New Syllabus Mathematics 7th Edition Book 2 Answers

Unformatted text preview: 7th EDITION NEW SYLLABUS MATHEMATICS TEACHER'S RESOURCE BOOK 2 1 Revision Exercise C1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 Revision Exercise C2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 Chapter 11: Congruence and Similarity Tests Teaching Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 Worked Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 Chapter 12: Area and Volume of Similar Figures and Solids Teaching Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 Worked Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 Chapter 13: Geometrical Properties of Circles Teaching Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 Worked Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 Revision Exercise D1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 Revision Exercise D2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 Problems in Real-World Contexts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 3 CONTENTS Syllabus Matching Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Scheme of Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Chapter 1: Direct and Inverse Proportions Teaching Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Worked Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Chapter 2: Linear Graphs and Simultaneous Linear Equations Teaching Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Worked Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Chapter 3: Expansion and Factorisation of Quadratic Expressions Teaching Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Worked Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Chapter 4: Further Expansion and Factorisation of Algebraic Expressions Teaching Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Worked Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Revision Exercise A1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Revision Exercise A2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 Chapter 5: Quadratic Equations and Graphs Teaching Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 Worked Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Chapter 6: Algebraic Fractions and Formulae Teaching Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 Worked Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 Chapter 7: Relations and Functions Teaching Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 Worked Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 Chapter 8: Congruence and Similarity Teaching Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 Worked Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 Chapter 9: Geometrical Transformation Teaching Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 Worked Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 Revision Exercise B1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 Revision Exercise B2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 Chapter 10: Pythagoras' Theorem Teaching Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 Worked Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 2 iii 1 Chapter 11: Trigonometric Ratios Teaching Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 Worked Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 Chapter 12: Volume and Surface Area of Pyramids, Cones and Spheres Teaching Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 Worked Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 Chapter 13: Symmetry Teaching Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 Worked Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 Revision Exercise C1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 Revision Exercise C2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 Chapter 14: Sets Teaching Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 Worked Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 Chapter 15: Probability of Single Events Teaching Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 Worked Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 Chapter 16: Statistical Diagrams Teaching Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 Worked Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 Chapter 17: Averages of Statistical Data Teaching Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 Worked Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 Revision Exercise D1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 Revision Exercise D2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 Problems in Real-World Contexts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 1 iv 3 Syllabus Matching Grid Cambridge O Level Mathematics (Syllabus D) 4024/4029. Syllabus for examination in 2018, 2019 and 2020. Theme or Topic Subject Content Reference 1. Number Identify and use: • Natural numbers • Integers (positive, negative and zero) • Prime numbers • Square numbers • Cube numbers • Common factors and common multiples • Rational and irrational numbers (e.g. p, 2 ) • Real numbers Book 1: Chapter 1 Chapter 2 2. Set language and notation • Use set language, set notation and Venn diagrams to describe sets and represent relationships between sets • Definition of sets: e.g. A = {x : x is a natural number}, B = {(x, y): y = mx + c}, C = {x : a < x < b}, D = {a, b, c, …} Book 2: Chapter 14 Book 4: Chapter 2 2. Squares, square roots, cubes and cube roots Calculate • Squares • Square roots • Cubes and cube roots of numbers Book 1: Chapter 1 Chapter 2 4. Directed numbers • Use directed numbers in practical situations Book 1: Chapter 2 5. Vulgar and decimal fractions and percentages • Use the language and notation of simple vulgar and decimal fractions and percentages in appropriate contexts • Recognise equivalence and convert between these forms Book 1: Chapter 2 6. Ordering • Order quantities by magnitude and demonstrate familiarity with the symbols =, ≠, <, >, <, >. Book 1: Chapter 2 Chapter 5 7. Standard form • Use the standard form A × 10n, where n is a positive or negative integer, and 1 < A < 10. Book 3: Chapter 4 8. The four operations Use the four operations for calculations with: • Whole numbers • Decimals • Vulgar (and mixed) fractions including correct ordering of operations and use of brackets. Book 1: Chapter 2 9. Estimation • Make estimates of numbers, quantities and lengths • Give approximations to specified numbers of significant figures and decimal places • Round off answers to reasonable accuracy in the context of a given problem Book 1: Chapter 3 10 Limits of accuracy • Give appropriate upper and lower bounds for data given to a specified accuracy • Obtain appropriate upper and lower bounds to solutions of simple problems given to a specified accuracy Book 3: Chapter 3 11 1 11. Ratio, proportion, rate 12. Percentages 13. Use of an electronic calculator • • • • Demonstrate an understanding of ratio and proportion Increase and decrease a quantity by a given ratio Use common measures of rate Solve problems involving average speed Book 1: Chapter 9 • • • • Calculate a given percentage of a quantity Express one quantity as a percentage of another Calculate percentage increase or decrease Carry out calculations involving reverse percentages Book 1: Chapter 8 • • • • Use an electronic calculator efficiently Apply appropriate checks of accuracy Enter a range of measures including 'time' Interpret the calculator display appropriately Book 1: Chapter 2 Chapter 4 Book 2: Chapter 1 Book 3: Chapter 5 Book 2: Chapter 11 Book 3: Chapter 10 Book 4: Chapter 4 14. Time • Calculate times in terms of the 24-hour and 12-hour clock • Read clocks, dials and timetables Book 1: Chapter 9 15. Money • Solve problems involving money and convert from one currency to another Book 3: Chapter 5 16. Personal and small business finance • Use given data to solve problems on personal and small business finance involving earnings, simple interest and compound interest • Extract data from tables and charts Book 3: Chapter 5 17. Algebraic representation and formulae • Use letters to express generalised numbers and express arithmetic processes algebraically • Substitute numbers for words and letters in formulae • Construct and transform formulae and equations Book 1: Chapter 4 Chapter 5 Book 2: Chapter 2 Book 3: Chapter 1 18. Algebraic manipulation 19. Indices 1 • • • • Manipulate directed numbers Use brackets and extract common factors Expand product of algebraic expressions Factorise where possible expressions of the form: ax + bx + kay + kby a2x2 – b2y2 a2 + 2ab + b2 ax2 + bx + c • Manipulate algebraic fractions • Factorise and simplify rational expressions Book 1: Chapter 4 • Understand and use the rules of indices • Use and interpret positive, negative, fractional and zero indices Book 3: Chapter 4 2 Book 2: Chapter 3 Chapter 4 Chapter 6 20. Solutions of equations and inequalities • • • • Solve simple linear equations in one unknown Solve fractional equations with numerical and linear algebraic denominators Solve simultaneous linear equations in two unknowns Solve quadratic equations by factorisation, completing the square or by use of the formula • Solve simple linear inequalities Book 1: Chapter 5 Book 2: Chapter 2 Chapter 5 Book 3: Chapter 1 Chapter 3 21. Graphical representation of inequalities • Represent linear inequalities graphically Book 4: Chapter 1 22. Sequences • Continue a given number sequence • Recognise patterns in sequences and relationships between different sequences • Generalise sequences as simple algebraic statements Book 1: Chapter 7 23. Variation • Express direct and inverse variation in algebraic terms and use this form of expression to find unknown quantities Book 2: Chapter 1 24. Graphs in practical situations • Interpret and use graphs in practical situations including travel graphs and conversion graphs • Draw graphs from given data • Apply the idea of rate of change to easy kinematics involving distance-time and speed-time graphs, acceleration and deceleration • Calculate distance travelled as area under a linear speed-time graph Book 1: Chapter 6 • Construct tables of values and draw graphs for functions of the form axn where a is a rational constant, n = –2, –1, 0, 1, 2, 3, and simple sums of not more than three of these and for functions of the form kax where a is a positive integer • Interpret graphs of linear, quadratic, cubic, reciprocal and exponential functions • Solve associated equations approximately by graphical methods • Estimate gradients of curve by drawing tangents Book 1: Chapter 6 • Use function notation, e.g. f(x) = 3x – 5; f : x 3x – 5, to describe simple functions • Find inverse functions f–1(x) Book 2: Chapter 7 • Demonstrate familiarity with Cartesian coordinates in two dimensions • Find the gradient of a straight line • Calculate the gradient of a straight line from the coordinates of two points on it • Calculate the length and the coordinates of the midpoint of a line segment from the coordinates of its end points • Interpret and obtain the equation of a straight line graph in the form y = mx + c • Determine the equation of a straight line parallel to a given line • Find the gradient of parallel and perpendicular lines Book 1: Chapter 6 25. Graphs in practical situations 26. Function notation 27. Coordinate geometry 33 Book 2: Chapter 2 Book 3: Chapter 7 Book 2: Chapter 1 Chapter 2 Chapter 5 Book 3: Chapter 1 Chapter 7 Book 3: Chapter 2 Book 2: Chapter 2 Book 3: Chapter 6 1 • Use and interpret the geometrical terms: point; line; plane; parallel; perpendicular; bearing; right angle, acute, obtuse and reflex angles; interior and exterior angles; similarity and congruence • Use and interpret vocabulary of triangles, special quadrilaterals, circles, polygons and simple solid figures • Understand and use the terms: centre, radius, chord, diameter, circumference, tangent, arc, sector and segment Book 1: Chapter 10 Chapter 11 • Measure lines and angles • Construct a triangle, given the three sides, using a ruler and a pair of compasses only • Construct other simple geometrical figures from given data, using a ruler and protractor as necessary • Construct angle bisectors and perpendicular bisectors using a pair of compasses as necessary • Read and make scale drawings • Use and interpret nets Book 1: Chapter 12 Chapter 14 • Solve problems and give simple explanations involving similarity and congruence • Calculate lengths of similar figures • Use the relationships between areas of similar triangles, with corresponding results for similar figures, and extension to volumes and surface areas of similar solids Book 2: Chapter 8 • Recognise rotational and line symmetry (including order of rotational symmetry) in two dimensions • Recognise symmetry properties of the prism (including cylinder) and the pyramid (including cone) • Use the following symmetry properties of circles: (a) equal chords are equidistant from the centre (b) the perpendicular bisector of a chord passes through the centre (c) tangents from an external point are equal in length Book 2: Chapter 13 • Calculate unknown angles and give simple explanations using the following geometrical properties: (a) angles at a point (b) angles at a point on a straight line and intersecting straight lines (c) angles formed within parallel lines (d) angle properties of triangles and quadrilaterals (e) angle properties of regular and irregular polygons (f) angle in a semi-circle (g) angle between tangent and radius of a circle (h) angle at the centre of a circle is twice the angle at the circumference (i) angles in the same segment are equal (j) angles in opposite segments are supplementary Book 1: Chapter 10 Chapter 11 33. Loci • Use the following loci and the method of intersecting loci for sets of points in two dimensions which are: (a) at a given distance from a given point (b) at a given distance from a given straight line (c) equidistant from two given points (d) equidistant from two given intersecting straight line Book 4: Chapter 8 34. Measures • Use current units of mass, length, area, volume and capacity in practical situations and express quantities in terms of larger or smaller units Book 1: Chapter 13 Chapter 14 28. Geometrical terms 29. Geometrical constructions 30. Similarity and congruence 31. Symmetry 32. Angles 1 4 Book 2: Chapter 8 Book 3: Chapter 9 to Chapter 13 Book 2: Chapter 8 Book 4: Chapter 8 Book 3: Chapter 11 Chapter 12 Book 3: Chapter 13 Book 3: Chapter 13 35. Mensuration 36. Trigonometry 37. Vectors in two dimensions • Solve problems involving: (a) the perimeter and area of a rectangle and triangle (b) the perimeter and area of a parallelogram and a trapezium (c) the circumference and area of a circle (d) arc length and sector area as fractions of the circumference and area of a circle (e) the surface area and volume of a cuboid, cylinder, prism, sphere, pyramid and cone (f) the areas and volumes of compound shapes Book 1: Chapter 13 Chapter 14 • Interpret and use three-figure bearings • Apply Pythagoras' theorem and the sine, cosine and tangent ratios for acute angles to the calculation of a side or an angle of a right-angled triangles • Solve trigonometrical problems in two dimensions involving angles of elevation and depression • Extend sine and cosine functions to angles between 90° and 180° • Solve problems using the sine and cosine rules for any triangle and the 1 formula area of triangle = ab sin C 2 • Solve simple trigonometrical problems in three dimensions Book 2: Chapter 10 Chapter 11 ⎛ x⎞ → • Describe a translation by using a vector represented by ⎜⎝ y ⎟⎠ , AB or a • Add and subtract vectors • Multiple a vector by a scalar ⎛ x⎞ • Calculate the magnitude of a vector ⎜⎝ y ⎟⎠ as x 2 + y 2 • Represent vectors by directed line segments • Use the sum and difference of two vectors to express given vectors in terms of two coplanar vectors • Use position vectors Book 2: Chapter 12 Book 3: Chapter 10 Book 3: Chapter 8 Chapter 9 Book 4: Chapter 7 38. Matrices • Display information in the form of a matrix of any order • Solve problems involving the calculation of the sum and product (where appropriate) of two matrices, and interpret the results • Calculate the product of a matrix and a scalar quantity • Use the algebra of 2 × 2 matrices including the zero and identity 2 × 2 matrices • Calculate the determinant |A| and inverse A–1 of a non-singular matrix A Book 4: Chapter 5 39. Transformations • Use the following transformations of the plane: reflection (M), rotation (R), translation (T), enlargement (E) and their combinations • Identify and give precise descriptions of transformations connecting given figures • Describe transformations using coordinates and matrices Book 2: Chapter 9 • Calculate the probability of a single event as either a fraction or a decimal • Understand that the probability of an event occurring = 1 – the probability of the event not occurring • U...
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