File Download Area
Information about "2008 Australian Mathematics Competition AMC Senior Years 11 and 12.pdf"
- Filesize: 107.29 KB
- Uploaded: 27/08/2019 15:35:32
- Status: Active
Free Educational Files Storage. Upload, share and manage your files for free. Upload your spreadsheets, documents, presentations, pdfs, archives and more. Keep them forever on this site, just simply drag and drop your files to begin uploading.
Download Urls
-
File Page Link
https://www.edufileshare.com/67b3e8a9805c3c57/2008_Australian_Mathematics_Competition_AMC_Senior_Years_11_and_12.pdf
-
HTML Code
<a href="https://www.edufileshare.com/67b3e8a9805c3c57/2008_Australian_Mathematics_Competition_AMC_Senior_Years_11_and_12.pdf" target="_blank" title="Download from edufileshare.com">Download 2008 Australian Mathematics Competition AMC Senior Years 11 and 12.pdf from edufileshare.com</a>
-
Forum Code
[url]https://www.edufileshare.com/67b3e8a9805c3c57/2008_Australian_Mathematics_Competition_AMC_Senior_Years_11_and_12.pdf[/url]
[PDF] 2008 Australian Mathematics Competition AMC Senior Years 11 and 12.pdf | Plain Text
A u s t rAl i An M At h e M At i c s c o M p e t i t i o n a n a c t i v i t y o f t h e a u s t r a l i a n m a t h e m a t i c s t r u s t thursday 31 July 2008 sEnIor dIvIsIon comPEtItIon PaPEr InstructIons and InformatIon GEnEraL 1. Do not open the booklet until told to do so by your teacher. 2. NO calculators, slide rules, log tables, maths stencils, mobile phones or other calculating aids are permitted. Scribbling paper, graph paper, ruler and compasses are permitted, but are not essential. 3. Diagrams are NOT drawn to scale. They are intended only as aids. 4. There are 25 multiple-choice questions, each with 5 possible answers given and 5 questions that require a whole number between 0 and 999. The questions generally get harder as you work through the paper. There is no penalty for an incorrect response. 5. This is a competition not a test; do not expect to answer all questions. You are only competing against your own year in your own State or Region so different years doing the same paper are not compared. 6. Read the instructions on the answer sheet carefully. Ensure your name, school name and school year are filled in. It is your responsibility that the Answer Sheet is correctly coded. 7. When your teacher gives the signal, begin working on the problems. tHE ansWEr sHEEt 1. Use only lead pencil. 2. Record your answers on the reverse of the Answer Sheet (not on the question paper) by FULLY colouring the circle matching your answer. 3. Your Answer Sheet will be read by a machine. The machine will see all markings even if they are in the wrong places, so please be careful not to doodle or write anything extra on the Answer Sheet. If you want to change an answer or remove any marks, use a plastic eraser and be sure to remove all marks and smudges. IntEGrItY of tHE comPEtItIon The AMC reserves the right to re-examine students before deciding whether to grant official status to their score. australian school years 11 and 12 time allowed: 75 minutes ©amt P u b l i s h i n g 2008 a m t t l i m i t e d a c n 083 950 341 Indicate Quantity Required in Box australian mathematics competition BooKs 2007 amc soLutIons and statIstIcs PrImarY vErsIon – $a35.00 EacH 2007 amc soLutIons and statIstIcs sEcondarY vErsIon – $a35.00 EacH 2007 amc soLutIons and statIstIcs PrImarY and sEcondarY vErsIons – $a57.00 for botH Two books are published each year for the Australian Mathematics Competition for the Westpac Awards, a Primary version for the Middle and Upper Primary divisions and a Secondary version for the Junior, Intermediate and Senior divisions. The books include the questions, full solutions, prize winners, statistics, information on Australian achievement rates, analyses of the statistics as well as discrimination and difficulty factors for each question. The 2007 books will be available early 2008. austraLIan matHEmatIcs comPEtItIon book 1 (1978-1984) – $a40.00 EacH austraLIan matHEmatIcs comPEtItIon book 2 (1985-1991) – $a40.00 EacH austraLIan matHEmatIcs comPEtItIon book 3 (1992-1998) – $a40.00 EacH austraLIan matHEmatIcs comPEtItIon book 3-cd (1992-1998) – $a40.00 EacH austraLIan matHEmatIcs comPEtItIon book 4 (1999-2005) – $a40.00 EacH These four books contain the questions and solutions from the Australian Mathematics Competition for the Westpac Awards for the years indicated. They are an excellent training and learning resource with questions grouped into topics and ranked in order of difficulty. BooKs For Further deVelopment oF mathematical sKills ProbLEms to soL vE In mIddLE scHooL matHEmatIcs – $a50.oo EacH This collection of problems is designed for use with students in Years 5 to 8. Each of the 65 problems is presented ready to be photocopied for classroom use. With each problem there are teacher’s notes and fully worked solutions. Some problems have extension problems presented with the teacher’s notes. The problems are arranged in topics (Number, Counting, Space and Number, Space, Measurement, Time, Logic) and are roughly in order of difficulty within each topic. ProbLEm soLvInG vIa tHE amc – $a40.00 EacH This book uses nearly 150 problems from past AMC papers to demonstrate strategies and techniques for problem solving. The topics selected include Geometry, Motion and Counting Techniques. cHaLLEnGE! (1991–1995) – $a40.00 EacH This book reproduces problems, solutions and extension questions from both Junior (Years 7 and 8) and Intermediate (Years 9 and 10) versions of the Mathematics Challenge for Young Australians, Challenge Stage. It is a valuable resource book for the classroom and the talented student. the above prices are current to 31 december 2008. details of other amt publications are available on the australian mathematics trust’s web site www.amt.canberra.edu.au/amtpub.html. all BooKs can Be ordered online @ www.amt.edu.au/amtpub.htnl a selection oF australian mathematics trust puBlications payment details payment must accompany orders. please allow up to 14 days for delivery. please forward publications to: (print clearly) NAME: ADDRESS: COUNTRY: POSTCODE: Postage and Handling - within Australia, add $A3.00 for the first book and $A1.00 for each additional book - outside Australia, add $A13.00 for the first book and $A5.00 for each additional book TOTAL: Cheque/Bankdraft enclosed for the amount of $A Please charge my Credit Card ( Visa, Mastercard) Amount authorised:$A Cardholder’s Name (as shown on card): Cardholder’s Signature: Expiry Date: Date: Tel (bh): Card Number: All payments (cheques/bankdrafts, etc) should be in Australian currency, made payable to austraLIan matHEmatIcs trust and sent to: australian mathematics trust, university of canberra act 2601, australia. tel: 02 6201 5137 Fax: 02 6201 5052Senior Division Questions 1 to 10, 3 marks each 1.The value of 8002−2008 is (A) 200 (B) 8 (C) 6006 (D) 1060 (E) 5994 2.The difference between1 20and2 10is (A) 0 (B)1 10(C)3 5(D)3 10(E)3 20 3.In the diagram,xequals ..................................................................................................... ..................................................................................................... ..................................................................................................... ................................................ .. .. . .. .. . .. .. .. . .. .. . .. .. . .. .. . .. .. . .. .. . .. .. . .. .. . .. .. . .. . . . .. .. . .. .. . .. .. . .. .. . .. .. .. . .. .. . .. .. . .. .. . ................................................................ ..................................................................................................... ............ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x◦ 100 ◦ 110 ◦ 80 ◦ (A) 100 (B) 110 (C) 120 (D) 130 (E) 140 4.The value of200×8 200÷8is (A) 1 (B) 8 (C) 16 (D) 64 (E) 200 5.The smallest value thatx 2−4x+3 can have is (A)−1(B)−3(C)1(D)3(E)2 6.$3 is shared between two people. One gets 50 cents more than the other. The ratio of the larger share to the smaller share is (A) 6 : 1 (B) 7 : 5 (C) 4 : 3 (D) 5 : 3 (E) 7 : 4
S2 7.When 1000 2008 is written as a numeral, the number of digits written is (A) 2009 (B) 6024 (C) 6025 (D) 8032 (E) 2012 8.A semicircle is drawn on one side of an equilateral triangle. The ratio of the area of the semicircle to the area of the triangle is (A) 1 : 1 (B)π:2√ 3(C)π:√ 3 (D)√ 3:π(E) 3 :π ..................................................................................................... ..................................................................................................... ................................................................................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ......... ...................................... ....... .... .... ... ... ... .. ... .. .. .. .. . .. .. . .. . .. . . .. . . .. . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.Given that cosx=0.5and0 ◦
S3 13.What is the smallest whole number which gives a square number when multiplied by 2008? (A) 2 (B) 4 (C) 251 (D) 502 (E) 2008 14.A cross is made up of five squares, each with side length 1 unit. Two cuts are made, the first fromXtoYand the second fromZtoT,sothatZT Xis a right angle. The three pieces are then arranged to form a rectangle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....................... ..................................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............................................................................ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...................................................................... ...... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .............. ........................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .............................. .............................................. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... ......................................................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............................................................................ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. .. ... .. .. .. .. . . .. .. .. ... .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. . . .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. . .. .. .. .. .. .. Z YX T III III . . . . . . . . . . ... .. ... ..................................................................................... ......................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .......... ........................................................ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II III I What is the ratio of the length to the width of the rectangle? (A) 3 : 1 (B)√ 10:1 (C)2:1 (D)2√ 3:1 (E)5:2 15.A function is said to be a toggle function on (p, q , r)iff(p)=q,f(q)=rand f(r)=p. The functionf(x)=ax 2+bx+cis a toggle function on (1,2,3). What is the value ofc? (A)−2(B)0(C)3(D)9(E)14 16.Two conical rollers with perpendicu- lar axes touch on a line that is 30 ◦to the axis of the smaller roller and 60 ◦ to the axis of the larger roller. If the larger roller makes 1 revolution per sec- ond and there is no slipping, how many revolutions per second does the smaller roller make? (A)1 2(B) 1 (C)√ 2 (D)√ 3(E)2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . .. . .. . .. .. . .. .. .. . .. .. .. . .. .. .. ... .. ... ... .. ... ... .... ... ... .... .... .... ..... .... ...... ..... ....... ....... ........ .......... . ............... ...................................................................................... ..................................................................................................... ............ ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .. . ... .. .... .... ...... ....... ............................................ . . . . . . . .. .. .. .. ... .... ..... ........ ...................... ........................ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .. ................. ... ... ... .. .. .. .. . .. .. . .. .. . .. . .. . . .. . . .. . . .. . . .. . . . . .. . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....... .. .. .. . . .. . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . .. . . . . . .. . . . . . .. . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . .. . . . . . .. . . . . . .. . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . .. . . . . . .. . . . . . .. . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . .. . . . . . .. . . . . . .. . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . .. . . . . . .. . . . . . .. . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . .. . . . . . .. . . . . . .. . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . ............................... ..................................................................................................... ......................................................................... ............................ ....................... ................................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ................ ....... . ...... ...... .... .... .... ... ... ... ... ... ... .. .. ... .. .. .. ... .. .. .. . .. .. .. .. . .. .. .. . .. . .. .. . .. . .. . . .. . . . . . .. . . .. . . .. . . . . .. . . . . .. . . . . . .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............... . . .. . . . . . . . .. . . . . . . . . . . . . . . . . . .. . .. .. .. . . . . . . . . . . . .. .. . . .. . . . .. . . . . . . . . . . . . . . . ............... ................ ................................................ ............................................................ ................................................................................................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ◦ 30 ◦
S4 17.Consider the setX={1,2,3,4,5,6}. How many subsets ofX, with at least one element, do not contain two consecutive integers? (A) 16 (B) 18 (C) 20 (D) 21 (E) 24 18.Farmer Taylor of Burra has two tanks. Water from the roof of his farmhouse is collected in a 100 kL tank and water from the roof of his barn is collected in a 25 kL tank. The collecting area of his farmhouse roof is 200 square metres while that of his barn is 80 square metres. Currently, there are 35 kL in the farmhouse tank and 13 kL in the barn tank. Rain is forecast and he wants to collect as much water as possible. He should: (A) empty the barn tank into the farmhouse tank (B) fill the barn tank from the farmhouse tank (C) pump 10 kL from the farmhouse tank into the barn tank (D) pump 10 kL from the barn tank into the farmhouse tank (E) do nothing 19.A sequence{u 1,u 2,...,u n}of real numbers is defined by u 1=√ 2,u 2=π, u n=u n−1 −u n−2 forn≥3. What isu 2008 ? (A)−√ 2 (B) 2008(√ 2−2008π) (C) 1003√ 2−1004π(D)π(E)√ 2 20.In the diagram,RUis equal in length toST. What is the ratio of the area of QRUto the area ofQS T? (A)√ 3:1(B)2:1(C)√ 6:1 (D)√ 3:2 (E)√ 6:2 ..................................................................................................... ..................................................................................................... ........................................................ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............. . . . . . . . . . . . . . ............. . . . . . . . . . . . . . 45 ◦ 30 ◦ U T QR S Questions 21 to 25, 5 marks each 21.P,Q,R,SandTare consecutive vertices of a regular polygon. When extended, the linesPQandTSmeet atUwith QU S= 160 ◦.Howmanysideshasthe polygon? (A) 36 (B) 42 (C) 48 (D) 52 (E) 54
S5 22.How many numbers from 1, 2, 3, 4,..., 2008 have a cubic number other than 1 as a factor? (A) 346 (B) 336 (C) 347 (D) 251 (E) 393 23.The numbers 828 and 313 are 3-digit palindromes where 828−313 = 515, which is also a palindrome. How many pairs (a, b) of 3-digit palindromes are there with a>band witha−balso a 3-digit palindrome? (A) 1972 (B) 1980 (C) 1988 (D) 1996 (E) 2008 24.The centres of all faces of a cube are joined to form an octahedron. The centres of all faces of this octahedron are now joined to form a smaller cube. What is the ratio of an edge of the smaller cube to an edge of the original cube? (A) 1 :√ 2(B)1:√ 3(C)1:2(D)1:3(E)1:4 25.In the figure, all line segments are par- allel to one of the sides of the equi- lateral trianglePQRwhich has side length 1 unit. How long shouldPX be to maximise the smallest of the ten areas defined? (A)1 3(B)4−√ 2 14(C)1 4 (D)1 5(E)1 √ 10 ..................................................................................................... ..................................................................................................... ..................................................................................................... ........................................................................................ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............. ............. ....... ...... ............. .............. ............. ............. ............. ............. ............. ..... ......... ............. ............. ............. ............. ............. .............. ............. ...... ....... ............. ............. ............. ..... .................................................. ..................................................................................................... ..................................................................................................... .......................................................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .......................................... ........................ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .......... ........... ... ............... .............. ....... P QRX For questions 26 to 30, shade the answer as an integer from 0 to 999 in the space provided on the answer sheet. Question 26 is 6 marks, question 27 is 7 marks, question 28 is 8 marks, question 29 is 9 marks and question 30 is 10 marks. 26.All possible straight lines joining the vertices of a cube with mid-points of its edges are drawn. At how many points inside the cube do two or more of these lines meet?
S6 27.Let us call a sum of integerscoo lif the first and last terms are 1 and each term differs from its neighbours by at most 1. For example, the sum 1 + 2 + 3 + 4 + 3 + 2+3+3+3+2+3+3+2+1 is cool. How many terms does it take to write 2008 as a cool sum if we use no more terms than necessary? 28.The positive integersxandysatisfy 3x 2−8y 2+3x 2y2= 2008. What is the value ofxy? 29.ApointOis inside an equilateral triangle PQRand the perpendicularsOL,OMand ONare drawn to the sidesPQ,QRandRP respectively. The ratios of lengths of the perpendiculars OL:OM:ONis 1 : 2 : 3. Ifarea ofLONP area ofPQR=a b,whereaandbare integers with no common factors, what is the val u e ofa+b? ..................................................................................................... ..................................................................................................... ..................................................................................................... ............................ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ................................................... ................................. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R PQ LM N O .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.What is the smallest value that 49 +a 2−7√ 2a+ a2+b 2−√ 2ab+√ 50 +b 2−10b can have for positive real numbersaandb? ***