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[分享] 會考故事之十六：AMaths風暴（第二集，開拓荒野）

Doraemon

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1^# 大中小發表於 2007-5-25 08:19 PM 只看該作者

會考故事之十六：AMaths風暴（第二集，開拓荒野）

經過上集的歷史洗禮，今集開始正式踏入Past Papers的世界。

Paper 1和2，劈頭一來，是古古怪怪的instruction......

"Emphasis should be placed on accurate working."

在當年的Maths，由於沒有計數機的關係，一些計算型的題目，只要求出正確的答案，就可以得分的了，但AMaths因為Syllabus的關係，步驟變得很重要了。

Paper 1，開始的，不是一些First Principle，或者是differentiation，而是一道Remainder Theorem的題目：

1967 I 1. When 14x^4 – 27x^3 + ax^2 + 12x + 9 is divded by (x - 2) the remainder is 5. What is the value of a?

驚訝嗎？現在對Maths和AMaths的Syllabus之觀念，在史前時期是行不通的。

還有啊......

1967 I 3. The sum of an Arithmetic Progression is 630 and the first and last terms are 5 and 55 respectively. How many
terms are there?
1967 I 4. There are 5 boys and 5 girls at a dance. In how many different ways can they pair off for dancing, if everyone
is dancing with a member of the opposite sex?

A.S.（以前稱作A.P.的）和Probability，竟在AMaths出現，令人感慨萬千。

短題目的怪異，何只這麼少呢？

1967 I 10. (a) From the above graph, answer the following questions, giving your reasons:
                  (i) How do you think the person was travelling? (Walking, running, cycling or driving a car?)
                  (ii) The journey represented by O to A was either uphill or downhill. Which is it?
               (b) Give a very brief description of the journey.

(b)於現在看來，還是一道較為合理的Physics題，但(a) part，則完全是聞所未聞，見所未見的題目，怪異程度，令人側目。

那麼，還有一些較為正常的題目嗎？

1967 I 7. Find dy/dx where y = tan (2x + a) if a is a constant.
是一道Standard的微分題目。

長題目了，除了頭尾兩題較為正常外（分別是一元二次方程和微積分的題目），其餘的都是令人感到疑惑，或者是現今在數學科出現的題目，反映出第一年出題的不定性。

1967 I 12. In order to make a closed tin in the form of a right circular cylinder of radius "r"and height "k.r",
               a manufacturer has to cut each circular end from a square of tin plate of side 2r, of which the remainder is
               wasted. The curved surface of the cylinder is obtained without waste.Find the value of k, correct to 3
               significant figures, for which the least amount of tin is required for a given volme.

是一道沒有圖的題目，看完文字以後，還真的有點一頭霧水的感覺。

1967 I 13. If the population of Hong Kong is 4 million in 1967 and it increases by 3% each year, what will be
the estimated the population in 1987? Give your answer correct to 4 significant figures.

這道題目，於現在看來，似乎是數學科的一道MC或者是選擇題。可是，不要忘記，1967年的香港，一部計數機可不便宜呢！（而且體積龐大......）故此，這道其實是Binomial Theorem（二項式定理）的題目，充分考驗考生的四則運算能力。

第14題，為一道很簡單的3D題，關鍵在於，題目給的單位是yards，但計算出來的答案，卻要是feets，某程度上，是包含了換算技巧在內的。

第15題是一道Co-Geo的題目，頭四部分比較正常，但接著的，卻非常過癮的......
......
(a) (v) Find what type of parallelogram ABCD is (giving your reasons).
(b) Confirm the result of (v) by an entirely different method.

有趣吧！本來已經prove了的，還要用另一方法再prove，變相同時考核你的Geometry，高明是也。

[ 本帖最後由 Doraemon 於 2007-5-26 10:02 AM 編輯 ]

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Doraemon

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2^# 大中小發表於 2007-5-25 08:20 PM 只看該作者

Paper 2，驟眼看來，大部分是微積分或Co-Geo的題目，只有第四、六兩題為其他類型的題目。第六題，是一道較typical的trigo題目，所以我們會集中在第四題之上。

1967 II 4. Indicate, by drawing sketches, whether or not it is possible to draw two non-congruent triangles
            ABC from each of the following sets of data:
            (a) a = 1, b = sqrt(3), A = 30^o
            (b) a = 2, c = 5, A = 30^o
            (c) a = 3. c = 4, B = 30^o
            (d) a = 4, b = 2, A = 120^o
            (e) a = 4, b = 3, B = 30^o
            （附注：^o代表degree）

是一道Geometry的題目，卻又似披著trigo的外表。

第二頁的三道短題目，分別是log、不定積分和trigo，當中以trigo最具該時代的特質，亦同時成為了現今部分AMaths書的例子。

1967 II 10. Express tan 2x in terms of tan x, and hence find the value of tan 22.5^o, without using tables.

長題目，題目類型主要有三種：Probability、Co-Geo和Mechanics。在1967年來說，Mechanics的地位還未有被奠定，故稍後再談AMaths中的Mechanics。

至於其他類型的題目，有些都不是太obvious的。
1967 II 12. (i) Give three completely different formulae for the area of a triangle in terms of its sides, angles,
                     and other associated distances.
               (ii) Using one of the above formulae, find the area of the triangle of which the angular points are
                     A(1, 2); B(4, 5); C(8, 1).
               (iii) Using the formula for the area of a triangle in terms of the coordinates of the angular points,
                     confirm your previous answer. (Show all your working in the question)

你可能會問，為甚麼會有這麼多的confirm型題目，原因仍然是：
當時是沒有計數機的。

1967 II 16. (i) In a bag there are 12 balls, 5 red, 4 blue and 3 yellow. If a blind man picks up 3 balls,
                  what is the probability that there is not a red one among these 3?
               (ii) If n is a positive integer, prove, by induction that
                     1^3 + 2^3 + 3^3 + … + n^3 = [1/2 n (n+1)]^2
                     Explain clearly your reasons why this applies to all positive integral values of n.

Probability和MI相遇的地方......同時亦反映出新數對AMaths的產生的影響，是如何的顯著。至於MI中Explain clearly的這一部分，可謂考起現在的學生了。