# Quod Erat Demonstrandum

## 2018/04/17

### 互斥與獨立

Filed under: NSS — johnmayhk @ 12:37 下午
Tags: , ,

## 2018/01/23

### 某類三角恆等式記法

Filed under: mathematics,NSS — johnmayhk @ 10:24 下午
Tags: ,

$\sin(-\theta)\equiv -\sin\theta$
$\cos(-\theta)\equiv \cos\theta$
$\tan(-\theta)\equiv -\tan\theta$

## 2017/09/21

### core math 某題：標準差

Filed under: mathematics,NSS — johnmayhk @ 4:49 下午
Tags:

http://www.wolframalpha.com/input/?i=standard+deviation+of+1,1,1,2,2,2,2,2,3,3,4,4,4,5

## 2016/07/26

### 相同特徵值及凱萊哈密頓

Filed under: NSS,Pure Mathematics — johnmayhk @ 10:29 上午
Tags: , ,

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
Tags: ,

https://teacher.desmos.com/browse

## 2015/02/21

### 表面面積

Filed under: Fun,Junior Form Mathematics — johnmayhk @ 7:47 下午
Tags: ,

(more…)

## 2015/02/17

### 數算球入盒

Filed under: mathematics,NSS,Pure Mathematics — johnmayhk @ 11:40 上午
Tags: ,

（一） (more…)

## 2015/02/01

### 黑白球

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

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
Tags: , ,

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
Tags: ,

１１
１２１
１３３１
１４６４１

## 2014/11/28

### 某 m2 題

Filed under: NSS — johnmayhk @ 4:34 下午
Tags: ,

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
Tags: ,

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

## 2011/04/05

### arctan

Filed under: Pure Mathematics — johnmayhk @ 9:56 下午
Tags: , ,

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

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

$\frac{d}{dx}\tan^{-1}x$ 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…)