Math_HL Workbook_Water Marked

INTERNATIONAL
BACCALAUREATE
DIPLOMA PROGRAMME
WORKBOOK
MATHEMATICS-HIGHER LEVEL
“If knowledge is light, then education is its intensity.”
(In conjunction with the syllabus of first
examinations, 2014)
VOLUME-I
GOODSHEPHERD INTERNATIONAL SCHOOL
TECHNICAL SUPPORT: ANGELIN MADHUSOODHANAN (M.Sc in Statistics)
AUTHOR: LENIN KUMAR GANDHI (M.Sc, M.Phil in Mathematics)
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KEY POINTS
1.) This book is meant for knowledge build-up and prepares the students for IBDP Math-HL
examinations.
2.) The questions in each content of the workbook are tested from the basic introductory level to
the uplifted level which requires highly critical thinking.
3.) Special care is taken to break the uniqueness method of testing the problems. The questions
are tested with various differing command terms and terminology which would surely make
the understanding process challenging and thus making thorough preparations for the
exams.
4.) Making one acquire holistic education is the main goal of this workbook and hence the
questions in each content of the syllabus are proportionately developed, meeting through all
the key requirements of the IBDP pedagogy.
5.) The questions, where so ever developed in whole or part-of are nowhere picked from the
external resources directly or indirectly, thus claiming for full copyright claims of the entire
workbook designed & created.
6.) With feedbacks and opinions taken into account from all across the globe, we hope for
improvisation in the quality of the book and the standard of the questions in forthcoming
editions, meeting through the requirements of the competitive global challenges.
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Math4life...
This book is dedicated to all the famous mathematicians
of history who have poured in their knowledge & life
itself, to bring in various fascinating inventions &
discoveries through their esteemed efforts…..
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1.) CIRCULAR FUNCTIONS & TRIGONOMETRY
2.) BINOMIAL THEOREM & EXPANSIONS
3.) SEQUENCES & SERIES
TOPICS COVERED IN VOLUME-I OF THE SYLLABUS:

ESSENTIAL KEY FACTORS OF THE PEDAGOGY:
This workbook is designed into preparing the students for future
university courses and is a pre-university knowledge built-up, keeping in
mind the changes as made in the Mathematics-Higher Level syllabus
starting from first examinations, May 2014. This workbook includes the
key essential factors as expected by IB to enhance the students’ skills
in acquiring holistic education and thus making one a life-long learner,
proudly meeting through the school’s vision. The following are the key
factors into which the whole syllabus is sub-divided into:
1.) Enhancing the problem-solving skills (tested on both the
GDC & non-GDC skills)
2.) Testing on the reverse process of understanding the content
approach.
3.) Consistent practice of inducing the critical thinking skills.
4.) Connecting the topics to real-life applications.
5.) Different ways of knowing the contents or the subject itself-
Connecting to Theory of Knowledge.
6.) Historical facts behind the inventions & discoveries in
Mathematics-
Connecting to related contents of the topic.
7.) Enhancing the investigative & research skills- Investigative
topics related to the contents of the topic are included as
‘Mini-Explorations’ at the end of each content.
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INTERNATIONAL BACCALAUREATE DIPLOMA PROGRAMME
Mathematics-higher level
ESSENTIAL CONTENTS OF THE WORKBOOK
I.) ENHANCING THE PROBLEM-SOLVING SKILLS:
Each part of the workbook exercise initially includes testing
the students’ level of solving the problems (tested both
through the GDC & non-GDC usage). The use of GDC is
included at the problems thereof (if required while solving).
Problems where critical thinking is required are also part of
the inclusions
PROCESS OF TESTING:
Reverse process of testing the students’ approach has been
identified to be the most effective practice in the learning
process and therefore every content of the workbook is
identified with the reverse process of testing the content.
For example, given the function, graphing its curve would be
a forward process, but given the graph of the function on an
XY Co-ordinate System, identifying its function is equally
an essential practice required in the form of reverse process
II.) REVERSE
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of learning. Further, it also makes the learning process more
deeper and stronger(a 2-way learning approach).
III.) REAL-LIFE APPLICATIONS:
Proper care is also taken to connect every content of the
syllabus with few situations demanding for real-life
applications of math. Through this, the students are mainly
intended to apply the learned contents in the classroom
sessions into the outer world of real-life situations and thus
learn the concepts more strongly and realistically. It also
gives an opportunity for the learners to appreciate the
applications of mathematics in real-life & thus making a
journey into the life-long learning. At some instances, it
even opens-up the opportunity for demonstrating one’s own
hands-on learning.
IV.) CONNECTING TO TOK:
TOK which stands for ‘Theory of Knowledge’ is defined as the
different ways of knowing the subject and is an essential key
factor in the learning process. In pursuit of holistic education,
it is essential that the subject is learnt not only in terms of
problem-solving skills but also through the different ways of
‘understanding’ & ‘justifying’ the means of evaluating the
learnt subject. For example, learning mathematics in the
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present class immediately after an English class would definitely
raise the doubt in the students’ mind with a fair TOK question-
“Is Mathematics a language?” & thus creates an inquiry
platform in researching the question raised in mind and hence is
very essential for ‘knowing’ the various ‘ways of knowing’ the
subject, all in pursuit of holistic education & life-long learning.
Therefore the workbook has also stressed in involving various
contents of the topic connected with different ways of knowing,
which in turn is expected to make the students in better
understanding of TOK connected with mathematics.
V.) HISTORICAL FACTS:
The history behind the inventions & discoveries in
Mathematics have always been hidden treasures since ages &
its time now to react to the opening-up of the actual facts
behind various inventions & discoveries which have impacted
many researches & downfalls and hence in international-
mindedness it is very essential that the students investigate
the facts while learning the contents. The impact of
historical facts has been so tremendous & violent that one
episode of history has taken away the life of a great
Physicist & a Mathematician due to his realistic inventions
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& discoveries. Copernicus who contradicted Pluto’s
assumption that the Earth is the center of the solar system
had made new discoveries and more importantly has
courageously stated that Earth is not the center of the solar
system and that the Sun was the center of the solar system
and the statement and the discovery has put him to death-
bed due to the false conflicts of prestige and selfishness of
the kings-times. Many contents in the workbook have opened
up information as investigative questions relating to various
historical facts in mathematics.
VI.) EXPLORATION TASKS:
The “Exploration” which is an Internal Assessment
component, contributes to 20% of the total weightage of
marks and hence this concept is included at the end of each
workbook content, to meet through the essential
requirements of the actual exploration task. It is expected
that the students investigate the Mini-Explorations, meeting
through all the five criteria of the latest Internal
Assessment guidelines. Through this consistent practice of
investigations & inquiries, the students will be able to
collaborate their topics to other subjects, thus expanding
their knowledge to the wider globe. It provides a learning
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platform for thoroughly preparing for the actual explorations
and is also a way of selecting the best topic suitable for
‘Exploration’ through choices of the Mini-Explorations as
already done. It also gives a wide opportunity for the
students to present their mathematical communication and
use of correct mathematical notations & terminology, the
decline of which is a major concern in the mathematical
society of the recent times, never to forget the importance
of mathematics in education system and its contribution to
real-life.
VII.) COMMAND TERMS:
The command terms which play a vital role in the questions
asked in final examinations need continuous practice of
understanding the different forms of command terms and
their individual rules of approach while solving the problems
and hence are included in the regular class-sessions. For
example, the key difference between ‘sketch’ & ‘draw’
applies itself with many rules for the former and the latter
accordingly, failing which the students would be penalized in
examinations, as per the strict marking scheme rules of IB.
Most of the students are bound to make mistakes in
understanding of the key difference between ‘Hence’ &
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‘Hence or otherwise’ and end up being penalized for not
applying the correct approach. Hence, each content is taken
care with differing command terms to make the practice
thorough through command term-based approach. It is also a
way of enhancing the students’ ability to improvise on their
mathematical communication skills and getting well prepared
for final examinat
Note: All the images & mathematical diagrams have been created using the
mathematical softwares ‘Geogebra’ & ‘Autograph’ and the images wheresoever
copied have been cited with the reference links thereof. The coloring,
designing, outlining and other special effects have been done using the
Microsoft Office-2013. Each content of the following pages have been created
using my qualifications, self-knowledge and the experiences of IB teaching in
conjunction with the IBDP Mathematics-Higher Level syllabus and no
information whatsoever has been picked from external resources.
ions.
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Imp: Until and unless otherwise stated, all numerical answers as obtained
from solving the problems must be given exactly or correct to 3 significant
figures.

CONSISTENT PRACTICE IS
THE KEY TO SUCCESS IN
MATHEMATICS….
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TOPIC:CIRCULAR FUNCTIONS AND
TRIGONOMETRY
CONTENT: PERIMETER AND AREA OF SECTOR
NON GDC QUESTIONS
1. Make the following conversions
(i) 500
= … … … . 𝑐
.
(ii) 12 𝑐
= … … … … 𝑜
(iii) 1 𝑐
= … … … … . . 𝑜
(iv) 1 𝑜
= … … … … . . 𝑐
(v)
11𝜋
6
= … … … … . 𝑜
2. Find the area and perimeter of the following shaded region
(i)
(ii)
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IBDP Mathematics-Higher LevelWorkbook .
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3. Find the area of the unshaded region and hence find its respective length of the
arc.
(i)
(ii)
With radius 3 cm
3c
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4. Find the radius of the circle whose minor sector area is
88
21
𝑐𝑚2
and with central
angle of 120o .Hence find the length of the major arc.
GDC BASED QUESTIONS
5. Find the area of the following shaded region whose length of the minor arc is 2.3
m .
Reverse
Process
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6. Find the radius of the following circles
(i)
4c
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(ii)
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Real life applications :-
The radius of the inner circle is 30 m and that of the outer circle is 50 m with the
common central angle to be 40 o.
Find the area of the estimated region and also the entire boundary estimated by
the engineer.
7. An engineer makes an estimate of a circular cricket ground for painting a 3D-
advertising logo picture on the pitch in the sectional area as represented by
the sector region COD as follows:
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Connecting to TOK
8. Connect the topic “SECTOR OF A CIRCLE “ to TOK .
Question: “Will the change of central ideas of a person at the core of his heart
affect his/her outer personality?”
Hint:-
Historical facts :-
9. Investigate on the real historical facts behind the central angle of a circle being
fixed at a value of 360 degrees.
Exploration :-
10.To what extent does the length of the Arc differ with its varied central angle?
Does the radius of the circle play a vital role in the process? Use your exploration
to explain the complete process of investigations.
Hint: A real-life activity & the use of effective Math Softwares are helpful for this
exploration.
Central angle of a sector
Length of the arc
l
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TOPIC : SEQUENCE AND
SERIES
ARITHMETIC SEQUENCE (or ARITHMETIC PROGRESSION)
 𝑎 − 𝑑, 𝑎, 𝑎 + 𝑑 , … … … … … ….
𝑎, 𝑎 + 𝑑 , 𝑎 + 2𝑑 , 𝑎 + 3𝑑 ,………………..
𝑈 𝑛 = 𝑢1 + (𝑛 − 1)𝑑
𝑆 𝑛 =
𝑛
2
(𝑢1 + 𝑢 𝑛) 𝑜𝑟
𝑆 𝑛 =
𝑛
2
[2𝑢1 + (𝑛 − 1)𝑑]
GEOMETRIC SEQUENCE ( GEOMETRIC PROGRESSION)
𝑎, 𝑎𝑟 , 𝑎𝑟2
, …………
𝑎
𝑟
, 𝑎 , 𝑎𝑟 , … … … … … … …
𝑈 𝑛 = 𝑢1 𝑟 𝑛−1
𝑆 𝑛 =
𝑢1(1 − 𝑟 𝑛)
1 − 𝑟
𝑜𝑟
𝑢1(𝑟 𝑛
− 1)
𝑟 − 1
, 𝑟 ≠ 1
𝑆∞ =
𝑢1
1−𝑟
if−1 < 𝑟 < 1.
𝐴 = 𝑃 (1 +
𝑅
100
)
𝑁
R – rate of interest
P- Initial amount
N- Time periods
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Problems:-
1) Write down the first 5 terms of the following sequences :
a) 5𝑛
b) 3𝑛 − 2
c) 2 −
1
𝑛
2) Find a formula for the nth term of the arithmetic sequence
7.5, 6.6, 5.7 , … … … … … .
Which term of the sequence will have the value −4.2 ?
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3) The 5th term of an arithmetic sequence is 15 and the 10th term is 45. Find the first
three terms of the sequence. Also find an expression for the nth term.
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4) Find the number of positive terms of the arithmetic sequence
59.2 , 58.4 , 57.6 , … … …
Hence or otherwise, find the value of the first negative term.
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5) Given that
24, 5𝑥 + 1, 𝑥2
− 1
are three consecutive terms of an arithmetic progression, find the possible values
of ‘𝑥’ and the numerical value of the fourth term for each value of ‘𝑥’ found.
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6) Three numbers are in arithmetic progression. Find the numbers if their sum is 30
and the sum of their squares is 332.
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7) Evaluate ∑ 3𝑟 + 25
𝑟=1
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8) Find the sum of first nine terms of the arithmetic series
−12 − 5 + 2 + ⋯ … … … … …
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9) Find the first term and the common difference of the arithmetic sequence in which
𝑢10 = −29 𝑎𝑛𝑑 𝑆10 = −110.
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10)The sum of first eight terms of an arithmetic series is 100, and the sum of first 15
terms is 555. Find the first term and the common difference.
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11)Consider the arithmetic series for which
𝑈 𝑛 = 72 − 6𝑛
If the sum of first n terms of the series is 378 then find ‘n’. Give reasons as to why
there are two possible values of ‘n’.
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12)Find the sum of all the multiples of 11 which are less than 1000.
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13)Consider the series
29.8 + 29.1 + 28.4 + ⋯ … … … … … …
Find the sum of all positive terms.
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14)Prove that the series for which 𝑆 𝑛 = 2𝑛2
+ 9𝑛 is arithmetic. Also find the first four
terms of the given series.
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15)Find the first 3 terms of the geometric sequence in which the common ratio is −
1
3
and the 7th term is −
2
81
.
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16)Prove that the sequence defined by 𝑈 𝑛 = 3 (−2) 𝑛
is geometric .
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17)a) If a, b, c are three consecutive terms of a geometric sequence, then show
that 𝑏2
= 𝑎𝑐.
b) If 𝑎 − 4, 𝑎 + 8, 54 are three consecutive terms of a geometric
sequence then find the possible values of ‘a’. Also find the numerical value of the
next term for each of the obtained values of ‘a’.
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18)The product of three consecutive numbers in a geometric progression is 27. The
sum of the first two numbers and nine times of the third number is −79. Find the
numbers.
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19)Find the sum of the first 8 terms of the geometric series
32 − 16 + 8 − ⋯ … … … … … ….
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20) Evaluate
∑ 0.99 𝑛
50
𝑛=1
.
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21)Find the first term and the common ratio of the geometric series for which
𝑆 𝑛 =
5 𝑛−4 𝑛
4 𝑛−1
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APPLICATIONS OF GEOMETRIC
SEQUENCE IN “BANKING”
22)A woman makes an annual deposit of $ 1000 into an account for which the bank
pays 5% interest, compounded annually. How much money should have been
accumulated into the account at the end of 10 years ?
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23)Calculate the amount in an account after 1 year, if $1000 is invested at 6% per
annum compound interest, and interest is paid,
a) Annually
b) Every 6 months
c) Quarterly
d) Monthly
e) Weekly
f) Daily
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24)Consider the infinite geometric series
∑ 10 ( 1 −
3𝑥
2
) 𝑛∞
𝑛=1
a) For what value of ‘x’ does the above series sum up to infinity?
b) Find the sum of the series for x= 1.3 .
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25) APPLICATIONS IN THE BOUNCING BALL:
A ball is dropped from a height of 10 meters and after each bounce from the
ground returns to a height which is 84% of the previous height. Calculate the total
distance travelled by the ball before coming to rest.
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26)a) Find the range of values of ‘x’ for which the geometric series
10 + 10 (2 𝑥) + 10 (2 𝑥)2
+ 10 (2 𝑥)3
+ ⋯ … … … … …
has a sum to infinity .
(b) Find the sum to infinity of the geometric series of part (a) if 𝑥 = −0.1 ,
and the smallest value of ‘n’ for which the sum of first n terms
exceeds 99% of the sum to infinity.
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Geometrical
patterns:
Find the number of dots in the nth stage of the following patterns. Also find
the number of dots in the 23rd stage of the diagram.
I.) Triangular Numbers:
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of 198.
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II.) Stellar Numbers:
Supportive Link:
https://www.google.co.in/search?q=Stellar+diagrams-
Clear+pictures&espv=2&biw=1152&bih=763&site=webhp&source=lnms&tbm=isch&sa=X&ei=PCor
VdqJIpPkasu-
gJAG&ved=0CAYQ_AUoAQ#tbm=isch&q=stellar+numbers+formula&imgrc=3BIfAmYDnX85bM%25
3A%3BBab12v9kK_Ul7M%3Bhttps%253A%252F%252F1millionmonkeystyping.files.wordpress.co
m%252F2014%252F12%252Fstellar1.jpg%3Bhttp%253A%252F%252Fibmathsresources.com%25
2F2015%252F01%252F20%252Fstellar-numbers-investigation%252F%3B819%3B419
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REVERSE PROCESS
Given the below graphs, find their corresponding sequences. Hence find an
expression for the nth dot of the graphs:
I.) Also identify the nth dot co-ordinates for the below graph:
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(II.) Also identify the nth dot co-ordinates for the below graph:
NOTE: The above graphs have been created using ‘Autograph’ software.
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REAL-LIFE APPLICATIONS
OF SEQUENCES
How to arrange things?
Situation:Imagine I had an oval shaped stadium where I need to arrange the chairs
for the seating, a day in advance for the cricket match, then I had a sense of estimating
the number of chairs required for the perfect arrangement through the patterns I
observed. In row1, I observed that I could perfectly fit 200 chairs (aligned in an oval
shape) of a suitable size to fit the maximum, followed by the row2 with 250 chairs &
continued with 300 chairs in the third row,
Thus observing a pattern with every elliptical outer row requiring additional 50 chairs to
that of the previous row. On the whole, if I had 74 rows covering the entire
stadium, then counting by the rows would make no sense to me as a Mathematical
student. I would seek for a tailored formula to estimate on the number of chairs I
need in order, for the cricket match to be a grand success for the day, & not to
forget the comforts needed for the spectators. At this stage, I think of sequences as
the sequential patterns for the real-life. Now the task is to
(i.) Estimate the number of chairs I need to fit in the last row.
(ii.) Estimate the possible number of chairs I need in the entire stadium
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ToK
Question: Can sequential life follow
the same pattern of successes or
failures throughout the life span?
i.) 0,0,0,0,0,………………
The above sequence is a continuous pattern with the element zero in each
successive term followed. Will this be an Arithmetic Sequence
as the common difference in each pair is un-un-1=0?
If ‘0’ is the total failure & ‘1’ is the complete success, then will there
be a total failure(since ‘0’ in each term of the above sequence) forever, even
after the nth stage, when the real-life is followed with the above sequence?
(or)
Will the complete success(i.e. ‘1’) as achieved in the initial stage(u1)
be followed forever even after the nth stage, when the real-life is
followed with the sequence:
1, 1, 1, 1, 1,…………..?
The common difference between each pair being ‘0’, will have no
differences at all(since ‘0’)?
ii.) Can the universal sequence,
1/0, 1/0, 1/0, 1/0,……………..
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be accepted as an arithmetic sequence with a well-defined common
difference?
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iii.) 1/0=?
0/0=?
iv.) X0, X1, X2, X3, X4,………….. can be observed for a pattern with the ‘X’
multiplied each time.
Initial term u1= X0
Hence, X=0 => The initial term of the above sequence reduces to
. u1=00=?
HINT:
In real life, it is very difficult to measure failure and success in terms
of a pattern. Failure and success are very dependent on
circumstances and are also relative i.e. for a person, a success may be
an initial failure (Steve Jobb) and a failure could be an overwhelming
success in the beginning.
Added, the critics always say that
“Failure is the stepping stone for
success”
So who knows,
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The general sequence 0,0,0,…….. may not be a pattern
continued with all zeroes(failures). It may end up with ‘1’(success) at
certain stage. Hence, maths in real-life may not always be a pattern.
NOTE: Explain your reasoning with other examples of life connecting the referred questions.
HISTORICAL FACTS
1.) Explore the historical facts of the ten avataras of the God (dashavataras) in one
kalpa yuga forming Arithmetic Sequence in each of the yuga followed with,
according to Hindu mythology.
2.) Explore the historical facts of the life span of human being depreciating ina multiple
of 10 (Geometric Sequence) in each of the Satya Yuga, Tretha Yuga, Dwapar
Yuga & Kali Yuga, according to Hindu mythology.
EXPLORATION
Investigate the “GOLDEN RATIO” associated with the Fibonacci Sequence.
OTHER INVESTIGATIONS:
1.) Find the number of possible handshakes between ‘n’
people in the conference room.
2.) Find the number of diagonals of a pentagon with ‘n’
sides.
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3.) Find the maximum number of pieces which can be
made by cutting a cube ‘n’ times.
4.) Find the maximum number of pieces which are
obtained by cutting a pizza for ‘n’ times.
INVESTIGATIVE
QUESTION:
Can sequences help us in winning certain
games for sure?
Hint: Connect with the match stick game- “A sure win
game”
Page 197 of 198.
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EXEMPLAR
INVESTIGATIONS:
I.)Investigate the areas & perimeters of
Vonkoch Snowflake patterns.
II.) Investigate the Recurrence relation
of Fibonacci Sequence.
III.) Explore the Lucas Numbers
IV.)Explore Serpeinski’s Triangle
V.) Explore on how fractals are
connected to the sequences topic.
TRY THIS PUZZLE: A fruit doubles in the
basket every one hour. It takes 10 hours for
the basket to fill completely. After how many
hours will the basket be filled exactly to half?Page 198 of 198.
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BIBLIOGRAPHY
https://www.google.co.in/search?q=Stellar+diagrams-
Clear+pictures&espv=2&biw=1152&bih=763&site=webhp&source=ln
ms&tbm=isch&sa=X&ei=PCorVdqJIpPkasu-
gJAG&ved=0CAYQ_AUoAQ#tbm=isch&q=stellar+numbers+formula&
imgrc=3BIfAmYDnX85bM%253A%3BBab12v9kK_Ul7M%3Bhttps%2
53A%252F%252F1millionmonkeystyping.files.wordpress.com%252F
2014%252F12%252Fstellar1.jpg%3Bhttp%253A%252F%252Fibmath
sresources.com%252F2015%252F01%252F20%252Fstellar-
numbers-investigation%252F%3B819%3B419
2.) Use of softwares:
Autograph 3.8
Geogebra
3.) Google site images
4.) Math equations & symbols- Microsoft Office-2013 (Math Equation
Editor)
5.) Math equations & symbol- ‘Math Type’ software
1.)Supportive Links:N
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NOTE: This copy is only the specimen copy of actual
workbook of International Baccalaureate-Mathematics Higher
Level syllabus.

Lenin K Gandhi, M.Sc., M.Phil., Mathematics Facilitator.
32
GSIS-IBDP PROGRAMME
To become active, compassionate and lifelong learners, let us abide by :-
“THE LEARNER PROFILES”
 Inquirers
 Knowledgeable
 Thinkers
 Communicators
 Principled
 Open-minded
 Caring
 Risk-takers
 Balanced
 Reflective
“It is not certain that everything is uncertain” – Blaise Pascal
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