Mathematics Gr11 SG PDF
Preview Mathematics Gr11 SG
R.Cloete cm Via Afrika Mathematics Grade 11 * Contents Introduction . ‘Chapter Exponents and surds .. Unit The quadratic formula... Unit 4 Quadratic Inequalities. ‘Units Simultaneous equations. Unit Word problem: UUnit7 The nature of oots.. Questions. Chapter 3 Number pattern: OVERMEN a Units Number pattern witha constant ceconddifeence ‘Questions. Chapter 4 Analytical Geometry. overnew. Units The inclination of in Unit2 The equation of straight ine Questions. Chapter 5 Functions... overvew. Unita Investigating the effec of the paramet ‘Unit 2 Average gradient between two polnts on a curve. Unit3 Teigonometrc gra Questions. Chapter 6 Trigonometry Overnew. Unita Tigonometi dent Unit2 Applying the trigonometric identities Unit3 Reduction formulae. Unit Negative anges. Unies Sobingtrgonometi equations. Questions... © Via Afrika » Mathematics Grade +4 CChapter7 Measurement. overnew. Units Combined objets Questions. Chapter 8 Euclidean Geometry. overnew. Units Circles. Unit 2 Cyelle quadrilateral Unita Tangents toa cirt.. Questions on Chapter 9 Trigonometry (area, sine, cosine TUl€S)naninnininennnmnnnnnnenns 9 overnew. Units The area rule. ‘Unit 2 The sine rule Unita The cose rte Unie Solving problems nwo dimensions Questions. ‘Chapter 10 Finance, growth and decay overvew.. nies Compound growth ‘unit2 Decay. Question Chapter 11 Probability, OVERMEN Units Combinations of events Unit 2 Dependent and independent events. Unit3 Tree diagram: Questions. Chapter 2 Statistics Overnew. Units Histograms. Unit Variance and standard deviction of ungrouped data. Units Symmetrical and skewed data... Unit Identifying outliers. © Via Afrika » Mathematics Grade +4 Introduction to Via Afrika Mathematics Grade 11 Study Guide ‘Woohoo! You made it! Ifyou're reading this it means that you made it through Grade 10, and are now in Grade 11. But I guess you are already well aware ofthat... Italso means that your teacher was brilliant enough to get the Via Affka Mathematics Grade 11 Learner's Book. This study guide contains summaries of each chapter, and should bbe used side-by-side with the Leamer's Book. It also contains lots of extra questions to help you master the subject matter. ‘Mathematics - not for spectators ‘You won't learn anything if you don’t involve yourself in the subject-matter actively. Do ‘the maths, feel the maths, and then understand and use the maths. Understanding the principles + Listen during class, This study guide is brilliant but it is not enough. Listen to your teacher in class as you may learn a unique or easy way of doing something. * Study the notation, properly. Incorrect use of notation will be penalised in tests and exams, Pay attention to notation in our worked examples, + Practise, Practise, Practise, and then Practise some more, You have to practise as much as possible. The more you practise, the more prepared and confident you will feel for exams. This guide contains lots of extra practice ‘opportunities. + Persevere. We can'tall be Einsteins, and even old Albert had difficulties learning some of the very advanced Mathematics necessary to formulate his theories. Ifyou don't understand immediately, work at it and practise with as many problems from this study guide as possible. You will find that topics that seem baffling at first, suddenly make sense. + Have the proper attitude, You can doit! ‘The AMA of Mathematics ABILITY is what you're capable of doing. MOTIVATION determines what you do. ATTITUDE determines how well you do it. ‘© Via Afvika » Mathematics Grade 1 “Give me a place to stand, and I will move the earth!" Archimedes (© Via Afrika » Mathematics Grade 11 2 Exponents and Surds Overview + Types of surds + Multiplying and dividing surds + Equations with surds In this chapter we review the laws of exponents and exponential equations. When we've covered that, we will have a look at rational exponents and surds. You will also learn how to solve exponential equations, simplify surds and solve equations containing surds, (© Via Afrika » Mathematics Grade 1 Rational exponents 1.1 Exponents and surds ‘+ Theexponent ofa number ells you how many times the number has to be multiplied by itself. + Asurd is a number that cannot be simplified further to xe ‘remove the root, They are irrational numbers. he * We always assume that a root without a number in front of it is a square root. ‘The square root ofa number a can be written as Vor in exponential form a**, ‘The cube root ofa number b can be written as Vb, or in exponential form b™*, ‘The nth root ofa number can be written as Vé, or in exponential form ¢ ”, In the expression 4/64, the 3s the order of the radical ‘and 64 is the radicand. We read V64 as the 3rd root of 64. ":!\. + Inthe exponential form, the base of the expression is the Nae radicand. ‘+ Remember, if we have the equation Va - a" and we raise both sides to the power 4k, we have (Ya)* = (a°/*), this simplifies to Ya" = a", ‘Simplify the following equations without using a calculator: 1 278 = (Bt) 3 Chatbiety2 = 2tatpitesyi/2 = 2a*htet (© Via Afrika » Mathematics Grade 1 4 on Ga) 1.2 Exponential equations ‘+ Some exponential equations have only one solution, while others have more. *+ Remember, this can be generalised as: Ifx*/> = c, where cis constant, then © {fais odd, there is only one solution. (© fais even, there are two solutions. One will be positive, and one will be negative, Solve the following equations without using a calculator: bey = (ay? ee Ee? ax = 432 (© Via Afrika » Mathematics Grade 1 5 Unit Surds 2a Types of surds {A surd is an irrational number and itcontains a radical. Wecan use the following laws to help us simplify expressions: © Product rule: 'Va.¥5~=Va.b © Quotient rule: "Va/"Vb = "/a75_ ‘Note that these laws only apply to multiplication and division, and a > 0 and b > °. ‘When we simplify surds, we write the numbers as the product of perfect squares and other numbers, eg. V8 = G2) = 2vE ‘Simplify the following without using a calculator: 3 2V5 + OVS — 16V3 + 6V3 = VE 2VB + OVE — 16V3 + 6V2 = 15V2 — 143 VIS ~ VIB = VI53-V9Z = VEE - VE =5V3 — 32 (HB + V2T)/N7S = (WIGS + V9B)/V253 = (TENE + VNB) VENT = (4N3 + 3V3)/5V3 = NB/SNE (© Via Afrika » Mathematics Grade 1 6 Ga) 2.2 Multiplying and dividing surds ‘+ Tomultiply or divide surds, we often need to use the distributive property: a(b +c) = ab + be ‘+ Some problems need to be solved by rationalisation. Rationalisation is the process ‘where we convert a denominator/numerator with an irrational number to a rational ‘number. We do this by multiplying the expression by the surd divided by itself. ‘Simplify the following without using a calculator: 1 5QVE +9) = 10VS + 45 2 (W5-6)Q2V6 + 8) = 2V30- 12V6 + BVE- 48 3 (WE +3)(5-a)= 92-4 2.3 Equations with surds ‘+ Tosolve equations that contain surds, we first have to remove the surd. ‘+ Toremove the surd, we have to raise both sides to the order of the radical, e.g. if we have to solve an equation with a square root, we first square both sides. + Check your solution! [Example 5s ‘Solve the following without the use of a calculator: 1 VEtS = resco ar 2 vet2-x=0 vet2 =x (© Via Afrika » Mathematics Grade 1 7
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