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內容節錄：

INSTRUCTIONS

Attempt ALL questions in this section. Write your answers to this section in the spaces

provided in this Question-Answer Book. Do not write in the margins. Answers written in the

margins will not be marked.

St. Paul's College

F.6 Internal Examination 2021-2022

MATHEMATICS Extended Part

Module 1 (Calculus and Statistics)

Section A Question-Answer Book

Time Allowed: 2 hours 30 minutes

Unless otherwise specified, all working must be clearly shown.

Marks will be deducted for poor and untidy presentation.

Unless otherwise specified, numerical answers should be either exact or given to 4 decimal

F.6 Internal Exam 2021-2022 Math MIA

Section Total (50)

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Answers written in the margins will not be marked.

F.6 Internal Exam 2021-2022 Math MIA

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Answers written in the margins will not be marked.

Let f(x) be a function such that f'(x) =

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F.6 Internal Exam 2021-2022 Math M1A

The straight line 6x - 5y -4 = 0 touches the curve y = f(x) at the point A.

It is given that the x-coordinate of A is 2.

where k is a constant.

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Answers written in the margins will not be marked.

END OF SECTION A

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F.6 Internal Exam 2021-2022 Math M1A

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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 p

q are constants.

It is given that Var(5X + 1) = 21.

(a) Find p and 9.

(b) Find E(5X + 1).

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F.6 Internal Exam 2021-2022 Math M1A

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Answers written in the margins will not be marked.

On a particular day, 80% of students in a school take MTR to school and 5% of the students

taking MTR are late to school. It is given that 94% of the students in the school arrive at school

on time. A student from the school is chosen at random.

(a) Find the probability that the chosen student takes MTR to school and arrives on time.

(b) Find the probability that the chosen student does not take MTR and is late to school.

Find the probability that the student arrives school on time given that the chosen student

does not take MTR to school.

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F.6 Internal Exam 2021-2022 Math M1A

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Answers written in the margins will not be marked.

Let A and B be two events. Denote the complementary event of A by A'. It is given that

and P(B) = P(A' | B).

Are A and B mutually exclusive? Explain your answer.

Show that ² ≤P(A) ≤ ²7.

If A and B are independent, find P(A).

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F.6 Internal Exam 2021-2022 Math M1A

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Answers written in the margins will not be marked.

A factory manufactures a batch of crystal balls. The diameters of the crystal balls follow a

normal distribution with a mean of 10 cm and a standard deviation of 0.2 cm. A crystal ball is

classified as undersized if its diameter is less than 9.7cm.

(a) Find the probability that a randomly selected crystal ball from the batch is undersized.

(b) 12 crystal balls are randomly selected and packed into a carton. Let X be the random

variable representing the number of undersized crystal balls in the carton. Find

(ii) P(X=2|X<3)

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F.6 Internal Exam 2021-2022 Math M1A

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Answers written in the margins will not be marked.

Let f(x) = (In x)².

(a) Let g(u) = e" (u² − 2u + 2), where u = ln x. Show that

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F.6 Internal Exam 2021-2022 Math M1A

(b) Express, in terms of e, the area of the region bounded by the curve y = f(x), the x-axis and

the straight line x = e.

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Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

F.6 Internal Exam 2021-2022 Math MIA

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Answers written in the margins will not be marked.

(a) Expand (1-3x²)5 in ascending power of x.

(b) Find the constant k such that the coefficient of x4 in the expansion of ex(1 - 3x²)5

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F.6 Internal Exam 2021-2022 Math M1A

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Answers written in the margins will not be marked.

(a) Let f(x)=x²-3x² - 9x +32 for 0≤x≤10.

Show that f(x) > 0 for 0≤x≤10.

(b) Let g(x) = ln (x³ - 3x² - 9x +32) for 0≤x≤10.

Find the greatest value of g(x) for 0≤x≤10.

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F.6 Internal Exam 2021-2022 Math M1A

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