Engage Education Foundation

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1 2015 End of Year Seminar Exam for 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 practiceexams@ee.org.au.

2 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: [1] Question 1g Number of fallen trees Question 1h Fallen trees = 14 [2] Page 1

3 The Engage Education Foundation Units 3 and 4 Further Maths: Exam 2: End of Year Seminar Exam = [1]. rainfall = = 3 mm [1] Question 1i R 2 = % 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] Page 2

4 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 = (t 1) = = 248 m [1] Question 1c 216 = (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 = = 382 [1] L6 = = 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] , 6.116, therefore it will switch electricity providers after seven years [1] Page 3

5 The Engage Education Foundation Units 3 and 4 Further Maths: Exam 2: End of Year Seminar Exam Module 2 Geometry and Trigonometry Question 1a AB = [1] Question 1b A = m m 2 [2] Question 1c sin sin 61 [1] Question 1d = 29 so bearing is S 29 E or 209 T [1] Question 1e tan tan 45 CB = 524m [1] Page 4

6 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 = = 816m Using the cosine rule we can find BE cos 61 CD = m [1]. BE = m [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] 2tan m [1] Question 2d tan 70. [1]. = [1] Distance between brooms = 4-2( ) = 2.73m [1] Page 5

7 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: [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 = = $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 [1] Question 2b 50 [1] mark for the equation. [1] mark for the correct inequality. Question 2c 45 [1] 40 [1] Page 6

8 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

9 The Engage Education Foundation Units 3 and 4 Further Maths: Exam 2: End of Year Seminar Exam Module 4 Business-related Mathematics Question 1a $ [1] Question 1b 3.88% [1] Question 1c = $ [1] Question 2a $132 interest [1] Question 2b r = 6% [1] Question 2c R = 10.7% [1] Question 2d t = 7 years [1] Question 3a $4425 per month [1] N=25x12=300 I%=5.8 PV=70000 PMT =? FV=0 P/Y=12 C/Y=12 Page 8

10 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=? P/Y=12 C/Y=12 Principal still owing = $ = $ paid off principal [1] Page 9

11 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=? P/Y=12 C/Y=12 Then interest rate changes and so does repayment N=? I%=4.7 PV= 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] Page 10

12 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= PMT =-4150 FV=? P/Y=12 C/Y=12 $ [1] Question 5a N=4x12=48 I%=4.3 PV= PMT =-210 FV=? P/Y=12 C/Y=12 $ [1] Page 11

13 The Engage Education Foundation Units 3 and 4 Further Maths: Exam 2: End of Year Seminar Exam Question 5b $ in account after 1 year [1] N=1x12=12 I%=4.3 PV= PMT =-210 FV=? P/Y=12 C/Y=12 Interest = ( x 12) = $ [1] Question 6a N=? I%=5.7 PV= 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= PMT =3450 FV=? P/Y=2 C/Y=2 $ [1] Page 12

14 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 [2, -0.5 for every incorrect element] Question 1b 5 [1] Question 2a = 8.7 km [1] Question 2b ABCFED, ABFCED, ACBFED [1] Question 2c ABCFED = 19.8 km [1] Question 2d HAMILTONIAN path [1] Question 3a Maximum flow = Minimum cut Cut passes through these edges [1] = 11.1 km [1] Question 3b = 20.5 [1] Question 3c = 24.3 [1] Page 13

15 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: [1] Take away smallest element from each row from that row (row reduce): [1] Since allocation does not work here continue with column reduction, subtract smallest element in column from that column: [1] [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 = 8 minutes [1] Page 14

16 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] 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 [1] 3 15 E = 2, W = 5, B = 3, X = [1] Question 1cii = 25 cans of energy drink in total [1] Question 1ciii = 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 [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 $ [1] Question 2bii FreshFirst [1] Question 3a [1] 46 Page 15

17 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 The class will first be held indoors in the sixth week. [1] Question 3biii The steady state matrix indicates that 48 people will attend the class in the long term [1] Page 16