How to Pass SQA Advanced Higher Maths

146 Pages·2021·8.344 MB·English

by  Robert Barclay

How to Pass
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© Robert Barclay 2021
First published in 2021 by
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Contents
Introduction iv
The list of formulae v
Part 1 Calculus
Chapter 1 Differentiation 1: Rules and standard derivatives 1
Chapter 2 Differentiation 2: Implicit functions and parametric equations 8
Chapter 3 Integration 15
Chapter 4 Differential equations 26
Part 2 Algebra, proof and number theory
Chapter 5 Partial fractions 35
Chapter 6 Functions and graphs 39
Chapter 7 Binomial theorem 52
Chapter 8 Sequences and series 57
Chapter 9 Proof 64
Chapter 10 Euclid’s algorithm 70
Part 3 Matrices, vectors and complex numbers
Chapter 11 Systems of equations 74
Chapter 12 Matrices 79
Chapter 13 Vectors 89
Chapter 14 Complex numbers 101
Appendix 1 Answers 112
Appendix 2 Some tips for improving your grade 137
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Introduction
Welcome to Advanced Higher Maths
This book is a revision guide aimed at candidates preparing for the SQA Advanced Higher
Mathematics examination. It should be used to supplement your course notes, core texts
and class teaching. It can be used throughout the year, but especially in key revision
periods and in the lead up to the examination to enable you to achieve the best grade you
possibly can.
The book is fully up to date with the 2020 course assessment and contains:
●● a summary of the course content and the structure of the examination
●● a list of formulae provided in the examination; you must remember all other formulae not
included in this list
●● helpful hints and tips for tackling questions
●● detailed coverage of the course content arranged into chapters.
Each chapter contains the features outlined below.
Features of each chapter
●● Key points – a list of skills (with checkboxes) which will be assessed in the exam
●● Key links – identifies skills from across the course which are often linked in exam questions
●● Summary notes – concise explanatory notes for each skill
●● Examples – worked examples to show how to answer exam-style questions
●● Hints and tips – hints and tips on how to achieve top marks and avoid common mistakes
●● Check-up questions – mainly short questions covering the core skills
●● Practice questions – exam-style questions
●● Answers to check-up and practice questions are provided at the end of the book – with
intermediate steps indicating where marks are awarded in the exam.
Summary of Advanced Higher Mathematics
course content
Calculus Algebra, proof and number theory Matrices, vectors and complex numbers
Differentiation Partial fractions Systems of equations
Integration Functions and graphs Matrices
Differential equations Binomial theorem Vectors
Sequences and series Complex numbers
Proof
Euclid’s algorithm
iv
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Introduction
Advanced Higher Mathematics examination
Paper 1 (non – calculator) Paper 2
Time 1 hour 2 hours 30 minutes
Marks 35 80
Proportion of Approximately 65% of the marks are available for level C responses.
level C questions
Approximately 35% of the marks are available for responses beyond level C.
Balance of skills Approximately 65% of the marks are available for questions assessing only operational skills.
Approximately 35% of the marks are available for questions assessing both operational and
reasoning skills.
More information about the course and the exam is available on the SQA website –
www.sqa.org.uk
For tips on improving your grade, see Appendix 2 on page 137.
Formulae list
Standard derivatives Standard integrals
f (x) f '(x) f (x) ∫f(x)dx
1
1
sin–1x sec2 (ax) tan(ax)+c
1− x2 a
1 1  x
cos–1x − sin−1 + c
1− x2 a2 − x2  a
1 1 1  x
tan–1x tan−1 + c
1+ x2 a2 + x2 a a
1
tan x sec2 x ln|x| + c
x
1
cot x –cosec2 x eax eax + c
a
sec x sec x tan x
cosec x –cosec x cot x
1
ln x
x
ex ex
v
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Introduction
Summations
1
(Arithmetic series) S = n[2a+(n−1)d]
n 2
a(1−rn)
(Geometric series) S = , r ≠ 1
n 1−r
n n(n+1) n n(n+1)(2n+1) n n2(n+1)2
∑r = ,∑r2 = ,∑r3 =
2 6 4
r=1 r=1 r=1
Binomial theorem
n n n n!
(a+b)n = ∑ an−rbr where = nC =
   
r r r r!(n−r)!
r=0
Maclaurin expansion
f"(0)x2 f"'(0)x3 f iv(0)x4
f(x)= f(0)+ f'(0)x+ + + +...
2! 3! 4!
De Moivre’s theorem
[r (cos θ + isin θ )]n = rn (cos nθ + isin nθ )
Vector product
i j k
a a a a a a
a×b =|a||b|sinθnˆ = a a a = i 2 3 − j 1 3 +k 1 2
1 2 3 b b b b b b
b b b 2 3 1 3 1 2
1 2 3
Matrix transformation
cosθ −sinθ
Anti-clockwise rotation through an angle, θ, about the origin,
 
sinθ cosθ 
vi
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Part 1 Calculus
Chapter 1
Differentiation 1: Rules and standard derivatives
Key points
!
Differentiate exponential and natural logarithmic functions
✽● differentiate functions involving ex and ln x
Differentiate functions using the chain rule
✽● apply the chain rule to differentiate the composition of at most three functions
Differentiate functions given in the form of a product or a quotient
f(x)
✽● differentiate functions of the form f (x) g (x) and
g(x)
✽● know the definitions and apply the derivatives of tan x, cot x, sec x and cosec x
✽● derive and use the derivatives of tan x, cot x, sec x and cosec x
✽● differentiate functions that require more than one application or a combination of applications
of the chain rule, product rule and quotient rule
dy 1
✽● apply = where appropriate
dx  dx
 
dy
 
Differentiate inverse trigonometric functions
✽● differentiate expressions of the form sin–1[f (x)], cos–1[f (x)] and tan–1[f (x)]
Apply differentiation to problems in context
✽● apply differentiation to problems in context
✽● apply differentiation to optimisation

Key links
Differentiate implicit functions and parametric equations (Chapter 2), investigate stationary
points and points of inflection (Chapter 6) and find Maclaurin expansions (Chapter 8) using
the content of this chapter.
The Chain rule
The chain rule for differentiation states that ( f (g(x))' = f '(g(x)).g'(x).
Alternatively, using Leibniz notation, treat y = f (g(x)) as y = f (u) where u = g(x)
dy dy du
then = × .
dx du dx
1
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Chapter 1 Differentiation 1: Rules and standard derivatives
f
Example
1 –1 –1 x
f(x)= x2 +5 =(x2 +5)2 ⇒ f'(x)= 1(x2 +5) 2.2x= x(x2 +5) 2 =
2
x2 +5
1 –1 –1 x
f(x)= x2 +5 =(x2 +5)2 ⇒ f'(x)= 1(x2 +5) 2.2x= x(x2 +5) 2 =
2
x2 +5
f
Example
(cid:31)
Hints & tips
dy
y = cos3(1− x) ⇒ = 3cos2(1− x).−sin (1− x).−1=3cos2(1− x) sin(1− x)
dx Remember ( f(g))' = f '(g) × g'.
dy Give final answers in their
y = cos3(1− x) ⇒ = 3cos2(1− x).−sin (1− x).−1=3cos2(1− x) sin(1− x)
simplest form.
dx
The Product rule
The product rule is a formula used to find the derivative of a product of functions.
It may be stated as (f(x)g(x))' = f'(x)g(x) + f(x)g'(x).
f
Example
f (x) = x2cos x ⇒ f '(x) = 2x.cos x + x2. – sin x = 2x cos x – x2sin x
f
Example
dy
y = x3(2− x)4 ⇒ = 3x2.(2− x)4 + x3.4(2− x)3.−1
(cid:31)
dx Hints & tips
= 3x2(2− x)4 −4x3(2− x)3
Remember ( fg)' = f 'g + fg'. Give
= x2(2− x)3[3(2− x)−4x]
final answers in their simplest
= x2(2− x)3(6−7x) form.
The Quotient rule
The quotient rule is a formula used to find the derivative of a quotient of functions.
  ′
f(x) f'(x)g(x)− f(x)g'(x)
 
It may be stated as  g(x) = ( )2 .
  g(x)
f
Example
1+sinx cosx.(1−sinx)−(1+sinx).−cosx 2cosx
f(x)= ⇒ f′(x)= =
1−sinx ( )2 ( )2
1−sinx 1−sinx
2
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Derivatives of exponential and natural logarithmic functions
f
Example (cid:31)
Hints & tips
2x+1 2x+1  ′
y = = f f'g − fg'
 
3− x2 (3− x2)12 Remember  g = g2 .
 
Look for a common factor on the
1 1 −1
2.(3− x2)2 −(2x+1). (3− x2)2.−2x numerator when simplifying.
dy
2
⇒ =
dx 3− x2
1
(3− x2)−2 2(3− x2)+x(2x+1) Check-up ?
 
=
3− x2
1−x
6+ x 6+ x 1 Differentiate f(x)= . 3
= = x2 +1
3 (3− x2)3
(3− x2)2
Derivatives of exponential and natural
f (x) f '(x)
logarithmic functions ex (or exp(x)) ex
1
ln x (or logx)
The derivatives of ex and ln x are given in the Formulae list. e x
f
Example (cid:31)
Hints & tips
f (x) = x3e2x+1 ⇒ f '(x) = 3x2.e2x+1 + x3.e2x+1.2 = x2e2x+1(3 + 2x)
Remember
d 1
(lnx)=
x
dx
f is only valid for the
Example
natural logarithmic
function.
dy 1 sinx
y = ln(cosx) ⇒ = .−sinx = − = −tanx
dx cosx cosx
?
Check-up
2 Differentiate
a) f(x)= e4x +1 b) y = x3ln2x. 2, 2
Derivatives of trigonometric functions
The reciprocal trigonometric functions secant, cosecant and tangent are defined by:
1 1 1
cotx= , secx= and cosecx= .
tanx cosx sinx
The derivatives of these functions and of tan x can be obtained by expressing each function in
terms of sin x and/or cos x then using the quotient rule.
3
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