Quod Erat Demonstrandum

2016/07/26

相同特徵值及凱萊哈密頓

Filed under: NSS,Pure Mathematics — johnmayhk @ 10:29 上午
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免插聲明:本文只是中學程度的討論,高手見諒。

續上個 post:

https://johnmayhk.wordpress.com/2016/07/22/flf-and-matrix/

M=\left(\begin{array}{rcl}a& b\\c& d\\\end{array}\right) 的特徵方程為 \det(M-\lambda I)=0,即

\lambda^2-(a+d)\lambda+(ad-bc)=0

留意上式係數,

(a+d) 就是矩陣 M 的跡(trace),即對角元的和,也是特徵值的和(sum of roots);而

(ad-bc) 就是矩陣 M 的行列式(determinant),也是特徵值的積(product of roots)。

有時出題目,想弄一個 2×2 矩陣,其特徵值是(比方說)2 和 8,可以先寫 (more…)

2015/12/17

好玩的二次圖

Filed under: Fun,NSS — johnmayhk @ 3:23 下午
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教中四數學的同工,可以給學生玩玩以下網上遊戲,他們要運用有關二次圖像(quadratic graph)的知識「取星星」。這不是幼童遊戲,有難度的。

先去:

https://teacher.desmos.com/browse

johnmayhk-marbleslides-01

選 Marbleslides: Parabolas,見 (more…)

2015/02/21

表面面積

Filed under: Fun,Junior Form Mathematics — johnmayhk @ 7:47 下午
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古語有云:「我鍾意食水果,每一個國家必須要食。」過年期間更要多吃水果,有助消化。

受網友啟發,兼剛開始教球體表面面積,於是找來一個乜乜乜橙:

johnmayhk-orange-sphere-1 (more…)

2015/02/17

數算球入盒

Filed under: mathematics,NSS,Pure Mathematics — johnmayhk @ 11:40 上午
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基本問題:把若干球放入若干盒子,共多少種放法?下表是總結:

johnmayhk-balls-and-boxes-01

以下 3 個情況屬 core mathematics 的範圍:

(一) (more…)

2015/02/01

黑白球

Filed under: mathematics,NSS — johnmayhk @ 10:33 上午
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同事擬某基本概率題:

In a game, Evan has to draw balls from a bag containing 2 black balls and 3 white balls one by one without replacement. If he gets two consecutive black balls, he wins; otherwise he loses. Find the probability that he wins.

標準答案如下:

P(wins)
=P(BB)+P(WBB)+P(WWBB)+P(WWWBB)
=\frac{2}{5}\frac{1}{4}+\frac{3}{5}\frac{3}{4}\frac{2}{3}+\frac{3}{5}\frac{2}{4}\frac{2}{3}\frac{1}{2}+\frac{3}{5}\frac{2}{4}\frac{1}{3}
=\frac{2}{5}

有沒有留意,盒內共有 5 球,黑球 2 個,\frac{2}{5} 就是從盒取 1 球,得黑球之機會。

試用別的例:盒內共有 7 球,黑球 3 個,Evan 取勝之機會,按標準答案之做法:

\frac{3}{7}\frac{2}{6}+\frac{4}{7}\frac{3}{6}\frac{2}{5}+\frac{4}{7}\frac{3}{6}\frac{3}{5}\frac{2}{4}+\frac{4}{7}\frac{3}{6}\frac{2}{5}\frac{3}{4}\frac{2}{3}+\frac{4}{7}\frac{3}{6}\frac{2}{5}\frac{1}{4}=\frac{3}{7}

看,又是等於在盒子取 1 球,得黑球之機會。

black white balls

這逼使我想:是否存在簡單方法處理原問題及其一般情況?結果如下。 (more…)

2015/01/23

某數算題

Filed under: mathematics,NSS — johnmayhk @ 5:34 下午
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Just reply to a F.5C student on a basic core mathematics question (on P.5.38):

There are 8 outstanding students from junior forms and 9 outstanding students from senior forms in a school this year. 5 out of these 17 students are now selected for an overseas exchange programme. Find the number of combinations of selecting at least 1 student from junior forms and 1 from senior forms.

Here is the ‘so-called’ solution from a student:

_8C_1\times _9C_1\times _{15}C_3

as the student claimed, select 1 from junior, _8C_1 ways; select 1 from senior, _9C_1 ways; then select the remaining 3 students from the remaining 15 students, _{15}C_3 ways, hence, the total number of combination should be _8C_1\times _9C_1\times _{15}C_3, right?

Sorry, it is incorrect. (more…)

2015/01/09

推特老題

Filed under: HKALE,Pure Mathematics — johnmayhk @ 3:26 下午
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早前在推特見:

johnmayhk-question-in-twitter

他們討論著,並表示不知道如何得出如此美妙式子。

如何美妙?修 M1,M2 的同學應該不難察覺,式子包含二項系數(binomial coefficients),

11
121
1331
14641

並且,把等號左邊的項加起來,得右邊單項式作分子云云。

但有修純數的人,相信很快察覺,那不過是一道有關部份分式(partial fractions)的基本題目(搖頭中)。Okay, (more…)

2014/11/28

某 m2 題

Filed under: NSS — johnmayhk @ 4:34 下午
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堂上給學生做習題,是來自 2009 年的 hkdse math(M2) sample paper Q.9

johnmayhk-M2-sample-paper-Q9

Part (c) 的建議答案,考慮

考慮 \overrightarrow{v}\overrightarrow{OA}\overrightarrow{OB} 的線性組合(linear combination),即 (more…)

2012/05/01

無聊談通項

那天觀課,同事開始教等差數列(arithmetic sequence),(估計是隨便)問學生:

2,1,4,\frac{1}{2},8,\frac{1}{4},\dots

的通項(general term)是甚麼?

關於通項,之前也談過,除非先有「特殊規定」,否則所謂通項是無定義的。

談回上題,假設同事的「特殊規定」是把兩個等比數列(geometric sequence)併在一起。當然,學生在中一時接觸過數型(number pattern),但要寫出上述數列的通項,似乎不是一蹴而就的事。因那是引起動機的其中一例,同事沒有繼續討論,但卻引發我思考一些東西。

只要時間許可,相信學生不難得到: (more…)

2011/11/18

polar form

Filed under: Pure Mathematics — johnmayhk @ 11:46 上午
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基礎題:

Convert z=-\cos \theta -i\sin\theta into polar form.

問:「在何象限,cosine 和 sine 值也是負?」 (這是極不精確的說法,但學生又明我說甚麼。)

答:「第三。」 (more…)

2011/04/05

arctan

Filed under: Pure Mathematics — johnmayhk @ 9:56 下午
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多星期前,剛開始談求導技巧,見書中某例:

\frac{d}{dx}\tan^{-1}(\frac{x-1}{x+1})

當得出

\frac{d}{dx}\tan^{-1}(\frac{x-1}{x+1})=\frac{1}{1+x^2}

後,堂上我問:同學感到有趣嗎?因為

\frac{d}{dx}\tan^{-1}x 也是 \frac{1}{1+x^2}

當然,堂上沒空做「無聊」的東西,唯現在從上述「發現」出發,胡說幾句。

首先,大家看看以下步驟有沒有問題:

\frac{d}{dx}\tan^{-1}x\equiv \frac{d}{dx}\tan^{-1}(\frac{x-1}{x+1})

\Rightarrow \tan^{-1}x\equiv \tan^{-1}(\frac{x-1}{x+1})+C

Put x=1 (more…)

2011/03/03

Polynomial identity


Is the following an identity? Prove or disprove your claim:

(x+1)(x-3)+1=(x+2)(x-1)

This is a trivial (more…)

2010/11/04

克萊姆法則

Filed under: NSS,Pure Mathematics — johnmayhk @ 6:04 下午
Tags:

早前,黎同學問了一個問題。

推論克萊姆法則(Cramer’s rule),我們經歷一些步驟,見下:

對於線性方程組

\left \{ \begin{array}{ll} ax + by + cz = k_1\\dx + ey + fz = k_2\\gx + hy + iz = k_3\end{array}\right.

(more…)

2010/05/27

just answer a textbook question from my student

Filed under: Additional / Applied Mathematics,HKALE — johnmayhk @ 12:45 下午
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To Lau (my student)

This is the question you’d asked

“A man lives at A and works at C, and is due at work at 8:30 a.m. He always catches the first train from A to B, which is scheduled to arrive at B at 8:15 a.m. Buses leaves B for C every 20 minutes, and the bus which leaves B at 8:20 a.m. is scheduled to arrive outside the factory at C at 8:27 a.m. The train is, on the average, one minute late and has a standard deviation of 4 minutes. The bus always leaves on time, but is, on the average, 1.5 minutes late with a standard deviation of 2 minutes. The man’s employer leaves home in his car at 8:15 a.m. and the time for his journey has mean value 13 minutes with a standard deviation of 3 minutes. Find the probability that the employer arrives before the employee." (more…)

2009/09/25

Find dy/dx at a point not on the curve

Filed under: Additional / Applied Mathematics,HKCEE — johnmayhk @ 4:50 下午
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When distributing the marked test paper to students, one student, Carman, reminded me that there was a ‘question’ in the following question:

If x^3 - 4x^2y + 3xy^2 - y^5 = 10, find \frac{dy}{dx} at the point (-2,1).

Carman said, ‘the point does NOT lie on the curve.’

Good observation! I had to say thank you to him. Although I’m not the setter, I should bear the responsibility of checking the paper.

But a natural follow-up question turns up: what is the meaning of the number \frac{dy}{dx}|_{(-2,1)} = \frac{31}{33} we are obtaining? Is the number meaningless or standing for something?

Let’s consider a simple example. (more…)

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