PREFACE. It is hoped that this book will help students to gain confidence in the subject and be better equipped to face the examinations.

Transcription

1 PREFACE Chapter 1 Prime Numbers, Factors and Multiples Secondary 2 Mathematics Tutorial 2A and 2B are designed to prepare Secondary 2 students in their understanding and application of mathematical concepts, skills and processes. What s covered in this book? Written in accordance with the latest syllabus, each chapter includes Objectives, Key Concepts and Formulae and Worked Examples to supplement and complement the lessons taught in school. Practice questions are structured as core, consolidation and challenging to ensure steady improvement and quick mastery of concepts. Books 2A and 2B cover all the topics for the entire school year. Additional feature Teacher s Desk provides important notes, tips, examples and highlights common student errors. Important concepts are highlighted to enhance understanding and retention. Tests and Assessments There are 4 assessment papers: Mid Year Examination Paper 1 and 2 (found in Book 2A) and End of Year Examination Paper 1 and 2 (found in Book 2B). Answer Key Fully worked solutions are provided for students to understand better how each problem is solved. These also serve as a tool for self study and assessment. It is hoped that this book will help students to gain confidence in the subject and be better equipped to face the examinations. The Editorial Team

2 CONTENTS Chapter Objectives Page 1 Maps and Scales Calculate actual distance and map distance Calculate actual area and map area of a region 2 Direct and Inverse Proportion Determine whether two quantities are in direct or inverse proportion Solve direct and inverse proportion problems Expansion and Factorisation of Algebraic Expressions Expanding products of algebraic expressions Expanding and factorising algebraic expressions using special products Factorisation of algebraic expressions by grouping Quadratic Expressions and Equations Factorise a quadratic expression ax 2 + bx + c using multiplication frame Solve quadratic equations ax 2 + bx + c = 0 Solve problems involving quadratic equations Algebraic Manipulation Perform addition, subtraction, multiplication and division of algebraic fractions Change the subject of a formula Solve simple equations involving algebraic fractions Graphs of Quadratic Functions Draw the graph of a quadratic function State the y-intercept, maximum/minimum point and the line of symmetry of a quadratic graph Simultaneous Linear Equations Solve simultaneous equations in two variables by the elimination method Solve simultaneous equations in two variables by the substitution method Solve simultaneous equations in two variables by the graphical method Solve word problems involving simultaneous linear equations Mid Year Examination Paper Mid Year Examination Paper Fully worked solutions S1 S81 Complete the course with Secondary 2 Mathematics Tutorial 2B (Chapters 8 13)

3 CHAPTER 1 Maps and Scales OBJECTIVES Calculate actual distance and map distance Calculate actual area and map area of a region Key Concepts and Formulae 1 Map scale = map length : actual length = 1 : r or 1 r 2 Linear scale = 1 : r 1 cm on map represents r cm on the actual piece of land 1 r is called the representative fraction (R.F.). 3 Area scale is the square of the linear scale, i.e. 1 2 : r 2 1 cm 2 on the map represents r 2 cm 2 on the actual piece of land. 4 Changing between Linear scale and Area scale: Linear scale square square root Area scale 1 Chapter 1 Maps and Scales

4 WORKED EXAMPLE 1 Suppose that l cm on a map represents 30 m on the ground. (a) Express the scale of the map in the form of 1 : n. (b) Find the actual length of a road, in metres, for a map distance of 32 cm. (c) The actual area of a lake is m 2. Find the area of a lake, in cm 2, on the map. Solution: Linear scale square Area scale square root 1 cm : 30 m is used to find the area scale (instead of 1 : 3000), as the numbers are simpler. (a) (b) (c) 1 cm : 30 m = 1 cm : 3000 cm = 1 : 3000 Actual length of road = m = 960 m Squaring each quantity on the linear scale, Area scale = 1 cm 2 : 30 2 m 2 = 1 cm 2 : 900 m 2 Area of lake on the map = cm2 = 200 cm 2 A map scale can be represented as 1 : r or 1 r. WORKED EXAMPLE 2 1 The scale of a map is Find the (a) map distance, in centimetres, for an actual distance of 15 km, (b) actual distance, in metres, for a map distance of 3.5 cm. Solution: Alternatively, map distance 1 = km = 75 cm Alternatively, actual distance = cm = 700 m (a) Scale of map = 1 : = 1 cm : cm = 1 cm : 0.2 km Map distance = cm = 75 cm (b) Actual distance = km = 0.7 km = 700 m 2 Chapter 1 Maps and Scales

5 WORKED EXAMPLE 3 A distance of 3 cm on a map represents an actual distance of 2.1 km. (a) Find the scale of the map in the form 1 r. (b) Find the area of a field on the map if the actual area is km 2. Solution: (a) Map scale = 3 cm : 2.1 km = 3 cm : cm = 3 : = 1 : The scale of the map is (b) Area scale = 3 2 cm 2 : km 2 = 9 cm 2 : 4.41 km 2 Area of field on the map = = 5.4 cm 2 1 r is also known as the representative fraction (R.F.). 2.1 km = 2100 m = cm It is advisable not to convert 2.1 km to cm as the number is large after squaring. WORKED EXAMPLE 4 The actual area of a town, 36 km 2, is represented by 100 cm 2 on a map. (a) Find the scale of the map in the form 1 : n. (b) Find the area of the town on another map with the scale 1 : In this question, you are given area scale first. Solution: (a) Area scale = 100 cm 2 : 36 km 2 Square rooting each quantity, Linear scale = 100 cm : 36 km = 10 cm : 6 km = 5 cm : 3 km = 5 : = 1 : (b) Linear scale = 1 : = 1 cm : 4 km Squaring each quantity, Area scale = 1 cm 2 : 4 2 km 2 = 1 cm 2 : 16 km 2 Area of town on the new map = cm2 = 2.25 cm 2 Area scale square root Linear scale It is neater and simpler when you express cm as 4 km. 1 km = 1000 m = cm 3 Chapter 1 Maps and Scales

6 Core Practice Maps and Scales 1 Express each of the following map scales in the form 1 : n. Convert both quantities to the same unit before finding 1 : n. (a) 2 cm : 50 cm (b) 2 cm : 3 m (c) 5 cm : 60 m (d) 4 cm : 1 km (e) 10 m : 12 km (f) 3 mm : 12 m (g) 0.5 cm : 300 m (h) 0.75 cm : 1.5 km 4 Chapter 1 Maps and Scales

7 2 Find the representative fraction, 1 r, of each of the map scales. Convert both quantities to the same unit before finding 1 r. (a) 3 cm : 60 cm (b) 12 cm : 216 m (c) 8 m : 3 km (d) 7 cm : 28 km (e) 5 mm : 200 cm (f) 4 mm : 2 m (g) 200 cm : 12.5 km (h) 0.4 mm : 0.2 km 5 Chapter 1 Maps and Scales

8 Chapter 1 Core Practice 1. (a) 2 cm : 50 cm = 1 cm : 25 cm = 1 : 25 (b) 2 cm : 3 m = 2 cm : 300 cm = 1 : 150 (c) 5 cm : 60 m = 5 cm : 6000 cm = 1 : 1200 (d) 4 cm : 1 km = 4 cm : cm = 1 : (e) 10 m : 12 km = 10 m : m = 1 : 1200 (f) 3 mm : 12 m = 3 mm : mm = 1 : 4000 (g) 0.5 cm : 300 m = 0.5 cm : cm = 1 : (h) 0.75 cm : 1.5 km = 0.75 cm : cm = 1 : (a) R.F. = 3 cm 60 cm = cm (b) R.F. = 216 m 12 cm = cm 1 = 1800 (c) R.F. = 8 m 3 km 8 m = 3000 m = (d) R.F. = 7 cm 28 km 7 cm = cm 1 = (e) R.F. = 5 mm 200 cm 5 mm = 2000 mm = (f) R.F. = 4 mm 2 m 4 mm = 2000 mm = cm (g) R.F. = 12.5 km 200 cm = cm = mm (h) R.F. = 0.2 km 0.4 mm = mm = (a) Actual length of road = 8 cm 2 cm 6 km = 24 km (b) 0.5 m = 50 cm 50 cm Actual length of road = 2 cm 6 km = 150 km 4. (a) Actual length of road = cm = cm = 750 m (b) Actual length of road = cm = cm = 1300 m 5. (a) Map length of road = 200 m 100 m 4 cm = 8 cm (b) 5 km = 5000 m Map length of road = 5000 m 100 m 4 cm = 200 cm 6. (a) Map length of road = 4500 m = 0.3 m = 30 cm 60 km (b) Map length of road = = km = 400 cm 7. (a) Area scale = 1 2 : 40 2 = 1 : 1600 (b) Area scale = 1 2 : 50 2 = 1 : 2500 (c) Area scale = 1 2 : = 1 : (d) Area scale = 1 2 : = 1 : (a) Area scale = 1 2 cm 2 : 3 2 km 2 = 1 cm 2 : 9 km 2 (b) Area scale = 2 2 cm 2 : 5 2 km 2 = 4 cm 2 : 25 km 2 Secondary 2 Mathematics Tutorial 2A S1 Chapter 1

9 (c) Area scale = 5 2 cm 2 : 12 2 km 2 = 25 cm 2 : 144 km 2 (d) Linear scale = 3 cm : 2 km Area scale = 3 2 cm 2 : 2 2 km 2 = 9 cm 2 : 4 km 2 9. (a) Area scale = 2 2 cm 2 : 3 2 m 2 = 4 cm 2 : 9 m 2 (b) Area scale = 7 2 cm 2 : 2 2 m 2 = 49 cm 2 : 4 m 2 (c) Linear scale = 5 cm : 200 cm = 5 cm : 2 m Area scale = 5 2 cm 2 : 2 2 m 2 = 25 cm 2 : 4 m 2 (d) Linear scale = 3 cm : 800 cm = 3 cm : 8 m Area scale = 3 2 cm 2 : 8 2 m 2 = 9 cm 2 : 64 m 2 (e) Linear scale = 30 cm : 0.05 km = 30 cm : 50 m Area scale = 30 2 cm 2 : 50 2 m 2 = 900 cm 2 : 2500 m 2 (f) Linear scale = 3000 cm : cm = 3 cm : 150 cm = 3 cm : 1.5 m = 6 cm : 3 m Area scale = 6 2 cm 2 : 3 2 m 2 = 36 cm 2 : 9 m (a) Linear scale = 1 cm : 3 km Area scale = 1 cm 2 : 9 km 2 Actual area of field = 2 9 km 2 = 18 km 2 (b) Actual area of field = km 2 = 4.5 km (a) Linear scale = 1 cm : cm = 1 cm : 5 km Area scale = 1 cm 2 : 25 km 2 Actual area of field = km 2 = 250 km 2 (b) Actual area of field = km 2 = 625 km (a) Area scale = 2 2 cm 2 : 5 2 m 2 = 4 cm 2 : 25 m 2 Map area of pool = cm2 = 64 cm 2 (b) Map area of pool = cm2 = 200 cm (a) Linear scale = 1 cm : 400 cm = 1 cm : 4 m Area scale = 1 cm 2 : 16 m 2 Map area of pool = cm2 = 25 cm 2 (b) 0.08 km 2 = ( ) m 2 = m Map area of pool = Consolidation Practice 16 1 cm2 = 5000 cm 2 1. (a) Actual length of railway = km = 15 km (b) Actual perimeter of lake = km = 6.25 km 2. (a) Actual length of bridge = 3 40 m = 120 m (b) Actual length of footpath = m = 268 m 3. (a) 1 cm : 250 cm = 1 cm : 2.5 m Map length of bridge = = 14 cm (b) Map length of track = = 32 cm 4. (a) 1 cm : cm = 1 cm : 5 km Map distance = 15 5 = 3 cm (b) Area scale = 1 cm 2 : 25 km 2 Map area = = 3 cm 2 5. (a) Area scale = 2 2 cm 2 : 50 2 m 2 = 4 cm 2 : 2500 m 2 Actual area of hospital = m2 = 5000 m (b) Map area of farm = cm2 = 19.2 cm 2 6. (a) Actual length of river = m = 1000 m or 1 km (b) Length on 2 nd map = 1000 m 500 = 2 m or 200 cm Secondary 2 Mathematics Tutorial 2A S2 Chapter 1

10 7. (a) Area scale = 1 2 cm 2 : 3 2 km 2 = 1 cm 2 : 9 km 2 Actual area of plantation = km 2 = 42.3 km 2 (b) Area scale of 2 nd map = 1 2 cm 2 : 2 2 km 2 = 1 cm 2 : 4 km 2 Map area on 2 nd map = = cm 2 8. (a) Area scale = 0.8 cm 2 : 3.2 km 2 = 1 cm 2 : 4 km 2 Linear scale = 1 cm : 4 km = 1 cm : 2 km (b) Map distance = = 1.25 cm 9. (a) Area scale = 25 cm 2 : 225 km 2 = 1 cm 2 : 9 km 2 Linear scale = 1 cm : 9 km = 1 cm : 3 km = 1 cm : cm = 1 : (b) Perimeter of forest on map = 72 3 (1 cm : 3 km) = 24 cm (c) Linear scale of 2 nd map = 1 cm : cm = 1 cm : 0.5 km Area scale of 2 nd map = 1 2 cm 2 : km 2 = 1 cm 2 : 0.25 km 2 area of forest on 2 nd map = = 900 cm (a) Area scale = 120 cm 2 : 270 m 2 = 4 cm 2 : 9 m 2 Linear scale = 4 cm : 9 m = 2 cm : 3 m Actual distance of road = m = 120 m (b) Actual area of plot = m2 (4 cm 2 : 9 m 2 ) = m (a) Linear scale = 1 cm : cm = 1 cm : 200 km Map distance between Singapore and Kuala Lumpur = = 1.65 cm (b) Area scale = 1 2 cm 2 : km 2 = 1 cm 2 : km 2 Map area of Indonesia = = 47.5 cm (a) Linear scale of Map A = 1 cm : 200 cm = 1 cm : 2 m Map length of path on Map A = 30 2 = 15 cm Linear scale of Map B = 1 cm : 250 cm = 1 cm : 2.5 m Map length of path on Map B = = 12 cm Difference in length = 15 cm 12 cm = 3 cm (b) Area scale on Map A = 1 2 cm 2 : 2 2 m 2 = 1 cm 2 : 4 m 2 Map area of the field on Map A = = 1800 cm 2 Area scale on Map B = 1 2 cm 2 : m 2 = 1 cm 2 : 6.25 m 2 Map area of the field on Map B = = 1152 cm 2 Difference in area = = 648 cm (a) Area scale = 0.5 cm 2 : 18 km 2 = 1 cm 2 : 36 km 2 Linear scale = 1 cm : 36 km = 1 cm : 6 km = 1 cm : cm = 1 : representative fraction = (b) 16 km 500 m = 16.5 km Map distance = (1 cm : 6 km) = 2.75 cm (c) Linear scale of 2 nd map = 1 cm : cm = 1 cm : 2.5 km Area scale of 2 nd map = 1 2 cm 2 : km 2 = 1 cm 2 : 6.25 km 2 Map area of village on 2 nd map = = 2.88 cm 2 Challenging Practice 1. Distance from Town A to Town B = cm = cm = 30 km Distance from Town B to Town C = cm = cm = 48 km Total distance travelled = = 78 km 2. Linear ratio of Map A : Map B : Map C = 3 : 4 : x Area ratio of Map A : Map B : Map C = 9 : 16 : x 2 (squaring 3 : 4 : x ) Secondary 2 Mathematics Tutorial 2A S3 Chapter 1