Microsoft Word Math M1 Sample of Candidates' Performance.doc

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1 目錄 TABLE OF CONTENTS 前言第五級示例及評語第四級示例及評語第三級示例及評語第二級示例及評語第一級示例及評語 Introduction Level 5 exemplar and comments Level 4 exemplar and comments Level 3 exemplar and comments Level exemplar and comments Level 1 exemplar and comments 01-DSE-MATH-EP(M1) 1 1

2 前言第一屆香港中學文憑考試的成績已於 01 年 7 月 0 日發布為讓教師學生以及公眾人士進一步了解中學文憑考試考生的水平, 本局選取個別考生的答卷, 輯錄為本套示例, 以助說明各等級考生的典型表現延伸部分的各單元設有一試卷, 本套示例取材於考生答卷, 以便說明各等級考生的典型表現參閱這套示例時, 讀者亦宜同時參考該年試卷及評卷參考, 以了解試題及評分要求有關文件刊於 01 年香港中學文憑考試 -- 考試報告及試題專輯內 01-DSE-MATH-EP(M1)

3 香港中學文憑考試數學延伸部分單元一 ( 微積分與統計 ) 第五級示例及評語乙部第 1 題示例及評語 1. 某公司為旅客提供吊車服務旅客投訴輪候吊車的時間過長根據過往經驗, 一名隨機選取的旅客的輪候時間 ( 以分鐘為單位 ) 服從平均值為 µ 標準差為 9 的正態分佈 (a) 該公司的客戶服務經理就輪候時間進行調查以便估算 µ (i) 現隨機選取 16 名旅客組成一樣本, 他們的輪候時間記錄如下 : 求 µ 的 90% 置信區間 (ii) 求最小樣本容量使 µ 的 90% 置信區間的寬度小於 6 分鐘 ( 7 分 ) (b) 假設 µ = 公司的客戶服務經理會見旅客, 並向每名輪候時間超過 65 分鐘的旅客贈送一張優惠券 (i) 求經理向他所會見的首 10 名旅客送出少於張優惠券的率 (ii) 求第 5 張優惠券是贈予經理會見的第 0 名旅客的率 ( 6 分 ) 01-DSE-MATH-EP(M1) 3 3

4 評語考生能精準地運用數學公式求取樣本平均值及平均值的置信區間考生在 (a)(ii) 的複雜情境中, 成功以不等式建構數學模型, 並得出最小樣本容量考生在 (b) 的複雜情境中, 首先使用正態分佈以求取贈送優惠劵的率, 繼而運用二項分佈正確地建構數學模型, 以求取情境中所需的率, 顯示考生對於統計分佈的念有深入的認識和理解總括而言, 考生能純熟地運用統計念及數學技巧解決問題, 也能精準有效地運用數學語言及符號表達意念 01-DSE-MATH-EP(M1) 4 4

5 第四級示例及評語乙部第 13 題 13. 在某城市中醉酒駕駛是違法行為警方於晚上在某條公路的入口處設置檢查路障, 以拘捕醉駕司機根據過往經驗, 被捕的醉駕司機數目服從泊松分佈, 其平均值為每小時.3 (a) 求在某小時內有至少名醉駕司機被捕的率 ( 分 ) (b) 給定在某小時內有至少名醉駕司機被捕的條件, 求不多於 4 名醉駕司機被捕的率 ( 3 分 ) (c) 在某星期內, 警方在三個晚上設置檢查路障, 時間均為凌晨 1 時至時已知各個晚上被捕醉駕司機的數目相互獨立 (i) 求第三晚為首個有至少名醉駕司機被捕的晚上的率 (ii) 求在三晚中的每一晚均有至少名醉駕司機被捕並且合共有 10 名醉駕司機被捕的率 ( 5 分 ) 01-DSE-MATH-EP(M1) 5 5

6 評語考生於 (a) (b) 及 (c)(i) 各不同的情境中, 成功地運用泊松分佈條件率及幾何分佈以建構相關的數學模型, 並準確得出所需的率, 顯示考生對於有關的統計念有紮實的認識和理解在 (c)(ii) 的複雜情境下, 考生亦能以數學模型清楚列舉所有的四種情況, 惟考生未能數出全部個案, 並誤以 (a) 的答案為 P(X = ), 以致未能完全解答題目總括而言, 考生能有效運用統計念解決問題, 也能準確運用數學語言及符號表達意念 01-DSE-MATH-EP(M1) 6 6

7 第三級示例及評語甲部第 6 題 6. 某學校學生的體重 ( 以 kg 為單位 ) 可用平均值為 67 標準差為 15 的正態分佈模擬現隨機選取 36 名學生組成一樣本 (a) 求該 36 名學生的平均體重超過 70 kg 的率 (b) 已知該樣本中的 9 名學生喜歡吃炸薯條求該學校中喜歡吃炸薯條的學生所佔比例的近似 95% 置信區間 ( 5 分 ) 評語考生能在熟悉的情境中適當地建構樣本平均值的模型, 並成功使用正態分佈求取所需的率, 顯示考生對該統計念有足夠的認識和理解考生亦能正確計算樣本比例, 惟未能使用正確的公式求取所需的置信區間總括而言, 考生能成功運用一些統計念解決問題, 也能適當運用數學語言及符號表達意念 01-DSE-MATH-EP(M1) 7 7

8 第二級示例及評語甲部第 4 題 3x 1 4. 設 = 3 x y, 其中 > x (a) 使用對數微分法, 以 x 表 1 dy y dx (b) 利用 (a) 的結果, 求當 = 3 d y x 時 dx 的值 ( 6 分 ) 評語考生能運用對數微分法求取所需的數式考生雖然在書寫答案時出錯, 但仍能顯示對該微分念有基礎的認識和理解考生在 (b) 嘗試運用積法則求取二階導數, 但因運算出錯而未能得出正確答案總括而言, 考生能成功運用微分法去解答簡單的運算問題, 也能運用數學語言及符號表達意念 01-DSE-MATH-EP(M1) 8 8

9 第一級示例及評語甲部第 1 題 1. 設 n 為正整數 (a) 依 x 的升冪次序展開 (b) 已知在 n ( 1+ 3x) 到含 x n 的展式中 e (1 + 3x) x 的項為止 x 的係數為 6, 求 n 的值 ( 4 分 ) n 評語考生能正確展開 ( 1+ 3x), 惟在 (b) 未能展開以求的係數總括而言, 考生對基礎課題有一定的認識和理解, 亦能運用數學語言及符號表達意念 e x x 01-DSE-MATH-EP(M1) 9 9

10 INTRODUCTION The results of the first HKDSE Examination were released on 0 July 01. To enhance the understanding of the standards of the HKDSE Examination, authentic samples of candidates scripts are selected to form this set of exemplars to illustrate the typical performance standards at different levels. There is one paper in each Module of the Extended Part. This set of exemplars is selected from candidates scripts to illustrate the typical performance standards at different levels. It is advisable to read this set of exemplars together with the question paper and the marking scheme in order to understand the requirements of the questions and the marking criteria. The question paper and the marking scheme are published in the Examination Report and Question Papers for 01 HKDSE Examination. 01-DSE-MATH-EP(M1) 10 10

11 HONG KONG DIPLOMA OF SECONDARY EDUCATION EXAMINATION Level 5 exemplar and comments Section B Question 11 MATHEMATICS EXTENDED PART MODULE 1 (CALCULUS AND STATISTICS) EXEMPLARS AND COMMENTS 11. In a research of the radiation intensity of a city, an expert modelled the rate of change of the radiation intensity R (in suitable units) by dr a(30 t) + 10 = dt ( t 35) + b 0 is the number of days elapsed since the start of the research, a, b and T are positive constants. where t ( t T ) It is known that the intensity increased to the greatest value of 6 units at t = 35, and then decreased to the level as at the start of the research at t = T. Moreover, the decrease of the intensity from t = 40 to 61 t = 41 is ln units. 50 (a) Find the value of a. ( marks) (b) Find the value of T. (4 marks) (c) Express R in terms of t. (4 marks) (d) For 0 t 35, when would the rate of change of the radiation intensity attain its greatest value? (4 marks) 01-DSE-MATH-EP(M1) 11 11

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14 Comments The candidate presents the solution in a clear and concise way and shows an ability to communicate precisely and logically using mathematical language and notations. Also, the candidate shows he/she understands the given conditions very well and possesses an excellent ability to integrate knowledge, understanding and skills from the topics of logarithm, differentiation and integration. 14

15 Level 4 exemplar and comments Section B Question Let I = e t t d t. (a) (i) Use the trapezoidal rule with 6 sub-intervals to estimate I. (ii) Is the estimate in (a)(i) an over-estimate or under-estimate? Justify your answer. (7 marks) (b) Using a suitable substitution, show that I = 1 e x d x. (3 marks) (c) Using the above results and the Standard Normal Distribution Table on page 14, show that π < 3.5. (3 marks) 15

16 Comments In (a) and (b), the candidate is able to present the solution in a clear and concise way and shows an ability to communicate accurately using mathematical language and notations. Also, the candidate demonstrates how to work out the trapezoidal rule accurately with an ability to conclude the over-estimated nature of the approximated value using the second derivative of the integrand. The proof in (b) is also done with full details given and to a degree of high accuracy. In (c), the candidate is able to use the given Standard Normal Distribution Table. However, the candidate is unable to merge the results in (b) with the results from the Normal Distribution Table to get the remaining marks. 16

17 Level 3 exemplar and comments Section A Question 8 8. Let X be a discrete random variable with probability function shown below: x P( X = x) 0.1 a b 0.05 where a and b are constants. It is known that E ( X ) = (a) Find the values of a and b. (b) Let F be the event that X 4 and G be the event that X < 8. (i) Find P( F G). (ii) Are F and G independent events? Justify your answer. (6 marks) Comments The candidate demonstrates adequate knowledge and understanding on the concept of expectation and basic probability laws by solving the unknowns correctly in (a). However, the candidate wrongly assumes the independence of the events F and G in (b)(i) and hence gets an incorrect answer. The candidate has the idea of using a fraction of two decimals to calculate the conditional probability in (b)(ii), but the candidate uses incorrect probabilities in the fraction. Generally, the candidate is capable of using some mathematical techniques to solve problems and using mathematical language and notations to present the workings in an appropriate way. 17

18 Level exemplar and comments Section A Question 5 5. The slope of the tangent to a curve S at any point ( x, y) on S is given by L be the tangent to S at the point A (0,1) on S. d y = x. Let dx e (a) Find the equation of S. (b) Find the equation of L. (c) Find the area of the region bounded by S, L and the line x = 1. (7 marks) Comments The candidate demonstrates basic knowledge and understanding of calculus by integrating the given derivative to obtain the equation of S. However, the dependent variable y is not clearly stated and the integration constant is not found by using the given information. In (b), the candidate finds the required equation of L correctly. In (c), the candidate is unable to formulate the required integral for the area of the bounded region. Conclusively, the candidate can use mathematical language, notations and graph to express some ideas. 18

19 Level 1 exemplar and comments Section A Question dv t. The rate of change of the value V (in million dollars) of a flat is given by =, where dt 4t + 1 t is the number of years since the beginning of 01. The value of the flat is 3 million dollars at the beginning of 01. Find the percentage change in the value of the flat from the beginning of 01 to the beginning of 014. (5 marks) Comments The candidate demonstrates elementary knowledge of calculus by using the method of substitution. However, the candidate misses a multiple of 1/4 when changing variables. Though the candidate does not attempt to find the required percentage change, ideas can still be expressed using mathematical language and notations. 19

Microsoft Word - DSE PHY Samples.doc

Microsoft Word - DSE PHY Samples.doc 目錄 TABLE OF CONTENTS 前言... 1 卷一乙部第五級示例及評語... 2 第四級示例及評語... 3 第三級示例及評語... 5 第二級示例及評語... 7 第一級示例及評語... 8 卷二第五級示例及評語... 10 第四級示例及評語... 12 第三級示例及評語... 14 第二級示例及評語... 16 第一級示例及評語... 18 Introduction... 20