ADDITIONAL MATHEMATICS PROJECT WORK 2016jpnperak.moe.gov.my/ppdkinta/attachments/article/2901/f....

PERAK STATE EDUCATION DEPARTMENT

PERAK STATE

ADDITIONAL MATHEMATICS

PROJECT WORK

2016

PERAK STATE Additional Mathematics Project Work 2016

1

Assignment 1: Writing history drawing poster

Exponential numbers got powers

And so awesome are logarithms

Magnificient like the twin towers

Singing together in amazing rhythms

PART 1 : Writing a brief history

1. John Napier is often associated with logarithms.

Write a brief history on John Napier and his contributions in developing the concept

and applications of logarithms.

PART 2 : Drawing a poster

1. Define clearly indices and logarithms.

Illustrate your answers with examples.

2. Describe briefly a real life application involving

(a) an exponential function,

(b) a logarithmic function.

Hence, draw a poster that shows these two real life applications of exponential and

logarithmic functions.

Show clearly the above mentioned exponential function and logarithmic function in

your poster.


2

Assignment 2: The graphical perspective

Graphs brings out the beauty of functions. Properties of a function can be easily obtained

from a correctly-drawn graph.

PART 1 : Graphs of exponential functions

(a) On the same axes, draw the graphs of

(i) y = 2 x,

(ii) y = ( ½ ) x.

(b) Based on the graphs that you have drawn, state

(i) how the graphs can be related,

(ii) three properties of the exponential function a x, a > 0.

PART 2 : Graphs of logarithmic functions

(a) On the same axes, draw the graphs of

(i) y = x2log ,

(ii) y = x

2

1log .

(b) Based on the graphs that you have drawn, state

(i) how the graphs can be related,

(ii) three properties of the logarithmic function xalog , a > 0.


3

Assignment 3: Logarithm is fun!

PART 1 : Making wonder rulers based on logarithm

1. Complete Table 1 by writing the values of log10 N correct to three decimal places.

N Log10 N N Log10 N

1 6

1.5 6.5

2 7

2.5 7.5

3 8

3.5 8.5

4 9

4.5 9.5

5 10

5.5

Table 1

2. From a piece of 20 cm 24 cm graph paper, cut out 12 strips of graph papers, each

measuring 20 cm 2 cm.

By using a scale of 2 cm to 0.1 unit, mark the values of log10 N on the first strip by

writing down the corresponding value of N only. Label this strip as Ruler A. [Refer

diagram below.]

Ruler A

In the same manner, make ruler B. [Refer diagram below.]

Ruler B

Make 5 more sets of ruler A and ruler B.

PART 2 : Let’s play with logarithm

By pasting the wonder rulers that you have made, show and explain how you can find the following

values.

(a) 2 3 (b) 1.5 4 (c) 3.5 2

(d) 9 ÷ 2 (e) 10 ÷ 4 (f ) 7.5 ÷ 3

Relate your explanation to the relevant laws of logarithm.

2 1.5 1

2 1.5 1


4

Assignment 4: Let’s get acquainted with the number e

Many real life applications of exponential functions involve the number e as the base. The

number e is named in honour of the great Swiss mathematician Leonhard Euler. In this

assignment, you are to determine the value of this awesome number e using two methods.

METHOD 1 : Using definition

The number e is defined as follows: n

n ne )

11(lim

By choosing 10 suitable values of n from 1 to 1 000 000, determine the value of e correct to

3 decimal places. Present your answer clearly and neatly in the form of a table. You are

encouraged to use ICT.

METHOD 2 : Using a series

The number e can be written in the form of a series as follows:

e = 1 + ..............)4)(3)(2(1

1

)3)(2(1

1

)2(1

1

1

1

By adding enough terms, determine the value of e correct to 3 decimal places. Present your

answer systematically.


5

Assignment 5: Exponential challenges

Challenge 1 : The greenhouse effect

Emissions of gases such as carbon dioxides, methane and chlorofluorocarbons (CFCs) have

the potential to alter the earth’s climate as well as destroying the ozone layer. The

concentrations of CFCs, measured in parts per billion (ppb), can be modelled by the

exponential function to the base e

f(x) = 0.5e 0.06x

where x = 0 represents the year 2000, x = 1 represents the year 2001, x = 2 represents the year

2002 and so on.

(a) Use this exponential function to estimate the concentration of CFCs in the year 2015.

(b) Estimate the percentage increase in CFCs from the year 2015 to the year 2020.

Challenge 2 : Estimating world population

World population is usually modelled by an exponential function. Suppose the world

population, P billions of people, is given by

P = 6.8 ( 1.012) T

where T is the number of years after the year 2000.

Based on this formula, estimate

(a) the world population in the year 2000,

(b) the world population in the year 2020,

(c) by which year the world population will be doubled that in the year 2000.


6

Assignment 6: Logarithmic challenges

Challenge 1 : Natural disaster – the earthquake

Challenge 2 : pH of liquids

A logarithmic function that is used to measure the acidity of a liquid is the pH of the liquid.

The formula for pH is

(a) Table 2 gives the concentrations of hydrogen ions in four liquids.

Liquid Concentration of hydrogen ions ( mol cm –3 )

P 7.95 10 – 9

Q 3.16 10 – 6

R 5.01 10 – 8

S 3.98 10 – 3

Table 2

Classify the liquids as acidic or alkali. Justify.

(b) A glass of Coca Cola and a glass of orange juice have a pH of 2.5 and 3.5

respectively.

What is the difference, in mol cm – 3, in the concentration of hydrogen ions in both

drinks?

MS

I10log

M = magnitude of the earthquake,

I = intensity of the earthquake

S = intensity of a standard earthquake

pH = – 10log [ H + ]

[ H + ] = concentration of hydrogen ions in mol cm – 3

A logarithmic function that is used to measure the

magnitude of earthquakes is the Richter scale. It is

defined as follows:

Early in the year, an earthquake that occurred in Town X registered 7.5 on the Richter

scale.

(a) In the middle of the year, another earthquake 5 times stronger occurred.

Calculate the magnitude of this earthquake.

(b) At the end of the year, another earthquake registered 6.8 on the Richter scale.

Compare the intensity of the earthquake that occurred early in the year with the

intensity of this earthquake.


7

Assignment 7: Bacteria versus Antibiotic

Table 3

(a) Plot log10 N against t.

Hence,

(b) find

(i) the initial bacterial population,

(ii) the range of values of t when the bacterial population is less than 80,

(iii) the percentage decrease in the bacterial population from t = 4.5 to t = 16.5,

(c) express N in terms of t.

Time t ( h) Population N

3 1 372

6 941

9 646

12 443

15 304

18 208

21 143

24 98

27 67

30 46

Antibiotic is often taken to kill bacteria. A patient

takes a dose of a particular antibiotic every 3 hours.

Table 3 shows the population of bacteria, N, for

intervals of 3 hours after the patient is treated with the

antibiotic.


8

Assignment 8: SPM CHALLENGE – I’m ready!

1. It is given that 8 x = h and 2 y = k.

(a) If y = x – 1, determine the relationship between h and k.

(b) Express 2k

h8log 4 in terms of x and y.

2. (a) Without using calculator, find the value of

(i) 5.0log 4 , (ii) 27log 9 .

(b) Find the exact value of

(i) 7log 5 + 7log 49 – 7log 35,

(ii) 3log 5 9

1log 25 .

3. It is given that xk 3log and yk 4log .

Express in terms of x and y,

(a) 22 9log

4

1k ,

(b) kk 75.012 loglog .

4. Given 2log x = p and 4log y = q, express 2 q + 3 ( 8 p – 1 ) in terms of x and y.

5. Given the progression: 3log 2 , 9log 2 , 27log 2 , 81log 2 , …………….....................

find the sum of the first 10 terms.


9

Assignment 9: SPM CHALLENGE – I can do it!

1. Solve the following equations :

(a) 16(4 3x – 1 ) = 2

1,

(b) 3( 2x ) – 2 x + 1 = 8,

(c) 2 ( 4x ) – 5 ( 2x ) = 3.

2. Solve

(a) 3log1)12(log 44 x ,

(b) 2

1log)32(log 24 xx .

3. Express xx 93 log4)52(log as a single logarithm to the base 3.

Hence, solve

(a) xx 93 log4)52(log = )2(log2log 33 x ,

(b) xx 93 log4)52(log = 1.

4. Solve the simultaneous equations

yxxy 333 log2log43log ,

32

yx

xy.

5. Solve the simultaneous equations

1log2log 9327 mnnm .


10

Assignment 10: Mistake mistake everywhere

Four solutions done by a student in an Additional Mathematics exercise on the topic ‘Indices

and Logarithms’ are given below.

(a) Identify every mistake done by stating the corresponding line number and the

corresponding mistake.

(b) Hence, solve each problem correctly.

Line Problem 1 : Simplify ( 3m 3 n ) 2 ÷ mn – 4 .

Solution : ( 3m 3 n ) 2 ÷ mn – 4

1 = 3m 6 n 2 ÷ mn – 4

2 = 3m 6 – 1 n 2 – 4

3 = 3m 5 n – 2

Line Problem 2 : Solve the exponential equation 2 ( 3 x ) = 36.

Solution : 2 ( 3 x ) = 36

1 6 x = 36

2 6 x = 6 2

3 x = 2

Line Problem 3 : Solve 2)1(3log 3 x .

Solution : 2)1(3log 3 x

1 233log 3 x

2 13log 3 x

3 3x = 3

4 x = 1

Line

Problem 4 : Solve .2log

)127(log

2

2

x

x

Solution : .2log

)127(log

2

2

x

x

1 2)127(log 2 xx

2 2)126(log 2 x

3 6x – 12 = 4

4 6x = 16

5 3

8x


11

Assignment 11: Reflection

After carrying out this project work, you should have realised that exponential and

logarithmic functions do have a lot of real life applications that you might not have imagined

before.

Reflect on these applications and show creatively this awesome aspect of Additional

Mathematics.

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