Full Colour Pages! Mathematics Chiam S.P. & Dr. Chiang K.W. & Samantha Neo QR Code PELANGI Form 1.2.3 KSSM KC117234
Mathematics Form 1 Chapter 1 Rational Numbers Form 1 1 Positive Integers 1, 2, 3, … Zero, 0 Negative Integers …, –3, –2, –1 Positive Fractions 1 2 , 3 2 Negative Fractions – 1 4 , – 5 2 Positive Decimals 0.5, 2.7 Negative Decimals –1.9, –3.25 Integers Fractions Decimals Rational Numbers Laws of Basic Arithmetic Operations Commutative Law a + b = b + a a × b = b × a Associative Law (a + b) + c = a + (b + c) (a × b) × c = a × (b × c) Distributive Law Identity Law a × (b + c) = a × b + a × c a × (b − c) = a × b − a × c a + 0 = a a × 0 = 0 a × 1 = a a + (−a) = 0 a × 1 a = 1 CHAPTER RATIONAL NUMBERS Form 1 1 CONCEPT MAP Learning Area: Numbers and Operations 1
Mathematics Form 1 Chapter 1 Rational Numbers Form 1 2 1.1 Integers 1. Our real-life situation often involves positive and negative numbers. (a) Positive number is greater than zero and written with a symbol ‘+’. For example, +1, + 1 2 and +2.5. (b) Negative number is less than zero and written with a symbol ‘−’. For example, −2, − 1 2 and −7.25. 2. The table below shows a few examples that can be represented using positive and negative numbers. Situation Positive numbers Negative numbers Movement To the right To the left Upwards Downwards Distance Above sea level Below sea level Business Profit Loss Temperature Higher than 0°C Lower than 0°C 3. Integer is a group of numbers involving positive whole numbers, negative whole numbers and zero, 0. (a) Positive integer is greater than zero. For example, +1, +13 and +29. (b) Negative integer is less than zero. For example, −2, −34 and −625. 4. Integers can be represented on a horizontal number line or a vertical number line. –3 –2 –1 0 1 2 3 Value increasing Value decreasing Negative integer Positive integer –3 –2 –1 0 1 2 3 Value increasing Value decreasing 5. On the number line, the value of integers which are on the right or above the number line, is greater than the value of integers on the left or below the number line.
Mathematics Form 1 Chapter 1 Rational Numbers Form 1 3 Example 1 Represent each of the following situation using positive numbers or negative numbers. (a) Amir dives 5.5 m into the sea. (b) Puan Salmah gains 90 sen from each pack of fried noodles sold. (c) Rahim pays RM200 using his credit card. Solution (a) −5.5 (b) +90 or 90 (c) −200 Example 2 Determine whether each of the following is an integer or a non-integer. 13, −2.5, −36, 7 4 , −4.50, −90 Solution Integer: 13, –36, −90 Non-integer: −2.5, 7 4 , – 4.50 Example 3 List all the integers from −3 to 6. Solution −3, −2, −1, 0, 1, 2, 3, 4, 5, 6 CAUTION ! Zero, 0 is an integer. Example 4 Determine and mark the position of each of the following integers on a number line. −8, 6, −5, 1, −2 Solution –8 –5 –2 1 6 Example 5 Compare the following integers and arrange in descending order. −7, 6, −3, 4, 1, −2 Solution 6, 4, 1, −2, −3, −7 1. Addition (a) of a positive integer can be represented on a number line by moving from left to right. (b) of a negative integer can be represented on a number line by moving from right to left. –3 –2 Addition of positive integer Addition of negative integer –1 0 1 Basic Arithmetic Operations Involving Integers 1.2
Mathematics Form 1 Chapter 1 Rational Numbers Form 1 4 2. Subtraction (a) of a positive integer can be represented on a number line by moving from right to left. (b) of a negative integer can be represented on a number line by moving from left to right. –3 –2 Subtraction of positive integer Subtraction of negative integer –1 0 1 3. Sign of the product of integers: (+) × (+) = (+) (+) × (−) = (−) (−) × (+) = (−) (−) × (−) = (+) 4. Sign of the quotient of integers: (+) ÷ (+) = (+) (+) ÷ (−) = (−) (−) ÷ (+) = (−) (−) ÷ (−) = (+) 5. Steps to calculate combined operations of integers are as follows. Brackets ( ) × or ÷ Start from left to right + or − Start from left to right 6. Laws of basic arithmetic operations. (a) Commutative Law a + b = b + a a × b = b × a (b) Associative Law (a + b) + c = a + (b + c) (a × b) × c = a × (b × c) (c) Distributive Law a × (b + c) = a × b + a × c a × (b − c) = a × b − a × c (d) Identity Law a + 0 = a a × 0 = 0 a × 1 = a a + (−a) = 0 a × 1 a = 1 Example 6 Calculate each of the following. (a) 2 + (+4) (b) 3 + (−2) (c) −1 − (+4) (d) −4 − (−3) Solution (a) 2 + (+4) = 2 + 4 = 6 (b) 3 + (−2) = 3 – 2 = 1 2 3 4 5 6 Move 4 units to the right 1 2 3 Move 2 units to the left (c) −1 − (+4) = −1 – 4 = −5 (d) −4 − (−3) = −4 + 3 = −1 Example 7 Solve each of the following. (a) 2 × (+4) (b) 3 × (−2) (c) −9 ÷ (+3) (d) −12 ÷ (−3) –5 –4 –3 –2 –1 Move 4 units to the left –4 –3 –2 –1 Move 3 units to the right
Mathematics Form 1 Chapter 1 Rational Numbers Form 1 5 Solution (a) 2 × (+4) = 2 × 4 = 8 (b) 3 × (−2) = − (3 × 2) = − 6 (c) −9 ÷ (+3) = −(9 ÷ 3) = –3 (d) −12 ÷ (−3) = +(12 ÷ 3) = 4 Example 8 Solve each of the following. (a) −2 × (−3 + 5) (b) –12 + (–24) –8 – (–2) Solution (a) −2 × (−3 + 5) = −2 × 2 = −4 (b) –12 + (–24) –8 – (–2) = –36 –6 = 6 Example 9 Solve each of the following using efficient computations. (a) 25 × 120 × 4 (b) 8 × 2 060 Solution (a) 25 × 120 × 4 = 120 × 25 × 4 = 120 × (25 × 4) = 120 × 100 = 12 000 (b) 8 × 2 060 = 8 × (2 000 + 60) = 8 × 2 000 + 8 × 60 = 16 000 + 480 = 16 480 Example 10 A diver wants to dive 60 m below the sea level. If he can dive 6 m into the sea in 5 minutes, calculate the time taken to reach 60 m below the sea level. Solution 60 ÷ 6 × 5 = 10 × 5 = 50 minutes He took 50 minutes to reach 60 m below the sea level. Commutative Law Associative Law Distributive Law 1.3 Positive and Negative Fractions 1. Fractions can be represented on a horizontal number line or a vertical number line. 0 Value increasing Value decreasing Negative fraction Positive fraction 4 3 – 2 1 – 4 1 – 4 1 2 1 4 3 2. Like integers, positive fractions are fractions greater than zero, whereas negative fractions are fractions less than zero.
Mathematics Form 1 Chapter 1 Rational Numbers Form 1 6 Example 11 Compare and arrange the following fractions in descending order, and represent on a number line. 1 3 , – 2 3 , – 1 3 , 1 2 , 3 4 Solution Descending order: 3 4 , 1 2 , 1 3 , – 1 3 , – 2 3 Number line: 3 2 – 3 1 – 3 1 2 1 4 3 TIPS The order of operations with fractions. Brackets ( ) × or ÷ Start from left to right + or – Start from left to right Example 12 Calculate each of the following. (a) 5 6 + 1– 2 3 2 × 1– 3 142 (b) 4 7 ÷ 1– 11 12 + 3 4 2 Solution (a) 5 6 + 1– 2 3 2 × 1– 3 142 Perform × operation first = 5 6 + 6 42 = 35 + 6 42 = 41 42 (b) 4 7 ÷ 1– 11 12 + 3 4 2 = 4 7 ÷ 1 –11 + 9 12 2 = 4 7 ÷ 1 –2 122 = 4 7 × 1 12 –2 2 = – 24 7 HOTS Example 1 Zuraidah has 2 litres of cooking oil. Salmah gives 1 4 of her cooking oil to Zuraidah. Zuraidah has 3.5 litres of cooking oil after giving 3 8 litre of cooking oil to Mila. Calculate Salmah’s initial amount of cooking oil. Solution Let Salmah’s intial amount of cooking oil = S litres 12 + 1 4 × S 2 − 3 8 = 3.5 12 + 1 4 × S 2 = 3.5 + 3 8 1 4 × S = 3.5 + 3 8 − 2 Hence, S = 13.5 + 3 8 – 22 ÷ 1 4 = 15 8 ÷ 1 4 = 15 8 × 4 = 7 1 2 Salmah’s initial amount of cooking oil is 7 1 2 litres. VIDEO Perform operation in brackets first
Mathematics Form 1 Chapter 1 Rational Numbers Form 1 7 2. Like integers, positive decimals are decimals greater than zero, whereas negative decimals are decimals less than zero. 1. Decimals can also be represented on a horizontal number line or a vertical number line. Value increasing Value decreasing Negative decimal Positive decimal –0.3 –0.2 –0.1 0 0.1 0.2 0.3 Example 13 Compare and arrange the following decimals in ascending order. −5.4, 5.7, 5.342, 5.197, −5.91 Solution −5.91, −5.4, 5.197, 5.342, 5.7 Example 14 State the values of P and Q in the number line below. –0.32 P –0.08 0.04 Q Solution P = −0.2 Q = 0.16 Example 15 Calculate each of the following. (a) 2.54 + (−2.57) – (−2.9) (b) −0.4 × (6.15 – 0.6 ÷ 0.2) Solution (a) 2.54 + (−2.57) – (−2.9) = 2.54 – 2.57 + 2.9 = −0.03 + 2.9 = 2.87 (b) −0.4 × (6.15 – 0.6 ÷ 0.2) = −0.4 × (6.15 – 3) = −0.4 × 3.15 = −1.26 TIPS The order of operations with decimals. Brackets ( ) × or ÷ Start from left to right + or – Start from left to right HOTS Example 2 The stock price of a company dropped 55 sen at a moment. The price then increased by 35 sen for two consecutive days. If the current price is RM5.60, find the initial stock price of the company. Solution 5.60 − 2 × 0.35 − (−0.55) = 5.60 − 0.7 + 0.55 = 4.9 + 0.55 = 5.45 Initial stock price of the company is RM5.45. Perform × operation first 1.4 Positive and Negative Decimals Perform ÷ operation in the brackets first
Mathematics Form 1 Chapter 1 Rational Numbers Form 1 8 1. Rational numbers are numbers that can be written in fraction form, which is a b , such that a and b are integers and b ≠ 0. Alternative Method (a) 4 + 1–2 3 4 2 × (–2.5) = 4 + 1–2 3 4 2 × 1–2 1 2 2 = 4 + 1– 11 4 2 × 1– 5 2 2 = 4 + 55 8 = 32 + 55 8 = 87 8 = 10 7 8 (b) –0.4 × 1 7 8 – 0.62 ÷ 8 = – 4 10 × 1 7 8 – 6 10 2 ÷ 8 = – 4 10 × 1 35 – 24 40 2 ÷ 8 = – 2 5 × 11 40 ÷ 8 = – 11 100 × 1 8 = – 11 800 TIPS The order of operations with rational numbers. Brackets ( ) × or ÷ Start from left to right + or – Start from left to right Example 16 Determine whether the following numbers are rational numbers. (a) 5 (b) 2 2 3 (c) −0.72 Solution (a) 5 = 5 1 5 is a rational number. (b) 2 2 3 = 8 3 2 2 3 is a rational number. (c) −0.72 = – 72 100 = – 18 25 –0.72 is a rational number. Example 17 Calculate each of the following. (a) 4 + 1–2 3 4 2 × (–2.5) (b) –0.4 × 1 7 8 – 0.62 ÷ 8 Solution (a) 4 + 1–2 3 4 2 × (–2.5) = 4 + (–2.75) × (–2.5) = 4 + 6.875 = 10.875 Express it to improper fraction. Express it to simplest fraction. 1.5 Rational Numbers
Mathematics Form 1 Chapter 1 Rational Numbers Form 1 9 Example 18 Rosmah has 16 boxes of gifts that need to be tied with ribbon. Each box needs 2 1 4 m of ribbon. Hence, she bought a 50 m long ribbon and the remaining of the ribbon is cut into 8 equal parts for other purposes. Calculate the length, in m, of each ribbon that has been cut. (Give your answer in decimals.) Solution 150 −16 × 2 1 4 2 ÷ 8 = 150 −16 × 9 4 2 ÷ 8 = (50 − 36) ÷ 8 = 14 ÷ 8 = 1.75 Thus, the length of each ribbon that has been cut is 1.75 m. HOTS Example 3 The diagram below shows the positions of Saleha and Salim from a point O. Saleha 4.5 m 9.25 m O Salim If Saleha and Salim move 4 m to the left and then 0.5 m to the right respectively, who will be the one further from point O? Solution Saleha: −4.5 − 4 + 0.5 = −8.5 + 0.5 = −8 m Salim: 9.25 − 4 + 0.5 = 5.25 + 0.5 = 5.75 m Thus, Saleha is further from point O compared to Salim. Section A 1. Which of the following is not a rational number? A 0.2 B 3.141592654… C 1.92 D –1 1 2 2. Which of the following integers, is arranged in descending order? A −101, −1, 1, 10 B −110, −101, 11, 1 C 1 , −1, −10, −100 D −111, −11 , −1 , −1.1 3. Wei Ming has 125 books in his bookshelves. He donates 3 5 of the books to an orphanage. Calculate the number of books left in his bookshelves. A 25 C 75 B 50 D 105 4. The diagram below shows a number line. –12 X 0 6 Y Calculate the value of X + Y. A -9 C 3 B 0 D 9 UASA PRACTICE
Mathematics Form 1 Chapter 1 Rational Numbers Form 1 10 Section B 5. (a) The diagram below shows a number line. –1.25 1 0 1 2 – 1 2 1 4 – 3 4 Based on the number line, state the positive integers and negative integers. (b) The diagram below shows some numbered cards. –5.9 –7 –8 1 3 5 5 –1 3.4 2 7 Find the difference between the largest integer value and the smallest integer value. Section C 6. (a) (i) The diagram below shows some balls labelled with numbers. –4 –13 9 5 Use the numbers above to fill in the empty space below to arrange integers in descending order. 7, , -3, , -12 (ii) The diagram below shows a number line. X –0.8 –0.4 Y 0.4 State the values of X and Y. (b) In the following diagram, fill in the empty space with the correct number in order to obtain the value as given in the circle. + 138 19 × 304 ÷ 3 (c) 3 5 of the students in a class are girls. 1 3 of the girls wear white shoes. If there are 9 girls who wear white shoes, calculate the number of boys in the class. HOTS Analysing 7. (a) Complete the empty boxes below. –16 – (–14) –2 = –16 14 –2 = –2 = (b) Ahmad gives RM80.80 to his 2 sons, Naim and Aiman equally. Aiman keeps all his money whereas Naim spends 3 4 of the money and keeps the rest. Pak Osman gives Naim RM22 and he keeps all the money. Calculate the total amount of money kept by Naim and Aiman. (c) Fisherman A rides a boat from a lighthouse to north for 12 m and rides back to south for 27 m. Fisherman B rides a boat from the lighthouse to south for 21 m and rides back to north to meet fisherman A. HOTS Analysing (i) Calculate the distance needed by fisherman B to meet fisherman A. (ii) State the distance of fisherman B from the lighthouse when he meets fisherman A.
Form 2 113 Repeated design or rule Ordered list Example: • Even numbers 2, 4, 6, 8, … +2 +2 +2 • Odd numbers 1, 3, 5, 7, … +2 +2 +2 • Pascal’s Triangle 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 • Fibonacci Numbers 1, 1, 2, 3, 5, … Example: 2, 5, 8, 11, 14, … 2, 4, 8, 16, 32, … nth term in a sequence can be written as Tn . Patterns in a sequence can be determined by addition, subtraction, multiplication or division of the previous number. Patterns Sequences Patterns and Sequences CHAPTER PATTERNS AND SEQUENCES Form 2 1 CONCEPT MAP Learning Area: Number and Operations 113
Mathematics Form 2 Chapter 1 Patterns and Sequences Form 2 114 1.1 Patterns 1. Pattern is defined as repeated design. 2. Number pattern is the arrangement of numbers through the operation of addition, subtraction, multiplication or division. 3. Examples of number patterns: (a) Even numbers (b) Odd numbers (c) Pascal’s Triangle (d) Fibonacci Numbers 4. 2, 4, 6, 8, 10, … +2 +2 +2 +2 Pattern for even numbers as above is adding 2 to the preceding number. 5. 23, 21, 19, 17, 15, … –2 –2 –2 –2 Pattern for odd numbers as above is subtracting 2 from the preceding number. 6. Pascal’s Triangle is a number pattern with geometrical arrangement in a triangle shape. 7. Every row in the Pascal’s Triangle starts and ends with number 1. The other number is obtained by adding two numbers from the preceding row. 1 1 1 1 1 1 1 1 5 10 10 5 1 4 6 4 3 3 1 2 1 + + + + + + + + + + + + + + + 8. Number pattern 1, 1, 2, 3, 5, 8, … is known as Fibonacci Numbers. 9. For Fibonacci Numbers, every number after the second number is obtained by adding two previous numbers in the sequence. 1, 1, 2, 1+1 3, 1+2 5, 2+3 8, 3+5 … 10. Apart from the number sequences, patterns can be found in the list of shapes, alphabets and objects. 11. Patterns for the list of shapes can be determined by observing the arrangement of preceding shapes. Example 1 The diagram below shows a Pascal’s Triangle. Find the values of X, Y and Z. 1 1 X 1 2 1 1 3 Y 1 1 Z 6 4 1 Solution 1 + X = 2 X = 2 – 1 X = 1 2 + 1 = Y Y = 3 1 + 3 = Z Z = 4
Mathematics Form 2 Chapter 1 Patterns and Sequences Form 2 115 Example 2 Describe the pattern of the following number sequence. −3, 6, −12, 24, … Solution −3, 6, −12, 24, … ×(–2) ×(–2) ×(–2) Multiply the preceding number with −2. TIPS Pattern in a number sequence can be identified by addition, subtraction, multiplication or division with preceding number. Example 3 Complete the following Fibonacci Numbers sequence. 1, 1, 2, ___, 5, ___, 13, ___, ___ Solution 1, 1, 2, 3 , 5, 8 , 13, 21 , 34 1+2 3+5 8+13 13+21 Example 4 The diagram above shows a list of shapes. What is the next shape? Sketch the shape. Solution The next shape is a triangle. Repeat alternatively Example 5 A E I O U A E I O U A E I The diagram above shows a list of alphabets. What are the next three letters? Solution The list of alphabets starts with A, E, I, O, U and is repeated. Hence, the next three letters are O U A Example 6 Determine whether each set of the numbers is a sequence. (a) 40, 50, 60, 70, 80, … (b) 2, 6, 18, 36, 40, … (c) 64, 32, 16, 8, 4, … Solution (a) 40, 50, 60, 70, 80, … +10 +10 +10 +10 The number set above is a sequence because its arrangement follows a pattern, that is adding 10 to the previous number. 1.2 Sequences 1. Sequence is a list of numbers or objects which has a certain pattern. 2. Sequence can be extended by determining its pattern beforehand.
Mathematics Form 2 Chapter 1 Patterns and Sequences Form 2 116 (b) 2, 6, 18, 36, 40, … ×3 ×3 ×2 +4 The number set above is not a sequence because its arrangement does not follow a pattern. (c) 64, 32, 16, 8, 4, … ÷2 ÷2 ÷2 ÷2 The number set above is a sequence because its arrangement follows a pattern, that is dividing the preceding number by 2. Example 7 Complete the following number sequences. (a) 20, 29, 38, 47, ___, ___, ... (b) ___, 60, 56, ___, 48, 44, ... (c) 4, ___, 36, 108, 324,___, ... (d) 128, 64, ___, 16, ___, 4, ... Solution (a) 20, 29, 38, 47, 56 , 65 , ... +9 +9 +9 +9 +9 (b) 64 , 60, 56, 52 , 48, 44, ... −4 −4 −4 −4 −4 (c) 4, 12 , 36, 108, 324, 972 , ... ×3 ×3 ×3 ×3 ×3 (d) 128, 64, 32 , 16, 8 , 4, ... ÷2 ÷2 ÷2 ÷2 ÷2 Example 8 State the patterns for the following sequences and extend the sequences by giving the next two numbers in the sequences. (a) 5, 14, 23, 32, 41, ... (b) 1, 3, 9, 27, 81, ... (c) 90, 80, 70, 60, 50, ... (d) 2 000, 1 000, 500, ... Solution (a) 5, 14, 23, 32, 41, 50, 59, ... +9 +9 +9 +9 The pattern for the sequence is adding 9 to the previous number. (b) 1, 3, 9, 27, 81, 243, 729, ... ×3 ×3 ×3 ×3 The pattern for the sequence is multiplying the previous number by 3. (c) 90, 80, 70, 60, 50, 40, 30, ... −10 −10 −10 −10 The pattern for the sequence is subtracting 10 from the previous number. (d) 2 000, 1 000, 500, 250, 125, ... ÷2 ÷2 The pattern for the sequence is dividing the previous number by 2. 1. Numbers arranged in a certain pattern is known as a sequence. 2. Pattern for a certain sequence can be ascertained by addition, subtraction, multiplication or division of the 1.3 Patterns and Sequences previous number in the number sequence. 3. Pattern for a certain sequence can be generalised in terms of numbers, words and algebraic expressions.
Mathematics Form 2 Chapter 1 Patterns and Sequences Form 2 117 Example 9 Represent the pattern for the following sequence by using number. 1, 2, 4, 8, 16, ... Solution 1, 2, 4, 8, 16, ... ×2 ×2 ×2 ×2 The pattern is ×2. Example 10 State the pattern for the number sequence 2 000, 400, 80, 16, … in words. Solution 2 000, 400, 80, 16, ... ÷5 ÷5 ÷5 The pattern is dividing the previous number by 5. Example 11 Determine the algebraic expressions to represent the following sequences. (a) 8, 11, 14, 17, 20, ... (b) 90, 85, 80, 75, 70, ... Solution (a) 8, 11, 14, 17, 20, ... +3 +3 +3 +3 8 = 8 + 3(0) 11 = 8 + 3(1) 14 = 8 + 3(2) 17 = 8 + 3(3) 20 = 8 + 3(4) The pattern is 8 + 3n where n = 0, 1, 2, 3, 4, ... . (b) 90, 85, 80, 75, 70, ... −5 −5 −5 −5 90 = 90 − 5(0) 85 = 90 − 5(1) 80 = 90 − 5(2) 75 = 90 − 5(3) 70 = 90 − 5(4) The pattern is 90 – 5n where n = 0, 1, 2, 3, 4, ... . Example 12 Find the 5th term for the following number sequences. (a) 1, 8, 15, ... (b) 10, 9, 8, ... (c) 80, 40, 20, ... Solution (a) 1, 8, 15, (15 + 7) , (22 + 7) +7 +7 5th term The 5th term is 29. (b) 10, 9, 8, (8 − 1) , (7 − 1) –1 –1 5th term The 5th term is 6. (c) 80, 40, 20, ... ÷2 ÷2 T1 = 80 T2 = 40 T3 = 20 T4 = 20 ÷ 2 = 10 T5 = 10 ÷ 2 = 5 The 5th term is 5.
Mathematics Form 2 Chapter 1 Patterns and Sequences 118 Example 13 Determine which term is 64 in the sequence –2, 4, –8, ... . Solution –2, 4, –8, ... ×(–2) ×(–2) T1 = −2 T4 = 16 T2 = 4 T5 = −32 T3 = −8 T6 = 64 64 is the 6th term. VIDEO General term of a number sequence Example 14 Raden sold a computer branded AXYTECH for RM1 500 on Monday, two computers of same brand on Tuesday and three on Wednesday. If the sales pattern continues, what are the total sales, in RM, of Raden on Thursday? Solution Pattern is determined beforehand to find the total sales of Raden on Thursday. Pattern: Add 1 500 to the previous number. Day Number of Computers Sold Total Sales (RM) Monday 1 1 500 Tuesday 2 3 000 Wednesday 3 4 500 Thursday 4 6 000 Hence, the total sales of Raden on Thursday is RM6 000. Section A 1. 2, 9, 16, 23, 30 Determine the pattern of the number sequence above. A Add 7 to the previous number. B Minus 7 from the previous number. C Multiply the previous number by 7. D Divide the previous number by 7. 2. The diagram below shows a part of Pascal’s Triangle. 1 1 1 1 P 3 1 1 2 1 1 4 Q 4 1 Calculate the value of P + Q. A 3 C 7 B 6 D 9 Form 2 Write a suitable algebraic expression to represent the sequence 9, 12, 15, 18, ... Quiz Quiz UASA PRACTICE
Mathematics Form 2 Chapter 1 Patterns and Sequences Form 2 119 3. Which of the following patterns is a sequence? A 2.5, 3.5, 4.5, 7.5, … B 45, 35, 15, 5, … C 7, 27, 27, 77, ... D 3.0, 2.8, 2.6, 2.4, ... 4. The table below shows a pattern using squares. First Second Third State the number of coloured squares of the fourth pattern. A 12 B 14 C 16 D 18 Section B 5. (a) Match the number sequences with the correct pattern. Number sequence Pattern 2, 5, 10, 17, … • • n3 , n = 1, 2, 3, 4, … • n2 + 1, n = 1, 2, 3, 4, … 1, 3, 5, 7, … • • 3n + 1, n = 1, 2, 3, 4, … • 2n – 1, n = 1, 2, 3, 4, … (b) Draw the next object of each of the following patterns. (i) (ii) a b g d π a b g d Section C 6. (a) Yong Jin will be having assessment test in a week. He plans to increase the duration time of revision starts with 25 minutes and additional 5 minutes every day. (i) List the number sequence of the duration of time, in minutes, of Yong Jin’s revision in the week. (ii) Write the number sequence using words. (b) The diagram below shows a number sequence. 1 3 , 2 5 , 3 7 , 4 9 , 5 11 (i) Write the algebraic expression of the number sequence. (ii) Hence, find the 10th term of the sequence. (c) Norain donated a part of her money to Yayasan Azwa every month for 6 months. Within the 6 months, she added RMk to the amount donated in previous month. She donated RM100 for the first month and RM350 for the sixth month. Find the value of k. HOTS Analysing
Form 3 228 Zero Index a0 = 1, a ≠ 0 Negative Indices a–n = 1 an , a ≠ 0 Fractional Indices a 1 —n = n a , a ≠ 0 a m —n = (am) 1 —n = (a 1 —n ) m a m —n = n am = (n a ) m Law of Indices am × an = am + n am ÷ an = am – n (am)n = amn (am × bn)q = amq × bnq 1 am bn 2 q = amq bnq Index Base n factors Index Form an = a × a × a × … × a CHAPTER INDICES Form 3 1 CONCEPT MAP Learning Area: Number and Operations 228
Mathematics Form 3 Chapter 1 Indices Form 3 229 1. A number multiplied by itself repeatedly can be written in index notation as follows. an = a × a × a × … × a n factors 1.1 Index Notation Example 1 Express 7 × 7 × 7 × 7 × 7 in index notation. Solution 7 × 7 × 7 × 7 × 7 = 75 1442443 5 factors Example 2 Express 1 1 4 2 6 as repeated multiplication. Solution 1 1 4 2 6 = 1 4 × 1 4 × 1 4 × 1 4 × 1 4 × 1 4 Example 3 Evaluate each of the following. (a) 34 (b) 1– 4 5 2 3 Solution (a) 34 = 3 × 3 × 3 × 3 = 81 (b) 1– 4 5 2 3 = 1– 4 5 2 × 1– 4 5 2 × 1– 4 5 2 = – 64 125 Alternative Method Using scientific calculator, press ( (–) 4 ab/c 5 ) ^ 3 = CAUTION ! (–a)n ≠ –(a)n Example 4 Express each of the following in index notation using the base in the bracket. (a) 64 [base 2] (b) 625 [base 5] a is known as the base and n is known as the index. 2. A number in index notation can be written as repeated multiplication.
Mathematics Form 3 Chapter 1 Indices Form 3 230 Solution (a) 64 = 26 2 64 2 32 2 16 2 8 2 4 2 2 1 (b) 625 = 54 5 625 5 125 5 25 5 5 1 1.2 Law of Indices 1. The multiplication of numbers in index form with a common base can be simplified by adding the indices. For example, am × an = am + n 2. The division of numbers in index form with a common base can be simplified by subtracting the indices. For example, am ÷ an = am – n 3. A number in index form that is raised to a power can be simplified by multiplying the indices. For example, (am)n = amn 4. (am × bn ) q = amq × bnq 1 am bn 2 q = amq bnq 5. a0 = 1 ; a ≠ 0 6. a–n = 1 an ; a ≠ 0 7. a 1 —n = n a ; a ≠ 0 8. a m —n = (am) 1 —n = (a 1 —n )m a m —n = n am = (n a)m Example 5 Simplify each of the following. (a) 24 × 22 (b) 3a2 × 4a3 × (–2a) Solution (a) 24 × 22 = 24 + 2 am × an = am + n = 26 (b) 3a2 × 4a3 × (–2a) = 3 × 4 × (–2) × (a2 × a3 × a1 ) = –24a2 + 3 + 1 = –24a6 Example 6 Simplify each of the following. (a) 4 × 52 × 46 (b) 4q3 × 2p × 5q2 × p3 Express (2p)3 as repeated multiplication. Quiz Quiz
Mathematics Form 3 Chapter 1 Indices Form 3 231 Solution (a) 4 × 52 × 46 = (41 × 46 ) × 52 = 47 × 52 (b) 4q3 × 2p × 5q2 × p3 = (4 × 2 × 5) × (p1 × p3 ) × (q3 × q2 ) = 40 × p1 + 3 × q3 + 2 = 40p4 q5 Example 7 Simplify each of the following. (a) 46 ÷ 43 (b) 45e6 ÷ 3e4 Solution (a) 46 ÷ 43 = 46 – 3 am ÷ an = am – n = 43 (b) 45e6 ÷ 3e4 = 45 3 × e6 – 4 = 15e2 Example 8 Simplify each of the following. (a) (53 ) 7 (b) (y4 )3 Solution (a) (53 )7 = 53 × 7 (am) n = amn = 521 (b) (y4 )3 = y4 × 3 = y12 Example 9 Simplify each of the following. (a) (4 × 64 )8 (b) (3x3 y4 z2 )6 Solution (a) (4 × 64 )8 = 41 × 8 × 64 × 8 (am × bn ) q = amq × bnq = 48 × 632 (b) (3x3 y4 z2 )6 = 31 × 6 x3 × 6 y4 × 6 z2 × 6 = 36 x18 y24 z12 Example 10 Simplify each of the following. (a) 1 42 53 2 2 (b) (54 ÷ t 2 )3 Solution (a) 1 42 53 2 2 = 42 × 2 53 × 2 1 am bn 2 q = amq bnq = 44 56 (b) (54 ÷ t 2 )3 = 54 × 3 ÷ t 2 × 3 = 512 ÷ t 6 = 512 t 6 Example 11 Simplify each of the following. (a) 65 × (43 )2 44 (b) (2a3 b2 ) 2 8a2 b Solution (a) 65 × (43 )2 44 = 65 × 43 × 2 44 = 65 × 46 44 = 65 × 46 – 4 = 65 × 42
Mathematics Form 3 Chapter 1 Indices Form 3 232 (b) (2a3 b2 ) 2 8a2 b = 21 × 2a3 × 2b2 × 2 8a2 b1 = 22 a6 b4 8a2 b1 = 4a6 b4 8a2 b1 = 4 8 a6 – 2 b4 – 1 = 1 2 a4 b3 Example 12 Simplify (3xy3 )3 × (y3 )4 ÷ xy8 . Solution (3xy3 )3 × (y3 )4 ÷ xy8 = 33 x3 y3 × 3 × y3 × 4 ÷ x1 y8 = 27x3 y9 × y12 ÷ x1 y8 = 27x3 – 1y9 + 12 – 8 = 27x2 y13 Example 13 State each of the following in the form 1 an . (a) 3–1 (b) 8–4 Solution (a) 3–1 = 1 3 a–n = 1 an (b) 8–4 = 1 84 Example 14 State each of the following in index notation. (a) 1 9 (b) 1 56 Solution (a) 1 9 = 9–1 (b) 1 56 = 5–6 Example 15 Evaluate each of the following. (a) 2–3 × 2–2 (b) 35 × (3–2 × 2–1)2 Solution (a) 2–3 × 2–2 = 2–3 + (–2) = 2–5 = 1 25 = 1 32 (b) 35 × (3–2 × 2–1)2 = 35 × 3–2 × 2 × 2–1 × 2 = 35 × 3–4 × 2–2 = 35 + (–4) × 1 22 = 31 × 1 4 = 3 4 Example 16 Express each of the following in the form n a . (a) 64 1 —3 (b) 243 1 —5 Solution (a) 64 1 —3 = 3 64 a —1 n = n a (b) 243 1 —5 = 5 243
Mathematics Form 3 Chapter 1 Indices Form 3 233 Example 17 State each of the following in index notation. (a) 144 (b) 4 1 296 Solution (a) 144 = 144 1 —2 (b) 4 1 296 = 1 296 1 —4 Example 18 Evaluate 2 401 1 —4 . Solution 2 401 1 —4 = 4 2 401 2 401 = 7 × 7 × 7 × 7 = 7 or 2 401 1 —4 = (74 ) 1 —4 = 74 × 1 —4 = 7 Alternative Method Using scientific calculator, press 2 4 0 1 ^ ( 1 ab/c 4 ) = Example 19 Express 8 2 —3 in the form (am) 1 —n , (a 1 —n ) m, n am and (n a )m. Solution (am) 1 —n (a 1 —n )m n am (n a )m (82 ) 1 —3 (8 1 —3 )2 3 82 ( 3 8 )2 Example 20 Evaluate each of the following. (a) 4 5 —4 × 4 3 —4 ÷ 14 1 —2 2 3 (b) (a2 ) 1 —4 ÷ a3 × (a5 ) 1 —2 Solution (a) 4 5 —4 × 4 3 —4 ÷ 14 1 —2 2 3 = 4 5 —4 + 3 —4 – 3 —2 = 4 1 —2 = (22 ) 1 —2 = 2 (b) (a2 ) 1 —4 ÷ a3 × (a5 ) 1 —2 = a 2 × 1 —4 ÷ a3 × a 5 × 1 —2 = a 1 —2 ÷ a3 × a 5 —2 = a 1 —2 – 3 + 5 —2 = a0 = 1 Example 21 Find the value of 64 2 —3 . Solution 64 2 —3 = (43 ) 2 —3 = 43 × 2 —3 = 42 = 16
Mathematics Form 3 Chapter 1 Indices Form 3 234 Alternative Method Using scientific calculator, press 6 4 ^ ( 2 ab/c 3 ) = Example 22 Find the value of 2– 1 —2 × 24 1 —2 × (35 ) 1 —2 . Solution 2 – 1 —2 × 24 1 —2 × (35 ) 1 —2 = 2– 1 —2 × (3 × 8) 1 —2 × 35 × 1 —2 = 2– 1 —2 × 13 1 —2 × 23 × 1 —2 2 × 3 5 —2 = 2– 1 —2 + 3 —2 × 3 1 —2 + 5 —2 = 21 × 33 = 2 × 27 = 54 Example 23 Simplify each of the following. (a) 1a 1 —3 b–22 2 × a 1 —3 b6 (b) (4p–2q) 1 —2 × 3pq 3 —2 Solution (a) 1a 1 —3 b–22 2 × a 1 —3 b6 = a 1 —3 × 2 b –2 × 2 × a 1 —3 b6 = a 2 —3 b –4 × a 1 —3 b6 = a 2 —3 + 1 —3 b –4 + 6 = ab2 (b) (4p–2q) 1 —2 × 3pq 3 —2 = 4 1 —2 p –2 × 1 —2 q 1 —2 × 3pq 3 —2 = (22 ) 1 —2 p –1 q 1 —2 × 3pq 3 —2 = 2 × 3 × p–1 + 1q 1 —2 + 3 —2 = 6p0 q2 = 6q2 HOTS Example 1 Given 3x ÷ 33 = 32x – 6, find the value of x. Solution 3x ÷ 33 = 32x – 6 3x – 3 = 32x – 6 If am = an , then m = n. x – 3 = 2x – 6 2x – x = 6 – 3 x = 3 Example 24 6m3 A 4m12 B C The diagram above shows a triangle ABC. Calculate the area of the triangle. Solution Area of triangle ABC = 1 2 × base × height = 1 2 × 4m12 × 6m3 = 1 2 × 4 × 6 × m12 + 3 = 12m15
Mathematics Form 3 Chapter 1 Indices Form 3 235 Section A 1. The term k4 is equal to A k + 4 C k × k × k × k B 4 × k D k + k + k + k 2. 5 √16 = 2 p q A p = 4, q = 5 C p = 5, q = 16 B p = 5, q = 4 D p = 16, q = 5 3. Calculate 8 2 3 × 27 2 3 . A 18 C 36 B 24 D 48 4. Given that 23 + 2n = 24 , find the value of n. A 1 C 3 B 2 D 4 5. Evaluate 1 16. A 2–4 C 42 B 24 D 8–2 6. (3m2 )2 = A 3m2 C 6m4 B 3m4 D 9m4 Section B 7. (a) Fill the blanks with the correct number. (i) 1 7 × 1 7 × 1 7 × 1 7 × 1 7 = 1 1 7 2 (ii) (–10) = 1 (b) Complete the following operation. (16 – √81)2 = (16 – ) 2 = 2 8. (a) Marks 7 for the term that is not the same with n5 . n2 × n3 n–7 × n12 (n2 )3 n2 ÷ n–3 (b) Given that 2x + 4 = 1 8 , calculate the value of x. Section C 9. (a) Match the following with the correct answers. x2 × x3 = x2 ÷ x3 = (x2 )3 = x6 x5 1 x (b) (i) Calculate the value of 4 1 —2 × 52 × 8 2 —3 . (ii) Given that 343n = 7, find the value of n. (c) Simplify m– 3 4 × m 1 2 m 1 4 10. (a) Simplify (i) (p3 q2 )5 (pq)3 (ii) 6e4 × 2e3 × 1 8 e–2 (b) Given 32x – 1 = 81 9 , find the value of x. HOTS Analysing (c) Given 2m × 7n 22 × 75 = 14, find the value of m + n. HOTS Analysing UASA PRACTICE
Form 1.2.3 KSSM RANGER Form 1.2.3 series is published specifically to fulfil the UASA (Ujian Akhir Sesi Akademik) format. The content of this series is comprehensive and incorporates all the latest syllabus. This book can serve as a quick revision for students to increase their understanding prior to their preparation for UASA assessment. • Bahasa Melayu • English • Matematik ✓ Mathematics • Sains • Science • Sejarah • Geografi • Reka Bentuk dan Teknologi • Pendidikan Islam i-THINK Gallery Answers Formative Practice Concise Notes Info & Video QR Code UASA Model Papers Form 1/2/3 & Answers QR Code Purchase eBook here! W.M: RM19.95 / E.M: RM19.95 KC117234 ISBN: 978-629-470-109-0 KC117234