2015 End of Year Seminar Exam for 2006-15 VCE study design Engage Education Foundation Units 3 and 4 Further Maths: Exam 2 Practice Exam Solutions Stop! Don t look at these solutions until you have attempted the exam. Any questions? Check the Engage website for updated solutions, then email practiceexams@ee.org.au.
Units 3 and 4 Further Maths: Exam 2: End of Year Seminar Exam The Engage Education Foundation Marks allocated are indicated by a number in square brackets, for example, [1] indicates that the line is worth one mark. Section A Core: Data Analysis Question 1a The independent variable is rainfall. [1] Question 1b Number of fallen trees is the discrete variable, because it can only take discrete, whole number values. [1] Question 1c Students are expected to refer to the Table 1 to answer this question R = 0.804, which suggests a strong positive linear relationship. [1] But the residual plot has a pattern to it, which suggests the data is more suitable to a nonlinear model. [1] Question 1d log [1] Question 1e Missing values are 0.07 and 0.05 [1] Question 1f The completed graph should look like: 60 50 40 30 20 10 0 0.00 0.10 0.20 0.30 0.40 0.50 0.60 [1] Question 1g Number of fallen trees 55.9 108.9 Question 1h Fallen trees = 14 [2] Page 1 www.engageeducation.org.au
The Engage Education Foundation Units 3 and 4 Further Maths: Exam 2: End of Year Seminar Exam 1 14 55.9 108.9 =. 0.385 [1]. rainfall = 2.599 = 3 mm [1] Question 1i R 2 = 0.9276083 92.8% of the variation in the number of fallen trees can be explained by the variation in [2] Question 1j No, because when data is plotted against time, the best way to linearize it to apply time series smoothing. [2] www.engageeducation.org.au Page 2
Units 3 and 4 Further Maths: Exam 2: End of Year Seminar Exam The Engage Education Foundation Section B Module 1 Number Patterns Question 1a Show 2 successive differences 56m 24m = 32m and 88m 52m = 32m[1] Question 1b August 2013, t = 8 [1] Hn = 24 + 32 (t 1) = 24 + 32 7 = 248 m [1] Question 1c 216 = 24 + 32 (t 1), t = 7 [1] July 2013 [1] Question 1d November 2013 [1] Question 2a Ln+1 = 2Ln +2 [1] L1 =10 [1] Question 2b Ln+1 = 2Ln +2 L1 =10 Hence L6 = 2 190 + 2 = 382 [1] L6 = 2 382 + 2 = 766 [1] Question 3a $3524 [1] No marks if the question is not answered to the nearest dollar Question 3b [1] $12706 [1] Question 3c 1.12 2000 [1] 4000 1.12 2000, 6.116, therefore it will switch electricity providers after seven years [1] Page 3 www.engageeducation.org.au
The Engage Education Foundation Units 3 and 4 Further Maths: Exam 2: End of Year Seminar Exam Module 2 Geometry and Trigonometry Question 1a 524 600 AB = 85424 292 [1] Question 1b 600 524 292 2 708 708 708 600708 524708 292 708 108184416 A = 76503.99m 2 76504 m 2 [2] Question 1c sin sin 61 [1] Question 1d 90 61 = 29 so bearing is S 29 E or 209 T [1] Question 1e tan 45 524 tan 45 CB = 524m [1] www.engageeducation.org.au Page 4
Units 3 and 4 Further Maths: Exam 2: End of Year Seminar Exam The Engage Education Foundation Question 1f ACD and ABE are similar triangles. AC = AB + BC = 292 + 524 = 816m Using the cosine rule we can find BE. 600 816 2600816 cos 61 CD = 742.3815864m [1]. BE = 265.6561559m [1] CD BE = 477m. John s pitch is 477m longer. [1] Question 2a Isosceles [1] Question 2b cos 70 DA = Question 2c tan 70 = 5.85m [1] 2tan70 5.50m [1] Question 2d tan 70. [1]. = 0.63694791 [1] Distance between brooms = 4-2(0.63694791) = 2.73m [1] Page 5 www.engageeducation.org.au
The Engage Education Foundation Units 3 and 4 Further Maths: Exam 2: End of Year Seminar Exam Module 3 Graphs and Relations Question 1a Complete graph should look like: 120 100 80 60 40 20 0 1 2 3 4 [2] Question 1b According to the pattern the additional charge doubles for each additional hour. So for 5 hours the additional charge would be $160. With the $30 flat fee, total charge for 5 hours should be $190. [1] Question 1c If the dogs were washed together it would take Dana 4 hours. Cost = 30 + 80 = $110 [1] If the dogs were washed separately 2 sets of 2 hours. Cost = 2 x 50 = $100. Trey should ask Dana to come on two separate days to wash each of the dogs. [1] Question 1d Trey saves $10 by getting the dogs washed separately. [1] Question 2a 15 10 [1] Question 2b 50 [1] mark for the equation. [1] mark for the correct inequality. Question 2c 45 [1] 40 [1] www.engageeducation.org.au Page 6
Units 3 and 4 Further Maths: Exam 2: End of Year Seminar Exam The Engage Education Foundation Question 2d Corners: A (0,40) B (10,40) C (45, 5) D (45,0) Profit at A = (0 x 15) + (40 x 10) = $400 Profit at B = (10 x 15) + (40 x 10) = $550 Profit at C = (45 x 15) + (5 x 10) = $725 Profit at D = (45 x 15) + (0 x 10) = $675 Maximum profit Dana can make is $725. [3] Question 2e To maximise profit Dana has to wash 50 animals (45 dogs and 5 cats). [1] To get the mark students must give the total number of animals not the individual number of cats and dogs. Page 7 www.engageeducation.org.au
The Engage Education Foundation Units 3 and 4 Further Maths: Exam 2: End of Year Seminar Exam Module 4 Business-related Mathematics Question 1a $151 200 [1] Question 1b 3.88% [1] Question 1c 840 000 1 3.2 100 = $923248.01 [1] Question 2a $132 interest [1] Question 2b 100132 1100 2 r = 6% [1] Question 2c 6 28 81 R = 10.7% [1] Question 2d 828 1500 t = 7 years [1] Question 3a $4425 per month [1] 1500 6.4 100 N=25x12=300 I%=5.8 PV=70000 PMT =? -4424.93 FV=0 P/Y=12 C/Y=12 www.engageeducation.org.au Page 8
Units 3 and 4 Further Maths: Exam 2: End of Year Seminar Exam The Engage Education Foundation Question 3b N=5x12=60 I%=5.8 PV=70000 PMT =-4425 FV=? -627695.2138 P/Y=12 C/Y=12 Principal still owing = $627695.21 700 000-627 695.21 = 72 304.79 $72 304.79 paid off principal [1] Page 9 www.engageeducation.org.au
The Engage Education Foundation Units 3 and 4 Further Maths: Exam 2: End of Year Seminar Exam Question 4a N=8x12=96 I%=5.8 PV=70000 PMT =-4425 FV=? -573136.5744 P/Y=12 C/Y=12 Then interest rate changes and so does repayment N=? 199.1583888 I%=4.7 PV=573136.5744 PMT =-4150 FV=0 P/Y=12 C/Y=12 Already paid 96 months, require another 200 months to pay off loan 296 months total to pay off the loan [1] www.engageeducation.org.au Page 10
Units 3 and 4 Further Maths: Exam 2: End of Year Seminar Exam The Engage Education Foundation Question 4b 199 payments find out how much is remaining N=199 I%=4.7 PV=573136.5744 PMT =-4150 FV=? -655.8265665 P/Y=12 C/Y=12 $655.83 [1] Question 5a N=4x12=48 I%=4.3 PV=-46000 PMT =-210 FV=? 65593.78577 P/Y=12 C/Y=12 $65 593.79 [1] Page 11 www.engageeducation.org.au
The Engage Education Foundation Units 3 and 4 Further Maths: Exam 2: End of Year Seminar Exam Question 5b $50 587.71 in account after 1 year [1] N=1x12=12 I%=4.3 PV=-46000 PMT =-210 FV=? 50587.71552 P/Y=12 C/Y=12 Interest = 50 587.72 (460 000 + 210 x 12) = $2067.72 [1] Question 6a N=? 35.72 I%=5.7 PV=-76700 PMT =3450 FV=0 P/Y=2 C/Y=2 36 half yearly payments [1] Question 6b N=30 (15 x 2) I%=5.7 PV=-76700 PMT =3450 FV=? 18002.56567 P/Y=2 C/Y=2 $18 002. 57 [1] www.engageeducation.org.au Page 12
Units 3 and 4 Further Maths: Exam 2: End of Year Seminar Exam The Engage Education Foundation Module 5 Networks and Decision Mathematics Question 1a P E M F P E M F 1 1 1 1 1 0 0 1 1 0 0 0 1 1 0 0 [2, -0.5 for every incorrect element] Question 1b 5 [1] Question 2a 1.2 + 3.2 + 3.1 + 1.2 = 8.7 km [1] Question 2b ABCFED, ABFCED, ACBFED [1] Question 2c ABCFED 1.2 + 3.2 + 9.2 + 1.2 + 5 = 19.8 km [1] Question 2d HAMILTONIAN path [1] Question 3a Maximum flow = Minimum cut Cut passes through these edges 1.2 + 5.6 + 3.1 + 1.2 [1] = 11.1 km [1] Question 3b 10.1 + 9.2 + 1.2 = 20.5 [1] Question 3c 5 + 9.2 +10.1 = 24.3 [1] Page 13 www.engageeducation.org.au
The Engage Education Foundation Units 3 and 4 Further Maths: Exam 2: End of Year Seminar Exam Question 4 Start with Bipartite graph and make into a matrix: 4 4 2 1 5 7 8 2 [1] 2 5 4 1 7 2 2 3 Take away smallest element from each row from that row (row reduce): 3 3 1 0 3 5 6 0 [1] 1 4 3 0 5 0 5 1 Since allocation does not work here continue with column reduction, subtract smallest element in column from that column: 2 3 0 0 2 5 5 0 [1] 0 3 2 0 4 0 4 1 2 3 0 0 2 5 5 0 [1] 0 3 2 0 4 0 4 1 A does O (2 minutes) B does P (2 minutes) C does M (2 minutes) D does N (2 minutes) Allocation can now be made, therefore, minimum time is 2 + 2 + 2 + 2 = 8 minutes [1] www.engageeducation.org.au Page 14
Units 3 and 4 Further Maths: Exam 2: End of Year Seminar Exam The Engage Education Foundation Module 6 Matrices Question 1a Buzz Ener Gee [1] Question 1bi AB = 4306 [1] 490 90 30 418 80 25 Question 1bii Matrix AB contains the total amount of energy from energy drinks that Samuel consumed in the week leading up to his first exam (contained in 2 cans of Ener-Gee, 5 cans of Wings, and 3 cans of Buzz energy drink). [1] Question 1ci 418 400 490 150 200 210 225 225 80 83 90 90 25 23 30 8 2 5 [1] 3 15 E = 2, W = 5, B = 3, X = 15 6556 5500 [1] 2195 375 Question 1cii 2 + 5 + 3 + 15 = 25 cans of energy drink in total [1] Question 1ciii 15 150 225 90 8 = 2250 3375 1350 120 Samuel consumed 15 cans of Brand X with 8 grams of sugar per can, totalling to 120 grams of sugar from Brand X drink [1] Question 2a 2 x 3 [1] Question 2bi 0.3 20 13 38 23 11 42 0.5 16.3 [1] 16.6 0.1 This tells us that the total cost of buying all the vegetables from FreshFirst would be $16.30 and, from GoGreen the total would be $16.60. [1] Question 2bii FreshFirst [1] Question 3a 0.7 0.2 0.3 0.8 100 20 74 [1] 46 Page 15 www.engageeducation.org.au
The Engage Education Foundation Units 3 and 4 Further Maths: Exam 2: End of Year Seminar Exam Question 3bi Sn = T n-1 x S1 [1] Question 3bii 0.7 0.2 0.3 0.8 100 20 51.25 68.75 0.7 0.2 0.3 0.8 100 20 49.625 70.375 The class will first be held indoors in the sixth week. [1] Question 3biii 0.7 0.2 100 0.3 0.8 20 48 72 0.7 0.2 100 0.3 0.8 20 48 72 The steady state matrix indicates that 48 people will attend the class in the long term [1] www.engageeducation.org.au Page 16