Topic outline
UNIT1:Trigonometric Formulae and Equations
Introductory activity
The height h of the cathedral is 485 m. The angle of elevation
of the top of the Cathedral from a point 280 m away from the
base of its steeple on level ground is θ . By using trigonometricconcepts, find the value of this angle θ in degree.
Trigonometry studies relationship involving lengths and angles
of a triangle. The techniques in trigonometry are used for
finding relevance in navigation particularly satellite systems
and astronomy, naval and aviation industries, land surveying, in
cartography and (creation of maps) and medicine. Even if those
are the scientific applications of the concepts in trigonometry,
most of the mathematics we study would seem to have little reallife application.
Trigonometry is really relevant in our day to dayactivities.
Objectives
By the end of this unit, a student will be able to:
ᇢ Solve trigonometric equations.
ᇢ Use trigonometric formulae and equations inreal life.
1.1. Trigonometric formulae
Activity 1.1
Deduce the formula of
tan(A B + − ),sin(A B),cos(A B − − ), tan(A B)
Deduce the formula oftan(A B + − ),sin(A B),cos(A B − − ), tan(A B)
1.1.1. Addition and subtraction formulae
From activity 1.1:
The addition and subtraction formulae are
Addition and subtraction formulae are useful for finding
trigonometric number of some angles.
1.1.2. Double angle formulae
Activity 1.2
1.1.3. Half angle formulae
Activity 1.3
Solution
Solution
Example 1.10
Transform in product the sum cos 2 cos 4 x x
Activity 1.6
Activity 1.9
Discuss how trigonometric theory is used in refraction of light.
In optics, light changes speed as it moves from one medium to
another (for example, from air into the glass of the prism). This
speed change causes the light to be refracted and to enter the new
medium at a different angle (Huygens principle).
A prism is a transparent optical element with flat, polished surfacesthat refract light.
1.3.3. 1.3.3 Application in medicine
Activity 1.10
Conduct research in different books of the library or on the
internet to discover the application of trigonometry in the field
of medicine such as heart’s electrical activities, trucking the
rhythms of lungs capacity, testing electrical activities of the
brain and abnormalities in the brain...
Trigonometric concepts such as sinusoidal waves contribute to
various medical testing and interpretation of those test results.
1. Electrocardiography: The measurement of electrical activities
in the heart. Through this process, it is possible to determine
how long the electrical wave takes to travel from one part of
the heart to the next by showing if the electrical activity is
normal or slow, fast or irregular.
2. Pulmonary function testing: a spirometer is used to measure
the volume of air inhaled and exhaled while breathing by
recording the changing volume over time. The output of a
spirogram can be quantified using trigonometric equations
and generally, it is possible to describe any repeating rhythmsby a sine wave:
In humans, when each breath is completed, the lung still
contains a volume of air, called the functional residual capacity
(approximately 2200 mL in humans). Each inhalation adds from
500 mL of additional air for normal (resting) breathing. Each
exhalation removes approximately the same volume as was
inhaled. The volume of air inhaled and exhaled normally is called
the tidal volume. It takes approximately 5 seconds to complete a
single cycle. With this information, we can develop an equation for
lung capacity as a function of time for normal (resting) breathing.
Since sy describes the midpoint of the sine wave, we take the
functional residual capacity plus 1/2 the amplitude to give us
sy=2200+250=2450 mL. Our period is the time for a single cycle,
or p=5 seconds. The volume of air inhaled gives the amplitude, or
A=1000 mL. We therefore have the following equation for normalbreathing:
Unit Summary
1. The addition and subtraction formulae:
UNIT 2:Sequences
Introductory activity
Suppose that an insect population is growing in such a way
that each new generation is 2 times as large as the previous
generation. If there are 126 insects in the first generation, on a
piece of paper, write down the number of insects that will bethere in second, third, fourth,…nth generation.
2.1. Arithmetic and harmonic sequences
2.1.1. Definition
Activity 2.1
Solution
2.1.4. Sum of arithmetic sequence
Activity 2.4
2.1.5. Harmonic sequences
Activity 2.5
Remark
To find the term of harmonic sequence, convert the sequence
into arithmetic sequence then do the calculations using the
arithmetic formulae. Then take the reciprocal of the answer
in arithmetic sequence to get the correct term in harmonicsequence.
2.2. Geometric sequences
2.2.1. Definition
Activity 2.6
Take a piece of paper which is in a square shape.
1. Cut it into two equal parts.
2. Write down a fraction corresponding to one part
according to the original piece of paper.
3. Take one part obtained in step (2) and cut, repeat step (1)
and then step 2).
4. Continue until you remain with a small piece of paper
that you are not able to cut it into two equal parts.
5. Observe the sequence of numbers you obtained and give
the relationship between any two consecutive numbers.
2.2.2. General term of a geometric sequence
Activity 2.7
Example 2.28
2.2.4. The sum of n terms of a geometric sequenceActivity 2.9
2.3. Convergent or divergent sequences
Activity 2.11
2.4. Applications of sequences in real life
Activity 2.12
20. The third, fifth and seventeenth terms of an arithmetic
progression are in geometric progression. Find the common
ratio of the geometric progression.
21. A mathematical child negotiates a new pocket money deal
with her unsuspecting father in which he/she receives
1 pound on the first day of the month, 2 pounds on the
second day, 4 pounds on the third day, 8 pounds on the
fourth day, 16 pounds on the fifth day, … until the end of
the month. How much would the child receive during the
course of a month of 30 days? (Give your answer to the
nearest million pounds.)
22. Find the common ratio of a geometric progression that has
a first term of 5 and sum to infinity of 15.
23. The sum of the first two terms of a geometric progression
is 9 and the sum to infinity is 25. If the common ratio is
positive, find the common ratio and the first term.
24. A culture of bacteria doubles every 2 hours. If there are
500 bacteria at the beginning, how many bacteria will
there be after 24 hours?
25. You complain that the hot tub in your hotel suite is not
hot enough. The hotel tells you that they will increase the
temperature by 10% each hour. If the current temperature
of the hot tub is 75º F, what will be the temperature of the
hot tub after 3 hours, to the nearest tenth of a degree?
26. The sum of the interior angles of a triangle is 180º, of a
quadrilateral is 360º and of a pentagon is 540º. Assuming
this pattern continues, find the sum of the interior angles
of a dodecagon (12 sides).UNIT 1:Logarithmic and Exponential Equations
Objectives
By the end of this unit, a student will be able to:
ᇢ solve exponential equations.
ᇢ solve logarithmic equations.
ᇢ apply exponential and logarithmic equations inreal life problems.
3.1. Introduction to Exponential and logarithmic functions
Activity 3.1
Activity 3.2
Solution
Activity 3.3
3.3. Applications
Exponential growth
Activity 3.4
Example 3.15
Jack operates an account with a certain bank which pays a
compound interest rate of 13.5% per annum. He opened the account
at the beginning of the year with 500,000 Frw and deposits the
same amount of money at the beginning of every year. Calculate
how much he will receive at the end of 9 years. After how longwill the money have accumulated to Frw 3.32 million?
Application activity 3.4
1. A certain cell culture grows at a rate proportional to the
number of cells present. If the culture contains 500 cells
initially 800 after 24h , how many cells will be there after
a further 12h ?
2. A radioactive substance decays at a rate proportional to
the amount present. If 30% of such a substance decays in
15 years, what is the half-life of the substance?
3. A colony of bacteria is grown under ideal conditions in
laboratory so that the population increases exponentially
with time. At the end of three hours there are 10,000
bacteria. At the end of 5 hours there are 40,000. How many
bacteria were present initially?
4. A radioactive material has a half-life of 1,200 years.
a) What percentage of the original radioactivity of a
sample is left 10 years?
b) How many years are required to reduce the
radioactivity by 10%?
5. Scientists who do carbon-14 dating use a figure of 5700
years for its half-life. Find the age of a sample in which
10% of the radioactive nuclei originally present have
decayed.
6. How much money needs to be invested today at a nominal
rate of 4% compounded continuously, in order that it
should grow to Frw 10,000 in 7 days?
7. The number of people cured is proportional to the number
y that is infected with the disease.
a) Suppose that in the course of any given year the
number of cases of disease is reduced by 20 %. If
there are 10,000 cases today, how many years will it
take to reduce the number to 1000?
b) Suppose that in any given year the number of cases
can be reduced by 25% instead of 20% .
(i) How long will it take to reduce the number of
cases to 1000?
(ii) How long will it take to eradicate the disease,
that is, to reduce the number of cases to lessthan 1?
Unit Summary
UNIT4:Trigonometric Functions and their Inverses
Objectives:
By the end of this unit, a student will be able to:
ᇢ find the domain and range of trigonometric
functions and their inverses.
ᇢ study the parity of trigonometric functions.
ᇢ study the periodicity of trigonometric functions.
ᇢ evaluate limits of trigonometric functions.
ᇢ differentiate trigonometric functions and theirinverses.
4.1. Generalities on trigonometric functions and
their inverses
4.1.1. Domain and range of six trigonometric
functions
Activity 4.1
4.1.2. Domain and range of inverses of
trigonometric functions
Activity 4.2
Remark
The inverses of the trigonometric functions are not functions,
they are relations. The reason they are not functions is that for a
given value of x , there is an infinite number of angles at which the
trigonometric functions take on the value of x . Thus, the range of
the inverses of the trigonometric functions must be restricted to
make them functions. Without these restricted ranges, they areknown as the inverse trigonometric relations.
Example 4.2
tan y x = are equivalent
Consider the following case:
If a function repeats every 2 units, then it will also repeat every
6 units. So if we have one function with fundamental period 2,
and another function with fundamental period 8, we have got no
problem because 8 is a multiple of 2, and both functions will cycle
every 8 units.
So, if we can patch up the periods to be the same,we know that
if we combine them, we will get a function with the patched up
period.
What we have to do is to find the Lowest Common Multiple (LCM)
of two periods.
What about if one function has period 4 and another has period 5?
We can see that in 20 units, both will cycle, so they are fine.
4.2. Limits of trigonometric functions and their
inverses
4.2.1. Limits of trigonometric functions
Activity 4.6
We do this until the indeterminate case is removed. Other methods
used to remove indeterminate cases are also applied.
4.3. Differentiation of trigonometric functions
and their inverses
4.3.1. Derivative of sine and cosine
Activity 4.8
4.4. Applications
Simple harmonic motion
Activity 4.15
Solution
UNIT5:Vector Space of Real Numbers
Introductory activity
A vector space (also called a linear space) is a collection of
objects called vectors, which may be added together and
multiplied by numbers, called scalars in this context.
To put it really simple, vectors are basically all about directions
and magnitudes. These are critical in basically all situations.
In physics, vectors are often used to describe forces, and forces
add as vectors do.
a) Discuss the properties of addition of vectors.
b) What happens when a vector is multiplied by a
scalar(real number)?c) Give at least 3 examples of vectors in real life.
Objectives
By the end of this unit, a student will be able to:
ᇢ define and apply different operations on vectors.
ᇢ Calculate the scalar and vector product of two
vectors.
ᇢ calculate the angle between two vectors.
ᇢ apply and transfer the skills of vectors to otherarea of knowledge
5.1. Vectors and operations in R3
Activity 5.1
5.3. Magnitude (or norm or length) of a vector
Activity 5.3
UNIT 6:Matrices and Determinant of Order 3
6.1. Square matrices of order three
6.1.1. Definitions
6.1.3. Operations on matrices
Activity 6.2
Multiplying matrices
Activity 6.4
Solution
Notice
Properties of a determinant
Activity 6.7
6.3. Application
6.3.1. System of 3 linear equations
Activity 6.9
Let use another useful method.
Solution:
Let C and M , B, represent the number of serving of chicken, baked
potatoes, and milk respectively.
UNIT 7:Bivariate Statistics
Introductory activity
In Kabeza village, after her 9 observations about farming,
UMULISA saw that in every house observed, where there are
a number x of cows there are also y domestic ducks, and then
she got the following results of (x,y) pairs: (1,4), (2,8), (3,2),
(4,12), (5,10), (6,14), (7,16), (8,6), (9,18)
a) Represent this information graphically in a
( x y coordinates , ) − .
b) Chose two points, find the equation of a line joining
them and draw it in the same graph. How are the positions
of remaining points vis-a -vis this line?
c) According to your observation from (a), explain in
your own words if there is any relationship between the
variation of the number x of cows and the number y ofdomestic ducks.
Until now, we know how to determine the measures of central
tendency in one variable. In this unit, we will use those measures
in two quantitative variables known as double series. In statistics,
double series includes technique of analyzing data in two variables,
when we focus on the relationship between a dependent variable-y
and an independent variable-x. The linear regression method will
be used in this unit. The estimation target is a function of the
independent variable called the regression function which will bea function of a straight line.
Objectives
By the end of this unit, a student will be able to:
ᇢ find measures of variability in two quantitative
variables.
ᇢ draw the scatter diagram of given statistical
series in two quantitative variables.
ᇢ determine the linear regression line of a given
series,
ᇢ calculate a linear coefficient of correlation of agiven double series and interpret it
In case of two variables, say x and y, there is another important
result called covariance of x and y, denoted cov , ( x y), which is a
measure of how these two variables change together.
The covariance of variables x and y is a measure of how these
two variables change together. If the greater values of one variable
mainly correspond with the greater values of the other variable,
and the same holds for the smaller values, i.e. the variables tend to
show similar behavior, the covariance is positive. In the opposite
case, when the greater values of one variable mainly correspond
to the smaller values of the other, i.e. the variables tend to show
opposite behavior, the covariance is negative. If covariance is zero
the variables are said to be uncorrelated, meaning that there is no
linear relationship between them.
Therefore, the sign of covariance shows the tendency in the linear
relationship between the variables. The magnitude of covariance
is not easy to interpret.Covariance of variables x and y, where the summation of frequencies
Method of ranking
Ranking can be done in ascending or descending order.
Example 7.6
Suppose that we have the marks, x, of seven students in this order:
12, 18, 10, 13, 15, 16, 9
We assign the rank 1, 2, 3, 4, 5, 7 such that the smallest value of xwill be ranked 1.
a) It is required to estimate the value of X for Nepal, where the
value of Y is 450.
i) Find the equation for a suitable line of regression.
Simplify your answer as far as possible, giving the
constants correct to three significant figures
ii) Use your equation to obtain the required estimate
b) Use your equation to estimate the value of x for North Korea,
where the value of Y was 858. Comment on your answer.
Solution
a) i) Neither variable has been controlled in the given data and
since you are required to estimate the life expectancy, X
years, when the GDP per head, Y dollars is 160 dollars, it is
sensible to use the regression line of X on YThe regression line of X on Y has equation
Unit Summary
r = 0.95End of Unit Assessment
1. For each set of data, find:
a) equation of the regression line of y on x.
b) equation of the regression line of x on y
Find both equations of the regression lines. Also estimate the value ofy for x = 30 .
5. The following results were obtained from records of age (x) and
systolic blood pressure
of a group of 10 men:
a) Calculate the Spearman’s coefficient of rank correlation
between position in the league and average attendance.
b) Comment on your results
21.A company is to replace its fleet of cars. Eight possible models
are considered and the transport manager is asked to rank them,
from 1 to 8, in order of preference. A saleswoman is asked to use
each type of car for a week and grade them according to their
suitability for the job (A-very suitable to E-unsuitable).The price is also recorded:
23.To test the effect of a new drug twelve patients were examined
before the drug was administered and given an initial score (I)
depending on the severity of various symptoms. After taking the
drug they were examined again and given a final score (F). A
decrease in score represented an improvement. The scores for
the twelve patients are given in the table below:
UNIT8:Conditional Probability and Bayes Theorem
Introductory activity
A box contains 3 red pens and 4 blue pens. One pen is taken
from the box and is not replaced. Another pen is taken from the
box. Let A be the event “the first pen is red” and B be the event
“the second pen is blue.”
Is the occurrence of event B affected by the occurrence of event
A? Explain.
Give more other examples of real life problems involving
probability.
Some academic fields based on the probability theory are statistics,
communication theory, computer performance evaluation, signal
and image processing, game theory...
In medical decision-making, clinical estimate of probability
strongly affects the physician’s belief as to whether or not a patient
has a disease, and this belief, in turn, determines actions: to rule
out, to treat, or to do more tests, doctors may use conditional
probability to calculate the probability that a particular patient
has a disease, given the presence of a particular set of symptoms....
Some applications of the probability theory are character
recognition, speech recognition, opinion survey, missile control
and seismic analysis, etc.
In addition, the game of chance formed the foundations ofprobability theory
Objectives
By the end of this unit, a student will be able to:
ᇢ use tree diagram to find probability of events.
ᇢ find probability of independent events.
ᇢ find probability of one event given that the other
event has occurred.ᇢ use and apply Bayes theorem.
8.1. Independent events
Activity 8.1
A bag contains books of two different subjects. One book is
selected from the bag and is replaced. A book is also selected
in the bag. Is the occurrence of the event for the secondselection affected by the event for the first selection? Explain
Example 8.2
A factory runs two machines. The first machine operates for 80%
of the time while the second machine operates for 60% of the time
and at least one machine operates for 92% of the time. Do thesetwo machines operate independently?
Application activity 8.1
1. A dresser drawer contains one pair of socks with each of
the following colors: blue, brown, red, white and black.
Each pair is folded together in a matching set.
You reach into the sock drawer and choose a pair of socks
without looking. You replace this pair and then choose
another pair of socks. What is the probability that you
will choose the red pair of socks both times?
2. A coin is tossed and a single 6-sided die is rolled. Find
the probability of landing on the head side of the coin androlling a 3 on the die.
8.2. Conditional probability
Activity 8.2
A bag contains books of two different subjects. One book is
selected from the bag and is not replaced. Another book is
selected in the bag. Is the occurrence of the event for the second
selection affected by the event for the first selection? Explain.
When the outcome or occurrence of the first event affects the
outcome or occurrence of the second event in such a way that
the probability sample space is changed, the events are said to be
dependent.
The probability of an event B given that event A has occurredis called the conditional probability of B given A and is written
Notice: contingency table
Contingency table (or Two-Way table) provides a different way
of calculating probabilities. The table helps in determining
conditional probabilities quite easily. The table displays
sample values in relation to two different variables that may be
dependent or contingent on one another.
Below, the contingency table shows the favorite leisure activities
for 50 adults, 20 men and 30 women. Because entries in thetable are frequency counts, the table is a frequency table.
Entries in the total row and total column are called marginal
frequencies or the marginal distribution. Entries in the body of
the table are called joint frequencies.
Example 8.6
Suppose a study of speeding violations and drivers who use carphones produced the following fictional data:
d) P(person is a car phone user OR person had no violation
in the last year).
e) P(person is a car phone user GIVEN person had a violation
in the last year).
f) P(person had no violation last year GIVEN person was nota car phone user).
Application activity 8.2
1. A coin is tossed twice in succession. Let A be the event
that the first toss is heads and let B be the event that the
second toss is heads. Find P(B\A)
2. Calculate the probability of a 6 being rolled by a die if it is
already known that the result is even.
3. A jar contains black and white marbles. Two marbles are
chosen without replacement. The probability of selecting
a black marble and then a white marble is 0.34, and the
probability of selecting a black marble on the first draw is
0.47. What is the probability of selecting a white marble
on the second draw, given that the first marble drawn wasblack?
8.3. Bayes theorem and its applications
Thus, the most likely diagnosis in the presence of rebound
tenderness/ painfulness is acute appendicitis. Since surgery
is generally not required for either pancreatitis or NSAP, but is
indicated for appendicitis, it is very useful to know the probability
that the patient showing rebound tenderness has appendicitis as
compared to the probability that s/he has one of the other twoconditions.
3. The probability of having an accident in a factory that
triggers an alarm is 0.1. The probability of it sounding
after the event of an incident is 0.97 and the probability
of it sounding after no incident has occurred is 0.02. In
an event where the alarm has been triggered, what is the
probability that there has been no accident?
End of Unit Assessment
1. The probability that it is Friday and that a student is
absent is 0.03. Since there are 5 school days in a week, the
probability that it is Friday is 0.2. What is the probability
that a student is absent given that today is Friday?
2. At Kennedy Middle School, the probability that a student
takes Technology and Spanish is 0.087. The probability that
a student takes Technology is 0.68. What is the probability
that a student takes Spanish given that the student is taking
Technology?
3. A car dealership is giving away a trip to Rome to one of
their 120 best customers. In this group, 65 are women,
80 are married and 45 married women. If the winner is
married, what is the probability that it is a woman?
4. A card is chosen at random from a deck of 52 cards. It is
then replaced and a second card is chosen. What is the
probability of choosing a jack and then an eight?
5. A jar contains 3 red, 5 green, 2 blue and 6 yellow marbles.
A marble is chosen at random from the jar. After replacing
it, a second marble is chosen. What is the probability of
choosing a green and then a yellow marble?
6. A school survey found that 9 out of 10 students like pizza.
If three students are chosen at random with replacement,
what is the probability that all three students like pizza?
7. A nationwide survey found that 72% of people in the
United States like pizza. If 3 people are selected at random,
what is the probability that all three like pizza?
8. A bag contains 2 red, 3 green and 2 blue balls. Two balls
are drawn at random. What is the probability that none of
the balls drawn is blue?
9. For married couples living in a certain suburb, the probability
that the husband will vote on a bond referendum is 0.21,
the probability that his wife will vote in the referendum is
0.28, and the probability that both the husband and wife
will vote is 0.15. What is the probability that:
a) at least one member of a married couple will vote?
b) a wife will vote, given that her husband will vote?
c) a husband will vote, given that his wife does not
vote?
10.In 1970, 11% of Americans completed four years of college,
43% of them were women. In 1990, 22% of Americans
completed four years of college; 53% of them were women.
(Time, Jan.19, 1996).
a) Given that a person completed four years of college
in 1970, what is the probability that the person was
a woman?
b) What is the probability that a woman would finish
four years of college in 1990?
c) What is the probability that in 1990 a man would
not finish college?
11.If the probability is 0.1 that a person will make a mistake
on his or her state income tax return, find the probability
that:
a) four totally unrelated persons each make a mistake
b) Mr. Jones and Ms. Clark both make a mistake andMr. Roberts and Ms. Williams do not make a mistake.
12.The probability that a patient recovers from a delicate
heart operation is 0.8. What is the probability that:
a) exactly 2 of the next 3 patients who have this
operation will survive?
b) all of the next 3 patients who have this operation
survive?
13.In a certain federal prison, it is known that 2/3 of the
inmates are under 25 years of age. It is also known that
3/5 of the inmates are male and that 5/8 of the inmates are
female or 25 years of age or older. What is the probability
that a prisoner selected at random from this prison is
female and at least 25 years old?
14.A certain federal agency employs three consulting firms
(A, B and C) with probabilities 0.4, 0.35, 0.25, respectively.
From past experience it is known that the probabilities
of cost overrun for the firms are 0.05, 0.03, and 0.15
respectively. Suppose a cost overrun is experienced by the
agency.
a) What is the probability that the consulting firm
involved is company C?
b) What is the probability that it is company A?
15.In a certain college, 5% of the men and 1% of the women
are taller than 180 cm. Also, 60% of the students are
women. If a student is selected at random and found to be
taller than 180 cm, what is the probability that this student
is a woman?
References
[1] A. J. Sadler, D. W. S. Thorning: Understanding Pure
Mathematics, Oxford University Press 1987.
[2] Arthur Adam Freddy Goossens: Francis Lousberg.
Mathématisons 65. DeBoeck,3e edition 1991.
[3] David Rayner, Higher GCSE Mathematics, Oxford
University Press 2000
[4] Direction des Progammes de l’Enseignement Secondaire.
Géometrie de l’Espace 1er Fascule. Kigali, October 1988
[5] Direction des Progammes de l’Enseignement Secondaire.
Géometrie de l’Espace 2ème Fascule. Kigali, October 1988
[6] Frank Ebos, Dennis Hamaguchi, Barbana Morrison & John
Klassen, Mathematics Principles & Process, Nelson Canada
A Division of International Thomson Limited 1990
[7] George B. Thomas, Maurice D. Weir & Joel R. Hass,
Thomas’ Calculus Twelfth Edition, Pearson Education, Inc.
2010
[8] Geoff Mannall & Michael Kenwood, Pure Mathematics 2,
Heinemann Educational Publishers 1995
[9] H.K. DASS... Engineering Mathematics. New Delhi, S.
CHAND&COMPANY LTD, thirteenth revised edition 2007.
[10] Hubert Carnec, Genevieve Haye, Monique Nouet,
ReneSeroux, Jacqueline Venard. Mathématiques TS
Enseignement obligatoire. Bordas Paris 1994.
[11] James T. McClave, P.George Benson. Statistics for Business
and Economics. USA, Dellen Publishing Company, a
division of Macmillan, Inc 1988.
[12] J CRAWSHAW, J CHAMBERS: A concise course in
A-Level statistics with worked examples, Stanley Thornes
(Publishers) LTD, 1984.
[13] Jean Paul Beltramonde, VincentBrun, ClaudeFelloneau,
LydiaMisset, Claude Talamoni. Declic 1re S Mathématiques.
Hachette-education, Paris 2005.
[14] JF Talber & HH Heing, Additional Mathematics 6th Edition
Pure & Applied, Pearson Education South Asia Pte Ltd 1995
[15] J.K. Backhouse, SPTHouldsworth B.E.D. Copper and P.J.F.
Horril. Pure Mathematics 2. Longman, third edition 1985,
fifteenth impression 1998.
[16] M.Nelkon, P. Parker. Advanced Level Pysics, Seventh
Edition. Heinemann 1995
[17] Mukasonga Solange. Mathématiques 12, Analyse
Numérique. KIE, Kigali2006.
[18] N. PISKOUNOV, Calcul Différential et Integral tom II 9ème
édition. Editions MIR. Moscou, 1980.
[19] Paule Faure- Benjamin Bouchon, Mathématiques
Terminales F. Editions Nathan, Paris 1992.
[20] Peter Smythe: Mathematics HL & SL with HL options,
Revised Edition, Mathematics Publishing Pty. Limited,
2005.
[21] Rwanda Education Board (2015), Subsidiary Mathematics
Syllabus S4-S6, Ministry of Education: Rwanda.
[22] Robert A. Adms & Christopher Essex, Calculus A complete
course Seventh Edition, Pearson Canada Inc., Toronto,
Ontario 2010
[23] Seymour Lipschutz. Schaum’s outline of Theory and
Problems of linear algebra. McGraw-Hill 1968.
[24] Shampiyona Aimable : Mathématiques 6. Kigali, Juin 2005.
[25] Tom Duncan, Advanced Physics, Fourth edition. John
Murray (Publishers) Ltd, London 1998
[26] Yves Noirot, Jean–Paul Parisot, Nathalie Brouillet. Cours de
Physique Mathématiques pour la Physique. Paris, DUNOD,
1997.
[27] http://www.tiem.utk.edu/~gross/bioed/webmodules/
lungcapacity.html
[28] https://prezi.com/kdrzbkm8yuye/trigonometry-and-medicin
e/?frame=b5001c61a02311bb8f2927d84e8643b207d0c72c
[29] file:///C:/Users/IT%20ZONE/Desktop/Reference%20books/Good%20Applied%20calculus.pd