Topical Revision Paper 8 Unit 8 Matrices

Topical Revision Paper 8 Term 2 Unit 8: Matrices Question 1: Given that 7 1 1 1 1 1 1 0 0 3 0 3 4 3 0 1 0 3 0 1 3 3 4 0 0 1 m                       , find the value of m . [2] Answer: 1m Solution:        3 1 4 0 3 1 4 3 0 1 44 1 m m m m          Question 2: It is given matrices M 0 1 x y   and N 10 2 3 1y   . (i) Find N – M and 2M in terms of x and y. [2] (ii) If 2M M N , find the value of x and of y. [3] Solution: (i) 1 0 0 2 3 1 1 1 32 x yy x y               NM 2 00 11 1 xx yy xy x y xy         M (ii)

2 2 1 3 2 1 3 1 32,23 x xy x y y xy yy xy yx                       M M N M N M Question 3: The price of ticket in each category at Cheap Cinema is as follows: Child - $3, Adult - $8 and Senior citizen - $5. (i) Write down the column matrix C represents the above information. [1] The number of tickets sold on one weekend is given by the matrix N as follows: Children Adult Senior Citizen Saturday 20 45 70 Sunday 50 90 40 Given that T NC , (ii) evaluate T. [3] (iii) What information can be derived from T ? [1] To improve revenue during weekends, two proposals are examined: Proposal 1: Double the price on Sunday only. Proposal 2: Increase the price by 25% on each day. The matrix P is such that PT gives the revenue for the weekend under proposal 1. The matrix Q is such that QT gives the revenue for the weekend under proposal 2. (iv) Evaluate PT and QT and state which proposal would be more profitable? [3] Solution: () 3 8 5 i C   

() 330 45 68 860 95 38 5 3(30) 45(8) 68(5) 60(3) 95(8) 38(5) 790 1130 ii T T T        (iii) T represents the amount collected by selling tickets on Saturday and Sunday (iv)             790 ; 1 2 ;Q 1.25 1.251130 79012 1130 3050 7901.25 1.25 1130 2400 TP PT PT QT QT          Proposal 1 will be more profitable Question 4: Given that P = 30 05   and Q = 0 ab c   , find the values of a, b and c such that 5PQ = 8P – 3Q. [4] Solution:

3 0 3 05 8 30 5 0 0 5 0 3 3 24 3 35 0 5 0 40 3 15 24 3 4 3 0 25 40 3 40 10 28 7 4 10, 0, 37 a b a b cc a b a b cc aa a b cc c a b c                                        Question 5: The following table shows the amount of flour, butter and sugar needed in making 12 chocolate cupcakes and 12 lemon cupcakes. Flour Butter Sugar Chocolate cupcakes 125 g 125 g 96 g Lemon cupcakes 225 g 125 g 128 g The amount of the 3 main ingredients used in making these cupcakes can be represented by the matrix M = 125 125 96 225 125 128   . (i) Given that N = 1 1 1    , evaluate MN. [2] (ii) What do the elements in the product matrix MN represent? [1] The cost of 100 g of flour is $0.80, 100 g butter is $2.40 and 100 g of sugar is $0.20. (iii) It is decided that the selling price of the cupcakes should include an increase of 450% of the total cost of using these 3 main ingredients. Write down the matrix

S in order to calculate the selling price of 12 chocolate cupcakes and 12 lemon cupcakes respectively. Calculate the respective selling price using MATRIX method. [4] (iv) A secret recipe of 12 luxurious lemon cupcakes requires the same amount of flour and sugar used as the lemon cupcakes, but with an unknown amount of butter. Fortunately, it is known that the total cost of flour, butter and sugar for making the luxurious lemon cupcakes is 10% more than that of the lemon cupcakes. Find the amount of butter needed for this secret recipe to make 12 luxurious cupcakes, to the nearest gram. [2] (i) 1125 125 96 1225 125 128 1 125 125 96 225 125 128 346 478        MN MN MN (ii) The total mass of the 3 main ingredients used in 12 chocolate cupcakes and 12 lemon cupcakes. (iii) 125 125 965.5 225 125 128   S

0.008125 125 965.5 0.024225 125 128 0.002 4.1925.5 5.056 23.056= 27.808         The selling price of 12 chocolate cupcakes is $23.06 and of 12 lemon cupcakes is $27.81. (iv)Let the amount of butter be x g for 12 luxurious lemon cupcakes. 225 0.008 0.024 128 0.002 5.056 1.1x       146.1g 146 gx (3 s. f.)