EUROPEAN UNION

UNITED KINGDOM OF GREAT BRITAIN

AND NORTHERN IRELAND

PASSPORT TO ADVANCED HIGHER MATHEMATICS

Name:

Tracking and Evaluation – Advanced Higher Maths

Unit Knowledge Assessment Standards BLOCK/TOPIC Passed (- )

Methods in Algebra and

Calculus

1.1 Applying algebraic skills to partial fractions 1/1

1.2 Applying calculus skills through techniques of differentiation 1/3 & 2/1

1.3 Applying calculus skills through techniques of integration 1/4 & 2/2

1.4 Applying calculus skills to solving differential equations 2/3 & 3/1

Applications of Algebra

and Calculus

1.1 Applying algebraic skills to the binomial theorem and to complex

numbers

1/2 & 2/4

1.2 Applying algebraic skills to sequences and series 2/6 & 3/2

1.3 Applying algebraic skills to summation and mathematical proof 2/7

1.4 Applying algebraic and calculus skills to properties of functions 1/5

1.5 Applying algebraic and calculus skills to problems 1/6

Geometry, Proofs and Systems of Equations

1.1 Applying algebraic skills to matrices and systems of equations 1/7

1.2 Applying algebraic and geometric skills to vectors 3/3

1.3 Applying geometric skills to complex numbers 2/5

1.4 Applying algebraic skills to number theory 3/4

1.5 Applying algebraic and geometric skills to methods of proof 2/8

October score: % Grade: Band:

Prelim score: % Grade: Band:

On the next pages are all sub skills in the entire course. You can expect to be assessed on most of these

at some point. Complete the table to keep track of your progress and performance through the course.

The school website has notes, worksheets, answers, video tutorials and past paper questions for all topics.

Key Note

LEVEL C Subskill is at unit assessment level, a similar question could appear in a unit assessment or an exam.

LEVEL A/B Subskill is at exam level and a question of similar difficulty will only appear in an exam.

Unit 1

Topic SUBSKILL

Content/Explanation of standard LEVEL C

Content/Explanation of standard LEVEL A/ B

Additional support questions from AH Past

Papers 2010 to 2014

1. Algebraic Skills Methods in Algebra and Calculus Expressing rational functions as a sum of partial fractions (denominator of degree at most 3 and easily factorised)

Applications in Algebra and Calculus Expanding expressions using the binomial theorem

Express a proper rational function as a sum of partial fractions where the denominator may contain: distinct linear factors, an irreducible quadratic factor, a repeated linear factor

Use the binomial theorem

(𝑎 + 𝑏)𝑛 = ∑ (𝑛𝑟

) 𝑎𝑛−𝑟𝑏𝑟𝑛𝑟=0 for r, n∈ N

Expand an expression of the form (𝑎𝑥 + 𝑏)𝑛

where 𝑛 ≤ 5 a,b ∈ Z

Reduce an improper rational function to a polynomial and a proper rational function by division or otherwise

Expand an expression of the form (𝑎𝑥𝑝 + 𝑏𝑦𝑞)𝑛 where a,b ∈ Q; p,q ∈ Z; n≤7

Using the general term for a binomial expansion, find a specific term in an expression

2010 Q7 first part 2011 Q1 first part 2012 Q15a) 2012 Q16b) first part 2010 Q5 2011 Q2 2012 Q4 2013 Q1 2014 Q2

2. Differentiation Methods in Algebra and Calculus Differentiating exponential and logarithmic functions

Differentiating functions using the chain rule

Differentiating functions given in the form of a product and in the form of a quotient Applying differentiation to problems, in context

Differentiate functions involving 𝑒𝑥, 𝑙𝑛𝑥

Apply the chain rule to differentiate the composition of at most 3 functions

Differentiate functions of the form 𝑓(𝑥)𝑔(𝑥)and 𝑓(𝑥)

𝑔(𝑥)

Know the definitions and use the derivatives of of tan x and

cot x

Apply Differentiation to rates of change, e.g. rectilinear motion and optimisation

Apply differentiation to problems in context eg A particle moves a distance s metres in t seconds. The

distance travelled by the particle is given by 𝑠 = 2𝑡3 −23

2𝑡2 +

3𝑡 + 5. Find the acceleration of the particle after 4 seconds. Apply differentiation to optimisation

Use the derivative of tan x

Know the definitions of sec x, cosec x

Candidates should be able to derive and use derivatives of

tan x, cot x, sec x, cosec x

Differentiating functions which require more than one application or combination of applications of chain rule, product rule and quotient rule eg

Know and use 𝑑𝑦

𝑑𝑥=

1𝑑𝑥

𝑑𝑦

Use logarithmic differentiation; recognise when it is appropriate in extended products, quotients, and in functions where the variable occurs in an index

2010 Q1a) 2011 Q7 2012 Q1b) 2013 Q2 2010 Q1a) 2011 Q3b), Q7 2012 Q1b) 2013 Q2 2010 Q1a) & b) 2011 Q3b), Q7 2012 Q1a) & b) 2013 Q2 2014 Q1a) 2012 Q12 2013 Q4a) 2013 Q13

3. Integration by Substitution Methods in Algebra and Calculus Integrating expressions using standard results Integrating by substitution Applications in Algebra and Calculus Applying integration to problems, in context

Use ∫ 𝑒𝑥𝑑𝑥 , ∫𝑑𝑥

𝑥, ∫ 𝑠𝑒𝑐2𝑥 𝑑𝑥

Use the integrals of 1

√𝑎2−𝑥2,

1

𝑎2+𝑥2

Use partial fractions to integrate proper rational functions where the denominator may have:

two separate or repeated linear factors

three linear factors with constant numerator

Integrate where the substitution is given eg use the substitution 𝑢 = 𝑙𝑛𝑥 to obtain

∫1

𝑥𝑙𝑛𝑥𝑑𝑥, where x > 1

Apply integration to volumes of revolution where the volume generated is by the rotation of the area under a single curve about the x axis.

Recognise and integrate expressions of the form

∫ 𝑔(𝑓(𝑥))𝑓′(𝑥)𝑑𝑥 and ∫𝑓′(𝑥)

𝑓(𝑥)𝑑𝑥

Use partial fractions to integrate proper rational functions where the denominator may have:

three linear factors with non-constant numerator

a linear factor and an irreducible quadratic factor of the form 𝑎𝑥2 + 𝑏𝑥 + 𝑐

Apply integration to volumes of revolution where the volume generated is by the rotation of the area under a single curve about the y axis. Use calculus to determine corresponding connected integrals Apply integration to the evaluation of areas including

integration with respect to y

2010 Q7 2011 Q1, Q11a) 2013 Q6 2010 Q3a) 2011 Q11b) 2012 Q8 2013 Q6 2014 Q12 2010 Q15 2013 Q4b) 2013 Q16 2013 Q17 2014 Q10

4. Properties of Functions Applications in Algebra and Calculus Finding the asymptotes to the graphs of rational functions

Investigating features of graphs and sketching graphs of functions

Find the vertical asymptote to the graph of a rational function

Find the non-vertical asymptote to the graph of a rational function

Investigate points of inflection , eg establish the coordinates of the point of inflection on the graph of y = x

3 + 3x

2 + 2x

Sketch related functions:

modulus functions

inverse functions

differentiated functions

translations and reflections

Investigate other features:

stationary points

domain and range

symmetry (odd/even)

continuous/discontinuous

extrema of functions: the maximum and minimum values

of a continuous function f defined on a closed interval [a,b]

can occur at stationary points, end points or points where f ' is not defined

eg calculate the maximum value, 0 ≤ 𝑥 ≤ 4, of 𝑓(𝑥) = 𝑒𝑥𝑠𝑖𝑛2𝑥 Sketch graphs using features given or obtained

2014 Q11a) & b) 2010 Q10 2011 Q6 2012 Q7a) & b) 2013 Q13a)-c) 2014 Q11

5. Matrices Geometry, Proof and Systems of Equations Using Gaussian elimination to solve a 3 x 3 system of linear equations Understanding and using matrix algebra

Find the solution to a system of equations Ax = b where A is a 3x3 matrix and where the solution is unique Candidates should understand the term augmented matrix

Perform matrix operations (at most order 3): addition, subtraction, multiplication by a scalar, multiplication of matrices

Candidates should know the meaning of the terms: matrix, element, row, column, order,

identity matrix, inverse, determinant, singular, non-singular, transpose.

Show that a system of equations has no solutions (inconsistency) Show that a system of equations has an infinite number of solutions (redundancy) Compare the solutions of related systems of two equations in two unknowns and recognise ill-conditioning Know and apply the properties of matrix addition and multiplication:

Know and apply key properties of the transpose, identity matrix and the inverse:

2010 Q14 2012 Q14 2014 Q3 2010 Q14 part 4 and 5 2011 Q4b) 2012 Q9 2013 Q3a) & c)

Calculating the determinant of a matrix

Finding the inverse of a matrix

Find the determinant of a 2x2 matrix and a 3x3 matrix Determine whether a matrix is singular

Know and use the inverse of a 2x2 matrix

Know and apply det(𝐴𝐵) = 𝑑𝑒𝑡𝐴𝑑𝑒𝑡𝐵

Find the inverse of a 3x3 matrix Use 2x2 matrices to carry out geometric transformations in the plane:

The transformations should include rotations, reflections and dilatations

Apply combinations of transformations

2011 Q4a) 2013 Q3b) 2010 Q4

UNIT 2

Topic SUBSKILL

Content/Explanation of standard LEVEL C

Content/Explanation of standard LEVEL A/ B

Additional support questions from AH

Past Papers 2010 to 2014

1. Further Differentiation Methods in Algebra and Calculus Differentiating inverse trigonometric functions Finding the derivative of functions defined implicitly

Finding the derivative of functions defined parametrically

𝑣𝑥 =𝑑𝑥

𝑑𝑡, 𝑣𝑦 =

𝑑𝑦

𝑑𝑡

Differentiating expressions of the form eg 𝑠𝑖𝑛−1𝑘𝑥, (learners should know how to derive this.) Use differentiation to find the first derivative of a function defined implicitly including in context

eg𝑥3𝑦 + 𝑥𝑦3 = 4 Apply differentiation to related rates in problems where the functional relationship is given implicitly, for example, the ‘falling ladder’ problem. Use differentiation to find the first derivative of a function defined parametrically including in context eg Apply parametric differentiation to motion in a plane If the position is given by 𝑥 = 𝑓(𝑡), 𝑦 = 𝑔(𝑡) , then i) velocity components are given by

ii) speed = √(𝑑𝑥

𝑑𝑡)

2

+ (𝑑𝑦

𝑑𝑡)

2

eg Apply differentiation to related rates in problems where the functional relationship is

given explicitly 𝑉 =1

3𝜋𝑟2ℎ; given

𝑑ℎ

𝑑𝑡 find

𝑑𝑉

𝑑𝑡

𝑡𝑎𝑛−1[𝑓(𝑥)], (differentiate with the aid of the formulae sheet.)

Use differentiation to find the second derivative of a function defined implicitly

Use differentiation to find the second derivative of a function defined parametrically Solve practical related rates by first establishing a functional relationship between appropriate variables

2011 Q3a) 2012 Q11a) 2013 Q11 2014 Q1b), Q6 2010 Q13 2012 Q13 2014 Q4

2. Further Integration Methods in Algebra and Calculus Integrating by parts

Use integration by parts with one application

Use integration by parts involving repeated applications

Eg ∫ 𝑥2𝑐𝑜𝑠𝑥𝑑𝑥𝜋

0

Eg ∫ 𝑥2𝑒3𝑥𝑑𝑥

2010 Q3b) 2011 Q16 2012 Q11b) 2013 Q8 2014 Q15

3. Differential Equations (1) Methods in Algebra and Calculus Solving first order differential equations with variables separable

Solving first order linear differential equations using an integrating factor

Solve equations that can be written in the form 𝑑𝑦

𝑑𝑥= 𝑔(𝑥)ℎ(𝑦) or

𝑑𝑦

𝑑𝑥=

𝑔(𝑥)

ℎ(𝑦)

Eg 𝑑𝑦

𝑑𝑥= 𝑦(𝑥 − 1)

Eg 𝑣𝑑𝑣

𝑑𝑥= −𝜔𝑥

Solve equations written in the standard form 𝑑𝑦

𝑑𝑥+ 𝑃(𝑥)𝑦 = 𝑓(𝑥)

Eg 𝑑𝑦

𝑑𝑥+

3𝑦

𝑥=

𝑒𝑥

𝑥3

Find general and particular solutions given suitable information

Eg 1

𝑥

𝑑𝑦

𝑑𝑥= 𝑦𝑠𝑖𝑛𝑥 given that when 𝑥 =

𝜋

2, 𝑦 = 1

Solve equations by first writing linear equations in the standard

form 𝑑𝑦

𝑑𝑥+ 𝑃(𝑥)𝑦 = 𝑓(𝑥)

Eg 𝑥2 𝑑𝑦

𝑑𝑥+ 3𝑥𝑦 = 𝑠𝑖𝑛𝑥

Find general and particular solutions given suitable information

2011 Q9 2012 Q15b)

4. Complex Numbers Applications in Algebra and Calculus Performing algebraic operations on complex numbers

Geometry, Proof and Systems of Equations Performing geometric operations on complex numbers

Perform all of the operations of addition, subtraction, multiplication and division Plot complex numbers in the complex plane (an Argand diagram) Know the definition of modulus and argument of a complex number Convert a given complex number from Cartesian to polar form and vice-versa

Finding the square root eg√8 − 6𝑖 Find the roots of a quartic with real coefficients when one complex root is given Solve equations involving complex numbers Eg solve 𝑧 + 𝑖 = 2𝑧̅ + 1

Eg solve 𝑧2 = 2𝑧̅ Use de Moivre’s theorem with integer and fractional indices eg expand (𝑐𝑜𝑠𝜃 + 𝑖𝑠𝑖𝑛𝜃)4 Apply de Moivre’s theorem to multiple angle trigonometric formulae Eg express 𝑠𝑖𝑛5𝜃 in terms of 𝑠𝑖𝑛𝜃

Eg express 𝑠𝑖𝑛5𝜃 in terms of sin/cos of multiples of 𝜃 Apply de Moivre’s theorem to find the nth roots of a complex number eg solve z

6=1

Interpret geometrically certain equations or inequalities in the complex plane Eg |𝑧 − 𝑖| = |𝑧 − 2|, |𝑧 − 𝑎| > 𝑏

2010 Q16 2012 Q3, Q16b) 2010 Q10 2012 Q2 part 2 2013 Q7, Q10 2014 Q16

5. Sequences Applications in Algebra and Calculus Finding the general term and summing arithmetic and geometric progressions Applying summation formulae

Apply the rules on sequences and series to find

the nth term

sum to n terms

common difference of arithmetic sequences

common ratio of geometric sequences Know and use sums of certain series and other straightforward results and combinations thereof (2, 4 and 5 appear on formulae sheet and are therefore ‘use’ only)

Apply the rules on sequences and series to find

sum to infinity of geometric series

determine the condition for a geometric series to converge

eg 1 + 2𝑥 + 4𝑥2 + 8𝑥3 + ⋯has a sum to infinity if and only

if |𝑥| <1

2

2010 Q2 2011 Q13 2012 Q2 2013 Q17 first part 2014 Q14 2011 Q8

6. Proofs Applications in Algebra and Calculus Using proof by induction

Geometry, Proof and Systems of Equations Disproving a conjecture by providing a counter-example Using indirect or direct proof in straightforward examples

∑ 𝑟3 =𝑛2(𝑛 + 1)2

4

𝑛

𝑟=1

Use mathematical induction to prove summation formulae eg

Disprove a conjecture by providing a counter-example Eg for all real values of a and b

𝑎 − 𝑏 > 0 → 𝑎2 − 𝑏2 > 0 . A counter-example is a=3, b=-4 Prove a statement by contradiction

Eg √2 is irrational

Eg if a and b are real then a2 + b

2 ≥ 2ab

Teaching examples can focus on classic results,

eg the infinitude of primes; the irrationality of √2

Use direct proof in straightforward examples Eg Prove that the product of any 3 consecutive natural numbers is divisible by 6

Eg Prove 𝑚2 + 𝑛2 < (𝑚 + 𝑛)2 ∀𝑚, 𝑛 ∈ 𝑁

Use proof by induction

Eg show that 1 + 2 + 22 + ⋯ + 2𝑛 = 2𝑛+1 − 1, 𝑛 ∈ 𝑁 Eg 8𝑛 is a factor of (4𝑛)!, 𝑛 ∈ 𝑁

Eg 𝑦 = 𝑥𝑛, then 𝑑𝑦

𝑑𝑥= 𝑛𝑥𝑛−1, 𝑛 ∈ 𝑁

Know and be able to use the symbols ∃ and ∀ Write down the negation of a statement Use further proof by contradiction Use proof by contrapositive Eg Prove that is n

2 is even then n is even

Eg If 𝑥, 𝑦 ∈ 𝑅: 𝑥 + 𝑦 is irrational then at least one of 𝑥, 𝑦 is irrational

2010 Q8b) 2011 Q12 2012 Q16a) 2013 Q9 2014 Q7 2013 Q12 2012 Q8a), Q12 2013 Q12

UNIT 3

1. Differential Equations (2) Methods in Algebra and Calculus Solving second order differential equations

Find the general solution and particular solution of second order homogeneous ordinary differential equations of the form

𝑎𝑑2𝑦

𝑑𝑥2 + 𝑏𝑑𝑦

𝑑𝑥+ 𝑐𝑦 = 0 with constant coefficients

where the roots of the auxiliary equation are real

𝑎𝑑2𝑦

𝑑𝑥2+ 𝑏

𝑑𝑦

𝑑𝑥+ 𝑐𝑦 = 𝑓(𝑥)

Find the general solution and particular solution of second order homogeneous ordinary differential equations of the form

𝑎𝑑2𝑦

𝑑𝑥2 + 𝑏𝑑𝑦

𝑑𝑥+ 𝑐𝑦 = 0 with constant coefficients where the roots

of the auxiliary equation are real or complex conjugates

Eg 𝑑2𝑦

𝑑𝑥2 − 4𝑑𝑦

𝑑𝑥+ 13𝑦 = 0

Solve second order non-homogeneous ordinary differential equations of the form

with constant coefficients using the auxiliary equation and particular integral method

2010 Q11 2011 Q14 2013 Q14 2014 Q8

2. Maclaurin’s Theorem Applications in Algebra and Calculus Using the Maclaurin expansion to find a stated number of terms of the power series for a simple function

Use the Maclaurin expansion to find a power series for a simple non-standard function

Eg 1

1+𝑥2

Candidates should be familiar with the standard power series expansions of 𝑒𝑥, 𝑠𝑖𝑛𝑥, 𝑐𝑜𝑠𝑥 and ln (1 ± 𝑥). For unit assessment power series

should be derived and not quoted.

Use the Maclaurin expansion to find a power series eg

𝑒𝑠𝑖𝑛𝑥 , 𝑒𝑥𝑐𝑜𝑠𝑥

2010 Q9 2011 Q5 2012 Q6 2013 Q17 (could be done using Maclaurin series) 2014 Q9

3. Vectors Geometry, Proof and Systems of Equations Calculating a vector product Working with lines in three dimensions

Working with planes

Use a vector product method in three dimensions to find the vector product Find the equation of a line in parametric, symmetric or vector form, given suitable defining information Find the equation of a plane in vector form, parametric form or Cartesian form, given suitable defining information

Evaluate the scalar triple product 𝒂 ∙ (𝒃 × 𝒄) This could be used to show learners how to calculate the volume of a parallelepiped.

Find the angle between two lines in three dimensions Determine whether or not two lines intersect and, where possible, find the point of intersection Find the point of intersection of a plane with a line which is not parallel to the plane Determine the intersection of two or three planes Find the angle between a line and a plane or between two planes

2010 Q6 2014 Q5 2011 Q15 2012 Q5 2013 Q15

4. Euclidean Algorithm Geometry, Proof and Systems of Equations Using Euclid’s algorithm to find the greatest common divisor of two positive integers

Use Euclid’s algorithm to find the greatest common divisor of two positive integers, ie use the division algorithm repeatedly

Express the greatest common divisor (of two positive integers) as a linear combination of the two integers Express integers in bases other than ten Know and use the Fundamental Theorem of Arithmetic

2012 Q10 2013 Q5