PREFACE SUCCESS IN MATHEMATICS is designed with the purpose of assisting students in their preparation for important school and state examinations. Students requiring revision of the concepts covered in earlier years of Secondary school would also find this guide useful. Teachers and tutors too could use this to develop a well-paced study program based on the sequence, coverage and the suggested pre-requisites for the various topics. Based on the latest syllabus, this guide is organized into 28 topics, with the sequence aligned as closely as possible to that generally followed by the schools. Each topic starts with the essential pre-requisite skills to be acquired in order that students could have a better grasp of the concepts covered. The topic is then broken down into sub-topics for a more comprehensive coverage. A topic would have its fundamental methodologies clarified with detailed steps and examples. Where appropriate, steps in the examples are further explained in greater details. Some questions are arranged to stretch the students academic abilities by including skills relevant to other topics. Detailed answers are provided for all questions and where necessary, there are explanations on how the steps are linked. Students are to focus on how to approach questions and acquiring a certain skill rather than merely getting the correct answers for the questions. It is indeed our desire that students reap the utmost benefit of this book, expanding their grasp of the subject while attaining their examination objectives. C. Sivakumaran B. Eng
CONTENTS 1 Numbers... 1 whole numbers highest common factor (HCF) lowest common multiple (LCM) prime factorisation HCF and LCM using prime factorisation fi nding square roots and cube roots fractions conversions of recurring decimals into fractions integers operations involving negative integers rational / irrational numbers numbers in standard form real numbers rounding off whole numbers and decimals 2 Basic Algebra... 14 basic rules in algebra basic evaluations of algebraic expressions simplifying algebraic expressions opening brackets for simplifying types of common algebraic expressions solving common algebraic equations solving a linear equation 3 Arithmetic Problems... 21 ratio and percentage average rate direct / inverse proportion profit and loss utilities interest hire purchase money exchange taxation commission 4 Basic Geometry... 33 common terms common angle formations at a vertex common angle correlations 5 Linear Inequalities... 40 basic rules in inequalities solving inequalities solving simultaneous inequalities 6 Indices... 45 basic rules of indices standard form numbers 7 Variations... 51 direct proportions inverse proportions 8 Expansion And Factorisation... 54 basic identities in algebra expanding algebraic expressions factorisation using identities factorisation involving four terms quadratic factorisation 9 Quadratic Equations... 61 simplifying techniques factorisation completing the square quadratic formula method graphical method
10 Algebraic Manipulations... 71 reducing algebraic fractions multiplying and dividing algebraic fractions HCF and LCM of algebraic terms evaluating algebraic fractions equations involving algebraic fractions changing subject of formulae 11 Co-ordinate Geometry... 79 length of a line segment gradient of a line (or line segment) equation of a line (or line segment) 12 Pythagoras Theorem... 88 pythagoras theorem applications 13 Triangles, Quadrilaterals And Polygons... 93 triangles quadrilaterals polygons rules applicable for regular polygons 14 Basic Trigonometry... 102 right-angled triangles radian angles 15 Functions And Graphs... 110 graphs with n = 0 graphs with n = 1 graphs with n = 2 graphs with n = 3 graphs with n = 1 graphs with n = 2 fi nding the value a in a graph with n value known exponential functions gradient of graphs sketching quadratic functions solving equations using graphs 16 Simultaneous Equations... 124 elimination method 17 Practical Graphs... 129 simple conversion graph distance-time graph speed-time graph 18 Congruency And Similarity... 138 congruent triangles similar triangles applications of similar triangles similar fi gures in general map scale 19 Set Language... 148 basics involving a set subsets complement set union set intersection set venn diagram 20 Matrices... 158 order of a matrix elements of a matrix addition and subtraction of matrices multiplication of a matrix by a scalar multiplication of two matrices special matrices and rules
21 Circle Properties... 166 perpendicular radius and chord angles in the same segment angles in a cyclic quadrilateral properties of tangents 22 Trigonometry... 176 obtuse-angled triangles area of triangle sine rule cosine rule bearings 23 Mensuration... 187 basic plane fi gures area of sectors and length of arcs prisms spheres pyramids cones 24 Vectors... 200 basics of vectors addition of vectors numerical values of vectors modulus of a vector numerical addition of vectors multiplication of a vector by a scalar geometrical applications of vectors 25 Probability... 215 basic definitions definition of probability complementary events tree diagram mutually exclusive events independent events events in without replacement situations 26 Basic Statistics... 223 constructing tables bar graph pictogram line graph pie chart histograms dot diagrams stem-and-leaf diagrams mean mode median 27 Cumulative Frequency... 236 cumulative frequency distribution box-and-whisker plot standard deviation comparison of two sets of data 28 Number Patterns And Problem Solving... 248 sum of positive integers arithmetic progression other sequences some problem solving techniques Solutions... S1-S97
1 NUMBERS Pre-requisite: Primary Level Mathematics GENERAL Ability with numbers is an imperative practically in any field of study, whether it is as out of the world as astronomy or as down to earth as agriculture. For the purpose of understanding the fundamentals of operating numerical values, numbers are located on a notional, imaginary line known as the number line. 0 1 2 3 4 5 6 Fig. 1.1 A number line showing the fi rst few whole numbers. When operating on numerical values, the order of operations has to be adhered to with the level of bracketed operations taken into account. Besides the usual order of operations, students should also be adept at carrying out calculations involving powers and roots (covered later in detail under the topic Indices). WHOLE NUMBERS Note The presence of a number in a particular category does not necessarily exclude it from the other categories. These are the basic numerical quantities that are used in daily applications like commercial transactions. Whole numbers could be further categorised as follows: (a) Natural numbers (or counting numbers) : 1, 2, 3, 4, 5, (basically whole numbers with 0 excluded) (b) Odd numbers : 1, 3, 5, 7, 9, (whole numbers which do not give a whole number value when divided by 2) (c) Even numbers : 0, 2, 4, 6, 8, (whole numbers which give a whole number value when divided by 2) (d) Prime numbers : 2, 3, 5, 7, 11, (whole numbers, excluding 0 and 1, which give a whole number value only when divided by 1 and by itself) 1 Topic 1 Numbers
Note 0 and 1 are neither prime nor composite numbers. (e) Composite numbers : 4, 6, 8, 9, 10, 12, (whole numbers, excluding 0 and 1, which give a whole number value when divided by 1, by itself and by at least another whole number) (f) Perfect square numbers : 0, 1, 4, 9, 16, 25, (whole numbers which are obtained by multiplying a whole number by itself, ie. 0 2, 1 2, 2 2, 3 2, 4 2, 5 2,...) (g) Perfect cube numbers : 0, 1, 8, 27, (whole numbers which are obtained by multiplying a whole number by itself 3 times, ie. 0 3, 1 3, 2 3, 3 3, 4 3, 5 3,...) HIGHEST COMMON FACTOR (HCF) For two or more whole numbers, the largest of the common factors would be the HCF. To determine the HCF, the following method could be used: (i) the numbers are divided by their common factors (need not be prime factors) until there are no more common factors, except 1; and (ii) obtain the HCF which is the product of the common factors used in the division. Find the HCF of 24, 36 and 42. 2 24, 36, 42 3 12, 18, 21 4, 6, 7 (no more common factors except 1) HCF of 24, 36 and 42 is 2 3 = 6 (Note that the process could have been shorter if the numbers were divided by 6 in the first step.) LOWEST COMMON MULTIPLE (LCM) For two or more whole numbers, the smallest of the common multiples would be the LCM. To determine the LCM, the following method could be used: (i) the numbers are divided by their prime factors, starting from the smallest prime factor as long as it divides at least one of the numbers; 2 Topic 1 Numbers
(ii) if a prime factor is not a factor of any one of the numbers, that number is just carried over without dividing; (iii) the process is repeated until all the numbers are reduced to 1 by division; and (iv) the LCM is obtained as the product of the prime factors. Find the LCM of 24, 36 and 42. 2 24, 36, 42 2 12, 18, 21 2 6, 9, 21 (Since 2 is not a factor of 21, it is not divided.) 3 3, 9, 21 (Since 2 is not a factor of 9 and 21, they are not divided.) 3 1, 3, 7 7 1, 1, 7 (Since 3 is not a factor of 1 and 7, they are not divided.) 1, 1, 1 LCM of 24, 36 and 42 is 2 2 2 3 3 7 = 504 PRIME FACTORISATION Prime factorisation is the process where a composite number is written out as a product of its prime factors. This process is useful in determining the HCF and LCM of numbers as well as for finding the roots of numbers. For prime factorisation of a number, start by dividing the number by its smallest prime factor and continuing until the quotient obtained by the division is a prime number. Obtain the prime factors of 504. 504 = 2 252 = 2 2 126 = 2 2 2 63 = 2 2 2 3 21 = 2 2 2 3 3 7 (no more composite number to divide) Prime factors of 504 are 2 2 2 3 3 7 The prime factors could also be stated in index notation as: 504 = 2 3 3 2 7 3 Topic 1 Numbers
HCF AND LCM USING PRIME FACTORISATION The earlier examples of finding the HCF and LCM of 24, 36 and 42 using prime factorisation is illustrated and students are to note the relative ease of the prime factorisation method in finding the HCF and LCM. First state the prime factors of the numbers in index notation: 24 = 2 3 3 1 36 = 2 2 3 2 42 = 2 1 3 1 7 1 For HCF, take the common prime factors with the lowest power. HCF of 24, 36, 42 is 2 1 3 1 = 6 2 is a common prime factor and 1 is its lowest power. 3 is a common prime factor and 1 is its lowest power. 7 is not a common prime factor and is not taken. For LCM, take all prime factors of the numbers with their highest powers. LCM of 24, 36, 42 is 2 3 3 2 7 1 = 504 2 has a highest power of 3. 3 has a highest power of 2. 7 is only present in 42 and the power is 1. FINDING SQUARE ROOTS AND CUBE ROOTS Note A perfect square has all the powers of its prime factors as multiples of 2 while a perfect cube has all the powers of its prime factors as multiples of 3. (i) To find the square root of 441, the number is prime factorised and written as follows: 441 = 3 2 7 2 The square root is the value obtained by dividing the powers by 2: 441 = 3 1 7 1 = 21 (ii) To find the cube root of 1728, the number is prime factorised and written as follows: 1728 = 2 6 3 3 The cube root is the value obtained by dividing the powers by 3: 3 1728 = 2 2 3 1 = 12 4 Topic 1 Numbers
FRACTIONS As numerical quantities cannot always be stated as whole numbers, they have to be stated frequently as fractions. While a fraction could be read as the number of portions when a whole quantity is divided equally, it is best seen at this stage as the value obtained by dividing the numerator by the denominator. Fractions could be in the form of proper fractions, improper fractions or mixed numbers. Decimals and percentages are alternative ways of stating fractional quantities. Some fractions, when converted into decimals, could not be expressed as decimals in the usual manner and hence, have to be approximated by rounding off or stated as a recurring decimal. 1 3 has to be approximated as 0.3 or 0.33 if written as correct to one or two decimal places respectively. If written as a recurring decimal, it is to be written as 0. 3. It implies that the digit 3 is repeated indefinitely. 1_ 2 Shown below are some fractions on a number line appropriately positioned according to their values. 1 2_ 3 4 9 10 6 1_ 5 0 1 2 3 4 5 6 7 Fig. 1.2 A number line showing some fractions, as indicated by arrows, among the whole numbers. CONVERSIONS OF RECURRING DECIMALS INTO FRACTIONS (1) 0.333333 = 0. 3 (2) 2.777777 = 2. 7 (3) 1.2080808 = 1.2 08 (4) 1.512512 = 1. 51 2 5 Topic 1 Numbers