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A Level Maths Exam Questions By Topic

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Although the following questions are predominantly from the OCR, OCR MEI and Edexcel exam boards, they are suitable practice for all UK A Level Maths qualifications unless otherwise stated . 

A Level Maths

Related Pages A Level Exam Practice Edexcel 2019 Pure Maths Paper 1 GCSE/IGCSE Maths

Need some A Level Maths help? We have a collection of videos, solutions, activities, worksheets and past year papers that are suitable for A Level Maths. Some materials here are also relevant for IGCSE/GCSE Additional Mathematics.

We will look at:

• Worked solutions for some A-level maths past year examinations.
• Concepts that you will need to know for the A-level Maths: Algebra, Transformation of Graphs, Functions, Indices, Logarithms & Exponential, Polynomials, Geometry. Coordinate Geometry, Differentiation, Integration, Sequences & Series, Binomial Expansion, Trigonometry, Vectors, Statistics & Probability, Mechanics and Further Pure Mathematics.

Edexcel Past Papers, Mark Schemes, Videos & Solutions

Questions & answers for edexcel mathematics past papers.

Pure Maths Paper 1 October 2020, 8MA0/01 Statistics Paper 21 October 2020, 8MA0/21 Mechanics Paper 22 October 2020, 8MA0/22

Pure Maths Paper 1 May/June 2019, 8MA0/01 Statistics Paper 21 May/June 2019, 8MA0/21 Mechanics Paper 22 May/June 2019, 8MA0/22

Pure Maths Paper 1 October 2020, 9MA0/01 Pure Maths Paper 1 June 2019, 9MA0/01 Pure Maths Mock Paper 1 2019 Pure Maths Paper 1 June 2018 Pure Maths Specimen Paper 1 2018 Pure Maths Sample Paper 1 2017

Pure Maths Paper 2 October 2020, 9MA0/02 Pure Maths Paper 2 June 2019, 9MA0/02 Pure Maths Mock Paper 2 2019 Pure Maths Paper 2 June 2018 Pure Maths Specimen Paper 2 2018 Pure Maths Sample Paper 2 2017

Statistics & Mechanics October 2020, 9MA0/31, 9MA0/32 Statistics & Mechanics Paper 3 October 2020, 9MA0/03 Statistics & Mechanics June 2019, 9MA0/31, 9MA0/32 Statistics & Mechanics Mock Paper 3 2019 Statistics & Mechanics Paper 3 June 2018 Statistics & Mechanics Specimen Paper 3 2018 Statistics & Mechanics Sample Paper 3 June 2017

Further Maths Paper 1 May 2020, 9FM0/01 Further Maths Paper 1 May 2020, 9FM0/02

Questions & Answers for Edexcel IAL Mathematics Past Papers

Edexcel Jan 2022 IAL Pure Maths WMA11/01 Edexcel Oct 2021 IAL Pure Maths WMA11/01 Edexcel May/June 2021 IAL Pure Maths WMA11/01 Edexcel Jan 2021 IAL Pure Maths WMA11/01 Edexcel Oct 2020 IAL Pure Maths WMA11/01 Edexcel Jan 2020 IAL Pure Maths WMA11/01 Edexcel Oct 2019 IAL Pure Maths WMA11/01

Edexcel Jan 2022 IAL Pure Maths WMA12/01 Edexcel Oct 2021 IAL Pure Maths WMA12/01 Edexcel May/June 2021 IAL Pure Maths WMA12/01 Edexcel Jan 2020 IAL Pure Maths WMA12/01

Edexcel Jan 2022 IAL Pure Maths WMA13/01 Edexcel May/June 2021 IAL Pure Maths WMA13/01 Edexcel Jan 2021 IAL Pure Maths WMA13/01 Edexcel Oct 2020 IAL Pure Maths WMA13/01 Edexcel Jan 2020 IAL Pure Maths WMA13/01 Edexcel 2020 IAL Pure Maths WMA13/01 Specimen

Edexcel Jan 2022 IAL Pure Maths WMA14/01 Edexcel Jan 2021 IAL Pure Maths WMA14/01

Edexcel Jan 2022 IAL Statistics WST01/01 Edexcel Jan 2021 IAL Statistics WST01/01 Edexcel Jan 2020 IAL Statistics WST01/01

Edexcel Jan 2022 IAL Mechanics and Materials WPH11/01

Cambridge International Exams (CIE) Past Papers & Solutions

9709 past papers past papers & solutions.

May 2022 Pure Maths 1 9709/11 May 2022 Pure Maths 1 9709/12 May 2022 Pure Maths 1 9709/13 Mar 2022 Pure Maths 1 9709/12

Oct 2021 Pure Maths 1 9709/11 Oct 2021 Pure Maths 1 9709/12 Oct 2021 Pure Maths 1 9709/13 May 2021 Pure Maths 1 9709/11 May 2021 Pure Maths 1 9709/12 May 2021 Pure Maths 1 9709/13 Mar 2021 Pure Maths 1 9709/12

Oct 2020 Pure Maths 1 9709/11 Oct 2020 Pure Maths 1 9709/12 Oct 2020 Pure Maths 1 9709/13 May 2020 Pure Maths 1 9709/11 May 2020 Pure Maths 1 9709/12 May 2020 Pure Maths 1 9709/13 Mar 2020 Pure Maths 1 9709/13

Mar 2022 Pure Maths 2 9709/22

May 2021 Pure Maths 2 9709/23 Mar 2021 Pure Maths 2 9709/22

May 2020 Pure Maths 2 9709/21 May 2020 Pure Maths 2 9709/23 Mar 2020 Pure Maths 2 9709/22

Mar 2022 Pure Maths 3 9709/32

Oct 2021 Pure Maths 3 9709/31 Oct 2021 Pure Maths 3 9709/32 May 2021 Pure Maths 3 9709/31 May 2021 Pure Maths 3 9709/32 May 2021 Pure Maths 3 9709/33 Mar 2021 Pure Maths 3 9709/32

Oct 2020 Pure Maths 3 9709/32 Oct 2020 Pure Maths 3 9709/33 May 2020 Pure Maths 3 9709/31 May 2020 Pure Maths 3 9709/32 Mar 2020 Pure Maths 3 9709/32

May 2022 Mechanics 4 9709/43 Mar 2022 Mechanics 4 9709/42

Oct 2021 Mechanics 4 9709/41 Oct 2021 Mechanics 4 9709/42 May 2021 Mechanics 4 9709/41 May 2021 Mechanics 4 9709/42 May 2021 Mechanics 4 9709/43 Mar 2021 Mechanics 4 9709/42

Oct 2020 Mechanics 4 9709/41 Oct 2020 Mechanics 4 9709/42 May 2020 Mechanics 4 9709/41 May 2020 Mechanics 4 9709/42 Mar 2020 Mechanics 4 9709/42

May 2022 Prob & Stats 1 9709/53

Oct 2021 Prob & Stats 1 9709/51 Oct 2021 Prob & Stats 1 9709/52 Oct 2021 Prob & Stats 1 9709/53

May 2021 Prob & Stats 1 9709/51 May 2021 Prob & Stats 1 9709/52 May 2021 Prob & Stats 1 9709/53 Mar 2021 Prob & Stats 1 9709/52

Oct 2020 Prob & Stats 1 9709/51 Oct 2020 Prob & Stats 1 9709/52 May 2020 Prob & Stats 1 9709/51 May 2020 Prob & Stats 1 9709/52 Mar 2020 Prob & Stats 1 9709/52

Mar 2022 Prob & Stats 2 9709/62

Oct 2021 Prob & Stats 1 9709/62 May 2021 Prob & Stats 2 9709/61 May 2021 Prob & Stats 2 9709/63 Mar 2021 Prob & Stats 2 9709/62

Oct 2020 Prob & Stats 2 9709/61 Oct 2020 Prob & Stats 2 9709/62 May 2020 Prob & Stats 2 9709/61 May 2020 Prob & Stats 2 9709/62 Mar 2020 Prob & Stats 2 9709/62

May 2022 Physics 1 9702/13 Mar 2022 Physics 1 9702/12 2022 Physics Specimen Paper 1 9702/1

Oct 2021 Physics 1 9702/13 May 2021 Physics 1 9702/12 May 2021 Physics 1 9702/13 Mar 2021 Physics 1 9702/12

Oct 2020 Physics 1 9702/11 Oct 2020 Physics 1 9702/12 Oct 2020 Physics 1 9702/13

Mar 2021 Physics 2 9702/22 Oct 2020 Physics 2 9702/22 Oct 2020 Physics 2 9702/23

Questions & Answers for Edexcel Core Mathematics

May 2017 C1 May 2016 C1 May 2015 C1

May 2015 C2 Exam Questions May 2015 C2 Solutions

Questions January 2015 C12 Solutions January 2015 C12 Mark Scheme January 2015

Questions January 2015 C34 Solutions January 2015 C34 Mark Scheme January 2015 C34

Questions & Solutions For Edexcel Further Pure Maths FP1 Advanced/Advanced Subsidiary

Questions June 2015 Solutions June 2015

Edexcel International Advanced Level Mathematics

Edexcel has redeveloped their AS/A level in Mathematics for international centres, called Edexcel International AS/Advanced Level in Mathematics. Students sitting exams from January 2014 will be awarded the new qualification.

What are the changes to the International Advanced Level in Mathematics specification?

The Core units have been combined. C1 and C2 have been combined to create one unit. C3 and C4 have been combined to create one unit.

Further Pure Mathematics 1 has been updated with some new content.

Application units available include: – Statistics 1, 2, and 3 – Mechanics 1, 2 and 3 – Decision Mathematics 1.

Further Pure Mathematics units available are Further Pure Mathematics 1, 2 & 3.

Questions May 2014 Mark Scheme May 2014 Solutions May 2014 Questions June 2013 Mark Scheme June 2013 Solutions June 2013 Questions January 2013 Mark Scheme January 2013 Solutions January 2013 Questions June 2012 Mark Scheme June 2012 Solutions June 2012 Questions January 2012 Mark Scheme January 2012 Solutions January 2012 Questions June 2011 Mark Scheme June 2011 Solutions June 2011 Questions January 2011 Mark Scheme January 2011 Solutions January 2011 Questions June 2010 Mark Scheme June 2010 Solutions June 2010 Questions January 2010 Mark Scheme January 2010 Solutions January 2010 Questions June 2009 Mark Scheme June 2009 Solutions June 2009 Questions January 2009 Mark Scheme January 2009 Questions June 2008 Mark Scheme June 2008 Questions January 2008 Mark Scheme January 2008 Questions June 2007 Mark Scheme June 2007 Questions January 2007 Mark Scheme January 2007

Questions May 2014 Mark Schemes May 2014 Solutions May 2014 Questions June 2013 Mark Schemes June 2013 Solutions June 2013 Questions January 2013 Mark Schemes January 2013 Solutions January 2013 Questions June 2012 Mark Schemes June 2012 Solutions June 2012 Questions January 2012 Mark Schemes January 2012 Solutions January 2012 Questions June 2011 Mark Schemes June 2011 Solutions June 2011 Questions January 2011 Mark Schemes January 2011 Solutions January 2011 Questions June 2010 Mark Schemes June 2010 Solutions June 2010 Questions January 2010 Mark Schemes January 2010 Solutions January 2010 Questions June 2009 Mark Schemes June 2009 Solutions June 2009 Questions January 2009 Mark Schemes January 2009 Solutions January 2009 Questions June 2008 Mark Schemes June 2008 Questions January 2008 Mark Schemes January 2008 Questions June 2007 Mark Schemes June 2007 Questions January 2007 Mark Schemes January 2007 Solutions January 2007

Questions May 2014 Mark Scheme May 2014 Solutions May 2014 Questions June 2013 Mark Scheme June 2013 Solutions June 2013 Questions January 2013 Mark Scheme January 2013 Solutions January 2013 Questions June 2012 Mark Scheme June 2012 Solutions June 2012 Questions January 2012 Mark Scheme January 2012 Solutions January 2012 Questions June 2011 Mark Scheme June 2011 Solutions June 2011 Questions January 2011 Mark Scheme January 2011 Solutions January 2011 Questions June 2010 Mark Scheme June 2010 Solutions June 2010 Questions January 2010 Mark Scheme January 2010 Solutions January 2010 Questions June 2009 Mark Scheme June 2009 Questions January 2009 Mark Scheme January 2009 Questions June 2008 Mark Scheme June 2008 Questions January 2008 Mark Scheme January 2008 Questions June 2007 Mark Scheme June 2007 Questions January 2007 Mark Scheme January 2007

A-level Exam Q1 - Q5 A-level Exam Q6 - Q10 A-level Exam Paper 2 Q1 - Q4 A-level Exam Paper 2 Q5 - Q8

A-level Examination Q1 - Q4 A-level Examination Q5 - Q9

Questions & Answers for OCR 4721 Core Mathematics 1

Questions for Jan 2010 Core Mathematics 1 January 2010 Q1-Q4 Core Mathematics 1 January 2010 Q5-Q8 Core Mathematics 1 January 2010 Q9-Q11

Questions & Answers For IGCSE/GCSE Additional Mathematics

Additional Maths 2008 Q1 - Q4 Additional Maths 2008 Q5 - Q8 Additional Maths 2008 Q9 - Q13

Bridging The Gap Between Gcse Maths & A-Level Maths

Fractions & Indices Surds Straight Line Graphs, Parallel & Perpendicular Lines Linear Algebra, Simultaneous Equations, Rearranging Equations Factoring & Solving Quadratic Equations, Completing the Square Solving Quadratics with Fractions, Simultaneous Equations (Linear & Non-Linear) Graph Sketching & Transformations Pythagoras Theorem, Sine Rule, Cosine Rule, Area of non-right Triangle

Pure Mathematics

Linear Inequalities (Review) Simultaneous Linear Equations Elimination Simultaneous Linear Equations Substitution Simultaneous Linear Equations - Graphical Factorising & Solving Quadratic Equations I Factorising & Solving Quadratic Equations II Completing the Square I Completing the Square II Sketching a Quadratic Graph by Completing the Square Completing the Square & Graph Transformation Derivation of the Quadratic Formula Solving Quadratic Equations using the Quadratic Formula Using the Quadratic Formula without a Calculator Discriminant of a Quadratic Equation I Discriminant of a Quadratic Equation II Disguised Quadratic Equations Quadratic Inequalities I Quadratic Inequalities II Fractional (or Rational) Inequalities Simultaneous Inequalities Simultaneous Equations - Linear & Non-Linear I Simultaneous Equations - Linear & Non-Linear II Simultaneous Equations - Logarithms Simplifying Algebraic Fractions I Simplifying Algebraic Fractions II Simplifying Algebraic Fractions III Partial Fractions Tutorial I Partial Fractions Tutorial II Partial Fractions Tutorial III Partial Fractions Exam Questions I Partial Fractions Exam Questions II Modulus Graphs I Modulus Graphs II Solving Modulus Equations

Graph Transformations (Translation, Reflection, Stretch) Graph Transformations- Proofs Sketch Quadratic Graphs by Completing the Square Exponential Graphs Sketching Exponential Graphs Modulus Graphs I Modulus Graphs II Graph Sketching Transformation of Trig. Graphs I Transformation of Trig. Graphs II More Graph Transformations Graph Transformation Exam Questions

Functions: Domain, Range, Mappings Inverse Functions Domain & Range of Inverse Functions Combined or Composite Functions Composite Function Exam Questions Even & Odd Functions Modulus Functions Function Exam Questions I Function Exam Questions II Function Exam Questions III Function Exam Questions IV Inverse & Self-Inverse Functions Exam Questions

Rules of Indices (Review) Simplifying Indices Indices Exam Questions Indices & Differentiation Exam Questions

Surds Tutorial I Surds Tutorial II Surds Tutorial III Surds Tutorial IV

Logarithms Tutorial Solving Logarithmic Equations Solving Logarithmic Equations Solving Logarithmic Inequalities Change of Base Log Equations Logarithm Simplification More Logarithm Simplification Quadratic Log Equations Solving Equations using Logs Logs & Exponential Equations I Logs & Exponential Equations II Logarithmic Equations Exam Questions ln(x) & e x equations I ln(x) & e x equations II Exponential Growth & Decay

Adding & Subtracting Polynomials Multiplying Polynomials Dividing Polynomials (Algebraic Long Division) Solving Cubics using Polynomial Long Division The Remainder Theorem Remainder Theorem Exam Questions The Factor Theorem Factorising Cubics using the Factor Theorem I Factorising Cubics using the Factor Theorem II Factor Theorem Exam Questions Using the Remainder Theorem & Factor Theorem Remainder & Factor Theorem I Remainder & Factor Theorem II

Pythagoras’ Theorem (Review) Trigonometric Ratios (Review) Three-dimensional Applications (Review) Sine Rule Cosine Rule The Ambiguous Case for the Sine Rule Area of a Triangle using Sine Trigonometric Questions with Bearings Area of Quadrilaterals (Review) Area of Circle (Review) Radian, Area of Sector & Segment Area of Sectors & Triangles Area of Circles, Sectors & Triangles Questions on Triangles I Questions on Triangles II

Midpoint & Distance Formula Parallel & Perpendicular Lines Equation of a Line I Equation of a Line II Distance of a Point from a Line Intersection of two straight lines Angle between two straight lines Equation of Circle Tutorial 1 Equation of Circle Tutorial 2 Equation of Circle Tutorial 3 Equation of Circle Tutorial 4 Find Equation of a Circle given Centre & Radius Find Centre & Radius given the Equation of a Circle Find Equation of a Circle given Three Points at the Circumference Equation of Circle & Distance Formula Equation of a Tangent to a Circle Equation of a Circle & the Tangent Equation of Circle Problems I Equation of Circle Problems II Centre of Circles & Chords Intersection of Two Circles

Differentiate ax n Differentiation Chain Rule Differentiation Product Rule Differentiation Quotient Rule Using the Chain Rule: Exponential, Log & Trig Functions Chain Rule (Core 3) Differentiate the Natural Log Product Rule Examples Quotient Rule Examples Chain Rule Examples Differentiation Combining Chain, Product & Quotient Rules I Differentiation Combining Chain, Product & Quotient Rules II Differentiation Combining Chain, Product & Quotient Rules III Product Rule to find Tangents & Normals Quotient Rule Stationary Points, Tangents & Normals Second Derivatives Tangents & Normal I Tangents & Normal II Increasing & Decreasing Functions Stationary Points I Stationary Points II Differentiation Lessons I Differentiation Lessons II Differentiation Lessons III Differentiation Exam Questions I Differentiation Exam Questions II Differentiation Exam Questions III Differentiation Exam Questions IV Differentiation Exam Questions V Differentiation: Connected Rates of Change Differentiation: Rates of Change Problems Differentiation: Maximum Volume Differentiation: Optimisation Lessons on Implicit Differentiation Differentiate Implicit Equations Implicit Differentiation Exam Questions Cubic Differentiation Tangents & Stationary Points to Implicit Curves Differentiating y = a x Parametric Equations I Parametric Equations II Parametric Equations & Cartesian Differentiate Parametric Equations Differentiate Parametric Equations I Differentiate Parametric Equations II Differentiate Parametric Equations III Differential Equations I Differential Equations II Differential Equations III Differential First Order equation

Introduction to Integration Basic Integration Exam Questions Integration Help I Integration Help II Integration Help III Integration Help IV Integration by Substitution Tutorial I Integration by Substitution Tutorial II Integration by Substitution Exam Questions Integration by Parts Tutorial Integration by Parts Exam Questions I Integration by Parts Exam Questions II Trapezium Rule Trapezium or Trapezoid Rule Integration of Exponential & Natural Log Functions Integration of Trigonometric Functions I Integration of Trigonometric Functions II Integration: Mixed Examples I Integration: Mixed Examples II Definite Integral Area bounded by a Curve I Area bounded by a Curve II Integration: Area under a Graph Questions I Integration: Area under a Graph Questions II Integration: Area under a Graph Questions III Area within a Parametric Graph Volume of Revolution I Volume of Revolution II Parametric Integration

Recurrence Relations Tutorial Sequences: Recurrence Relations Sigma Notation Arithmetic Sequences & Series Tutorial Arithmetic Progressions Arithmetic Sequences & Series I Arithmetic Sequences & Series II Arithmetic Sequences & Series III Arithmetic Sequences & Series IV Geometric Series Tutorial Geometric Sequences & Series Tutorial I Geometric Sequences & Series Tutorial II Geometric Sequences & Series Tutorial III Geometric Sequences Geometric Series I Geometric Series II Geometric Series III Geometric Series IV Geometric Series V Geometric Series VI Geometric Sequences & Series

Binomial Expansion Beginners Tutorial I Binomial Expansion Beginners Tutorial II Binomial Expansion Beginners Tutorial III Binomial Expansion Tutorial Binomial Expansion Formula Finding Terms in a Binomial Expansion Binomial Expansion C2 Exam Questions I Binomial Expansion C2 Exam Questions II Binomial Expansion C2 Exam Questions III Binomial Expansion C4 Exam Questions I Binomial Expansion C4 Exam Questions II Binomial Expansion Approximations & Estimations Binomial Expansions with Surds Binomial Expansion Examples I Binomial Expansion Examples II

Trigonometry Tutorial 1 Trigonometry Tutorial 2 Trigonometry Tutorial 3 Trigonometry Tutorial 4 Trigonometry Tutorial 5 Quadrant Method Solving Sin θ = + value Solving Sin θ = − value Solving Cos θ = + value Solving Cos θ = − value Solving Tan θ = + value Solving Tan θ = − value Solving Sin 2x = + value Trig Ratios for multiples of 30, 45 & 60 degrees Trigonometric Basics - Reciprocal Functions Trigonometric Equations - Factorising Types Trigonometric Equations - Identity Types Trigonometric Equations - Double Angle Identity Types Trig. Equations Examples using CAST Diagrams Solving Equations using Identities Trig. Equations : A sin x - B cos x = C Type Solving equations using reciprocal trig identities More Reciprocal Trig Identities Trig Identities Examples Evaluating Inverse Trig Values Quadratic Trigonometric Equations Proof of A sin x + B cos x = R sin (x + alpha) Proof of A cos x + B sin x = R cos (x - alpha) Proof of A sin x - B cos x = R sin (x - alpha) Proof of A cos x - B sin x = R cos (x + alpha) Proving Trig Identities I Proving Trig Identities II Proving Trig Identities III Proving Trig Identities IV Proving Trig Identities V Trigonometry C2 Exam Questions I Trigonometry C2 Exam Questions II Trigonometry C2 Exam Questions III Trigonometry C3 Exam Questions I Trigonometry C3 Exam Questions II Trigonometry C3 Exam Questions III Trigonometry C3 Exam Questions IV

Vector Equation of a Line Vectors: Shortest Distance between point & line Vector Exam Questions I Vector Exam Questions II Vector Exam Questions III

Statistics & Probability

Permutation & Combination Tutorial Conditional Probability & Probability Tree Diagrams Probability & Venn Diagrams Probability Density Functions Correlation & Regression Continuous Random Variable: Mode, Mean & Variance Continuous Random Variable: Median, Quartiles & Percentiles Normal Distribution: Mean & Standard Deviation Normal Distribution: Probability in a given range Normal Distribution: Probability Example Using Normal Distribution to find Probabilities Using Normal Distribution as an Approximation to Binomial Distribution Cumulative Distribution Function Probability Distribution Table Discrete Random Variables Binomial Distribution Tutorial Binomial Distribution: Cumulative Probability Tables Poisson Approximation to the Binomial Distribution Binomial Distribution: Hypothesis Testing Binomial Distribution: Find Critical Values Poisson Distribution Tutorial Poisson Distribution: Hypothesis Testing Probability Exam Questions: Conditional Probability & Tree Diagrams Statistics Exam Questions: Correlation & Regression Statistics Exam Questions: Grouped Frequency Table Statistics Exam Questions: Normal Distribution Statistics Exam Questions: Discrete Probability Distribution

Mechanics Tutorial Force & Moments Mechanics Newton’s Law of Motion, F=ma Resolving Forces Splitting a Force into Components Mechanics Exam Questions Forces Mechanics Exam Questions Force Vectors Mechanics Exam Questions Gravity & Acceleration Inclined Planes Finding Coefficient of Friction & Missing Angles Conservation of Momentum Mechanics Exam Questions Impulse, Momentum Mechanics Exam Questions Moments I Mechanics Exam Questions Moments II Mechanics Exam Questions Velocity-Time Graphs Mechanics Exam Questions Connected Particles I Mechanics Exam Questions Connected Particles II Mechanics Exam Questions Connected Particles III Friction, Limiting Equilibrium, Coefficient of Friction Mechanics Exam Questions Friction Mechanics Exam Questions Position & Velocity Vectors Mechanics Exam Questions Velocity Vectors Kinematics SUVAT equations I Kinematics SUVAT equations II Kinematics SUVAT equations III Kinematics SUVAT equations IV Kinematics Vertical Motion I Kinematics Vertical Motion II Kinematics Acceleration-Time & Speed-Time Graphs I Kinematics Acceleration-Time & Speed-Time Graphs II Moments 1 Basic example of taking moments Equilibrium - taking moments Moments 2 Example of non-uniform rod Finding the centre of a non-uniform rod Hanging a non-uniform bar Moments 3 Taking moments at an angle Sum of moments about a point Hanging rod in equilibrium

Further Pure Mathematics

Proof by Induction I Summation of series, Divisibility Proof by Induction II Recurrence Relations, Matrices Matrices Dimensions, Adding & Subtracting Matrices Matrices Scalar, Multiplying Matrices Matrices Basic Linear Transformations Matrices Finding Linear Transformations Matrices Multiple Linear Transformations Matrices Inverse & Determinants Matrices Area of image using determinants Matrices 2x2 Matrix to solve simultaneous equations Summing Finite Series I with limits, with geometric series, proof by induction Summing Finite Series II logs, unknown coefficients, finding 2n

Complex Numbers 1 Adding & subtracting complex numbers Simplifying purely imaginary parts Quadratics with complex roots Complex Numbers 2 Powers of i Multiplying Complex Numbers Complex Numbers 3 Conjugates Dividing Complex numbers Solving Equation with conjugate Complex Numbers 4 Sum & product of roots Complex Numbers 5 Plotting Argand Diagram Complex Numbers 6 Argand Diagram modulus Argand Diagram parallelogram law Finding Modulus & Argument Basics Complex Numbers 7 Polar Form r(cos0 + isin0) Polar Form to rectangular form Complex Numbers 8 Comparing imaginary numbers Square roots of a complex number Complex Numbers 9 Finding a quadratic Finding a cubic Complex Numbers 10 Finding a quartic equation Quartic equation on Argand Diagram

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A-Level Edexcel Pure Maths Past Paper Questions by Topic

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Here you'll find pure maths past paper questions by topic that are applicable to the latest Edexcel specification. As with your exams, these questions cover every difficulty level from E to A* - we recommend not skipping the easier questions as they often trip students in exams. If you're stuggling, we recommend viewing our collection of A-Level Pure Maths Videos or even returning to the basics with our GCSE Maths content.

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Other resources, coordinate geometry, sequences and series, trigonometry, exponential and logarithms, differentiation, integration, numerical methods.

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As statistics, as mechanics.

• A-Level Pure
• A-Level Statistics
• A-Level Mechanics

AS Pure Mathematics

• Expanding Brackets
• Factorising
• Negative and Fractional Indices
• Question Set 1: Algebraic Expressions
• Solving Quadratics
• Completing the Square
• Sketching Quadratic Graphs
• The Discriminant
• Modelling with Quadratics
• Question Set 2: Quadratics
• Linear Simultaneous Equations
• Quadratic Simultaneous Equations
• Simultaneous Equations on Graphs
• Linear Inequalities
• Quadratic Inequalities
• Inequalities on Graphs
• Question Set 3: Equations and Inequalities
• Cubic Graphs
• Quartic Graphs
• Reciprocal Graphs
• Points of Intersection
• Translating Graphs
• Stretching Graphs
• Transforming Functions
• Question Set 4: Graphs and Transformations
• Equations of Straight Lines
• Parallel and Perpendicular Lines
• Length and Area
• Modelling with Straight Lines
• Question Set 5: Straight Line Graphs
• Midpoints and Perpendicular Bisectors
• Equation of a Circle
• Intersections of Straight Lines and Circles
• Use Tangent and Chord Properties
• Circles and Triangles
• Question Set 6: Circles
• Algebraic Fractions
• Dividing Polynomials
• The Factor Theorem
• Mathematical Proof
• Methods of Proof
• Question Set 7: Algebraic Methods
• Pascal's Triangle
• Factorial Notation
• The Binomial Expansion
• Solving Binomial Problems
• Binomial Estimation
• Question Set 8: The Binomial Expansion
• The Cosine Rule
• The Sine Rule
• Areas of Triangles
• Solving Triangle Problems
• Graphs of Sine, Cosine and Tangent
• Transforming Trigonometric Graphs
• Question Set 9: Trigonometric Ratios
• Angles in all four Quadrants
• Exact Values of Trigonometric Ratios
• Trigonometric Identities
• Simple Trigonometric Equations
• Harder Trigonometric Equations
• Equations and Identities
• Question Set 10: Trigonometric Identities and Equations
• Representing Vectors
• Magnitude and Direction
• Position Vectors
• Solving Geometric Problems
• Modelling with Vectors
• Question Set 11: Vectors
• Gradients of Curves
• Differentiation from first principles
• Differentiating x^n
• Differentiating Quadratics
• Differentiating functions with two or more terms
• Gradients, Tangents and Normals
• Increasing and Decreasing Functions
• Second Order Derivatives
• Stationary Points
• Sketching Gradient Functions
• Modelling with Differentiation
• Question Set 12: Differentiation
• Integrating x^n
• Indefinite Integrals
• Finding Functions
• Definite Integrals
• Areas under Curves
• Areas under the x-axis
• Areas between curves and lines
• Question Set 13: Integration
• Exponential Functions
• Exponential Modelling
• Laws of Logarithms
• Solving Equations using Logarithms
• Working with Natural Logarithms
• Logarithms and Non-Linear Data
• Question Set 14: Exponentials and Logarithms
• Populations and Samples
• Non-Random Sampling
• Types of Data
• The Large Data Set
• Question Set 1: Data Collection
• Measures of Central Tendency
• Other Measures of Location
• Measures of Spread
• Variance and Standard Deviation
• Question Set 2: Measures of Location and Spread
• Cumulative Frequency
• Comparing Data
• Question Set 3: Representations of Data
• Correlation
• Linear Regression
• Question Set 4: Correlation
• Calculating Probabilities
• Venn Diagrams
• Mutually Exclusive and Independent Events
• Tree Diagrams
• Question Set 5: Probability
• Probability Distributions
• The Binomial Distribution
• Cumulative Probabilities
• Question Set 6: Statistical Distribution
• Hypothesis Testing
• Finding Critical Values
• One-Tailed Tests
• Two-Tailed Tests
• Question Set 7: Hypothesis Testing
• Constructing a Model
• Modelling Assumptions
• Quantities and Units
• Working with Vectors
• Question Set 1: Modelling in Mechanics
• Displacement-Time Graphs
• Velocity-Time Graphs
• Constant Acceleration Formulae 1
• Constant Acceleration Formulae 2
• Vertical Motion Under Gravity
• Question Set 2: Constant Acceleration
• Force Diagrams
• Forces as Vectors
• Forces and Acceleration
• Motion in 2 Dimensions
• Connected Particles
• Question Set 3: Forces and Motion
• Functions of Time
• Using Differentiation
• Maxima and Minima Problems
• Using Integration
• Constant Acceleration Formulae
• Question Set 4: Variable Acceleration

A Level Pure Mathematics

• Proof by Contradiction
• Partial Fractions
• Repeated Factors
• Algebraic Division
• Question Set 1: Algebraic Methods
• The Modulus Function
• Functions and Mappings
• Composite Functions
• Inverse Functions
• y=|f(x)| and y=f(|x|)
• Combining Transformations
• Solving Modulus Problems
• Question Set 2: Functions and Graphs
• Arithmetic Sequences
• Arithmetic Series
• Geometric Sequences
• Geometric Series
• Sum to Infinity
• Sigma Notation
• Recurrence Relations
• Modelling with Series
• Question Set 3: Sequences and Series
• Expanding (1 + x)^n
• Expanding (a + bx)^n
• Using Partial Fractions
• Question Set 4: Binomial Expansion
• Radian Measure
• Areas of Sectors and Segments
• Solving Trigonometric Equations
• Small Angle Approximations
• Question Set 5: Radians
• Secant, Cosecant and Cotangent
• Graphs of Sec x, Cosec x and Cot x
• Using Sec x, Cosec x and Cot x
• Inverse Trigonometric Functions
• Question Set 6: Trigonometric Functions
• Addition Formulae
• Using the Angle Addition Formulae
• Double Angle Formulae
• Simplifying a cos x +- b sin x
• Proving Trigonometric Identities
• Modelling with Trigonometric Functions
• Question Set 7: Trigonometry and Modelling
• Parametric Equations
• Using Trigonometric Identities
• Curve Sketching
• Modelling with Parametric Equations
• Question Set 8: Parametric Equations
• Differentiating sin x and cos x
• Differentiating exponentials and logarithms
• The Chain Rule
• The Product Rule
• The Quotient Rule
• Differentiating Trigonometric Functions
• Parametric Differentiation
• Implicit Differentiation
• Using Second Derivatives
• Rates of Change
• Question Set 9: Differentiation
• Locating Roots
• The Newton-Raphson Method
• Applications to Modelling
• Question Set 10: Numerical Methods
• Integrating Standard Functions
• Integrating f(ax + b)
• Reverse Chain Rule
• Integration by Substitution
• Integration by Parts
• Finding Areas
• The Trapezium Rule
• Solving Differential Equations
• Modelling with Differential Equations
• Question Set 11: Integration
• 3D Coordinates
• Vectors in 3D
• Application to Mechanics
• Question Set 12: Vectors

A Level Statistics

• Exponential Models
• Measuring Correlation
• Hypothesis Testing for Zero Correlation
• Question Set 1: Regression, Correlation and Hypothesis Testing
• Set Notation
• Conditional Probability
• Conditional Probabilities in Venn Diagrams
• Probability Formulae
• Question Set 2: Conditional Probability
• The Normal Distribution
• Finding Probabilities for Normal Distributions
• The Inverse Normal Distribution Function
• The Standard Normal Distribution
• Finding the mean and standard deviation
• Approximating a Binomial Distribution
• Hypothesis Testing with the Normal Distribution
• Question Set 3: The Normal Distribution

A level Mechanics

• Resultant Moments
• Equilibrium
• Centres of Mass
• Question Set 1: Moments
• Resolving Forces
• Inclined Planes
• Question Set 2: Forces and Friction
• Horizontal Projection
• Horizontal and Vertical Components
• Projection at Any Angle
• Projctile Motion Formulae
• Question Set 3: Projectiles
• Static Particles
• Modelling with Statics
• Friction and Static Particles
• Static Rigid Bodies
• Dynamics and Inclined Planes
• Question Set 4: Applications of Forces
• Vectors in Kinematics
• Vector Methods with Projectiles
• Variable Acceleration in One Dimension
• Differentiating Vectors
• Integrating Vectors
• Question Set 5: Further Kinematics

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Algebra and Functions

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Understand your world with Maths and Science

Cheat Sheets || Questions by Topic || Worksheets

This topic is included in Paper 1 for AS-level Edexcel Maths and Papers 1 & 2 for A-level Edexcel Maths .

Cheat Sheets

• Ch.1 Algebraic Expressions
• Ch.2 Quadratics
• Ch.3 Equations and Inequalities
• Ch.4 Graphs and Transformations
• Ch.7 Algebraic Methods
• Ch.1 Algebraic Methods
• Ch.2 Functions and Graphs

Questions by Topic

Year 1 (2018-2021 papers), question papers & ms.

• Factor Theorem
• Inequalities
• Surds and Indices
• Transformations and Graphs

Model Answers

• Factor Theorem MA
• Inequalities MA
• Quadratics MA
• Surds and Indices MA
• Transformations and Graphs MA

Video Solutions

• Factor Theorem VS
• Inequalities VS
• Quadratics VS
• Surds and Indices VS
• Transformations and Graphs VS

Year 2 (2018-2021 papers)

• Composite and Inverse Functions
• Modulus of Functions
• Partial Fractions
• Composite & Inverse Functions MA
• Modulus of Functions MA
• Partial Fractions MA
• Composite and Inverse Functions VS
• Modulus of Functions VS
• Partial Fractions VS

Year 1 (pre-2018 papers)

• Remainder and Factor Theorem
• Simultaneous Equations

Year 2 (pre-2018 papers)

• Modulus Functions
• Rational Functions

These are Solomon Press worksheets. They were written for the outgoing specification but we have carefully selected ones which are relevant to the new specification.

• 01a. Indices
• 01b. Indices - Answers
• 02b. Surds - Answers
• 03a. Indices and surds − further questions
• 03b. Indices and surds − further questions - Answers
• 04a. Quadratics – the quadratic formula and the discriminant
• 04b. Quadratics – the quadratic formula and the discriminant - Answers
• 05a. Quadratics – completing the square
• 05b. Quadratics – completing the square - Answers
• 06a. Quadratics – factorising
• 06b. Quadratics – factorising - Answers
• 07a. Quadratics – further questions
• 07b. Quadratics – further questions - Answers
• 08a. Simultaneous equations
• 08b. Simultaneous equations - Answers
• 09a. Inequalities
• 09b. Inequalities - Answers
• 10a. Inequalities - further questions
• 10b. Inequalities - further questions - Answers
• 11a. Polynomials
• 11b. Polynomials - Answers
• 12a. Algebraic division, the factor theorem and the remainder theorem
• 12b. Algebraic division, the factor theorem and the remainder theorem - Answers
• 13a. The factor theorem and the remainder theorem − further questions
• 13b. The factor theorem and the remainder theorem − further questions - Answers
• 14a. Graphs of functions
• 14b. Graphs of functions - Answers
• 15a. Transformations of graphs
• 15b. Transformations of graphs - Answers
• 16a. Mixed exam-style questions on algebra
• 16b. Mixed exam-style questions on algebra - Answers
• 17a. Mixed exam-style questions on algebra
• 17b. Mixed exam-style questions on algebra - Answers
• 18a. Mixed exam-style questions on algebra
• 18b. Mixed exam-style questions on algebra - Answers
• 19a. Mixed exam-style questions on algebra
• 19b. Mixed exam-style questions on algebra - Answers
• 20a. Mixed exam-style questions on graphs of functions
• 20b. Mixed exam-style questions on graphs of functions - Answers
• 01a. Domain, range and values of functions
• 01b. Domain, range and values of functions - Answers
• 02a. Rational expressions
• 02b. Rational expressions - Answers
• 03a. Rational expressions − further questions
• 03b. Rational expressions − further questions - Answers
• 04a. The modulus function
• 04b. The modulus function - Answers
• 05a. Composite functions
• 05b. Composite functions - Answers
• 06a. Inverse of a function
• 06b. Inverse of a function - Answers
• 07a. Transformations of functions
• 07b. Transformations of functions - Answers
• 08a. Transformations of functions − further questions
• 08b. Transformations of functions − further questions - Answers
• 09a. Partial fractions
• 09b. Partial fractions - Answers
• 10a. Mixed exam-style questions on functions
• 10b. Mixed exam-style questions on functions - Answers
• 11a. Mixed exam-style questions on functions
• 11b. Mixed exam-style questions on functions- Answers
• 12a. Mixed exam-style questions on partial fractions
• 12b. Mixed exam-style questions on partial fractions - Answers
• 13a. Mixed exam-style questions on rational expressions
• 13b. Mixed exam-style questions on rational expressions - Answers

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GCSE, IGCSE, A-Level MAT, STEP and Maths Challenges

AS and Year 1 A-level Maths Exam Questions by Topic

Linked below you can find collections of exam questions for the new A-level syllabus, with the questions arranged according to the chapters of the most popular textbooks.

Note: I don’t currently have mark-schemes for these papers and I have currently abandoned the project to make video solutions available but will look to provide alternative resources to help in future. All the questions are from official and freely available past papers and so solutions to individual questions can be found at the websites of the relevant exam boards.

Get ready for A-level maths – a bundle of 5 short courses to help you master the end of GCSE+IGCSE maths and get started with A-level maths

• Indices, Surds and Logarithms
• Quadratic equations + inequalities (includes sketching general polynomials)
• Co-ordinate geometry + differentiation
• Trigonometry + trigonometric equations
• Sequences + series – arithmetic + geometric progressions

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Resources for A level Mathematics teachers

Home > Teachers > A level Mathematics > A level Mathematics resources

Transition to A level: Essential Skills

Our Essential Skills resources have been developed to support your students with the transition from GCSE to AS and A level Mathematics, and will be particularly useful during the autumn term. The y focus on developing six essential skills and explore how these are applied across a key topics in A level Mathematics. Included in each section are a skills check, a chance to ‘practise and explore’, and extension and enrichment ideas for further investigation.

Printable versions of the resources are available for you to give to students with limited access to technology.

Topic plans

The AS/A level Mathematics topic plans cover all the AS/A level Mathematics specifications and provide ready access to easy-to-use resources, along with identification of pre-requisite knowledge, areas where students struggle and suggested tasks, activities and questions for use with all your students.

Integral ® Resources

If your school/college has registered with the AMSP, you will be able to access resources for the Large Data Set as well as A level Further Mathematics in Integral ® using your registration account. A suite of A level Mathematics and Further Mathematics videos for use by students as part of a blended learning approach or to support remote learning is also available using the shared student account provided when schools and colleges register with the AMSP.

Where maths meets… the world of work!

These videos and activities are designed to use in class with your students. The activities all contain contextualised teaching points linked to careers and will help you to achieve Gatsby Benchmark 4 (linking careers to the curriculum). Some of the activities are written by employers and so are particularly authentic.

FMSP resources

The Further Mathematics Support Programme (FMSP) produced a wealth of resources to support the teaching and learning of A level Mathematics. They are available within the FMSP Resources Archive .

• A collection of A level Problem Solving Resources which includes material for use with both AS/A level Mathematics and AS/A level Further Mathematics students.
• Mechanics resources which include practical activities for Modelling with the Dynakar .

Schemes of work

MEI has produced some suggested schemes of work for teaching AS and A level Mathematics.

Using Desmos for the large data sets

These workbooks show how to use Desmos to investigate the large data sets, workbooks can be downloaded from the MEI website.

Four separate workbooks: one for each A level specification

• AQA: Using Desmos for the AQA large data set
• Edexcel: Using Desmos for the Edexcel large data set
• MEI: Using Desmos for the MEI large data set
• OCR: Using Desmos for the OCR large data set

Additional materials

Additional materials are available from:

• Underground Mathematics
• MEI/IET AS and A level mechanics practical videos
• The NRICH Mathematics Enrichment Project
• STEM Learning resource collection
• Mathcentre collection

Promotional materials

The AMSP is developing materials to help you to promote AS and A level Mathematics to your students and their parents/carers.

We’ve developed a 30 minute presentation for students and parents/carers about the importance of studying maths post-GCSE. It outlines the options available, and provides compelling reasons for choosing to take an advanced maths qualification.

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• The Number System and Place Value
• Calculations and Numerical Methods
• Fractions, Decimals, Percentages, Ratio and Proportion
• Properties of Numbers
• Patterns, Sequences and Structure
• Algebraic expressions, equations and formulae
• Coordinates, Functions and Graphs

Geometry and measure

• Angles, Polygons, and Geometrical Proof
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Working mathematically

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For younger learners

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Advanced Mathematical Problem Solving Resources

Introduction to Advanced Problem Solving

How to Use Advanced Problem Solving

Worked Examples

• STEP Support resources

Welcome to the NRICH Advanced Problem Solving pages.

This is an archive resource. For our current STEP preparation programme please visit maths.org/step . These resources are designed to help potential university applicants develop their advanced problem-solving skills and prepare for sitting examinations such as STEP . The material has been carefully selected to provide an accessible and supportive introduction to advanced problem-solving, and to help you build your confidence, fluency and speed. Start by reading about the materials , or scroll down to get started on the modules. The 24 modules below begin with general problem-solving skills and gradually introduce new techniques, ideas and advice to help you to solve advanced problems. You may also be interested in Stephen Siklos's book,  Advanced Problems in Mathematics: Preparing for University , which can be downloaded for free.

Advanced Problem Solving Module 1

Our first Advanced Problem Solving module provides a gentle introduction to being a problem-solver.

Advanced Problem Solving Module 2

The second Advanced Problem Solving module introduces you to mathematical notation and logical thinking.

Advanced Problem Solving Module 3

This Advanced Problem Solving module introduces some important aspects of mathematical proof.

Advanced Problem Solving Module 4

Our fourth module aims to give you a clearer understanding of the concepts underpinning calculus.

Advanced Problem Solving Module 5

Our fifth module explains some key calculus skills.

Advanced Problem Solving Module 6

In our sixth Advanced Problem Solving module, clever substitutions are the key to solving problems.

Advanced Problem Solving Module 7

This module contains advice and resources to help you prepare for university interviews.

Advanced Problem Solving Module 8

This module will help you to improve your trigonometry skills and understanding.

Advanced Problem Solving Module 9

In this module, you will learn about proof by induction and apply it to STEP questions.

Advanced Problem Solving Module 10

This module highlights the importance of integration.

Advanced Problem Solving Module 11

This module offers advice and practice on how to sketch graphs.

Advanced Problem Solving Module 12

This module looks at how to solve problems involving vectors.

Advanced Problem Solving Module 13

This module helps you to understand how to approach advanced geometry questions.

Advanced Problem Solving Module 14

This module explores what you need to know in order to tackle advanced Mechanics questions.

Advanced Problem Solving Module 15

This module looks at Statistics questions and what you need to know in order to solve them successfully.

Advanced Problem Solving Module 16

In this module, we explore how to tackle advanced Probability questions.

Advanced Problem Solving Module 17

This module offers advice and practice on dealing with questions involving Differential Equations.

Advanced Problem Solving Module 18

This module covers some techniques for dealing with complex numbers.

Advanced Problem Solving Module 19

This module offers some tricks and tips for dealing with questions involving equations and inequalities.

Advanced Problem Solving Module 20

This module looks at some of the things you need to understand about polar coordinates.

Advanced Problem Solving Module 21

This module offers advice on getting your head round "unusual" questions.

Advanced Problem Solving Module 22

This module looks at hyperbolic functions.

Advanced Problem Solving Module 23

This module looks at some of the key concepts involving series and sums.

Advanced Problem Solving Module 24

This module offers last-minute hints, advice and practice for preparing for advanced problem solving examinations.

A-LEVEL Maths

• Maths Categories

Browse The General Certificate of Education Advanced Level (A-Level) Maths Categories

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Problem solving guidance and practice - AS and A Level Maths

Subject: Mathematics

Age range: 16+

Resource type: Assessment and revision

Last updated

11 April 2022

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Guidance for students as to how to attack difficult and unfamiliar problems, and practice in doing so.

The resource includes:

• “Methods for attacking unfamiliar problems” - a collection of strategies that I (a pretty decent mathematician) use when tackling hard problems, laid out clearly for students
• Collections of tricky exam problems, organised by AS/A-level and by Pure/Statistics/Mechanics.
• An overview/instructions sheet so that the resource may be easily set for independent practice

Most of the problems have been sourced from OCR sample assessment materials, but align with any other A-level specification. Other problems were written by me.

The problems may be used as good revision/practice before exams, after spec content has been covered. The “methods” sheet may be introduced at any point in the year, to support students’ problem solving.

If using the problems in class, try not to help students with the maths! Instead, talk to them about their thought processes, direct them to relevant “methods” from the sheet, and encourage them to persevere.

pdf and editable docx versions included.

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Practice Problems

This page contains question sheets which are sent out to new students by many colleges before they arrive to start their undergraduate degree. These questions make suitable bridging material for students with single A-level Mathematics as they begin university - the material is partly revision, partly new material. All 11 sheets cover material relevant to the Mathematics, Mathematics & Statistics and Maths and Philosophy courses; sheets 8, 9 and 10 are not relevant to the Mathematics and Computer Science degree.

For each sheet the subject matter is briefly described, and there is some recommended reading material; the chapter numbers refer to the fourth edition of D.W.Jordan and P.Smith's book Mathematical Techniques, published by Oxford University Press in 2008.

• Sheet 1: Standard Functions and Techniques,  PDF Reading: §§ 1.3, 1.6-1.8, 1.10-1.16
• Sheet 2: Differentiation, PDF Reading: Chapter 2
• Sheet 3: Further Differentiation,  PDF Reading: §§ 3.1-3.5, 3.9-3.10
• Sheet 4: Applications of Differentiation, PDF Reading: §§ 4.1-4.4
• Sheet 5: Taylor Series, PDF Reading: §§ 5.1-5.4
• Sheet 6: Complex Numbers, PDF Reading: Chapter 6
• Sheet 7: Matrices, PDF Reading: Chapter 7
• Sheet 8: Vectors, PDF Reading: §§ 9.1-9.4, 9.6
• Sheet 9: The Scalar 'Dot' Product, PDF Reading: §§ 10.1-10.3, 10.9
• Sheet 10: The Vector 'Cross' Product, PDF Reading: §§ 11.1-11.2
• Sheet 11: Integration, PDF Reading: §§ 14.1-15.4, 15.8
• All the above 11 sheets as one file: PDF
• All the above 11 sheets as one webpage: Questions
• Induction 1: PDF Reading: R.B.J.T. Allenby Numbers and Proof , Chapter 7
• Induction 2:  PDF Reading: R.B.J.T. Allenby Numbers and Proof , Chapter 7
• Algebra 1: PDF Reading: No pre-requisites
• Algebra 2: PDF Reading: Chapters 7 and 8
• Calculus 1 - Curve Sketching: PDF Reading: §§ 4.1-4.4
• Calculus 2 - Numerical Methods and Estimation: PDF Reading: §4.6, §5.2
• Calculus 3 - Techniques of Integration: PDF Reading: §§17.5-17.7
• Calculus 4 - Differential Equations: PDF Reading: §§ 22.3-22.4, Chapter 18
• Calculus 5 - Further Differential Equations:  PDF Reading: Chapter 19, §22.5
• Complex Numbers: PDF Reading: Chapter 6
• Geometry: PDF Reading: §10.1, §10.9, §11.1, §16.1
• The second 11 sheets as one file:  PDF
• The second 11 sheets as one webpage: More challenging Questions
• Dynamics 1 - Basic Definitions. Newton's Second Law  PDF
• Dynamics 2 - Oscillations and Further Examples.  PDF
• These two sheets as one webpage: Further Sheets

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Free Math Worksheets — Over 100k free practice problems on Khan Academy

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High school statistics, ap®︎/college statistics, precalculus, differential calculus, integral calculus, ap®︎/college calculus ab, ap®︎/college calculus bc, multivariable calculus, differential equations, linear algebra.

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• Prepare for the 2020 AP®︎ Statistics Exam
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• Derivatives: definition and basic rules
• Derivatives: chain rule and other advanced topics
• Applications of derivatives
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• Differentiation: definition and basic derivative rules
• Differentiation: composite, implicit, and inverse functions
• Contextual applications of differentiation
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• Integration and accumulation of change
• Applications of integration
• AP Calculus AB solved free response questions from past exams
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• Integrals review
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• Green’s, Stokes’, and the divergence theorems
• First order differential equations
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• Laplace transform
• Vectors and spaces
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• Alternate coordinate systems (bases)

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Watch CBS News

Teens come up with trigonometry proof for Pythagorean Theorem, a problem that stumped math world for centuries

By Bill Whitaker

May 5, 2024 / 7:00 PM EDT / CBS News

As the school year ends, many students will be only too happy to see math classes in their rearview mirrors. It may seem to some of us non-mathematicians that geometry and trigonometry were created by the Greeks as a form of torture, so imagine our amazement when we heard two high school seniors had proved a mathematical puzzle that was thought to be impossible for 2,000 years. 

We met Calcea Johnson and Ne'Kiya Jackson at their all-girls Catholic high school in New Orleans. We expected to find two mathematical prodigies.

Instead, we found at St. Mary's Academy , all students are told their possibilities are boundless.

Come Mardi Gras season, New Orleans is alive with colorful parades, replete with floats, and beads, and high school marching bands.

In a city where uniqueness is celebrated, St. Mary's stands out – with young African American women playing trombones and tubas, twirling batons and dancing - doing it all, which defines St. Mary's, students told us.

Junior Christina Blazio says the school instills in them they have the ability to accomplish anything. 

Christina Blazio: That is kinda a standard here. So we aim very high - like, our aim is excellence for all students. 

The private Catholic elementary and high school sits behind the Sisters of the Holy Family Convent in New Orleans East. The academy was started by an African American nun for young Black women just after the Civil War. The church still supports the school with the help of alumni.

In December 2022, seniors Ne'Kiya Jackson and Calcea Johnson were working on a school-wide math contest that came with a cash prize.

Ne'Kiya Jackson: I was motivated because there was a monetary incentive.

Calcea Johnson: 'Cause I was like, "$500 is a lot of money. So I-- I would like to at least try."

Both were staring down the thorny bonus question.

Bill Whitaker: So tell me, what was this bonus question?

Calcea Johnson: It was to create a new proof of the Pythagorean Theorem. And it kind of gave you a few guidelines on how would you start a proof.

The seniors were familiar with the Pythagorean Theorem, a fundamental principle of geometry. You may remember it from high school: a² + b² = c². In plain English, when you know the length of two sides of a right triangle, you can figure out the length of the third.

Both had studied geometry and some trigonometry, and both told us math was not easy. What no one told  them  was there had been more than 300 documented proofs of the Pythagorean Theorem using algebra and geometry, but for 2,000 years a proof using trigonometry was thought to be impossible, … and that was the bonus question facing them.

Bill Whitaker: When you looked at the question did you think, "Boy, this is hard"?

Ne'Kiya Jackson: Yeah. 

Bill Whitaker: What motivated you to say, "Well, I'm going to try this"?

Calcea Johnson: I think I was like, "I started something. I need to finish it." 

Bill Whitaker: So you just kept on going.

Calcea Johnson: Yeah.

For two months that winter, they spent almost all their free time working on the proof.

CeCe Johnson: She was like, "Mom, this is a little bit too much."

CeCe and Cal Johnson are Calcea's parents.

CeCe Johnson:   So then I started looking at what she really was doing. And it was pages and pages and pages of, like, over 20 or 30 pages for this one problem.

Cal Johnson: Yeah, the garbage can was full of papers, which she would, you know, work out the problems and-- if that didn't work she would ball it up, throw it in the trash. 

Bill Whitaker: Did you look at the problem? 

Neliska Jackson is Ne'Kiya's mother.

Neliska Jackson: Personally I did not. 'Cause most of the time I don't understand what she's doing (laughter).

Michelle Blouin Williams: What if we did this, what if I write this? Does this help? ax² plus ….

Their math teacher, Michelle Blouin Williams, initiated the math contest.

Bill Whitaker: And did you think anyone would solve it?

Michelle Blouin Williams: Well, I wasn't necessarily looking for a solve. So, no, I didn't—

Bill Whitaker: What were you looking for?

Michelle Blouin Williams: I was just looking for some ingenuity, you know—

Calcea and Ne'Kiya delivered on that! They tried to explain their groundbreaking work to 60 Minutes. Calcea's proof is appropriately titled the Waffle Cone.

Calcea Johnson: So to start the proof, we start with just a regular right triangle where the angle in the corner is 90°. And the two angles are alpha and beta.

Bill Whitaker: Uh-huh

Calcea Johnson: So then what we do next is we draw a second congruent, which means they're equal in size. But then we start creating similar but smaller right triangles going in a pattern like this. And then it continues for infinity. And eventually it creates this larger waffle cone shape.

Calcea Johnson: Am I going a little too—

Bill Whitaker: You've been beyond me since the beginning. (laughter) 

Bill Whitaker: So how did you figure out the proof?

Ne'Kiya Jackson: Okay. So you have a right triangle, 90° angle, alpha and beta.

Bill Whitaker: Then what did you do?

Ne'Kiya Jackson: Okay, I have a right triangle inside of the circle. And I have a perpendicular bisector at OP to divide the triangle to make that small right triangle. And that's basically what I used for the proof. That's the proof.

Bill Whitaker: That's what I call amazing.

Ne'Kiya Jackson: Well, thank you.

There had been one other documented proof of the theorem using trigonometry by mathematician Jason Zimba in 2009 – one in 2,000 years. Now it seems Ne'Kiya and Calcea have joined perhaps the most exclusive club in mathematics. 

Bill Whitaker: So you both independently came up with proof that only used trigonometry.

Ne'Kiya Jackson: Yes.

Bill Whitaker: So are you math geniuses?

Calcea Johnson: I think that's a stretch. 

Bill Whitaker: If not genius, you're really smart at math.

Ne'Kiya Jackson: Not at all. (laugh) 

To document Calcea and Ne'Kiya's work, math teachers at St. Mary's submitted their proofs to an American Mathematical Society conference in Atlanta in March 2023.

Ne'Kiya Jackson: Well, our teacher approached us and was like, "Hey, you might be able to actually present this," I was like, "Are you joking?" But she wasn't. So we went. I got up there. We presented and it went well, and it blew up.

Bill Whitaker: It blew up.

Calcea Johnson: Yeah. 

Ne'Kiya Jackson: It blew up.

Bill Whitaker: Yeah. What was the blowup like?

Calcea Johnson: Insane, unexpected, crazy, honestly.

It took millenia to prove, but just a minute for word of their accomplishment to go around the world. They got a write-up in South Korea and a shout-out from former first lady Michelle Obama, a commendation from the governor and keys to the city of New Orleans. 

Bill Whitaker: Why do you think so many people found what you did to be so impressive?

Ne'Kiya Jackson: Probably because we're African American, one. And we're also women. So I think-- oh, and our age. Of course our ages probably played a big part.

Bill Whitaker: So you think people were surprised that young African American women, could do such a thing?

Calcea Johnson: Yeah, definitely.

Ne'Kiya Jackson: I'd like to actually be celebrated for what it is. Like, it's a great mathematical achievement.

Achievement, that's a word you hear often around St. Mary's academy. Calcea and Ne'Kiya follow a long line of barrier-breaking graduates. 

The late queen of Creole cooking, Leah Chase , was an alum. so was the first African-American female New Orleans police chief, Michelle Woodfork …

And judge for the Fifth Circuit Court of Appeals, Dana Douglas. Math teacher Michelle Blouin Williams told us Calcea and Ne'Kiya are typical St. Mary's students.  

Bill Whitaker: They're not unicorns.

Michelle Blouin Williams: Oh, no no. If they are unicorns, then every single lady that has matriculated through this school is a beautiful, Black unicorn.

Pamela Rogers: You're good?

Pamela Rogers, St. Mary's president and interim principal, told us the students hear that message from the moment they walk in the door.

Pamela Rogers: We believe all students can succeed, all students can learn. It does not matter the environment that you live in. 

Bill Whitaker: So when word went out that two of your students had solved this almost impossible math problem, were they universally applauded?

Pamela Rogers: In this community, they were greatly applauded. Across the country, there were many naysayers.

Bill Whitaker: What were they saying?

Pamela Rogers: They were saying, "Oh, they could not have done it. African Americans don't have the brains to do it." Of course, we sheltered our girls from that. But we absolutely did not expect it to come in the volume that it came.  

Bill Whitaker: And after such a wonderful achievement.

Pamela Rogers: People-- have a vision of who can be successful. And-- to some people, it is not always an African American female. And to us, it's always an African American female.

Gloria Ladson-Billings: What we know is when teachers lay out some expectations that say, "You can do this," kids will work as hard as they can to do it.

Gloria Ladson-Billings, professor emeritus at the University of Wisconsin, has studied how best to teach African American students. She told us an encouraging teacher can change a life.

Bill Whitaker: And what's the difference, say, between having a teacher like that and a whole school dedicated to the excellence of these students?

Gloria Ladson-Billings: So a whole school is almost like being in Heaven. 

Bill Whitaker: What do you mean by that?

Gloria Ladson-Billings: Many of our young people have their ceilings lowered, that somewhere around fourth or fifth grade, their thoughts are, "I'm not going to be anything special." What I think is probably happening at St. Mary's is young women come in as, perhaps, ninth graders and are told, "Here's what we expect to happen. And here's how we're going to help you get there."

At St. Mary's, half the students get scholarships, subsidized by fundraising to defray the $8,000 a year tuition. Here, there's no test to get in, but expectations are high and rules are strict: no cellphones, modest skirts, hair must be its natural color.

Students Rayah Siddiq, Summer Forde, Carissa Washington, Tatum Williams and Christina Blazio told us they appreciate the rules and rigor.

Rayah Siddiq: Especially the standards that they set for us. They're very high. And I don't think that's ever going to change.

Bill Whitaker: So is there a heart, a philosophy, an essence to St. Mary's?

Summer Forde: The sisterhood—

Carissa Washington: Sisterhood.

Tatum Williams: Sisterhood.

Bill Whitaker: The sisterhood?

Voices: Yes.

Bill Whitaker: And you don't mean the nuns. You mean-- (laughter)

Christina Blazio: I mean, yeah. The community—

Bill Whitaker: So when you're here, there's just no question that you're going to go on to college.

Rayah Siddiq: College is all they talk about. (laughter) 

Pamela Rogers: … and Arizona State University (Cheering)

Principal Rogers announces to her 615 students the colleges where every senior has been accepted.

Bill Whitaker: So for 17 years, you've had a 100% graduation rate—

Pamela Rogers: Yes.

Bill Whitaker: --and a 100% college acceptance rate?

Pamela Rogers: That's correct.

Last year when Ne'Kiya and Calcea graduated, all their classmates went to college and got scholarships. Ne'Kiya got a full ride to the pharmacy school at Xavier University in New Orleans. Calcea, the class valedictorian, is studying environmental engineering at Louisiana State University.

Bill Whitaker: So wait a minute. Neither one of you is going to pursue a career in math?

Both: No. (laugh)

Calcea Johnson: I may take up a minor in math. But I don't want that to be my job job.

Ne'Kiya Jackson: Yeah. People might expect too much out of me if (laugh) I become a mathematician. (laugh)

But math is not completely in their rear-view mirrors. This spring they submitted their high school proofs for final peer review and publication … and are still working on further proofs of the Pythagorean Theorem. Since their first two …

Calcea Johnson: We found five. And then we found a general format that could potentially produce at least five additional proofs.

Bill Whitaker: And you're not math geniuses?

Bill Whitaker: I'm not buying it. (laughs)

Produced by Sara Kuzmarov. Associate producer, Mariah B. Campbell. Edited by Daniel J. Glucksman.

Bill Whitaker is an award-winning journalist and 60 Minutes correspondent who has covered major news stories, domestically and across the globe, for more than four decades with CBS News.

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History Daily

New Orleans teens solve 2,000-year-old math problem

Posted: May 11, 2024 | Last updated: May 11, 2024

Some ingenuity

Teacher Michelle Blouin Williams initiated a math competition with a bonus question tasking students to develop a new proof for the Pythagorean theorem using trigonometry, without anticipating that anyone would successfully tackle the challenge.  

She said, “I was just looking for some ingenuity.”   

Williams' expectations

Calcea Johnson and Ne’Kiya Jackson exceeded Williams' expectations by successfully solving the challenge in 2023.   

St. Mary's Academy

These two teenagers, who were seniors at St. Mary's Academy in New Orleans, a renowned Catholic school for girls with impressive college acceptance and graduation rates, were featured on CBS News' "60 Minutes" to discuss their accomplishment.   

$500 reward

Initially enticed by the math competition's $500 reward, their determination to complete the task intensified as they delved into the complex bonus question.   

Over a period of two months, these high school seniors dedicated themselves to completing their proof.  

Pages and pages

During an interview with "60 Minutes," CeCe Johnson, the mother of Calcea, said “It was pages and pages and pages of, like, over 20 or 30 pages for this one problem.”  

The garbage can

Her father, Cal Johnson, added, “Yeah, the garbage can was full of papers, which she would, you know, work out the problems and — if that didn’t work she would ball it up, throw it in the trash.”  

Upon completion

Upon completion, the teachers at St. Mary's Academy acknowledged the significance of Calcea and Ne'Kiya's achievement and submitted their proof to the American Mathematical Society for recognition at a conference in March 2023, where the students presented their findings.  

The Pythagorean theorem

The Pythagorean theorem, at its core, states that knowing the lengths of two sides of a right triangle allows you to determine the length of the third side using the formula a² + b² = c².   

While commonly attributed to the Greek mathematician Pythagoras, historical evidence suggests its existence in Babylon and Iron Age India. This theorem finds practical applications in construction, architecture, navigation, and surveying.  

A mathematical proof

A mathematical proof is a logical argument that demonstrates the truth of a mathematical theorem. American mathematician Daniel Kane likens proofs to essays, but rooted in mathematical concepts.  

Using trigonometry

As per the "60 Minutes" segment, “there had been more than 300 documented proofs of the Pythagorean Theorem using algebra and geometry, but for 2,000 years a proof using trigonometry was thought to be impossible.”  

Mathematician Elisha Loomis

Back in 1927, mathematician Elisha Loomis made a statement in his book "The Pythagorean Proposition." Loomis contended that a trigonometric proof of the theorem was impossible as it would create a circular argument.  

Stuart Anderson

Stuart Anderson, a retired mathematics professor from Texas A&M University-Commerce, mentioned to Scientific American, “A lot of the basic trig ‘identities’ are nothing more than Pythagoras’ theorem.”  

Trigonometric functions

Loomis contended that attempting to use trigonometric functions to prove the Pythagorean theorem would result in a circular reasoning loop, as the functions themselves are built upon the theorem. This, according to Loomis, would constitute a fundamental mathematical mistake.  

The law of sines

As reported by Scientific American, the teenagers challenged this notion during their presentation in 2023 and stated that “a trigonometric identity called the law of sines didn’t depend on the Pythagorean theorem and that they could use it to prove the theorem.”  

Final peer review

Calcea and Ne'Kiya are now part of a select few who have achieved a similar milestone, such as mathematician Jason Zimba, who developed a new proof in 2009. They submitted their proof for the final peer review earlier this year and are actively working on crafting additional proofs.  

In recognition

In recognition of their accomplishment, the teenagers were honored with the keys to the city of New Orleans and received commendations from the governor of Louisiana, along with other public acknowledgments.  

Widespread recognition

Despite the widespread recognition of their accomplishment which “blew up,” as Ne'Kiya expressed it, the two students maintain their humility and even chuckled at being labeled as geniuses.  

Surprised and skeptical

Upon the revelation of their achievement, there were individuals who appeared surprised and skeptical, dismissing the news as untrue, as mentioned by St. Mary's president Pamela Rogers during the interview.  

African Americans

“They were saying, ‘Oh, they could not have done it. African Americans don’t have the brains to do it.’ ... People — have a vision of who can be successful. And — to some people, it is not always an African American female. And to us, it’s always an African American female.”  

The significant reaction

When questioned by interviewer Bill Whitaker about their thoughts on the significant reaction, Ne’Kiya said, “Probably because we’re African American, one. And we’re also women. So I think — oh, and our age. Of course our ages probably played a big part.”  

Great mathematical achievement

“I’d like to actually be celebrated for what it is. Like, it’s a great mathematical achievement,” she continued.  

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