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La Salle College

Mid-Year Examination 2016-2017

Mathematics

Extended Part

(Calculus and Statistics)

Time allowed: 2 hours 30 minutes

Question - Answer Book

Instructions

Write your examination number in the spaces provided on

the top right corner of the cover page.

This paper consists of Section A and Section B. Each section

carries 50 marks.

Answer ALL questions in this paper.

All working must be clearly shown.

Unless otherwise specified, numerical answers should be

either exact or correct to 4 decimal places.

The diagrams in this paper are not necessarily drawn to scale.

Mid-Year Examination 2016-2017

F.5 Module 1 - Calculus and Statistics

The curve y = x³ - 3x² - 10x+24 intersects the x-axis at points A, B and C. Find the shaded area

enclosed by the curve and the x-axis as shown in the figure.

Mid-Year Examination 2016-2017

and Statistics

y=x² − 3x² −10x + 24

(a) Show that [(x+1)+¹(x−1)″-¹] = 2(nx − 1)(x+1)″ (x−1)″-², where n is a constant.

Given that the slope at any point (x, y) of a curve C is given by

2(2017x-1)(x+1) 2017(x-1) 20¹5. If the y-intercept of C is 2, find the equation of C.

Mid-Year Examination 2016-2017

and Statistics

Section B [50 marks]

Answer ALL questions in this section and write your answers in the spaces provided.

A researcher models the rate of change of antibodies in the bloodstream of a patient t hours after

injection of a vaccine by the following formula:

(a) Prove that A = 40ln

where A units is the amount of antibodies in the bloodstream. When t = 0, A = 300.

dt t² - 6t+10

Mid-Year Examination 2016-2017

and Statistics

t² - 6t+10

0≤t≤300,

Prove that t²-6t+10>0 for all real values of t.

When will the amount of antibodies in the bloodstream reach the minimum?

What is the minimum amount of antibodies in the bloodstream?

The patient will be protected when the amount of antibodies is not less than 500 units. The

researcher claims that the patient can be protected within 40 hours after the injection of the

vaccine. Is the claim correct? Explain your answer.

Mid-Year Examination 2016-2017

and Statistics

The market research department of a smartphone manufacturer estimates the total sales (in

after t months of its launch,

thousands) of a new phone model will increase at a rate S'(t)=

where 0

first 5 months after its launch.

(i) Find the estimate.

Is the estimate above an under-estimate? Justify your answer.

The researcher re-estimates the total sales by expanding

powers of t.

Mid-Year Examination 2016-2017

and Statistics

in ascending powers of t as far as the term in 1³.

(ii) Using the result of (b)(i), re-estimate the total sales for the first 5 months after its

as a series in ascending

Mid-Year Examination 2016-2017

and Statistics

Let f(x)= x² Inx.

Find the range of value(s) of x such that f(x)≥0.

Using the result of (b)(i), or otherwise, show that

x + C, where C is a constant.

(x³ In x).

√ x² In x dx

(iii) Hence, show that the area bounded by the x-axis, the lines x = 1 and x = 2 as well as the

curve y = x ln x is

Mid-Year Examination 2016-2017

and Statistics

Using the results of (b)(iii) and (c)(i), show that

1.014 In 1.01 +1.024 In 1.02+1.034 In 1.03+...+In 1.994 In 1.99 > 632 ln 2-124.

Mid-Year Examination 2016-2017

and Statistics

A researcher models the number of people infected by bird flu x days after the outbreak of the

disease in a region with the following formula:

1+ a[e¹(x-1)]

where a and b are constants.

Mid-Year Examination 2016-2017

and Statistics

as a linear function of x.

It is found that the slope and the x-intercept are −3 and

Find a and b.

Show that y = P(x) is an increasing function.

(iii) Will the number of infected people exceed 500? Explain your answer.

+1 respectively.

Let a be the root of P"(x)=0. Find a.

Briefly describe the behaviour of P'(x) before and after x = a.

Mid-Year Examination 2016-2017

and Statistics

Section A [50 marks]

Answer ALL questions in this section and write your answers in the spaces provided.

Solve the following equations.

(a) ex (e*+2)=8

(b) In(x+2)+In(x-1)=0

Mid-Year Examination 2016-2017

and Statistics

Mid-Year Examination 2016-2017

and Statistics

Supplementary Answer Sheet

- End of Paper

2. Evaluate lim

Mid-Year Examination 2016-2017

and Statistics

In the expansion of

(a) Find the value of n.

Mid-Year Examination 2016-2017

and Statistics

Hence, find the constant term.

in ascending powers of x, the coefficient of the 3rd term is 420.

4. Find the equation of the tangent to the curve y = (x

y=(x² − 3)e¹-x at x=2.

Mid-Year Examination 2016-2017

and Statistics

It is given that f(x)= x³ − 2x² − 4x+5. Find the range of values of x such that the curve

y = f(x) is

(a) decreasing;

(b) concave upwards.

Mid-Year Examination 2016-2017

and Statistics

Anson deposited a certain amount of money in a bank at the beginning of 2016. The manager

claimed that Anson's profit, P(t) (in thousand dollars) after t months, followed the equation

= 3e 100. Find the profit obtained from March 2016 to August 2016, correct to the nearest

thousand dollars.

Mid-Year Examination 2016-2017

and Statistics

Adrian goes from Town A to Town B. He first rides on boat with a speed of 10 km/h to cross the

river to reach C and then rides on bicycle with a speed of 20 km/h to Town B as shown in the

figure. It is given that AD L BD with AD = 5 km and BD = 30 km. Let x km be the length of CD

and t hours be the time taken for Adrian to travel from Town A to Town B.

Prove that t=

Mid-Year Examination 2016-2017

and Statistics

2√x² +25-x+30

Adrian claims that he can travel from Town A to Town B within 2 hours. Is the claim correct?

Explain your answer.

Evaluate the following definite integrals.

(a) √₁²x-³e-x¯²dx

•1 2 + √x

Mid-Year Examination 2016-2017

and Statistics