Rujukan Custom-made Matematik (Dwibahasa) Tingkatan 2 ALAF SANJUNG SDN. BHD. (516756-V) Wisma Alaf Sanjung No. 23, Jalan Sungai Besi Indah 5/2, Taman Sungai Besi Indah, 43300 Seri Kembangan, Selangor Darul Ehsan. Tel : 03-8941 0411 / 03-8941 0611 Faks : 03-8941 0041 E-mel : [email protected] © Alaf Sanjung Sdn. Bhd. (516756-V) Semua hak cipta terpelihara. Sebarang bahagian dalam buku ini tidak boleh diterbitkan semula, disimpan dalam cara yang boleh dipergunakan lagi, ataupun dipindahkan, dalam sebarang bentuk atau cara lain tanpa kebenaran terlebih dahulu daripada Alaf Sanjung Sdn. Bhd. ISBN 978-629-7512-45-7 Dicetak oleh : Central Printing Low Ming Yike Alaf Sanjung Sdn Bhd
KANDUNGAN / CONTENTS Bab 1 : Pola dan Jujukan Patterns and Sequences 1 – 7 Bab 2 : Pemfaktoran dan Pecahan Algebra Factorisation and Algebraic Fractions 8 – 15 Bab 3 : Rumus Algebra Algebraic Formulae 16 – 18 Bab 4 : Poligon Polygon 19 – 22 Bab 5 : Bulatan Circles 23 – 31 Bab 6 : Bentuk Geometri Tiga Dimensi Three-Dimensional Geometrical Shapes 32 – 46 Bab 7 : Koordinat Coordinates 47 – 54 Bab 8 : Graf Fungsi Graphs of Functions 55 – 60 Bab 9 : Laju dan Pecutan Speed and Acceleration 61 – 65 Bab 10 : Kecerunan Garis Lurus Gradient of a Straight Line 66 – 69 Bab 11 : Transformasi Isometri Isometric Transformations 70 – 84 Bab 12 : Sukatan Kecenderungan Memusat Measures of Central Tendencies 85 – 90 Bab 13 : Kebarangkalian Mudah Simple Probability 91 – 99 Jawapan Alaf Sanjung Sdn Bhd
© Alaf Sanjung Sdn. Bhd. (516756-V) 1 Matematik T2 Bab 1 Pola dan Jujukan Chapter 1 Patterns and Sequences 1.1 Pola / Patterns A. Apakah pola? What is patterns? Pola ialah peraturan atau corak tertentu dalam senarai nombor atau objek. A pattern is a specific rule or pattern in a list of numbers or objects. Suatu pola dalam senarai nombor ditentukan dengan menambah, menolak, mendarab atau membahagi nombor sebelumnya. A pattern in a list of numbers is determined by adding, subtracting, multiplying or dividing the previous number. Suatu pola dalam objek ditentukan dengan memerhatikan susunan objek sebelumnya. A pattern in an object is determined by observing the order of the previous object. Sebagai contoh / For example, (1) ialah set nombor genap. is a set of even numbers. Set nombor genap bermula dengan 2 dan polanya ialah tambah 2 untuk mendapat setiap nombor yang berikutnya. The set of even numbers starts with 2 and the pattern is to add 2 to get each subsequent number. Nombor dalam set nombor genap boleh dibahagi tepat dengan 2. Numbers in a set of even numbers can be divided exactly by 2. (2) ialah set nombor ganjil. is a set of odd numbers. Set nombor ganjil bermula dengan 1 dan polanya ialah tambah 2 untuk mendapat setiap nombor yang berikutnya. The set of odd numbers starts with 1 and the pattern is to add 2 to get each subsequent number. Nombor dalam set nombor ganjil tidak boleh dibahagi tepat dengan 2. Numbers in a set of odd numbers cannot be divided exactly by 2. Pemahaman konsep pola Understanding of the concept of patterns Perihalkan pola bagi set nombor berikut. Describe the pattern for the following set of numbers. (a) 5, 11, 17, 23, 29, ... _______________________________ _______________________________ (b) 64, 60, 56, 52, 48, ... _______________________________ _______________________________ (c) –23, –69, –207, –621, ... _______________________________ _______________________________ (d) 2 401, 343, 49, 7, ... _______________________________ _______________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ 2, 4, 6, 8, 10, … 2, 4, 6, 8, 10, … +2 +2 +2 +2 +2 1, 3, 5, 7, 9, … +2 +2 +2 +2 +2 1, 3, 5, 7, 9, … Customise your own notes Alaf Sanjung Sdn Bhd
© Alaf Sanjung Sdn. Bhd. (516756-V) 2 Matematik T2 _______________________________________________________________________________________ _______________________________________________________________________________________ _______________________________________________________________________________________ _______________________________________________________________________________________ _______________________________________________________________________________________ (3) ialah set nombor Fibonacci. is a set of Fibonacci numbers. Bermula dengan 0, 1, 1 dan diikuti dengan menambah dua sebutan sebelumnya. Starts with 0, 1, 1 and followed by adding the previous two terms. (4) Rajah di bawah menunjukkan suatu Segi Tiga Pascal. The diagram below shows a Pascal’s Triangle. Semua baris bermula dan diakhiri dengan nombor 1. All the rows start and end with the number 1. Setiap nombor dalam segi tiga ialah hasil tambah dua nombor pada baris sebelumnya. Each number in the triangle is the sum of two numbers in the previous row. (5) Pola dalam set bentuk. Patterns in a set of shapes. Suatu pola dalam objek ditentukan dengan memerhatikan susunan objek sebelumnya. A pattern in an object is determined by observing the order of the previous object. (6) Pola dalam set huruf. Patterns in a set of letters. Suatu pola dalam huruf ditentukan dengan memerhatikan susunan huruf sebelumnya. A pattern in letters is determined by observing the order of the previous letter. Lengkapkan pola nombor. Complete the number pattern. (a) 0, 1, 1, , , , 8, 13, 21, ... (e) v, v, Q, R, v, , , R, , ... (b) 0, 2, 2, , , , 16, 26, 42, ... (f) (c) 0, 5, 5, , , , 40, 65, 105, ... (d) 0, 1, 1, 2, 3, 5, 8, … 0 + 1 0, 1, 1, 2, 3, 5, 8, … 1 + 2 3 + 5 1 + 1 2 + 3 1 1 1 1 1 1 1 1 2 1 3 3 4 6 4 ABooABooABooAB 5 + 8 Alaf Sanjung Sdn Bhd
© Alaf Sanjung Sdn. Bhd. (516756-V) 3 Matematik T2 Pemahaman konsep jujukan Understanding of the concept of sequences Tentukan sama ada urutan nombor yang berikut suatu jujukan atau bukan. Determine whether the following number sequence is a sequence or not. (a) 1, 3, 5, 7, 9, 11, … Jawapan / Answer : ___________________ (b) 1, 4, –2, 3, 5, 9, … Jawapan / Answer : ___________________ (c) 2, 3, 5, 7, 11, 13, … Jawapan / Answer : ___________________ (d) 1, 6, 24, 96, 288, 864, … Jawapan / Answer : ___________________ (e) 2, 6, 18, 54, 162, 486, … Jawapan / Answer : ___________________ (f) Jawapan / Answer : ___________________ _______________________________________________________________________________________ _______________________________________________________________________________________ _______________________________________________________________________________________ _______________________________________________________________________________________ _______________________________________________________________________________________ _______________________________________________________________________________________ _______________________________________________________________________________________ _______________________________________________________________________________________ Bab 1 Pola dan Jujukan Chapter 1 Patterns and Sequences 1.2 Jujukan / Sequences A. Apakah jujukan? What is sequences? Jujukan ialah suatu set nombor atau objek yang disusun mengikut suatu pola. A sequence is a set of numbers or objects that are arranged according to a pattern. Sebagai contoh / For example, (a) –5, 0, 5, 10, 15, … Jujukan sebab set nombor mengikut pola tertentu. Sequence because the set of numbers is according to a certain pattern. Pola jujukan ini ialah tambah 5 untuk mendapat setiap nombor yang berikutnya. The pattern of this sequence is to add 5 to get each subsequent number. (b) 17, 13, 10, 5, –5, … Bukan jujukan sebab set nombor tidak mengikut pola tertentu. Not a sequence because the set of numbers does not according to a certain pattern. –5, 0, 5, 10, 15, … +5 +5 +5 +5 +5 17, 13, 10, 5, –5, … –4 –3 –5 –10 –? Customise your own notes Alaf Sanjung Sdn Bhd
© Alaf Sanjung Sdn. Bhd. (516756-V) 4 Matematik T2 B. Mengenal pasti dan menghuraikan corak urutan, dan seterusnya melengkapkan dan melanjutkan urutan. Identifying and describing the pattern of a sequence, and then completing and extending the sequence. Suatu jujukan nombor boleh dibentuk dengan menambah, menolak, mendarab atau membahagi. A sequence of numbers can be formed by adding, subtracting, multiplying or dividing. Suatu pola dalam bentuk geometri atau objek ditentukan dengan memerhatikan susunan objek sebelumnya. A pattern in the form of geometry or an object is determined by observing the order of the previous object. Contoh 1 / Example 1 Lengkapkan urutan nombor berikut. Complete the following number sequence. 2, 7, , , 22, 27, … Penyelesaian / Solution : 2, 7, , , 22, 27, … 2, 7, , , 22, 27, … ○1 Kenal pasti pola. Identify the pattern. +5 +5 ○2 Tentukan pola dan lengkapkan. Determine the pattern and complete it. 12 17 Contoh 2 / Example 2 Tulis dua nombor yang seterusnya dalam jujukan berikut. Write the next two numbers in the following sequence. 3, 6, 12, 24, , , … Penyelesaian / Solution : 3, 6, 12, 24, , , … 3, 6, 12, 24, , , … ×2 ×2 ×2 ○1 Kenal pasti pola. Identify the pattern. ○2 Tentukan pola dan tuliskan. Determine the pattern and write. 48 96 Contoh 3 / Example 3 Lukis dua rajah yang seterusnya dalam jujukan berikut. Draw the next two diagrams in the following sequence. Penyelesaian / Solution : ________________________ ________________________ Pola Sequence Keempat Fourth Kelima Fifth Pertama First Kedua Second Ketiga Third _______________________________________________________________________________________ _______________________________________________________________________________________ _______________________________________________________________________________________ _______________________________________________________________________________________ _______________________________________________________________________________________ _______________________________________________________________________________________ ______________________________________________________________ Alaf Sanjung Sdn Bhd
© Alaf Sanjung Sdn. Bhd. (516756-V) 5 Matematik T2 Kenal pasti dan perihalkan pola Identify and describe the pattern Lengkapkan jujukan berikut. Complete the following sequence. (a) 36, 32, , , 20, 16, , , … (b) 10, 7, 4, , , –5, , , … (c) Ketiga / Third Kedua / Second Keempat / Fourth Kedua / Second Kedua / Second Kelima Fifth Keenam Sixth __________________________________ __ __________________________________ __ Pertama / First _______________________________________________________________________________________ _______________________________________________________________________________________ _______________________________________________________________________________________ _______________________________________________________________________________________ _______________________________________________________________________________________ _______________________________________________________________________________________ _______________________________________________________________________________________ _______________________________________________________________________________________ _______________________________________________________________________________________ _______________________________________________________________________________________ _______________________________________________________________________________________ _______________________________________________________________________________________ _______________________________________________________________________________________ _______________________________________________________________________________________ _______________________________________________________________________________________ _______________________________________________________________________________________ _______________________________________________________________________________________ _______________________________________________________________________________________ _______________________________________________________________________________________ _______________________________________________________________________________________ _______________________________________________________________________________________ _______________________________________________________________________________________ _______________________________________________________________________________________ _______________________________________________________________________________________ Alaf Sanjung Sdn Bhd
© Alaf Sanjung Sdn. Bhd. (516756-V) 6 Matematik T2 Bab 1 Pola dan Jujukan Chapter 1 Patterns and Sequences 1.3 Pola dan Jujukan / Patterns and Sequences A. Membuat generalisasi tentang pola suatu jujukan menggunakan nombor, perkataan dan ungkapan algebra. Making generalisation about the pattern of a sequence using numbers, words and algebraic expressions. Pola suatu jujukan menggunakan nombor Pattern of a sequence using numbers Pola suatu jujukan menggunakan perkataan Pattern of a sequence using words Pola suatu jujukan menggunakan ungkapan algebra Pattern of a sequence using algebraic expression 1 1 = 1 + 3(0) 1 +3 4 4 = 1 + 3(1) 1 +3 +3 7 7 = 1 + 3(2) 1 +3 +3 +3 10 10 = 1 + 3(3) 1 +3 +3 +3 +3 13 13 = 1 + 3(4) Buat generalisasi tentang pola bagi jujukan nombor dengan menggunakan nombor, perkataan dan ungkapan algebra. Make generalisations about patterns for number sequences using numbers, words and algebraic expressions. (a) Nombor / Numbers __________________________________ __________________________________ (b) Perkataan / Words __________________________________ __________________________________ __________________________________ __________________________________ (c) Ungkapan algebra / Algebraic expression Penyelesaian / Solution : 1, 4, 7, 10, 13, ... 1, 4, 7, 10, 13, … +3 5 +3 5 +3 5 +3 5 Pola ialah +3. / The pattern is +3. 1, 4, 7, 10, 13, ... Penyelesaian / Solution : Pola ialah menambah 3 kepada nombor sebelumnya. The pattern is adding 3 to the previous number. 1, 4, 7, 10, 13, ... Penyelesaian / Solution : 1 + 3n dengan keadaan / such that n = 0, 1, 2, 3, 4, … 10, 6, 2, –2, –6, … 10, 6, 2, –2, –6, … a 10 10 = 10 – 4(0) 10 –4 6 6 = 10 – 4(1) 10 10 10 Customise your own notes Alaf Sanjung Sdn Bhd
© Alaf Sanjung Sdn. Bhd. (516756-V) 7 Matematik T2 B. Menentukan suatu sebutan. Determining a term. Apakah sebutan? What is a term? Setiap unsur dalam suatu jujukan dikenali sebagai sebutan. Each element in a sequence is known as a term. Sebutan sesuatu jujukan dikenali sebagai sebutan ke-n. A term of a sequence is known as nth term. Sebutan ke-n nth term = Tn Sebutan Term Kedudukan sebutan Term position Jujukan / Sequence Sebutan ke-n / nth term 11, 22, 33, 44, …, n, … a T1, T2, T3, T4, …, Tn, … a Berdasarkan rajah di atas, Based on the diagram above, (a) Nyatakan sebutan ke-7. State the 7th term. Contoh / Example Diberi jujukan nombor / Given the number sequence: –34, –28, –22, –16, … (i) Nyatakan sebutan ke a enam bagi jujukan nombor itu. State the sixth term of the number sequence. (ii) Tentukan nombor 32 ialah sebutan yang keberapa. Determine the number 32 is located at which term. Penyelesaian / Solution : Penyelesaian / Solution : –34, –28, –22, –16, … +6 5 +6 5 +6 Maka / Thus : 5T5 = –16 + 6 = –10 T6 = –10 + 6 = –4 T7 = –4 + 6 = 2 T8 = 2 + 6 = 8 T9 = 8 + 6 = 14 T10 = 14 + 6 = 20 T11 = 20 + 6 = 26 T12 = 26 + 6 = 32 Maka, 32 ialah sebutan ke-12. Thus, 32 is the 12th term. Sebutan ke-6 ialah –4. The 6th term is –4. Pola / Pattern (b) Berapakah segi tiga yang ada apabila dibina dengan menggunakan 54 biji bola? How many triangles are there when built using 54 balls? ______________________________ ______________________________ ______________________________ ______________________________ ______________________________ ______________________________ ______________________________ ______________________________ ______________________________ ______________________________ ______________________________ ______________________________ Alaf Sanjung Sdn Bhd
© Alaf Sanjung Sdn. Bhd. (516756-V) 8 Matematik T2 Bab 2 Pemfaktoran dan Pecahan Algebra Chapter 2 Factorisation and Algebraic Fractions 2.1 Kembangan / Expansion Kembangan ungkapan algebra ialah hasil pendaraban satu atau dua ungkapan dalam kurungan. Expansion of algebraic expression is a product of one or two expressions in brackets. Mengembangkan adalah untuk membuang kurungan daripada ungkapan algebra. Expanding is to remove brackets from an algebraic expression. Kembangan dua ungkapan algebra dilaksanakan dengan mendarab setiap sebutan dalam ungkapan pertama dengan setiap sebutan dalam ungkapan kedua dalam tanda kurung. The development of two algebraic expressions is implemented by multiplying each term in the first expression by each term in the second expression in parentheses. Kembangkan setiap yang berikut. Expand each of the following. (a) 3(3 + x) (b) x(8 + y) (c) (5 + 3x)(2 + 5x) (d) (x + 4y)(7x + y) ________________________________________________ ________________________________________________ ________________________________________________ ________________________________________________ ________________________________________________ ___________________________________ ___________________________________ ________________________________________________ ______________________ Hukum Kalis Agihan Distributive Law × ( + ) = × + × = + Hasil darab + dikenali sebagai kembangan bagi pendaraban × ( + ). The product of + is known as the expansion of the multiplication of × ( + ). Ungkapan algebra ialah gabungan nombor, pemboleh ubah dan tanda operasi ( +, –, ×, ÷ ) untuk mewakili suatu situasi. An algebraic expression is a combination of numbers, variables and operations ( +, –, ×, ÷ ) to represent a situation. Secara umumnya, In general, () ( + ) = 2 + () ( + ) = + () ( + )( + ) = + + + Customise your own notes Alaf Sanjung Sdn Bhd
© Alaf Sanjung Sdn. Bhd. (516756-V) 9 Matematik T2 ________________________________ ________________________________ ________________________________ ________________________________ ________________________________ ________________________________ ________________________________ ________________________________ ________________________________ ________________________________ ________________________________ ________________________________ ________________________________ ________________________________ ________________________________ ________________________________ ____ Contoh / Example Permudahkan ungkapan. Simplify the expressions. 6(3a – bc) – (5a + 4bc) = 18a – 6bc – 5a – 4bc = 18a – 5a – 6bc – 4bc = 13a – 10bc ○1 Buang tanda kurung dengan mendarab setiap sebutan di dalamnya. Remove the brackets by multiplying each term inside. ○2 Susun sebutan serupa. Arrange the like term. ○3 Permudahkan. Simplify. Bagaimana mengingat urutan yang betul? How to remember the correct order? Lakukan kurungan dahulu; kemudian kuasa atau indeks; kemudian × atau ÷ (dari kiri ke kanan); akhir sekali + atau – (dari kiri ke kanan). Do brackets first; then power or indices; then × or ÷ (from left to right); finally + and – (from left to right). B I D M A S Tip Permudahkan ungkapan. Simplify the expressions. 3m(m – 4n) – 8m(5m + n) Penyelesaian: Solution: • (+a) × (+b) = +ab • (+a) × (–b) = –ab • (–a) × (+b) = –ab • (–a) × (–b) = +ab Tip Alaf Sanjung Sdn Bhd
© Alaf Sanjung Sdn. Bhd. (516756-V) 10 Matematik T2 Bab 2 Pemfaktoran dan Pecahan Algebra Chapter 2 Factorisation and Algebraic Fractions 2.2 Pemfaktoran / Factorisation Pemfaktoran ialah proses mengenal pasti faktor sebutan dan ungkapan algebra. Factorisation is the process of identifying the term and algebraic expression factor. Pemfaktoran ialah proses songsangan kepada kembangan. Factorisation is an inverse process of expansion. Kita boleh memfaktorkan suatu ungkapan algebra dengan kaedah pendaraban silang. We can factorise an algebraic expression by the method of cross multiplication. Pemfaktoran juga boleh dilakukan bagi ungkapan algebra dalam bentuk berikut. Factorisation can also be done for algebraic expressions in the following forms. Kembangan Expansion ( + 2)( + 3) = 2 + 3 + 2 + 6 = 2 + 5 + 6 Pemfaktoran Factorisation 2 + 5 + 6 = ( + 2)( + 3) Faktor / Factor + 2 = 2 + 2 + 2 2 + 2 + 2 = + 2 − 2 = 2 − 2 + 2 2 − 2 + 2 = − 2 + − = 2 − 2 2 − 2 = + − Contoh / Example Kaedah pendaraban silang / Method of cross multiplication × − − −1 × 21 1 × −21 −3 × 7 3 × −7 (a) Faktorkan / Factorise: − − Langkah / Step ○1 Langkah / Step ○2 Darabkan secara mencancang. Multiply vertically. Maka / Thus, 2 −4 − 21 = +3 − 7 Customise your own notes 3 + −7 = −4 + 3 + 3 − 7 − 7 2 − 21 − 4 Alaf Sanjung Sdn Bhd
© Alaf Sanjung Sdn. Bhd. (516756-V) 11 Matematik T2 Faktorkan setiap ungkapan berikut. Factorise each of the following expressions. (a) 2 − 100 Penyelesaian / Solution: (b) 121 2 − 1 Penyelesaian / Solution: (c) 2 2 + 11 + 12 Penyelesaian / Solution: _________________________________ _________________________________ _________________________________ _________________________________ _________________________________ _________________________________ _________________________________ _________________________________ _________________________________ _________________________________ _________________________________ _________________________________ _________________________________ _________________________________ _________________________________ _________________________________ _________________________________ _________________________________ _________________________________ _________________________________ _________________________________ _________________________________ _________________________________ _________________________________ _________________________________ _________________________________ _________________________________ _________________________________ ____ Kaedah ungkapan algebra / Method of algebraic expressions (b) Faktorkan / Factorise: − − = 4 2 − 5 2 = 4 +5 4 − 5 Kedua-dua sebutan ialah kuasa dua sempurna. Both terms are perfect squares. Operasi: (–) Operation: (–) Alaf Sanjung Sdn Bhd