48 Form 4 CHAPTER 4 (a) Draw a Venn diagram to represent the elements in the universal set ξ, set A and set B. (b) (i) List the elements of A ∪ B. (ii) Shade the region which represents A ∪ B. Solution (a) ξ = {2, 3, 4, 5, 6, 7, 8, 9, 10} A = {3, 4, 5, 6, 7, 8, 9, 10} B = {4, 9} A 2 5 6 7 8 3 10 4 9 B (b) (i) A ∪ B = {3, 4, 5, 6, 7, 8, 9, 10} (ii) A 2 5 6 7 8 3 10 4 9 B Example 11 It is given that the universal set, ξ = {x : 11 x 20, x is an integer}, set A = {x : x is an even number} and set B = {x : x is an odd number}. (a) Draw a Venn diagram to represent the elements in the universal set ξ, set A and set B. (b) (i) List the elements of A ∪ B. (ii) Shade the region that represents A ∪ B. Solution (a) A = {12, 14, 16, 18, 20} B = {11, 13, 15, 17, 19} A B 12 14 16 18 20 11 15 17 13 19 (b) (i) A ∪ B = {11, 12, 13, 14, 15, 16, 17, 18, 19, 20} (ii) A B 12 14 16 18 20 11 15 17 13 19 Example 12 The following Venn diagram shows set M and set N such that ξ is the universal set. On the Venn diagram, shade the set M' ∪ N'. M N Solution Step 1: Shade the region which represents the set M' using the pattern . M N Step 2: Shade the region which represents the set N' using the pattern . M N Step 3: The union of sets is represented by ALL the regions which are shaded. These include the patterns of , and . M N Analysis SPM Maths 2023 Eng F4 C4 2nd.indd 48 17/2/2023 9:27:31 PM PENERBIT ILMU BAKTI SDN. BHD.
49 Form 4 CHAPTER 4 Region listing method Label each region using Roman numerals. I II III M N IV Set M’ = {III, IV} Set N’ = {I, IV} M’ ∪ N’ = {I, III, IV} Shade the regions I, III and IV, as follows: I II III M N IV Alternative method Example 13 It is given that the universal set, ξ = {x : 3 x 13, x is an integer}, set C = {4, 5, 6, 9, 10} and set D = {5, 7, 10, 13}. (a) List the elements of (C ∪ D)'. (b) Draw a Venn diagram to represent the elements of the universal set ξ, set C and set D. Hence, shade the region that represents (C ∪ D)'. Solution (a) ξ = {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13} C = {4, 5, 6, 9, 10} D = {5, 7, 10, 13} C ∪ D = {4, 5, 6, 7, 9, 10, 13} (C ∪ D)' = {3, 8, 11, 12} (b) C D 11 12 10 5 4 6 9 7 13 3 8 Example 14 In a group of 64 students of a school, 16 students like to drink both barley and soy milk. The number of students who like to drink barley only is four times the number of students who like to drink soy milk only. The number of students who do not like to drink barley nor soy milk is 8. Find the number of students who like to drink barley or do not like to drink soy milk. Solution B K 16 8 4h h The number of students who like to drink both barley and soy milk. The number of students who like to drink soy milk only. The number of students who like to drink barley only. The number of students who do not like to drink barley nor soy milk. n(ξ) = 64 4h + 16 + h + 8 = 64 5h = 40 h = 8 Using h = 8, the complete Venn diagram is as follows. 8 81632 B K SMART TIP The number of students who like to drink barley or do not like to drink soy milk is given by n(B ∪ K'). Step 1: Shade the region which represents set B using the pattern . B K Analysis SPM Maths 2023 Eng F4 C4 2nd.indd 49 17/2/2023 9:27:33 PM PENERBIT ILMU BAKTI SDN. BHD.
50 Form 4 CHAPTER 4 Step 2: Shade the region which represents the set K' using the pattern . B K Step 3: The union set is represented by ALL the shaded regions. These include the patterns of , and . B K 81632 8 Hence, the number of students who like to drink barley or do not like to drink soy milk = n(B ∪ K') = 32 + 16 + 8 = 56 4.3 Combined Operation on Sets Example 15 Let the three words in the above photograph be represented by set V = {P, E, R, A, K}, set W = {D, A, R, U, L} and set X = {R, I, D, Z, U, A, N}. (a) Determine the elements of the set V∪ (W∩X). (b) Draw a Venn diagram and shade the region which represents the set V ∪ (W ∩ X). Solution (a) V ∪ (W ∩ X) = (W ∩ X) ∪ V = [{D, A, R, U, L} ∩ {R, I, D, Z, U, A, N}] ∪ {P, E, R, A, K} = {D, A, R, U} ∪ {P, E, R, A, K} = {P, E, R, A, K, D, U} Implement the operation in the brackets first. Implement the operation from left to right. (b) I Z N U D R A P E K W L V X Example 16 Given the universal set, ξ = {x : 1 x 9, x is an integer}, set H = {2, 3, 5, 6, 7}, set K = {1, 3, 4, 5, 9} and set M = {3, 4, 6, 8}. List the elements of H ∩ (K ∪ M'). Solution ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9} K = {1, 3, 4, 5, 9} M' = {1, 2, 5, 7, 9} K ∪ M' = {1, 2, 3, 4, 5, 7, 9} H = {2, 3, 5, 6, 7} Hence, H ∩ (K ∪ M') = {2, 3, 5} Example 17 HOTS Applying The Venn diagram shows set P, set Q and set R such that the universal set, ξ = P ∪ Q ∪ R. Q P R Shade the set (a) P ∩ R', (b) P ∩ (Q ∪ R'). Solution (a) Q P R P ∩ R' means that the required region lies outside of set R but inside of set P. Analysis SPM Maths 2023 Eng F4 C4 2nd.indd 50 17/2/2023 9:27:34 PM PENERBIT ILMU BAKTI SDN. BHD.
51 Form 4 CHAPTER 4 (b) Step 1: Shade the region which represents the set Q ∪ R' using the pattern . Q P R Q ∪ R' includes the whole set Q and all the region outside set R. Step 2: Shade the region which represents set P using the pattern . Q P R Region Listing Method Label each region with the Roman numerals. Q P R III II I IV VII V VI Set P = {I, II, V, VI} Set Q = {II, III, IV} Set R' = {I, II, III, IV, V} Q ∪ R' = {I, II, III, IV, V} ∴ P ∩ (Q ∪ R') = {I, II, V} Shade the regions I, II and V, as follows: Q P R III II I IV VII V VI Alternative Method Step 3: Shade the intersection region between the two different patterns at step 1 and step 2. Q P R Example 18 The following Venn diagram shows set A, set B and set C such that the universal set, ξ = A ∪ B ∪ C. A B C State the set which is represented by the shaded region. Solution SMART TIP The shaded region lies outside set B and outside set C but lies inside set A. Hence, the set which is represented by the shaded region is (B ∪ C)' ∩ A. Example 19 HOTS Applying HOTS Analysing Draw a Venn diagram that has the following properties: A ⊂ B, B ∩ C ≠ φ, A ∩ C = φ Hence, shade the set (A' ∩ B) ∪ C. Solution SMART TIP • B ∩ C ≠ φ means there is intersection between set B and set C. • A ∩ C = φ means set A does not intersect with set C. The required Venn diagram is as shown below. B A C Analysis SPM Maths 2023 Eng F4 C4 2nd.indd 51 17/2/2023 9:27:36 PM PENERBIT ILMU BAKTI SDN. BHD.
52 Form 4 CHAPTER 4 The shaded region which represents the set (A' ∩ B) ∪ C is as shown below: B A C Example 20 The class of Form 4 Theta has 41 students. They read types of newspapers, X, Y and Z. It is given that 30 of the students read newspaper X, 32 students read newspaper Y, 27 students read newspaper Z and 18 students read all three types of newspapers. The number of students who read newspapers X and Y, newspapers X and Z and newspapers Y and Z are the same. Find the number of students who read at least two types of newspapers. HOTS Applying HOTS Analysing HOTS Evaluating Solution Let X = {students who read newspaper X}, Y = {students who read newspaper Y}, Z = {students who read newspaper Z} It is given that n(X ∩ Y ∩ Z) = 18. Let n(X ∩ Y) = n(X ∩ Z) = n(Y ∩ Z) = k, n(students who read newspaper X only) = a, n(students who read newspaper Y only) = b n(students who read newspaper Z only) = c. The Venn diagram that represents the given problem is as follows. X Y Z a b c k – 18 k – 18 k – 18 18 • 30 students read newspaper X, a = 30 – 2(k – 18) – 18 = 48 – 2k • 32 students read newspaper Y, b = 32 – 2(k – 18) – 18 = 50 – 2k • 27 students read newspaper Z, c = 27 – 2(k – 18) – 18 = 45 – 2k n(X ∪ Y ∪ Z) = a + b + c + 3(k – 18) + 18 = (48 – 2k) + (50 – 2k) + (45 – 2k) + 3(k – 18) + 18 = 48 – 2k + 50 – 2k + 45 – 2k + 3k – 54 + 18 = 107 – 3k Thus, 107 – 3k = 41 The class of Form 4 Theta has 41 students. 3k = 107 – 41 3k = 66 k = 22 Thus, a = 48 – 2(22) = 4 b = 50 – 2(22) = 6 c = 45 – 2(22) = 1 Hence, the complete Venn diagram is as follows. X Y Z 4 4 4 4 18 6 1 Hence, the number of students who read at least two types of newspapers = 18 + 4 + 4 + 4 = 30 Analysis SPM Maths 2023 Eng F4 C4 2nd.indd 52 17/2/2023 9:27:37 PM PENERBIT ILMU BAKTI SDN. BHD.
53 Form 4 CHAPTER 4 A survey was conducted on a group of students regarding their favourite sports. Let the universal set, ξ = {students involve in the survey}, set A = {students who like badminton}, set B = {students who like tennis} and set C = {students who like ping pong}. The following Venn diagram shows the outcome of the survey where x is a variable. A B C 15 25 – x x – 2 19 – x 9 8 4 12 (a) Find the maximum value of x and the minimum value of x. (b) If the actual number of students involved in the survey is 81, find the actual value of x. (c) Calculate the number of students who do not like tennis but like badminton or ping pong. HOTS Applying HOTS Analysing HOTS Evaluating Solution (a) SMART TIP The number of elements in any region cannot be negative. 25 – x 0 ⇒ x 25 (…, 24, 25) x – 2 0 ⇒ x 2 (2, 3, …) 19 – x 0 ⇒ x 19 {…, 18, 19} Hence, the maximum value of x is 19 and the minimum value of x is 2. (b) n(ξ) = 81 15 + 9 + 8 + 4 + 25 + 19 + 12 – 2 – x + x – x = 81 90 – x = 81 x = 9 (c) The number of students who do not like tennis but like badminton or ping pong. = 15 + 9 + (19 – x) = 43 – x = 43 – 9 = 34 1 The following Venn diagram shows the universal set ξ, set A and set B. A B The set which represents the shaded region is A A' ∪ B' C A' ∩ B' B A' ∪ B D A ∩ B 2 The following Venn diagram shows the universal set ξ, set P, set Q and set R. P Q R Which of the following sets represents the shaded region? A P' ∩ Q' ∪ R' C P ∪ Q ∪ R' B P' ∩ Q ∪ R' D P' ∩ R' ∩ Q Objective Questions HOTS Zone SPM Practice 4 Analysis SPM Maths 2023 Eng F4 C4 2nd.indd 53 17/2/2023 9:27:38 PM PENERBIT ILMU BAKTI SDN. BHD.
91 Form 4 CHAPTER 6 Objective Questions 1 B 2 B 3 A 4 C Subjective Questions 1 6x + 3y 50 2 35x + 30y 390 3 (a), (b), (c) Point y 2x – 6 Conclusion A(2, –2) –2 –2 y = 2x – 6 B(–3, 2) 2 –12 y > 2x – 6 C(2, –5) –5 –2 y < 2x – 6 (d) (i) y > 2x – 6 –2–4 O 2 4 2 y x –2 –4 –6 B (–3, 2) y = 2x – 6 (ii) y < 2x – 6 –2–4 2 4 2 y x –2 –4 –6 y = 2x – 6 C (2, –5) O 6 4 (a) O y x 3 2 3 (b) O y x 3 1 (c) O y x 6 2 (d) O y x –6 3 (e) O y x –2 – 3 4 Answers Analysis SPM Maths 2023 Eng F4 C6 3rd.indd 91 17/2/2023 9:30:46 PM PENERBIT ILMU BAKTI SDN. BHD.
92 Form 4 CHAPTER 6 (f) O y x 20 –5 (g) O y x 12 6 (h) O y x 9 6 5 (a) O y x 2 –6 (b) O – y x 1 2 1 (c) O y x 1 –2 (d) O y x 4 4 6 (a) O y x 4 2 (2, 4) (b) O y x 2 4 (4, 2) (c) O y x 5 5 (d) O y x 3 –3 7 5x + 3y 150, y – x 10, y 1 10x 8 x 10, y 2x, 2x + 3y 180 9 x + y 90, x 2y, y – x 10 10 (a) 2 4 6 8 2 4 6 8 y x R x = 2 y = –x + 8 y = x 2 1 O (b) –2 2 4 6 2 4 6 y x R x = 2 x + y = 6 y = x – 2 2 1 O (c) 2 4 6 8 2 4 6 y x y = – x + 3 y = x + 3 2 1 R x = 6 2 1 O (d) 2 4 6 2 4 6 y x y = x x + y = 6 R y = – x + 3 2 1 O Analysis SPM Maths 2023 Eng F4 C6 3rd.indd 92 17/2/2023 9:30:49 PM PENERBIT ILMU BAKTI SDN. BHD.
93 Form 4 CHAPTER 6 (e) –1 2468 2 4 6 8 y x O R y = 2x y = –x + 8 y = x – 1 2 1 (f) –2 2 4 6 2 4 6 y x –2 O R y = 2x + 2 2y = x x + y = 6 (g) 2 4 6 2 4 6 y x O R y = x y = –x + 6 y = – x + 7 2 1 11 (a) x + 2y 50, 3x + 8y 120, y – 2x 10 (b) 10 20 30 40 50 10 20 30 40 50 y – 2x = 10 x + 2y = 50 R y x 3x + 8y 120 O (c) 30 12 (a) x + y 80, x 3y, y 2x (b) 10 20 30 40 50 60 70 80 10 20 30 40 50 60 70 80 x + y = 80 y = 2x R x y x = 3y O (c) 20, 40 13 (a) 7x + 4y 420, x + 4y 80, x 2y (b) 10 20 30 40 50 60 70 80 10 20 30 40 50 60 70 80 7x + 4y = 420 x + 4y = 80 x = 2y R x y O 90 100 105 (c) 15, 17 Analysis SPM Maths 2023 Eng F4 C6 3rd.indd 93 17/2/2023 9:30:50 PM PENERBIT ILMU BAKTI SDN. BHD.
94 Form 4 CHAPTER 6 14 (a) x + y 70, y 2x, 3x + 2y 180 (b) 10 20 30 40 50 60 70 10 20 30 40 50 60 70 80 90 3x + 2y = 180 y = 2x R x y x + y = 70 O (c) 20, 30 15 (a) x 2y, x + y 200, y 1 5 x (b) 20 40 60 80 100 120 140 160 20 40 60 80 100 120 140 160 180 200 x + y = 200 x = 2y R x y y = x 5 1 O (c) 80, 160 16 (a) x + y 10, y 3x, 5x + 3y 60 (b) 2 4 6 8 10 12 2 4 6 8 10 12 14 16 18 20 y = 3x 5x + 3y = 60 R x y x + y = 10 O (c) 4, 10 Analysis SPM Maths 2023 Eng F4 C6 3rd.indd 94 17/2/2023 9:30:51 PM PENERBIT ILMU BAKTI SDN. BHD.
95 7.1 Distance-Time Graphs Solving distance-time graph problems 1 O 100 200 300 1 2 3 4 Distance (km) Time (hours) (a) The quantity represented by the vertical axis is the distance travelled, in km. (b) The quantity represented by the horizontal axis is the time taken, in hours. (c) Gradient = Change in distance (in km) Change in time (in hours) Hence, the quantity represented by the gradient of the graph is the rate of change of distance with respect to time, i.e. speed, in km h–1. 2 O Distance (m) Time (s) The particle is stationary. The particle travels from a point (let it be A) to another point (let it be B). The particle travels from point B back to point A. 95 EXPRESS NOTES Graphs of Motion Chapter 7 Learning Area: Relationship and Algebra Analysis SPM Maths 2023 Eng F4 C7 3rd.indd 95 17/2/2023 9:35:11 PM PENERBIT ILMU BAKTI SDN. BHD.
96 Form 4 CHAPTER 7 3 Average speed = Total distance travelled Total time taken 7.2 Distance-Time Graphs Determining distance and acceleration from a speed-time graph 4 O Speed (m s–1) Time (s) The area under a speed-time graph is the product of speed and time, i.e. distance. 5 Rate of change of speed (or acceleration) = Gradient of speed-time graph Solving problems involving speed-time graphs 6 O Speed (m s–1) Time (s) A B C D The horizontal part of the graph (AB) indicates that the particle is moving with a uniform speed. The part of the graph with a positive gradient (BC) indicates that the speed of the particle is increasing with respect to time, i.e. acceleration, such that the rate of change of speed is positive. The part of the graph with a negative gradient (CD) indicates that the speed of the particle decreases with respect to time, i.e. deceleration, such that the rate of change of speed is negative. Analysis SPM Maths 2023 Eng F4 C7 3rd.indd 96 17/2/2023 9:35:12 PM PENERBIT ILMU BAKTI SDN. BHD.
97 Form 4 CHAPTER 7 7.1 Distance-Time Graphs Example 1 At 0700, a lorry is at 20 km from town P. It departs from town P heading to town Q. When it reaches town Q, the lorry travels back to town P. The following distance-time graph shows the whole journey of the lorry. 20 40 60 80 100 120 0700 0800 0900 1000 1100 1200 1300 Distance (km) O Time Calculate the speed of the lorry, in km h–1, for the journey (a) from 0700 to 0900, (b) from 0900 to 1000, (c) from 1000 to 1230. Solution The horizontal distance of the distance-time graph in the question can be relabelled as follows: D 1 3 2 4 5 6 20 40 60 80 100 120 Distance (km) Time A B C O (a) Speed from 0700 to 0900 = Gradient of AB = (120 – 20) km (2 – 0) h A = (x1 , y1 ) = (0, 20) B = (x2 , y2 ) = (2, 120) Gradient of AB = y2 – y1 x2 – x1 = 100 km 2 h = 50 km h–1 (b) Speed from 0900 to 1000 = Gradient of BC = (120 – 120) km (3 – 2) h B = (x1 , y1 ) = (2, 120) C = (x2 , y2 ) = (3, 120) Gradient of BC = y2 – y1 x2 – x1 = 0 km 1 h = 0 km h–1 SMART TIP The horizontal part (with gradient 0) of a distance-time graph shows an object is stationary. (c) Speed from 1000 to 1230 = Gradient of CD = (0 – 120) km (5.5 – 3) h C = (x1 , y1 ) = (3, 120) D = (x2 , y2 ) = (5.5, 0) Gradient of CD = y2 – y1 x2 – x1 = –120 km 2.5 h = –48 km h–1 SMART TIP The negative speed means the object is travelling in a direction opposite to the original direction. Hence, the speed of the lorry from 1000 to 1230 is 48 km h–1, from town Q back to town P. Example 2 The graph ABCD represents the journey of a taxi and the graph ACE represents the journey of a van. Both vehicles depart from town R at the same time and travel along the same road. O 45 108 160 Distance from town R (km) Time (hour) 0.5 1.0 2.4 A B C D E Analysis SPM Maths 2023 Eng F4 C7 3rd.indd 97 17/2/2023 9:35:13 PM PENERBIT ILMU BAKTI SDN. BHD.
98 Form 4 CHAPTER 7 (a) State the period of time, in minutes, the taxi is stationary. (b) Calculate the average speed, in km h–1, of the taxi for a period of 2.4 hours. (c) At a moment during the journey, both vehicles meet at the same location, S. (i) State the distance, in km, between the location S and town R. (ii) State the time, in hours, taken by both vehicles to meet at location S from town R. Solution (a) Period of time the taxi is stationary = (1.0 – 0.5) hours Referring to the horizontal part of the distancetime graph. = 0.5 hours = (0.5 × 60) minutes = 30 minutes (b) Average speed = Total distance travelled Total time taken = 160 km 2.4 h = 66 2 3 km h–1 (c) (i) Distance between location S and town R = 45 km (ii) Time taken by both vehicles to meet at location S from town R = 1 hour Location S is the point C on the graph, i.e. the point of intersection between the graph ABCD (for the taxi) and the graph ACE (of the van). 7.2 Speed-Time Graphs Example 3 The following diagram shows a speed-time graph of a particle for a period of 25 s. O u 30 Speed (m s–1) Time (s) 8 15 25 Calculate (a) the period of time, in seconds, the particle is travelling at a uniform speed, (b) the rate of change of speed, in m s–2, in the last 10 seconds, (c) the value of u if the total distance travelled in the first 15 s is 386 m. Solution (a) The period of time when the particle is travelling at a uniform speed = (15 – 8) s = 7 s This refers to the horizontal part of the speedtime graph O u 30 Speed (m s–1) Time (s) 8 15 25 Uniform speed (b) O u 30 Speed (m s–1) Time (s) 8 15 (15, 30) (25, 0) Rate of change of speed in the last 10 s = Gradient of graph = y2 – y1 x2 – x1 = 0 – 30 25 – 15 = –30 10 = –3 m s–2 Note that the gradient of the graph is negative. SMART TIP • If the question asks for ‘rate of change of speed, the answer is –3 m s–2, i.e. with a negative sign. • If the question asks for ‘rate of decrease of speed or deceleration’, its answer is 3 m s–2, i.e. without the negative sign. Analysis SPM Maths 2023 Eng F4 C7 3rd.indd 98 17/2/2023 9:35:13 PM PENERBIT ILMU BAKTI SDN. BHD.
255 1 The area of a gigantic Malaysia flag for the National Day march is 9.9 × 103 m2 . The width of the flag is 9 000 cm. Calculate the length, in cm, of the flag. A 1.1 × 103 C 1.1 × 105 B 1.1 × 104 D 1.1 × 106 2 Which of the following is the weakness of using a credit card? I Easier to shop online II Will be charged interest if late in paying the money used III Need not bring a big sum of cash IV Could be overspent A I and II C II and III B I and III D II and IV 3 Encik Hamzah saves RM6 500 in a bank with an annual interest rate of 3.5% which is compounded 2 times a year. Calculate the maturity value at the end of the fourth year. A RM6 910 C RM7 468 B RM7 410 D RM15 600 4 An online tuition charges a deposit of RM100 and a monthly fee of RM50. If f(n) represents the total money charged, in RM, after n months, the linear function that can model the situation is A f(n) = 100 + 50n C f(n) = 50n B f(n) = 50 + 100n D f(n) = 100 × 50n 5 Maheran dine in a hotel. The price of the meal is RM55. The total payment, including service tax is A RM57.75 C RM58.85 B RM58.30 D RM59.40 6 Diagram 1 shows a speed-time graph of the journey of Encik Selvam from school to his house. Speed (km h–1) k 80 0 10 22 30 Time (minutes) Diagram 1 If the rate of change of speed from the 10th minute to the 22nd minute is 40 km h–2, find the value of k. A 85 C 87 B 86 D 88 7 Diagram 2 shows a box plot for the length, in cm, of the fish caught by a fisherman. 15 20 25 30 35 40 45 Length (cm) Diagram 2 Which of the following statements is true? A The length distribution is left-skewed. B The length distribution is right-skewed. C The length distribution is symmetrical. D The interquartile range is 31 cm. Instructions: 1 This question paper consists of 40 questions. 2 Each question is followed by four answer options, A, B, C and D. Choose the correct option. 3 Answer all questions. 4 The diagrams in the questions provided are not drawn to scale unless stated otherwise. 5 You may use a scientific calculator. Paper 1 [40 marks] Time: 1 hour 30 minutes SPM Model Test Analysis SPM Maths 2023 Eng MT 4th.indd 255 17/2/2023 10:22:32 PM PENERBIT ILMU BAKTI SDN. BHD.
256 Model Test SPM 8 Diagram 3 shows a quadratic graph. y x A B O Diagram 3 The equation of the graph is y = x2 – 8x + 12. If the straight line AB is the axis of symmetry of the graph, determine the equation of AB. A x = 3 C x = 5 B x = 4 D y = 4 9 Coulomb’s Law states that F ∝ q1 q2 r2 , where F is the electrical force, in N, q1 and q2 are electrical charges, in C, while r is the distance, in m, between the two electrical charges. Calculate the constant of variation, k, based on the following information. F = 125 N q2 = 2.0 × 10–5 C q1 = 1.0 × 10–5 C r = 0.12 m A 9 × 106 C 9 × 108 B 9 × 107 D 9 × 109 10 Find the gradient of the straight line x –6 + y 3 = 1. A – 1 2 C 2 B 1 2 D –2 11 In Diagram 4, the quadrilateral PQRS is the image of the quadrilateral ABCD. A B D C P Q S R Diagram 4 State the scale factor of the enlargement. A –3 C 1 3 B – 1 3 D 3 12 Which of the following graphs represents y = cos 2x for 90° x 270°? A 1 –1 90° 180° 270° y x B 1 –1 90° 180° 270° y x C 1 –1 90° 180° 270° y x D 1 –1 90° 180° 270° y x 13 In Diagram 5, X is a moving point such that its distance from point R is always 6 cm. Y is a moving point such that its distances from the line SR and the line PQ are always the same. A B C D S G R H F P 3 cm 3 cm Q 3 cm 3 cm Diagram 5 Analysis SPM Maths 2023 Eng MT 4th.indd 256 17/2/2023 10:22:34 PM PENERBIT ILMU BAKTI SDN. BHD.
257 Model Test SPM Which of the points, A, B, C or D, is the intersection point between the loci of point X and point Y. 14 Which of the following drawing, is not a tessellation? A B C D 15 Diagram 6 shows a graph. P Q R S U T Diagram 6 Which of the following statements is true? A The graph is a simple graph. B The degree of the vertex P is 3. C The number of edges of the graph is 8. D V = {P, Q, R, S, T} E = {(P, P), (P, Q), (P, R), (R, S), (S, T), (S, T)} 16 If –2x > –10, then x < 5. The inverse of the implication is: A If x < 5, then –2x > –10. B If –2x –10, then x 5. C If x 5, then –2x –10. D If –2x –10, then x 5. 17 Diagram 7 is a stem-and-leaf plot for the age distribution of the teachers in a school. Stem Leaf 2 3 4 5 5 7 9 1 2 3 4 7 5 6 8 9 9 9 0 5 6 9 Key: 2|5 means 25 Diagram 7 Calculate the mean of the age distribution. A 47 1 8 C 44 8 17 B 47 1 4 D 41 8 9 18 Darul Aman Company send two tenders for two road construction projects, H and K. The events of the company winning the two tenders are independent. If the probabilities of the company winning the tender for project H and project K are 1 5 and 2 7 respectively. Calculate the probability that the company wins the project H or project K. A 2 35 C 17 35 B 4 35 D 3 7 19 A jar contains 10 chocolate sweets and 16 strawberry sweets. Calculate the number of chocolate sweets that have to be added to the jar so that the probability of selecting a chocolate sweet at random becomes 7 15 . A 2 B 3 C 4 D 5 Analysis SPM Maths 2023 Eng MT 4th.indd 257 17/2/2023 10:22:34 PM PENERBIT ILMU BAKTI SDN. BHD.
258 Model Test SPM 20 A bag contains 4 grey buttons, 5 white buttons and 6 black buttons. Two buttons are drawn at random, one after the other, without replacement. Calculate the probability that the two same colour buttons are drawn. A 74 105 C 5 21 B 16 105 D 31 105 21 Diagram 8 shows a triangle ABC. A C B 1.5 cm Diagram 8 Given that the gradient of AC is 1 4 , find ∠CAB and the length, in cm, AC. A 14.47°, 6.00 B 14.04°, 6.18 C 14.47°, 6.18 D 14.04°, 6.20 22 Calculate 6307 – 4467 . A 1317 C 1337 B 1327 D 1517 23 Calculate the mean of 1001102 , 22103 and 7158 in base 10. A 190 C 191 1 3 B 191 D 191 2 3 24 Diagram 9 shows a solid. A B C D Q S U P R T E K L F 1 cm 6 cm 2 cm 3 cm Diagram 9 The plan of the solid is A 1 cm 2 cm 3 cm B 6 cm 1 cm 3 cm C 6 cm 1 cm 2 cm D 6 cm 3 cm 25 Simplify 4x–4y – 2 3 –w 2 3 × y2 w3 . A 64x12 y4 C – 64x12 y4 B 64 x12 D – 64 x12 26 Given that p = 6 + 5q q – 3 , then q = A p + 5 3(2 – p) C 3(2 – p) p + 5 B p – 5 3(2 + p) D 3(2 + p) p – 5 27 Find the roots of the quadratic equation 2x – 1 x – 3 = 5 – x 2x – 5 . A 1 C 3 B 2 D 4 Analysis SPM Maths 2023 Eng MT 4th.indd 258 17/2/2023 10:22:35 PM PENERBIT ILMU BAKTI SDN. BHD.
259 Model Test SPM 28 In Diagram 10, PQR is a tangent to the circle QST at point Q. S T Q P R 76° s° Diagram 10 The arc length of QS is equal to the arc length of ST. Find the value of s. A 76 C 28 B 38 D 26 29 Cik Audrey drawn a water tank which is to be constructed using scale. The actual dimension of the water tank is 2 m wide and 6 m long. The measurement of the drawing is 8 cm wide and 24 cm long. Calculate the scale used by her. A 1 : 25 C 1 : 4 B 25 : 1 D 4 : 1 30 30 A circle with an area of 616 cm2 is drawn using a scale of 1 : 1 4 . Find the area, in cm2 , of the drawing. A 856 C 24 649 B 2 464 D 9 856 31 Diagram 11 shows a cylinder with a height of 18 cm and a base-radius of 7 cm. A cone STU is cut and removed. The volume of the remaining solid is 2 310 cm3 . T S U Diagram 11 Calculate the height, in cm, of the cone. 3Use π = 22 7 4 A 6 C 8 B 7 D 9 32 It is given that the universal set ξ = {x : 10 x 40, x is an integer} set M = {x : x is a multiple of 5} and set N = {x : x is a multiple of 4}. Which of the following Venn diagrams shows correctly the number of elements in each region? A 7 4 1 M N 19 B 6 4 1 M N 20 C 6 4 1 M N 20 D 5 6 2 M N 18 HOTS Applying HOTS Analysing HOTS Evaluating 33 Table 1 shows a survey regarding the favourite food for breakfast for 50 students. Food Number of students Fried egg 24 Toasted bread 26 Fried egg and toasted bread 7 Fried egg and instant noodle only 3 Instant noodle and toasted bread only 9 Fried egg, toasted bread and instant noodle 5 Not fried egg nor toasted bread nor instant noodle 2 Table 1 Analysis SPM Maths 2023 Eng MT 4th.indd 259 17/2/2023 10:22:37 PM PENERBIT ILMU BAKTI SDN. BHD.
262 Model Test SPM 1 Diagram 1 shows two rectangles, BCDE and EFGH. BHE and EFD are straight lines. B C H E F D G 20 cm 8 cm a + 3 a Diagram 1 Given that the area of the rectangular shaded region is 60 cm2 , find the value of a. [4 marks] 2 (a) Diagram 2 is a Venn diagram which shows set X, set Y and set Z, such that the universal set ξ = X ∪ Y ∪ Z. X Y Z Diagram 2 Using set notations, state the operation of sets between set X, set Y and set Z of the shaded region. [1 mark] (b) Diagram 3 is a Venn diagram which shows the set P, set Q and set R, such that the universal set, ξ = P ∪ Q ∪ R. P Q R Diagram 3 Shade the set (P ∩ Q) ∪ R. [1 mark] 3 Diagram 4 shows three strings tied to support a flag pole such that PQ and RS are parallel. P R T S Q Diagram 4 The gradient of PQ is 2 5 . The horizontal distance of P from T is 5.5 m. The vertical height of point S from point T is 1.6 m. Calculate the distance of PR, in m. [4 marks] 4 The price of a new car P is RM54 6189 . while the price of a new car Q is RM75 4188 . Calculate the difference between the price of car P and car Q, in base 10. [4 marks] Instructions: 1 This question paper contains three sections: Section A, Section B and Section C. 2 The diagrams in the questions provided are not drawn to scale unless stated otherwise. 3 Working must be shown. Paper 2 [100 marks] Time: 2 hours 30 minutes Section A [40 marks] Answer all the questions. Analysis SPM Maths 2023 Eng MT 4th.indd 262 17/2/2023 10:22:38 PM PENERBIT ILMU BAKTI SDN. BHD.
263 Model Test SPM 5 Kamal bought a health insurance policy with a provision clause of the percentage of co-insurance 80/20 and a deductible of RM3 000. His operation expenses in a hospital is RMx. Given that the total cost paid by Kamal is RM7 400, find the value of x. [4 marks] HOTS Applying HOTS Analysing HOTS Evaluating 6 Ghazali’s yearly salary is RM60 000. He donated RM250 to a old folk’s home. Table 1 shows Ghazali’s tax relief. Tax relief Amount Individual 9 000 Live insurance and EPF (restricted to RM7 000) 1 600 Medical insurance (restricted to RM3 000) 3 600 Table 1 Table 2 shows the rate of income tax. Range of Taxable Income Calculation (RM) Rate (%) Tax (RM) 35 001 – 50 000 First 35 000 Next 15 000 8 600 1 200 Table 2 Calculate Ghazali’s income tax. [4 marks] 7 In Diagram 5, GH, HK and KL are straight lines. Point H lies on the x-axis. The straight line GH is parallel to the straight line KL. The straight line HK is parallel to the y-axis, It is given that the equation of GH is 2x + y = 6. G O H K y L (10, -4) x Diagram 5 (a) Determine the equation of the straight line HK. [1 mark] (b) Find the equation of the straight line KL and hence, find the x-intercept of KL. [4 marks] 8 A bag contains x red beads and 5 blue beads. If a bead is drawn randomly from the bag, the probability to draw a red bead is 7 12 , find the value of x. [3 marks] 9 Daud is a teacher in a pre-U college. His income and expenses are RM4 200 dan RM2 950 respectively. He wants to renovate his house which needs a cost of RM15 500 in a year. (a) Can Daud achieve his goal? Explain why. (b) Suggest two ways for Daud to increase his net income. [4 marks] 10 Table 3 shows the mass distribution of watermelons sold at a fruit shop. Mass Frequency 1.0 – 1.4 6 1.5 – 1.9 h 2.0 – 2.4 12 2.5 – 2.9 14 3.0 – 3.4 8 Table 3 (a) It is given that the mean mass is 2.28, find the value of h. [3 marks] (b) Hence, calculate the standard deviation of the mass distribution. [3 marks] Analysis SPM Maths 2023 Eng MT 4th.indd 263 17/2/2023 10:22:39 PM PENERBIT ILMU BAKTI SDN. BHD.
264 Model Test SPM 11 Diagram 6 shows quadrilaterals ABCD, EKLM and EFGH drawn on a Cartesian plane. y x O −2 2 4 6 8 10 12 2 4 6 8 10 12 E H M K L GF A B C D Diagram 6 (a) The quadrilateral EFGH is the image of the quadrilateral ABCD under a combined transformation VU. Describe in full the transformation (i) U, (ii) V. [5 marks] (b) It is given that ABCD represents a region that has an area of 210 m2 . Calculate the area, in m2 , of the region represented by the shaded region. [2 marks] (c) Draw another two quadrilaterals which are congruent to the quadrilateral EFGH to form a tessellation on the following square grid: [2 marks] 12 A courier service company uses x trucks and y vans to send parcels based on the following constraints: I The total number of vehicles used is at least 10. II The number of vans used is at most 3 times of the number of trucks used. III The total number of parcels that have to be delivered in a day is 1 800 units such that the capacity of a truck and a van are 150 and 90 parcels respectively. (a) Write three inequalities, other than x > 0 and y > 0, that satisfy all the given constraints. [3 marks] (b) Using a scale of 2 cm to 2 vehicles on both axes, construct and shade a region R that satisfies all the given constraints. [3 marks] (c) If the ratio of the number of vans and the number of trucks used is 5:3, (i) write an equation, (ii) draw a straight line to represent the equation in (c)(i). [2 marks] 13 Diagram 7 shows a combined solid which consists of two right prisms with the horizontal rectangular base AMND and the square base BCNM. The plane AMBHGFE is the uniform cross section of the combined solid. The square EFKL and the rectangle GHIJ are horizontal planes. The rectangles FGKJ and BCIH are inclined planes. It is given that GM = 4 cm and GM = JN. 3 cm 3 cm 2 cm 8 cm 4 cm 3 cm E L K F A D G N H I C M B J Y X Diagram 7 Draw to full scale, (a) the plan of the solid, [3 marks] (b) the front elevation of the solid at a vertical plane that is parallel to AMB as seen from X, [3 marks] Section B [45 marks] Answer all the questions. Analysis SPM Maths 2023 Eng MT 4th.indd 264 17/2/2023 10:22:39 PM PENERBIT ILMU BAKTI SDN. BHD.
265 Model Test SPM (c) the side elevation of the solid at a vertical plane that is parallel to BC as seen from Y. [3 marks] 14 A boutique advertise the selling prices of a shirt and a pair of shoes at the prices of RM60 and RM85 respectively. The cost price of two shirts and four pairs of shoes is RM200 while the cost price of four shirts and two pairs of shoes is RM160. (a) Using the matrix method, find (i) the cost prices of a shirt and a pair of shoes, (ii) the profits of a shirt and a pair of shoes. [6 marks] (b) Encik Khairul has RM260 cash in his wallet. Is the cash enough to buy 3 shirts and a pair of shoes. Justify your answer using the matrix method. [3 marks] 15 Diagram 8 is a Venn diagram which shows the number of students from the class 5A1 who attend tuition classes for different subjects. E S M Diagram 8 Set S = {Students who attend tuition classes for Science} Set M= {Students who attend tuition classes for Mathematics} Set E = {Students who attend tuition classes for English} It is given that the universal set, ξ = S ∪ M ∪ E. It is given that the total number of students in the class 5A1 is 36. There are 12 students who attend tuition classes for all the three subjects. The number of students who attend tuition classes for Mathematics and Science only is 3. The number of students who attend tuition classes for Science and English only is 6. The number of students who attend tuition classes for Science only is 7. The number of students who attend tuition classes for Mathematics and English only is 4. The number of students who attend tuition classes for Mathematics only is the same as the number of students who attend tuition classes for English only. (a) Complete the Venn diagram. [6 marks] (b) Determine the number of students who (i) do not attend tuition classes for Mathematics or English, (ii) attend tuition classes for two subjects only, (iii) who attend tuition classes for Science or Mathematics but not English. [3 marks] Analysis SPM Maths 2023 Eng MT 4th.indd 265 17/2/2023 10:22:40 PM PENERBIT ILMU BAKTI SDN. BHD.
266 Model Test SPM 16 The distance-time graph in Diagram 9 shows the journey of two busses from the main bus terminal to Station Z using the same roads. OPQRS represents the journey of bus A while TRU represents the journey of bus B. Bus A departs at 6:30 a.m. During its journey, the bus stops at a location to carry a few passengers. Bus B departs at 6:50 a.m. Both busses meet at Town R at the time m. 55 32 20 O P Q R SU T 15 20 m 35 45 Time (min) Distance from the main bus terminal (km) Diagram 9 (a) (i) State the period of time, in minutes, where bus A stops to take passengers. (ii) State the distance, in km, travelled by bus A when it meet bus B at Town R. (iii) Find the value of m. [6 marks] (b) Calvin takes bus to school. The probability that Calvin take bus A is 3 5 . If he takes bus A, the probability that he is late for school is 1 10 . If he takes bus B, the probability that he is late for school is 1 8 . (i) Complete the tree diagram by filling the probability in each box. BA BB L L′ L L′ Bus A – BA Late – L Bus B – BB No late – L’ (ii) Calculate the probability that Calvin is not late for school. [4 marks] (c) Olahraga club in a school wants to select either Calvin or Raymond to represent the school in the 100 m race event based on a more consistent performance. Athlete Time (s) Trial 1 2 3 4 5 Raymond 11.18 11.19 11.73 11.25 11.30 Calvin 11.20 11.30 11.15 11.26 11.72 Who should be selected? Give your justification. 17 (a) Siva, Aaron and Jailani sell food in the school Canteen Day. The sales by them are in the ratio 2 : 7 : 3. The difference between the sales of Aaron and Siva is RM200. Calculate the sales by Jailani. [2 marks] (b) Diagram 10 shows Jailani standing at a horizontal stage. The points P, Q and R lies at the same vertical plane. The angle of depression of R from Jailani’s eye level are 40° dan 60° when he at the locations P and Q respectively. Section C [15 marks] This section contains two questions. Answer one question. Analysis SPM Maths 2023 Eng MT 4th.indd 266 17/2/2023 10:22:40 PM PENERBIT ILMU BAKTI SDN. BHD.
267 Model Test SPM Paper 1 1 B 2 D 3 C 4 A 5 B 6 D 7 A 8 B 9 D 10 B 11 C 12 D 13 B 14 D 15 D 16 B 17 D 18 D 19 C 20 D 21 B 22 A 23 C 24 D 25 D 26 D 27 B 28 C 29 A 30 D 31 D 32 D 33 A 34 A 35 A 36 C 37 B 38 A 39 C 40 B Paper 2 1 a = 12 2 (a) (X ∪ Z)’ ∩ Y (b) P Q R 3 1.5 m 4 RM4 720 5 x = 25 000 6 RM1 492 7 (a) x = 3 (b) y = –2x + 16, 8 8 x = 7 9 (a) Since Daud’s net income [RM15 000] is less that RM15 500, Daud cannot achieve his goal. (b) Two ways for Daud can increase his net income are: (i) Reduce his variable expenses, such as utility bills and entertainment expenses. (ii) Find passive income such as giving online tuitions and be an e-hailing driver. 10 (a) h = 100 (b) 0.6274 kg 11 (a) (i) U is a reflection in the straight line y = 6. (ii) V is an enlargement at the centre E with a scale factor of 1 2 . (b) 157.5 unit2 (c) 12 (a) I x + y 10 II y 3x III 5x + 3y 60 P Q R Stage 5 m Diagram 10 Find the distance, in m, of PQ. [6 marks] (c) Siva, Aaron and Jailani are from the class 5 Bestari. The masses of the students in the class 5 Bestari are shown in Table 4. Mass (kg) Frequency 40 – 49 4 50 – 59 6 60 – 69 12 70 – 79 8 80 – 89 4 90 – 99 2 Table 4 Using a scale of 2 cm to 10 kg on the horizontal axis and 2 cm to 5 students at the vertical axis, draw an ogive for the data. Hence, (i) determine the interquartile range, (ii) find the percentage of students who will join the Healthy Lifestyle Programme for those who have masses of more than 86.5 kg. [7 marks] Answers Analysis SPM Maths 2023 Eng MT 4th.indd 267 17/2/2023 10:22:41 PM PENERBIT ILMU BAKTI SDN. BHD.
268 Model Test SPM (b) 0 2468 10 12 2 4 6 8 10 12 14 16 18 20 y x R y = 3x 5x + 3y = 60 y = 5 3 x x + y = 10 (c) (i) y = 5 3 x (ii) Refer to the above graph. 13 45° 45° 4 cm 3 cm B/M/A 3 cm 4 cm 4 cm 3 cm 1 cm 2 cm 1 cm 8 cm 3 cm 3 cm 2 cm 1 cm C/N/D A/D L/D J K I C M/N H/I B/C F/K H/I F/E K/L E/L H/G E/A F G/M H B I/J G/J (a) (c) (b) 14 (a) (i) RM20, RM40 (ii) RM40, RM45 (b) RM265 Encik Khairul’s wallet has RM260 cash. It is not enough for three shirts and a pair of shoes (RM265). 15 (a) E S M 1 3 2 12 6 4 2 (b) (i) 7 students (ii) 13 students (iii) 12 students 16 (a) (i) 5 minutes (ii) 32 km (iii) m = 28.73 (b) BA BB L L′ L L′ 9 10 1 10 7 8 1 8 2 5 3 5 11 100 (c) Raymond – x = 11.330 s σx = 0.2046 s Calvin – y = 11.326 s σy = 0.2035 s Since – y < – x and σy < σx , Calvin is faster and more consistent. Hence, Calvin scould be selected. 17 (a) RM120 (b) 5.321 m 0 39.5 49.5 59.5 69.5 79.5 89.5 99.5 5 10 15 20 25 30 35 40 Cumulative frequency 36 33 27 9 Mass (kg) 58.5 (Q1 ) 74.5 (Q3 ) 86.5 (i) 16 (ii) 8 1 3 % Analysis SPM Maths 2023 Eng MT 4th.indd 268 17/2/2023 10:22:42 PM PENERBIT ILMU BAKTI SDN. BHD.
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