2022 AMC Paper SEN OMR [PDF]

Citation preview

2020



2022



AUSTRALIAN MATHEMATICS COMPETITION



Senior Years 11–12 (AUSTRALIAN SCHOOL YEARS)



Instructions and Information General



DATE



3–5 August



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. 5. This is a competition and not a test so don’t worry if you can’t answer all the questions. Attempt as many as you can — there is no penalty for an incorrect answer. 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 Your answer sheet will be scanned. To make sure the scanner reads your paper correctly, there are some DOs and DON’Ts: DO: • use only a lead pencil • record your answers on the answer sheet (not on the question paper) • for questions 1–25, fully colour the circle matching your answer — keep within the lines as much as you can • for questions 26–30, write your 3-digit answer in the box — make sure your writing does not touch the box • use an eraser if you want to change an answer or remove any marks or smudges. DO NOT: • doodle or write anything extra on the answer sheet • colour in the QR codes on the corners of the answer sheet.



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.



Copyright © 2022 Australian Mathematics Trust | ACN 083 950 341



TIME ALLOWED



75 minutes



2022 AUSTRALIAN MATHEMATICS COMPETITION SENIOR



Senior Division Questions 1 to 10, 3 marks each 1.



The temperature in the mountains was 4 ◦ C but dropped overnight by 7 ◦ C. What was the temperature in the morning? (A) 3 ◦ C



2.



(B) 11 ◦ C



(B) 9



5.



(E) 15



(B)



1 42



(C) −



(D) 50



1 440



(D) −



(E) 65



1 44



(E)



Three vertices of a rectangle are the points (1, 4), (7, 4) and (1, 8). At which point do the diagonals of the rectangle cross? (A) (4, 6)



(B) (3, 2)



(C) (3, 1)



(D) (5, 6)



6.



h



15



The value of 51 + 42 + 33 + 24 + 15 is (A) 20 (B) 30 (C) 35



1 1 − = 20 22 1 (A) 2



(E) −11 ◦ C



Area = 135



(C) 11



(D) 13



4.



(D) −4 ◦ C



The rectangle shown has area 135. What is the value of h in the diagram? (A) 7



3.



(C) −3 ◦ C



(E) (7, 8)



1 1 1



What fraction of this trapezium is shaded? (A)



1 2



(B)



1 3



(C)



1 4



(D)



1 5



(E)



4 15



2



1



2



1 220



2022 AUSTRALIAN MATHEMATICS COMPETITION SENIOR



√ 2 7



√



√ −4 7



√ −3 7



√ 5 7



In this diagram, what is the value of x? (B) 70



(C) 75



(D) 80



x◦



(E) 85 x◦ 140◦



40◦



The smaller of these dice has three zeroes and three ones on its faces and the larger has the numbers 1, 3, 5, 7, 9, and 11 on its faces. Both dice are rolled once and the numbers showing on top are added. What is the probability of obtaining a sum of 12? 1 6



(B)



− 12



10. The value of 24 (A) 0.2



(C)



+ 17



− 14



(B) 0.5



1 9



(D)



1 12



(E)



1 36



0



(A) 0



3



(A) 65



9.



√ 3 5



5



7



8.



The sum √of the numbers on these six cards is 4 5. Lily removes one of them. What is the largest possible sum of the remaining five cards? √ √ √ √ (A) 4 5 + 3 7 (B) 4 5 + 7 7 √ √ √ √ (C) 4 5 − 2 7 (D) 4 5 − 5 7 √ √ (E) 4 5 + 4 7



11



1



7.



1



is closest to (C) 0.7



(D) 1



(E) 1.2



Questions 11 to 20, 4 marks each 11. The operation  is defined by p  q = 2p − q. Each of p and q is an integer from −6 to +6. How many pairs of values (p, q) will have p  q = q  p? (A) 6



(B) 13



(C) 36



12. In the diagram, the lengths of the sides of the triangle are 8, 9 and 13 centimetres. The centres of the circles are at the vertices of the triangle and the circles just touch. The radius, in centimetres, of the largest circle is (A) 6 (B) 6.5 (C) 7 (D) 7.5 (E) 8



(D) 49



(E) 169



2022 AUSTRALIAN MATHEMATICS COMPETITION SENIOR



13. The four integers 3, 4, 8, 11 have their mean and range calculated. A fifth integer is then included that is different from the other four. This doesn’t change the range, but the mean is now an integer. What is this new mean? (A) 4 (B) 5 (C) 6 (D) 7 (E) 8



B D



>



14. Two triangles ABC and BCD are both right angled as shown. Lines AB and CD are parallel. Also AC = 1 and AB = 2. What is the length of CD? √ √ 3 (A) 2 (B) (C) 3



>



2



2



16 (D) 9



(E)



5 3



1



A



C



√ 15. The value of x in the equation 3x + 3x+1 + 3x+2 = 13 3 is √ √ 1 13 3 − 9 (A) 0 (B) (C) (D) 3 2



√ 13 3 − 3 (E) 9



9



16. Four numbers are equally spaced on the number line, in the given order 1 , 20



1 , 22



X,



Y



What is the value of Y ? (A)



1 26



(B)



2 21



(C)



3 44



(D)



17. A small equilateral triangle sits inside a larger equilateral triangle as shown. What is the ratio of the areas of the smaller and larger equilateral triangles? (A) 1 : 3



(B) 7 : 16 (D) 37 : 64



(C) 9 : 16 (E) 21 : 32



2 55



(E)



2 6 6 2 6



2



3 26



2022 AUSTRALIAN MATHEMATICS COMPETITION SENIOR



18. The square pyramid shown is divided into 3 pieces, P , Q and R, by two planes that are parallel to the square base. Each of the 3 pieces has the same height. The volume of the piece P is 5 cm3 . What is the volume of piece R, in cm3 ? (A) 15



(B) 25



(C) 40



(D) 95



P Q R



(E) 125



19. A sequence of values a1 , a2 , . . . , a100 is calculated as follows: a1 = 1 ,



a2 = 2 ,



a3 =



a2 + 1 , a1



...



an =



an−1 + 1 an−2



,



...



a100 =



a99 + 1 a98



What is a100 ? (B) 2



(C) 3



(D) 5



20. A triangular ramp is in the shape of a right-angled tetrahedron. The horizontal base is an equilateral triangle with sides 8 metres. The apex is 1 metre directly above one corner of the base, so that two faces are vertical. In square metres, what is the area of the sloping face? √ 65 √ (A) 16 3 (B) 28 (C) 3 4 √ (D) 4 33 (E) 32



(E) 50



1 | |



8



|



(A) 1



Questions 21 to 25, 5 marks each 21. A cuboctahedron is a solid formed by joining the midpoints of the edges of a cube as shown. What is the volume of a cuboctahedron of side length 2? √ √ √ 10 2 40 2 (A) (B) (C) 8 2 3 3 √ √ √ 8√ (E) 8 2 + 3 (D) 6 2 + 8 3 3



2



2



2



2



2022 AUSTRALIAN MATHEMATICS COMPETITION SENIOR



22. In the two equations ax − b = c and dy + e = f , each of the letters a, b, c, d, e, and f is replaced by a different digit from 1 to 9. When the two equations are solved for x and y, the lowest possible value of x + y is (A) less than −7



(B) between −7 and −5



(D) between −3 and −1



(C) between −5 and −3



(E) greater than −1



23. A semicircle is inscribed in a right-angled isosceles triangle and a square is inscribed in the semicircle as shown. What is the ratio of the area of the square to the area of the triangle? √ (A) 1 : 3 (B) 1 : 2 (C) 2 : 5 √ (E) 3 : 8 (D) 1 : 2 2



24. In the grid shown, the numbers 1 to 8 are placed so that when joined in ascending order they make a trail. The trail moves from one square to an adjacent square but does not move diagonally. In how many ways can the numbers 1 to 8 be placed in the grid to give such a trail? (A) 12



(B) 20



(C) 24



2



3



8



7



1



4



5



6



(D) 28



(E) 36



25. When I cycled around the lake yesterday, my children Sally and Wally decided to ride the same route in the opposite direction. We all set off at the same time, from the same point, and finished at that same spot. We each rode at our own steady speed. It took me 77 minutes. Sally and I passed each other, waving, exactly 42 minutes after we started. Precisely 2 minutes later, Wally and I passed each other, puffing. To the nearest minute, how much longer did Wally take than Sally to ride around the lake? (A) 2 (B) 4 (C) 6 (D) 8 (E) 10



2022 AUSTRALIAN MATHEMATICS COMPETITION SENIOR



For questions 26 to 30, shade the answer as an integer from 0 to 999 in the space provided on the answer sheet. Questions 26–30 are worth 6, 7, 8, 9 and 10 marks, respectively.



26. Horton has a regular hexagon of area 60. For each choice of three vertices of the hexagon, he writes down the area of the triangle with these three vertices. What is the sum of the 20 areas that Horton writes down?



27. An even square number is multiplied by an odd cube greater than 1, resulting in a fourth power. If the fourth power is as small as possible, what is the sum of the square and the cube?



28. In an infinite sequence, the first two terms are 2 and 6, and apart from the first term, each term is one less than the average of its two neighbours. What is the largest term less than 1000?



29. Wasteful Wayne takes one sheet of paper with ‘My Document’ printed on it. He runs it through the photocopier to make two copies which he then stamps with his ‘COPY’ stamp. Wayne then takes the original and the two copies, runs all three through the photocopier to make two copies of each, stamps the six new copies with his ‘COPY’ stamp, and adds them to the top of the pile. He repeats this process by making two copies of each sheet of paper in his existing pile, stamping the new copies, then adding them to the pile. So the pile triples in size each time. After Wayne has done this eight times in total, the pile is 6561 sheets high. How many sheets have exactly 2 ‘COPY’ stamps on them?



30. Ali, Beth, Chen, Dom and Ella finish a race in alphabetical order: Ali in first place, then Beth, Chen, Dom and Ella. The next week, they run another race and their placings all change. Two of the runners receive a placing higher than the week before, and the other three runners receive a placing lower than the week before. Given this information, in how many orders could the five runners have finished this second race?



Senior Years 11–12 (AUSTRALIAN SCHOOL YEARS)



CORRECTLY RECORDING YOUR ANSWER (QUESTIONS 1–25) Only use a lead pencil to record your answer. When recording your answer on the sheet, fill in the bubble completely. The example below shows the answer to Question 1 was recorded as ‘B’.







Correct



DO NOT record your answers as shown below. They cannot be read accurately by the scanner and you may not receive a mark for the question.



Incorrect







Incorrect







Incorrect



Incorrect



 



this one!



Incorrect



 



Incorrect



Use an eraser if you want to change an answer or remove any pencil marks or smudges. DO NOT cross out one answer and fill in another answer, as the scanner cannot determine which one is your answer.



CORRECTLY WRITING YOUR ANSWER (QUESTIONS 26–30) For questions 26–30, write your answer in the boxes as shown below. 1 digit



2 digits



5



2+3=



3 digits



4 l



20 + 21 =



2 3 8



200 + 38 =



WRITING SAMPLES







0



0



0



Correct



3



3



3



Correct



6



6



6



Correct



9



9



9



 







Correct







1



l



4



4



4



Correct



7



7



7



Correct



1



1



Correct



 



2



3 6







2



2



2



Correct



5



5



5



Correct



8



8



8



Correct



5 0 4



5



8



 







Incorrect



Your numbers MUST NOT touch the edges of the box or go outside it. The number one must only be written as above, otherwise the scanner might interpret it as a seven. DO NOT doodle or write anything extra on the answer sheet or colour in the QR codes on the corners of the answer sheet, as this will interfere with the scanner.