Core: Recursion and Financial modelling FURTHER MATHEMATICS. Student Name: Ms. Cambridge Mr. Dang Ms. Gugger Mr. Hutchison Mr. Ly

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1 SCHOOL SSESSED COURSEWORK 2016 (2017 Practice SC) Ttal Writing Time 115 minutes with 5 minutes Reading time FURTHER MTHEMTICS Student Name: Subject Teacher s Name (circle): Ms. Cambridge Mr. Dang Ms. Gugger Mr. Hutchisn Mr. Ly Mr. Sliman Mr. Williams Mr. Weiser Equipment Permitted: 1 bund reference (ntes r text bk NOT BOTH), TI- nspire CS Calculatr, scientific calculatr, Ruler, Pencil and Pens. Outcmes: Outcme 1: On cmpletin f this unit the student shuld be able t define and explain key cncepts and apply related mathematical techniques and mdels as specified in rea f Study 1 in rutine cntexts. Outcme 2: On cmpletin f this unit the student shuld be able t select and apply the mathematical cncepts, mdels and techniques as specified in rea f Study 1 in a range f cntexts f increasing cmplexity. Outcme 3: On cmpletin f this unit the student shuld be able t select and apprpriately use numerical, graphical, symblic and statistical functinalities f technlgy t develp mathematical ideas, prduce results and carry ut analysis in situatins requiring prblem- slving, mdelling r investigative techniques r appraches. SC Rules FULL SCHOOL UNIFORM MUST BE WORN TO LL SCS Students must bring crrect equipment t each SC, fr example: pens, pencils, rulers, calculatrs and ther materials as infrmed by staff N fd r drink is t be cnsumed in the SC rm (bttled water excepted) Students are nt t cmmunicate with each ther in any way during a SC N mbile phnes, ipds, mp3 players etc are allwed in the SC rm Students must remain fr the duratin f the SC. Students are t behave apprpriately at all times disruptive behaviur will be cnsidered a serius breach f SC rules. Page 1 f 9

2 SESSION ONE T start saving fr her hme, Natalie has invested $8000 in a special savings accunt that pays 6.0% simple interest, with the interest being calculated annually. a) Calculate the amunt f interest, crrect t the nearest cent, earned each year. b) Using V n t represent the value f the investment after n years, write a recurrence relatin that mdels this investment situatin. c) Use yur calculatr t determine recursively the value f Natalie s investment after fur years. The table belw has been given t assist yu. Year number Calculatin detail Value f investment ($) Page 2 f 9

3 Trent als started saving fr his hme. He started an accunt with $2500 and each mnth added anther $150. The accunt pays interest at 6.0% per annum, cmpunded mnthly. The mney is added t the accunt after the interest is calculated each mnth. d) Using V n t represent the value f the investment after n years, write a recurrence relatin that mdels this investment situatin. e) Use yur calculatr t determine recursively the value f Tent investment after six mnths. The table belw has been given t assist yu. Year number Calculatin detail Value f investment ($) (3 marks) f) What will be the value f Trent s investment after fur years, crrect t the nearest cent? Shw details f any Financial Slver calculatins in the table prvided belw. (1mark) g) Find the cmbined ttal value f Trent and Natalie s investments at the end f the sixth year f their investment. Page 3 f 9

4 SESSION TWO Trent runs his wn business frm an ffice at the back f his huse. PROBLEM ONE He recently purchased new furniture fr his hme ffice, spending a ttal f $8500. He will depreciate the value f the furniture using prime cst r flat rate depreciatin, where the annual depreciatin will be 12.5% f the purchase price. a) By what amunt, crrect t the nearest cent, will the furniture be depreciated each year. b) Using V n t represent the value f the furniture after n years, write a recurrence relatin that mdels this depreciatin situatin. c) Use yur calculatr t determine recursively the value f the furniture, each year, fr the first five years, and fill in the table belw. Year number Calculatin detail Value f investment ($) (3 marks) d) Trent will replace this furniture when its value has depreciated belw $3500. If he purchased the furniture at the end f 2013, in which year will it reach its replacement value? Page 4 f 9

5 PROBLEM TWO He has a laptp cmputer which will need replacing sn. He bught it at the end f March fr $4 400 and its value has been depreciating by 3% per mnth since then. e) Using V n t represent the value f the cmputer after n mnths, write a recurrence relatin that mdels this depreciatin situatin. Trent s accuntant has advised him t trade in the cmputer fr a new ne when its value has drpped belw a certain amunt. Trent has determined that this value will be reached in June f) What is the likely replacement value f the cmputer, written t tw significant figures? Page 5 f 9

6 PROBLEM THREE Trent purchased a new small car t use fr his business at a cst f $ He is allwed t depreciate it in value at a rate f 35 cents fr every kilmetre driven. g) Using V n t represent the value f the car after travelling n kilmetres, write a recurrence relatin that mdels this depreciatin situatin. h) What is the value f the car after it has travelled kilmetres? i) If the scrap value f the car is $15 000, hw many kilmetres will it have travelled crrect t the nearest hundred kilmetre, t reach its scrap value? Page 6 f 9

7 SESSION THREE Trent and Natalie have fund a huse that they like, and t buy it they will have t brrw $ frm their bank at 6.575% per annum, repayable mnthly ver 20 years. a) What is the effective interest rate, crrect t three significant figures, that Trent and Natalie will be paying n their lan at 6.575% per annum cmpunded mnthly? b) Shw, using the Finance Slver functin n yur calculatr that Trent and Natalie s mnthly payment, crrect t the nearest cent will be $ Write the values using the Finance Slver in the table belw. The bank has decided t make the mnthly payment $ (3 marks) c) Calculate the balance f Trent and Natalie s lan, crrect t the nearest cent, after five years, if the mnthly payments are $ Write the values used in the Finance Slver in the table belw. d) Hw much has been paid ff the lan in five years? e) Hw much interest has been paid in five years? f) What percentage, crrect t the nearest whle number, des the interest represent f the ttal mney paid after five years? Page 7 f 9

8 g) pprximately hw much interest will be paid, crrect t the nearest dllar, if the lan is amrtised ver the 20 years at the same interest rate (6.575% p.a.) with the same mnthly payment ($ )? h) In actual fact, the final repayment n Trent and Natalie s lan will be less than $2700, since their mnth payment as set by the back is slightly higher than the actual required payment (see part (a)). ssuming that the interest rate des nt change fr the duratin f their lan remaining at 6.575% p.a. calculate crrect t the nearest cent the amunt f their final payment. Shw details f any Finance Slver calculatins in the table prvided belw. Page 8 f 9

9 Five years after taking ut their lan, Trent and Natalie inherited $ frm a relative and they decided t use it t help pay dwn their lan. fter the payment was made, the back ffered tw chices cntinue t pay ff the lan with the mnthly repayment as previusly calculated (and amrtise the lan sner), r cntinue t pay ff the lan ver the fifteen remaining years with a lwer mnthly repayment. i) Which ptin will cst Trent and Natalie the least mney verall? OPTION ONE Shw detail f any Finance Slver calculatins in the table prvided belw. OPTION TWO Shw detail f any Finance Slver calculatins in the table prvided belw. (5 marks) Page 9 f 9