49 9 Graf laju-masa dalam Rajah 8 menunjukkan gerakan suatu zarah. The speed-time graph in Diagram 8 shows the motion of a particle. O 30 25 2 5 Masa (s) Time (s) Laju / Speed (m s –1) Rajah 8/Diagram 8 Antara berikut, yang manakah betul bagi menerangkan gerakan zarah itu dalam 3 saat terakhir? Which of the following correctly describes the motion of the particle in the last 3 seconds? A Zarah itu memecut pada kadar 6 m s–2 The particle is accelerating at the rate of 6 m s–2 b Zarah itu memecut pada kadar 10 m s–2 The particle is accelerating at the rate of 10 m s–2 c Zarah itu mengalami nyahpecutan pada kadar 10 m s–2 The particle is decelerating at the rate of 10 m s–2 d Zarah itu mengalami nyahpecutan pada kadar 6 m s–2 The particle is decelerating at the rate of 6 m s–2 TP 3 BT ms.200 10 Graf laju-masa dalam Rajah 9 menunjukkan gerakan sebuah kereta api dalam masa 20 saat. The speed-time graph in Diagram 9 shows the motion of a train for a period of 20 seconds. O 60 30 5 15 20 Masa (s) Time (s) Laju / Speed (m s–1) Rajah 9/Diagram 9 Hitung kadar perubahan laju, dalam m s–2, untuk tempoh 5 saat pertama. Calculate the rate of change of speed, in m s–2, in the first 5 seconds. A −6 c 6 b 3 d 225 TP 4 BT ms.200 – 201 11 Graf laju-masa dalam Rajah 10 menunjukkan gerakan sebuah kenderaan untuk tempoh 15 saat. The speed-time graph in Diagram 10 shows the motion of a vehicle for a period of 15 seconds. Aras S Aras S Aras S O 20 14 t 15 Masa (s) Time (s) Laju / Speed (m s–1) Rajah 10/Diagram 10 Jarak yang dilalui untuk tempoh t saat pertama ialah 85 m. Apakah nilai t? The distance travelled in the first t seconds is 85 m. What is the value of t? A 2 b 5 c 10 d 15 TP 5 BT ms.203 – 205 12 Graf laju-masa dalam Rajah 11 menunjukkan gerakan sebuah objek dalam tempoh 15 saat. The speed-time graph in Diagram 11 shows the motion of an object for a period of 15 seconds. O 30 25 5 10 15 Masa (s) Time (s) Laju / Speed (m s–1) Rajah 11/Diagram 11 Antara pernyataan berikut, yang manakah betul menerangkan gerakan objek itu dari saat ke-5 ke saat ke-10? Which of the following statements correctly describes the motion of the object from 5th second to the 10th second? A Objek itu tidak bergerak untuk tempoh 5 saat The object is stationary for a period of 5 seconds b Objek itu memecut pada kadar 5 m s–2 untuk tempoh 5 saat The object is accelerating at the rate of 5 m s–2 for a period of 5 seconds c Objek itu bergerak dengan kelajuan seragam 25 m s–1 untuk tempoh 5 saat The object moves at a uniform speed of 25 m s–1 for a period of 5 seconds d Objek itu mengalami nyahpecutan pada kadar 5 m s–2 untuk tempoh 5 saat The object is deccelerating at the rate of 5 m s–2 for a period of 5 seconds TP 3 BT ms.200 Aras S 07 PT SPM Mate Tg4_Bab 7_47-54_Final.indd 49 2/6/25 7:43 PM PENERBIT ILMU BAKTI SDN. BHD.
50 13 Rajah 12 menunjukkan gerakan dua objek. Graf OAB mewakili gerakan objek X, manakala graf CD mewakili gerakan objek Y. Diagram 12 shows the motion of two objects. Graph OAB represents the motion of object X while graph CD represents the motion of object Y. O A B C v D v – 10 7 20 Masa (s) Time (s) Laju / Speed (m s–1) Rajah 12/Diagram 12 Aras T Beza di antara jarak yang dilalui oleh objek X dengan objek Y dalam tempoh 20 saat ialah 35 m. Hitung nilai v. KBAT Menilai The difference between the distance travelled by object X and object Y for a period of 20 seconds is 35 m. Calculate the value of v. A 10 m s–1 b 18 m s–1 c 20 m s–1 d 25 m s–1 TP 5 BT ms.204 7.1 Graf Jarak-Masa/ Distance-Time Graphs 1 Pergerakan suatu objek diberi oleh persamaan s = 60t. BT ms.184 – 185 The motion of an object is given by the equation s = 60t. (a) Lengkapkan jadual jarak-masa bagi objek tersebut. TP 1 Aras R Complete the distance-time table for the object. [1 markah/mark] Masa, t (jam) / Time, t (hours) 0 1 2 3 4 Jarak, s (km) / Distance, s (km) 0 60 120 240 (b) Pada graf di bawah, lukiskan graf jarak-masa untuk mewakili gerakan objek tersebut. TP 3 Aras S On the graph below, draw a distance-time graph to represent the motion of the object. [2 markah/marks] 300 — 250 — 200 — 150 — 100 — 50 — 0 1 2 3 4 Jarak (km) Distance (km) Masa (jam) Time (hours) Kertas 2 / Paper 2 Bahagian A / Section A 07 PT SPM Mate Tg4_Bab 7_47-54_Final.indd 50 2/6/25 7:43 PM PENERBIT ILMU BAKTI SDN. BHD.
51 2 Rajah 1 menunjukkan graf jarak-masa bagi perjalanan dua buah kenderaan, A dan B. Kedua-dua kenderaan menuju ke destinasi yang sama. BT ms.191 – 193 Diagram 1 shows the distance-time graph for the journey of two vehicles, A and B. Both vehicles travel to the same destination. O A B 28 12 50 Masa (s) Time (s) Jarak (m) Distance (m) Rajah 1/Diagram 1 (a) Kenderaan yang manakah akan sampai ke destinasi dahulu? TP 2 Aras R Which vehicle reaches the destination first? [1 markah/mark] (b) Hitung laju, dalam m s–1, bagi kenderaan A dan kenderaan B. TP 4 Aras S Calculate the speed, in m s–1, of vehicle A and of vehicle B. [3 markah/marks] Jawapan/Answer: 7.2 Graf Laju-Masa / Speed-Time Graphs 3 Perubahan kelajuan objek diberi oleh persamaan v = 20 – 5t. BT ms.195 – 196 The change of speed of an object is given by the equation v = 20 – 5t. (a) Lengkapkan jadual laju-masa bagi objek tersebut. TP 2 Aras R Complete the speed-time table for the object. Masa, t (jam) / Time, t (hours) 0 1 2 3 4 Laju, v (km j–1) / Speed, v (km h–1) 20 15 10 [2 markah/marks] (b) Pada graf di bawah, lukiskan graf laju-masa untuk mewakili gerakan objek tersebut. TP 4 Aras S On the graph below, draw a speed-time graph to represent the motion of the object. 20 — 15 — 10 — 5 — 0 1 2 3 4 5 Masa (jam) Time (hours) Laju (km j–1) Speed (km h–1) [2 markah/marks] 07 PT SPM Mate Tg4_Bab 7_47-54_Final.indd 51 2/6/25 7:43 PM PENERBIT ILMU BAKTI SDN. BHD.
52 Bahagian B / Section B 4 Graf jarak-masa dalam Rajah 2 menunjukkan gerakan kereta Elina dari Johor Bahru ke Kuala Lumpur. BT ms.187 – 193 The distance-time graph in Diagram 2 shows the motion of Elina’s car from Johor Bahru to Kuala Lumpur. (a) Huraikan pergerakan kereta itu dari jam ke-2 hingga jam ke-3. TP 2 Aras R Describe the motion of the car from the 2nd hour till the 3rd hour. [1 markah/mark] (b) Jika kelajuan kereta dari Melaka ke Kuala Lumpur ialah 85 km j–1, cari nilai t. TP 5 Aras S If the speed of the car from Melaka to Kuala Lumpur is 85 km h-1, find the value of t. [3 markah/marks] (c) Hitung masa yang diambil, dalam jam, dari Melaka ke Kuala Lumpur. TP 3 Aras S Calculate the time taken, in hours, from Melaka to Kuala Lumpur. [2 markah/marks] (d) Hitung laju purata, dalam km j–1, kereta itu bagi keseluruhan perjalanan. TP 4 Aras S Calculate the average speed, in km h–1, of the car for the whole journey. [2 markah/marks] Jawapan/Answer: 5 Yong Xuan berbasikal sejauh 880 m dari rumahnya ke perpustakaan. Jadual 1 menunjukkan catatan perjalanannya. BT ms.191 – 193 Yong Xuan cycles for a distance of 880 m from his house to the library. Table 1 shows the note of his journey. Masa Time 8 Februari 2025 / 8 February 2025 Sabtu / Saturday 8:30 a.m. Bertolak dari rumah / Departs from house 8:40 a.m. Berhenti di gerai makan setelah berbasikal sejauh 240 m Stops at a food stall after cycling for 240 m 8:55 a.m. Meneruskan perjalanan sejauh 640 m lagi Continues the journey for another 640 m 9:10 a.m. Tiba di perpustakaan / Arrives at library Jadual 1/Table 1 (a) Rajah 3 menunjukkan graf jarak-masa bagi perjalanannya. Aras S Diagram 3 shows the distance-time graph for his journey. (i) Nyatakan nilai m dan nilai n. State the value of m and of n. 300 215 O 2 3 t Masa (jam) Time (hours) Jarak (km) Distance (km) Kuala Lumpur Melaka Johor Bahru Rajah 2/Diagram 2 07 PT SPM Mate Tg4_Bab 7_47-54_Final.indd 52 2/6/25 7:43 PM PENERBIT ILMU BAKTI SDN. BHD.
53 (ii) Lengkapkan graf itu untuk mewakilkan keseluruhan perjalanan Yong Xuan. Complete the graph to represent Yong Xuan’s whole journey. [5 markah/marks] (b) Hitung laju purata, dalam km j–1, bagi keseluruhan perjalanan. TP 4 Aras S Calculate the average speed, in km h–1, for the whole journey. [3 markah/marks] Jawapan/Answer: 6 Rajah 4 menunjukkan graf laju-masa bagi gerakan suatu objek dalam tempoh 36 saat. BT ms.200 – 205 Diagram 4 shows the speed-time graph for the movement of an object for a period of 36 seconds. (a) Hitung tempoh masa, dalam s, objek itu bergerak dengan laju seragam. TP 2 Aras R Calculate the duration of time, in s, for which the object moves with uniform speed. [1 markah/mark] (b) Nyatakan pecutan objek itu dari saat ke-8 hingga saat ke-26. TP 2 Aras R State the acceleration of the object from the 8th second till the 26th second. [1 markah/mark] (c) Hitung kadar perubahan laju, dalam m s–2, objek itu untuk 10 saat terakhir. TP 3 Aras S Calculate the rate of change of speed, in m s–2, of the object in the last 10 seconds. [2 markah/marks] (d) Hitung nilai v, jika jumlah jarak yang dilalui dalam tempoh 36 saat ialah 1 200 m. TP 5 Aras S Calculate the value of v, if the total distance travelled for the period of 36 seconds is 1 200 m. [4 markah/marks] Jawapan/Answer: O 880 m 10 n 40 Masa (minit) Time (minutes) Jarak (m) Distance (m) Rajah 3/Diagram 3 O 42 v 8 26 36 Laju / Speed (m s–1) Masa (s) Time (s) Rajah 4/Diagram 4 07 PT SPM Mate Tg4_Bab 7_47-54_Final.indd 53 2/6/25 7:43 PM PENERBIT ILMU BAKTI SDN. BHD.
54 1 Graf jarak-masa dalam rajah di bawah menunjukkan perjalanan dua buah bas dari terminal bas pusat ke stesen Z dengan menggunakan laluan yang sama. OPQRS mewakili perjalanan bas A, manakala TRU mewakili perjalanan bas B. Bas A bertolak pada pukul 1:40 p.m. Dalam perjalanan, bas A berhenti untuk mengambil beberapa penumpang. Bas B bertolak pada pukul 2:00 p.m. Kedua-dua bas bertemu di bandar R pada waktu m. BT ms.191 – 195 Aras T KBAT Menganalisis The distance-time graph in the diagram below shows the journey of two buses from the central bus terminal to Station Z by using the same route. OPQRS represents the journey of bus A while TRU represents the journey of bus B. Bus A departs at 1:40 p.m. During the journey, bus A stops to pick up some passengers. Bus B departs at 2:00 p.m. Both buses meet at town R at time m. O P Q T R U S 55 32 20 15 20 m 35 45 Jarak dari terminal bas pusat (km) Distance from central bus terminal (km) Masa (min) Time (min) (a) Nyatakan tempoh masa, dalam minit, bas A berhenti untuk mengambil penumpang. TP 2 State the duration, in minutes, bus A stops to pick up the passengers. (b) Hitung jarak, dalam km, yang dilalui oleh bas A apabila ia bertemu dengan bas B di bandar R. TP 2 Calculate the distance, in km, travelled by bus A when it meets bus B at town R. (c) Cari laju, dalam km j–1, bagi bas A selepas melalui bandar R. Beri jawapan dalam 1 titik perpuluhan. Find the speed, in km h–1, of bus A after passing through town R. Give the answer in 1 decimal point. TP 5 Jawapan/Answer: Zon KBAT 07 PT SPM Mate Tg4_Bab 7_47-54_Final.indd 54 2/6/25 7:43 PM PENERBIT ILMU BAKTI SDN. BHD.
55 8.1 Serakan / Dispersion 1 Rajah 1 ialah sebuah plot titik yang menunjukkan bilangan kereta yang dijual oleh seorang ejen dalam tempoh 5 minggu. Diagram 1 is a dot plot showing the number of cars sold by an agent in 5 weeks. Bilangan Kereta yang Dijual Number of Cars Sold Minggu/ Week 1 2 3 4 5 Rajah 1/Diagram 1 Pada minggu berapakah paling banyak kereta terjual? In which week were the cars sold the most? A Minggu pertama/ First week B Minggu kedua/ Second week C Minggu ketiga/ Third week D Minggu keempat/ Fourth week TP 2 BT ms.213 – 215 2 Rajah 2 menunjukkan tinggi 25 orang murid Tingkatan 5 Murni yang diwakili oleh plot batang-dan-daun. Diagram 2 shows the height of 25 students of Form 5 Murni represented by a stem-and-leaf plot. Batang/Stem Daun/Leaf 14 7 8 9 9 15 0 1 2 3 5 5 5 7 7 8 9 16 3 4 4 5 7 8 8 17 0 1 1 Kekunci: 14|7 bermaksud 147 cm Key: 14|7 means 147 cm Rajah 2/Diagram 2 Berapakah beza ketinggian antara murid tertinggi dengan murid terendah? What is the height difference between the tallest and the shortest student? A 9 cm C 24 cm B 12 cm D 30 cm TP 2 BT ms.214 – 215 Aras R Aras R 3 Plot batang-dan-daun dalam Rajah 3 menunjukkan markah Sejarah yang diperoleh oleh murid-murid Kelas 4B dan Kelas 4C. The stem-and-leaf plot in Diagram 3 shows the History marks obtained by students in Class 4B and Class 4C. Kelas 4B /Class 4B Kelas 4C /Class 4C 8 8 5 5 7 8 2 5 6 3 0 1 1 4 3 0 0 0 0 4 5 6 7 8 9 2 0 2 3 1 4 3 0 5 4 1 5 4 1 5 5 3 8 7 4 5 Kekunci: 5|4|2 bermaksud 45 markah untuk kelas 4B dan 42 markah untuk kelas 4C Key: 5|4|2 means 45 marks for class 4B and 42 marks for class 4C Rajah 3/Diagram 3 Cari peratus murid yang memperoleh 60 markah dan ke bawah. Find the percentage of students who scored 60 marks and below. A 35% c 55% b 52.5% d 70% TP 3 BT ms.216 8.2 Sukatan Serakan / Measure of Dispersion 4 Markah suatu Ujian Bahasa Inggeris untuk 11 orang murid ditunjukkan dalam Rajah 3. The marks of English Test for 11 students are shown in Diagram 3. 74, 55, 60, 56, 72, 58, 65, 66, 70, 78, 69 Cari julat antara kuartil markah itu. Find the interquartile range of the marks. A 10 C 20 B 14 D 23 TP 2 BT ms.219 – 220 5 Jadual 1 menunjukkan bilangan jam yang digunakan oleh 30 orang murid untuk mengulang kaji selama seminggu. Table 1 shows the number of hours that 30 students spent on studying in a week. Aras S Aras R Aras S Kertas 1 / Paper 1 Bidang Pembelajaran: Statistik dan Kebarangkalian BAB 8 Sukatan Serakan Data Tak Terkumpul Measure of Dispersion for Ungrouped Data 08 PT SPM Mate Tg4_Bab 8_55-63_Final.indd 55 2/6/25 7:51 PM PENERBIT ILMU BAKTI SDN. BHD.
56 Bilangan jam Number of hours 9 10 11 12 13 14 Bilangan murid Number of students 1 3 7 10 4 5 Jadual 1/Table 1 Cari julat antara kuartil bagi data di atas. Find the interquartile range of the data above. A 2 C 4 B 3 D 5 TP 2 BT ms.220 – 221 6 Rajah 4 menunjukkan sebuah plot kotak yang mewakili ketebalan beberapa buah buku di sebuah perpustakaan mini. Diagram 4 shows a box plot which represents the width of some books in a mini library. 5 10 15 20 25 30 35 40 Ketebalan/Width (mm) 10 15 20 25 30 x Rajah 4/Diagram 4 Tentukan julat ketebalan buku di perpustakaan mini itu. Determine the range of the width of books in the mini library. A 17 mm C 26 mm B 18 mm D 32 mm TP 2 BT ms.226 – 227 7 Plot kotak dalam Rajah 5 menunjukkan masa menunggu, dalam minit, untuk menaiki bas semasa waktu puncak oleh sekumpulan murid. The box plot in Diagram 5 shows the waiting time, in minutes, to get on the bus during peak hour by a group of students. p q r s t Masa / Time (min) Rajah 5/Diagram 5 Diberi bahawa m% murid menunggu antara r minit hingga s minit untuk menaiki bas. Nyatakan nilai m. It is given that m% of the students wait between r minutes to s minutes to get on the bus. State the value of m. A 20 c 50 b 25 d 75 TP 3 BT ms.226 – 227 Aras R Aras S 8 Rajah 6 menunjukkan plot batang dan daun. Diagram 6 shows a stem-and-leaf plot. Batang /Stem Daun /Leaf 4 5 6 7 2 3 2 0 6 3 2 6 3 5 Kekunci: 4 | 2 bermaksud 42 Key: 4 | 2 means 42 Rajah 6/Diagram 6 Tentukan julat antara kuartil bagi data itu. Determine the interquartile range of the data. A 16 c 20 b 19 d 28 TP 3 BT ms.214 – 215, 220 9 Jadual 2 menunjukkan skor yang diperoleh 11 orang murid dalam satu kuiz Matematik. Table 2 shows the scores obtained by 11 students in a Mathematics quiz. Skor Score 3 p 7 8 q 10 Bilangan murid Number of students 2 1 2 3 1 2 Jadual 2/Table 2 Skor tersebut disusun mengikut tertib menaik. Kuartil ketiga ialah q dan min skor ialah 7. Cari nilai p. KBAT Mengaplikasi The scores are arranged in ascending order. The third quartile is q and the mean score is 7. Find the value of p. A p = 4, q = 9 c p = 5, q = 9 b p = 4, q = 8 d p = 5, q = 10 TP 4 BT ms.220 – 221 10 Plot batang-dan-daun dalam Rajah 7 menunjukkan panjang, dalam m, bagi lima batang rod. The stem-and-leaf plot in Diagram 7 shows the lengths, in m, of five pieces of rod. Batang /Stem Daun /Leaf 1 2 3 1 4 0 6 5 Kekunci: 1 | 1 bermaksud 1.1 m Key: 1 | 1 means 1.1 m Rajah 7/Diagram 7 Hitung sisihan piawai bagi data itu. Calculate the standard deviation of the data. A 0.4616 c 2.120 b 0.6794 d 4.494 TP 4 BT ms.221 – 223 Aras S Aras S Aras S 08 PT SPM Mate Tg4_Bab 8_55-63_Final.indd 56 2/6/25 7:51 PM PENERBIT ILMU BAKTI SDN. BHD.
57 11 Plot kotak dalam Rajah 8 mewakili satu set data. The box plot in Diagram 8 represents a set of data. p 2q 4q 5q Rajah 8/Diagram 8 Diberi julat dan julat antara kuartil masingmasing ialah 150 dan 80. Cari nilai p. Given the range and interquartile range are 150 and 80 respectively. Find the value of p. KBAT Mengaplikasi A 40 c 60 b 50 d 70 TP 4 BT ms.226 – 227 12 Jumlah nilai bagi tujuh nombor ialah 63. Diberi varians bagi tujuh nombor itu ialah 13. Cari jumlah kuasa dua bagi nombor-nombor itu. The sum of seven numbers is 63. Given the variance of the seven numbers is 13. Find the sum of squares of the numbers. A 476 c 658 b 570 d 757 TP 3 BT ms.236 Aras S Aras S 13 Suatu set 12 nombor positif mempunyai varians 46 dan jumlah kuasa dua nombor-nombor positif itu ialah 2 280. Cari min bagi set nombor itu. A set of 12 positive numbers has a variance of 46 and the sum of squares of the positive numbers is 2 280. Find the mean of the set of numbers. KBAT Mengaplikasi A 8 c 11 b 10 d 12 TP 5 BT ms.236 14 Diberi satu set data x1 , x2 , x3 , ..., xn mempunyai min 20 dan sisihan piawai 8. Setiap nilai dalam set data itu didarabkan dengan 2 dan kemudian ditambahkan dengan 5. Cari min baharu dan sisihan piawai baharu. KBAT Menilai Given a set of data x1 , x2 , x3 , ..., xn has a mean of 20 and standard deviation of 8. Each value of the set of data is multiplied by 2 and then is added by 5. Find the new mean and new standard deviation. Min baharu New mean Sisihan piawai baharu New standard deviation A 25 16 B 25 13 C 45 21 D 45 16 TP 5 BT ms.236 Aras T Aras T 8.1 Serakan / Dispersion 1 Plot titik dalam Rajah 1 menunjukkan saiz kasut yang dipakai oleh dua kumpulan orang yang berbeza. The dot plots in Diagram 1 show the sizes of shoes worn by two different groups of people. BT ms.217 Kumpulan P / Group P Kumpulan Q / Group Q 0 2 4 6 8 10 12 0 2 4 6 8 10 12 Saiz kasut Size of shoes Saiz kasut Size of shoes 0 2 4 6 8 10 12 0 2 4 6 8 10 12 Saiz kasut / Size of shoes Saiz kasut / Size of shoes Rajah 1/Diagram 1 (a) Bandingkan bentuk kedua-dua plot titik tersebut. TP 4 Aras R Compare the shapes of the two dot plots. [2 markah/marks] (b) Apakah maklumat yang boleh diperoleh daripada plot titik tentang saiz kasut kedua-dua kumpulan? TP 4 Aras S What is the information that can be obtained from the dot plots about the sizes of shoes of the two groups? [2 markah/marks] Jawapan/Answer: Kertas 2 / Paper 2 Bahagian A / Section A 08 PT SPM Mate Tg4_Bab 8_55-63_Final.indd 57 2/6/25 7:51 PM PENERBIT ILMU BAKTI SDN. BHD.
58 8.2 Sukatan Serakan / Measure of Dispersion 2 Cari varians dan sisihan piawai bagi set data yang berikut: TP 3 BT ms.221 – 223 Aras S Find the variance and standard deviation for the following sets of data: 23, 24, 24, 26, 29 Jawapan/Answer: x x 2 23 529 24 576 24 26 29 sx = 126 sx 2 = [4 markah/marks] 3 Wakilkan set data yang berikut menggunakan plot kotak. TP 4 BT ms.226 – 227 Aras S Represent the following set of data using a box plot. 20, 27, 23, 50, 33, 37, 26 [4 markah/marks] Nilai minimum/ Minimum value = _______________ Nilai maksimum/ Maximum value = _______________ Kuartil pertama/ First quartile, Q1 = _______________ Median = _______________ Kuartil ketiga/ Third quartile, Q3 = _______________ 10 20 30 40 50 4 Tentukan julat, kuartil pertama, kuartil ketiga dan julat antara kuartil bagi plot kotak di bawah. TP 3 BT ms.226 – 227 Aras S Determine the range, first quartile, third quartile and interquartile range for the box plot below. [4 markah/marks] Julat/ Range = _______________ Kuartil pertama/ First quartile, Q1 = _______________ Kuartil ketiga/ Third quartile, Q3 = _______________ Julat antara kuartil/ Interquartile range = Q1 – Q3 = _______________ = _______________ 10 15 20 25 30 08 PT SPM Mate Tg4_Bab 8_55-63_Final.indd 58 2/6/25 7:52 PM PENERBIT ILMU BAKTI SDN. BHD.
59 Bahagian B / Section B 5 Jadual 1 menunjukkan markah yang diperoleh dua orang murid, Nurul Ain dan Jasmine, dalam ujian Bahasa Inggeris. BT ms.224 – 225 Table 1 shows the marks obtained by two students, Nurul Ain and Jasmine, in an English test. (a) Cari min dan sisihan piawai bagi markah Nurul Ain dan markah Jasmine. TP 4 Aras S Find the mean and standard deviation of Nurul Ain’s marks and Jasmine’s marks. [6 markah/marks] (b) Pencapaian murid yang manakah lebih konsisten? Terangkan. TP 3 Aras T KBAT Mengaplikasi Whose performance is more consistent? Explain. [2 markah/marks] Jawapan/Answer: 6 Data dalam Rajah 2 menunjukkan nilai pH yang diperoleh daripada 15 jenis sabun tangan. The data in Diagram 2 shows the pH values obtained from 15 types of hand soaps. 8.9 10.3 9.4 9.3 10.5 9.0 9.2 9.1 10.0 9.8 8.8 9.0 9.3 9.5 9.2 Rajah 2/Diagram 2 (a) Lukis plot kotak untuk mewakili data di atas. TP 4 BT ms.226 – 227 Aras S Draw a box plot to represent the above data. [4 markah/marks] (b) Penunjuk semesta ialah penunjuk pH yang mengubah warnanya mengikut keasidan atau kealkalian bahan tersebut. Jika pH bahan itu lebih daripada 10, warna penunjuk ialah biru. Jika pH adalah antara 8 hingga 10, warna penunjuk ialah hijau. Nyatakan warna penunjuk semesta apabila ia ditambahkan ke dalam cecair sabun tangan yang diperoleh daripada sabun tangan yang sepadan dengan kuartil ketiga. TP 5 BT ms.236 – 237 Aras T KBAT Menganalisis A universal indicator is a pH indicator that changes its colour according to the acidity or alkalinity of a substance. If the pH of the substance is more than 10, the colour of the indicator is blue. If the pH is between 8 and 10, the colour of the indicator is green. State the colour of the universal indicator when it is added to the hand soap liquid that obtained from the hand soap corresponding to the third quartile. [4 markah/marks] Jawapan/Answer: Murid Student Markah Marks Nurul Ain 83, 65, 61, 56, 63, 66 Jasmine 64, 73, 59, 60, 61, 77 Jadual 1/Table 1 08 PT SPM Mate Tg4_Bab 8_55-63_Final.indd 59 2/6/25 7:52 PM PENERBIT ILMU BAKTI SDN. BHD.
60 Bahagian C / Section C 7 Rajah 3.1 ialah dua plot kotak yang menunjukkan taburan pendapatan bulanan yang diperoleh dua kumpulan pekerja, P dan Q di Penerbitan Maju. BT ms.227 Diagram 3.1 is two box plots showing the distribution of monthly incomes earned by two groups of workers, P and Q in Penerbitan Maju. Pendapatan bulanan (× RM1 000) Monthly income (× RM1 000) 6543210 Kumpulan/Group Q Kumpulan/Group P Rajah 3.1/Diagram 3.1 (a) Cari julat antara kuartil bagi TP 3 Aras R Find the interquartile range of (i) Kumpulan/Group P: (ii) Kumpulan/Group Q: [2 markah/marks] (b) Untuk setiap pernyataan di bawah, nyatakan sama ada anda bersetuju atau tidak bersetuju. Mengapa? TP 4 Aras T KBAT Menganalisis For each of the statements below, state whether you agree or disagree. Why? Pernyataan I : Secara purata, pekerja daripada Kumpulan P dibayar lebih tinggi. Statement I : On average, the workers from Group P are paid higher. Pernyataan II : Lebih ramai pekerja dalam Kumpulan Q dibayar lebih daripada RM4 500 sebulan berbanding dengan pekerja dalam Kumpulan P. Statement II : There is a greater proportion of the workers in Group Q who are paid more than RM4 500 per month compared to the workers in Group P. [5 markah/marks] (c) Penerbitan Maju ingin mengambil seorang pekerja baharu untuk jabatan suntingan. Tiga orang calon disenarai pendek dan perlu membuat ujian bertulis sebelum dipanggil untuk ditemu duga. Markah yang diperoleh ketiga-tiga orang calon itu diwakili oleh plot kotak dalam Rajah 3.2. Penerbitan Maju wants to hire a new worker for the editorial department. Three candidates are shortlisted and required to take a writing test before being called for an interview. The marks obtained by the three candidates are represented in the box plots in Diagram 3.2. BT ms.236 – 237 Aras T Calon A Candidate A Calon B Candidate B Calon C Candidate C Markah/Marks (%) 50 55 60 65 70 Rajah 3.2/Diagram 3.2 08 PT SPM Mate Tg4_Bab 8_55-63_Final.indd 60 2/6/25 7:52 PM PENERBIT ILMU BAKTI SDN. BHD.
Praktis Topikal SPM Matematik Tingkatan 4 – Jawapan J1 Fungsi dan Persamaan Kuadratik dalam Satu Pemboleh Ubah Bab 1 Kertas 1 1 B 2 D 3 b 4 c 5 c 6 D 7 C 8 A 9 C 10 B 11 A 12 C 13 B 14 D 15 C 16 C Kertas 2 Bahagian A 1 5 + m(5) – 2(5)2 = 0 5m – 45 = 0 m = 9 5 + 9x – 2x 2 = 0 2x 2 – 9x – 5 = 0 (2x + 1)(x – 5) = 0 x = – 1 2 ; x = 5 [ Penyelesaian yang satu lagi ialah/ Another solution is x = – 1 2 . 2 a = 1 > 0, graf berbentuk/ the graph is y = x 2 – 6x = x(x – 6) Apabila/ When y = 0 ; x(x – 6) = 0 x = 0 ; x = 6 Graf itu melalui titik-titik (0, 0) dan (6, 0). The graph passes through points (0, 0) and (6, 0). Garis x = 3 merupakan paksi simetri bagi graf. Line x = 3 is the axis of symmetry of the graph. Apabila/ When x = 3, y = (3)2 – 6(3) = –9 0 10 — 5 — — –5 — –10 — × y x (3, –9) x = 3 × 2 4 6 8 Titik minimum/ Minimum point = (3, –9) 3 3x 2 + 11x – 20 = 9x 2 – 24x + 16 6x 2 – 35x + 36 = 0 (3x – 4)(2x – 9) = 0 x = 4 3 , x = 9 2 4 (a) Harga kos sebatang pen/ Cost price for each pen = 48 x (b) Harga jual sebatang pen/ Selling price for each pen = 48 x + 3 (c) (48 x + 3)(x – 3) = 48 + 15 (48 x + 3)(x – 3) = 63 (48 + 3x)(x – 3) = 63x 3x 2 – 24x – 144 = 0 x 2 – 8x – 48 = 0 (x – 12)(x + 4) = 0 x = 12, x = –4 (x > 0) ∴ x = 12 5 w(w + 5)(40) = 12 000 w(w + 5) = 300 w2 + 5w – 300 = 0 (w – 15)(w + 20) = 0 w = 15, w = –20 (w > 0) ∴ w = 15 6 (a) n 2 (n – 3) = 90 n2 – 3n – 180 = 0 (n – 15)(n + 12) = 0 n = 15; n = –12 (n 0) ∴ n = 15 Poligon mempunyai 15 sisi/ The polygon has 15 sides. (b) Tiada/ No n 2 (n – 3) = 90 n2 – 3n –100 = 0 Tiada nombor bulat yang positif untuk n. There is no positive whole number for n. 7 (a) Luas kawasan berlorek = Luas PQRS – Luas WQV – Luas STU Area of shaded region = Area of PQRS – Area of WQV – Area of STU = 14(3x + 4) – 1 2(x)(x) – 1 2(2x)(2x) = 42x + 56 – 1 2x2 – 2x2 = – 5 2x2 + 42x + 56 (b) – 5 2 x2 + 42x + 56 = 184 5x2 – 84x + 256 = 0 (x – 4)(5x – 64) = 0 x = 4, x = 64 5 (x = nombor bulat/ whole number) ∴ x = 4 Bahagian B 8 (a) f(x) = x2 + 8x + 13 x –8 –5 –4 –1 0 1 2 3 y 13 –2 –3 6 13 22 33 46 (b) x y 50 40 30 20 10 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 y = x 2 + 8x + 13 (i) Bentuk U/ U-shaped (ii) Titik minimum/ Minimum point: (–4, –3) (iii) Persamaan paksi simetri/ Equation of axis symmetry: x = –4 9 (a) Andaikan bilangan ahli Persatuan Matematik sebagai x Assume the number of members in Mathematics Society is x 360 x – 360 x + 10 = 6 360(x + 10) – 360x = 6x(x + 10) 6x 2 + 60x – 3600 = 0 x 2 + 10x – 600 = 0 (x – 20)(x + 30) = 0 x = 20, x = –30 (x > 0) Bilangan ahli Persatuan Matematik ialah 20 orang. The number of members in Mathematics Society is 20. Jawapan 11 PT SPM Mate Tg4_Jawapan_J1-12_Final.indd 1 2/6/25 8:20 PM PENERBIT ILMU BAKTI SDN. BHD.
Penerbit Ilmu Bakti Sdn. Bhd. (732516-M) 2025 J2 (b) RM360 (20 + 10) = RM12 (c) RM360 20 = RM18 atau/or RM12 + RM6 = RM18 Bahagian C 10 (a) t 0 1 2.5 3 5 6 s 95 160 197.5 194 100 5 (b) s t s = 20t s = 95 + 81t – 16t 2 0 1 2 3 4 5 6 7 200— 150— 100— 50— × × × × × × (c) (i) Lebih daripada 6 saat atau 6.05 saat. More than 6 seconds or 6.05 seconds. (ii) 5.1 saat/ seconds (iii) 197.5 m , x = 2.5 s (d) (i) x 0 5 y 0 100 (ii) 5 saat/ seconds Batu dan roket berada pada ketinggian 100 m dari tanah pada saat ke-5. The stone and the rocket were at 100 m above the ground at the 5th seconds. Bab 2 Asas Nombor Kertas 1 1 C 2 D 3 a 4 d 5 a 6 d 7 B 8 c 9 c 10 a 11 b 12 a 13 b 14 b 15 a 16 c 17 B 18 d 19 b 20 d 21 b 22 c 23 a 24 c 25 d 26 b 27 c Kertas 2 Bahagian A 1 49 + 21 + 2 = 7210 2 3 1 3 1 3 1 1 1 0 2 0 0 4 2 0 1102003 = 4209 3 [ p = 2020 1 1 1 1441 6 + 135 6 2020 6 4 42 41 40 RM1324 = 1 × 42 + 3 × 41 + 2 × 40 = 16 + 12 + 2 = 3010 Harga jual bagi 1 malt coklat Selling price for 1 packet of chocolate malt = RM30 + RM5 = RM3510 RM2314 = 2 × 42 + 3 × 41 + 1 × 40 = 32 + 12 + 1 = 4510 Harga jual bagi 1 kampit beras wangi Selling price for 1 bag of fragrant rice = RM45 + RM7 = RM5210 5 2348 – 578 = 1558 82 81 80 1 5 58 = 1 × 82 + 5 × 81 + 5 × 80 = 64 + 40 + 5 = 10910 Bil elektrik/ Electricity bill = 109 × RM0.65 = RM70.85 Bahagian B 6 (a) 334 = 21 20 21 20 1 1 1 12 334 = 11112 (b) 23 22 21 20 1 1 1 12 = 1 × 23 + 1 × 22 + 1 × 21 + 1 × 20 = 8 + 4 + 2 + 1 = 1510 51 50 2 35 = 2 × 51 + 3 × 50 = 10 + 3 = 1310 41 40 3 24 = 3 × 41 + 2 × 40 = 12 + 2 = 1410 32 31 30 1 2 23 = 1 × 32 + 2 × 31 + 2 × 30 = 9 + 6 + 2 = 1710 Maka, susunannya Therefore, the arrangement is: 235 , 324 , 11112 , 1223 Zon KBAT 1 34 33 32 31 30 1 2 1 2 03 = 1 × 34 + 2 × 33 + 1 × 32 + 2 × 31 + 0 × 30 = 81 + 54 + 9 + 6 +0 = 15010 Harga 1 kg durian/ Price of 1 kg of durian = RM15010 ÷ 5 = RM3010 Harga 8 kg durian/ Price of 8 kg of durian = RM3010 × 8 = RM24010 Baki wang/ Money change = RM300 – RM40 = RM6010 Bab 3 Penaakulan Logik Kertas 1 1 D 2 A 3 D 4 B 5 A 6 b 7 A 8 C 9 B 10 C 11 D Kertas 2 Bahagian A 1 (a) (i) Palsu/ False (ii) Benar/ True (b) (i) 71 bukan nombor perdana/ 71 is not a prime number (ii) Hasil tambah 2 nombor ganjil bukan nombor genap The sum of 2 odd numbers is not an even number 11 PT SPM Mate Tg4_Jawapan_J1-12_Final.indd 2 2/6/25 8:20 PM PENERBIT ILMU BAKTI SDN. BHD.
Praktis Topikal SPM Matematik Tingkatan 4 – Jawapan J3 2 (a) Antejadian/ Antecedent Akibat/ Consequent (b) x < 0 x2 > 0 Sudut sepadan pada dua garis lurus adalah sama The corresponding angles of two straight lines are equal Garis lurus adalah selari The straight lines are parallel 3 (a) Nilai kebenaran Truth value Penyangkal Counter-example (b) Palsu/ False 6 ialah gandaan 3 tetapi bukan 9 6 is a multiple of 3 but not a multiple of 9 Palsu/ False Faktor untuk 2 (nombor perdana) ialah 1 dan 2 sahaja The factors for 2 (prime number) are 1 and 2 only 4 Akas Converse Songsangan Inverse Kontrapositif Contrapositive Jika/ If x >14, maka/ then x +10 > 24 Jika/ If x +10 < 24, maka/ then x < 14 Jika/ If x <14, maka/ then x +10 < 24 5 (a) (i) Lemah dan tidak meyakinkan/ Weak and not cogent (ii) Kuat dan meyakinkan/ Strong and cogent (b) 1 2n ; n = 1, 2, 3, 4, … 6 (a) Jika luas segi empat sama PQRS bukan 49 cm2 , maka sisi segi empat sama PQRS bukan 7 cm. Benar. If the area of a square PQRS is not 49 cm2 , then the side of square PQRS is not 7 cm. True. (b) ABC ialah sebuah segi tiga sama sisi. Sah dan munasabah. ABC ia an equilateral triangle. Valid and sound. (c) n = 7, 180° × 7 – 360° = 900° Bahagian B 7 (a) 57 ÷ 3 = 19 (Palsu/ False) (b) Songsangan: Jika 5 bukan faktor n, maka n tidak boleh dibahagi tepat dengan 5 Inverse: If 5 is not a factor of n, then n is not divisible by 5 (c) 15 bukan integer/ is not an integer (d) n = 6. (6 – 2) × 180° = 720°. (e) Akas: Jika a – b < 0, maka a < b. Akas itu benar. Converse: If a – b < 0, then a < b. The converse is true. (f) (i) d = n – 1 (ii) Corak untuk segi tiga/ Pattern for a triangle: 8 (a) Benar/ True (b) Kontrapositif: Jika garis lurus AB bukan tangen kepada bulatan C, maka garis lurus AB tidak menyentuh bulatan C pada satu titik. Benar. Contrapositive: If straight line AB is not a tangent to the circle C, then the straight line AB does not touch the circle C at exactly one point. True. (c) (i) atau/ or (ii) (a) Benar/ True (b) Benar/ True (d) 4n + n2 , n = 1, 2, 3, 4, … (e) PQRS ialah sisi empat/ PQRS is a quadrilateral (f) (i) c2 = a2 + b2 (ii) c2 = 82 + 152 = 289 c = 17 9 RM1, RM4, RM(4 + 3), RM(4 + 3 + 3), … (a) RM1 = 3 × 1 – 2 RM4 = 3 × 2 – 2 RM7 = 3 × 3 – 2 . . . Simpanan pada hari ke-n,/ Savings on nth day, Tn = 3n – 2 (b) Sebutan ke-100/ 100th term, T100 = 3(100) – 2 = 298 1, 4, 7, 10... 292, 295, 298 299 299 299 Jumlah simpanan selepas 100 hari/ Total savings after 100 days = 100 2 × RM299 = RM14 950 10 (a) n = 3n + 2; n = 1, 2, 3, 4,… (b) Tahap Level Bilangan kepingan Number of pieces Bilangan kepingan terkumpul Number of accumulated pieces 1 3(1) + 2 = 5 5 2 3(2) + 2 = 8 5 + 8 = 13 3 3(3) + 2 = 11 13 + 11 = 24 4 3(4) + 2 = 14 24 + 14 = 38 5 3(5) + 2 = 17 38 + 17 = 55 6 3(6) + 2 = 20 55 + 20 = 75 7 3(7) + 2 = 23 75 + 23 = 98 [ 100 keping domino mencukupi untuk membina struktur 7 tingkat/ aras. ∴ 100 pieces of dominoes are sufficient to build a 7-levels structure. (c) Bilangan domino yang diperlukan untuk tapak struktur ialah 23. The number of dominoes needed for the base of the structure is 23. 11 (a) Jumlah bilangan ternakan pada 31 Disember 2028 Total number of livestock on 31 December 2028 = 190(42 + 4 + 330) = 66 500 (b) Bilangan ternakan/ The number of livestock Tahun Year Bilangan ternakan Number of livestock 1 190(12 + 1 + 330) = 63 080 2 190(22 + 2 + 330) = 63 840 3 190(32 + 3 + 330) = 64 980 4 190(42 + 4 + 330) = 66 500 5 190(52 + 5 + 330) = 68 400 6 190(62 + 6 + 330) = 70 680 Jumlah ternakan akan melebihi 70 000 ekor pada tahun keenam. The total number of livestock will exceed 70 000 by the sixth year. Bab 4 Operasi Set Kertas 1 1 C 2 D 3 d 4 C 5 D 6 C 7 d 8 A 9 c 10 A 11 D Kertas 2 Bahagian A 1 (a) C D = {c, r, d} (b) G H I = {2, 3, 5} 2 (a) A B 11 PT SPM Mate Tg4_Jawapan_J1-12_Final.indd 3 2/6/25 8:20 PM PENERBIT ILMU BAKTI SDN. BHD.
Penerbit Ilmu Bakti Sdn. Bhd. (732516-M) 2025 J4 (b) A B C 3 (a) (i) A B = {1, 2, 3, 4, 5} (ii) A B C = {a, b, c, d, e, f } (b) (i) A B = {2, 4, 6, 7, 8} (ii) (A B) = {3, 5, 9} 4 (a) (i) A B = [ (ii) A C = {11, 13, 15, 17, 19} (iii) A B C9 = {10, 12, 14, 16, 18, 20} 5 (a) Y , X (b) A C B 6 (a) M N (b) (P R) Q 7 (a) n(A C) = 3 + 5 = 8 (b) n(B9) = 6 + 3 + 8 = 17 (c) n[(A B) C] = 3 + 5 + 2 = 10 Bahagian B 8 (a) 6 orang murid/ students (b) Jumlah bilangan murid/ the number of students = 80 x + y + 52 = 80 x + y = 28 ------ y + 12 + 6 + 8 = 2(x + 12 + 6 + 3) y + 26 = 2x + 42 –2x + y = 16 ------ – 3x = 12 x = 4 12 + 6 + 3 + 4 = 25 (c) Dari/From 4 + y = 28; x = 4 y = 24 Maka/So, x + y = 4 + 24 = 28 9 x = {jumlah bilangan murid the total number of students} P = {murid yang lulus Fizik students who passed Chemistry} C = {murid yang lulus Kimia students who passed biology} n(x) = 40, n(P) = 32, n(C) = 35 n(P C)9 = 3 (a) n(P C9) = 40 – 3 – 35 = 2 n(P C) = 32 – 2 = 30 n(P9 C) = 35 – 30 = 5 P 2 30 5 3 C ξ (b) (i) Bilangan murid yang lulus satu subjek sahaja The number of students who passed one subject only = 2 + 5 = 7 (ii) Bilangan murid yang lulus kedua-dua mata pelajaran The number of students who passed both subjects. = 30 (c) 85% × 40 = 34 [ n(B) = 34 n(P) = 32, n(C) = 35 Bilangan murid yang lulus Kimia adalah yang paling ramai. Maka, Kimia mempunyai peratus lulus yang tertinggi. The number of student who passed Chemistry is the highest. Thus, Chemistry has the highest pass percentage. Zon KBAT 1 36 36 14 46 50 10 A A B B x x Nilai minimum Minimum value n(a B) = 14 Nilai maksimum Maximum value n(a B) = 50 Bab 5 Rangkaian dalam Teori Graf Kertas 1 1 A 2 C 3 D 4 D 5 b 6 a 7 c 8 b 9 c 10 b 11 B Kertas 2 Bahagian A 1 (a) Bukan graf berkait The graph is not connected (b) Tidak/ No (c) Graf mudah kerana tidak mengandungi gelung atau berbilang tepi A simple graph because without loop or multiple edges 2 (v1 , v2 ) a b c d e a 5 4 b 5 8 6 c 8 2 3 d 2 2 e 4 6 3 2 3 b a d c c f e j Subgraf/ Subgraph 1 Subgraf/ Subgraph 2 e j f e g h h i Subgraf/ Subgraph 3 Subgraf/ Subgraph 4 4 (a) Ya, ini pokok. Terdapat satu dan hanya satu tepi yang mengaitkan mana-mana dua titik. Yes, this is a tree. There is one and only one edge joining any two vertices. (b) Tidak, ini bukan pokok. Terdapat berbilang tepi di antara C dengan O. No, this is not a tree. There are multiple edges between C and O. 5 (a) Graf tidak berpemberat/ Unweighted graph (b) E = {(a, b), (a, c), (c, b), (c, d), (d, a), (d, b)} n(E) = 6 11 PT SPM Mate Tg4_Jawapan_J1-12_Final.indd 4 2/6/25 8:20 PM PENERBIT ILMU BAKTI SDN. BHD.