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1 A Free Exam for VCE study design Engage Education Foundation Units 3 and 4 Maths Methods (CAS): 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 Maths Methods (CAS): Exam 2: Free Exam A The Engage Education Foundation Section A Multiple-choice questions Question 1 Substitute in the two y-values, and then find the matching domain The domain value will include / exclude that point just as the corresponding value from the range does. Question 2 The correct answer is B Use the discriminant and set it to 0 to find the value of that will produce just one solution: and 8 will both satisfy the equation, but only 0 is an option. Question Let : and 1 5 and 1 log log 5 3 log log 5 0 Page 1
3 The Engage Education Foundation Units 3 and 4 Maths Methods (CAS): Exam 2: Free Exam A Question 4 The correct answer is E. Expand the matrix multiplication and addition ( and represent the transformed coordinates, and are the original coordinates): Therefore: Now substitute these into the given equation ( 4 ): Question 5 The correct answer is E. sin0 0 and sin 0, so the average rate of change is 0. Question 6 The correct answer is E. Question 7 The correct answer is A. sin 3sin 0 sin sin 3 0 sin 0 sin 3 0 sin 3 The only difficulty with this question is the notation. It is asking for the set that contains the same values as the solutions to the given equations, however, the sets themselves are given in terms of equations. Question 8 The correct answer is B. 2 log log 2 log 2 log Page 2
4 Units 3 and 4 Maths Methods (CAS): Exam 2: Free Exam A The Engage Education Foundation Question 9 The asymptote comes from the fact that the denominator (bottom) of a fraction cannot be 0: 20 2 The asymptote comes from the limit of the fraction as approaches infinity: lim lim 1 (since lim 0) Question 10 The correct answer is D. (divide through by ) Strictly positive excludes 0. Question 11 The correct answer is D. 4 sin3 3 cos3 when 0: 41sin0 3 1 cos0 033 Question 12 The correct answer is D. log is defined only for 0. Question 13 The correct answer is D. This is an application of the chain rule. Question 14 The correct answer is E. 21 1, which has one x-intercept. Therefore there is only one stationary point. Furthermore, 0 for all, so must always be stationary or increasing. Hence, the sole stationary point is a stationary point of inflection. Question 15 The correct answer is A (by the properties of integrals) Question 16 1 gives the average value of the function over Page 3
5 The Engage Education Foundation Units 3 and 4 Maths Methods (CAS): Exam 2: Free Exam A Question 17 The correct answer is B. 10 log log 210 log log Question 18. Pr Question The probability of getting a head is 0.5, and there are 11 trials. This is a binomial experiment, so: Question 20 The correct answer is B. The only set of events that can possibly happen at the same time (ie. aren t mutually exclusive) is B. D only occurs when two dice is simultaneously rolled. Question 21 The correct answer is A. Pr 11.5 Pr. (use ) Pr 0.5 Pr 0.5, by the symmetry properties of the normal distribution Question 22 The correct answer is B. If you are unsure about a question similar to this, draw a diagram such as the following, where the lines represent the slope of the curve around the point: 5 x f (x) +ve 0 -ve Page 4
6 Units 3 and 4 Maths Methods (CAS): Exam 2: Free Exam A The Engage Education Foundation Section B Marks allocated are indicated by a number in square brackets, for example, [1] indicates that the line is worth one mark. For any question worth more than one mark, students must show working to receive any marks for that question. Question 1a sin [2] Question 1b 2 2 cos [1] 0 sin [1] 42 2 cos [1] Question 1c Use second equation from 1b: 0, since this would make the other equations false. sin 2 0 2, 2, since 0 [1] Use first and third equations from 1b: 2 2 cos 42 2 cos Rearrange for : Combine and solve for : [1] Substitute in to solve for : Page 5
7 The Engage Education Foundation Units 3 and 4 Maths Methods (CAS): Exam 2: Free Exam A [1] Question 1d 2 cos 4 2 (or using values of, and found in 1c) 2 cos 4 2 [1] 4 2 cos 2 2 cos where x [2 2,6 2] [1] Domain is required for full mark Question 1e (or using values of, and found in 1c) [1] 2 cos for all [1] domain [1] Question 1f 2 cos [1] 2 cos 2 0 cos 2 0, [1], [1] Question 2a Use Pythagoras theorem: km [1] Question 2b 5 20 [1] 020 [1] Page 6
8 Units 3 and 4 Maths Methods (CAS): Exam 2: Free Exam A The Engage Education Foundation Question 2c 0, , 25 [1 mark for shape, 1 mark for each point, for a total of 3 marks] Question 2d If Marty can only walk 45 km in a day, that means each trip can be a maximum of 22.5 km. [1] Solve by hand or using calculator to find km. [1] Question 2e Using similar triangles, [2 marks for diagram and explanation] [1] Substitute into equation for to find km. [1] Question 3a Area given by model m [1] error actual predicted % error error % [1] actual. Page 7
9 The Engage Education Foundation Units 3 and 4 Maths Methods (CAS): Exam 2: Free Exam A Question 3b x-axis intercepts are 0 and area [1] m [1] Question 3c i 0 is a solution (by inspection). So 0, 0 is a point of intersection. [1] 3 [1] At, 3 Question 3c ii area 8.43 m [1] 3 [1] Question 3d 3 [2] 3. So the other point of intersection is,3 [1] Question 3e Use calculator to solve 3 5 [1] 2.42 m [1] Question 4a PrClare wins 7 games in a row [1] Question 4b Let X the number of games out of 10 won by Clare Pr 7 Pr 7 Pr 8 Pr 9 Pr 10 [1] [1] Question 4c PrSally wins next 2 points [1] Page 8
10 Units 3 and 4 Maths Methods (CAS): Exam 2: Free Exam A The Engage Education Foundation Question 4d PrSally wins 2 of next 3 points PrSSC PrSCS PrCSS, where C represents Clare winning, and S represents Sally winning [1] [1] [1] Question 4e , 1 0 [1] [1] Therefore, PrClare wins 12 th point [1] Question 4f Therefore, Clare wins 43.33% of points in the long term. Question 4g Want 1 [1] lim 1 lim100 1 lim lim [1] Question 4h lim lim Question 4i Pr 50 lim [2] [1] 100 [1] [1] Page 9
Engage Education Foundation
B Free Exam for 2006-15 VCE study design Engage Education Foundation Units 3 and 4 Specialist Maths: Exam 2 Practice Exam Solutions Stop! Don t look at these solutions until you have attempted the exam.
Engage Education Foundation
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Engage Education Foundation
E Free 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
Engage Education Foundation
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
Engage Education Foundation
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