SECTION B Extended response questions. Question 1: Part (a) Working. TI-Nspire CAS screenshot(s) Part (b) Working
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1 Explanatory notes: Note that the VCAA only supplies multiple-choice answers to sample papers. Every effort has been made to ensure that these solutions are correct. The author of these solutions has no affiliation with the VCAA. SECTION B Extended response questions Question 1: Part (a) f Part (b) 1 f is the reflection of f in the line y x f and,. 1 f intersect at, and On TI-Nspire CAS, graph x f y using the relation graphing feature. Part (c) Solving k arccos for k 6 gives k. 1
2 Part (d) 1 A f x f x dx So A (correct to 3 dp).
3 Parts (e) (i) and (ii) 1 where L f x dx x sin 6 f x x 3cos 6 So L 3.67 (correct to 3 dp). The Notes application can be used to calculate the arc length of a curve. Question : Part (a) (i) u cis 3 The Notes application can be used to perform complex number arithmetic. 3
4 Part (a) (ii) 6 u cis6 3 cis. 1 This question part (a one mark show that question) is best done without the use of TI- Nspire CAS. Part (a) (iii) We require 6 points located at: z 1, i, i And we require that u and w are labelled correctly. Part (b) (i) Circle centre, and radius 1. Diagram is below in (b) (ii). Part (b) (ii) Straight line passing through,. 4
5 Part (b) (iii) By reading the answers from the Argand diagram we obtain 3 1, and 3 1,. Alternatively (but not as efficiently): Solving z 1 and z u z u for x and y gives the above intersection points. Question 3: Part (a).4t loge N6 3e d dn 1 dn log e N and dn dt N dt d.4t.4t 6 3e 1.e dt 1 dn.4t So 1.e. N dt Substituting into the LHS of the differential equation gives:.4t.4t 1.e.4 6 3e.4 This question part is best attempted without using CAS. Part (b) N (correct to the nearest integer) 5
6 Part (c) When t, loge N 6 and so N 43 (correct to the nearest integer). Part (d) (i) d N d dn dn dt dn dt dt 1dN.46 loge N.4N N dt.16n 5log N 6 log N e e This question part is best attempted without using CAS. Part (d) (ii) d N Solving for N gives dt N 148 (correct to the nearest integer). This occurs at t.7 (years) (correct to 1 dp). 6
7 Part (e) TI-Nspire CAS can be used to help graph the function. See the above right screenshot. Note that the differential equation graphing feature can graph the solution to the differential equation dn.4n6 log e N. dt Question 4: Part (a) r 1 cos 6 i 1sin 6 j 6i 6 3j 7
8 Part (b) t t r t i gt j c r 6i 6 3j and so c 6i 6 3j t t r t 6 i 6 3 gt j 3 t r t 6t i 6 3 gt t 6 3t j d 6 r and so d 3 t r t 6t i 6 3 gt t 6 3t j 6 The Notes application can be used to perform vector calculus calculations. Part (c) At t T, the skier lands on the down-slope represented by the equation y x. Solving T T 6 3T gt T 6 6 for T with T gives 1 T 3 1. g 8
9 Part (d) 1 r g (correct to 1 dp) (m/s) The Notes application can be used to perform vector calculus calculations. Question 5: Part (a) Solving 3g T1 3a and T1 g a for a and T 1 gives g a (ms - ). The Notes application can be used to solve equations of motion. Part (b) Solving the system of equations in 3g (a) gives T1 (N). See the above screenshot. Part (c) 3gsin 3T 3b and Solving T g b for b (and T ) gives g b (ms - ). 8 9
10 Part (d) b so T g Solving 3gsin g for gives 19.5 (correct to 1 dp). Part (e) Solving 3g 3 T 3gsin 1 4 and g 3 gt 1 4 for (and T ) gives 45. Question 6: Part (a) H : 1, H 1 : 1 TI-Nspire CAS functionality does not offer any assistance here. 1
11 Part (b) p Pr X (correct to 3 dp) The Notes application can be used to solve probability and statistics exam questions. Part (c) Since p.5, H is not rejected at the 5% level of significance. TI-Nspire CAS functionality does not offer any assistance here. Part (d) Using the inverse normal feature 1 with p.5, 1 and s 5 we obtain C (correct to 3 dp). 11
12 Part (e) (i) Pr X (correct to 3 dp). Part (e) (ii) This represents a type II error as it is the same as not rejecting H when it is false. In other words, X results in H not being rejected even though 9.5. TI-Nspire CAS functionality does not offer any assistance here. 1
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1 w z k k States or implies that 4 i TBC Uses the definition of argument to write 4 k π tan 1 k 4 Makes an attempt to solve for k, for example 4 + k = k is seen. M1.a Finds k = 6 (4 marks) Pearson Education
Thursday 14 June 2012 Morning
Thursday 14 June 01 Morning A GCE MATHEMATICS (MEI) 4757 Further Applications of Advanced Mathematics (FP3) QUESTION PAPER *471575061* Candidates answer on the Printed Answer Book. OCR supplied materials:
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