Quod Erat Demonstrandum

2008/02/20

Additional Mathematics 補底

Filed under: Additional / Applied Mathematics,HKCEE — johnmayhk @ 11:01 上午

更新日期:2008-02-21 (希望不斷更新。)
免插聲明:本文為修附加數學科的中五同學純粹補底之用,高手勿插,謝謝。

1. 口訣:一冇 trigo,即寫 general

有關 general solution,考生常犯毛病見下

\sin(2\theta) = \sin30^o
2\theta = 30^o
\theta = 15^o
\therefore \theta = 180^on + (-1)^n(15^o)

錯!最後一步才寫 general solution 的形式是錯的!

不要到最後一步,而是要一開始便寫,即是

\sin(2\theta) = \sin30^o
2\theta = 180^on + (-1)^n(30^o)

即口訣的:一冇 trigo function 符號(本例是 sin),立即 (係立即!!!) 要寫 general solution 的形式。於是,我們有

\sin(2\theta) = \sin30^o
2\theta = 180^on + (-1)^n(30^o)
\therefore \theta = 90^on + (-1)^n(15^o)

才是正確答案。

2. 又 degree 又 radian;個樣衰過 Edit 神

即係呢,例如你寫

\theta = n\pi + (-1)^n30^o

不堪入目,死哂。

\pi is in radian but 30^o is in degree, don’t mix them up! You may write either

\theta = n\pi + (-1)^n\frac{\pi}{6}

or

\theta = 180^on + (-1)^n30^o

3. 向量:寫箭咀,冇得除,有點要露點,寫 square 好危險!

(a) 寫箭咀
即係記得寫箭咀符號,例如 \overrightarrow{a}。就算題目用『粗體』表示向量,例如 Let \overrightarrow{AB} = a,作為考生的你,千祈唔好學佢用『粗體』,一定要用箭咀!即是寫 \overrightarrow{AB} = \overrightarrow{a}.

(b) 冇得除
千祈唔好出現『向量除向量』,例如,你寫 \frac{\overrightarrow{a}}{\overrightarrow{b}},錯!有時,題目有以下形式

Suppose A,B,C lie on the same line with B divides A and C internally.
Given \frac{AB}{BC} = 3, express \overrightarrow{AB} in terms of \overrightarrow{BC}

不要寫 \frac{AB}{BC} = 3 \Rightarrow \frac{\overrightarrow{AB}}{\overrightarrow{BC}} = 3 \Rightarrow \overrightarrow{AB} = 3\overrightarrow{BC}

雖然答案正確,但表達是錯呀!

應寫

\frac{AB}{BC} = 3 \Rightarrow AB = 3BC \Rightarrow \overrightarrow{AB} = 3\overrightarrow{BC}

(c) 有點要露點
做 dot product,露點唔少得,即
\overrightarrow{a} \bullet \overrightarrow{b},啱!
\overrightarrow{a}\overrightarrow{b},錯!
\overrightarrow{a}(\overrightarrow{b} + \overrightarrow{c}),都錯!

(d) 寫 square 好危險

同學問,下面寫法 OK 嗎?

\overrightarrow{a} = \overrightarrow{b} + \overrightarrow{c}
\overrightarrow{a}^2 = (\overrightarrow{b} + \overrightarrow{c})^2

唔得呀,\overrightarrow{a}^2 意義不明。要寫

\overrightarrow{a} = \overrightarrow{b} + \overrightarrow{c}
\overrightarrow{a}\bullet\overrightarrow{a} = (\overrightarrow{b} + \overrightarrow{c})\bullet(\overrightarrow{b} + \overrightarrow{c})

才可。

4. unit vector 唔係 1,你寫 \overrightarrow{u} = 1 就死得。

嗱,unit vector 是長度(magnitude)等於 1 的向量。要寫就應該寫

|\overrightarrow{u}| = 1

5. 向量相減唔會零,點解講極都唔明

\overrightarrow{a} - \overrightarrow{a} = 0 錯!
\overrightarrow{a} - \overrightarrow{a} = \overrightarrow{0} 啱!

6. 求向量長度口訣:有 i,j 畢氏定理;冇 i,j 用 dot 無死

\overrightarrow{p} = 3\overrightarrow{i} + 4\overrightarrow{j}
|\overrightarrow{p}| = \sqrt{3^2 + 4^2}(這是畢氏定理也)

\overrightarrow{p} = 3\overrightarrow{a} + 4\overrightarrow{b}
那麼,千祈唔好以為
|\overrightarrow{p}| = \sqrt{3^2 + 4^2},錯錯錯錯錯錯錯錯錯錯錯錯錯!
這時,冇 \overrightarrow{i}, \overrightarrow{j},唔可以用畢氏定理,用咩?Dot。
即考慮

\overrightarrow{p} \bullet \overrightarrow{p} = |\overrightarrow{p}|^2

|\overrightarrow{p}|^2 = (3\overrightarrow{a} + 4\overrightarrow{b}) \bullet (3\overrightarrow{a} + 4\overrightarrow{b})

做些手腳,得

|\overrightarrow{p}|^2
= 9\overrightarrow{a} \bullet \overrightarrow{a} + 24\overrightarrow{a} \bullet \overrightarrow{b} + 16\overrightarrow{b} \bullet \overrightarrow{b}
= 9|\overrightarrow{a}|^2 + 24\overrightarrow{a} \bullet \overrightarrow{b} + 16|\overrightarrow{b}|^2

如無意外,題目應給出具體的 \overrightarrow{a}, \overrightarrow{b}。當知其長度與夾角,上述便能算之。

…寫住咁多先…想死中…

2 則迴響 »

  1. I see that the symbal will be taught at the end of the book…
    Chinese lesson is completely terrible, that situation has never happened in the worst primary school class, not that I experienced!
    So how’s the New Year?
    How comes there’s a little sheet of paper asking who on earth I think is the worst guy in class? Is that the class idea or…
    Sent you a letter, dunno if you’d seen…

    迴響 由 Ed' — 2008/02/20 @ 8:55 下午 | 回覆

  2. Yep.

    I and Ms JD will take the situation of Chi lesson seriously. Hopefully, Ms LTW will be back before Talentine.

    My holiday was fine because I’d left HK with my wife for 7 days without doing any work.

    The sheet is an idea of the class association, some kinds of statistics to be posted on board.

    The letter? Oops, forget about it.

    迴響 由 johnmayhk — 2008/02/21 @ 5:16 下午 | 回覆


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