ( 5) 4 ( ) 5 ± 41 x = 4 x =.851, No solution 1 solution solutions Solving Quadratic Equations Except for the simplest form of quadratic equation shown on the left the first move is always collect together terms on the left hand side with 0 on the right hand side. Factorisation: If the quadratic expression factorises, this is the simplest method of solution. Make sure you understand the connection between the factors and the x-intercepts (see previous section) since questions can link the equation to the graph. Formula: All quadratics can be solved using the formula, although it is most useful when the expression does not factorise. The solution of ax b ± b 4ac + bx + c is: x =. It is the ± a which leads to the two possible solutions. Make sure that you can use your calculator to find both solutions quickly. (As well as a straight calculation, you can use an APP or use the equation solver). Be careful to substitute correctly, particularly when there are minus signs around. Follow the example on the left carefully. The solutions to a quadratic equation are the points where the graph crosses the x-axis. This can lead to 0, 1 or solutions. These correspond to values of b 4ac which are <0, =0 and >0 respectively. b 4ac is called the discriminant since it discriminates between the number of solutions. Find the range of values of p for which x px + (p + 3) > 0 for all real x. Look at the small sketches above: a quadratic function with all positive values has no solutions, so its discriminant is negative. So, (-p) 4(p + 3) < 0 p 4p 1 < 0 This is a quadratic inequality (see page 36) which is solved by finding the critical values (ie which give 0). (p 6)(p + ) < 0 has critical values p = 6 and p = - We want the values of p to be less than 0 so < p < 6. The function f(x) = r + qx - px has the graph shown. a) How many solutions are there to the equation r + qx = px? b) Show that q + 4pr > 0 and hence find the minimum value of r if q = 4 and p =. a) b) r > - Solve the equation 4 x 1 = x + 8 First replace the 4 with an appropriate power of, then simplify (don t forget how to simplify expressions of the form a m - n. You should find you have a quadratic equation in x. x = 3 Page 34 IBDP Mathematics HL
5 PRACTICE QUESTIONS The questions which follow are not designed to cover every aspect of the syllabus, nor are they exam style questions. Their purpose is to give you some practice in the basics: if you cannot, for example, rearrange an equation with a log function in it, or correctly identify which integral technique to use, then you may be getting questions wrong simply because of a lack of basic techniques. You should answer all of these questions as part of your revision. If you get an answer wrong, find out why: then come back to it later, and see if you can get it right next time. ALGEBRA 1. Find the 5th term and the sum of the first 54 terms of the sequence which begins: 3, 8, 13, 18. An arithmetic sequence has first term 7 and common difference 3.5. How many terms are required for the sum of the sequence to be What is the 1th term and the sum to 18 terms of the sequence which begins 3, 1, 48, 19? 4. A geometric series has a first term 400, ten terms and a sum of What is the common ratio? 5. Why does the sum to infinity exist for the sequence 100, 80, 64, 51.? Find S 0 and S and also the percentage error in approximating S by S Write the recurring decimal as a fraction in its simplest form. 7. Write + 3log 10 x as a single logarithm. 8. Solve the equation 3.1 x = 10 x 1. Answer to 4 decimal places. 9. If s = e 0.4t, use find t in the form alnb when s = 15. x x 10. Solve the equation = 0 (Hint: Replace x with y) 11. Use your GDC to solve x + log 3 x = A team of 3 is to be chosen from 9 volunteers for a general knowledge contest. How many possible teams are there? If the 9 volunteers consist of 5 boys and 4 girls, how many of the possible teams will have more girls than boys? 13. There are 6 letters in the alphabet and 10 numerical digits. A car has a registration number consisting of 3 letters followed by digits. How many possibilities are there? And how many possibilities if all the letters must be different? If there is a free choice of letters, but the digits cannot begin with a 0, and must form an even number, how many possibilities are there? Find the constant term in the expansion of 3x x. 15. Use mathematical induction to prove that the sum of the first n square numbers is given by the formula S n = 1 n( n + 1)( n + 1) Convert to cis form: 1 + i, 3i, -6i, ( + i) 5π 5π 17. Convert to a + bi form: cis60º, 3(cos + i sin ), 4e iπ Solve z = 5 1i. 19. Given that + i is a root of the equation z 3 11z + 0 = 0, find the remaining roots. 0. Find the real numbers p for which 1 + pi is a solution of z - z + (p + 7) = Find a cubic equation (with real coefficients) which has 3 + i and - as two of its roots.. Work out ( i) 4 using De Moivre's theorem. Give your answer in both modulus-argument and Cartesian forms. (The argument should be in radians). 3. Find the fifth roots of 1 + i using De Moivre's theorem. 4. Find the value of a for which the following system of equations does not have a unique solution: 4x y + z = 1 x + 3y = 6 x y + az = k For this value of a, how many solutions are there if k = 3.5, and how many if k = 0? 6 Page 108 IB Math Higher
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