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Questions 1 to 10, 3 marks each 1. What is the area of this triangle in square
centimetres?
(A) 10 (B) 12 (C) 14 (D) 7 (E) 6
12 cm 2 cm
2. A movie lasts for 21
3 hours. The movie is shown in two equal sessions.
For how many minutes does each session last?
(A) 85 (B) 70 (C) 80 (D) 65 (E) 75
3. If p = 11 and q =−4, then p2− q2 equals
(A) 105 (B) 137 (C) 117 (D) 115 (E) 94
4. 2015− 20.15 equals
(A) 1984.85 (B) 1995.15 (C) 1994.85 (D) 1995.85 (E) 2035.15
5. What is the value of 2015 twenty-cent coins?
(A) $2015 (B) $107.50 (C) $17.50 (D) $403 (E) $43
6. Ana, Ben, Con, Dan and Eve are sitting around a ta-ble in that order. Ana calls out the number 1, then Ben calls out the number 2, then Con calls out the number 3, and so on. After a person calls out a num-ber, the next person around the table calls out the next number.
Anyone who calls out a multiple of 7 must immediately leave the table.
Who is the last person remaining at the table? (A) Ana (B) Ben (C) Con
(D) Dan (E) Eve
Ana
Ben
Con Dan
I 2
7. On a farm the ratio of horses to cows is 3 : 2 and ratio of cows to goats is 4 : 3. The ratio of goats to horses is
(A) 5 : 7 (B) 3 : 8 (C) 3 : 5 (D) 5 : 18 (E) 1 : 2
8. Warren the window washer starts on the 38th floor of a building that has 12 windows per floor. He washes all of the windows on each floor before moving down to the floor below. Which floor is Warren on after he has washed 141 windows?
(A) 25th (B) 24th (C) 28th (D) 27th (E) 26th
9. A packet of lollies contains 5 blue lollies, 15 yellow lollies and some red lollies. One-third of the lollies are red. What fraction of the lollies are yellow?
(A) 1 3 (B) 5 6 (C) 1 2 (D) 1 6 (E) 2 3
10. The diagram shows two small squares in opposite corners of a large square. The squares have sides of length 1 cm, 2 cm and 7 cm.
What is the area of the shaded pentagon?
(A) 18 cm2 (B) 16 cm2 (C) 22 cm2
Questions 11 to 20, 4 marks each
11. Jenna measures three sides of a rectangle and gets a total of 80 cm. Dylan measures three sides of the same rectangle and gets a total of 88 cm. What is the perimeter of the rectangle?
(A) 112 cm (B) 132 cm (C) 96 cm (D) 168 cm (E) 156 cm
12. A bar-tailed godwit was recorded by satellite tag in 2007 to have flown 11 500 km in eight days.
On average, approximately how many kilometres per hour is that?
(A) 120 (B) 6 (C) 1 (D) 24 (E) 60
13. A cube has the letters A, C, M, T, H and S on its six faces. Here are two views of this cube.
C
M
A
A
M
T
Which one of the following could be a third view of the same cube? (A)
M
H
T
(B)A
C
T
(C)T S
C
(D)H
T
A
(E)S
M
C
14. Two ordinary dice are rolled. The two resulting numbers are multiplied together to create a score. The probability of rolling a score that is a multiple of six is
(A) 1 6 (B) 5 12 (C) 1 4 (D) 1 3 (E) 1 2
I 4
15. The diagram shows part of a tiled floor. The tiles are all squares, of seven different sizes. The smallest tile is 1 cm× 1 cm and the next smallest is 3 cm× 3 cm.
What is the distance XY ? (A) √346 cm (C) √369 cm (B) 19 cm (D) √377 cm (E) 20 cm X Y
16. How many three-digit numbers have no two digits the same?
(A) 764 (B) 900 (C) 648 (D) 520 (E) 480
17. The diagram shows two weightless balance beams. Each tips as a result of an imbalance in the hanging weights. Each weight has mass X kg, Y kg or Z kg, as shown.
Which of the following lists the weights from lightest to heaviest? (A) X < Y < Z (B) X < Z < Y (C) Y < X < Z (D) Y < Z < X (E) Z < X < Y Y kg Y kg Z kg X kg Y kg X kg
18. A strip of paper 1 cm wide is folded 4 times to make a regular octagon as shown.
If the ends of the strip meet exactly when folded, how many centimetres long is the strip?
19. The country of Numismatica has six coins of the following denominations: 1 cent, 2 cents, 4 cents, 10 cents, 20 cents and 40 cents.
Using the coins in my pocket, I can pay exactly for any amount up to and including 200 cents.
What is the smallest number of coins I could have? (A) 12 (B) 10 (C) 11 (D) 9 (E) 8 2c 1c 4c 10c 20c 40c
20. What fraction of the large triangle is shaded?
(A) 1 6 (B) 1 3 (C) 4 9 (D) 1 2 (E) 2 5 1 2 1 2 1 2
Questions 21 to 25, 5 marks each
21. A student noticed that in a list of five integers, the mean, median and mode were consecutive integers in ascending order. What is the largest range possible for these five integers?
(A) 5 (B) 9 (C) 8 (D) 7 (E) 6
22. The square P QRS has sides of length 2 units and
J is the midpoint of P S. The line QJ intersects
the diagonal P R at L. The length of LP is (A) √ 2 3 (B) √ 3 3 (C) √ 2 2 (D) 2 √ 3 3 (E) 2√2 3 S Q 2 2 1 1 P R J L
I 6
23. For each integer from 0 to 999, Andr´e wrote down the sum of its digits. What is the average of the numbers that Andr´e wrote down?
(A) 13.5 (B) 15 (C) 12 (D) 12.5 (E) 10.5
24. Max’s journey around this grid starts on a grid point on side AB.
He visits a grid point on each of sides BC, CD and
DA in order before returning to his starting point,
forming a quadrilateral.
Max does not visit corner points A, B, C or D. How many journeys are possible which are not rect-angles? (Note that a square is a rectangle.)
(A) 256 (B) 252 (C) 64 (D) 248 (E) 76
A D
C B
25. It takes Nicolai one and a half hours to paint the walls of a room and two hours to paint the ceiling. Elena needs exactly one hour to paint the walls of the same room and one hour to paint the ceiling.
If Nicolai and Elena work together, what is the shortest possible time in minutes in which they can paint the walls and the ceiling of the room?
(A) 72 (B) 60 (C) 83 (D) 75 (E) 76
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. Mike has 2015 matches and uses them to build a triangular pattern like the one shown, but as big as possible. How many matches does he have left over?
27. How many positive integers n less than 2015 have the property that 1
3 + 1
n can be
simplified to a fraction with denominator less than n?
28. A rectangle has all sides of integer length. When 3 units are added to the height and 2 units to the width, the area of the rectangle is tripled. What is the sum of the original areas of all such rectangles?
29. At Berracan station, northbound trains arrive every three minutes starting at noon and finishing at midnight, while southbound trains arrive every five minutes starting at noon and finishing at midnight. Each day, I walk to Berracan station at a random time in the afternoon and wait for the first train in either direction. On average, how many seconds should I expect to wait?
30. In a 14× 18 rectangle ABCD, points P, Q, R and S are chosen, one on each side of ABCD as pictured. The lengths AP , P B, BQ, QC, CR, RD, DS and
SA are all positive integers and P QRS is a rectangle.
What is the largest possible area that P QRS could have? D C B A R Q P S
