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甲部8 題 · 50 分
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二項展式
指數函數和對數函數
函數的導數
函數的求導法
二階導數
求導法的應用
不定積分法及其應用
定積分法及其應用
運用梯形法則計算定積分的近似值
條件概率和貝葉斯定理
離散隨機變量
概率分佈、期望值和方差
二項分佈
泊松分佈
二項分佈和泊松分佈的應用
正態分佈的基本定義及其性質
正態變量的標準化及標準正態分佈表的運用
正態分佈的應用
抽樣分佈和點估計
總體平均值的置信區間

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2027-DSEMATH EPM1

STUDIFY MOCK EXAMINATION

HONG KONG DIPLOMA OF SECONDARY EDUCATION EXAMINATION 2027 · PRACTICE PAPER

MATHEMATICS Extended PartModule 1 (Calculus and Statistics)

Question-Answer Book

8.30 am – 11.00 am (2½ hours)
This paper must be answered in English

INSTRUCTIONS

(1)After the announcement of the start of the examination, you should first write your Candidate Number in the space provided on Page 1 and stick barcode labels in the spaces provided.
(2)This paper consists of TWO sections, A and B.
(3)Attempt ALL questions in this paper. Write your answers in the spaces provided in this Question-Answer Book. Do not write in the margins. Answers written in the margins will not be marked.
(4)Graph paper and supplementary answer sheets will be supplied on request. Write your Candidate Number, mark the question number box and stick a barcode label on each sheet, and fasten them with string INSIDE this book.
(5)Unless otherwise specified, all working must be clearly shown.
(6)Unless otherwise specified, numerical answers should be either exact or given to 4 decimal places.
(7)No extra time will be given to candidates for sticking on the barcode labels or filling in the question number boxes after the 'Time is up' announcement.

Studify practice paper. Not an official HKEAA examination paper.

Please stick the barcode label here.

Candidate Number

Answers written in the margins will not be marked.

SECTION A (50 marks)

The table below shows the probability distribution of a discrete random variable

X

, where

a

and

b

are constants. It is given that

\mathrm{E}(X)=3

. (a) Find

a

and

b

. (b) Find

\mathrm{Var}(X)

and

\mathrm{Var}(3-2X)

.(6 marks)

x	1	2	4	7
P(X = x)	0.2	a	b	0.1

Answers written in the margins will not be marked.

Let

A

and

B

be two events. Denote the complementary events of

A

and

B

A'

and

B'

respectively. It is given that

\mathrm{P}(A)=0.4

\mathrm{P}(B\mid A)=0.3

and

\mathrm{P}(A\cup B)=0.7

. (a) Find

\mathrm{P}(B)

. (b) Are

A

and

B

independent? Explain your answer. (c) Find

\mathrm{P}(A'\mid B')

.(5 marks)

Answers written in the margins will not be marked.

The number of customers arriving at a convenience store in a 10-minute interval follows a Poisson distribution with a mean of 1.8. (a) Find the probability that at least 2 customers arrive at the store in a 10-minute interval. (b) Find the probability that exactly 3 customers arrive at the store in a 20-minute interval. (c) Find the probability that, in 5 separate 10-minute intervals, there are exactly 3 intervals in each of which at least 2 customers arrive at the store.(7 marks)

Answers written in the margins will not be marked.

The weights of apples from an orchard are normally distributed with a standard deviation of 15 g. A random sample of 36 apples is taken and their mean weight is 182 g. (a) Construct a 95% confidence interval for the mean weight of the apples from the orchard. (b) Find the least sample size such that the width of a 99% confidence interval for the mean weight is less than 8 g.(6 marks)

Answers written in the margins will not be marked.

(a) Expand

e^{3x}

in ascending powers of

x

as far as the term in

x^{3}

. (b) Find the coefficient of

x^{3}

in the expansion of

e^{3x}(1-2x)^{5}

.(6 marks)

Answers written in the margins will not be marked.

Consider the curve

C: y=\ln(2x-3)+x^{2}

, where

\displaystyle x>\dfrac{3}{2}

. (a) Find

\displaystyle \dfrac{dy}{dx}

. (b) Find the equation of the tangent to

C

at the point where

x=2

. (c) The tangent in (b) cuts the

x

-axis and the

y

-axis at

P

and

Q

respectively. Let

O

be the origin. Find the area of

\triangle OPQ

.(7 marks)

Answers written in the margins will not be marked.

Let

g(x)

be a function such that

\displaystyle g'(x)=\dfrac{4x}{x^{2}+1}

for all real numbers

x

. It is given that

g(0)=3

. (a) Find

g(x)

. (b) Find the exact value of the area of the region bounded by the curve

y=g'(x)

, the

x

-axis and the straight line

x=2

.(6 marks)

Answers written in the margins will not be marked.

The profit

P

(in million dollars) of a company

t

years after its establishment is modelled by

P=20te^{-0.25t}

, where

0\leq t\leq 10

. (a) Find

\displaystyle \dfrac{dP}{dt}

. (b) Find the greatest profit of the company and the value of

t

at which it occurs. (c) Is the profit increasing when

t=6

? Explain your answer.(7 marks)

Answers written in the margins will not be marked.

SECTION B (50 marks)

A factory has two machines, A and B, which produce 60% and 40% of its chips respectively. 2% of the chips produced by machine A and 5% of the chips produced by machine B are defective.

(a)

Find the probability that a randomly selected chip produced by the factory is defective.(1 分)

(b)

Given that a randomly selected chip is defective, find the probability that it is produced by machine B.(2 分)

(c)

The chips are packed in boxes of 20. A box is accepted if it contains at most 1 defective chip. (i) Find the probability that a box is accepted. (ii) An inspector checks boxes one by one. Find the probability that the first box which is not accepted is the 4th box checked. (iii) Find the probability that exactly 8 out of 10 boxes are accepted.(9 分)

Answers written in the margins will not be marked.

10.

The daily commuting time of the workers in a city follows a normal distribution with a mean of 42 minutes and a standard deviation of 8 minutes.

Extract from the table of the area under the standard normal curve between 0 and z

z	1.00	1.28	1.50	1.96
Area	.3413	.3997	.4332	.4750

(a)

Find the probability that the daily commuting time of a randomly selected worker exceeds 50 minutes.(2 分)

(b)

Find

t

such that 10% of the workers have daily commuting times of more than

t

minutes.(2 分)

(c)

16 workers are randomly selected. (i) Find the probability that the mean daily commuting time of the 16 workers exceeds 45 minutes. (ii) Find the probability that exactly 3 of the 16 workers have daily commuting times of more than 50 minutes.(4 分)

(d)

A year later, a random sample of 64 workers has a mean daily commuting time of 44.5 minutes. Assume that the standard deviation is still 8 minutes. (i) Construct a 95% confidence interval for the mean daily commuting time

\mu

minutes of the workers. (ii) A reporter claims that the mean daily commuting time of the workers has increased. Using the result of (d)(i), do you agree? Explain your answer.(5 分)

Answers written in the margins will not be marked.

11.

Let

f(x)=\ln(x^{2}+1)

. Denote the region bounded by the curve

y=f(x)

, the

x

-axis and the straight lines

x=1

and

x=3

R

(a)

Using the trapezoidal rule with 4 sub-intervals, estimate

\displaystyle\int_{1}^{3}f(x)\,dx

.(3 分)

(b)

Is the estimate in (a) an over-estimate or an under-estimate? Explain your answer.(3 分)

(c)

Someone claims that the area of

R

exceeds 3.1. Do you agree? Explain your answer.(2 分)

(d)

(i) Find the equation of the tangent to the curve

y=f(x)

at the point where

x=2

. (ii) Hence, show that the area of

R

is less than

2\ln 5

.(4 分)

Answers written in the margins will not be marked.

12.

A fisheries officer models the number of fish

N

(in thousands) in a reservoir by

\displaystyle N=\dfrac{50}{1+4e^{-0.5t}}

, where

t

is the number of years elapsed since the start of the study.

(a)

Find the number of fish when

t=0

.(1 分)

(b)

Express

\displaystyle \ln\left(\dfrac{50}{N}-1\right)

as a linear function of

t

, and write down the slope and the intercept on the vertical axis of its graph.(2 分)

(c)

Find

\displaystyle \dfrac{dN}{dt}

.(2 分)

(d)

Find the value of

t

at which the rate of increase of the number of fish is the greatest.(4 分)

(e)

Estimate the number of fish in the reservoir after a very long time.(1 分)

(f)

The officer claims that the number of fish will exceed 45 000 within the first 6 years of the study. Do you agree? Explain your answer.(3 分)

Answers written in the margins will not be marked.

END OF PAPER

參考答案

揀一條題目就會顯示答案同評分指引。建議先自己試做。

第 1 題6 分

(a)

0.2+a+b+0.1=1

1(0.2)+2a+4b+7(0.1)=3

a=0.35

and

b=0.35

1A (b)

\mathrm{E}(X^{2})=1(0.2)+4(0.35)+16(0.35)+49(0.1)=12.1

\mathrm{Var}(X)=12.1-3^{2}=3.1

\mathrm{Var}(3-2X)=(-2)^{2}(3.1)=12.4

1A (6)

第 2 題5 分

(a)

\mathrm{P}(A\cap B)=0.4(0.3)=0.12

\mathrm{P}(B)=0.7-0.4+0.12=0.42

1A (b)

\mathrm{P}(A)\mathrm{P}(B)=0.168\neq 0.12=\mathrm{P}(A\cap B)

, so

A

and

B

are not independent. 1A f.t. (c)

\displaystyle \mathrm{P}(A'\mid B')=\dfrac{1-\mathrm{P}(A\cup B)}{1-\mathrm{P}(B)}=\dfrac{0.3}{0.58}

\displaystyle =\dfrac{15}{29}\approx 0.5172

1A (5)

第 3 題7 分

(a)

1-e^{-1.8}(1+1.8)

\approx 0.5372

1A (b) The number in a 20-minute interval follows a Poisson distribution with mean 3.6. 1M

\displaystyle \dfrac{e^{-3.6}(3.6)^{3}}{3!}\approx 0.2125

1A (c)

C^{5}_{3}(0.537163)^{3}(1-0.537163)^{2}

1M+1M

\approx 0.3320

1A (7)

第 4 題6 分

(a)

\displaystyle 182\pm 1.96\left(\dfrac{15}{\sqrt{36}}\right)

1M The confidence interval is

(177.1,\ 186.9)

. 1A (b)

\displaystyle 2(2.576)\left(\dfrac{15}{\sqrt{n}}\right)<8

1M+1M

n>93.3

1A The least sample size is 94. 1A (6) (Accept 2.575 for the critical value.)

第 5 題6 分

(a)

\displaystyle e^{3x}=1+(3x)+\dfrac{(3x)^{2}}{2!}+\dfrac{(3x)^{3}}{3!}+\cdots

\displaystyle =1+3x+\dfrac{9}{2}x^{2}+\dfrac{9}{2}x^{3}+\cdots

1A (b)

(1-2x)^{5}=1-10x+40x^{2}-80x^{3}+\cdots

1A Coefficient of

\displaystyle x^{3}=1(-80)+3(40)+\dfrac{9}{2}(-10)+\dfrac{9}{2}(1)

1M+1M

\displaystyle =-\dfrac{1}{2}

1A (6)

第 6 題7 分

(a)

\displaystyle \dfrac{dy}{dx}=\dfrac{2}{2x-3}+2x

1A (b) When

x=2

y=\ln 1+4=4

and

\displaystyle \dfrac{dy}{dx}=2+4=6

. 1M The equation of the tangent is

y-4=6(x-2)

, i.e.

y=6x-8

. 1A (c)

\displaystyle P=\left(\dfrac{4}{3},\,0\right)

and

Q=(0,\,-8)

1M+1A Area of

\displaystyle \triangle OPQ=\dfrac{1}{2}\left(\dfrac{4}{3}\right)(8)

\displaystyle =\dfrac{16}{3}

1A (7)

第 7 題6 分

(a) Let

u=x^{2}+1

\displaystyle g(x)=\displaystyle\int\dfrac{4x}{x^{2}+1}\,dx=\int\dfrac{2}{u}\,du

=2\ln(x^{2}+1)+c

g(0)=3

gives

c=3

, so

g(x)=2\ln(x^{2}+1)+3

. 1A (b)

g'(x)\geq 0

for

x\geq 0

, so the area is

\displaystyle \displaystyle\int_{0}^{2}\dfrac{4x}{x^{2}+1}\,dx

=\big[2\ln(x^{2}+1)\big]_{0}^{2}

=2\ln 5

1A (6)

第 8 題7 分

(a)

\displaystyle \dfrac{dP}{dt}=20e^{-0.25t}+20t(-0.25)e^{-0.25t}=20e^{-0.25t}(1-0.25t)

1M+1A (b)

\displaystyle \dfrac{dP}{dt}=0

gives

t=4

. 1M

\displaystyle \dfrac{dP}{dt}>0

for

0\leq t<4

and

\displaystyle \dfrac{dP}{dt}<0

for

4<t\leq 10

, so

P

is greatest when

t=4

. 1M Greatest profit

=80e^{-1}\approx 29.4304

million dollars 1A (c) When

t=6

\displaystyle \dfrac{dP}{dt}=20e^{-1.5}(1-1.5)<0

1M So the profit is not increasing. 1A f.t. (7)

第 9 題12 分

(a)

0.6(0.02)+0.4(0.05)=0.032

1A (b)

\displaystyle \dfrac{0.4(0.05)}{0.032}

=0.625

1A (c) (i)

(0.968)^{20}+C^{20}_{1}(0.032)(0.968)^{19}

1M+1M

\approx 0.8668

1A (ii)

(0.866798)^{3}(1-0.866798)

1M+1M

\approx 0.0867

1A (iii)

C^{10}_{8}(0.866798)^{8}(1-0.866798)^{2}

1M+1M

\approx 0.2544

1A (12)

第 10 題13 分

(a)

\mathrm{P}(X>50)=\mathrm{P}(Z>1)=0.5-0.3413=0.1587

1M+1A (b)

\mathrm{P}(Z>z)=0.1

gives

z\approx 1.28

, so

t\approx 42+1.28(8)=52.24

1M+1A (c) (i)

\displaystyle \bar X\sim N\left(42,\,\dfrac{8^{2}}{16}\right)

, so

\mathrm{P}(\bar X>45)=\mathrm{P}(Z>1.5)

=0.0668

1A (ii)

C^{16}_{3}(0.1587)^{3}(0.8413)^{13}

\approx 0.2367

1A (d) (i)

\displaystyle 44.5\pm 1.96\left(\dfrac{8}{\sqrt{64}}\right)

1M+1M The confidence interval is

(42.54,\ 46.46)

. 1A (ii) 42 is not in the confidence interval in (d)(i), 1M so there is evidence that the mean commuting time has increased. The claim is agreed. 1A f.t. (13)

第 11 題12 分

(a)

\displaystyle \displaystyle\int_{1}^{3}f(x)\,dx\approx\dfrac{0.5}{2}\big[\ln 2+2(\ln 3.25+\ln 5+\ln 7.25)+\ln 10\big]

1M+1M

\approx 3.1335

1A (b)

\displaystyle f'(x)=\dfrac{2x}{x^{2}+1}

\displaystyle f''(x)=\dfrac{2(x^{2}+1)-2x(2x)}{(x^{2}+1)^{2}}=\dfrac{2(1-x^{2})}{(x^{2}+1)^{2}}

1M For

1<x<3

f''(x)<0

, so the curve is concave downward and the estimate in (a) is an under-estimate. 1A f.t. (c) Area of

R>3.1335>3.1

, so the claim is agreed. 1M+1A f.t. (d) (i)

f(2)=\ln 5

and

\displaystyle f'(2)=\dfrac{4}{5}

, so the tangent is

\displaystyle y=\ln 5+\dfrac{4}{5}(x-2)

. 1A (ii) Since the curve is concave downward on

1\leq x\leq 3

, it lies below the tangent there. 1M Area of

\displaystyle R<\displaystyle\int_{1}^{3}\left[\ln 5+\dfrac{4}{5}(x-2)\right]dx=2\ln 5+\dfrac{4}{5}\left[\dfrac{(x-2)^{2}}{2}\right]_{1}^{3}=2\ln 5

1M+1A (12)

第 12 題13 分

(a)

\displaystyle N=\dfrac{50}{1+4}=10

1A (b)

\displaystyle \dfrac{50}{N}-1=4e^{-0.5t}

, so

\displaystyle \ln\left(\dfrac{50}{N}-1\right)=\ln 4-0.5t

1M The slope is

-0.5

and the intercept on the vertical axis is

\ln 4

. 1A (c)

\displaystyle \dfrac{dN}{dt}=\dfrac{-50(4)(-0.5)e^{-0.5t}}{(1+4e^{-0.5t})^{2}}=\dfrac{100e^{-0.5t}}{(1+4e^{-0.5t})^{2}}

1M+1A (d) Let

u=e^{-0.5t}

\displaystyle \dfrac{dN}{dt}=\dfrac{100u}{(1+4u)^{2}}

and

\displaystyle \dfrac{d}{du}\left[\dfrac{100u}{(1+4u)^{2}}\right]=\dfrac{100(1-4u)}{(1+4u)^{3}}

1M This is 0 when

\displaystyle u=\dfrac{1}{4}

; it is positive for

\displaystyle u<\dfrac{1}{4}

and negative for

\displaystyle u>\dfrac{1}{4}

. 1M

\displaystyle e^{-0.5t}=\dfrac{1}{4}

gives

t=2\ln 4\approx 2.7726

1A Since

u

decreases as

t

increases,

\displaystyle \dfrac{dN}{dt}

is greatest at

t=2\ln 4

. 1A (e) As

t\to\infty

e^{-0.5t}\to 0

, so

N\to 50

, i.e. about 50 000 fish. 1A (f) When

t=6

\displaystyle N=\dfrac{50}{1+4e^{-3}}\approx 41.6963

N

is increasing for all

t

, so

N<41.6963<45

for

0\leq t\leq 6

. 1M The claim is disagreed. 1A f.t. (13)