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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 B 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 MIB

Section Total (50)

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END OF PAPER

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F.6 Internal Exam 2020-2021 Math M1B

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.1915 .1950

.2881 .2910 .2939

0.9 .3159 .3186 .3212

.0080 .0120

.0438 .0478 .0517 .0557

.0832 .0871 .0910 .0948

.1217 .1255 .1293 .1331

.1554 .1591 .1628 .1664 .1700

Standard Normal Distribution Table

.4332 .4345

1.6 .4452 .4463

.4554 .4564

.1985 .2019 .2054

.2967 .2995

.2291 .2324

.2611 .2642

.3438 .3461

.3665 .3686

.4032 .4049

.4207 .4222

.4990 .4991 .4991

2.0 .4772 .4778 .4783 .4788

2.1 .4821 .4826 .4830 .4834

2.2 .4861 .4864 .4868

2.3 .4893 .4896 .4898

2.4 .4918 .4920 .4922

2.5 .4938 .4940

2.6 .4953 .4955 .4956

.4974 .4975 .4976

.4982 .4982

.4649 .4656 .4664 .4671

.4719 .4726 .4732

F 6 Internal Exam 2021-2022 Math M1B

.3238 .3264 .3289

.2704 .2734

.4901 .4904

.3485 .3508

.3554 .3577 .3599 .3621

.3810 .3830

.4115 .4131

.4251 .4265 .4279

.0987 .1026

.1368 .1406

.1736 .1772 .1808

.2123 .2157 .2190

.2486 .2517

.3023 .3051

.4959 .4960

.4968 .4969 .4970

.0675 .0714

.1443 .1480

.4878 .4881

.4906 .4909

.4929 .4931

.4978 .4979

.4984 .4984 .4985

.4418 .4429

.4394 .4406

.4505 .4515 .4525

.4599 .4608

.4625 .4633

.4744 .4750

.4756 .4761

.4989 .4989

.4989 .4990 .4990

.4992 .4992

.4992 .4993

3.2 .4993 .4993

.4994 .4995

.4995 .4995

.4996 .4996 .4996

.4997 .4997

.4997 .4997

.4997 .4997

.4998 .4998 .4998 .4998 .4998 .4998 .4998 .4998 .4998

.4808 .4812

.4850 .4854

.4911 .4913

.4932 .4934

.4979 .4980

An entry in the table is the area under the standard normal curve between x = 0 and x = z (z ≥ 0).

Areas for negative values of z can be obtained by symmetry.

A(=)=√ √ 2 e ² dr

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SECTION B (50 marks)

A manufacturer launches an incentive scheme to help to boost the productivity of his factory.

A worker can get 2 points if he takes less than 1 hour to complete a task, 1 point if he takes

between 1 and 2.3 hours, and 0 point if he takes longer than 2.3 hours.

Assume the time for a worker to complete a task is normally distributed with a mean of

u hours and a standard deviation of o hour, and the number of tasks assigned to a worker

follows a Poisson distribution with a mean of 1.6 tasks per day.

(a) Let p, be the probability of a worker getting i point(s) upon completing a task, where

i = 0,1,2. If P₂ = 0.1056 and p₁ = 0.8716, find u and o.

(b) Find the probability that a worker is assigned not more than 4 tasks on a certain day.

(c) Find the probability that a worker gets exactly 3 points on a certain day under each of the

following conditions:

(i) 3 tasks are assigned,

(ii) 4 tasks are assigned.

(d) It is given that a worker is assigned not more than 4 tasks on a certain day. Find the

probability that the worker gets exactly 3 points.

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F.6 Internal Exam 2020-2021 Math M1B

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

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F.6 Internal Exam 2020-2021 Math M1B

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

10. The weights of adult males a community follow a normal distribution with a mean of u kg

and a standard deviation of 3.6 kg.

(a) A survey is conducted in the community to estimate µ.

A sample of 16 adult males in the community is randomly selected and their weights

(in kg) are recorded as follows:

62 65 68 69 69 70 70 71 71 71 72 73 73 74 76 78

Find a 90% confidence interval for μ.

(ii) Find the least sample size to be taken such that the width of a 99% confidence

interval for u is less than 2.

(b) Suppose that µ = 70.86. If the weight of an adult male in the community is over 75 kg, he

can join a free cholesterol test.

Find the percentage of adult males in the community that can join the free

cholesterol test.

(ii) There are only five quotas for the free cholesterol test left today. Ten adult males are

randomly selected from the community and their weights are measured one by one,

find the probability that

(1) The quotas are filled with measuring at most seven adult males,

(2) The quotas are just filled by measuring the tenth adult males.

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F.6 Internal Exam 2020-2021 Math M1B

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

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F.6 Internal Exam 2020-2021 Math M1B

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11. A researcher studied the population of fruit fly F under limited food supply. Let t be the

number of days since the beginning of the study and N(t) (in thousand) be the number of

fruit fly F at time t and modelled the number of fruit fly F by

-1=-kt + Ina

where a and k are positive constants.

(10 In 4.5,- In 4.5) and (5 ln 4.5, 0). Find the values of a and k.

(b) Using the values of a and k found in (a), express N(t) in terms of t.

(c) Take a = 4.5 and k = 0.2.

(i) Find N'(t) and N"(t).

(a) It is given that the graph of In

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F.6 Internal Exam 2020-2021 Math M1B

Estimate the number of fruit fly F after a very long time. Hence, or otherwise,

describe the number of fruit fly F since the start of the study.

(iii) Find the number of fruit fly F when the rate of change of the number of fruit fly F is

the greatest.

against t passes through the points

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F.6 Internal Exam 2020-2021 Math M1B

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12. A factory has two machines, F and G, for producing steel. The two machines start production

at the same time. The manager of the factory models the rates of change of the amount of steel

produced (in thousand tonnes per month) by F and G respectively by

√t² +100

where t is the number of months elapsed since the production begins. Denote the total mount

of steel produced by F in the first 5 months by a thousand tonnes. Let a, be the estimate of

a by using the trapezoidal rule with 5 sub-intervals.

(a) Find a₁.

(b) Let ß thousand tonnes be the total amount of steel produced by G in

the first 5 months.

(i) Prove that

5(20.061-2-1

(i) Using the substitution u=5-22-0.061, find ß.

(ii) Find f"(t).

(iii) The manager estimates a by using the trapezoidal rule with 5 sub-intervals. He

then claims that the total amount of steel produced by G in the first 5 months must

exceed the total amount of steel produced by F in the first 5 months. Do you agree?

Explain your answer.

v= Int+ √² +100

(ii) Using the substitution v=

(0 ≤t≤5),

v=In(t + √² +100), find a.

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F.6 Internal Exam 2020-2021 Math M1B

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

F.6 Internal Exam 2020-2021 Math M1B

Answers written in the margins will not be marked.