Chapter 6. More about Probability Chapter 2. Chapter 7. Chapter 8. Equations of Straight Lines Chapter 4. Chapter 9 Chapter 10 Chapter 11

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1 Chapter Development of Number Sstems Chapter 6 More about Probabilit Chapter Quadratic Equations in One Unknown Chapter 7 Locus Chapter Introduction to Functions Chapter 8 Equations of Straight Lines Chapter Graphs of Quadratic Functions Chapter 9 Equations of Circles Chapter Variations Chapter 0 More about Graphs of Functions Chapter 6 More about Polnomials Chapter Transformation of Graphs of Functions Chapter 7 More about Inequalities Chapter Arithmetic Sequences Chapter 8 Linear Programming Chapter Geometric Sequences Chapter 9 Chapter 0 Chapter Chapter More about Trigonometr Eponential Functions Logarithmic Functions More about Equations Chapter Chapter More about Applications of Trigonometr More about Use and Misuse of Statistics Chapter Basic Properties of Circles (I) Chapter Basic Properties of Circles (II) Chapter Measures of Dispersion (I) Chapter Measures of Dispersion (II) Chapter Permutations and Combinations 00 Chung Tai Educational Press. All rights reserved.

2 . Shade the region which represents the graphical solution of each of the following inequalities. (a) (b) < The point (0 0) can be used to perform a test. = = O O. In each of the following the shaded region represents the graphical solution of an inequalit. Write down the inequalit. (a) (b) = = O O The inequalit is. The inequalit is Chung Tai Educational Press. All rights reserved.

3 . Solve the following inequalities graphicall. (a) (b) = = 0 0 The graphical solution of The graphical solution of is as follows. is as follows. O O (c) > (d) < = = 0 The graphical solution of > The graphical solution of < is as follows. is as follows. O O Chung Tai Educational Press. All rights reserved.

4 . In each of the following use arrows to indicate the graphical solution of each inequalit on the same rectangular coordinate plane then shade the region which represents the graphical solution of the sstem of inequalities. (a) (b) < The overlapping region of the graphical solutions of the inequalities is the graphical solution of the sstem of inequalities. = = = O O =. In each of the following the shaded region / dots represent(s) the graphical solution of a sstem of inequalities. Write down the sstem of inequalities. (a) = 6 = (b) = = O O = = The sstem of inequalities is The sstem of inequalities is. a nd are Chung Tai Educational Press. All rights reserved.

5 . Solve the following sstems of inequalities graphicall. > (a) (b) < and are integers = = 0 0 = = 0 0 The graphical solution of < is as follows. 6 O 6 The dots in the following graph represent the > graphical solution of. and are integers O Chung Tai Educational Press. All rights reserved.

6 . A shop sells two tpes of barbecue food sets A and B. The number of chicken wings and number of sausages in each set A and each set B are as follows. Chicken wing Sausage Set A 6 Set B 9 It is given that Ken requires at least 0 chicken wings and at most 80 sausages. If sets A and sets B are to be bought write down all the constraints about and. The constraints are.. A carpenter is going to use at most 0 units of wood and 80 working hours to make wardrobes and beds. It is given that making a wardrobe requires units of wood and working hours while making a bed requires 0 units of wood and working hours. (a) Write down all the constraints about and. The constraints are Chung Tai Educational Press. All rights reserved.

7 which are equivalent to. (b) Represent the feasible solutions on a rectangular coordinate plane O The ordered pairs representing all points with integral coordinates in the shaded region represent all feasible solutions Chung Tai Educational Press. All rights reserved.

8 . In each of the following the shaded region / dots represent(s) the feasible solutions of certain constraints. If ( ) is an point in the feasible region find the maimum and minimum values of f ( ). (a) f ( ) = 7 6 O = From the graph f ( ) attains its maimum / minimum values at the points ( ) and ( ). At the point ( ) f ( ) = ( ) ( ) = At the point ( ) f ( ) = ( ) ( ) = Maimum value = Minimum value = Chung Tai Educational Press. All rights reserved.

9 (b) f ( ) = O = 0 From the graph f ( ) attains its maimum / minimum values at the points ( ) and ( ). At the point ( ) f ( ) = = At the point ( ) f ( ) = = Maimum value = Minimum value = (c) f ( ) = = 0 6 O Chung Tai Educational Press. All rights reserved.

10 From the graph f ( ) attains its maimum / minimum values at the points ( ) and ( ). At the point ( ) f ( ) = = At the point ( ) f ( ) = = Maimum value = Minimum value =. (a) Draw the feasible region which represents the following constraints on a rectangular coordinate plane O The shaded region in the graph is the feasible region Chung Tai Educational Press. All rights reserved.

11 (b) Using the result of (a) find the maimum and minimum values of above constraints. f ( ) = subject to the Consider the vertices ( ) ( ) and ( ) of the feasible region. At the point ( ) f ( ) = = At the point ( ) f ( ) = = At the point ( ) f ( ) = = Maimum value = Minimum value =. A compan is going to spend at most $0 000 and 70 man-hours to produce items A and items B. The details of producing an item A and an item B are as follows. Item A Item B Cost ($) Production time (hour) (a) Write down all the constraints about and. The constraints are Chung Tai Educational Press. All rights reserved.

12 which are equivalent to. (b) Represent the feasible solutions on a rectangular coordinate plane O The ordered pairs representing all points with integral coordinates in the shaded region represent all feasible solutions. 009 Chung Tai Educational Press. All rights reserved. 8.

13 (c) If the profits of selling an item A and an item B are $00 and $00 respectivel how man items A and items B should the compan produce to obtain the maimum profit? Total profit $ P( ) = $( ) = $ Draw a straight line i.e. the straight line on the graph in (b). From the graph the profit is the maimum when = and =. The compan should produce items A and items B Chung Tai Educational Press. All rights reserved.

14 Shade the region which represents the graphical solution of each of the following inequalities.. (a) (b) < 6 = O = 6 O In each of the following the shaded region represents the graphical solution of an inequalit. Write down the inequalit.. (a) (b) O = O = Use shaded region / dots to represent the graphical solution of each of the following sstems of inequalities. ( ) > 0. (a) (b) 6 < 0 = 0 6 = 0 O = = 6 O = Chung Tai Educational Press. All rights reserved.

15 . (a) and are integers (b) > < 0 and are integers O = = 6 = = 6 O = In each of the following the shaded region / dots represent(s) the graphical solution of a sstem of inequalities. Write down the sstem of inequalities. ( 6). (a) = 0 O = = (b) 6 = 0 O = 6 = = 6. (a) = = O = (b) = 6 6 O = Chung Tai Educational Press. All rights reserved.

16 7. In the figure the shaded region represents the graphical solution of an inequalit. = 7 O (a) Write down the inequalit. (b) Determine whether the following ordered pairs are solutions of the inequalit. (i) ( 9) (ii) ( 0) (iii) ( ) Solve the following inequalities graphicall. (8 ) > 0. <. In the figure the shaded region is bounded b three inequalities. = 8 O = = (a) Write down the three inequalities. (b) List all integral solutions satisfing the three inequalities Chung Tai Educational Press. All rights reserved.

17 Chung Tai Educational Press. All rights reserved. Solve the following sstems of inequalities graphicall. ( 0).. < > > < < < < integers are and 6 0. > < integers are and 8. In the figure the shaded region represents the graphical solution of an inequalit. O 6 = (a) Write down the inequalit. (b) If ( k) is a solution of the inequalit where k is a real number find the smallest integral value of k b referring to the figure.

18 . In the figure the shaded region represents the graphical solution of an inequalit. O 6 7 = 6 (a) Write down the inequalit. (b) If (p ) is not a solution of the inequalit where p is a real number find the greatest integral value of p b referring to the figure. < 7. (a) Solve the sstem of inequalities 8 graphicall. 9 (b) Hence list all integral solutions of the sstem of inequalities. > 6 6. (a) Solve the sstem of inequalities graphicall. < (b) Hence list all integral solutions of the sstem of inequalities. <. (a) Solve the sstem of inequalities graphicall. < (b) Let k be a real number. If the sstem of inequalities has onl 9 integral solutions find k the smallest integral value of k. 009 Chung Tai Educational Press. All rights reserved. 8.7

19 0 6. (a) Solve the sstem of inequalities < graphicall. 0 0 < (b) It is given that ( ) and ( ) both satisf the sstem of inequalities where h and 0 h < < k k are real numbers. (i) Find the greatest integral value of h. (ii) Find the smallest integral value of k. (iii) At least how man integral solutions does the sstem of inequalities have? 7. Iris is organizing a birthda part and she is going to bu L of cola and L of orange juice according to the following constraints. I. The total capacit of cola and orange juice should not be less than 0 L. II. The total capacit of cola should be at least twice that of orange juice. Write down all the constraints about and. 8. A factor is going to produce tos P and tos Q. The details of producing a to P and a to Q are as follows. Material required (unit) Production time (hour) To P 6 To Q Given that there are 00 units of materials and 600 working hours available write down all the constraints about and Chung Tai Educational Press. All rights reserved.

20 9. There are 80 apples 0 mangoes and 00 oranges available in a fruit shop and a shopkeeper is going to sell the fruits b packing them into fruit baskets A and fruit baskets B. The details of each tpe of fruit basket are as follows. Apple Mango Orange Fruit basket A 8 6 Fruit basket B 6 8 If fruit baskets A and fruit baskets B are to be produced write down all the constraints about and. 0. A restaurant provides two tpes of soup vegetable soup and seafood soup each da. It is given that the cost of each litre of vegetable soup and each litre of seafood soup are $ and $0 respectivel and the restaurant prepares L of vegetable soup and L of seafood soup each da according to the following constraints. I. The total cost of preparing the two tpes of soup each da should not be more than $600. II. At most 80 L of soup should be prepared each da. (a) Write down all the constraints about and. (b) Represent the feasible solutions on a rectangular coordinate plane.. A grocer store sells toothpaste and toothbrushes in two tpes of packages famil set and econom set. The details of each tpe of package are as follows. Number of tubes of toothpaste Number of toothbrushes Famil set Econom set 6 Tomm wants to bu at least 0 tubes of toothpaste and 6 toothbrushes as gifts of an oral health promotional event. If famil sets and econom sets are to be bought (a) write down all the constraints about and. (b) represent the feasible solutions on a rectangular coordinate plane.. A beverage shop has 900 pears and 600 apples available to prepare jars of special drink A and jars of special drink B. It is given that preparing a jar of special drink A requires pears and apple while preparing a jar of special drink B requires pears and apples. (a) Write down all the constraints about and. (b) Represent the feasible solutions on a rectangular coordinate plane. 009 Chung Tai Educational Press. All rights reserved. 8.9

21 . An ice-cream shop provides two tpes of dessert sets A and B. The details of each tpe of dessert sets are as follows. Green tea ice-cream (ml) Chocolate ice-cream (ml) Mango ice-cream (ml) Dessert set A Dessert set B It is given that at most 0 L of green tea ice-cream 0 L of chocolate ice-cream and 8 L of mango ice-cream are available for preparing dessert set A and dessert set B each week. If dessert sets A and dessert sets B are prepared each week (a) write down all the constraints about and. (b) represent the feasible solutions on a rectangular coordinate plane.. There are 600 units of chemicals A 800 units of chemicals B and 00 units of chemicals C available in a laborator for producing ml of vaccine P and ml of vaccine Q. The details of ml of vaccine P and ml of vaccine Q are as follows. Chemical A (unit) Chemical B (unit) Chemical C (unit) Vaccine P 6 Vaccine Q (a) Write down all the constraints about and. (b) Represent the feasible solutions on a rectangular coordinate plane.. An organization is going to invite male guests and female guests to a ball. The number of souvenirs and the amounts of food and drink prepared for each male guest and each female guest are as follows. Souvenir Fruit tart Champagne (glass) Male guest Female guest It is given that the organization has prepared 00 souvenirs 0 fruit tarts and 0 glasses of champagne. (a) Write down all the constraints about and. (b) Represent the feasible solutions on a rectangular coordinate plane Chung Tai Educational Press. All rights reserved.

22 6. The owner of a pet shop is going to use at most $9 000 to bu some cats and dogs. Due to the limitation of space the pet shop can onl hold at most cats and dogs in total and the number of dogs should not be more than that of cats. It is given that the cost of each cat and each dog are $80 and $00 respectivel. (a) Write down all the constraints about the number of cats and the number of dogs to be bought. (b) Represent the feasible solutions on a rectangular coordinate plane. 7. A factor produces digital video discs (DVD) and video compact discs (VCD). For a bo of DVD $7 are spent on materials and $ on packaging. For a bo of VCD $ are spent on materials and $ on packaging. It is given that the material cost and packaging cost of producing the two tpes of discs each hour should not be more than $ 00 and $ 00 respectivel and the number of boes of DVDs produced each hour should not be more than. times that of VCDs. (a) Write down all the constraints about the number of boes of DVDs and the number of boes of VCDs produced each hour. (b) Represent the feasible solutions on a rectangular coordinate plane. 8. A train provides at most 0 m of floor area for passengers and carries at most 00 kg of luggage. There are two tpes of seats available first-class seats and econom-class seats where each of them occupies m and.6 m of floor area respectivel. It is given that the number of first-class seats should be less than of the number of econom-class seats and at most 6 kg and 0 kg of luggage is allowed for each passenger taking first-class seat and econom-class seat respectivel. (a) Write down all the constraints about the number of first-class seats and the number of econom-class seats provided. (b) Represent the feasible solutions on a rectangular coordinate plane. 9. A factor has 00 kg of white rice 7 kg of red rice and 0 kg of brown rice available to produce two rice mitures A and B. The details of each kg of miture A and miture B are as follows. White rice (g) Red rice (g) Brown rice (g) Miture A 6 0 Miture B It is given that the amount of miture B produced should not be more than twice that of miture A and the amount of miture B produced should be at least 00 kg. (a) Write down all the constraints about the amount of miture A and miture B to be produced. (b) Represent the feasible solutions on a rectangular coordinate plane. 009 Chung Tai Educational Press. All rights reserved. 8.

23 In each of the following if ( ) is an point in the shaded region find the values of and such that f ( ) attains its maimum and minimum values. (0 ) 0. f ( ) =. f ( ) = 6 O O 6 In each of the following the shaded region / dots represent(s) the feasible solutions of certain constraints. If ( ) is an point in the feasible region find the maimum and minimum values of f ( ). ( ). f ( ) =. f ( ) = O 6 O. f ( ) =. f ( ) = ( ) O (6 ) 6 O (6 ) ( ) Chung Tai Educational Press. All rights reserved.

24 Chung Tai Educational Press. All rights reserved. Find the maimum and minimum values of ) ( f subject to each of the following constraints. (6 ) f = ) ( f ) ( = ) ( = f ) ( = f 0. < integers are and 7 6. < > < integers are and f = ) ( 8 ) ( = f. (a) Draw the feasible region which represents the following constraints on a rectangular coordinate plane. 6 (b) Using the result of (a) find the maimum and minimum values of f ) ( = subject to the above constraints. (c) If an additional constraint is added find the maimum and minimum values of ) ( f.

25 . (a) Draw the feasible region which represents the following constraints on a rectangular coordinate plane. 6 0 > and are non-nega tive integers (b) Using the result of (a) find the maimum and minimum values of f ( ) = subject to the above constraints. (c) If the constraint 7 is removed find the maimum and minimum values of f ( ).. In the figure the shaded region is bounded b the following four straight lines. L : = 0 L L : = 0 L : 0 = 0 : = (a) Write down the sstem of inequalities with the shaded region as its graphical solution. 0 0 L L L L (b) Let P = 0 where ( ) is an point in the shaded region. O 0 0 (i) Find the maimum and minimum values of P. (ii) If P 0 0 find the range of values of b adding a suitable straight line in the figure.. In the figure R is the region bounded b the following four straight lines. L : 7 = 0 L : = 0 L : = 0 L : 9 = 0 L B R O C L D A L L (a) Find the coordinates of A B C and D. (b) Find the maimum and minimum values of where ( ) is an point in the region R Chung Tai Educational Press. All rights reserved.

26 6. A factor is going to produce items A and items B. The details of producing an item A and an item B are as follows. Item A Item B Material cost ($) 80 0 Production time (hour) 0 0 Given that there are $7 00 available for the materials and 00 working hours (a) write down all the constraints about and. (b) represent the feasible solutions on a rectangular coordinate plane. (c) If the profits of selling an item A and an item B are $00 and $0 respectivel how man items of each tpe should the factor produce to obtain the maimum profit? 7. Jack is organizing a part and he is going to bu packs of chocolate in package A and packs of chocolate in package B. The table below shows the details of a pack of chocolate in each package. Number of pieces of chocolate with nuts Number of pieces of milk chocolate Package A 8 8 Package B Given that at least 0 pieces of chocolate with nuts and 8 pieces of milk chocolate are required in the part (a) write down all the constraints about and. (b) represent the feasible solutions on a rectangular coordinate plane. (c) If the selling prices of a pack of chocolate in package A and package B are $6 and $.6 respectivel how man packs of chocolate in each package should Jack bu to minimize the ependiture? 8. A coordinator is going to take 0 performers to the performance venue b hiring coaches A and coaches B. It is given that each coach A can carr 6 passengers and each coach B can carr 8 passengers and the coach hire compan can arrange at most 8 drivers and coaches A. (a) Write down all the constraints about and. (b) Represent the feasible solutions on a rectangular coordinate plane. (c) If the rental for a coach A and a coach B are $00 and $800 respectivel how man coaches of each tpe should be hired so that the rental is kept at the minimum? Find the minimum rental. 009 Chung Tai Educational Press. All rights reserved. 8.

27 9. A carpenter is going to produce pieces of furniture A and pieces of furniture B. The details of producing a piece of furniture A and a piece of furniture B are as follows. Material required (unit) Production time (hour) Furniture A Furniture B 0 Given that there are 0 units of materials and 8 working hours available (a) write down all the constraints about and. (b) represent the feasible solutions on a rectangular coordinate plane. (c) If the profits of selling a piece of furniture A and a piece of furniture B are $0 and $90 respectivel how man pieces of furniture of each tpe should the carpenter produce to maimize the profit? Find the maimum profit. 60. A baker sells pancakes and cakes. Making each kg of pancake requires mangoes and peach while making each kg of cake requires mangoes and peaches. It is given that the baker has at most 0 mangoes and 0 peaches available to make kg of pancakes and kg of cakes each da. (a) Write down all the constraints about and. (b) Represent the feasible solutions on a rectangular coordinate plane. (c) If the profit of selling each kg of cake is. times the profit of selling each kg of pancake how man pancakes and cakes (in kg) should be made each da to obtain the maimum profit? 6. The owner of a stationer shop is going to sell at most 0 ball-point pens and 0 correction pens b packing them into packages A and packages B. The table below shows the number of ball-point pens and correction pens in each tpe of package. Ball-point pen Correction pen Package A Package B 6 (a) Write down all the constraints about and. (b) Represent the feasible solutions on a rectangular coordinate plane. (c) If the profits of selling a package A and a package B are in the ratio of : how man packages of each tpe should be produced to obtain the maimum profit? Chung Tai Educational Press. All rights reserved.

28 6. A food supplier provides students lunch sets with two tpes of food A and B. The table below shows the nutritional content of food A and food B. Carbohdrates (unit) Protein (unit) Fat (unit) Ever 00 g of food A 0 0 Ever 00 g of food B 0 6 It is given that each lunch set should contain at least 08 units of carbohdrates at least 7 units of protein and at most 6 units of fat. Suppose each lunch set includes g of food A and g of food B (a) write down all the constraints about and. (b) represent the feasible solutions on a rectangular coordinate plane. (c) If the cost for ever 00 g of food A and food B are $.8 and $ respectivel how much food A and food B (in g) should each lunch set include to minimize the cost? Find the minimum cost of each lunch set. 6. A factor is going to spend not more than $ 000 and 00 minutes of working hours in producing electronic components. The details of producing ever 00 electronic components b three production lines are as follows. Cost required ($) Production time (minute) Production line A 00 Production line B 600 Production line C 800 Let and z be the number of electronic components to be produced b production lines A B and C respectivel. (a) (i) Epress z in terms of and. (ii) Write down all the constraints about and. (b) Represent the feasible solutions on a rectangular coordinate plane. (c) To let production line C produce the greatest possible number of electronic components how man electronic components should production lines A and B produce? Find the greatest possible number of electronic components to be produced b production line C. 6. Ivan spends hours in doing part-time job A and hours in doing part-time job B each da under the following constraints. I. He should work at least hours and at most 0 hours each da. II. The time spent on part-time job B should be at most twice that on part-time job A. III. The time spent on part-time job A should not be more than that on part-time job B. 009 Chung Tai Educational Press. All rights reserved. 8.7

29 (a) Write down all the constraints about and. (b) Represent the feasible solutions on a rectangular coordinate plane. (c) It is given that the hourl wages of part-time job A and part-time job B are $7 and $ respectivel. (i) How should Ivan allocate his working hours so that his dail income attains its maimum? Find his maimum dail income. (ii) If Ivan works at most 8 hours each da instead how should he allocate his working hours so that his dail income attains its maimum? 6. The owner of a baker uses at most 0 kg of grade A flour 80 kg of grade B flour and 0 kg of grade C flour to prepare two flour mitures P and Q. Miture P is prepared b miing grade B flour and grade C flour in the ratio of :. Miture Q is prepared b miing grade A grade B and grade C flour in the ratio of : :. It is given that the owner prepares kg of miture P and kg of miture Q. (a) Write down all the constraints about and. (b) Represent the feasible solutions on a rectangular coordinate plane. (c) Given that the profit of making bread with each kg of miture Q is three times that with each kg of miture P (i) how much miture (in kg) of each tpe should the owner prepare to maimize the profit? (ii) When the profit is at the maimum how much flour (in kg) of each tpe is used to prepare the two flour mitures? Chung Tai Educational Press. All rights reserved.

30 Worksheet 8A - I (page 8.). (a) (b) Worksheet 8A - II (page 8.) 6. (a) (b) and are integers. (a) 8 (b) ( 0) ( ) ( ) ( ) ( ). (a) (b). (a) 6 (b) 6. (b) ( ) ( ) ( ) ( ). (b) ( ) ( ) ( ) ( ) ( ). (b) Worksheet 8B (page 8.) and ar e non-negative integers 0. (a) 60 and are non-negative integers Worksheet 8C (page 8.7). (a) Maimum value minimum value (b) Maimum value 0 minimum value (c) Maimum value minimum value 9. (b) Maimum value 8 minimum value 000. (a) 70 and ar e non-negative integers (c) 0 items A and 7 items B 6. (b) (i) (ii) (iii) Build-up Eercise 8B (page 8.8) and are non-negative integers and are non-negative integers (a) 0 0 Build-up Eercise 8A (page 8.). (a) (b). (a) 0 8 and are non-negative integers 0. (a) 6. (a) and are integers 7. (a) 7 (b) (i) No (iii) Yes (b) (b) 0 0 and are integers (ii) No. (a). (a). (a) and are non-negative integers and are non-negative integers Chung Tai Educational Press. All rights reserved.

31 00 0. (a) 0 and are non-negative integers 6. (a) Suppose cats and dogs are to be bought the constraints are 0. and are non-negative integers 7. (a) Suppose boes of DVDs and boes of VCDs are produced each hour the constraints are and are non-negative integers 8. (a) Suppose first-class seats and econom-class seats are provided the constraints are 8 00 < and are non-negative integers 9. (a) Suppose kg of miture A and kg of miture B are to be produced the constraints are Build-up Eercise 8C (page 8.) 0. Maimum value: = = 0 minimum value: = 0 =. Maimum value: = 0 = 0 minimum value: = 0 =. Maimum value =. minimum value =. Maimum value = minimum value =. Maimum value = minimum value =. Maimum value = minimum value = 6. Maimum value = minimum value = 8 7. Maimum value = 7 minimum value = 6 8. Maimum value = minimum value = 7 9. Maimum value = minimum value = 0. Maimum value = 6 minimum value = 0. Maimum value = minimum value = 0. (b) Maimum value = minimum value = (c) Maimum value = 6 minimum value =. (b) Maimum value = 8 minimum value = 6 (c) Maimum value = 6 minimum value = (a) 0 0 (b) (i) Maimum value = minimum value = (ii) 0. (a) A(6 ) B( ) C(0 ) D( ) (b) Maimum value = minimum value = 0 6. (a) 7. (a) 8. (a) 9. (a) and are non-negative integers (c) 0 items A and 0 items B 0 and are non-negative integers (c) Package A: package B: 8 and are non-negative integers (c) coaches A and coaches B; $ and are non-negative integers (c) pieces of furniture A and 6 pieces of furniture B or pieces of furniture A and piece of furniture B; $ (a) (a) (c) kg of pancakes and kg of cakes and are non-negative integers (c) packages A and 8 packages B (a) (c) 0 g of food A and 00 g of food B; $.6 6. (a) (i) z = (ii) and are non-negative integers (c) Production line A: 000 production line B: 000; Chung Tai Educational Press. All rights reserved.

32 0 6. (a) 0 0 (c) (i) hours of part-time job A and hours of part-time job B; $60 (ii) hours of part-time job A and hours of part-time job B (a) (c) (i) 60 kg of miture P and 80 kg of miture Q (ii) 0 kg of grade A flour 80 kg of grade B flour and 0 kg of grade C flour Chung Tai Educational Press. All rights reserved.