# 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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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

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

$(a+d)$ 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）。

## 2015/12/17

### 好玩的二次圖

Filed under: Fun,NSS — johnmayhk @ 3:23 下午
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https://teacher.desmos.com/browse

## 2015/02/21

### 表面面積

Filed under: Fun,Junior Form Mathematics — johnmayhk @ 7:47 下午
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(more…)

## 2015/02/17

### 數算球入盒

Filed under: mathematics,NSS,Pure Mathematics — johnmayhk @ 11:40 上午
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（一） (more…)

## 2015/02/01

### 黑白球

Filed under: mathematics,NSS — johnmayhk @ 10:33 上午
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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.

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

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

## 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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１１
１２１
１３３１
１４６４１

## 2014/11/28

### 某 m2 題

Filed under: NSS — johnmayhk @ 4:34 下午
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Part (c) 的建議答案，考慮

## 2012/05/01

### 無聊談通項

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

## 2011/11/18

### polar form

Filed under: Pure Mathematics — johnmayhk @ 11:46 上午
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Convert $z=-\cos \theta -i\sin\theta$ into polar form.

## 2011/04/05

### arctan

Filed under: Pure Mathematics — johnmayhk @ 9:56 下午
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$\frac{d}{dx}\tan^{-1}(\frac{x-1}{x+1})$

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

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

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

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

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

## 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…)