YORK UNIVERSITY Faculty of Science Department of Mathematics and Statistics MATH A Test #2 November 03, 2014

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1 Famly Name prnt): YORK UNIVERSITY Faculty of Scence Department of Mathematcs and Statstcs MATH A Test #2 November 0, 2014 Solutons Gven Name: Student No: Sgnature: INSTRUCTIONS: 1. Please wrte your name, student number and fnal answers n nk. 2. Ths s a closed-book test, duraton 50 mnutes.. Calculator of the model Texas Instruments BA II Plus or any other non programmable calculator s permtted. NO INTERNET CONNECTED DEVICES OR OTHER AIDS ARE PERMITTED. 4. There are sx questons on sx pages. Answer all questons. Fll n answers n desgnated spaces. Your work must justfy the answer you gve. Show your work on the space provded. If you need more space, use the back of a page and clearly ndcate ths fact on an orgnal page each tme when you use the back of a page for your work. 5. Reman seated untl we collect all the test papers. 6. There are 55 marks n the test. However, your grade wll be counted out of Do the easest questons frst, GOOD LUCK! Queston Ponts Scored Total: 55 0

2 pts) For the each questons below crcle the correct answer. No need to justfy your answer or show any work. a) Palmer wants to set up a fund to provde a half-yearly scholarshp of $2,500 n perpetuty. An amount s to be deposted nto the fund on December 17, 2014 to allow for ths. Scholarshps are pad half-yearly startng on December 17, 2018, and the fund can be assumed to earn.5% per year compoundng half-yearly. To the nearest dollar, the amount that Palmer should depost on December 17, 2014 to just cover ths s: A) $,822 B) $20,000 C) $26,167 D) $40,000 E) $61,145 C) b) Elane opens a bank account on the frst day of the year. Ths account pays nterest compounded yearly at a rate of 10% per year. To begn wth, there s no money n the account. However, at the end of each year Elane deposts money nto the account. The szes of these deposts ncrease each year. The frst payment deposted at the end of the frst year, s $400. Each subsequent payment s $400 more than the last. Ths contnues for 8 years. To the nearest dollar, the amount of money that s accumulated n the account on the day after the last payment s: A) $6,4 B) $8,545 C) $1,744 D) $18,18 E) $11,584 D) pts) a) If the nflaton rate s 2.1%, what effectve annual rate of nterest up to decmal places) should Smon obtan on hs nvestment n order to receve a real rate of return of 5% per year? Let r be annual rate of nflaton and be annual rate of nterest. Then the real rate of nterest real r 1 r 1 + r. 1

3 Substtutng r 1 and real 0.05, we obtan Solvng for, we obtan or 7.205%. 1 b) Derve the dentty: + 1. s n a n 1 s n ) n )n 1 + ) n 1 1+) n 1 1+) n ) 1. n a n c) Show that f > 0, then Ia) d a ä. Also, )a n nν n Ia) lm Ia) n lm n n lm a n 1 n lm n nνn 1 + a 1 lm n d 1 1 d a ä. n 1 + ) n. 9 pts) You have $5, 000 down-payment on an $18, 000 car. The dealer offers you the followng two optons: a) payng the balance wth end-of-month payments over the next three years at the rate ) 6%; b) a reducton of $500 n the prce of the car, the same down-payment of $5, 000 and bank fnancng of the balance after down-payment, over three years at ) 8%. Whch of the optons would you choose? Hnt: Compare the correspondng monthly payments. 2

4 a) , 000 5, 000 R a a , where R a s monthly payment for opton a). R a 1, 000 a , ) $ ) 6 b) , R b a 6, where R b s monthly payment for opton b). R b, 500 a 6, ) $ Hence, opton b) s better pts) a) Jason and Jeff would lke to open ther own busness n 4 years. They estmate that they wll need $50,000 at that tme. How much should they depost at the end of each quarter nto ther jont savngs account, f they have $5,000 n the account now and the nterest rate s 4) 6%? We have to fnd R such that per quarter). Rs , 000 5, ) ) )16 R 5, 000) 5, 000) s 1.015) ) $2, )16 75) 1.015) 16 1

5 b) Mchael deposts $100 per month for four years, nothng for one year, and then $200 per month for three years. Interest rates start at ) 10% and falls to ) 7% on the date of the frst $200 deposts. Fnd the accumulated value at the tme of the fnal $200 depost. The equvalent to ) 0.1 effectve monthly rate s m Lkewse, the equvalent monthly rate for j 0.07 s m The accumulated value S at the tme of the fnal $200 depost wll be ) S 100s ) ) ) s ) ) ) $16, ) pts) Stan elects to receve hs retrement beneft over 20 years at the rate of $2,000 per month begnnng one month from now. The monthly beneft ncreases by 5% each year. At a nomnal nterest rate of 6% convertble monthly, calculate the present value of the retrement beneft. Gven ) Interest converson pero s 1 month. So, the equvalent effectve monthly rate of nterest m and the equvalent effectve annual rate of nterest a 1 + m ) ) Hence, monthly payments of $2,000 to be made durng the frst year, accumulated on December 1 wll be: K 2, 000s , ) 1 24, Thus, the present value of the retrement beneft: 1+ )n P V K 1 1+r r where K 24, 671., r 0.05 and Therefore, P V 24, 671.) ) , $419, pts) A savngs account pays at ) 8% and bonus of $100 at the end of each year n whch the balance exceeds $5,000. You depost $200 at the end of each month for 10 years. How much wll you have on the account after 10 years? 4

6 Hnt: You may need to fnd the number of bonus payment perods. ) So, the equvalent effectve monthly rate m 0.08 and the equv- ) alent effectve annual rate a 1 + m ) 1 So, 200s n m > 5, 000 s n m > )n > ) n > n > ln1 1 6 ) ln1 + ) 2.2. Therefore, bonus starts at the end of the second year. Hence, the accumulated value S 200s 0 m + 100s 9 a 200) 1 + ) ) ) , ) $, The end. 5