Matematik T4

Matematik Tingkatan 4 Bab 3 29 9 Lengkapkan jadual yang berikut. SP: 3.1.4 TP4 Sederhana Complete the following table. Pernyataan Statement Implikasi 1 Implication 1 Implikasi 2 Implication 2 (a) Sebuah segi empat selari ialah sebuah rombus jika dan hanya jika pepenjurunya adalah berserenjang. A parallelogram is a rhombus if and only if its diagonals are perpendicular. Jika sebuah segi empat selari ialah sebuah rombus, maka pepenjurunya adalah berserenjang. If a parallelogram is a rhombus, then its diagonals are perpendicular. Jika pepenjuru sebuah segi empat selari adalah berserenjang, maka segi empat selari itu ialah sebuah rombus. If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. (b) B ⊂ A jika dan hanya jika unsur set A ialah unsur set B. B ⊂ A if and only if the elements of set A are elements of set B. Jika B ⊂ A, maka unsur set A ialah unsur set B. If B ⊂ A, then the elements of set A are elements of set B. Jika unsur set A ialah unsur set B, maka B ⊂ A. If the elements of set A are elements of set B, then B ⊂ A. (c) θ ialah sudut tirus jika dan hanya jika 0° N θ N 90°. θ is an acute angle if and only if 0° N θ N 90°. Jika θ ialah sudut tirus, maka 0° N θ N 90°. If θ is an acute angle, then 0° N θ N 90°. Jika 0° N θ N 90°, maka θ ialah sudut tirus. If 0° N θ N 90°, then θ is an acute angle. (d) Sisi sebuah segi tiga adalah sama panjang jika dan hanya jika sudut pedalamannya adalah sama. The sides of a triangle are equal if and only if its interior angles are equal. Jika sisi sebuah segi tiga adalah sama panjang, maka sudut pedalamannya adalah sama. If the sides of a triangle are equal, then its interior angles are equal. Jika sudut pedalaman sebuah segi tiga adalah sama, maka sisinya adalah sama panjang. If the interior angles of a triangle are equal, then its sides are equal. x2 = 25 jika dan hanya jika x = ±5. x2 = 25 if and only if x = ±5. Jika x2 = 25, maka x = ±5. If x2 = 25, then x = ±5. Jika x = ±5, maka x2 = 25. If x = ±5, then x2 = 25. Contoh 10 Tulis akas, songsangan dan kontrapositif bagi setiap implikasi yang berikut. SP: 3.1.5 TP4 Sederhana Write the converse, inverse and contrapositive for each of the following implications. (a) Jika 7 + 3 = 10, maka 10 – 7 = 3. If 7 + 3 = 10, then 10 – 7 = 3. Akas/ Converse : Songsangan/ Inverse : Kontrapositif/ Contrapositive : Jika q ialah 30, maka 3q < 100. If q is 30, then 3q < 100. Akas/ Converse : Songsangan/ Inverse : Kontrapositif/ Contrapositive : Jika 3q < 100, maka q ialah 30./ If 3q < 100, then q is 30. Jika q bukan 30, maka 3q M 100./ If q is not 30, then 3q M 100. Jika 3q M 100, maka q bukan 30./ If 3q M 100, then q is not 30. Jika 10 – 7 = 3, maka 7 + 3 = 10./ If 10 – 7 = 3, then 7 + 3 = 10. Jika 7 + 3 ≠ 10, maka 10 – 7 ≠ 3./ If 7 + 3 ≠ 10, then 10 – 7 ≠ 3. Jika 10 – 7 ≠ 3, maka 7 + 3 ≠ 10./ If 10 – 7 ≠ 3, then 7 + 3 ≠ 10. Contoh CONTOH

Matematik Tingkatan 4 Bab 3 30 (b) Jika c bernilai positif, maka pintasan-y mempunyai nilai positif. If c is a positive value, then the y-intercept has a positive value. Akas Converse : Songsangan Inverse : Kontrapositif Contrapositive : (c) Jika sebuah poligon mempunyai tiga sisi, maka poligon itu ialah sebuah segi tiga. If a polygon has three sides, then the polygon is a triangle. Akas Converse : Songsangan Inverse : Kontrapositif Contrapositive : Jika pintasan-y mempunyai nilai positif, maka c bernilai positif. If the y-intercept has a positive value, then c is a positive value. Jika c bukan bernilai positif, maka pintasan-y tidak mempunyai nilai positif. If c is not a positive value, then the y-intercept does not have a positive value. Jika pintasan-y tidak mempunyai nilai positif, maka c bukan bernilai positif. If the y-intercept does not have a positive value, then c is not a positive value. Jika sebuah poligon ialah sebuah segi tiga, maka poligon itu mempunyai tiga sisi. If a polygon is a triangle, then the polygon has three sides. Jika sebuah poligon tidak mempunyai tiga sisi, maka poligon itu bukan sebuah segi tiga. If a polygon does not have three sides, then the polygon is not a triangle. Jika sebuah poligon bukan sebuah segi tiga, maka poligon itu tidak mempunyai tiga sisi. If a polygon is not a triangle, then the polygon does not have three sides. 11 Lengkapkan jadual berikut dan seterusnya tentukan nilai kebenaran bagi pernyataan itu dengan menandakan bagi benar atau ✗ bagi palsu. SP: 3.1.6 TP5 Sederhana Complete the following table and hence determine the truth value for the statement by marking for true or ✗ for false. Pernyataan Statement Antejadian Antecedent Akibat Consequent Nilai kebenaran Truth value Jika 102 = 2, maka 1002 = 20. If 102 = 2, then 1002 = 20. ✗ ✗ Songsangan Inverse (a) Jika 102 ≠ 2, maka 1002 ≠ 20. If 102 ∙ 2, then 1002 ∙ 20. (i) ✗ (ii) (iii) Kontrapositif Contrapositive (b) Jika 1002 ≠ 20, maka 102 ≠ 2. If 1002 ∙ 20, then 102 ∙ 2. (i) (ii) ✗ (iii) ✗ Akas Converse (c) Jika 1002 = 20, maka 102 = 2. If 1002 = 20, then 102 = 2. (i) ✗ (ii) (iii) Jika 3 + 3 + 3 = 9, maka 9 ÷ 32 = 0. If 3 + 3 + 3 = 9, then 9 ÷ 32 = 0. ✗ ✗ Akas Converse Jika 9 ÷ 32 = 0, maka 3 + 3 + 3 = 9. If 9 ÷ 32 = 0, then 3 + 3 + 3 = 9. ✗ Contoh CONTOH

Matematik Tingkatan 4 Bab 3 31 12 Nyatakan contoh penyangkal bagi setiap pernyataan palsu yang berikut. SP: 3.1.6 TP4 Mudah State the counter-example for each of the following false statements. (a) Jika binatang itu bersayap, maka binatang itu boleh terbang. If the animal has wings, then the animal can fly. Burung unta tidak boleh terbang. Ostrich cannot fly. (b) Semua reptilia tidak mempunyai kaki. All reptiles do not have legs. Buaya mempunyai empat kaki. Crocodiles have four legs. (c) Semua nombor genap boleh dibahagi tepat dengan 4. All even numbers are divisible by 4. 6 tidak boleh dibahagi tepat dengan 4. 6 is not divisible by 4. 4 dan 7 ialah faktor bagi 21. 4 and 7 are factors of 21. 4 bukan faktor bagi 21. 4 is not a factor of 21. Contoh 3.2 Hujah Arguments Buku Teks m/s 71 – 88 1 Tentukan sama ada pernyataan-pernyataan berikut adalah khusus atau umum. SP: 3.2.1 TP1 Mudah Determine whether the following statements are specific or general. (a) Semua pokok mempunyai daun. All trees have leaves. Pernyataan umum/ General statement (b) Nombor genap boleh dibahagi tepat dengan 2. Even numbers are divisible by 2. Pernyataan umum/ General statement (c) 82 = 64 Pernyataan khusus/ Specific statement (d) Semua nombor gandaan 4 ialah nombor genap. All multiples of 4 are even numbers. Pernyataan khusus/ Specific statement (e) Nombor dalam asas dua melibatkan digit 0 dan 1. Numbers in base two involve digits 0 and 1. Pernyataan umum/ General statement Segi tiga tidak mempunyai pepenjuru. Triangles do not have diagonals. Pernyataan khusus/ Specific statement Contoh 2 Tentukan sama ada setiap hujah yang berikut ialah hujah deduktif atau hujah induktif. SP: 3.2.1 TP2 Sederhana Determine whether each of the following arguments is deductive argument or inductive argument. (a) Semua nombor genap boleh dibahagi tepat dengan 2. 30 ialah nombor genap. Maka, 30 boleh dibahagi tepat dengan 2. All even numbers are divisible by 2. 30 is divisible by 2. Therefore, 30 is an even number. Hujah deduktif Deductive argument (b) Jus epal perlu disimpan dalam peti sejuk. Jus oren perlu disimpan dalam peti sejuk. Maka, jus buah-buahan perlu disimpan dalam peti sejuk. Apple juice needs to be kept in the refrigerator. Orange juice needs to be kept in the refrigerator. Therefore, fruit juices need to be kept in the refrigerator. Hujah induktif Inductive argument (c) 20 boleh dibahagi tepat dengan 5. 50 boleh dibahagi tepat dengan 5. Maka, nombor gandaan 10 boleh dibahagi tepat dengan 5. 20 is divisible by 5. 50 is divisible by 5. Therefore, the multiples of 10 are divisible by 5. Hujah induktif Inductive argument Semua segi tiga mempunyai tiga sisi. DEF mempunyai tiga sisi. Maka, DEF ialah sebuah segi tiga. All triangles have three sides. DEF has three sides. Therefore, DEF is a triangle. Hujah deduktif Deductive argument Contoh Info Digital 3.2 CONTOH

Matematik Tingkatan 4 Bab 3 32 3 Berdasarkan premis dan kesimpulan yang diberi, tentukan sama ada setiap hujah yang berikut adalah sah dan munasabah atau tidak. Berikan justifikasi sekiranya tidak. SP: 3.2.2 TP3 TP4 Sukar Based on the given premises and conclusion, determine whether each of the following arguments is valid and sound or not. Justify if not. (a) Premis 1: Premise 1: Semua reptilia mempunyai sisik. All reptiles have scales. Premis 2: Premise 2: Ikan bawal mempunyai sisik. Pomfrets have scales. Kesimpulan: Conclusion: Ikan bawal ialah reptilia. Pomfrets are reptiles. Tidak sah kerana tidak mematuhi bentuk hujah deduktif yang sah. Tidak munasabah kerana kesimpulan palsu. Not valid because it does not comply with a valid form of deductive argument. Not sound because the conclusion is false. (b) Premis 1: Premise 1: Jika n tidak boleh dibahagi dengan sebarang nombor kecuali 1 dan n, maka n ialah nombor perdana. If n cannot be divided by any number except 1 and n, then n is a prime number. Premis 2: Premise 2: 17 tidak boleh dibahagi dengan sebarang nombor kecuali 1 dan 17. 17 cannot be divided by any number except 1 and 17. Kesimpulan: Conclusion: 17 ialah nombor perdana. 17 is a prime number. Sah dan munasabah. Valid and sound. (c) Premis 1: Premise 1: Jika binatang itu ada sepasang kaki, maka binatang itu ialah burung. If the animal has a pair of legs, then the animal is a bird. Premis 2: Premise 2: Ayam ialah burung. Chicken is a bird. Kesimpulan: Conclusion: Ayam ada sepasang kaki. Chicken has a pair of legs. Tidak sah kerana tidak mematuhi bentuk hujah deduktif yang sah. Maka, hujah ini tidak munasabah. Not valid because it does not comply with a valid form of deductive argument. Thus, the argument is not sound. (d) Premis 1: Premise 1: Jika y > 9, maka y > 3. If y > 9, then y > 3. Premis 2: Premise 2: 6 > 3. Kesimpulan: Conclusion: 6 > 9. Tidak sah dan munasabah kerana tidak mematuhi bentuk hujah deduktif yang sah dan kesimpulannya palsu. Not valid and sound because it does not comply with a valid form of deductive argument and its conclusion is false. Sah tetapi tidak munasabah kerana Premis 1 dan kesimpulan tidak benar. Valid but not sound because Premise 1 and conclusion is not true. Premis 1: Premise 1: Jika x ialah gandaan 4, maka x boleh dibahagi tepat dengan 8. If x is a multiple of 4, then x is divisible by 8. Premis 2: Premise 2: 20 ialah gandaan 4. 20 is a multiple of 4. Kesimpulan: Conclusion: 20 boleh dibahagi tepat dengan 8. 20 is divisible by 8. Contoh CONTOH

Matematik Tingkatan 4 Bab 3 33 4 Tuliskan satu kesimpulan untuk membentuk satu hujah deduktif yang sah dan munasabah. SP: 3.2.3 TP3 Sederhana Write a conclusion to form a valid and sound deductive argument. (a) Premis 1: Premise 1: Jika satu nombor ialah faktor bagi 6, maka nombor itu ialah faktor bagi 18. If a number is a factor of 6, then the number is a factor of 18. Premis 2: Premise 2: 2 ialah satu faktor bagi 6. 2 is a factor of 6. Kesimpulan: Conclusion: 2 ialah satu faktor bagi 18. 2 is a factor of 18. (b) Premis 1: Premise 1: Jika y = 5, maka (y + 1)(y – 5) = 0. If y = 5, then (y + 1)(y – 5) = 0. Premis 2: Premise 2: (y + 1)(y – 5) ≠ 0. Kesimpulan: Conclusion: y ≠ 0. (c) Premis 1: Premise 1: x2 = 64. Premis 2: Premise 2: 8 = x. Kesimpulan: Conclusion: 82 = 64. (d) Premis 1: Premise 1: Jika satu benda terdiri daripada sel-sel, maka benda itu ialah satu benda hidup. If a thing is made up of cells, then the thing is a living thing. Premis 2: Premise 2: Televisyen bukan benda hidup. Television is not a living thing. Kesimpulan: Conclusion: Televisyen tidak terdiri daripada sel-sel. Television is not made up of cells. Premis 1: Premise 1: Tumbuhan yang mempunyai klorofil dapat menghasilkan makanan sendiri. Plants which have chlorophyll can produce their own food. Premis 2: Premise 2: Tumbuhan hijau mempunyai klorofil. Green plants have chlorophyll. Kesimpulan: Conclusion: Tumbuhan hijau dapat menghasilkan makanan sendiri. Green plants can produce their own food. Contoh CONTOH

Matematik Tingkatan 4 Bab 3 34 5 Tuliskan satu premis untuk membentuk satu hujah deduktif yang sah. SP: 3.2.3 TP3 Sederhana Write a premise to form a valid deductive argument. (a) Premis 1: Premise 1: Jika hari itu hujan, maka Susie akan membawa payung. If the day rains, then Susie will bring an umbrella. Premis 2: Premise 2: Hari ini hujan. Today rains. Kesimpulan: Conclusion: Hari ini Susie membawa payung. Today Susie brings an umbrella. (b) Premis 1: Premise 1: Jika sebuah poligon mempunyai tiga sisi, maka poligon itu ialah sebuah segi tiga. If a polygon has three sides, then the polygon is a triangle. Premis 2: Premise 2: Pentagon bukan sebuah segi tiga. Pentagon is not a triangle. Kesimpulan: Conclusion: Pentagon tidak mempunyai tiga sisi. Pentagon does not have three sides. (c) Premis 1: Premise 1: Semua kucing mempunyai misai. All cats have whiskers. Premis 2: Premise 2: Si Comel ialah seekor kucing. Si Comel is a cat. Kesimpulan: Conclusion: Si Comel mempunyai misai. Si Comel has whiskers. (d) Premis 1: Premise 1: Jika Tina pergi melancong, maka dia telah mengambil cuti. If Tina goes on a holiday, then she had applied leave. Premis 2: Premise 2: Tina pergi melancong. Tina goes on a holiday. Kesimpulan: Conclusion: Tina telah mengambil cuti. Tina had applied leave. Premis 1: Premise 1: Semua piramid mempunyai lima permukaan. All pyramids have five surfaces. Premis 2: Premise 2: Pepejal C ialah sebuah piramid. Solid C is a pyramid. Kesimpulan: Conclusion: Pepejal C mempunyai lima permukaan. Solid C has five surfaces. Contoh CONTOH

Matematik Tingkatan 4 Bab 3 35 6 Tentukan kekuatan hujah induktif berikut dan sama ada hujah yang kuat adalah meyakinkan. Wajarkan jawapan anda. SP: 3.2.4 TP4 Sederhana Determine the strength of the following inductive arguments and whether strong argument is cogent. Justify your answers. (a) Premis 1: Premise 1: Segi tiga mempunyai 2 pepenjuru. Triangles have 2 diagonals. Premis 2: Premise 2: Pentagon mempunyai 5 pepenjuru. Pentagons have 5 diagonals. Premis 3: Premise 3: Heksagon mempunyai 9 pepenjuru. Hexagons have 9 diagonals. Premis 4: Premise 4: Heptagon mempunyai 14 pepenjuru. Heptagons have 14 diagonals. Kesimpulan: Conclusion: Semua poligon mempunyai sekurangkurangnya 2 pepenjuru. All polygons have at least 2 diagonals. Hujah ini lemah dan tidak meyakinkan kerana Premis 1 dan kesimpulan adalah palsu. This argument is weak and not cogent because Premise 1 and conclusion are false. (b) Premis 1: Premise 1: 10 ÷ 2 = 5 Premis 2: Premise 2: 20 ÷ 4 = 5 Kesimpulan: Conclusion: Nombor gandaan 10 boleh dibahagi tepat dengan 5. Multiples of 10 are divisible by 5. Hujah ini kuat dan meyakinkan kerana semua premis dan kesimpulan adalah benar. This argument is strong and cogent because all premises and conclusion are true. (c) Premis 1: Premise 1: John suka bermain bola sepak. John likes to play football. Premis 2: Premise 2: Faizal suka bermain bola sepak. Faizal likes to play football. Premis 3: Premise 3: Siva suka bermain bola sepak. Siva likes to play football. Premis 4: Premise 4: Seng Hwa suka bermain bola sepak. Seng Hwa likes to play football. Kesimpulan: Conclusion: Semua lelaki suka bermain bola sepak. All boys like to play football. Hujah ini lemah dan tidak meyakinkan kerana premis adalah benar tetapi kesimpulan mungkin palsu. This argument is weak and not cogent because the premises are true but the conclusion maybe false. (d) Premis 1: Premise 1: Katak boleh hidup di darat dan di air. Frogs can live on land and in water. Premis 2: Premise 2: Neut boleh hidup di darat dan di air. Newts can live on land and in water. Premis 3: Premise 3: Salamander boleh hidup di darat dan di air. Salamanders can live on land and in water. Kesimpulan: Conclusion: Amfibia boleh hidup di darat dan di air. Amphibians can live on land and in water. Hujah ini kuat dan meyakinkan kerana semua premis dan kesimpulan adalah benar. This argument is strong and cogent because all premises and conclusion are true. Hujah ini lemah dan tidak meyakinkan kerana premis adalah benar tetapi kesimpulannya palsu. This argument is weak and not cogent because the premises are true but the conclusion is false. Premis 1: Premise 1: Aiskrim berasa manis. Ice-cream tastes sweet. Premis 2: Premise 2: Kek berasa manis. Cake tastes sweet. Premis 3: Premise 3: Jeli berasa manis. Jelly tastes sweet. Kesimpulan: Conclusion: Makanan berasa manis. Food tastes sweet. Contoh CONTOH

Matematik Tingkatan 4 Bab 3 36 7 Bentuk satu kesimpulan induktif yang kuat berdasarkan pola nombor atau premis yang diberi. SP: 3.2.5 TP5 Form a strong inductive conclusion based on the given number patterns or premises. Sederhana (a) Pola/ Pattern: 11, 17, 35, 89, … 11 = 3 + 8 17 = 32 + 8 35 = 33 + 8 89 = 34 + 8 � 3n + 8, n = 1, 2, 3, 4, … (b) Pola/ Pattern: 99, 96, 91, 84, … 99 = 100 – 1(1) 96 = 100 – 2(2) 91 = 100 – 3(3) 84 = 100 – 4(4) � 100 – n2 , n = 1, 2, 3, 4, … (c) Pola/ Pattern: 0, 2, 6, 12, … 0 = 1 – 1 2 = 2(2) – 2 6 = 3(3) – 3 12 = 4(4) – 4 � n2 – n, n = 1, 2, 3, 4, … Pola/ Pattern: 5, 9, 13, 17, … 5 = 4(1) + 1 9 = 4(2) + 1 13 = 4(3) + 1 17 = 4(4) + 1 � 4n + 1, n = 1, 2, 3, 4, … Contoh (d) Premis 1: Premise 1: Segi tiga dibina daripada tiga garis lurus. Triangles are made of three straight lines. Premis 2: Premise 2: Sisi empat dibina daripada empat garis lurus. Quadrilaterals are made of four straight lines. Premis 3: Premise 3: Pentagon dibina daripada lima garis lurus. Pentagons are made of five straight lines. Kesimpulan: Conclusion: Poligon dibina daripada garis lurus. Polygons are made of straight lines. (e) Premis 1: Premise 1: Unit laju bagi zarah ialah meter per saat. The unit of speed for particles is metres per second. Premis 2: Premise 2: Unit laju bagi kereta ialah kilometer per jam. The unit of speed for cars is kilometres per hour. Premis 3: Premise 3: Unit laju bagi kapal terbang ialah batu per jam. The unit of speed for airplanes is miles per hour. Kesimpulan: Conclusion: Unit laju am ialah jarak per unit masa. The general unit of speed is distance per unit time. Premis 1: Premise 1: Burung merpati mempunyai bulu pelepah. Pigeons have feathers. Premis 2: Premise 2: Helang mempunyai bulu pelepah. Eagles have feathers. Kesimpulan: Conclusion: Burung ada bulu pelepah. Birds have feathers. Contoh CONTOH

Matematik Tingkatan 4 Bab 3 37 8 Selesaikan setiap masalah yang berikut. SP: 3.2.6 TP5 TP6 KBAT Sukar Solve each of the following problems. (a) Jadual berikut menunjukkan bilangan dan jisim bola sepak. The following table shows the number and mass of footballs. Bilangan Number 3 9 27 81 Jisim (g) Mass (g) 53 59 77 131 Tulis satu pernyataan umum untuk menunjukkan hubungan antara bilangan dan jisim bola sepak. Write a general statement to show the relation between the number and the mass of footballs. 3n + 50, n = 1, 2, 3, 4 (b) Diberi persamaan kuadratik ialah f(x) = ax2 + bx + c. Bina satu kesimpulan secara deduktif bagi fungsi kuadratik dengan keadaan a = 2, b = –2 dan c = 3. Given the quadratic equation is f(x) = ax2 + bx + c. Form a deductive conclusion for the quadratic function where a = 2, b = –2 and c = 3. f(x) = 2x2 – 2x + 3 (c) Mariam lebih tinggi daripada Ah Chong. Siva lebih tingggi daripada Ah Chong. Siva lebih tinggi daripada Mariam. Mariam is taller than Ah Chong. Siva is taller than Ah Chong. Siva is taller than Mariam. (i) Siapakah yang paling rendah? Who is the shortest? Ah Chong (ii) Bentuk satu kesimpulan berdasarkan Siva. Make a conclusion based on Siva. Maka, Siva lebih tinggi daripada Mariam dan Ah Chong. Then, Siva is taller than Mariam and Ah Chong. (d) Jadual berikut menunjukkan keputusan sesuatu eksperimen kimia. The following table shows the result of a chemical experiment. Bilangan titisan alkali Number of drops of alkali 4 8 12 16 Nilai pH pH value 7 x 9 10 (i) Bina rumus berdasarkan pola titisan alkali. Form a formula based on the pattern of drops of alkali. 4n, n = 1, 2, 3, 4, … (ii) Tentukan nilai x. Determine the value of x. 8 (iii) Tulis satu kesimpulan induktif bagi eksperimen kimia ini. Write one inductive conclusion for this chemical experiment. Nilai pH meningkat 1 pada setiap penambahan 4 titis alkali. The value of pH increases 1 at every addition of 4 drops of alkali. CONTOH

Matematik Tingkatan 4 Bab 3 38 (e) Jika Nadia mempunyai RM50, maka dia akan membeli sebuah novel. Jika Nadia mempunyai RM80, dia akan membeli sebuah beg tangan. If Nadia has RM50, then she will buy a novel. If Nadia has RM80, she will buy a handbag. (i) Nadia masih ada baki selepas membeli sebuah novel. Mengapakah Nadia tidak membeli beg tangan? Berikan justifikasi. Nadia still have some change after buying a novel. Why did Nadia not buy a handbag? Give a justification. Jika Nadia tidak mempunyai RM80, maka dia tidak akan membeli sebuah beg tangan. If Nadia does not have RM80, then she will not buy a handbag. Kesimpulan deduktif/ Deductive conclusion: Nadia mempunyai lebih daripada RM50 tetapi kurang daripada RM80. Nadia has more than RM50 but less than RM80. (ii) Hitungkan wang yang dimiliki oleh Nadia pada asalnya jika bakinya sekarang ialah RM20. Calculate the money Nadia originally had if her current change is RM20. RM50 + RM20 = RM70 Kertas 2 Jawab semua soalan./ Answer all questions. Bahagian A/ Section A 1 (a) Nyatakan sama ada ayat berikut ialah pernyataan atau tidak. State whether the following sentence is a statement or not. 5 adalah faktor bagi 24. 5 is a factor of 24. [1 markah/ mark] Pernyataan/ Statement (b) Tulis akas bagi pernyataan yang berikut dan seterusnya tentukan nilai kebenarannya. Write the converse for the following statement and hence determine the truth value. Jika (A ∩ B) � A, maka (A ∩ B) � B. If (A ∩ B) � A, then (A ∩ B) � B. [1 markah/ mark] Jika (A ∩ B) � B, maka (A ∩ B) � A. If (A ∩ B) � B, then (A ∩ B) � A. Benar/ True (c) Tulis Premis 1 untuk melengkapkan hujah berikut: Write down Premise 1 to complete the following argument: [1 markah/ mark] Premis 1: Premise 1: Kubus mempunyai 6 permukaan. Cube has 6 surfaces. Premis 2: Premise 2: Pepejal A ialah sebuah kubus. Solid A is a cube. Kesimpulan: Conclusion: Pepejal A mempunyai 6 permukaan. Solid A has 6 surfaces. Praktis Kendiri CONTOH

Matematik Tingkatan 4 Bab 3 39 2 (a) Nyatakan sama ada pernyataan berikut adalah benar atau palsu. State whether the following statement is true or false. 8 – 2 = 3 – 9 [1 markah/ mark] Palsu/ False (b) Tulis dua implikasi berdasarkan pernyataan majmuk berikut. Write down two implications based on the following compound statement. x < y jika dan hanya jika x + 4 < y + 4. x < y if and only if x + 4 < y + 4. [2 markah/ marks] Implikasi 1: Implication 1: Jika x < y, maka x + 4 < y + 4. If x < y, then x + 4 < y + 4. Implikasi 2: Implication 2: Jika x + 4 < y + 4, maka x < y. If x + 4 < y + 4, then x < y. (c) Buat satu kesimpulan induktif bagi urutan nombor 50, 45, 40, 35, … yang mengikut pola berikut. Make an inductive conclusion for the sequence of numbers 50, 45, 40, 35, … which follows the following pattern. 50 = 50 – 5(0) 45 = 50 – 5(1) 40 = 50 – 5(2) 35 = 50 – 5(3) � 50 – 5n, n = 0, 1, 2, 3, … [1 markah/ mark] 3 (a) Nyatakan nilai kebenaran bagi pernyataan berikut. State the truth value of the following statement. Jumlah bagi dua nombor positif adalah bernilai positif dan ialah satu nombor genap. The sum of two positive numbers is a positive value and an even number. [1 markah/ mark] Palsu/ False (b) Tulis satu kesimpulan bagi hujah deduktif berikut. Write a conclusion for the following deductive argument. Premis 1: Premise 1: Semua pelajar harus mematuhi peraturan sekolah. All students must follow the school rules. Premis 2: Premise 2: Abu ialah seorang pelajar. Abu is a student. [1 markah/ mark] Kesimpulan: Conclusion: Abu harus mematuhi peraturan sekolah. Abu must follow the school rules. (c) Tentukan kekuatan hujah induktif berikut dan sama ada hujah ini meyakinkan atau tidak. Determine the strength of the following inductive argument and whether this argument is cogent or not. Premis 1: Premise 1: Ayam mempunyai dua kaki. Chicken has two legs. Premis 2: Premise 2: Rusa mempunyai empat kaki. Deer has four legs. Premis 3: Premise 3: Manusia mempunyai dua kaki. Human has two legs. Kesimpulan: Conclusion: Semua haiwan mempunyai kaki. All animals have legs. [2 markah/ marks] Hujah ini lemah dan tidak meyakinkan kerana semua premis adalah benar tetapi kesimpulan adalah palsu. This argument is weak and not cogent because all premises are true but the conclusion is false. CONTOH

Matematik Tingkatan 4 Bab 3 40 Bahagian B/ Section B 4 (a) Tentukan nilai kebenaran bagi pernyataan-pernyataan berikut. Determine the truth value for the following statements. (i) Semua segi tiga mempunyai satu sudut tegak. All triangles have one right-angle. (ii) Jika x > 0, maka 2x < 0. If x > 0, then 2x < 0. (iii) Fungsi kuadratik mempunyai pemboleh ubah dengan kuasa dua. Quadratic functions have variables to the power of two. [3 markah/ marks] (b) Tulis dua implikasi bagi pernyataan majmuk berikut. Write two implications for the following compound statement. x M 0 jika dan hanya jika x ialah satu nombor asli. x M 0 if and only if x is a natural number. [2 markah/ marks] (c) Tulis akas dan songsangan bagi implikasi berikut. Write the converse and inverse of the following implication. Jika x2 + 2x = 0, maka x = 0 atau –2. If x2 + 2x = 0, then x = 0 or –2. [2 markah/ marks] (d) Berdasarkan maklumat di bawah, buat satu kesimpulan secara deduksi bagi isi padu silinder dengan jejari 7 cm dan tinggi 14 cm. Based on the information below, make a conclusion by induction for the volume of cylinder with a radius of 7 cm and a height of 14 cm. Isi padu bagi silinder dengan jejari r cm dan tinggi h cm ialah πr2 h. The volume of a cylinder with radius r cm and height h cm is πr 2 h. [2 markah/ marks] (a) (i) Palsu/ False (ii) Palsu/ False (iii) Benar/ True (b) Implikasi 1/ Implication 1: Jika x M 0, maka x ialah satu nombor asli. If x M 0, then x is a natural number. Implikasi 2/ Implication 2: Jika x ialah satu nombor asli, maka x M 0. If x is a natural number, then x M 0. (c) Akas/ Converse: Jika x = 0 atau –2, maka x2 + 2x = 0. If x = 0 or –2, then x2 + 2x = 0. Songsangan/ Inverse: Jika x2 + 2x ≠ 0, maka x ≠ 0 atau –2. If x2 + 2x ≠ 0, then x ≠ 0 or –2. (d) Isi padu silinder/ Volume of cylinder: π(72 )(14) = 686π cm3 CONTOH

41 Praktis Intensif 4.1 Persilangan Set Intersection of Sets Buku Teks m/s 96 – 105 1 Diberi P = {5, 7, 8, 9, 11, 12, 13, 15}, Q = {7, 9, 11, 12, 14} dan R = {5, 12, 17}. SP: 4.1.1 TP1 TP2 Mudah Given that P = {5, 7, 8, 9, 11, 12, 13, 15}, Q = {7, 9, 11, 12, 14} and R = {5, 12, 17}. (i) Senaraikan semua unsur bagi persilangan set yang berikut. List all the elements for the following intersection of sets. (ii) Nyatakan bilangan unsur bagi persilangan set yang berikut. State the number of elements for the following intersection of sets. (a) (i) P ∩ R = {5, 12} (ii) n(P ∩ R) = 2 P = { 5 , 7, 8, 9, 11, 12 , 13, 15} R = { 5 , 12 , 17} (b) (i) Q ∩ R = {12} (ii) n(Q ∩ R) = 1 Q = {7, 9, 11, 12 , 14} R = {5, 12 , 17} (c) (i) P ∩ Q ∩ R = {12} (ii) n(P ∩ Q ∩ R) = 1 P = {5, 7, 8, 9, 11, 12 , 13, 15} Q = {7, 9, 11, 12 , 14} R = {5, 12 , 17} (i) P ∩ Q = {7, 9, 11, 12} (ii) n(P ∩ Q) = 4 P = {5, 7 , 8, 9 , 11 , 12 , 13, 15} Q = { 7 , 9 , 11 , 12 , 14} Unsur sepunya/ Common elements Tip Bestari P ∩ Q ialah set yang mengandungi unsurunsur sepunya bagi kedua-dua set. P ∩ Q is a set containing the common elements for both sets. Contoh 2 Gambar rajah Venn berikut menunjukkan set P, Q dan R. Senaraikan semua unsur bagi persilangan set yang berikut. The following Venn diagram shows set P, set Q and set R. List all the elements for the following intersections of sets. SP: 4.1.1 TP2 Mudah •3 •5 •4 P Q R •8 •1 •2 •7 •6 •9 •10 (a) P ∩ Q (b) P ∩ R (c) Q ∩ R (d) P ∩ Q ∩ R (a) P ∩ Q (b) P ∩ R (c) Q ∩ R (d) P ∩ Q ∩ R •3 •5 •4 P Q R •8 •1 •2 •7 •6 •9 •10 •3 •5 •4 P Q R •8 •1 •2 •7 •6 •9 •10 •3 •5 •4 P Q R •8 •1 •2 •7 •6 •9 •10 •3 •5 •4 P Q R •8 •1 •2 •7 •6 •9 •10 {3, 4, 5, 6} {3, 4, 8} {3, 4} {3, 4} Contoh Operasi Set 4 Operations on Sets Bab Info Digital 4.1 CONTOH

Matematik Tingkatan 4 Bab 4 42 P Q R •18 •14 •16 •15 •19 •20 •17 (a) P ∩ Q (b) P ∩ R (c) Q ∩ R (d) P ∩ Q ∩ R (a) {14, 16, 20} (b) {15, 16, 17, 20} (c) {16, 20} (d) {16, 20} 3 Selesaikan setiap yang berikut. SP: 4.1.1 TP2 Mudah Solve each of the following. Diberi P = {1, 2, 3, 4, 5, 6}, Q = {nombor perdana kurang daripada 10} dan R = {8, 9, 10}. Given P = {1, 2, 3, 4, 5, 6}, Q = {prime number less than 10} and R = {8, 9, 10}. (a) Senaraikan semua unsur bagi persilangan set berikut. List all the elements for the following intersection of sets. (i) P ∩ Q (ii) P ∩ R (iii) Q ∩ R (b) Lukis gambar rajah Venn yang mewakili set P, set Q dan set R. Seterusnya, lorekkan kawasan yang mewakili persilangan set berikut. Draw a Venn diagram to represent set P, set Q and set R. Hence, shade the region represent the following intersection of sets. (i) P ∩ Q (ii) Q ∩ R (a) P = {1, 2, 3, 4, 5, 6} Q = {2, 3, 5, 7} R = {8, 9, 10} (i) P ∩ Q = {2, 3, 5} (ii) P ∩ R = { }/ Ø (iii) Q ∩ R = { }/ Ø (b) (i) •5 •3 •4 P Q R •8 •1 •2 •7 •6 •9 •10 Mewakili kawasan P ∩ Q Represents the region of P ∩ Q (ii) •5 •3 •4 P Q R •8 •1 •2 •7 •6 •9 •10 Set Q dan set R tidak mempunyai unsur-unsur sepunya Set Q and set R do not have common elements { }/ Ø mewakili set kosong, iaitu set yang tidak mengandungi sebarang unsur { }/ Ø represents an empty set, i.e. the set without any elements Contoh CONTOH

Matematik Tingkatan 4 Bab 4 43 Diberi bahawa set semesta, ξ = {x : x ialah integer, 10 N x N 30}, set A = {x : x ialah gandaan 5}, set B = {faktor bagi 30} dan set C = {x : x ialah nombor genap}. Given that the universal set, ξ = {x : x is an integer, 10 N x N 30}, set A = {x : x is a multiple of 5}, set B = {factor of 30} and set C = {x : x is an even number}. (a) Senaraikan unsur-unsur bagi List the elements of (i) A ∩ B (ii) A ∩ C (iii) B ∩ C (iv) A ∩ B ∩ C (b) Lukis sebuah gambar rajah Venn untuk mewakili semua data yang diberikan. Seterusnya, lorekkan kawasan yang mewakili A ∩ B ∩ C. Draw a Venn diagram to represent all the given data. Hence, shade the region that represents A ∩ B ∩ C. (a) ξ = {10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30} A = {10, 15, 20, 25, 30} B = {10, 15, 30} C = {10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30} (i) A ∩ B = {10, 15, 30} (ii) A ∩ C = {10, 20, 30} (iii) B ∩ C = {10, 30} (iv) A ∩ B ∩ C = {10, 30} (b) •25 •15 •20 A B C •12 •14 •16 •18 •22 •24 •26 •28 •10 •30 4 Senaraikan semua unsur dan nyatakan bilangan unsur bagi setiap yang berikut. SP: 4.1.2 TP3 Sederhana List all the elements and state the number of elements for each of the following. Diberi set semesta, ξ = {x : x ialah integer, 1 N x N 15}, set P = {3, 4, 5, 7, 8, 10, 11, 13}, Q = {1, 3, 5, 6, 11, 12, 15} dan R = {2, 3, 5, 7, 11, 13}. Given the universal set, ξ = {x : x is an integer, 1 N x N 15}, set P = {3, 4, 5, 7, 8, 10, 11, 13}, Q = {1, 3, 5, 6, 11, 12, 15} and R = {2, 3, 5, 7, 11, 13}. (a) (P ∩ Q) (b) (P ∩ R) (c) (P ∩ Q ∩ R) (a) (P ∩ Q) = {3, 5, 11} (P ∩ Q) = {1, 2, 4, 6, 7, 8, 9, 10, 12, 13, 14, 15} n(P ∩ Q) = 12 (b) (P ∩ R) = {3, 5, 7, 11, 13} (P ∩ R) = {1, 2, 4, 6, 8, 9, 10, 12, 14, 15} n(P ∩ R) = 10 (c) (P ∩ Q ∩ R) = {3, 5, 11} (P ∩ Q ∩ R) = {1, 2, 4, 6, 7, 8, 9, 10, 12, 13, 14, 15} n(P ∩ Q ∩ R) = 12 Contoh Tip Bestari (P ∩ Q)' bermaksud semua unsur yang bukan dalam persilangan set P dan set Q. (P ∩ Q)' means that all the elements which are not in the intersection of set P and set Q. Unsur sepunya bagi set P, set Q dan set R Common elements of set P, set Q and set R Unsur sepunya bagi set P dan set Q CONTOH Common elements of set P and set Q

Matematik Tingkatan 4 Bab 4 44 Diberi set semesta, ξ = {x : x ialah integer, 10 N x N 20}, set P = {12, 15, 18}, Q = {10, 12, 14, 16, 18, 20} dan R = {11, 12, 14, 17, 20}. Given the universal set, ξ = {x : x is an integer, 10 N x N 20}, set P = {12, 15, 18}, Q = {10, 12, 14, 16, 18, 20} and R = {11, 12, 14, 17, 20}. (a) (P ∩ Q) (b) (P ∩ R) (c) (Q ∩ R) (d) (P ∩ Q ∩ R) (a) (P ∩ Q) = {12, 18} (P ∩ Q) = {10, 11, 13, 14, 15, 16, 17, 19, 20} n(P ∩ Q) = 9 (b) (P ∩ R) = {12} (P ∩ R) = {10, 11, 13, 14, 15, 16, 17, 18, 19, 20} n(P ∩ R) = 10 (c) (Q ∩ R) = {12, 14, 20} (Q ∩ R) = {10, 11, 13, 15, 16, 17, 18, 19} n(Q ∩ R) = 8 (d) (P ∩ Q ∩ R) = {12} (P ∩ Q ∩ R) = {10, 11, 13, 14, 15, 16, 17, 18, 19, 20} n(P ∩ Q ∩ R) = 10 5 Selesaikan masalah berikut. SP: 4.1.3 TP4 TP5 KBAT Sukar Solve the following problems. (a) Satu soal selidik dijalankan terhadap 30 orang murid untuk mengetahui warna kegemaran mereka. 12 orang murid suka warna merah jambu, 8 orang murid suka warna ungu dan 5 orang murid suka kedua-dua warna merah jambu dan ungu. A survey is carried out on 30 students to find out their favourite colours. 12 students love pink, 8 students love purple and 5 students love both pink and purple. Hitung/ Calculate (i) bilangan murid yang menyukai warna merah jambu atau ungu. the number of students who love pink or purple. (ii) bilangan murid yang tidak menyukai kedua-dua warna tersebut. the number of students who do not love both colours. Katakan ξ = {jumlah murid}, M = {murid suka merah jambu}, U = {murid suka ungu} Let ξ = {total number of students}, M = {students who love pink}, U = {students who love purple} 15 7 5 3 M U ξ (i) Bilangan murid yang menyukai merah jambu atau ungu: Number of students who love pink or purple: 7 + 5 + 3 = 15 orang murid/ students (ii) Bilangan murid yang tidak menyukai kedua-dua warna tersebut: Number of students who do not love both colours: 30 – 15 = 15 orang murid/ students CONTOH

Matematik Tingkatan 4 Bab 4 45 (b) Satu kajian telah dijalankan terhadap 400 orang murid untuk mengetahui cara mereka datang ke sekolah. Daripada kajian tersebut, 140 orang menaiki bas, 100 orang menaiki basikal, 132 orang menaiki kereta, 16 orang menaiki kedua-dua bas dan basikal sahaja, 20 orang menaiki kedua-dua bas dan kereta sahaja, 12 orang menaiki kedua-dua basikal dan kereta sahaja dan 96 orang menaiki bas sahaja. A survey is conducted on 400 students to find out how they go to school. From the survey, 140 students used bus, 100 students used bicycle, 132 students used car, 16 students used both bus and bicycle only, 20 students used both bus and car only, 12 students used both bicycle and car only and 96 students used only bus. (i) Lukis gambar rajah Venn untuk mewakili maklumat di atas. Draw a Venn diagram to represent the above information. (ii) Hitung/ Calculate (a) bilangan murid yang menaiki ketiga-tiga kenderaan tersebut. the number of students who used three vehicles. (b) bilangan murid yang tidak menggunakan ketiga-tiga kenderaan tersebut. the number of students who do not used any of the three vehicles. (i) ξ 96 Bas Bus Kereta Car Basikal Bicycle 16 64 20 12 92 92 8 (ii) (a) 140 – 96 – 16 – 20 = 8 orang murid/ students (b) 400 – 96 – 16 – 64 – 20 – 8 – 12 – 92 = 92 orang murid/ students (c) Satu tinjauan tentang binatang kegemaran telah dijalankan terhadap 4 500 orang murid di sebuah sekolah. Berdasarkan tinjauan tersebut, 35% daripada jumlah murid suka kucing, 25% suka anjing manakala 33% suka arnab. Antara jumlah tersebut, 4% suka kucing dan anjing, 5% suka kucing dan arnab, 3% suka anjing dan arnab manakala 2% lagi suka ketiga-tiga jenis binatang. A survey on favourite animals was conducted on 4 500 students in a school. Based on the survey, 35% of the students like cats, 25% like dogs whereas 33% like rabbits. Among the total, 4% like cats and dogs, 5% like cats and rabbits, 3% like dogs and rabbits whereas the other 2% like all three types of animals. (i) Lukis gambar rajah Venn untuk mewakili keputusan tinjauan. Draw a Venn diagram to represent the results of the survey. (ii) Cari bilangan murid yang suka satu binatang sahaja. Find the number of students who like only one animal. (iii) Berapakah bilangan murid yang tidak suka ketiga-tiga jenis binatang itu? What is the number of students who do not like those three types of animals? (i) Katakan K sebagai murid yang suka kucing, D sebagai murid yang suka anjing dan A sebagai murid yang suka arnab. Let K be students who like cats, D be students who like dogs and A be student who like rabbits. ξ 28% 27% 2% 20% 2% 3% 1% A K D 17% (iii) Bilangan murid yang tidak suka ketiga-tiga jenis binatang itu: Number of students who do not like those three animals: 17 100 × 4 500 = 765 orang murid/ students (ii) Bilangan murid yang suka satu binatang sahaja: Number of students who like only one animal: 28 + 20 + 27 100 × 4 500 = 75 100 × 4 500 CONTOH = 3 375 orang murid/ students

Matematik Tingkatan 4 Bab 4 46 4.2 Kesatuan Set Union of Sets Buku Teks m/s 106 – 115 1 Diberi set P = {10, 11, 13, 14, 15, 17, 18, 19, 20}, set Q = {12, 15, 16, 19} dan set R = {13, 18}. SP: 4.2.1 TP1 TP2 Given set P = {10, 11, 13, 14, 15, 17, 18, 19, 20}, set Q = {12, 15, 16, 19} and set R = {13, 18}. Mudah (i) Senaraikan semua unsur bagi set berikut. List all the elements for the following set. (ii) Lukis gambar rajah Venn untuk mewakili set P, set Q dan set R dan seterusnya, lorekkan kawasan yang mewakili kesatuan set yang berikut. Draw a Venn diagram to represent the set P, set Q dan set R and hence, shade the region that represents the union of the following sets. (a) P ∪ R = {10, 11, 13, 14, 15, 17, 18, 19, 20} P Q R •13 •10 •11 •14 •15 •12 •17 •19 •16 •20 •18 (b) Q ∪ R = {12, 13, 15, 16, 18, 19} P Q R •13 •10 •11 •14 •15 •12 •17 •19 •16 •20 •18 (c) P∪ Q ∪ R = {10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20} P Q R •13 •10 •11 •14 •12 •17 •16 •20 •18 •19 •15 P ∪ Q = {10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20} P Q R •13 •10 •11 •14 •15 •12 •17 •19 •16 •20 •18 Contoh 2 Diberi set semesta, ξ = {x : x ialah integer, 1 N x N 10}, P = {2, 4, 5, 6, 9}, Q = {3, 5, 6, 7} dan R = {1, 3, 5, 7}. Given the universal set, ξ = {x : x is an integer, 1 N x N 10}, P = {2, 4, 5, 6, 9}, Q = {3, 5, 6, 7} and R = {1, 3, 5, 7}. Senaraikan semua unsur dan nyatakan bilangan unsur bagi set yang berikut. SP: 4.2.2 TP2 Sederhana List all the elements and state the number of elements for the following sets. (a) (P ∪ R) (P ∪ R) = {1, 2, 3, 4, 5, 6, 7, 9} (P ∪ R)� = {8, 10} n(P ∪ R)� = 2 (b) (Q ∪ R) (Q ∪ R) = {1, 3, 5, 6, 7} (Q ∪ R)� = {2, 4, 8, 9, 10} n(Q ∪ R)� = 5 (c) (P ∪ Q ∪ R) (P ∪ Q ∪ R) = {1, 2, 3, 4, 5, 6, 7, 9} (P ∪ Q ∪ R)� = {8, 10} n(P ∪ Q ∪ R)� = 2 (P ∪ Q) ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} (P ∪ Q) = {2, 3, 4, 5, 6, 7, 9} (P ∪ Q)� = {1, 8, 10} n(P ∪ Q)� = 3 Contoh Info Digital 4.2 CONTOH

Matematik Tingkatan 4 Bab 4 47 3 Senaraikan unsur-unsur bagi (M ∪ N), (M ∪ P), (N ∪ P) dan (M ∪ N ∪ P). Seterusnya, lukis gambar rajah Venn dan lorekkan kawasan untuk mewakili set (M ∪ N ∪ P). SP: 4.2.2 TP4 KBAT Sukar List the elements of (M ∪ N)’, (M ∪ P)’, (N ∪ P)’ and (M ∪ N ∪ P)’. Hence, draw a Venn diagram and shade the region to represent the set (M ∪ N ∪ P)’. Diberi set semesta ξ = {x : x ialah integer, 1 N x N 20}, M = {faktor bagi 20}, N = {gandaan 2} dan P = {nombor yang boleh dibahagi tepat dengan 4}. Given that the universal set, ξ = {x : x is an integer, 1 N x N 20}, M = {factor of 20}, N = {multiple of 2} and P = {number that is divisible by 4}. ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20} M = {1, 2, 4, 5, 10, 20} N = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20} P = {4, 8, 12, 16, 20} (M ∪ N) = {1, 2, 4, 5, 6, 8, 10, 12, 14, 16, 18, 20} (M ∪ N) = {3, 7, 9, 11, 13, 15, 17, 19} (M ∪ P) = {1, 2, 4, 5, 8, 10, 12, 16, 20} (M ∪ P) = {3, 6, 7, 9, 11, 13, 14, 15, 17, 18, 19} (N ∪ P) = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20} (N ∪ P) = {1, 3, 5, 7, 9, 11, 13, 15, 17, 19} (M ∪ N ∪ P) = {1, 2, 4, 5, 6, 8, 10, 12, 14, 16, 18, 20} (M ∪ N ∪ P) = {3, 7, 9, 11, 13, 15, 17, 19} ξ M •5 •1 •11 •3 •7 •9 •19 •13 •15 •17 •2 •4 •10 •20 •8 •12 N P •14 •6 •18 •16 Diberi set semesta ξ = {x : x ialah integer, 1 N x N 15}, M = {faktor bagi 10}, N = {nombor perdana} dan P = {nombor ganjil}. Given that the universal set, ξ = {x : x is an integer, 1 N x N 15}, M = {factor of 10}, N = {prime number} and P = {odd number}. ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} M = {1, 2, 5, 10} N = {2, 3, 5, 7, 11, 13} P = {1, 3, 5, 7, 9, 11, 13, 15} (M ∪ N) = {1, 2, 3, 5, 7, 10, 11, 13} (M ∪ N) = {4, 6, 8, 9, 12, 14, 15} (M ∪ P) = {1, 2, 3, 5, 7, 9, 10, 11, 13, 15} (M ∪ P) = {4, 6, 8, 12, 14} (N ∪ P) = {1, 2, 3, 5, 7, 9, 11, 13, 15} (N ∪ P) = {4, 6, 8, 10, 12, 14} (M ∪ N ∪ P) = {1, 2, 3, 5, 7, 9, 10, 11, 13, 15} (M ∪ N ∪ P) = {4, 6, 8, 12, 14} M N P •10 •12 •14 •11 •2 •5 •1 •3 •7 •9 •4 •6 •8 •15 •13 ξ Contoh CONTOH

Matematik Tingkatan 4 Bab 4 48 4 Selesaikan masalah berikut. SP: 4.2.3 TP5 Sukar Solve the following problem. Seramai 100 orang murid terlibat dalam satu kajian mengenai aktiviti sukan kegemaran mereka. 40 orang suka pingpong, 45 orang suka bola sepak, 22 orang suka badminton, 8 orang suka kedua-dua pingpong dan bola sepak, dan 2 orang suka kedua-dua bola sepak dan badminton. Berapakah bilangan murid yang tidak suka mana-mana aktiviti sukan tersebut? A total of 100 students are involved in a survey about their favourite sports. 40 students like ping-pong, 45 students like soccer, 22 students like badminton, 8 students like both ping-pong and soccer, and 2 students like both soccer and badminton. How many students do not like any of these sports? ξ = {jumlah murid/ total students} P = {pingpong/ ping-pong} S = {bola sepak/ socccer} B = {badminton/ badminton} ξ 32 8 35 2 20 P S B 3 Bilangan murid yang tidak suka mana-mana aktiviti: The number of students who do not like any activities: 100 – 32 – 8 – 35 – 2 – 20 = 3 orang murid / students Sebuah kelas mempunyai 40 orang murid. Daripada jumlah tersebut, 17 orang suka menyanyi, 20 orang suka menari dan 5 orang suka kedua-duanya. Berapakah bilangan murid yang suka menyanyi atau menari? A class has 40 students. From the total, 17 like singing, 20 like dancing and 5 like both. How many students like singing or dancing? ξ = {jumlah murid/ total students} S = {menyanyi/ singing} D = {menari/ dancing} ξ S D 17 – 5 = 12 5 20 – 5 = 15 40 – 12 – 15 – 5 = 8 Menyanyi sahaja/ Singing only: 17 – 5 = 12 orang murid/ students Menari sahaja/ Dancing only: 20 – 5 = 15 orang murid/ students Bilangan murid yang suka menyanyi atau menari: The number of students who like singing or dancing: 12 + 15 + 5 = 32 orang murid/ students Contoh 4.3 Gabungan Operasi Set Combined Operations on Sets Buku Teks m/s 116 – 122 1 Gambar rajah Venn berikut menunjukkan set semesta, ξ = P ∪ Q ∪ R. Senaraikan unsur-unsur bagi   SP: 4.3.1 TP1 The following Venn diagram shows the universal set, ξ = P ∪ Q ∪ R. List the elements of Mudah •5 •3 •4 P Q R •1 •7 •6 •9 •19 •15 •12 •11 •10 •20 •24 •14 •16 •17 •18 •13 •29 •25 (a) (P ∩ Q) ∪ R = {1, 3, 7, 10, 12, 14, 16, 17, 19, 20, 24} (b) P ∩ (Q ∪ R) = {10, 12, 20, 24} (c) (P ∩ R) ∪ Q = {4, 10, 12, 13, 16, 17, 18, 20, 24, 25, 29} (P ∪ Q) ∩ R = {10, 16, 17, 20, 24} (P ∪ Q) = {4, 5, 6, 9, 10, 11, 12, 13, 15, 16, 17, 18, 20, 24, 25, 29} R = {1, 3, 7, 10, 14, 16, 17, 19, 20, 24} Contoh Info Digital 4.3 CONTOH

Matematik Tingkatan 4 Bab 4 49 2 Lorekkan kawasan yang mewakili set yang diberi pada gambar rajah Venn yang berikut. SP: 4.3.1 TP4 Sederhana Shade the region representing the given set on the following Venn diagram. (a) A ∪ (B ∩ C) ξ B A C (b) (F ∩ H) ∪ G ξ H F G (c) (X ∪ Y) ∩ Z ξ X Y Z P ∩ Q ∪ R P Q R ξ Contoh 3 Lorekkan kawasan yang mewakili set yang diberi pada gambar rajah Venn yang berikut. SP: 4.3.2 TP4 Sederhana Shade the region representing the given set on the following Venn diagram. (a) (P ∪ Q) ∩ R ξ P Q R (b) P  ∩ (Q ∪ R) ξ Q P R (c) (P ∩ R) ∪ (Q ∩ R) ξ R P Q (P ∩ Q) ∩ R ξ P Q R Contoh R CONTOH P Q

Matematik Tingkatan 4 Bab 4 50 4 Selesaikan masalah berikut. SP: 4.3.3 TP5 KBAT Sukar Solve the following problems. (a) Gambar rajah Venn berikut menunjukkan set semesta, ξ, set X, set Y dan set Z. The following Venn diagram shows the universal set ξ, set X, set Y and set Z. Lorekkan kawasan yang mewakili set yang diberi pada gambar rajah Venn yang berikut. Shade the region representing the given set on the Venn diagram below. (i) X ∩ (Y � ∪ Z) (ii) (X ∪ Z)� ∩ Y ξ X Z Y ξ X Z Y (b) Dalam satu kajian, sekumpulan murid dari sebuah kelas ditanya tentang jenis sukan kegemaran. Gambar rajah Venn berikut menunjukkan bilangan murid yang minat dalam satu sukan atau lebih daripada satu sukan. In a survey, a group of students from a class are asked regarding their favourite sports. The following Venn diagram shows the number of students who are interested in a type of sport or more than a sport. Diberi/ Given ξ = {jumlah murid di dalam kelas/ total number of students in the class} A = {murid yang suka bermain badminton/ students who like playing badminton} B = {murid yang suka bermain bola sepak/ students who like playing football} C = {murid yang suka berenang/ students who like swimming} ξ A B 5 6 3 8 7 7 4 C Cari/ Find (i) jumlah bilangan murid di dalam kelas itu. the total number of students in the class. (ii) bilangan murid yang suka dua jenis sukan. the number of students who like two types of sports. (iii) bilangan murid yang suka sejenis sukan sahaja. the number of students who only like a type of sport. (iv) bilangan murid yang suka bermain badminton atau bola sepak. the number of students who like to play badminton or football. (iii) Bilangan murid yang suka sejenis sukan sahaja: The number of students who like only a type of sport: 5 + 8 + 7 = 20 orang murid/ students (iv) Bilangan murid yang suka bermain badminton atau bola sepak: The number of students who like to play badminton or football: 5 + 6 + 3 + 4 + 8 + 7 = 33 orang murid/ students (i) Jumlah bilangan murid: Total number of students: 5 + 6 + 3 + 4 + 8 + 7 + 7 = 40 orang murid/ students (ii) Bilangan murid yang suka dua jenis sukan: The number of students who like two types of sport: 6 + 4 + 7 = 17 orang murid/ students CONTOH

Matematik Tingkatan 4 Bab 4 51 (c) Satu soal selidik dijalankan terhadap 50 orang murid tentang mata pelajaran kegemaran mereka. Jadual berikut menunjukkan keputusan soal selidik tersebut. A survey is carried out on 50 students regarding their favourite subjects. The following table shows the results of the survey. Mata pelajaran Subject Bilangan murid Number of students Diberi/ Given that ξ = {murid-murid di dalam kelas} ξ = {students in the class} M = {murid yang suka Matematik} M = {students who like Mathematics} S = {murid yang suka Sains} S = {students who like Science} Matematik Mathematics 22 Sains Science 27 Matematik dan Sains Mathematics and Science x Tidak suka kedua-dua mata pelajaran Do not like both of the subjects 7 (i) Hitung nilai x. Calculate the value of x. (ii) Lukis sebuah gambar rajah Venn untuk menunjukkan hubungan antara ξ, M dan S. Draw a Venn diagram to show the relationship between ξ, M and S. (iii) Dari gambar rajah Venn di (c) (ii), hitung bilangan murid yang hanya suka satu mata pelajaran sahaja. From the Venn diagram in (c) (ii), calculate the number of students who only like one subject. (i) 22 + 27 – x = 50 – 7 49 – x = 43 x = 49 – 43 = 6 (iii) 16 + 21 = 37 orang murid/ students (ii) ξ M S 7 16 6 21 (d) Gambar rajah Venn berikut menunjukkan keputusan suatu kajian daripada 32 orang. The following Venn diagram shows the results of a survey from 32 people. C F S 12 x y 3 1 2 2 C = {orang yang suka makan ayam} = {people who like eating chicken} F = {orang yang suka makan ikan} = {people who like eating fish} S = {orang yang suka makan sayur} = {people who like eating vegetables} Diberi bilangan orang yang suka makan ikan adalah 2 kali bilangan orang yang suka makan sayur. Cari Given the number of people who like eating fish is 2 times the number of people who like eating vegetables. Find (i) nilai x dan nilai y, the values of x and y, (iii) bilangan orang yang hanya suka makan sejenis makanan. the number of people who only like to eat one type of food. (i) x + 3 + 2 + 2 = 2(y + 1 + 2 + 2) x + 7 = 2(y + 5) x + 7 = 2y + 10 x = 2y + 3 ...... ① 12 + 3 + 2 + 1 + x + 2 + y = 32 x + y + 20 = 32 x = 12 – y ...... ② ① = ②: 2y + 3 = 12 – y 3y = 9 y = 3 x = 12 – 3 = 9 (ii) 12 + 9 + 3 = 24 orang/ people CONTOH

Matematik Tingkatan 4 Bab 4 52 Kertas 1 Jawab semua soalan./ Answer all questions. 1 Rajah 1 menunjukkan gambar rajah Venn dengan set semesta, ξ = A ∪ B ∪ C. Diagram 1 shows a Venn diagram with the universal set, ξ = A ∪ B ∪ C. •3 A C •8 •2 •6 •10 •15 •11 •5 B Rajah 1/ Diagram 1 Senaraikan unsur-unsur bagi (A ∩ B ) ∪ C. List the elements of (A ∩ B’) ∪ C. A {3, 5, 10} B {2, 3, 5, 10, 15} C {2, 5, 6, 8, 11, 15} D {2, 3, 6, 8, 10, 11, 15} 2 Rajah 2 merupakan gambar rajah Venn yang menunjukkan bilangan unsur dalam set X, set Y dan set Z. Diagram 2 is a Venn diagram showing the number of elements in sets X, Y and Z. X Z Y 6 x 5 4 8 Rajah 2/ Diagram 2 Diberi set semesta, ξ = X ∪ Y ∪ Z dan n(ξ) = 30. Cari nilai n[X ∩ (Y ∩ Z) ]. Given that the universal set, ξ = X ∪ Y ∪ Z and n(ξ) = 30. Find the value of n[X ∩ (Y ∩ Z)’]. A 5 B 7 C 17 D 18 3 Diberi bahawa set A = {1, 2, 3, 5, 6, 7, 9, 10}, set B = {2, 3, 6, 7, 9, 11} dan set C = {5, 6, 7, 8, 9}. Given that set A = {1, 2, 3, 5, 6, 7, 9, 10}, set B = {2, 3, 6, 7, 9, 11} and set C = {5, 6, 7, 8, 9}. Senaraikan unsur-unsur bagi (A ∩ B) ∪ C. List the elements of (A ∩ B)’ ∪ C. A {1, 8, 10} B {1, 2, 3, 8, 10} C {1, 5, 6, 7, 8, 9, 10, 11} D {1, 2, 3, 5, 6, 8, 9, 10, 11} 4 Rajah 3 ialah gambar rajah Venn yang menunjukkan bilangan murid dalam set A, set B dan set C. Diagram 3 is a Venn diagram showing the number of students in set A, set B and set C. A B C 2 5 – x 9 – x x – 1 x 3x 3x Rajah 3/ Diagram 3 Diberi/ Given that ξ = A ∪ B ∪ C A = {murid yang suka minum susu} A = {students who like to drink milk} B = {murid yang suka minum coklat panas} B = {students who like to drink hot chocolate} C = {murid yang suka minum kopi} C = {students who like to drink coffee} Jika jumlah murid yang suka satu jenis minuman ialah 17, cari bilangan unsur dalam set B ∩ (A ∪ C)�. If the number of students who like one type of drink is 17, find the number of elements in set B ∩ (A ∪ C) . A 2 B 3 C 6 D 17 5 Rajah 4 ialah gambar rajah Venn yang menunjukkan bilangan unsur dalam set semesta, ξ, set X, set Y dan set Z. Diagram 4 is a Venn diagram showing the number of elements in the universal set, ξ, set X, set Y and set Z. X Z Y 2x x x 6 7 ξ Rajah 4/ Diagram 4 Diberi n(X ∩ Y ) = n[(Z ∪ Y) ∩ X ], cari n(ξ). Given n(X ∩ Y’) = n[(Z ∪ Y) ∩ X’], find n(ξ). A 5 B 11 C 13 D 17 Praktis Kendiri CONTOH

Matematik Tingkatan 4 Bab 4 53 6 Rajah 5 ialah satu gambar rajah Venn dengan set semesta, ξ, set P dan set Q. Diagram 5 is a Venn diagram with universal set, ξ, set P and set Q. P Q ξ Rajah 5/ Diagram 5 Antara berikut, yang manakah adalah sama dengan (P ∩ Q) ? Which of the following is equivalent to (P ∩ Q)’? A P  ∩ Q  C P  ∪ Q  B P  ∩ Q  D P  ∪ Q  7 Diberi set semesta, ξ = G ∪ H, set G = {Faktor bagi 18} dan set H = {Faktor bagi 12}. Cari nilai bagi n(ξ). Given universal set, ξ = G ∪ H, set G = {Factors of 18} and set H = {Factors of 12}. Find the value for n(ξ). A 4 C 8 B 6 D 12 8 Antara gambar rajah Venn berikut, yang manakah menunjukkan (P  ∩ R) ∪ Q? Which of the following Venn diagram shows (P’ ∩ R) ∪ Q? A Q R P B Q R P C Q R P D Q R P 9 Rajah 6 merupakan gambar rajah Venn dengan set semesta, ξ = A ∪ B ∪ C. Diagram 6 is the Venn diagram with the universal set, ξ = A ∪ B ∪ C. A C B Rajah 6/ Diagram 6 Apakah yang diwakili oleh kawasan berlorek? What is represented by the shaded region? A (A ∪ C) ∩ B B B ∩ (A ∩ C) C A ∪ B ∩ C  D (A ∪ C ) ∩ B 10 Diberi set semesta, ξ = X ∪ Y ∪ Z, X ∩ Z = Z dan Y = (X ∪ Z) . Given that the universal set, ξ = X ∪ Y ∪ Z, X ∩ Z = Z and Y = (X ∪ Z)’. Antara gambar rajah Venn berikut, yang manakah menunjukkan hubungan yang betul? Which of the following Venn diagram shows the correct relationship? A Y X Z B Y X Z C Y X Z D X Y Z CONTOH

Matematik Tingkatan 4 Bab 4 54 Kertas 2 Jawab semua soalan. / Answer all questions. Bahagian A/ Section A 1 Diberi set semesta, ξ = {x : 10 N x N 20, x ialah satu integer}, P = {10, 12, 14, 16, 18, 20} dan Q = {10, 11, 12, 13, 15, 16}. Given that the universal set, ξ = {x : 10 N x N 20, x is an integer}, P = {10, 12, 14, 16, 18, 20} and Q = {10, 11, 12, 13, 15, 16}. (a) Cari P ∩ Q. Find P ∩ Q. [1 markah/ mark] P = { 10 , 12 , 14, 16 , 18, 20} Q = { 10 , 11, 12 , 13, 15, 16 } P ∩ Q = {10, 12, 16} (b) Lukis sebuah gambar rajah Venn bagi mewakili maklumat di (a). Seterusnya, lorekkan kawasan yang mewakili P ∩ Q. Draw a Venn diagram to represent the information in (a). Hence, shade the region representing P ∩ Q. [2 markah/ marks] P Q ξ •19 •17 •14 •18 •20 •11 •13 •15 •10 •12 •16 2 (a) Diberi set semesta, ξ = P ∪ Q ∪ R. Lorekkan kawasan yang mewakili (P ∩ Q) ∩ (Q ∪ R). Given that the universal set, ξ = P ∪ Q ∪ R. Shade the region represent (P ∩ Q)’ ∩ (Q ∪ R)’. [1 markah/ mark] ξ R P Q (b) A, B dan C ialah tiga set dengan keadaan A ∪ B = A dan B ∩ C ≠ Ø. A, B and C are three sets such that A ∪ B = A and B ∩ C ≠ Ø. Lengkapkan dua gambar rajah Venn yang berkemungkinan untuk menunjukkan hubungan antara set A, set B dan set C seperti dinyatakan di atas. Complete two possible Venn diagrams to show the relationship between set A, set B and set C as stated above. [2 markah/ marks] (i) B C A (ii) B A C 3 Diberi/ Given that ξ = A ∪ B ∪ C ∪ D ∪ E ∪ F A = {x : 30 < x N 50, x ialah satu integer} {x : 30 < x N 50, x is an integer} B = {x : 40 N x N 50, x ialah satu integer} {x : 40 N x N 50, x is an integer} C = {nombor-nombor genap antara 31 dan 52} {even numbers between 31 and 52} D = {gandaan 5 yang kurang daripada 50} {multiples of 5 less than 50} E = {faktor-faktor bagi 100} {factors of 100} F = {kuasa dua sempurna kurang daripada 50} {perfect squares less than 50} (a) Senaraikan unsur-unsur bagi set C, D, E dan F. List the elements of the sets C, D, E and F. [2 markah/ marks] C = {32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52} D = {5, 10, 15, 20, 25, 30, 35, 40, 45} E = {1, 2, 4, 5, 10, 20, 25, 50, 100} F = {1, 4, 9, 16, 25, 36, 49} CONTOH

Matematik Tingkatan 4 Bab 4 55 (b) Cari/ Find (i) D ∩ E ∩ F (ii) (A ∩ D) ∪ E (iii) (B ∩ E)� ∩ (C ∪ D) (iv) (A ∩ B ∩ D) ∪ (D ∩ E ∩ F) [4 markah/ marks] (i) D = {5, 10, 15, 20, 25, 30, 35, 40, 45} E = {1, 2, 4, 5, 10, 20, 25, 50, 100} F = {1, 4, 9, 16, 25, 36, 49} D ∩ E ∩ F = {25} (ii) A = {31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50} D = {5, 10, 15, 20, 25, 30, 35, 40, 45} A ∩ D = {35, 40, 45} (A ∩ D) ∪ E = {1, 2, 4, 5, 10, 20, 25, 35, 40, 45, 50, 100} (iii) (B ∩ E)� = {1, 2, 4, 5, 9, 10, 15, 16, 20, 25, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 52, 100} (C ∪ D) = {5, 10, 15, 20, 25, 30, 32, 34, 35, 36, 38, 40, 42, 44, 45, 46, 48, 50, 52} (B ∩ E)� ∩ (C ∪ D) = {5, 10, 15, 20, 25, 30, 32, 34, 35, 36, 38, 40, 42, 44, 45, 46, 48, 52} (iv) (A ∩ B ∩ D) = {40, 45} (D ∩ E ∩ F) = {25} (A ∩ B ∩ D) ∪ (D ∩ E ∩ F) = {25, 40, 45} Bahagian B/ Section B 4 Sebuah kelas terdiri daripada 40 orang murid. Mereka dibahagikan kepada beberapa kumpulan mengikut warna kegemaran mereka. A class consists of 40 students. They are divided into few groups according to their favourite colours. Diberi/ Given that set semesta, ξ = {warna ungu, warna biru, warna putih} universal set, ξ = {purple colour, blue colour, white colour} set P = {warna ungu/ purple colour} set Q = {warna biru/ blue colour} set R = {warna putih/ white colour} Warna Colour Bilangan murid Number of students Warna ungu Purple colour 27 Warna biru Blue colour 1 orang murid kurang daripada warna ungu 1 student less than purple colour Warna putih White colour 14 orang murid kurang daripada warna biru 14 students less than blue colour Warna ungu atau warna putih Purple colour or white colour 27 Warna biru atau warna putih Blue colour or white colour 30 Warna ungu atau warna biru Purple colour or blue colour 40 Semua warna All colours x Jadual 1/ Table 1 K B A T CONTOH

Matematik Tingkatan 4 Bab 4 56 (a) Cari nilai x. Find the value of x. [2 markah/ marks] Q P R x n(Q ∩ R) = n(P ∩ Q ∩ R) = x n(R) + n(Q) – n(Q ∩ R) = n(Q ∪ R) (27 – 1 – 14) + (27 – 1) – x = 30 12 + 26 – x = 30 x = 38 – 30 = 8 (b) Lengkapkan unsur pada gambar rajah Venn berikut untuk mewakili data dalam Jadual 1. Complete the elements in the following Venn diagram to represent the data in Table 1. [2 markah/ marks] Q 10 4 8 5 13 P R (c) Hitung bilangan murid yang hanya suka satu warna. Calculate the number of students who only like one colour. [1 markah/ mark] 10 + 13 = 23 orang murid/ students (d) Cari bilangan murid dalam set yang berikut. Find the number of students in the following sets. (i) (P ∩ R)� ∪ Q (ii) (P ∩ R) ∪ Q� [4 markah/ marks] (i) Q P R 10 + 8 + 5 + 13 = 36 orang murid/ students (ii) 10 + 4 + 8 = 22 orang murid/ students P Q CONTOH R

Matematik Tingkatan 4 Bab 4 57 5 Dalam satu kajian yang melibatkan 48 orang, 6 orang tidak mempunyai haiwan peliharaan daripada mana-mana tiga jenis haiwan: kucing, ikan dan burung. In a survey that involved 48 people, 6 of them do not have any of the three types of pets: cat, fish and bird. Jadual berikut menunjukkan bilangan orang yang memelihara kucing, ikan dan burung. The following table shows the number of people who keep cat, fish and bird. Haiwan Animal Bilangan orang Number of people Kucing Cat 23 Ikan Fish 3 orang kurang daripada orang yang memelihara kucing 3 people less than people who keep cat Burung Bird 1 orang lebih daripada orang yang memelihara ikan 1 people more than people who keep fish Kucing dan ikan Cat and fish 11 Ikan dan burung Fish and bird 7 Kucing dan burung Cat and bird 9 Kucing, ikan dan burung Cat, fish and bird x Diberi set semesta, ξ = {kucing, ikan, burung}, set K = {kucing}, set I = {ikan} dan set B = {burung}. Given the universal set, ξ ={cat, fish, bird}, set K = {cat}, set I = {fish} and set B = {bird}. (a) Lukis sebuah gambar rajah Venn yang menunjukkan hubungan antara set K, set I, set B dan set semesta. Draw a Venn diagram that shows the relationships between set K, set I, set B and the universal set. [4 markah/ marks] ξ K I B 6 11 – x 9 – x 7 – x x + 5 x + 3 2 + x x n(K) = 23 – (11 – x) – x – (9 – x) = 23 – 11 + x – x – 9 + x = 3 + x n(I) = (23 – 3) – (11 – x) – x – (7 – x) = 20 – 11 + x – x – 7 + x = 2 + x n(B) = (20 + 1) – (9 – x) – x – (7 – x) = 21 – 9 + x – x – 7 + x = 5 + x (b) Hitung nilai x. Calculate value of x. [2 markah/ marks] (x + 3) + (11 – x) + x + (9 – x) + (2 + x) + (7 – x) + (x + 5) + 6 = 48 43 + x = 48 x = 5 (c) Cari bilangan orang yang memelihara satu atau kurang daripada satu haiwan peliharaan. Find the number of people who keep one or less than one pet. [2 markah/ marks] 6 + (5 + 3) + (2 + 5) + (5 + 5) = 31 CONTOH

58 Praktis Intensif 5.1 Rangkaian Network Buku Teks m/s 130 – 147 1 Namakan bahagian graf berdasarkan huraian berikut. SP: 5.1.1 TP1 Mudah Name the parts of the graph based on the following descriptions. Suatu graf yang mempunyai sekurang-kurangnya satu pasang bintik yang berkait. A graph which has at least a pair of linked dots. Rangkaian Network (a) Bilangan tepi yang berkait dengan sesuatu bintik pada graf. The number of edges linked to a dot on a graph. Darjah bucu Degree of vertex (b) Bucu-bucu pada graf. Vertices on the graph. Bintik Dot (c) Suatu siri bintik yang berkait melalui garis. A series of dots which is linked by lines. Graf Graph (d) Graf tak terarah yang tidak mengandungi gelung atau berbilang tepi. Undirected graph which does not have loops or multiple edges. Graf mudah Simple graph Contoh 2 Tandakan bagi rangkaian dan bagi bukan rangkaian. SP: 5.1.1 TP2 Mudah Mark for the network and for not a network. (a) (b) (c) Contoh 3 Senaraikan bucu, V dan tepi, E bagi setiap graf mudah yang berikut dan seterusnya nyatakan n(V), n(E) dan bilangan darjah. SP: 5.1.1 TP2 Mudah List vertices, V and edges, E for each of the following simple graphs and hence state n(V), n(E) and the sum of degrees. (a) 1 2 3 4 V = {1, 2, 3, 4} E = {(1, 2), (1, 4), (2, 3), (3, 4)} n(V) = 4, n(E) = 4 Bilangan darjah: 2(4) = 8 Sum of degrees (b) J K N L M V = {J, K, L, M, N} E = {(J, K), (J, L), (K, L), (L, M), (L, N), (M, N)} n(V) = 5, n(E) = 6 Bilangan darjah: 2(6) = 12 Sum of degrees (c) 1 2 3 4 7 6 5 V = {1, 2, 3, 4, 5, 6, 7} E = {(1, 2), (1, 7), (2, 3),(2, 7), (3, 4), (3, 6), (4, 5), (5, 6), (6, 7)} n(V) = 7, n(E) = 9 Bilangan darjah: 2(9) = 18 Sum of degrees V = {P, Q, R, S, T} E = {(P, Q), (P, T), (Q, R),(Q, S), (R, T), (R, S), (S, T)} n(V) = 5, n(E) = 7 Bilangan darjah: 2(7) = 14 Sum of degrees Q P T S R Contoh 5 Bab Info Digital 5.1 Rangkaian dalam Teori Graf Network in Graph Theory CONTOH

Matematik Tingkatan 4 Bab 5 59 4 Lengkapkan jadual yang berikut berdasarkan graf G(V, E) yang diberi. SP: 5.1.1 TP3 Sederhana Complete the following table based on the given graph G(V, E). A B C D E Graf 1/ Graph 1 A B C D E H F G Graf 2/ Graph 2 G F A B C D E Graf 3/ Graph 3 Graf 1 Graph 1 Graf 2 Graph 2 Graf 3 Graph 3 V (a) {A, B, C, D, E, F, G, H} (b) {A, B, C, D, E, F, G} E (c) {(A, B), (A, D), (B, C), (C, D), (D, E), (E, F), (E, H), (F, G), (G, H), (G, H)} (d) {(A, B), (A, G), (B, C), (B, F), (C, D), (C, F), (D, E), (E, F), (F, G)} n(V) (e) 8 (f) 7 d(A) (g) 2 (h) 2 d(B) (i) 2 (j) 3 d(C) (k) 2 (l) 3 d(D) (m) 3 (n) 2 d(E) (o) 3 (p) 2 d(F) (q) 2 (r) 4 d(G) (s) 3 (t) 2 d(H) (u) 3 – Ʃd(V) (v) 20 (w) 18 {A, B, C, D, E} {(A, B), (A, E), (B, C), (B, C), (C, D), (D, D), (D, E)} 5 2 3 3 4 2 – – – 14 Contoh CONTOH

Matematik Tingkatan 4 Bab 5 60 5 Lukiskan satu graf mudah bagi setiap maklumat yang diberi. SP: 5.1.1 TP3 Sederhana Draw a simple graph for each of the following given information. (a) V = {P, Q, R, S, T} E = {(P, Q), (Q, R), (Q, S), (R, S), (S, T)} Q S T R P (b) V = {1, 2, 3, 4, 5} E = {(1, 2), (1, 4), (1, 5), (2, 3), (2, 5), (3, 5), (3, 4), (4, 5)} 1 2 3 4 5 V = {1, 2, 3, 4} E = {(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)} 1 2 4 3 Contoh 6 Lukiskan satu graf yang berbilang tepi dan gelung berdasarkan maklumat yang diberi. SP: 5.1.1 TP3 Sederhana Draw a graph with multiple edges and loops based on the given information. V = {1, 2, 3, 4, 5} E = {(1, 2), (1, 5), (1, 1), (2, 3), (2, 5), (2, 5), (2, 2), (3, 4), (4, 5), (5, 5)} 1 5 2 4 3 V = {A, B, C} E = {(A, B), (A, B), (A, B), (B, C), (C, C)} A B C Contoh 7 Tentukan sama ada setiap set bilangan darjah yang diberi boleh membentuk satu graf. Jika boleh, lukiskan graf tersebut. SP: 5.1.1 TP3 Sukar Determine whether each given set of degrees of vertices can form a graph. If can, draw the graph. (a) 3, 3, 2 Jumlah/ Total: 3 + 3 + 2 = 8 Boleh/ Can (b) 3, 1, 3, 2, 2 Jumlah/ Total: 3 + 1 + 3 + 2 + 2 = 11 Tidak boleh/ Cannot (c) 2, 2, 2, 2 Jumlah/ Total: 2 + 2 + 2 + 2 = 8 Boleh/ Can 3, 1, 1, 1 Jumlah/ Total: 3 + 1 + 1 + 1 = 6 Boleh/ Can Contoh CONTOH

Matematik Tingkatan 4 Bab 5 61 8 Tandakan bagi graf terarah dan bagi graf tak terarah. SP: 5.1.2 (i) TP2 Midah Mark for directed graphs and for undirected graphs. (a) (b) (c) (d) (e) Contoh 9 Lukis sebuah graf terarah berdasarkan maklumat yang diberi. SP: 5.1.2 (i) TP3 Sederhana Draw a directed graph based on the given information. V = {1, 2, 3, 4} E = {(1, 2), (2, 4), (3, 2), (4, 2), (4, 4)} 1 3 2 4 V = {A, B, C, D, E} E = {(A, B), (A, D), (B, C), (D, B)} A B C E D Contoh 10 Lukis sebuah graf terarah berdasarkan maklumat yang diberi. SP: 5.1.2 (i) TP3 Sederhana Draw a directed graph based on the given information. • Mempunyai gelung pada bucu A dan B Has loops at vertices A and B • CD ialah berbilang tepi CD is multiple edges • din(A) = 2, dout(A) = 2 din(B) = 1, dout(B) = 3 din(C) = 2, dout(C) = 1 din(D) = 2, dout(D) = 1 A B D C • Mempunyai gelung pada bucu A Has loop at vertex A • BC ialah berbilang tepi BC is multiple edges • din(A) = 1, dout(A) = 3 din(B) = 2, dout(B) = 1 din(C) = 2, dout(C) = 1 A B C Contoh CONTOH

Matematik Tingkatan 4 Bab 5 62 11 Tandakan bagi graf berpemberat dan bagi graf tak berpemberat. SP: 5.1.2 (ii) TP2 Mudah Mark for weighted graphs and for unweighted graphs. (a) A B D E C 9 km 3 km 5 km 8.7km 5.5 km (b) (c) M N T (d) A M D S CH 1 2 3 5 (e) 2 3 4 5 1 P Q R T S 43 35 57 45 25 Contoh 12 Lukis sebuah graf berpemberat berdasarkan maklumat yang diberi. SP: 5.1.2 (ii) TP3 Sederhana Draw a weighted graph based on the given information. V = {P, Q, R} Pasangan bucu Pair of vertices Jarak (unit) Distance (unit) (P, Q) 4 (Q, R) 5 (P, R) 4 Q R P 4 unit/ units 4 unit/ units 5 unit/ units V = {A, B, C, D, E} Pasangan bucu Pair of vertices Jarak (km) Distance (km) (A, B) 10 (A, C) 8 (B, C) 17 (B, E) 14 (C, D) 8 (D, E) 3 A B E D C 10 km 17 km 14 km 8 km 8 km 3 km Contoh CONTOH

Matematik Tingkatan 4 Bab 5 63 13 Selesaikan setiap masalah yang berikut. SP: 5.1.2 (i), (ii) TP4 TP5 Sukar Solve each of the following problems. (a) Jadual berikut menunjukkan jarak antara lima bandar. The following table shows the distance between five cities. Bandar City Jarak (km) Distance (km) (Suria, Mahkota) 52 (Mahkota, Damai) 38 (Mahkota, Kati) 75 (Damai, Pusing) 55 (Damai, Kati) 60 (Pusing, Kati) 28 (i) Bina sebuah graf berpemberat dan tak terarah untuk menunjukkan rangkaian bandar-bandar tersebut. Construct a weighted and undirected graph to show the network of the cities. (ii) Encik Bakal yang tinggal di Bandar Suria hendak melawat adiknya di Bandar Pusing. Tentukan jarak terpendek bagi perjalanan Encik Bakal. Mr Bakal who lives in Suria City wants to visit his sister in Pusing City. Determine the shortest distance for Mr Bakal’s journey. (iii) Kemudian, Encik Bakal mengambil jarak terpanjang pulang ke Bandar Suria. Hitungkan jarak perjalanan Encik Bakal. After that, Mr Bakal took the longest distance to return to Suria City. Calculate the distance of Mr Bakal’s journey. (b) Jadual berikut menunjukkan panjang lebuh raya antara empat bandar. The following table shows the length of the highway between four towns. Bandar Town Panjang (km) Length (km) (E, F) 55 (F, G) 43 (G, H) 110 (H, G) 100 (G, E) 60 (G, F) 45 (i) Lukis sebuah graf berpemberat dan terarah untuk menunjukkan rangkaian lebuh raya antara bandar. Draw a weighted and directed graph to show the network of highways between towns. (ii) Encik Siva bergerak ke Bandar H dari Bandar G melalui Bandar E. Hitung jumlah jarak, dalam km, perjalanan Encik Siva. Mr Siva moved to town H from town G, passing through town E. Calculate the total distance, in km, of Mr Siva’s journey. (iii) Kemudian, Encik Siva perlu pergi ke Bandar F untuk mengambil barang sebelum pulang ke Bandar G. Nyatakan beza jarak, dalam km, jika Encik Siva tidak pergi mengambil barang. After that, Mr Siva went to town F to pick up his things before returning to town G. State the difference in distance, in km, if Mr Siva did not go to pick up his things. (i) Mahkota Suria Pusing Damai Kati 60 km 28 km 55 km 75 km 52 km 38 km (ii) Perjalanan Encik Bakal/ Mr Bakal’s journey: Suria Mahkota Damai Pusing 52 + 38 + 55 = 145 km (iii) Perjalanan Encik Bakal/ Mr Bakal’s journey: Pusing Kati Damai Mahkota Suria 28 + 60 + 38 + 52 = 178 km (i) H G F E 60 km 55 km 45 km 43 km 100 km 110 km (ii) G E F G H 60 + 55 + 43 + 110 = 268 km (iii) H G F G 100 + 45 + 43 = 188 km Beza jarak: Difference in distance: 188 – 100 = 88 km CONTOH

Matematik Tingkatan 4 Bab 5 64 14 Berdasarkan graf K, tandakan bagi subgraf dan bagi bukan subgraf. SP: 5.1.3 TP2 Mudah Based on graph K, mark for the subgraphs and for not subgraph. (a) 1 2 e3 e1 (b) 1 (c) 1 2 e1 e3 (d) 1 2 e1 (e) 1 2 e1 e2 e3 2 e3 1 2 e2 e1 e3 Graf K Graph K Contoh 15 Lukis dua subgraf yang dinyatakan berdasarkan graf di sebelah kiri. SP: 5.1.3 TP3 Sederhana Draw two stated subgraphs based on the graph on the left. e4 e2 e1 e3 e5 Q R S P (i) {P, Q, R} ⊂ {P, Q, R, S} e2 e4 e1 Q R P (ii) {e2, e3, e5} ⊂ {e1, e2, e3, e4, e5} e2 e3 e5 Q S R P (a) e1 e6 e4 e2 e3 e5 A E D C B (i) {A, B, D, E} ⊂ {A, B, C, D, E} e6 e1 e4 e5 A B E D (ii) {e2, e3, e4} ⊂ {e1, e2, e3, e4, e5, e6} e4 e3 e2 D B C (b) e6 e1 e7 e5 e2 e4 e3 J K L M N O (i) {J, K, L, O} ⊂ {J, K, L, M, N, O} e7 e2 e e1 6 J O K L (ii) {e5, e6, e7}⊂ {e1, e2, e3, e4, e5, e6, e7} e6 e7 e5 J N L O Contoh CONTOH

Matematik Tingkatan 4 Bab 5 65 16 Tentukan sama ada setiap graf yang berikut ialah satu pokok. SP: 5.1.3 TP2 TP3 Mudah Determine whether each of the following graphs is a tree. (a) (b) (c) (d) (e) Contoh 17 Kaitkan bintik-bintik berikut untuk membentuk satu pokok dan seterusnya, nyatakan bilangan bucu dan tepi bagi pokok tersebut. SP: 5.1.3 TP4 Sederhana Link the following dots to form a tree and hence, state the number of vertices and edges for the tree. Pokok Tree (a) (b) Bilangan bucu Number of vertices (i) 6 (i) 7 Bilangan tepi Number of edges (ii) 5 (ii) 6 5 4 Contoh 18 Bina satu graf yang sesuai untuk menunjukkan rangkaian bagi setiap situasi yang berikut. SP: 5.1.4 TP5 Sukar Construct a suitable graph to show the network for each of the following situations. Keluarga Amin terdiri daripada bapa, ibu dan seorang adik perempuan, Siti. Siti telah berkahwin dengan Ahmad dan mempunyai dua orang anak, Fauzi dan Faiz. Amin’s family consists of father, mother and a sister, Siti. Siti is married to Ahmad and has two children, Fauzi and Faiz. Contoh Fauzi Ahmad Faiz Bapa/ Father Ibu/ Mother Amin Siti CONTOH

Matematik Tingkatan 4 Bab 5 66 (a) Mei Li, Luna, Susie, Janet dan Kenny tinggal di satu kawasan perumahan yang sama. Jarak antara rumah mereka adalah seperti berikut: Mei Li, Luna, Susie, Janet and Kenny live in the same neighbourhood. The distance between their houses are as follows: Rumah/ House Jarak (m)/ Distance (m) Mei Li dan/ and Luna 300 Luna dan/ and Susie 220 Luna dan/ and Janet 150 Janet dan/ and Kenny 180 Kenny dan/ and Mei Li 125 Mei Li Luna Susie Kenny Janet 180 m 125 m 150 m 300 m 220 m (b) Jadual berikut menunjukkan empat jenis hobi bagi beberapa orang murid. The following table shows four hobbies of some students. Hobi/ Hobby Murid/ Students Badminton Shaufiq, Elly, Shane, David, Zukri Melukis/ Drawing Elly, Sophia Menyanyi/ Singing Shaufiq, Shane, Pau Shan Berkebun/ Gardening Sophia, Zukri, David, Pau Shan Badminton Sophia Pau Shan Melukis Drawing Berkebun Gardening Menyanyi Singing Elly David Zukri Shaufiq Shane 19 Selesaikan setiap soalan yang berikut. SP: 5.1.5 TP6 Sukar Solve each of the following questions. (a) Jadual berikut menunjukkan jenis pengangkutan yang sedia ada untuk menaiki Gunung Rapi. The following table shows the transportation available to go up Rapi Hill. Pengangkutan Transportation Masa Time Harga Fee Frekuensi Frequency Kereta kabel Cable car 0900 – 1230 1400 – 1630 RM50 Setiap 15 minit Every 15 minutes Basikal Bicycle 0830 – 1830 RM15 Sepanjang hari Whole day Bas Bus 1000 – 1700 RM35 Setiap 2 jam Every 2 hours Sekiranya Amir hendak menaiki Gunung Rapi awal dengan selesa, tentukan pilihan pengangkutan yang sesuai bagi dia. Wajarkan jawapan anda. If Amir wants to go up Rapi Hill early comfortably, determine the choice of transportation for him. Justify your answer. Kereta kabel, kerana kereta kabel tidak menggunakan tenaga manusia berbanding dengan basikal dan masa perjalanannya bermula lebih awal daripada bas. Cable car, because cable car does not require human strength compared to bicycle and its depature time starts earlier than the bus. (b) Rajah berikut menunjukkan subgraf bagi suatu rangkaian pelanggan-pembekal. The following diagram shows the subgraph of a network of customer-supplier. Peniaga Trader Pengguna Consumer Pembekal Supplier Pemborong Wholesaler Pemborong luar negara Overseas wholesaler Kos barang meningkat sebanyak 120% pada setiap peringkat agihan bekalan. Jika kos daripada kilang ialah RM2.50, cari harga pasaran yang paling tinggi. The cost of item increased by 120% at each stage of supply distribution. If the cost from the factory is RM2.50, find the highest market price. RM2.50 × 2.22 = RM12.10 K B A T CONTOH

Matematik Tingkatan 4 Bab 5 67 Kertas 1 Jawab semua soalan. / Answer all questions. 1 Antara berikut, yang manakah bukan rangkaian? Which of the following is not a network? A B C D 2 Diberi bilangan darjah pada sebuah graf bagi dua bucu ialah 4 dan dua bucu lagi ialah 3. Cari bilangan tepi pada graf tersebut. Given the sum of degrees on a graph for two vertices is 4 and another two vertices is 3. Find the number of edges on the graph. A 4 B 7 C 11 D 14 3 Rajah 1 menunjukkan sebuah graf. Diagram 1 shows a graph. Q R T S P Rajah 1/ Diagram 1 Tentukan jenis graf di Rajah 1. Determine the type of graph in Diagram 1. A Graf mudah Simple graph B Graf terarah Directed graph C Graf berpemberat Weighted graph D Graf berpemberat dan terarah Weighted and directed graph 4 Rajah 2 menunjukkan sebuah graf. Diagram 2 shows a graph. K L J Rajah 2/ Diagram 2 Cari bilangan darjah bagi graf di Rajah 2. Find the sum of degrees for the graph in Diagram 2. A 4 B 7 C 14 D 18 5 Rajah 3 menunjukkan sebuah graf. Diagram 3 shows a graph. Rajah 3/ Diagram 3 Antara berikut, yang manakah bukan subgraf bagi graf di Rajah 3? Which of the following is not a subgraph of the graph in Diagram 3? A C B D 6 Diberi satu pokok mempunyai empat bucu. Tentukan bilangan tepi bagi pokok tersebut. Given a tree has four vertices. Determine the number of edges of the tree. A 6 B 5 C 4 D 3 Praktis Kendiri CONTOH

Matematik Tingkatan 4 Bab 5 68 Kertas 2 Jawab semua soalan. / Answer all questions. Bahagian A/ Section A 1 Jadual 1 menunjukkan kod IATA bagi lapangan terbang dan Jadual 2 menunjukkan masa transit laluan penerbangan dari Kuala Lumpur ke London. Table 1 shows the IATA code of airports and Table 2 shows the transit time of flight route from Kuala Lumpur to London. Lapangan terbang Airport Kod IATA IATA code Kuala Lumpur KUL Abu Dhabi AUH Dubai DXB Dublin DUB London LHR Jadual 1/ Table 1 Laluan penerbangan Flight route Masa transit (jam) Transit time (hours) KUL – AUH 7.17 KUL – DXB 7.33 KUL – LHR 14.33 AUH – DUB 8.5 DUB – LHR 1.5 AUH – LHR 8.08 DXB – LHR 7.83 Jadual 2/ Table 2 (a) Lukis satu graf terarah untuk menunjukkan masa transit dalam rangkaian laluan penerbangan dari Kuala Lumpur ke London. Draw a directed graph to show the transit time in the network of flight route from Kuala Lumpur to London. [2 markah/ marks] AUH DUB LHR DXB 8.5 jam/ hours 1.5 jam/ hours 8.08 jam/ hours 7.33 jam/ hours 7.83 jam/ hours 7.17 jam/ hours 14.33 jam/ hours KUL (b) Hitung beza antara masa transit yang paling singkat dan masa transit yang paling lama, dalam jam dan minit. Find the difference between the shortest transit time and the longest transit time, in hours and minutes. [2 markah/ marks] Masa transit yang paling lama/ Longest transit time: 7.17 + 8.5 + 1.5 = 17.17 jam/ hours = 17 jam 10 minit = 17 hours 10 minutes Beza masa transit/ Difference in transit time: 17 jam 10 minit – 14 jam 20 minit 17 hours 10 minutes – 14 hours 20 minutes = 2 jam 50 minit/ 2 hours 50 minutes 2 Rajah 2 menunjukkan rangkaian ion-ion dalam suatu struktur kimia. Diagram 2 shows the network of ions in a chemical structure. H1 H6 H5 H4 C5 C6 C1 C2 H2 C4 C3 H3 Rajah 2/ Diagram 2 (a) Diberi tepi dalam rajah mewakili perkongsian sepasang elektron, cari bilangan elektron yang dikongsi antara ion dalam struktur kimia ini. Given the edge in the diagram represents the sharing of a pair of electrons, find the number of electrons being shared between the ions in this chemical structure. [2 markah/ marks] 15 × 2 = 30 elektron/ electrons (b) Perkongsian elektron merupakan satu ikatan struktur antara ion. Nyatakan bilangan maksimum ikatan yang terdapat pada setiap ion C. The sharing of electrons is a bonding of the structure between ions. State the maximum number of bonding found on each C ion. [1 markah/ mark] 4 CONTOH

Matematik Tingkatan 4 Bab 5 69 Bahagian B/ Section B 3 Jadual berikut menunjukkan pilihan pengangkutan, jarak, masa perjalanan dan perbelanjaan perjalanan. The following table shows the choices of transportation, distance, travel duration and travel expenses. Pengangkutan Transportation Destinasi Destination Jarak Distance Masa perjalanan Travel duration Perbelanjaan perjalanan Bertolak Travel expenses Departure Ketibaan Arrival Kereta sendiri Private car Bandar A Town A Rumah nenek Grandmother’s house 200 km 2 jam 15 minit 2 hours 15 minutes Bayaran tol/ Toll fare: RM5.90 Harga petrol/ Petrol price: RM0.18 per km Kereta api Train Bandar A Town A Bandar B Town B 190 km 2 jam 30 minit 2 hours 30 minutes RM30 Bas Bus Bandar A Town A Bandar B Town B 190 km 3 jam 3 hours RM13 Teksi Taxi Bandar B Town B Rumah nenek Grandmother’s house 15 km 30 minit 30 minutes RM20 Thomas tinggal di Bandar A. Dia hendak melawat neneknya yang tinggal di Bandar B. Berdasarkan jadual di atas, Thomas lives in Town A. He wants to visit his grandmother who lives in Town B. Based the above table, (a) lukis graf bagi setiap laluan yang mungkin boleh diambil oleh Thomas untuk sampai ke rumah neneknya, draw graphs for each possible route which can be taken by Thomas to reach his grandmother’s house, [3 markah/ marks] Laluan 1/ Route 1: Bandar A Town A Kereta sendiri Private car Rumah nenek Grandmother’s house Laluan 2/ Route 2: Bandar A Town A Bandar B Town B Kereta api Train Teksi/ Taxi Rumah nenek Grandmother’s house Laluan 3/ Route 3: Bandar A Town A Bas/ Bus Teksi/ Taxi Rumah nenek Grandmother’s house Bandar B CONTOH Town B

Matematik Tingkatan 4 Bab 5 70 (b) hitungkan jumlah jarak, perbelanjaan dan masa bagi setiap laluan di (a), calculate the total distance, expenses and time for each route in (a), [3 markah/ marks] Laluan 1/ Route 1: Jarak/ Distance: 200 km Jumlah perbelanjaan/ Total expenses: (200 × RM0.18) + RM5.90 = RM41.90 Masa/ Time: 2 jam 15 minit/ 2 hours 15 minutes Laluan 2/ Route 2: Jarak/ Distance: 190 + 15 = 205 km Jumlah perbelanjaan/ Total expenses: RM30 + RM20 = RM50 Masa: 2 jam 30 minit + 30 minit = 3 jam Time: 2 hours 30 minutes + 30 minutes = 3 hours Laluan 3/ Route 3: Jarak/ Distance: 190 + 15 = 205 km Jumlah perbelanjaan/ Total expenses: RM13 + RM20 = RM33 Masa: 3 jam + 30 minit = 3 jam 30 minit Time: 3 hours + 30 minutes = 3 hours 30 minutes (c) tentukan laluan yang terbaik bagi Thomas. Wajarkan jawapan anda. determine the best route for Thomas. Justify your answer. [2 markah/ marks] Laluan 1, kerana Laluan 2 terlalu mahal dan Laluan 3 mengambil masa yang agak lama. Route 1, because Route 2 is too expensive and Route 3 takes a very long time. CONTOH

71 Praktis Intensif 6.1 Ketaksamaan Linear dalam Dua Pemboleh Ubah Linear Inequalities in Two Variables Buku Teks m/s 156 – 164 1 Tentukan sama ada setiap yang berikut ialah ketaksamaan linear atau tidak. TP1 Mudah Determine whether each of the following is a linear inequality or not. (a) x = 5y – 2 Tidak/ No (b) 2y N 3x + 4 Ya/ Yes y > 2x + 3 Ya/ Yes Contoh 2 Wakilkan setiap situasi yang berikut dalam bentuk ketaksamaan linear dengan pemboleh ubah yang sesuai. Represent each of the following situations in the form of linear inequality with suitable variables. SP: 6.1.1 TP2 Mudah (a) James mempunyai 10 buah buku. Bilangan buku yang James masih ada selepas dia meminjamkan beberapa buah buku kepada kawannya. James has 10 books. The number of books that James still has after he had lent some books to his friends. y < 10 (b) Amirul mendapat 70 markah dalam ujian Matematik. Selepas Zaidi mendapat 5 markah tambahan, markah Zaidi menjadi lebih tinggi daripada markah Amirul. Zaidi mendapat berapa markah? Amirul obtained 70 marks in a Mathematical test. After Zaidi obtained 5 additional marks, Zaidi’s marks became higher than Amirul’s marks. How many marks does Zaidi obtain? x + 5 > 70 (c) John mempunyai RMx. Wang kepunyaan Jackson adalah dua kali ganda John. Walaupun John memberi RM3 kepada Jackson, wang kepunyaan Jackson masih tidak melebihi RM20. Berapakah wang kepunyaan Jackson? John has RMx. Jackson’s money is twice of John’s. Although John gave RM3 to Jackson, Jackson’s money is still not more than RM20. How much money does Jackson have? 2x + 3 N 20 Satu nombor yang lebih daripada kuasa dua empat. A number which is greater than the square of four. y > 42 y > 16 Contoh 3 Tentukan sama ada titik-titik berikut memenuhi y = x + 3, y > x + 3 atau y < x + 3. SP: 6.1.2 TP2 Mudah Determine whether the following points satisfy y = x + 3, y > x + 3 or y < x + 3. (a) (5, 5) y = 5, 5 + 3 = 8, (5 < 8) ∴ Titik (5, 5) memuaskan y < x + 3. Point (5, 5) satisfies y < x + 3. (b) (3, 8) y = 8, 3 + 3 = 6, (8 > 6) ∴ Titik (3, 8) memuaskan y > x + 3. Point (3, 8) satisfies y > x + 3. (c) (–5, 2) y = 2, –5 + 3 = –2, (2 > –2) ∴ Titik (–5, 2) memuaskan y > x + 3. Point (–5, 2) satisfies y > x + 3. (d) (–3, –2) y = –2, –3 + 3 = 0, (–2 < 0) ∴ Titik (–3, –2) memuaskan y < x + 3. Point (–3, –2) satisfies y < x + 3. (e) (–3, 0) y = 0, –3 + 3 = 0, (0 = 0) ∴ Titik (–3, 0) memuaskan y = x + 3. Point (–3, 0) satisfies y = x + 3. (1, 4) y = 4, 1 + 3 = 4, (4 = 4) ∴ Titik (1, 4) memuaskan y = x + 3. Point (1, 4) satisfies y = x + 3. Contoh Ketaksamaan Linear dalam Dua Pemboleh Ubah Linear Inequalities in Two Variables 6 Bab Info Digital 6.1 CONTOH

Matematik Tingkatan 4 Bab 6 72 4 Lorekkan rantau yang mewakili setiap ketaksamaan linear yang berikut. SP: 6.1.3 TP3 Mudah Shade the region which represents each of the following linear inequalities. (a) 3x + 2 > y y < 3x + 2 y = 3x + 2 y x 2 O – 2 3 (b) y M 3 y = 3 y x 3 O (c) y + 3x > 5 y > –3x + 5 y = –3x + 5 y x 5 O 5 3 (d) 2x + y < –4 y < –2x – 4 y = –2x – 4 y x –2 –4 O (e) x < –2 x = –2 y x –2 O y – x N 7 y N x + 7 y y = x + 7 x –7 7 O Contoh 5 Wakil dan lorekkan ketaksamaan linear berikut pada grid segi empat sama yang diberi. SP: 6.1.3 TP3 Sederhana Represent and shade the following linear inequalities on the given square grids. (a) y < –x + 3 x 0 3 y 3 0 x y y = –x + 3 3 3 O y M 3x – 6 x 0 2 y –6 0 x y y = 3x – 6 –6 O 2 Contoh CONTOH

Matematik Tingkatan 4 Bab 6 73 (b) 4x – 3y > 12 3y < 4x – 12 x 0 3 y –4 0 y < 4 3 x – 4 x y y = 4 3x – 4 3 –4 O (c) y M 3x + 9 x 0 –3 y 9 0 x y –3 9 O y = 3x + 9 6.2 Sistem Ketaksamaan Linear dalam Dua Pemboleh Ubah Systems of Linear Inequalities in Two Variables Buku Teks m/s 165 – 175 1 Tulis ketaksamaan linear berdasarkan setiap pernyataan yang berikut. SP: 6.2.1 TP2 Mudah Write the linear inequalities based on each of the following statements. (a) y tidak lebih daripada x. y is not more than x. y N x (b) y melebihi x sekurang-kurangnya w. y is at least w more than x. y – x M w (c) y selebih-lebihnya w kali x. y is at most w times of x. y N wx (d) Maksimum y ialah w. The maximum of y is w. y N w (e) y kurang daripada x. y is less than x. y < x (f) Beza y dan x kurang daripada w. Difference of y and x is less than w. y – x < w (g) Hasil tambah x dan y lebih besar daripada w. The sum of x and y is greater than w. x + y > w (h) Minimum y ialah w. The minimum of y is w. y M w (i) y tidak kurang daripada x. y is not less than x. y M x y lebih besar daripada x. y is greater than x. y > x Contoh Info Digital 6.2 CONTOH

Matematik Tingkatan 4 Bab 6 74 2 Tuliskan ketaksamaan linear berdasarkan syarat-syarat yang dinyatakan dalam setiap situasi yang berikut. Write the linear inequalities based on the constraints stated in each of the following situations. SP: 6.2.1 TP4 Sederhana (a) Sebuah bakul mempunyai x biji oren dan y biji epal. A basket has x oranges and y apples. (i) Jumlah bilangan buah-buahan dalam bakul adalah sekurang-kurangnya 20 biji. The total number of fruits in the basket is at least 20. (ii) Beza antara bilangan epal dan bilangan oren adalah lebih daripada 3. The difference between the number of apples and the number of oranges is more than 3. (iii) Bilangan maksimum oren ialah 8. The maximum number of oranges is 8. (i) x + y M 20 (ii) y – x > 3 (iii) x N 8 (b) Ketua kampung akan menganjurkan suatu pesta makanan. Pesta makanan tersebut mesti ada x buah gerai makanan dan y buah gerai permainan. The village chief is going to organise a food fiesta. The food fiesta must have x food stalls and y game stalls. (i) Bilangan maksimum gerai ialah 40. The maximum number of stalls is 40. (ii) Bilangan gerai permainan mesti selebihlebihnya 3 kali bilangan gerai makanan. The number of game stalls must be at most 3 times the number of food stalls. (iii) Beza bilangan gerai makanan dan bilangan gerai permainan mesti tidak lebih daripada 10. The difference in the number of food stalls and the number of game stalls must not be more than 10. (i) x + y N 40 (ii) y N 3x (iii) x – y N 10 (c) Pemarkahan suatu permainan melibatkan dua jenis bola yang berlainan warna. Sebiji bola merah mewakili 5 markah dan sebiji bola hijau mewakili 2 markah. Jumlah markah bergantung kepada hasil tambah daripada dua orang pemain. Untuk memenangi sebuah hadiah, John dan James perlu: The scoring of a game involves balls of two different colours. A red ball represents 5 marks while a green ball represents 2 marks. The total marks depend on the sum from two players. To win a prize, John and James need to: (i) skor lebih daripada 50 markah, score more than 50 marks, (ii) dapat sekurang-kurangnya 20 biji bola, get at least 20 balls, (iii) dapat lebih bola merah daripada bola hijau. get more red balls than green balls. (i) 5m + 2b > 50 (ii) m + b M 20 (iii) m > b (d) Ahmad diberi RM25 untuk membeli x batang pen merah dan y batang pen biru. Syarat-syarat pembeliannya adalah seperti berikut: Ahmad is given RM25 to buy x red pens and y blue pens. The conditions of his purchase are as follows: (i) Pen biru mesti lebih daripada pen merah sekurang-kurangnya 6 batang. Blue pens must be more than red pens at least by 6. (ii) Bilangan minimum pen merah ialah 10. The minimum number of red pens is 10. (iii) Beza harga pen biru dan harga pen merah adalah kurang daripada RM2 jika harga sebatang pen merah dan sebatang pen biru masing-masing ialah RM1 dan RM1.50. The difference in the price of blue pen and the price of red pen is less than RM2 if the prices of a red pen and blue pen are RM1 and RM1.50 respectively. Tuliskan ketaksamaan linear bagi bilangan pen yang dapat dibeli oleh Ahmad. Write the linear inequality for the number of pens which can be bought by Ahmad. (i) y – x M 6 (ii) x M 10 (iii) 1.5y – x < 2 x + 1.5y N 25 3y – 2x < 4 2x + 3y N 50 Harga sebatang pensel ialah RM2 dan harga sebuah buku ialah RM5. Hassim hendak membeli x batang pen dan y buah buku. The price of a pencil is RM2 and the price of a book is RM5. Hassim wants to buy x pencils and y books. (i) Perbelanjaannya tidak melebihi RM50. His expenditure must not be more than RM50. (ii) Bilangan pensel yang dibeli mesti tidak kurang daripada 10 batang. The number of pencils bought must not be less than 10 pieces. (iii) Bilangan pensel mesti 2 kali lebih daripada bilangan buku. The number of pencils must be 2 times more than the number of books. (i) 2x + 5y N 50 (ii) x M 10 (iii) 2x > y Contoh CONTOH

Matematik Tingkatan 4 Bab 6 75 3 Tentukan rantau yang memuaskan ketaksamaan linear x + 2y N 10 dan 6x – y > 8 dengan titik-titik yang diberi dalam rantau berlainan. SP: 6.2.2 TP3 Sederhana Determine the region which satisfies the linear inequalities x + 2y N 10 and 6x – y > 8 with the given points in different region. Rantau Region Koordinat titik Coordinates of points x + 2y N 10 / 6x – y > 8 / P (1, 7) 1 + 2(7) = 15 15 M 10 6(1) – 7 = –1 –1 < 8 Q (a) (6, 6) 6 + 2(6) = 18 18 M 10 6(6) – 6 = 30 30 > 8 R (b) (3, 2) 3 + 2(2) = 7 7 N 10 6(3) – 2 = 16 16 > 8 S (c) (1, 3) 1 + 2(3) = 7 7 N 10 6(1) – 3 = 3 3 < 8 x y 2 2 4 4 6 6 8 8 10 O P S R Q Contoh (d) Rantau R memuaskan ketaksamaan linear x + 2y N 10 dan 6x – y > 8. Region R satisfies the linear inequalities x + 2y N 10 and 6x – y > 8. 4 Lorekkan rantau yang memuaskan sistem ketaksamaan linear yang berikut. SP: 6.2.3 TP4 Sederhana Shade the region which satisfies the following systems of linear inequalities. (a) 5y – 4x M 1, x + y M 3 dan/ and x < 2 1 5 – 1 4 O 2 3 3 x = 2 5y – 4x = 1 x + y = 3 x y (b) x N y, x + y M 6 dan/ and y < 5 x y O 6 6 5 x + y = 6 y = 5 x = y (c) 3x – y < 4 dan/ and 3x – 5y M –15 x y O 3 4 3 3x – 5y = –15 3x – y = 4 x N 3, y < x dan/ and x + y M 4 y x O 4 3 4 x + y = 4 x = 3 y = x Contoh CONTOH

Matematik Tingkatan 4 Bab 6 76 5 Lakar dan lorekkan rantau yang memuaskan setiap sistem ketaksamaan linear yang berikut. SP: 6.2.3 TP5 Sukar Sketch and shade the region which satisfies each of the following systems of linear inequalities. (a) x M 0, y M 0, 6 – 5y > x dan/ and 2y M 3x – 5 6 – 5y = x 2y = 3x – 5 6 5 5 3 O 6 x y (b) y M 0, 5x – y M 10, y > 2x – 10 dan/ and y < 8 5x – y = 10 y = 2x – 10 y = 8 O 8 2 5 x y (c) x M 0, y M 0, 2x – 2y > –4 dan/ and x – 5 N –y x – 5 = –y 2x – 2y = –4 O 2 5 5 x y 2x + 3y < 5, 3x – y M 3 dan/ and y M 0 5 2 5 3 2x + 3y = 5 3x – y = 3 O 1 x y Contoh 6 Nyatakan semua ketaksamaan linear yang mentakrifkan rantau sepunya dalam setiap sistem ketaksamaan linear yang berikut. SP: 6.2.3 TP5 Sederhana State all the linear inequalities which define the common region in each of the following systems of linear inequalities. (a) x + y = 6 x = 6 y = x 6 6 x O y x + y > 6 y N x x N 6 x = 3y y = 2 y = –x 2 x O y y > –x x N 3y y N 2 Contoh CONTOH

Matematik Tingkatan 4 Bab 6 77 (b) x + 2y = 8 2y = x y = 1 8 1 4 x O y x + 2y N 8 2y < x y M 1 (c) y = –x – 3 2y – x = 4 –3 –4 –3 2 O x y x N 0 2y – x < 4 y > –x – 3 7 Selain daripada x M 0 dan y M 0, tuliskan semua ketaksamaan linear yang memuaskan rantau berlorek yang berikut. Other than x M 0 and y M 0, write all linear inequalities which satisfy the following shaded region. SP: 6.2.3 TP5 Sukar 1 2 6 12 –1 (2, 5) O y x Persamaan / Equation : Kecerunan/ Gradient, m: 5 – (–1) 2 – 0 = 3 Maka/ Thus, y = 3x – 1 Persamaan / Equation : Kecerunan/ Gradient, m: – 6 12 = – 1 2 Maka/ Thus, y = – 1 2 x + 6 Ketaksamaan linear yang memuaskan rantau berlorek ialah y < 3x – 1 dan y N – 1 2 x + 6. The linear inequalities which satisfy the shaded region are y < 3x – 1 and y N – 1 2 x + 6. 2 3 1 3 3 10 10 (12, 9) x O y Persamaan / Equation : Kecerunan/ Gradient, m: – 10 10 = –1 Maka/ Thus, y = –x + 10 Persamaan / Equation : Kecerunan/ Gradient, m: 9 – 3 12 – 0 = 1 2 Maka/ Thus, y = 1 2 x + 3 Persamaan / Equation : Kecerunan/ Gradient, m: 9 – 0 12 – 3 = 1 Dari/ From (3, 0): 0 = 3 + c c = –3 Maka/ Thus, y = x – 3 Ketaksamaan linear yang memuaskan rantau berlorek ialah y N –x + 10, y < 1 2 x + 3 dan y > x – 3. The linear inequalities which satisfy the shaded region are y N –x + 10, y < 1 2 x + 3 and y > x – 3. Contoh CONTOH

Matematik Tingkatan 4 Bab 6 78 8 Selesaikan setiap masalah yang berikut. SP: 6.2.4 TP6 KBAT Sukar Solve each of the following problems. (a) Rajah di bawah menunjukkan satu sistem ketaksamaan linear pada suatu satah Cartes. The diagram below shows a system of linear inequalities on a Cartesian plane. y x O E C D B A 2y = 5x – 10 7x + 4y = 28 (i) Nyatakan dua buah segi tiga yang terbentuk pada satah Cartes. State two triangles formed on the Cartesian plane. (ii) Nyatakan semua ketaksamaan linear yang memuaskan bagi rantau kedua-dua segi tiga tersebut. State all the linear inequalities which satisfy the region of both triangles. (b) Sebuah restoran menjual nasi lemak dan nasi goreng. Harga sepinggan nasi lemak ialah RM3 dan harga sepinggan nasi goreng ialah RM4. Restoran tersebut menjual x pinggan nasi lemak dan y pinggan nasi goreng sehari. Syarat-syarat bagi jualan makanan adalah seperti berikut: A restaurant sells nasi lemak and fried rice. The price of a plate of nasi lemak is RM3 and the price of a plate of fried rice is RM4. The restaurant sells x plates of nasi lemak and y plates of fried rice in a day. The conditions of the food sales are as follows: I Jualan kedua-dua nasi tidak lebih daripada 150 pinggan. The sales of both rice are not more than 150 plates. II Jumlah pendapatan daripada jualan tidak kurang daripada RM480. The total income from the sales is not less than RM480. (i) Senaraikan semua ketaksamaan linear selain daripada x M 0 dan y M 0 yang mewakili jualan makanan. List all the linear inequalities other than x M 0 and y M 0 which represent the food sales. (ii) Lukis dan lorek rantau sepunya bagi sistem ketaksamaan linear tersebut. Draw and shade the common region for the system of linear inequalities. (i) ΔABE dan/ and ΔBCD (ii) Ketaksamaan linear dalam rantau ABE: Linear inequalities in region ABE: x M 0, 2y > 5x – 10 dan/ and 7x + 4y N 28 Ketaksamaan linear dalam rantau BCD: Linear inequalities in region BCD: y M 0, 7x + 4y N 28 dan/ and 2y < 5x – 10 (i) x + y N 150 y N –x + 150 3x + 4y M 480 y M – 3 4 x + 120 (ii) x y 150 150 120 O 160 y = –x + 150 y = – 3 4x + 120 x 0 150 y 150 0 x 0 160 y 120 0 CONTOH