# Quod Erat Demonstrandum

## 2012/04/03

### HKALE 2012 Pure Mathematics

Filed under: HKALE,Pure Mathematics — johnmayhk @ 8:53 下午

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## 28 則迴響 »

1. 剛看到同事在 fb 說 Paper 2 Q.8 好好玩，那我也玩玩：

(b)
Suppose
f(a+b)=f(a)
f(a)f(b)-f(a)-f(b)+2=f(a)
[f(a)-1][f(b)-2]=0
Hence f(a)=1 or f(b)=2
But if f(a)=1,2=f(0)=f(a-a)=f(a)f(-a)-f(a)-f(-a)+2=1 (contradiction) … (*)
Thus, f(b)=2 => b=0 (by (a) & condition (2))
So, f is injective.
(c)
By (*), it is impossible to find a real no. a such that f(a)=1.
Hence f is not surjective.

迴響 由 johnmayhk — 2012/04/03 @ 8:54 下午 | 回應

2. 我在試場想了3小時也想不到
你這麼輕鬆就可以解決
厲害！

迴響 由 Lam — 2012/04/03 @ 9:15 下午 | 回應

3. Hi, Ng Sir, do u think it is one of the most difficult paper in Pure Math history? (like 1997 and 2005?)
Some of my students even can’t finish the paper (both paper 1 and 2)..and they lose many marks.
Also, what do you think about the curve this year will look like? Thx~

迴響 由 Sam — 2012/04/04 @ 4:28 下午 | 回應

4. 作為一個局外人做呢份卷 paperII唔計Q8 其他難度還可以吧，只不過對於一些打死唔做co-geom既人要做Q6同唔揀Q8而被逼做Q11的人應該頗為痛苦…

迴響 由 — 2012/04/04 @ 4:46 下午 | 回應

5. @Lam
我只是感到一般學生不易取這 6 分。回想我平日的教學，教導學生應付 injective 和 surjective 的證明或否證，也是頗大路的；處理多數是具體的函數的問題；豈料今次玩那種描象的，數學比賽才常見的 functional equation 之定性問題。估計學生陣亡者眾。

@Sam
In fact, I had not done all questions (I’m not a good teacher /_\). I just feel that this year is not easy, because some of my students said their performance was not good enough. The format looks similar to recent papers, however, the paper may involve tedious computations, candidates had undergone hard hours and they felt frustrated. The curve sketching is not comparable to the one in 2005, that was a monster, for me at least.

@楓
我估自己去考都應該有同樣痛苦… …

迴響 由 johnmayhk — 2012/04/04 @ 5:45 下午 | 回應

6. Agree. According to most of my students, they just show steps in the paper, and give up the (accuracy) marks for each question, especially Paper II Section A. They give up at least 2 marks in Q4, 5 etc…Yes, you are right, curve sketching is more straightforward in a sense, but other questions are really difficult to tackle all within 3 hours. I am sure most students wouldn’t do very good in both paper. (though format is similar, difficulties of questions are really different from recent years)

迴響 由 Sam — 2012/04/04 @ 7:46 下午 | 回應

7. 我個日考反而覺得paper 1 難過paper 2
paper 2 既format都叫做standard
paper 1 既 第三條C, LQ 既polynomial同埋inequalities 都十分有難度..
第三條C都要你諗唔continuous既function….考起唔少人…

迴響 由 Carmen — 2012/04/04 @ 8:14 下午 | 回應

8. Actually, f(x+y) = f(x)f(y) – f(x) – f(y) + 2, so f(x+y) – 1 = (f(x) – 1)(f(y) – 1). If we define g(x) = f(x) – 1, we have g(x+y) = g(x)g(y). By repeatedly applying this n times, we have g(nx) = g(x)^n for all positive integer n and real x. Putting x = y/m for any positive integer m, we have g(y n/m) = g(y/m)^n = g(y)^(1/m)^n = g(y)^(n/m). So for all rational number r, g(ry) = g(y)^r. Letting g(1) = c, we have g(r) = c^r. If a limit exists at any x, this must generalize to g(x) = c^x. The only remaining issue for the whole question is to identify c. At this point all parts of the problem becomes pretty trivial.

迴響 由 Isaac — 2012/04/04 @ 11:10 下午 | 回應

9. Any HKALE 2012 Applied Mathematics questions available?

迴響 由 Chow — 2012/04/05 @ 10:17 上午 | 回應

10. @Issac
Yes, g(x+y)=g(x)g(y) determines an exponential function, and it may be the original idea of the setter (wild guess).

@Chow
Surely, after 26/4/2012.

迴響 由 johnmayhk — 2012/04/05 @ 10:40 上午 | 回應

11. 我想講 同大陸嘎比起 呢d真係濕濕碎咯

迴響 由 daxi — 2012/04/07 @ 11:34 上午 | 回應

12. […] My solution for the 2012 Pure Math Paper 1. For the question paper, please refer to https://johnmayhk.wordpress.com/2012/04/03/hkale-pure-mathematics-2012/ […]

通告 由 HKALE 2012 Pure Mathematics Paper 1﻿﻿ Solutions - Benson's Blog — 2012/04/08 @ 2:20 上午 | 回應

13. […] My solution for the 2012 Pure Math Paper 2. For the question paper, please refer to https://johnmayhk.wordpress.com/2012/04/03/hkale-pure-mathematics-2012/ […]

通告 由 HKALE 2012 Pure Mathematics Paper 2 Solutions - Benson's Blog — 2012/04/08 @ 2:22 上午 | 回應

14. 我想問問如果做 paper 2 第 8 題
用 (d)(ii) 既答案黎 apply on (?) parts (b), (c)
雖然係證到 injective / not surjective
但係咪應該無分俾?

迴響 由 Myst — 2012/04/08 @ 1:13 下午 | 回應

• 應該不能給 full mark，但估計同學若懂 injective 或 surjective 的意義，也有些少分數吧？（瞎猜的）

迴響 由 johnmayhk — 2012/04/12 @ 7:06 上午 | 回應

15. John sir, 想問貴校有否作 M1/M2 / Applied 的試場?
因本人未能找到以上各卷，希望閣下(如不嫌煩)能夠相助。

迴響 由 路人甲 — 2012/04/28 @ 11:52 上午 | 回應

16. Ng Sir, 想問有否 M1/M2 / Applied各卷，如有，可否send給本人作參考？thanks a lot！

迴響 由 Sam — 2012/04/28 @ 3:18 下午 | 回應

迴響 由 johnmayhk — 2012/04/28 @ 6:18 下午 | 回應

• John sir, 請問可否send M1/M2/Applied的試卷給本人參考? 先謝!
看網上留言好多人都覺得今年Applied比以往3年淺，不知是否Pure出得太怪所以EAA於Applied “放生" 考生?

迴響 由 Hong — 2012/04/28 @ 10:56 下午 | 回應

• Pls check email。聞 Applied 淺了，但相信和 Pure 深了沒太大關係。

迴響 由 johnmayhk — 2012/04/29 @ 7:18 上午

18. JOHN SIR，我是他校同工，可否將M1/M2/ APPLIED卷 EMAIL給本人參考?勞煩你

迴響 由 溟天凱 — 2012/04/29 @ 10:15 上午 | 回應

19. Thank you so much, Ng Sir. 我覺得 M2 都系比較淺 （相比今年的pure卷），去不到舊制pure的程度，而M1跟練習卷format差不多，學生很容易掌握答題方式。

迴響 由 Sam — 2012/04/29 @ 1:42 下午 | 回應

20. JOHN SIR,可否send M2卷給我,我明年會考dse

迴響 由 jimmy — 2012/04/29 @ 7:50 下午 | 回應

21. 我也想要m1,m2卷
可否也send給我？
謝謝

迴響 由 Lam — 2012/05/01 @ 10:50 下午 | 回應

22. 我都想要今年ge m1/ m2/Pure math 卷
可唔可以send 比我?
萬分感謝!

迴響 由 George — 2012/05/06 @ 7:29 下午 | 回應

23. 可否send DSE core math/M1/M2卷給我？
謝謝！

迴響 由 Xeror — 2012/05/30 @ 1:19 上午 | 回應

24. 可否send DSE core math/M1/M2卷給我？Thank you very much!

迴響 由 paullee — 2012/06/26 @ 8:36 上午 | 回應

25. 你唔好再教壞D同學啦！
眼中只有數學
只向學生宣傳數學幾好…
其實數學不過是一門基礎學科而已！！

迴響 由 2009 SFXC F.7 Graduate — 2012/12/07 @ 6:41 下午 | 回應