Time allowed: 90 minutesTotal marks: 100Questions: 37
Instructions
• Answer ALL questions.
• Show your working where required — marks may be awarded for method.
• Write your answers clearly in the spaces provided.
• You may NOT use a calculator unless stated otherwise.
• Check your work if you finish early.
1. Q1(a): Find the value of (3 2/3 + 2/9) x 7 2/7, giving your answer as a reduced, mixed fraction.
[3 marks]
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2. Q1(b): Find the value of (68/19 / 17/76) / 6/7, giving your answer as a reduced, mixed fraction.
[4 marks]
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3. Q1(c): Find the value of 327 7/12 + 271 5/9, giving your answer as a reduced, mixed fraction.
[3 marks]
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4. Q1(d): Find the value of (4 - 3/4)^2, giving your answer as a reduced, mixed fraction.
[3 marks]
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5. Q2(a): Find the value of 0.035 x 0.0022, giving your answer as a decimal.
[3 marks]
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6. Q2(b): Find the value of 0.51 / 0.068, giving your answer as a decimal.
[3 marks]
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7. Q2(c): Find the value of (-1.1)^3, giving your answer as a decimal.
[3 marks]
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8. Q3(a): If a = 1 and b = -2, find the value of a/b - b/a, leaving your answer in simplified form.
[1 mark]
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9. Q3(b): If a = 1 and b = -2, find the value of (a^2 + b^2) / (a + b), leaving your answer in simplified form.
[2 marks]
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10. Q4(a): Simplify the following algebraic expression fully, leaving no brackets in your final answer: 2x - (3y + x) + {3x - (5y - 4x + 7y)}
[2 marks]
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11. Q4(b): Simplify the following algebraic expression fully, leaving no brackets in your final answer: a - [a - b - {d - c + (a - b + c - d)}]
[2 marks]
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12. Q5: Solve the following inequality, giving your final answer as a reduced, mixed fraction. In your final answer, x must appear on the left-hand side.
3 - 7x < 19 - 2x
[3 marks]
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13. Q6: I have a glass containing five and two fifteenths fluid ounces of wine. I pour out one and seven twelfths fluid ounces of the wine. Find the volume of wine remaining in the glass, in fluid ounces as a reduced, mixed fraction.
[4 marks]
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14. Q7: Six years ago, Alice was 5 times as old as Beatrice was, but now she is only twice as old. Find the difference between the ages of Alice and Beatrice.
[4 marks]
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15. Q8: In the following diagram, line segments AB and CD are parallel. There is a transversal from A (on line AB) going up to a point above line CD, where it meets another line. The angle at A (between line AB and the transversal) is 132 degrees. At the top intersection, the angle is 161 degrees. The angle x is at the point where the transversal crosses line CD. Calculate angle x. (Refer to original PDF for figure.)
[3 marks]
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16. Q9: A class contains 30 pupils. 14 are boys and the rest are girls. In a test, the average mark of the boys is 62%, and the average mark of the girls is 68%. Find the average mark of the entire class, leaving your answer as a percentage correct to 1 decimal place.
[4 marks]
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17. Q10: Solve the following equation, simplifying your final answer:
(2/3)(2x/3 - 3) - (1/6)(3x/2 - 8) = x/12
[3 marks]
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18. Q11(a): If one builder can build a wall in 5 hours, and a second builder can build one of the same size in 7 hours, how long will they take to build a wall working together? Give your answer in hours and minutes.
[2 marks]
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19. Q11(b): If one builder can build a wall in A hours, and a second builder can build one of the same size in B hours, how many hours will they take to build a wall working together? Leave your answer as a single fraction.
[2 marks]
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20. Q11(c): If p litres of paint are required to paint a rectangular wall of side lengths 5q by q, how many litres of paint are required to paint a rectangular wall of side lengths 3r by r?
[3 marks]
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21. Q11(d): A car travels at a rate of x feet in y seconds. How many hours does it take to travel z miles? You are given that 1 mile = 5280 feet. Leave your answer as a reduced fraction.
[3 marks]
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22. Q12(a): The volume of a cylinder is equal to the area of its circular base multiplied by its perpendicular height. Suppose a cylinder has eight times the perpendicular height of a second cylinder and has a circular base one tenth the diameter of the second cylinder. What is the ratio of the volume of the first cylinder to the volume of the second cylinder?
[3 marks]
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23. Q12(b): Given that a suitably sized rectangular piece of paper can be wrapped around the curved surface of a cylinder so as to cover it exactly once with no overlap, write down a formula for the curved surface area of a cylinder in terms of the radius r of its base and its perpendicular height h.
[2 marks]
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24. Q12(c): A child's toy is made of two solid cylinders joined together as illustrated. The larger cylinder has diameter 6p cm and perpendicular height 2p cm. The smaller cylinder has diameter 4p cm and perpendicular height 2p cm. Find a formula for the total exposed surface area of the toy, leaving your answer simplified and in terms of p and pi. (Refer to original PDF for figure.)
[5 marks]
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25. Q13(a)(i): Suppose that x = 7.53333... (where the 3 repeats). Write down the value of 10x and 100x.
[2 marks]
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26. Q13(a)(ii): Hence, given that x = 7.5333..., what is the value of 90x? Also express 7.5333... as a reduced, mixed fraction.
[2 marks]
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27. Q13(b): Suppose that y = 1.9999... (where the 9 repeats). Using a method similar to part (a), what is the value of y?
[2 marks]
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28. Q13(c): Suppose that z = 17.bccc... (where the digit c repeats), and b and c are digits between 0 and 9. Find whole numbers u, v and w such that z = (u + vb + c)/w.
[4 marks]
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29. Q14(a): In the diagram, length AE equals 2*sqrt(2) units and M is the mid-point of AE. Points B, C and D lie on a semicircle with diameter AE. Lengths AB, BC, CD and DE are all equal and angle ACE is a right angle. What is the length of AC? (Refer to original PDF for figure.)
[1 mark]
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30. Q14(b): In the same diagram, CLMN is a quadrilateral where L is between M and E, and N is between A and M. Prove that CLMN is a square. What is the side length of CLMN? (Refer to original PDF for figure.)
[2 marks]
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31. Q14(c): In the same diagram, what is the length DL? Express your answer in simplified surd form. (Refer to original PDF for figure.)
[2 marks]
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32. Q14(d): In the same diagram, what is the length DE? Express your answer in simplified surd form. (Refer to original PDF for figure.)
[3 marks]
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33. Q14(e): Hence, using the results from the previous parts, what value is pi greater than? Express your answer as 4 * sqrt(2 - sqrt(2)). In other words, verify that pi > 4 * sqrt(2 - sqrt(2)). (Refer to original PDF for figure.)
[2 marks]
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34. Q15(a): The Towers of Hanoi puzzle uses three poles (A, B, C) and discs of different sizes. Only one disc can be moved at a time, and a larger disc can never be placed on top of a smaller one. The goal is to move all discs from Pole A to Pole C. For 2 discs (labelled 1 and 2, where 2 is larger), the initial position has both on Pole A. After Move One, Pole A has disc 2, Pole B has disc 1. After Move Three, both discs are on Pole C. What is the position after Move Two?
[1 mark]
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35. Q15(b): For the Towers of Hanoi with 3 discs (labelled 1, 2, 3 in order of increasing size), some moves are given. The initial position is all on Pole A. Move 1: A={3,2}, B=empty, C={1}. Move 2: A={3}, B={2}, C={1}. Move 3: A={3}, B={2,1}, C=empty. Move 4: A=empty, B={2,1}, C={3}. Move 7: all on Pole C. Complete Moves 5 and 6.
[3 marks]
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36. Q15(c): For the Towers of Hanoi with 4 discs, given the first 3 moves and the final position (Move 15, all on Pole C), complete the table showing all 15 moves. The given entries are: Initial: A={4,3,2,1}. Move 1: A={4,3,2}, B={1}. Move 2: A={4,3}, B={1}, C={2}. Move 3: A={4,3}, C={2,1}. Move 15: C={4,3,2,1}. List the positions for Moves 4 through 14.
[5 marks]
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37. Q15(d): If the Towers of Hanoi game is now played with n discs, conjecture a formula, in terms of n, for the smallest number of steps necessary to complete the puzzle, giving a reason for your answer.