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11 students

ST. STEPHEN'S GIRLS' COLLEGE

Final Examination 2018 - 19

Mathematics Extended Part Module 1 (Calculus and Statistics)

Time allowed: 2 hours

Total marks: 80

Question/Answer Paper

Please read the following instructions very carefully.

1. Write your class, class number and name in the

spaces provided on this cover.

2. Attempt ALL questions in this paper. Write your

in the spaces provided in this

Question/Answer Paper.

3. Unless otherwise specified, all working must be

clearly shown.

4. Unless otherwise specified, numerical answers should

be exact or given to 4 decimal places.

Marker's Use Only

F.5 Mathematics Extended Part Module 1 Final Examination 2018-19

7. In a department store, the number of purchases in a minute follows a Poisson distribution

with a mean of 3.5.

(a) Find the probability that there are at most 3 purchases at the department store in a

certain minute.

(b) The manager of the department store launches a promotion plan. A customer who

spends $20 or more in a purchase can get stamps. The details are given in the

following table:

Purchase amount ($x)

20≤x < 30

Number of stamps

It is known that 40%, 25% and 15% of the customers each gets 1 stamp, 2 stamps and

4 stamps respectively in a purchase.

(i) Find the probability that the 9th purchase is the 3rd purchase in which 2 stamps are

given away by the department store.

(ii) If there are exactly 3 purchases at the department store in a certain minute, find

the probability that 6 stamps are given away by the department store in that

(iii) Given that there are at most 3 purchases at the department store in a certain

minute, find the probability that a total of 6 stamps are given away by the

department store in that minute.

F.5 Mathematics Extended Part Module 1 Final Examination 2018-19

F.5 Mathematics Extended Part Module 1 Final Examination 2018-19

F.5 Mathematics Extended Part Module 1 Final Examination 2018-19

8. A scientist, Cindy, studies the growth rate of a bacteria. She found that the rate of change

of the number of bacteria (in thousand per hour) is given by

C(t) = 50-In(4t +1),

where t (0 < t < 2) is the number of hours elapsed since the start of the study. Let N (in

thousand) be the growth of the number of bacteria from t = 0 to t = 2.

Denote the estimate of N by using the trapezoidal rule with 5 sub-intevals by Nc.

(ii) Determine whether Nc is an over-estimate or an under-estimate. Explain your

(b) Another scientist, Sally, estimates the rate of change of the number of bacteria (in

thousand per hour) by

where t (0 < t < 2) is the number of hours elapsed since the start of the study. Let Ns

(in thousand) be the growth of the number of bacteria from t = 0 to t = 2 under Sally's

estimation.

(i) Find Ns.

Sally claims that in order to estimate N, Ns is more accurate than Nc.

agree? Explain your answer.

F.5 Mathematics Extended Part Module 1 Final Examination 2018-19

F.5 Mathematics Extended Part Module 1 Final Examination 2018-19

F.5 Mathematics Extended Part Module 1 Final Examination 2018-19

9. During the morning peak hours, route A or route B mini-buses which are 16-seaters leave the

terminus whenever there are exactly 13 passengers on the mini-bus. Every morning,

Benny waits at a mini-bus stop which is the first stop on both route A and route B. He will

go to work in the first mini-bus that he can get on. From the past experience, 40% of the

arriving mini-buses belong to route A and 60% of the arriving mini-buses belong to route B.

It is assumed that the numbers of people standing in the queues for route A and route B

before Benny follow Poisson distributions with means 2.5 and 3 respectively. Suppose that

no passengers get off at the first stop and no one joins both queues at the first stop after

(a) Assume that the numbers of people standing in the queues are independent.

(i) Find the probability that Benny can get on the first arriving mini-bus, which is

Given that Benny goes to work in the first arriving mini-bus, find the probability

that it belongs to route A.

(iii) Find the probability that Benny goes to work in the second arriving mini-bus.

(b) The distribution of the travelling time of a route B mini-bus from its first stop to

Benny's company is shown in the following table.

Travelling time (X minute)

30 < X < 40

It is given that Benny has gone to work by travelling on route B on 5 consecutive days.

Find the probabilities that

(i) he spends more than 30 minutes on travelling to the company for more than 3 days,

(ii) he spends less than 30 minutes on travelling to the company for 1 day, more than

40 minutes for 1 day and in the range 30 minutes to 40 minutes for the remaining

F.5 Mathematics Extended Part Module 1 Final Examination 2018-19

F.5 Mathematics Extended Part Module 1 Final Examination 2018-19

End of Paper

F.5 Mathematics Extended Part Module 1 Final Examination 2018-19

Let A and B be two events. Suppose that P(A) = 0.4, P(B) = 0.33 and P(B'| A')=0.7,

where A' and B' are the complementary events of A and B respectively.

(a) Find P(A'B').

(b) Find P(AUB).

(c) Are A and B independent ? Explain your answer.

F.5 Mathematics Extended Part Module 1 Final Examination 2018-19

2. The table below shows the probability distribution of a discrete random variable X, where a

and b are positive constants.

It is given that E(X) = 0.4.

(a) Find a and b.

Hence, find E(10X + 34) and V(10X + 34).

F.5 Mathematics Extended Part Module 1 Final Examination 2018-19

Let k be a constant.

(a) (i) Expand e²kx + e

in ascending powers of x as far as the term in x².

Hence, or otherwise, expand (e

in ascending powers of x as far as the

terms in x².

(b) If the sum of the coefficients of x and x² in the expansion of (1-kx)6(e²kx +e-²kx)² is

52, find the value(s) of k.

F.5 Mathematics Extended Part Module 1 Final Examination 2018-19

In a company, 45% of the staff wear glasses. There are 3 departments and each

department has 5 managers.

(a) Find the probability that among all the managers, at least 3 of them wear glasses.

Given that there are exactly 4 managers wearing glasses, find the probability that each

department has at least one manager wearing glasses.

F.5 Mathematics Extended Part Module 1 Final Examination 2018-19

Consider the curve C: y=(x-14)√√x-k, where k is a constant and x > k. It is given that

the tangent to C at the point (3, p) is 9x + 2y + m = 0.

in terms of k.

Find the values of k, p and m.

F.5 Mathematics Extended Part Module 1 Final Examination 2018-19

It is found that a balloon filled with air is getting smaller and smaller.

Let V m³ be the volume of air inside the balloon. It is given that

where t (20) is the number of hours elapsed since the leaking begins, h and k are constants.

(a) Express In 4 as a linear function of t.

(b) It is given that the graph of the linear function obtained in (a) passes through the origin

and the point (2, 1). Find

(iii) the value of V when

It is found that S=V³, where S m² is the surface area of the balloon.

Find the value of

attains its least value.

attains its least value when

Is it true? Explain your answer.

(ii) It is believed that

attains its least value.

attains its least value.

F.5 Mathematics Extended Part Module 1 Final Examination 2018-19

F.5 Mathematics Extended Part Module 1 Final Examination 2018-19