會考故事之二十一：AMaths風暴（第七集，填充題）
71年的Paper 3，繼續是可愛的Mechanics。
短題目和現在的AL Physics，除了lamina以外，越來越有一脈相承之處。
Group III長題目，其中有一道是火車上斜路的題目，共有九卡列車，而斜路的「angle of inclination」，是一個unknown來的。
1971 III 12. A train consisting of locomotive and 9 carriages linked by light rigid links is travelling at a uniform speed up an
incline, which makes an angle A with the horizontal. The weights of the locomotive and each of the carriages are,
in oerder, W, W-K, …, W-9k kg weight, where 0 < 9K < w. All resistances may be neglected.
(i) Find the tractive force required to maintain the uniform motion.
(ii) If the tractive force is increased to P kg weight,
(a) prove that the acceleration of the whole train is {P/[5(2W-9K)]-sinA}g m/s^2, and
(b) find the tension in the link between the fourth and the fifth carriages.
踏入72年，AMaths繼續演化，開始出現一些locus的題目。
1972 I 4. A point P moves such that its ditance from the point (1,5) is equal to its distance from the line y = x. Write down
and simplify the equation of the locus of P.
而72年的Paper 1，除了兩道短題目和一道長題目，其餘全都是微積分和Co-Geo，嚴重的泛濫，當時的考生，可真是挺難吃得消。
即使Paper 1出現了微積分，Paper 2的短題目還有一道用Method of Substitution的積分題目，可見當年考生的相關技巧是需要極度熟悉的。
Paper 2短題目第一題，很趣怪的。
1972 II 1. Solve the equations 3a-2b = 2a+5b-11 = 5 for a and b.
原本題目是用希臘符號「phi」和「theta」的，一個不小心看錯，還以為考Trigo呢！
至於log、remainder theorem、A.S./G.S.，是繼續散落在題目之間；Binomial則是一道短題目，和一道應用於compound interest的長題目。
Paper 3，詳細的不多講了，只給大家一短一長的題目，給大家玩玩。
1972 III 8. Figure 7 shows two particles, A and B, connected by a light and inextensible string passing over a light and
frictionless pulley. The strign is 3 m long, excluding the part round the pulley, and initially, the portion between
the pulley and particle B is slack.
Will the heavier particle B be raised appreciably above the ground if
(a) the lighter particle A is placed on the hand and lowered gradually and very slowly until the string becomes taut?
(b) particle A is dropped from rest until the string becomes taut?
Give very briefly the reason in each case. (Calculation is NOT required.)
1972 III 9. Referring to the system in Question 8 (Figure 7), if particle A falls from rest, part of the subsequent motion is
described by the following passage. You are required to write in your answer book the answers corresponding to
the blanks (a), (b), …… (Whenever g is involved, you should leave it as such instead of replacing it by its numerical value.)
“Initially, particle A falls with an acceleration of ___(a)___m/s^2. When it has descended a distance of 0.5 m, its
velocity is ___(b)___ and the string then becomes taut. Particle B is then raised above the ground with an initial
velocity of ___(c)___ upwards and a deceleration of ___(d)___. The tension in the string is ___(e)___. B continues
to move upwards until it reaches a height of ___(f)___. It then returns to the ground, taking ___(g)___ seconds to
do so. At this moment, A has a velocity of ___(h)___ and will rise to a maximum height of ___(i)___ above the ground.”
不錯，是填充題，是長題目，更是一道與短題目連繫的長題目。
因此，老師經常教導我們，開考後先掃視整份卷，這就是活生生的例子。答第8題時，不看看第9題，就可能走了許多冤枉路了。
72年的題目在編排方面，是重拾上幾年的做法，Paper 1、2和3的長題目都需要分groups作答，每個group各三題。
可是，到了73年，Paper 2又再重獲自由，變回六選三。
73年Paper 1，並沒有72年的「微積分狂潮」，取而代之的是均衡題目，除了Trigo以外，還有一些另類的題目，如log graph，與及Percentage error的題目。
1973 I 6. The radius R, of the circular orbit of a satellite is related to the period of revolution, T, of the satellite by the
expression R = k[T^(2/3] where k is a constant. If the measurement of T is -0.5% in error, what is the
corresponding percentage error in R?
在Paper 2方面，開始出現一些standard的題目。
1973 II 1. Find the general solution of 20secA + 10cosA = 33.
1973 II 7. It is given that y = (x^2 - 3x + 4)/(x - 3).
對於今天的AMaths學生來說，可以說是毫無難度的。
當然，還有可愛的MI呢！
1973 II 5. If n^3 + an is divisble by 3 when n = k, find the least positive value of a so that n^3 + an is also divisble by 3 when n = k +1. (k is a positive integer.)
在長題目方面，沒有太大的驚喜，今年主要的特色是運用一些substiution的技巧。
下一集，亂局再起？
