(17) Jay, Joi Lam - HW17 - AI HL 2023 Nov Paper 2
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8823 – 7202© International Baccalaureate Organization 202311 pages31 October 2023Zone AafternoonZone BafternoonZone Cafternoon2 hoursMathematics: applications and interpretationHigher levelPaper 2Instructions to candidatesyDo not open this examination paper until instructed to do so.yA graphic display calculator is required for this paper.yAnswer all the questions in the answer booklet provided.yUnless otherwise stated in the question, all numerical answers should be given exactly or correct to three significant figures.yA clean copy of the mathematics: applications and interpretation formula booklet is required for this paper.yThe maximum mark for this examination paper is [110 marks].
– 2 –8823 – 7202Answer all questions in the answer booklet provided. Please start each question on a new page. Full marks are not necessarily awarded for a correct answer with no working. Answers must be supported by working and/or explanations. Solutions found from a graphic display calculator should be supported by suitable working. For example, if graphs are used to find a solution, you should sketch these as part of your answer. Where an answer is incorrect, some marks may be given for a correct method, provided this is shown by written working. You are therefore advised to show all working.1. [Maximum mark: 15]Madhu is designing a jogging track for the campus of her school. The following diagram shows an incomplete portion of the track.Madhu wants to design the track such that the inner edge is a smooth curve from point Ato point B,and the other edge is a smooth curve from point C to point D. The distance between points Aand Bis 50metres.diagram not to scale50 metresDBMACtracktrackTo create a smooth curve, Madhu first walks to M, the midpoint of [AB].(a) Write down the length of [BM]. [1](This question continues on the following page)
– 3 –8823 – 7202Turn over(Question 1 continued)Madhu then walks 20metres in a direction perpendicular to [AB]to get from point Mto point F. Point Fis the centre of a circle whose arc will form the smooth curve between points Aand Bon the track, as shown in the following diagram.diagram not to scaleDBMACF(b) (i) Find the length of [BF].(ii) Find BF̂M. [4](c) Hence, find the length of arc AB. [3]The outer edge of the track, from Cto D, is also a circular arc with centre F, such that the track is 2metres wide.(d) Calculate the area of the curved portion of the track, ABDC. [4]The base of the track will be made of concrete that is 12 cmdeep.(e) Calculate the volume of concrete needed to create the curved portion of the track. [3]
– 6 –8823 – 72023. [Maximum mark: 16]Tiffany wants to buy a house for a price of285 000 US Dollars (USD). She goes to a bank to get a loan to buy the house. To be eligible for the loan, Tiffany must make an initial down payment equal to 15 %of the price of the house.The bank offers her a 30-year loan for the remaining balance, with a 4 %nominal interest rate per annum, compounded monthly. Tiffany will pay the loan in fixed payments at the end of each month.(a) (i) Find the original amount of the loan after the down payment is paid. Give the exact answer.(ii) Calculate Tiffany’s monthly payment for this loan, to two decimal places. [5](b) Using your answer from part (a)(ii), calculate the total amount Tiffany will pay over the life of the loan, to the nearest dollar. Do notinclude the initial down payment. [2]Tiffany would like to repay the loan faster and increases her payments such that she pays 1300 USDeach month.(c) Find the total number of monthly payments she will need to make to pay off the loan. [2]This strategy will result in Tiffany’s final payment being less than 1300 USD.(d) Determine the amount of Tiffany’s final payment, to two decimal places. [4](e) Hence, determine the total amount Tiffany will save, to the nearest dollar, by making the higher monthly payments. [3]
– 7 –8823 – 7202Turn over4. [Maximum mark: 12]A plane takes off from a horizontal runway. Let point Obe the point where the plane begins to leave the runway and xbe the horizontal distance, in km, of the plane from O. The function hmodels the vertical height, in km, of the nose of the plane from the horizontal runway, and is defined byh xx( )..1011500 060 07e, x≥0.diagram not to scaleh(x)Orunway(a) (i) Find h(0).(ii) Interpret this value in terms of the context. [2](b) (i) Find the horizontal asymptote of the graph of y =h (x).(ii) Interpret this value in terms of the context. [2](c) Find h′(x)in terms of x. [4]A safety regulation recommends that h′(x)never exceed 0.2.(d) Given that this plane flies a distance of at least 200 kmhorizontally from point O, determine whether the plane is following this safety regulation. [4]
– 8 –8823 – 72025. [Maximum mark: 18]The following diagram is a map of a group of four islands and the closest mainland. Travel from the mainland and between the islands is by boat. The scheduled boat routes between the ports A, B, C, Dand Eare shown as dotted lines on the map.MainlandBCDAELet the undirected graph Grepresent the boat routes between the ports A, B, C, Dand E.(a) Draw graph G. [1](b) Graph Gcan be represented by an adjacency matrix P, where the rows and columns represent the ports in alphabetical order.(i) Given that P301241125225465464612562aa, find the value of a.(ii) Hence, write down the number of different ways that someone could start at port Band end at port C, using three boat route journeys. [3](c) Find a possible Eulerian trail in G, starting at port A. [2](This question continues on the following page)
– 9 –8823 – 7202Turn over(Question 5 continued)The cost of a journey on the different boat routes is given in the following table; all prices are given in USD. The cost of a journey is the same in either direction between two ports.ABCDEA10B2025C205045D10255030E4530Sofia wants to make a trip where she travels on each of the boat routes at least once, beginning and ending at port A.(d) Find the minimum cost of Sofia’s trip. [3]The boat company decides to add an additional boat route to make it possible to travel on each boat route exactly once, starting and ending at the same port.(e) (i) Identify between which two ports the additional boat route should be added.(ii) Determine the cost of the additional boat route such that the overall cost of the trip is the same as your answer to part (d). [2]The boat company plans to redesign which ports are connected by boat routes. Their aim is to have a single boat trip that visits all the islands and minimizes the total distance travelled, starting and finishing at the mainland, A.The following table shows the distances in kilometres between the ports A, B, C, Dand E.ABCDEA80905060B803070120C903045100D50704555E6012010055(f) (i) Use the nearest neighbour algorithm to find an upper bound for the minimum total distance.(ii) Use the deleted vertex algorithm on port Ato find a lower bound for the minimum total distance. [7]
– 10 –8823 – 72026. [Maximum mark: 14]François is a video game designer. He designs his games to take place in two dimensions, relative to an origin O. In one of his games, an object travels on a straight line L1with vector equationr1121.(a) Write down L1in the form x=x0+λland y=y0+λm, where l , m∈. [1]François uses the matrix T1771to transform L1into a new straight line L2. The object will then travel along L2.(b) Find the vector equation of L2. [4]François knows that the transformation given by matrix Tis made up of the following three separate transformations (in the order listed):yA rotation of π4, anticlockwise (counter-clockwise) about the origin OyAn enlargement of scale factor 52, centred at OyA reflection in the straight line y=mx, where m=tanα, 0≤α<π(c) Write down the matrix that represents(i) the rotation.(ii) the enlargement. [2](d) The matrix Rrepresents the reflection. Write down Rin terms of α. [1](e) Given that T =RX,(i) use your answers to part (c) to find matrix X.(ii) hence, find the value of α. [6]