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8/17/2019 Rubric KPMT 2016 (Dalam Bahasa Inggeris)
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PERAK STATE EDUCATION DEPARTMENT
PERAK STATE
ADDITIONAL MATHEMATICS
PROJECT WORK
2016
8/17/2019 Rubric KPMT 2016 (Dalam Bahasa Inggeris)
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PERAK STATE Additional Mathematics Project Work 2016
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Assignment 1: Writing history drawing poster
Exponential numbers got powers
And so awesome are logarithms
Magnificient like the twin towersSinging 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.
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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 = x2
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 xa
log , a > 0.
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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.
21.51
21.51
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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: nn n
e )1
1(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.
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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 0.06 x
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.
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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 bothdrinks?
M S
I 10log
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 Richterscale.
(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.
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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.
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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 9log4
1k ,
(b) k k 75.012 loglog .
4. Given 2log x = p and 4log y = q, express 2q + 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.
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Assignment 9: SPM CHALLENGE – I can do it!
1. Solve the following equations :
(a) 16(4 3 x – 1 ) =2
1,
(b) 3( 2 x ) – 2 x + 1 = 8,
(c) 2 ( 4 x ) – 5 ( 2 x ) = 3.
2. Solve
(a) 3log1)12(log 44 x ,
(b)2
1log)32(log 24 x x .
3. Express x x 93 log4)52(log as a single logarithm to the base 3.
Hence, solve
(a) x x
93 log4)52(log
=)2(log2log
33 x
,
(b) x x 93 log4)52(log = 1.
4. Solve the simultaneous equations
y x xy 333 log2log43log ,
32
y x
xy.
5. Solve the simultaneous equations
1log2log 9327 mnnm .
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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.
LineProblem 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 = 3m5
n – 2
LineProblem 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
LineProblem 3 : Solve 2)1(3log 3 x .
Solution : 2)1(3log 3 x
1 233log 3 x 2 13log 3 x
3 3 x = 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 x x 2 2)126(log 2 x
3 6 x – 12 = 4
4 6 x = 16
53
8 x
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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.