中六 數學試卷 (F6 Maths Past Paper)
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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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F.6 Internal Exam 2021-2022 Math MIA
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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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END OF SECTION A
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F.6 Internal Exam 2021-2022 Math M1A
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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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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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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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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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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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F.6 Internal Exam 2021-2022 Math MIA
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(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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(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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