# Quod Erat Demonstrandum

## 2017/05/28

### Transformation of graphs

Filed under: NSS,Teaching — johnmayhk @ 8:48 下午
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Share some points about the topic ‘transformation of graphs’. Nothing new.

Transformation of graphs in the HKDSE syllabus refers to

– translation
– reflection (with respect to the x-axis or the y-axis)
– enlargement or contraction (along the x-axis or the y-axis)

and all happen in the xy-plane.

(more…)

## 2016/07/22

### 有理函數和矩陣

Filed under: NSS,Pure Mathematics,Teaching,University Mathematics — johnmayhk @ 12:06 下午
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Core mathematics 介紹過 rational function（有理函數），即形如　$\frac{P(x)}{Q(x)}$　者（其中 $P(x)$$Q(x)$ 皆為多項式）。

$P(x)$$Q(x)$ 皆為線性（linear），即形如　$\frac{ax+b}{cx+d}$　者，稱之曰 fractional linear function（FLF）。

$f(f(x))$

$=\frac{af(x)+b}{cf(x)+d}$

$=\frac{a\times \frac{ax+b}{cx+d}+b}{c\times \frac{ax+b}{cx+d}+d}$

$=\frac{(a^2+bc)x+b(a+d)}{c(a+d)x+bc+d^2}$

## 2016/07/10

### 論商餘（三）

Filed under: mathematics,NSS,Teaching — johnmayhk @ 9:45 下午
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(a) 多項式與數字

## 2016/06/13

### 變易圖式

Filed under: Teaching — johnmayhk @ 2:44 下午
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(more…)

## 2014/11/03

### 存在非平凡解的齊次線性方程組

Filed under: NSS,Teaching — johnmayhk @ 4:08 下午
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$\left \{ \begin{array}{ll} 2x+(2+k)y+2z=0\\(4+k)x+2y+5z=0\\7x+3y+(6+k)z=0\end{array}\right.$

$k$ 值。

$\left|\begin{array}{ccc}2 & 2+k & 2\\4+k & 2 & 5\\7 & 3 & 6+k\end{array}\right|=0$

$k=1,-1,-12$

$\left(\begin{array}{cccc}2 & 2+k & 2 & 0\\4+k & 2 & 5 & 0\\7 & 3 & 6+k & 0\end{array}\right)$

~$\left(\begin{array}{cccc}1 & \frac{k}{2}+1 & 1 & 0\\7 & 3 & 6+k & 0\\4+k & 2+k & 5 & 0\end{array}\right)$

~$\left(\begin{array}{cccc}1 & \frac{k}{2}+1 & 1 & 0\\ 0 & -\frac{7k}{2}-4 & k-1 & 0\\ 0 & -\frac{k^2}{2}-3k-2 & 1-k & 0\end{array}\right)$

~$\left(\begin{array}{cccc}1 & \frac{k}{2}+1 & 1 & 0\\ 0 & -\frac{7k}{2}-4 & k-1 & 0\\ 0 & -\frac{k^2}{2}-\frac{13k}{2}-6 & 0 & 0\end{array}\right)$

$-\frac{k^2}{2}-\frac{13k}{2}-6=0$

$k=-1,-12$

::: 停一停，想一想 ::: (more…)

## 2013/11/14

### 上課偶拾之 sd

Filed under: mathematics,NSS,Teaching — johnmayhk @ 3:46 下午

1. Range 和 Inter-quartile range 作為量度離散程度的工具之不足。
2. 創作可以涉及全體數據的量度工具，由最簡單考慮

$\displaystyle{\frac{\sum_{i=1}^n(x_i-\overline{x})}{n}}$

## 2012/10/05

### Rationalization

Filed under: NSS,Teaching — johnmayhk @ 3:20 下午

## 2012/04/21

### 圓錐截線切線

Filed under: NSS,Teaching — johnmayhk @ 2:40 下午

Core Mathematics 習題：

Let $C:x^2+y^2-6x+2y-15=0$. Show that $P(6,3)$ lies on $C$ and find the equation of the tangent to $C$ at $P$.

Let $L: y-3=m(x-6) \Rightarrow y=mx+(3-6m)$ (more…)

## 2012/02/16

### Core Math 某題：數算 ABC

Filed under: NSS,Teaching — johnmayhk @ 8:58 上午

Select 3 letters from a set {A,A,B,B,B,C,C}. How many different 3-letter ‘words’ can be formed? (e.g. AAB, CBA etc.)

$\frac{P^7_3}{2!3!2!}$

## 2012/02/13

### Core Math 某題：概率

Filed under: NSS,Teaching — johnmayhk @ 11:14 上午
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In a lucky draw of a car, only 1 key out of 10 can open the door of the car. Chris, Rachel and Mike take turns to draw a key at random without replacement. The person who can open the car door will get the car. Find the probabilities of the following events happening.

(a) Chris will get the car.
(b) Rachel will get the car.
(c) Mike will get the car.

## 2012/02/07

### Integration by parts

Filed under: NSS,Pure Mathematics,Teaching — johnmayhk @ 9:29 上午

$\int u(x)v(x)dx$

（其中 $u(x)$ 是多項式，$v(x)$ 是容易計算積分的函數）

Evaluate $\int x^2\cos xdx$.

## 2011/10/18

### Chain rule

Filed under: HKALE,NSS,Pure Mathematics,Teaching — johnmayhk @ 4:29 上午
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## 2011/09/23

### second fundamental theorem of calculus

Filed under: HKALE,Pure Mathematics,Teaching — johnmayhk @ 11:42 上午

$\frac{d}{dx}\int_a^xf(t)dt=f(x)$

（其中 $f$ 在開區間 $I$ 內連續，$a\in I$ 是常數。）

$\frac{d}{dx}\int_0^x(x-t)^2dt$ 是甚麼？ (more…)

## 2011/08/19

### 某數算題

Filed under: Additional / Applied Mathematics,HKALE,NSS,Teaching — johnmayhk @ 6:34 下午

If three tickets are chosen at random without replacement from a set of 6n tickets numbered respectively 1, 2,…, 6n, what is the probability that the sum of the numbers on the numbers on the chosen tickets is 6n?

## 2011/08/17

### 還是數算

Filed under: Additional / Applied Mathematics,HKALE,NSS,Teaching — johnmayhk @ 11:29 下午

There are 10 empty boxes. 5 balls are going to put one by one into a randomly selected box. Find the probability that two of the boxes each contains 2 balls.