德威公学Dulwich College Year 9 数学入学考试笔试真题 内附标准答案哟!

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　　这篇笔试考试允许的时间:1小时30分钟

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　　•可能使用计算器。

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　　•这篇论文总共有100分。

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　　•每个问题的分数都在方括号内。

　　1. Use your calculator to work out the value of

　　(a) Write down the first 10 digits shown on your calculator.

　　(b) Write your answer to (a) rounded to 3 decimal places.

　　(c) Write your answer to (a) rounded to 2 significant figures

　　2. (a) A car was bought on January 1^st 2010 for £16,000. By January 1^st 2011, its value had fallen by 20%.

　　(i) Calculate the value of the car on January 1^st 2011.

　　(ii) In each subsequent year, the value of the car fell by 15%. Calculate its value on January 1^st 2013.

　　(b) A rare postage stamp was bought for £300 in 1980. Its value at that time was just 12% of its current value. Calculate the current value of the stamp.

　　3. Below is a sequence of numbers:

　　(c) David saves 5p on Sunday 1^st December, 8p on Monday 2nd December, 11p on Tuesday 3^rd December, and so on according to the sequence above. Calculate the day and date on which David will save 77p.

　　4. a) Adam and Brian share £544 in the ratio 7 ∶ 9. How much does Brian receive?

　　b) Last month a local shop found that its total revenue from selling calculators and pens could be expressed in the ratio 𝑐 ∶ 𝑝. That total revenue was £150, with the greater portion coming from calculator sales. In terms of c and p, what is the difference between the revenues for calculators and pens?

　　5. Simplify the following:

　　6. The diagram, which is not drawn to scale, shows a shape with one line of symmetry。

　　Calculate the area of this shape.

　　7. Calculate the length of the side marked x, giving your answer correct to 2 decimal places.

　　8. (a) On the axes below, draw the line given by the equation 𝑦 = 2 − 𝑥 .

　　9. Solve the following equations for x:

　　10. Find the values of 𝑎, 𝑏 and 𝑐 in the diagrams below.

　　(b) The diagram below shows two identical squares meeting at one of their corners.

　　11. Factorise the following expression fully:

　　12. While taking part in a 10 km race, a runner completed the first 6500 m in 26 minutes.

　　(a) Calculate the average speed of the runner, in km per hour, over this section of the course.

　　The runner’s target was to complete the entire race is under 40 minutes. For the remaining 3500 m his average speed was 16 km per hour.

　　(b) Show your working and conclusion clearly, determine whether the runner was successful in achieving his target.

　　13. The volume of the prism below is 2450 cm³

　　Calculate the length marked x in the diagram.

　　15. The diagram below, which is not drawn to scale, shows three sides of a regular polygon with n sides. Work out the value of n.

　　16. The mean of 8 numbers is m. When one of these numbers is discarded, the mean of the remaining 7 numbers falls to m - 4. In terms of m, what was the value of the discarded number?

　　17. A palindromic number is a number that remains the same when its digits are reversed.

　　Examples are 27572 and 5826285.

　　If S is the set of all whole numbers greater than 100 and less than 301, and one whole number is randomly selected from S:

　　(a) what is the probability that the selected number is palindromic?

　　(b) what is the probability that the selected number is palindromic or even?

　　(c) If this process of selection was repeated 500 times, how many times would you expect to select a palindromic number?

　　18. Given that a, b, c and d are points on the number line such that

　　• a > b

　　• b is halfway between a and c

　　• the distance between a and d is three times the distance between a and c

　　• d < c,

　　19. Jack spends half of his money and gives one fifth of what remains to his friend. Jack is then left with £24. How much money did he start with?

　　21. The diagram below shows five identical circles, each with radius r cm. A square is formed with its vertices (corners) at the centres of the four outer circles and the inner circle just touches each of the four outer circles. The unshaded area inside the square is (72 – 18π) cm²

　　Showing clear working, calculate the radius r of each circle

　　22. When written out in full, how many digits are there in the number 16250 × 25500?

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