Blogs about: Pure Mathematics
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利用圖像尋找非實根
在高中一 NSS 數學課，我開始教二次圖像和二次方程之根（roots）的關係。現在課程涉及複數，卻沒有教如何利用圖像尋找複根（complex roots）或非實根（unreal roots），我在此補充一下。 以下是在下用極速粗製濫造的 ETV，同學先看看： 解說： （甲）二次方程的情況 若 的複根是 ，則 ………………… more →
Quod Erat Demonstrandum
A nice coordinate geometry problem (suitable for Form 7 pure)
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koopakoo wrote 3 weeks ago: Let be a point on the curve Let be the tangent line at Given intersects the curve again at Prove tha … more →
For my IMO Students (Basic and Advanced)
koopakoo wrote 3 weeks ago: Here is good problem for my IMO students. Given a real number . Find the greatest area of triangle A … more →
利用圖像尋找非實根
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johnmayhk wrote 3 weeks ago: 在高中一 NSS 數學課，我開始教二次圖像和二次方程之根（roots）的關係。現在課程涉及複數，卻沒有教如何利用圖像尋找複根（complex roots）或非實根（unreal roots），我在此補 … more →
For my Form 7 Pure students: A proof of the Rational Root Theorem
koopakoo wrote 1 month ago: Let Suppose , where is a rational root. Then we have . Multiplying both sides by yields: Since , we … more →
逆函數未必連續
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johnmayhk wrote 1 month ago: 函數 連續，並不保證逆函數 也連續。 (在定義兩個拓樸空間同胚（homeomorphic）時，就是要求他們之間存在一一對應的連續函數 ，並 也要連續。) 舉例，設 並 （即 不過是 中的 unit c … more →
To my Form 7 students: I am impressed!
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koopakoo wrote 1 month ago: Just finished my CB class tonight and I am impressed by two students. They both expressed interests … more →
Selected solutions for my F7 Pure (Polynomials)
koopakoo wrote 1 month ago: Book 2, pg 14. (00IQ12)(c) (i) Let y = x^2, then we have ay^2 – by + a. Considering the discri … more →
Different formats of primitive functions
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johnmayhk wrote 2 months ago: Just take a rest from work, type something boring here… Students, you may regard the above as … more →
拼出 $100
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johnmayhk wrote 2 months ago: 老題：利用面值 $10，$20 及 $50 紙幣若干張，要拼出 $100，問有多少組合方式？ 可以拿 2 張 $50 紙幣，這是一種組合； 可以拿 3 張 $10，1 張 $20 及 1 張 $50， … more →
Finding general term by generating function
johnmayhk wrote 2 months ago: It is extremely easy to set up questions on number pattern, like 1, 3, 8, 19, 42, 89, ? for more det … more →
Challenging problem in Linear Algebra
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koopakoo wrote 3 months ago: Let Solve the following matrix equation where … more →
Mental exercises for my Pure Complex Class (1)
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koopakoo wrote 3 months ago: Dear Students, Please avoid as much calculations as possible. In ideal situation, you should not do … more →
Selected Solutions to My Pure Complex Class (1)
koopakoo wrote 3 months ago: Lesson 1: Page 26 where the last line follows from part a(ii). Page 27 (i) Hence Note that we have i … more →
Concept Check for my Linear Algebra Students
koopakoo wrote 3 months ago: Let (E) be the following system: Let be the augmented matrix. Determine whether the following statem … more →
[FW][YouTube] 維度 數學漫步 Dimensions Tour
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johnmayhk wrote 4 months ago: More at http://www.youtube.com/view_play_list?p=ED4CEC2CC684DDDD … more →
計到即存在?
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johnmayhk wrote 4 months ago: 初中時教解二次方程，我通常順便說一個無聊的例： = ? 要求出「答案」，我們可設 等號左右兩邊取平方，則 故此 解出 或 （不合），亦即 當然， 的表達有點兒「唔清唔楚」，清楚一點的表達，比如是 即 … more →
閒談一些基本東西：導數符號，函數，解釋
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johnmayhk wrote 4 months ago: 1. 高階導數的符號 同學問，為何 D 兩次（即求二階導數）的符號是 而不是 或 ？ 記得 CJ 老師曾經出上面的來考同學。 我不知其來源，只是靠估兼無聊地說： 比如求 的導數（w.r.t. x）， … more →
考試前後
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johnmayhk wrote 5 months ago: 前 考中二數學前夕，梁同學致電問數。我都「好野」，一邊行街赴父親節宴會，一邊做數講數。 問題太多，晚上回家，梁同學再問下半場。 他問什麼中二數學問題？列幾個： 1. Factorize . 2. 把 … more →
應用純數
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johnmayhk wrote 5 months ago: 有人強調數學在生活中的應用，有人指出所謂「數學在生活中的應用」是牽強的。經驗使我比較贊成後者。 無論如何，以趣味的角度向學生介紹（所謂的）數學在生活中應用的例子，或許可以達到某些教學的成效。 讓我在這 … more →
