Download the non-calculator paper

CONTENTS
ABOUT THIS BOOK ................................................................................. 3
THE NON-CALCULATOR PAPER ........................................................... 4
ALGEBRA ................................................................................................. 5
Sequences and Series ...............................................................5
Sequences and Series – Applications ........................................7
Exponents and Logarithms ........................................................8
Permutations and Combinations .............................................. 12
Binomial Expansion ................................................................. 13
Proof by Induction .................................................................... 15
Complex Numbers ................................................................... 17
The Complex Plane ................................................................. 18
De Moivre's Theorem ............................................................... 20
Systems of Equations .............................................................. 22
FUNCTIONS AND EQUATIONS ............................................................ 24
Basics of Functions .................................................................. 24
Graphs of Functions................................................................. 28
Reciprocal Functions ............................................................... 32
Quadratic Functions ................................................................. 33
Solving Quadratic Equations .................................................... 34
Sum and Product of Roots ....................................................... 35
Inequalities .............................................................................. 37
Polynomial Functions ............................................................... 38
Exponential and Logarithmic Functions.................................... 40
CIRCULAR FUNCTIONS AND TRIGONOMETRY ................................ 42
Definitions and Formulae ......................................................... 42
Trigonometric Formulae ........................................................... 46
Solving Trigonometric Equations ............................................. 47
The Solution of Triangles ......................................................... 49
Graphing Periodic Functions .................................................... 50
VECTORS ............................................................................................... 52
Basics of Vectors ..................................................................... 52
Scalar (Dot) Product ................................................................ 54
Vector (Cross) Product ............................................................ 55
Equations of Lines ................................................................... 56
Equations of Planes ................................................................. 57
Equations of Lines and Planes – Summary.............................. 58
Intersections ............................................................................ 59
Angles in Three Dimensions .................................................... 61
Miscellaneous Vector Questions .............................................. 62
STATISTICS AND PROBABILITY ......................................................... 63
Statistics .................................................................................. 63
Probability Notation and Formulae ........................................... 68
Lists and Tables of Outcomes.................................................. 69
Venn Diagrams ........................................................................ 70
Tree Diagrams ......................................................................... 71
Bayes’ Theorem....................................................................... 72
Discrete Probability Distributions ............................................. 73
Binomial Distribution ................................................................ 74
Poisson Distribution ................................................................. 77
Continuous Distributions .......................................................... 78
The Normal Distribution ........................................................... 80
CALCULUS ............................................................................................. 84
Differentiation – The Basics ..................................................... 84
Differentiation from First Principles .......................................... 85
The Chain Rule ........................................................................ 86
Product and Quotient Rules ..................................................... 87
IBDP Mathematics HL
Page 1
Second Derivative ....................................................................88
Applications of Differentiation ...................................................89
Implicit Differentiation ...............................................................91
Graphical behaviour of functions ..............................................93
Sketching Graphs – Examples .................................................94
Indefinite Integration ................................................................95
Definite Integration ...................................................................97
Integration By Substitution .......................................................98
Integration by Parts ..................................................................99
General Methods for Integration.............................................100
Further Kinematics .................................................................102
Volumes of Revolution ...........................................................103
Calculus – Non-Calculator Techniques ..................................104
MAXIMISING YOUR MARKS ............................................................... 105
ASSESSMENT DETAILS ..................................................................... 107
PRACTICE QUESTIONS ...................................................................... 108
Answers to Practice Questions ..............................................114
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IBDP Mathematics HL
THE NON-CALCULATOR PAPER
The format of the two exam papers is the same – a section A
consisting of short answer questions, and a section B involving
extended response questions. However, calculators are only
allowed to be used in the Paper 2.
It is not intended that Paper 1 will test your ability to perform
complicated calculations with the potential for careless errors. It is
more to see if you can analyse problems and provide reasoned
solutions without using your calculator as a prop. However, this
doesn’t mean that there are no arithmetic calculations. You should,
for example, be able to:
Add and subtract using decimals and fractions:
Examples:
18.43 + 12.87, 2 21 + 3 52
Multiply using decimals and fractions (brush up your
multiplication tables):
Examples:
 0.5 0.1 2 0.5 
432 × 14, 12.6 × 5, 21 × 52 + 32 × 41 , (2 × 106 ) × (5.1× 10−4 ), 


 −0.1 0.2  1 −2 
Carry out simple divisions using decimals and fractions
Examples:
14 ÷ 0.02, 121 ÷ 53 , find x as a fraction is simplest form if 999x = 324
And don’t forget that divisions can be written as fractions, eg:
9 ÷ 15 = 159 = 53 = 0.6
Fraction simplification can help with more complex calculations:
Convert 81km/h to m/s
81× 1000 81× 10 9 × 10 9 × 5 45
=
=
=
=
= 22.5m/s
3600
36
4
2
2
Percentage calculations:
Examples:
15% of 600kg, Increase 2500 by 12%, what is 150 as a percentage
of 500.
Quadratic equations
You will be called on to solve quadratic equations many times
in the papers. Solving by factorisation is easier than using the
formula when you are not using a calculator.
Examples:
2
2
Solve x + 7x – 60 = 0; 3x – 19x + 20 = 0
NOTE: The Revision Guide contains many boxed questions which
are either worked examples or practice questions. Any which would
be hard to solve without a calculator will be shown with a double line
(as in this box). For the remaining questions, calculator use is either
irrelevant (for example, differentiating a function), or the question
could be answered both with and without a calculator. In the latter
case, it would be sensible for you to answer the question without a
calculator, and then check your answer with a calculator.
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IBDP Mathematics HL
ALGEBRA
Sequences and Series
There are many different types of number sequence. You only
need to know about two: the arithmetic sequence (AP) and the
geometric sequence (GP). In an AP each number is the previous
number plus a constant. In a GP each number is the previous
number multiplied by a constant.
A series is the same as a sequence except that the terms are
added together: thus a series has a sum, whereas a sequence
doesn’t.
To answer most sequences and series questions, make sure you
are familiar with the formulae below. First, the notation:
u1 = the first term of the sequence (many formulae use a instead)
n = the number of terms in the sequence
l = the last term of the sequence
d = the common difference (the number added on in an AP)
r = the common ratio (the multiplier in a GP)
un = the value of the nth term
Sn = the sum of the first n terms
S∞ = the sum to infinity
Examples:
Arithmetic sequences:
3, 5, 7, 9 …..
1.1, 1.3, 1.5, 1.7 …..
11, 7, 3, -1, -5 ……
Geometric Sequences:
1, 3, 9, 27 …..
4, 6, 9, 13.5 …..
12, 6, 3, 1.5, 0.75 …..
2, -6, 18, -54 …….
I
use
AP
(Arithmetic
Progression)
and
GP
(Geometric Progression), but
these are not terms used in
IB.
The formulae:
For an AP:
The value of the nth term:
The sum of n terms:
For a GP:
The value of the nth term:
un = u1 + (n – 1)d
d = un + 1 - un
n
n
Sn = (u1 + l ) = (2u1 + (n − 1)d )
2
2
un = u1rn –1
u
r = n +1
un
u1( r n − 1)
r −1
And for GPs only there is a formula for “the sum to infinity.” If the
common ratio is has a value between -1 and 1 (ie -1 < r < 1) then
the terms get ever smaller and approach zero. In this case, the
sum of the series will converge on a particular value. To calculate
this value:
u1
S∞ =
1− r
Series questions often involve algebra as well as numbers. Note
that to find d given two consecutive terms in an AP, subtract the
first from the second; and to find r in a GP, divide the second by
the first.
The sum of n terms:
Sn =
Sigma Notation: Sigma notation is just a shorthand for defining a
series. The ∑ symbol means “the sum of” and will include a
general formula for the terms of the series. For example,
4
∑ (n
2
The sum formulae always
start from the first term. If you
wanted to sum, say, the 10th
to the 20th terms, you would
calculate s20 – s9. Think
about it!
Example: A GP has first two
terms 2 and k. What range of
values of k will ensure the
series converges?
The common ratio must be
between –1 and 1. The
k
common ratio is , so:
2
k
-1 < < 1 so -2 < k < 2
2
− 2) = (12 − 2) + (2 2 − 2) + (3 2 − 2) + (4 2 − 2) = 22
1
IBDP Mathematics HL
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