CONTENTS Form 4 CHAPTER 1 Quadratic Functions and Equations in One Variable ............1 CHAPTER 2 Number Bases .............................12 CHAPTER 3 Logical Reasoning .......................23 CHAPTER 4 Operations on Sets ......................33 CHAPTER 5 Network in Graph Theory.............43 CHAPTER 6 Linear Inequalities in Two Variables...............................56 CHAPTER 7 Graphs of Motion .........................69 CHAPTER 8 Measures of Dispersion for Ungrouped Data ..........................81 CHAPTER 9 Probability of Combined Events...97 CHAPTER 10 Consumer Mathematics: Financial Management ..............109 Form 5 CHAPTER 1 Variation.....................................116 CHAPTER 2 Matrices .....................................129 CHAPTER 3 Consumer Mathematics: Insurance ...................................144 CHAPTER 4 Consumer Mathematics: Taxation .....................................158 CHAPTER 5 Congruency, Enlargement and Combined Transformations........173 CHAPTER 6 Ratios and Graphs of Trigonometric Functions ............189 CHAPTER 7 Measures of Dispersion for Grouped Data ............................202 CHAPTER 8 Mathematical Modeling..............217 ii Contents GrabME Mathematics SPM F4.indd 2 28/10/2022 11:58 AM
1 CHAPTER QUADRATIC FUNCTIONS AND EQUATIONS IN ONE VARIABLE 1 • A quadratic expression is a quadratic algebra in the form of ax2 + bx + c, such that a, b and c are constants, a ≠ 0 and x is a variable. (a) Case 1: b = 0 such as ax2 + c. For example, 3x2 – 18. (b) Case 2: c = 0 such as ax2 + bx. For example, 3x2 + 5x. (c) Case 3: b = c = 0. For example, 3x2 . • The general form of a quadratic function: f(x) = ax2 + bx + c such that x is a variable, a, b and c are constants and a ≠ 0. A quadratic function must be: (a) Involving only one variable. (b) The highest power of the variable is two. REMEMBER • The following are the examples of the graph of quadratic functions, f(x) = ax2 + bx + c: O x p q (p, q) f(x) Shape of graph: Value a . 0 Point (p, q) is a minimum point Axis of symmetry: x = p O x p q (p, q) f(x) Shape of graph: Value a , 0 Point (p, q) is a maximum point Axis of symmetry: x = p 1.1 Quadratic Functions and Equations Relationship and Algebra 01 GrabME Mathematics SPM F4.indd 1 28/10/2022 11:38 AM
2 • Effects of changing the values of a, b and c on the graph of quadratic function, f(x) = ax2 + bx + c: y-intercept lies on the positive side of the y-axis Value of a affects the shape of the graph Value of b affects the position of the axis of symmetry Value of c affects the position of y-intercept x a = 1 a = –1 a = a = 3 O 1 – –2 f(x) The smaller the value of a, the wider the curved shape of the graph and vice versa a . 0 x O b < 0 f(x) x O b = 0 f(x) x O b > 0 f(x) Axis of symmetry lies on the right of the y-axis Axis of symmetry is the y-axis Axis of symmetry lies on the left of the y-axis a , 0 x O b < 0 f(x) x O b = 0 f(x) x b > 0 f(x) O Axis of symmetry lies on the left of the y-axis Axis of symmetry is y-axis Axis of symmetry lies on the right of the y-axis a . 0 x O f(x) c x O f(x) c a , 0 01 GrabME Mathematics SPM F4.indd 2 28/10/2022 11:38 AM
7 Example 8 Given one of the root for quadratic equation 2x2 + 3kx – 8 = 0 is 4. Find the value of k. Solution Substitute x = 4, 2(4)2 + 3k(4) – 8 = 0 32 + 12k – 8 = 0 12k = –24 k = –2 Example 9 Solve (x – 1)(x – 2) = 20. Solution (x – 1)(x – 2) = 20 x2 – 2x – x + 2 = 20 Expand x2 – 3x – 18 = 0 Express in general form (x + 3)(x – 6) = 0 Solve by factorisation x + 3 = 0 or x – 6 = 0 x = – 3 or x = 6 TIPS The solution of a quadratic function also known as x-intercept for the graph. Example 10 Solve the following equation. (2x – 5)2 – 36 = 0 Solution (2x – 5)2 – 36 = 0 4x2 – 10x – 10x + 25 – 36 = 0 4x2 – 20x – 11 = 0 (2x – 11)(2x + 1) = 0 2x – 11 = 0 or 2x + 1 = 0 x = 11 2 or x = – 1 2 Another Method Use formula a2 – b2 = (a – b)(a + b): (2x – 5)2 – 36 = 0 (2x – 5)2 – 62 = 0 ((2x – 5) – 6)((2x – 5) + 6) = 0 (2x – 11)(2x + 1) = 0 2x – 11 = 0 or 2x + 1 = 0 x = 11 2 or x = – 1 2 01 GrabME Mathematics SPM F4.indd 7 28/10/2022 11:38 AM
33 CHAPTER 4 OPERATIONS ON SETS • The intersection between sets A and B, can be written as A B, is a set that contains the common elements of both set A and set B. A B A B • The complement of the intersection of sets A and B refers to all the elements which are not in the intersection of sets A and B. • The complement of the intersection of sets is written by using the symbol “ ’ ”. A B ξ (A B) Example 1 It is given that set P = {factors of 30} and set Q = {prime numbers which are less than 10}. (a) Draw the Venn diagram and shade the region that represents the intersection P Q. (b) List all the elements of the intersection P Q. (c) Determine the value of n(P Q). Solution P = {1, 2, 3, 5, 6, 10, 15, 30} Q = {2, 3, 5, 7} (a) P Q ● 5 ● 2 ● 3 ● 7 ● 1 ● 6 ● 10 ● 15 ● 30 (b) P Q = {2, 3, 5} (c) n(P Q) = 3 The symbol for intersection of two or more sets is . REMEMBER 4.1 Intersection of Sets Discrete Mathematics 04 GrabME Mathematics SPM F4.indd 33 28/10/2022 11:46 AM
37 4.2 Union of Sets • The union of sets A and B, can be written as A B, is a set that represents all the elements in set A or set B or in both sets A and B. A B A B • The complement of the union of sets A and B refers to all the elements which are not present in set A or set B. • The complement of the union of sets is written by using symbol “ ’ ”. A B ξ (A B) Example 7 It is given that set M = {factors of 12} and set P = {prime numbers which are less than 12}. (a) Draw the Venn diagram and shade the region that represent the union of sets M and P. (b) List all the elements for M P. (c) Determine the value of n(M P). Solution M = {1, 2, 3, 4, 6, 12} P = {2, 3, 5, 7, 11} (a) M P ●1 ●4 ●6 ●12 ●2 ●3 ●5 ●7 ●11 (b) M P = {1, 2, 3, 4, 5, 6, 7, 11, 12} (c) n(M P) = 9 The symbol for union of two or more sets is . REMEMBER 04 GrabME Mathematics SPM F4.indd 37 28/10/2022 11:46 AM
43 CHAPTER 5 NETWORK IN GRAPH THEORY • A graph that represents a network consists of vertices and edges with their own characteristics. Vertex (V) Edge (E) • V is a set of dots or vertices. V = {v1 , v2 , v3 , … , vn } • E is a set of edges or lines linking each pair of vertices. E = {e1 , e2 , e3 , …, en } • The degree, d, of a vertex is the number of edges that connect it to another vertices. Sum of degrees, ∑d(v) = 2E; v V • The following are the characteristics of a graph that has multiple edges and loops: 1 6 7 3 4 5 2 Multiple edges Involve two vertices. The vertices are connected by more than one edge. The sum of degrees is twice the number of edges. P Q R Loops Involve one vertex. The edge is in the form of an arc that starts and ends at the same vertex. Each loop adds 2 to the degree. 5.1 Network Discrete Mathematics 05 GrabME Mathematics SPM F4.indd 43 28/10/2022 2:49 PM
44 • The differences between undirected graph, directed graph, unweighted graph and weighted graph: A graph drawn without any direction is assigned to the edge that connecting two vertices. A B C D A graph in which a direction is assigned to the edge that connecting two vertices. A B C D Undirected graph Directed graph (a) Simple graph (b) Graph with loops and multiple edges Unweighted graph Not associated with a value or a weight. For example, job hierarchy in an organisation chart. Weighted graph Associated with a value or a weight. For example, distance between two cities. A B C D a2 a1 a3 a4 a5 A B C D E 12 21 23 15 11 13 17 05 GrabME Mathematics SPM F4.indd 44 28/10/2022 2:49 PM
75 Solution (a) (i) Speed = 1.5 km min–1 47 – 20 t – 30 = 1.5 Speed = Distance Time 27 t – 30 = 1.5 t – 30 = 18 t = 48 (ii) Distance from Sungai Buloh to Kuala Selangor = 47 – 20 = 27 km (b) Average speed = 47 km ( 48 60 ) h Average speed = Total distance Total time = 58.75 km h–1 7.2 Speed-Time Graphs • Speed-time graph is a graph that shows the changes of speed of a particle or object in a certain of time. • In a speed-time graph: (a) the vertical axis represents the speed of a movement. (b) the horizontal axis represents the total time taken. (c) the gradient of the graph represents the rate of changes in speed with respect to time, that is acceleration. • In speed-time graph, Area under the graph = Total distance travelled Rectangle: Speed Time t v O Area = tv Triangle: Speed Time t v O Area = 1 2 tv Trapezium: Area = 1 2 (v1 + v2 )t Speed Time t v2 v1 O 07 GrabME Mathematics SPM F4.indd 75 28/10/2022 3:05 PM
76 • Interpretation of speed-time graph, OP is a graph with a positive gradient that shows the rate of changes in speed of a particle or an object increases or accelerates in the duration of t 1 . PQ is a graph with zero gradient that shows a particle or an object moving at a uniform speed in the duration of t 1 to t 2 . QR is a graph with a negative gradient that shows the rate of changes in speed of a particle or an object decreases or decelerates in the duration of t 2 to t 3 . Speed Time V O P Q R t 1 t 2 t 3 Example 8 The table below shows the changes of speed of Encik Daniel’s car for a period of 8 seconds. Time (second) 0 1 2 3 4 5 6 7 8 Speed (m s−1) 0 15 30 45 60 75 90 90 90 Draw a speed-time graph based on the given table. Solution Choose an appropriate scale on vertical axis and horizontal axis Plot the points which represent the speed and its corresponding time Join all plotted points by using a ruler The straight line formed represents the speed-time graph · OP adalah graf dengan kecerunan positif menunjukkan kadar perubahan laju bagi suatu zarah atau objek meningkat atau memecut dalam tempoh t 1 saat · PQ adalah graf dengan kecerunan sifar menunjukkan suatu zarah atau objek bergerak dengan kelajuan seragam dalam tempoh t 1 hingga t 2 saat · QR adalah graf dengan kecerunan negatif menunjukkan kadar perubahan laju bagi suatu zarah atau objek menurun atau nyahpecut dalam tempoh t 2 hingga t 3 saat 07 GrabME Mathematics SPM F4.indd 76 28/10/2022 3:05 PM
77 Speed is represented at vertical axis Time is represented at horizontal axis Speed (m s–1) Time (s) 30 60 90 2 4 6 8 O Example 9 Calculate the distance, in km, travelled by a motion for each of the following speed-time graphs. (a) (b) Speed (km h–1) Time (hour) 60 40 1.5 2 O Speed (km h–1) Time (hour) 70 30 2.5 4 O 0.75 Solution For a speed-time graph, the area under the graph is the same as the distance travelled. REMEMBER (a) Distance = 40 1.5 + 60 40 0.5 = [1.5 × 40] + [ 1 2 × (40 + 60) × 0.5] = 60 + 25 = 85 km (a) Distance = 0.75 30 + 1.75 30 + 1.5 30 70 = [ 1 2 × 0.75 × 30] + [1.75 × 30] + [ 1 2 × (30 + 70) × 1.5] = 11.25 + 52.5 + 75 = 138.75 km 07 GrabME Mathematics SPM F4.indd 77 28/10/2022 3:05 PM
CHAPTER 1 VARIATION • Direct variation is a relation in which one variable varies directly to another variable. For example: (a) If variable y increases, then variable x also increases. (b) If variable y decreases, then variable x also decreases. • In direct variable, “y varies directly as xn ” can be written as: y fi xn (Variation relation) or y = kxn Equation form where n = 1, 2, 3, 1 2, 1 3 and k is a constant. • Joint variation is a direct variation in which one variable varies directly as a product of two or more variables. • The following shows the examples of direct variation and joint variation: Direct variation Variation relation Equation form y varies directly as x2 y fi x2 y = kx2 y varies directly as ! x y fi ! x y = k! x Joint variation Variation relation Equation form y varies directly as x and z y fi xz y = kxz y varies directly as ! x and z2 y fi ! x z2 y = k! x z2 116 Relationship and Algebra 1.1 Direct Variation B01 GrabME Mathematics SPM F5.indd 116 28/10/2022 12:21 PM
119 Example 3 The mass, m g, of a solid varies directly as its volume, p cm3 . The mass of 20 cm3 of solid is 60 g. Find the relation between m and p. Solution m fi p m = kp Write the relation in equation form Substitute p = 20 and m = 60, 60 = k(20) k = 3 º Thus, m = 3p. Substitute k = 3 into the equation form Example 4 Given y varies directly as x. If y = 5 6 when x = 15, calculate the value of y when x = 45. Solution y fi x y = kx Substitute y = 5 6 and x = 15, 5 6 = k(15) k = 5 6 × 1 15 k = 1 18 Thus, y = x 18. When x = 45, y = 45 18 Substitute x = 45 into y = x 18 = 2.5 Another Method Using proportional concept: y1 x1 = y2 x2 , (5 6) 15 = y2 45 1 18 = y2 45 y2 = 45 18 y2 = 2.5 B01 GrabME Mathematics SPM F5.indd 119 28/10/2022 12:21 PM
121 Substitute T = 48, x = 4 and y = 9, 48 = k(4)!9 48 = 12k k = 4 º Thus, T = 4x! y. Example 7 The table below shows two sets of the values y, g and H. y g H 60 2 3 w 3 4 It is given that y varies directly as the square of g and H, find the value of w. Solution y fi g2 H y = kg2 H Substitute y = 60, g = 2 and H = 3, 60 = k(2)2 (3) 12k = 60 k = 5 Thus, y = 5g2 H. Substitute y = w, g = 3 and H = 4, w = 5(3)2 (4) w = 180 Another Method Using proportional concept: Let y1 = 60, g1 = 2, H1 = 3, y2 = w, g2 = 3, H2 = 4 y1 g2 1H1 = y2 g2 2H2 , 60 (22 )3 = w (32 )4 60 12 = w 36 w = 60 12 × 36 w = 180 Example 8 The table below shows two sets of the values L, m and v. L m v 100 5 4 16 2 d B01 GrabME Mathematics SPM F5.indd 121 28/10/2022 12:21 PM
CHAPTER CONSUMER MATHEMATICS: INSURANCE 3 • Risk is the possibility of disaster happening which is unavoidable. • Insurance is a form of risk management used for covering the possible risk that occurred, in terms of monetary loss. • The following is the system of insurance: agaushgduaudufysiyfsidyfidsidfa gheihdfiighrhuthut9uhuruuruurue nbiehtghhlbhytr8ugruirgeygeuu9 ie9t8ge9iihbihrtgurut9ug9iugiuvh wergpwgioj[o09 Policyholder Insurance contract Paying compensation for insured losses Paying premium Insurance company Insurance company A party who agrees to pay compensation on the loss that insured. Policy holder A party who claims and receives the compensation for the losses occured. Insurance contract A policy as an evidence of agreement made between the insurance company and the policyholder. Premium A specific amount of money payable by the policyholder to the insurance company. • The principle of indemnity applied in this system acts as a function to restore the finance of the policyholder to the pre-loss condition. 144 3.1 Risk and Insurance Coverage Number and Operations B03 GrabME Mathematics SPM F5.indd 144 28/10/2022 12:26 PM
145 • The following is the types of main insurances: • Life insurance guarantees the payment of monetary benefits to the policyholder upon the death of the insured person. The risks covered by life insurance includes death, loss of ability or critical illness. • General insurance provides coverage against any loss or damage incurred. • Group insurance is given to the employees in a company, pupils in school or students in educational institutions in order to provide coverage to the group. For example: (a) Group insurance for an organisation (b) Group insurance for pupils Type of insurance Life insurance General insurance Motor insurance Medical and health insurance • Hospitalisation and surgical insurance • Critical illness insurance • Disability income insurance • Hospital income insurance Fire insurance Travel insurance Personal accident insurance Group insurance B03 GrabME Mathematics SPM F5.indd 145 28/10/2022 12:26 PM
195 The Graphs of Sine, Cosine and Tangent Functions 6.2 • The following is the characteristics of the graphs of trigonometric functions for 0° < x < 360°: (a) Sine graph, y = sin x 1 0 90° 180° 270° 360° –1 y y = sin x x • Maximum value is 1 when x = 90° • Minimum value is −1 when x = 270° • x-intercept is 0°, 180° and 360° • y-intercept = 0 (b) Cosine graph, y = cos x • Maximum value is 1 when x = 0° and x = 360° • Minimum value is −1 when x = 180° • x-intercept is 90° and 270° • y-intercept = 1 1 O 90° 180° 270° 360° –1 y y = cos x x (c) Tangent graph, y = tan x 0 90° 180° 270° 360° y y = tan x x • Maximum value is • Minimum value is − • The value is undefined when x = 90° and x = 270° • x-intercept is 0°, 180° and 360° • y-intercept = 0 TIPS • Undefined: An expression which is not assigned an interpretation or a value. • Infinity: A quantity with the limitless value. • The effects of changes in constants a, b and c on the graphs of trigonometric functions: Changes Effects Value of a • Graph of y = a sin bx + c and y = a cos bx + c – Maximum and minimum values change – Amplitude of function = a B06 GrabME Mathematics SPM F5.indd 195 28/10/2022 3:25 PM
196 Changes Effects Value of a • Graph of y = a tan bx + c – The curve of the graph change – No amplitude Value of b • Graph of y = a sin bx + c and y = a cos bx + c – Period of the function changes, period of the graph = 360° b – x-intercept changes • Graph of y = a tan bx + c – Period of the function changes, period of the graph = 180° b – x-intercept changes Value of c All the graphs of trigonometric functions • When c . 0, the graph moves upward with translation 0 c • When c , 0, the graph moves downward with translation 0 –c In the graph of y = a sin bx + c: Period y x Amplitude Shift horizontally 0 Shift vertically REMEMBER Example 7 Complete the following tables. Hence, sketch the graphs and state the maximum value and minimum value. (a) y = sin x x 0° 90° 180° 270° 360° y (b) y = cos x x 0° 90° 180° 270° 360° y B06 GrabME Mathematics SPM F5.indd 196 28/10/2022 3:25 PM
197 (c) y = tan x x 0° 63.5° 116.5° 180° 243.5° 296.5° 360° y Solution (a) y = sin x x 0° 90° 180° 270° 360° y 0 1 0 –1 0 1 0 90° 180° 270° 360° –1 y y = sin x x Thus, the maximum value is 1 and the minimum value is −1. (b) y = cos x x 0° 90° 180° 270° 360° y 1 0 –1 0 1 1 0 90° 180° 270° 360° –1 y x y = cos x Thus, the maximum value is 1 and the minimum value is −1. (c) y = tan x x y 0° 0 63.5° 2 116.5° –2 180° 0 243.5° 2 296.5° –2 360° 0 Thus, the maximum value is and the minimum value is −. 1 0 90° 180° 270° 360° –1 y x 2 –2 y = tan x B06 GrabME Mathematics SPM F5.indd 197 28/10/2022 3:25 PM