section a(1) (35 marks)data.pakkau.edu.hk/~chanchiho/數學/hkdse/mock paper_jack/mock_paper… ·...

Mock Paper_2013-2014_version A Paper I Jack CHAN

1

SECTION A(1) (35 marks)

1. Simplify (𝑥3𝑦−2)

−2

𝑥3𝑦2 and express your answer with positive indices.

化簡 (𝑥3𝑦−2)

−2

𝑥3𝑦2 ，並以正指數表示答案。 (3 marks)

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2. Make h the subject of the formula =3(ℎ−2)

ℎ .

令 h 成為公式 𝑘 =3(ℎ−2)

ℎ 的主項。 (3 marks)

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3. Factorize

因式分解

(a) 25𝑎2 − 9𝑏2

(b) 25𝑎2 − 9𝑏2 − 15𝑎 − 9𝑏

(3 marks)

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4. The weight of 3 balls and 4 books is 7.2 kg while the weight of 4 balls and 3 books is 6.8 kg. Find the weight

of a ball.

3 個籃球和 4 本書的重量為 7.2 kg，而 4 個籃球和 3 本書的重量為 6.8 kg。求一個籃球的重

量。 (4 marks)

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5. (a) Solve the inequality 4−9𝑥

5> 2(𝑥 − 10) .

(b) Find all integers satisfying both the inequality 4−9𝑥

5> 2(𝑥 − 10) and 16 − 8𝑥 ≤ 0 .

(a) 解不等式 4−9𝑥

5> 2(𝑥 − 10) 。

(b) 求所有能同時滿足不等式 4−9𝑥

5> 2(𝑥 − 10) 及 16 − 8𝑥 ≤ 0 的整數。

(4 marks)

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6. The marked price of a smartphone was $6000 and the seller can make a profit of 20% if it was sold at that

price. In a sale, the smartphone was sold at a discount of 20%.

(a) Find the cost of the smartphone.

(b) Did the seller still make a profit? Explain your answer.

一部智能電話的標價為 $6000，若以此價格售出，賣家可獲得 20% 的利潤。在一個特賣會內，這

電話以八折出售。

(a) 求話電話的成本。

(b) 在那特賣會內，賣家仍可獲利嗎？試解釋你的答案。

(4 marks)

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4

7. The box-and-whisker diagram below shows the distribution of the exam marks of the students of School D.

下面的框線圖顯示某學校的一大群學生的考試成績的分佈：

The inter-quartile range and the range of the above distribution are 41 and 59.

(a) Find 𝑎 and 𝑏.

(b) The students joined Jack’s tutorial and the lowest mark was increase by 5 marks in the second

exam. Jack held the belief that at least 25% of the students enhanced their exam performance after

the tutorial. Do you agree? Explain your answer.

該分佈的四分位數間距及分佈域分別為 41 及 59。

(a) 求 𝑎 和 𝑏.。

(b) 該些學生參與積克的補習班。現知在補習後，該些學生在第二次考試中，最低的分數比之前

上升了 5 分。積克宣稱在補習後，至少 25% 的學生的成績有改善。你是否同意？試解釋你

的答案。

(4 marks)

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Marks (分數) 𝑎 𝑏 22 49 78


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8. The length of a string is measured as 2.0 m correct to the nearest 0.1 m.

(a) Find the longest possible length of the string with a unit of cm.

(b) If the string is divided into 46 equal parts. Is it possible that each part is measured as 5 cm correct

to the nearest 1 cm? Explain your answer.

一條繩子的長度量得 2.0 m 準確至最接近的 0.1 m。

(a) 求一條繩子的最長可取長度，答案以 cm 為單位。

(b) 若那條繩子分為 46 條相同長度的小繩子，每一條小繩子有沒有可能量得 5 cm 準確至最接近

的 cm？試解釋你的答案。

(5 marks)

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9. In figure 1, the volume of the prism ABCDEFGH is 340 cm3. The base ABCD of the prism is a trapezoid and

AD is parallel to BC. It is known that ∠BAD = 90°, AB = 4 cm, AD = 10 cm and DE = 10 cm.

圖 1 中，實心直立角柱體 ABCDEFGH 的體積為 340 cm3 。該柱體的底 ABCD 為一梯形，其中 AD

平行於 BC 。已知 ∠BAD = 90°， AB = 4 cm， AD = 10 cm 及 DE = 10 cm。

Find

(a) The length of BC,

(b) The total surface area of the prism ABCDEFGH.

求

(a) BC 的長度，

(b) 角柱體 ABCDEFGH 的總表面面積。

(5 marks)

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Figure 1 圖 1


7

SECTION A(2) (35 marks)

10. Jack is studying how many hours the secondary students spent on playing games in a week. He received 30

questionnaires. The stem-and-leaf diagram below shows data of the number of hours.

積克進行一項中學生在某星期內用於打機的時數的調查。在發出的問卷中，有三十份回覆。下面的

幹葉圖顯示該三十份問卷記錄得的時數：

(a) Find the mean and median of the data above.

(b) Jack then received four questionnaires more. He found that the mean of the four data is 24. It is

known that two of the data are 25 and 26.

(i) Find the mean of the 34 data.

(ii) Is the median of the 34 data as same as the median found in (a)? Please explain your

answer.

(a) 求該三十份問卷記錄得的時數的平均值及中位數。 (2 marks)

(b) 積克再收到四份問卷。他得知這四份問卷記錄得的時數的平均值為 24。現知這四份問卷其中

兩份記錄得的時數為 25 及 26。

(i) 寫出該三十四份問卷記錄得的時數的平均值。

(ii) 該三十四份問卷記錄得的時數的中位數與 (a) 所求得的中位數有沒有可能相同？試

解釋你的答案。

(4 marks)

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0 0 0 1 1 1 2 2 8 8 8 8 1 1 2 2 3 4 5 6 8 8 9

5 6 7 7 8 9 9


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11. Let A cm2 be the area of playground with a width 𝑙 cm. It is known that A is the sum of two parts. One

part varies directly as 𝑙 and another part varies directly as 𝑙2. When 𝑙 = 10, A = 1000. At the same

time, when 𝑙 = 20, 𝐴 = 1600.

(a) Find the area of playground when its width is 35 cm.

(b) Find the maximum possible value of the area of playground.

設 A cm2 為一闊度是 𝑙 cm 的運動場的面積。已知 A 為兩部分之和，一部分隨 𝑙 正變，而另一部

分隨 𝑙2 正變。當𝑙 = 10 時， A = 1000；當𝑙 = 20 時， A = 1600。

(a) 求一闊度為 35 cm 的運動場的面積。 (4 marks)

(b) 求運動場的面積的最大可能值。 (2 marks)

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9

12. Let 𝑓(𝑥) = 𝑥3 + 𝑥2 − 4𝑥 + 𝑘 where k is a constant. It is known that 𝑓(𝑥) = (𝑥 − 2)(𝑎𝑥2 + 𝑏𝑥 + 𝑐)

where a, b and c are constants.

(a) Find a, b and c.

(b) Jack claims that 𝑦 = 𝑥3 + 𝑥2 − 4𝑥 + 1 intercepts 𝑦 = 5 at three points. Do you agree? Please

explain your answer.

設 𝑓(𝑥) = 𝑥3 + 𝑥2 − 4𝑥 + 𝑘 ，其中 k 為一常數。已知 𝑓(𝑥) = (𝑥 − 2)(𝑎𝑥2 + 𝑏𝑥 + 𝑐) ，其中 a、

b 及 c 均為常數。

(a) 求 a、b 及 c 。 (4 marks)

(b) 積克宣稱 𝑦 = 𝑥3 + 𝑥2 − 4𝑥 + 1 與 𝑦 = 5 有三個交點。你是否同意？試解釋你的答案。

(3 marks)

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10

13. In a factory, two congruent metal cones with the radius of R cm are melt and reconstructed to 27 smaller

congruent metal cones with the radius of r cm and the height of 5 cm. It is known that the base area of

larger cone is 9 times as the base area of smaller cone.

(a) Find

(i) r : R ,

(ii) The height of the larger cone.

(b) Someone claims that smaller cone is similar to larger cone. Do you agree? Please explain your

answer.

在某工場內，把 2 個底半徑均為 R cm 的完全相同的實心金屬圓錐體熔化，並重鑄成 27 個底半

徑均為 r cm 及高均為 5 cm 的完全相同的較小的實心圓錐體。已知較大的圓錐體的底面積為教小

的 9 倍。

(a) 求

(i) r : R ，

(ii) 較大的圓錐體的高。

(5 marks)

(b) 某人宣稱較小的圓錐體與較大的圓錐體相似。你是否同意？試解釋你的答案。 (2 marks)

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11

14. The y-intercept of two parallel line 𝐿1 and 𝐿2 are 2 and -2 and x-intercept of 𝐿1 is -3. P is a moving

point so that the shortest distance between P and 𝐿1 is equal to the shortest distance between P and 𝐿2.

Let Γ be the locus of P.

(a) (i) Describe the geometry relationship between Γ and 𝐿1.

(ii) Find the equation of Γ.

(b) The equation of circle C is (𝑥 − 3)2 + (𝑦 − 2)2 = 25. Let Q be the centre of C.

(i) Does Γ pass through Q? please explain your answer.

(ii) If 𝐿1 intercepts C at point A and B and Γ intercepts C at point H and K.

Find ∠AQH ∶ ∠BQK.

兩平行線 𝐿1 and 𝐿2 的 y 軸截距分別為 2 及 -2 ，且 𝐿1 的 x 軸截距為 -3。 P 為直角座標平面

上的一動點使得由 P 至 𝐿1 的最短距離等於由 P 至 𝐿2 的最短距離。將 P 的軌跡記為 Γ 。

(a) (i) 描述 Γ 與 𝐿1 之間的幾何關係。

(ii) 求 Γ 的方程。

(5 marks)

(b) 圓 C 的方程為 (𝑥 − 3)2 + (𝑦 − 2)2 = 25。將 C 的圓心記為 Q。

(i) Γ 是否通過 Q？試解釋你的答案。

(ii) 若 𝐿1 與 C 相交於 A 及 B 而 Γ 與 C 相交於 H 及 K，

求 ∠AQH ∶ ∠BQK。

(4 marks)

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12

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13

SECTION B (35 marks)

15. The mean of exam marks of a group of students in an examination are 60. Jack believed the exam paper is

too difficult and adjusted their exam marks. All is increased by 10% and then added by 3.

(a) Find the mean of the students’ exam marks after the adjustment.

(b) A parent complained Jack that the adjustment is unfair and the students’ standard scores will be

changed. Do you agree? Please you answer.

一班學生在某數學測驗得分的平均分為 60 分。積克相信考試卷的程度太深，故此將每名學生的測

驗得分調整，使每個得分均增加 10%然後額外加 3 分。

(a) 求得分調整後，測驗得分的平均分。 (1 mark)

(b) 一名家長投訴積克這做法不公平，認為學生的標準分因得分調整而改變。你同意這位家長的

說法嗎？試解釋你的答案。 (2 marks)

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14

16. There are 20 balls in a bag. 7 of them are back and 13 of them are white. 9 balls are randomly selected.

(a) Find the probability of that at least 5 black balls will be selected.

(b) Find the probability of that at least 5 white balls will be selected.

某個袋子內有二十個波，其中有 7 個是黑色的和 13 個是白色的。9 個波隨機由那個袋子被抽出。

(a) 求最少有 5 個黑色波被抽出的概率。 (2 marks)

(b) 求最少有 5 個白色波被抽出的概率。 (2 marks)

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15

17. The coordinates of the centre of a circle C are (4, 6). It is known that y-axis is the tangent of C.

(a) Find the equation of C.

(b) The slope of a straight line L and x-intercept are 1 and k. Find the coordinates of the mid-point of AB

in terms of k if L intercepts C at point A and B.

圓 C 的圓心的座標為 (4, 6)。已知 y 軸為 C 的切線。

(a) 求 C 的方程。 (2 marks)

(b) 直線 L 的斜率及 x 截距分別為 1 及 k。若 L 與 C 相交於 A 及 B，以 k 表 AB 的中點座標。

(5 marks)

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17

18. (a) In figure 2(a), it shows a triangle card ABC and AB = 25cm, BC = 18cm and AC = 32cm. Let M be

a point on AC so that ∠BMC = 60°.

圖 2(a) 顯示一張三角形紙卡 ABC，其中 AB = 25cm, BC = 18cm and AC = 32cm。設 M 為 AC

上的一點使得 ∠BMC = 60°。

Find

求

(i) ∠BCM ,

(ii) BM .

(3 marks)

18. (b) Jack folded the triangle card in (a) by following BM so that AB and BC both lie on the horizontal

plane, shown in figure 2(b). Let N be a point on plane ABC so that MN is perpendicular to plane ABC

and MN = 10cm.

積克將 (a) 所描述的三角形紙卡沿 BM 摺起，使得 AB 及 BC 均位於水平地面上，如圖 3(b)

所示。設 N 為平面 ABC 上的一點使得 MN 垂直於平面 ABC 和 MN = 10cm。

(i) Find the angle between plane MBC and plane ABC.

(ii) Let O be a point on BC. Please describe how ∠MON changes when O moves from B to C.

(i) 求平面 MBC 與平面 ABC 的夾角。

(ii) 設 O 為 BC 上的一點，試描述當 O 由 B 移到 C， ∠MON 如何改變。

(5 marks)

Figure 2(a) 圖 2(a)

Figure 2(a) 圖 2(a)


18

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19

19. The development of population in a city is under study. It is given that the total number of elderly people

at the end of the 1st year is 2 × 106 and the subsequent years, the total number of elderly people

increased each year is 𝑟 % of the total number of elderly people at the end of the precious year ,where r

is a constant, and the total number of the elderly died each year is 8 × 104. It is found that the total

number of the elderly people at the end of the 3rd year is 2.252 × 106.

現對某城市人口的發展進行研究。已知第 1 年年終時所有老人的數目為 2 × 106，並且在隨後幾年

裹，每年老人的數目會增加數目為前一年年終時老人的數目的 𝑟 %，其中 r 為一常數，而每年老人

的又會減少 8 × 104。現知第 3 年年終時老人的數目為 2.252 × 106。

(a) (i). Express in terms of r , the total number of the elderly people at the end of the 2nd year.

(ii) Find r .

(a) (i). 以 r 表第 2 年年終時老人的數目。

(ii) 求 r 。

(4 marks)

(b) (i). Express in terms of n , the total number of the elderly people at the end of the nth year.

(ii) At the end of which year will the total number of the elderly people first exceed 3 × 106?

(b) (i). 以 n 表第 n 年年終時老人的數目。

(ii) 哪一年年終時老人的數目會首次超過 3 × 106 ？

(5 marks)

(c) It is assumed that the total number of the manpower in the city at the end of the nth year is

(𝑎(1.21)𝑛 + 𝑏), where a and b are constants. Some research results reveal the following

information:

現假設第 n 年年終時勞動人口的數目為 (𝑎(1.21)𝑛 + 𝑏) ，其中 a 及 b 均為常數。某些研究

結果顯示下列資料：

n The total number of the manpower at the end of the nth year

第 n 年年終時勞動人口的數目

1 4 × 106

2 4.64 × 106


20

A research assistant claims that based on the above assumption, the total number of the elderly

people will be greater than the total number of the manpower at the end of a certain year. Is the

claim correct? Explain your answer.

一研究助理宣稱基於上述假設，某年年終時老人的數目會大於勞動人口的數目。該宣稱是否

正確？試解釋你的答案。 (4 marks)

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section a(1) (35 marks)data.pakkau.edu.hk/~chanchiho/數學/hkdse/mock paper_jack/mock_paper… ·...

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