答客問

Q：
Let y=-x/(x^2-2x+4), where x is real. a)Find the range of values of y for all real values of x. b)hence,find the value of x such that the value of y is maximum.

Sol：
a) yx^2+(1-2y)x+4y = 0
delta >= 0
(1-2y)^2 - 16y^2 >= 0
(1-6y)(1+2y) >= 0
-1/2 <= y<= 1/6

b) sub. y = 1/6
x^2 + 4x + 4 = 0
x = -2

Test Solution (Maths Ch 1 & A.M. Ch.1)

Question 1

Question 2

Question 3

Question 3 Lines added on the given figure

Question 4

Question 5

Question 6

Question 7

Question 8

恢復無期

網絡供應故障，停用至今已超過一星期。

不上學的日子，與網上世界中止了一切連繫。

身邊人發生的事原來還真不少。

出卷、網上張貼教材都出現問題。

抱歉了，四茂的同學。

執到寶

這天在船山版上看到1089系列，驚嘆美妙教材無處不在。
忍不了抄它過來：

一、1089 之一

這是上學年Fun Fair Day數學學會讓來客猜問的遊戲。

任意選一個三位數。哪個也行，只要首尾的兩個數字相差 2 或以上就行.

把這個三位數左右倒轉，拿來跟原初的數相減 (以大數減去小數)。例如：

782 - 287 = 495

把得到的結果再左右倒轉一次，跟倒轉前的相加。

現在，我可猜到你心中的神秘數字了。當然就是……

呵，知道何解嗎？


二、1089之二

最美的畢氏定理證明。三、1089之三

π 和單數（奇數）1，3，5，7……竟然扯得上關係，你可知道？

四、1089之四

拓撲, 可說是超級橡皮糖的幾何:左圖兩環相扣著的那一塊超級橡皮糖，有沒有法子 smoothly（不裂開不拉斷，孔可以移位但不會消失）拉長縮短，變成右圖解開的那一塊？

五、1089之五

何用費神論證？直覺歸納有何不好？這裡有一例。

經典拆地雷遊戲

數字表示比鄰八格的地雷總數。問地雷在哪裡？

Interesting Result from Equation Solving

Consider

x - 1 = 0

The only root of the above equation is 1.

However, when we take
x = 1
and take square of both sides, then

x² = 1
x² - 1 = 0
(x - 1)(x + 1) = 0

Consequently, x = 1 or -1

TWO roots are found, but actually one of them SHOULD BE REJECTED.

*********************

Don't Be Afraid

More than One root found. Don't Be Afraid.

Not all the roots are real. Don't Be Afraid.

Among the real roots, some should be rejected.

Just do it.


More than One girl friend got. Don't Be Afraid.

Not all the girl friends are real. Don't Be Afraid.

Among the real friends, some should be rejected.

Just say it.


More challenging problems ahead. Don't Be Afraid.

Not all of them are real challenges. Don't Be Afraid.

Most of them will be eventually solved.

So just believe it.

We just Don't Need To Be Afraid.

Events which are "Equally Likely to Happen"

Determine which of the following is true:

1. Mr. Chiu plays chess with Fat Chuen. He will either win, lose, or get a draw. As a result the probability that Mr. Chiu win is 1/3.

2. Mr. Chiu plays the famous lottery style game "Mark Six". He selects 6 numbers out of a possible 49 and it costs him $5. He will either win or lose. The probability of getting a prize is 1/2.

3. There are 2 red and 3 white balls in a bag. One ball is drawn out randomly. The probability of getting a red ball is 1/2.

4. A coin is tossed 2 times. Denote a head by H and a tail by T. There are 3 possible outcomes: 2H, 1H and 1T, 2T. Consequently the chance of getting 2 heads is 1/3.

5. 2 dice are thrown. The sum of the 2 numbers faced up is lying between 2 and 12. The probability of getting a 'double six' is hence 1/11.

5A 0708 Revision MC Ch16 Answers

5A 0708 Revision Test 5 Answers

Question 1a


Question 1b, 2 and 3


Question 4


Question 5

5A 0708 Revision MC Ch12 & 13 Answers

Chapter 12 Mensuration

Chapter 13 Basic Properties of Circle

5A 0708 Revision Test 4 Answers

1a, 1b至6之解答。



第7題之解答。

5A 0708 Revision Test 3 Answers

Question 1 and 2a
Question 2b
Question 3
Question 4, 5 and 6

5A 0708 Revision Test 2 Answers

一至四題之解答。

第五第六題之解答。

Review: Polynomials

1. Write down the sum of
a) (x - 3) and (3x + 2)

b) (4x² - 3x - 2) and (2x² +5x - 6)

2. Write down the difference when
a) (3x - 2) is subtracted by (2x - 5)

b) (2x - 7) is subtracted by (5x + 4)

c) (3x + 6) is subtracted by (2x - 5)

d) (4a - b) is subtracted by (2a + b)

e) (3a + 2b) is subtracted by (a - 4b)

3. Write down the product of
a) (x + 2) and (x + 3)

b) (x + 5) and (x + 1)

c) (x - 2) and (x - 1)

d) (x + 4) and (x - 3)

e) (x - 5) and (x + 2)

f) (2x - 1) and (x - 2)

g) (2x + 3) and (x - 1)

h) (3x - 2) and (2x - 3)

4. Expand
a) (x - 3)²

b) (x + 4)²

c) (2x - 3)²

d) (3x - 1)²

e) (4a - 5)(a + 2)

f) (3a - b)(2a + b)

g) (5a - 2b)(a + b)

5A 0708 Revision Test 1 Answers

第一至四題之解答。
第五、六題之解答。
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