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[PDF] 2019 AMC Senior Years 11 and 12 Questions Australian Mathematics Competition [PDF]
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Copyright © 2019 Australian Mathematics Trust AMTT Limited ACN 083 950 341 AUSTRALIAN MATHEMATICS COMPETITION Senior Years 11 & 12 (Australian school years) NAME: TIME ALLOWED: 75 minutes INSTRUCTIONS AND INFORMATION General 1 Do not open the booklet until told to do so by your teacher. 2 NO calculators, 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 requiring a single answer, and 5 questions that require a whole number answer 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 country/Australian state 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 entered. It is your responsibility to correctly code your answer sheet. 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 scanned. The optical scanner will attempt to read 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 AMT reserves the right to re-examine students before deciding whether to grant official status to their score. Reminder: You may sit this competition once, in one division only, or risk no score. THURSDAY 1 AUGUST 2019
Senior Division Questions 1 to 10, 3 marks each 1. What is the value of 201 ×9? (A) 189 (B) 1809 (C) 1818 (\b) 2001 (E) 2019 2. What is the area of the shaded triangle? (A) 8 m 2 (B) 12 m 2 (C) 14 m 2 (\b) 20 m 2 (E) 24 m 2 6m 4m 4m 3. What is 19% of $20? (A) $20.19 (B) $1.90 (C) $0.19 (\b) $3.80 (E) $0.38 \b. What is the value of z? (A) 30 (B) 35(C) 45 (\b) 50 (E) 55 50 ◦ 45 ◦ z◦ 60 ◦ 5.The value of 2 0+1 9is (A) 1 (B) 2(C) 3 (\b) 10 (E) 11 6.Letf(x)=3 x 2−2x. Then f(−2) = (A) −32 (B)−8 (C) 16 (\b) 32 (E) 40 2019 AMC — Senior
S2 7.This kite has angles θ, θ, θand θ 3. What is the size of the angle θ? (A) \b20 ◦ (B) \b05 ◦ (C) 90 ◦ (D) \b\b2 ◦ (E) \b08 ◦ θ θ 3 θθ 8. Consider the undulatingnumber sequence \b, 4,7,4,\b,4,7,4,\b,4,... , which repeats every four terms. The running total of the first 3 terms is \b2. The running total of the first 7 terms is 28. Which one of the following is also a running total of this sequence? (A) 6\b (B) 62(C) 67 (D) 66 (E) 65 9.Mia walks at \b.5 metres per second. Her friend Crystal walks at 2 metres per second. They walk in opposite directions around their favourite bush track, starting together from the same point. They first meet again after 20 minutes. How long, in kilometres, is the track? (A) 3.5 (B) 4.2(C) 6(D) 7 (E) 8.4 10.\b 1+2 2+3 3+4 4 \b1+2 2+3 3 = (A) 2 3 (B) 3 2 (C) \b\b (D) 4 3 (E) 259 Questions 11 to 20, 4 marks each 11. The 5-digit number P679Q is divisible by 72. The digit Pis equal to (A) \b (B) 2(C) 3 (D) 4 (E) 5 12.The altitude of a right-angled triangle divides the hypotenuse into lengths of 4 and 6. What is the area of the triangle? (A) \b0 √ 6 (B) 24 (C) 25 (D) \b2 (E) 6√ \b0 - ˆ 3 2019 AMC — Senior
4 ffffi ffffi 3 7 \b 5 8 \b ffi? ? ffiffi? ffi? ? ?? ? ?? ff ? ? ?\b ? ?? \b? 1 2 ? 2 3 ? 1 4 ? 3 4 ? ff2 2 ff ? \b ? ? ? ??? ff ? ??? ff ? ??? ff ? ?ffi? ff ? ??? ff ff ? ?? ? ? ?? ? ??? \b? ?ff?ff??ff ??ff ffiffiffi ? ffi ffiff ?ff??ff ??ff ffiffiffi ? ? \b \b ?? ? ?\b\b ??? ? ffi??? ??? ? ??? ? ?? ff ? ?? ?? ? ? ? ?? ?? ??? ? ? ? ? ? ? ? ?? 48 25ff ? ff3 ff ? 2 ff ? 8 ff 2 25ff ? 8 5ff 2019 AMC — Senior
S4 18.For what values of xdoes the triangle with side lengths 5, 5 and xhave an obtuse angle? \bA) 0
6 ffffi ffffffi \b \b \b ? ? ff ??? ?ffiffi ??? ? ff??? ? ??? ? ? ff 2 ff ffi ??? ?? ?? ? ? ?? ?? \b ?\b ? ? ? ?? ? ??? ? ? ??? ffi? ? ??? ? ? ??? ?? ? ff ffi ff \b ff ff ff ff ffi ff ffi ff ff ffffi ff ff 2 ff ff ff ffffi ffi \b \b ? ? ? \b \b \b ? ?\b \b \b ? ? \b ? ffffi 1 2 ff 1 3 ff ff2 2 ff ff ffi ff ff5ffi 1 2 2019 AMC — Senior 2019 AMC — Senior
S6 For questions 26 to 30, shade the answer as an integer from 0 to 999in the s\bace \brovided on the answer sheet. Questions 26–30 are worth 6, 7, 8, 9 and 10 marks, res\bectively. 26. The number 35 has the property that when its digits are both increased by \b, and then multiplied, the result is 5 ×7 = 35, equal to the original number. Find the sum of all two-digit numbers such that when you increase both digits by \b, and then multiply these numbers, the product is equal to the original number. 27. In a list of numbers, an odd-sum tripleis a group of three numbers in a row that add to an odd number. For instance, if we write the numbers from 1 to 6 in this order, 64\b135 then there are exactly two odd-sum triples: (4, \b,1) and (1, 3,5). What is the greatest number of odd-sum triples that can be made by writing the numbers from 1 to 1000 in some order? 28. Terry has a solid shape that has four triangular faces. Three of these faces are at right angles to each other, while the fourth face has side lengths 11, \b0 and \b1. What is the volume of the solid shape? 29. The diagram shows one way in which a 3 ×10 rectangle can be tiled by 15 rectangles of size 1 ×\b. Since this tiling has no symmetry, we count rotations and reflections of this tiling as different tilings. How many different tilings of this 3 ×10 rectangle are possible? 30. A function f, defined on the set of positive integers, has f(1) = \b and f(\b) = 3. Also f (f (f(n))) = n+ \b if nis even and f(f (f(n))) = n+ 4 if nis odd. What is f(777)? 7 2019 AMC — Senior
2019 AMC — SENIOR SOLVE PROBLEMS. CREATE THE FUTURE. amt.edu.au Problems are part of life and we’ve made it our mission to equip young students with the skills to solve more of them. Problem solving is a life skill and by developing it, students can create more choices for themselves and the future.