PAPA Cambridge: AS/Alevel

The following collection of worksheets cover the realms of Pure Mathematics, Statistics, and Discrete Mathematics at both AS and A-level (including Further Mathematics) and can be downloaded and photocopied as appropriate.
Some of the worksheets are VERY extensive in their coverage of the underlying topic(s).

A.
Application of calculus: stationary points, related rates of change, volumes.

Arrangements: basic counting, permutations, combinations (extensive!)
Arrangements: exam questions.

B.
Binomial expansions: positive integer index only.

C.
Central limit theorem (includes binomial and poisson distributions).

Circular measures: arcs & sectors practice.
Circular measures: exam practice.
Circular measures: radians, arcs, sectors etc. (quite extensive).

Complex numbers: basic practice. Includes conjugates, modulus argument form etc.
Complex numbers: De Moivre's theorem; nth roots, trig. id's., series etc.
Complex numbers: loci.

Confidence intervals: population means (using the Central limit).

Continuous random variables: (including means and variances).

Co-ordinate geometry (all the usual! - Quite extensive.) Updated.
Co-ordinate geometry: circles (finding equations etc.)

Co-ordinate geometry: tangents & normals to circles (does not require differentiation!)

Correlation: product moment and Spearmans rank correlation.

D.
Data representation: cumulative frequencies, histograms etc.

Differential equations: a worked example.
Differential equations: two difficult worked examples (Further Mathematics!)
Differential equations: forming and solving (includes related rates of change).
Differential equations: integrating factors.
Differential equations: separating the variables.

Second order differential equations: a worked example (with full solution).
Second order differential equations: linear, constant coefficients.

Differentiation (quite an extensive first exercise).
Differentiation: chain / product / quotient rules. Parametric & implicit differentiation.
Differentiation: an updated version of the above sheet.

Differentiation: implicit and inverse differentiation.
Differentiation: Further Mathematics; inverse funtions, hyperbolic functions etc.

Discrete random variables: expectation and variance.
Discrete random variables: binomial distributions.
Discrete random variables: geometric distributions.
Discrete random variables: binomial and geometric distributions.
Discrete random variables: poisson distributions (includes various approximations).

E.
Errors in measurements: absolute and relative errors.

F.
Functions: domains, ranges, one-one, inverses etc. Updated.

G.
Graph theory: basic theory. Includes Eulerian and Planar graphs.

Group theory: algebra in groups.
Group theory: basic group theory.
Group theory: a list of the basic isomorphism classes.
Group theory: subgroups and cyclic groups.

H.
Hyperbolic functions.

Hypothesis testing: binomial and poisson distributions; large / small samples; type I / II errors etc. (very extensive).
Hypothesis testing: normal distribution questions (using Central limit).
Hypothesis testing: type I / II errors (normal distribution and large sample binomials).

I.
Indices and surds.

Applications of integration: arc lengths and surface areas of rev. Cartesian and parametric equations.
Applications of integration (more extensive version of above sheet - includes approximating areas).
Integration: areas between graphs.
Integration (quite an extensive first exercise).
Integration by substitution.
Integration: (involves inverse trig. functions etc.)
Integration: an 'advanced' worksheet using mostly substitution.

Iteration (involves some radian measure).
Iteration: staircase / cobweb diagrams. Error terms. Newton-Raphson method.

L.
Linear programming: graphical solutions.
Linear programming: the Simplex algorithm.

Logarithms: logarithms and exponential growth / decay.

M.
Matrices: 2 by 2 matrices.
Matrices: 3 by 3 matrices.
Matrices: 2 by 2 and 3 by 3 matrices. Determinants, adjoints, inverses and transformations.

Modulus functions.

Modulus functions & basic graphs.

N.
Networks: minimum connector problems (Prim's and Kruskal's algorithms).
Networks: route inspection problems (the Chinese postman algorithm).
Networks: shortest route problems (Djikstra's algorithm).

Normal distribution: basic questions.
Normal distribution homework: exam questions.

Numerical solution of differential equations: Eulers method (including his modified method).

P.
Polynomial functions: factor theorem etc.

Probability (includes tree diagrams and conditional probability).

Q.
Quadratic functions (very extensive!)

R.
Rational functions and asymptotes.

Reduction formulae.

S.
Sample proportions: confidence intervals and hypothesis testing of binomial proportions (includes type I / II errors).

Sequences and series: A.P.'s, G.P.'s, sigma notation, covergence etc. (very extensive!)
Sequences, series and induction.
Series expansions: binomial expansions with fractional indices. Maclaurin series.

Stationary values (a few questions.)

T.
Trigonometry: no radians. Graphs, equations, Pythagoras etc. (very extensive!)
Trigonometrical functions: identities, equations, compound angle formulae etc. Includes some radians. (Very extensive!)

V.
Vectors: (Quite extensive. Includes lines in 3-d and scalar product etc.)
Vectors and plane geometry. (Thoroughly comprehensive coverage!)

Pure Mathematics	Statistics	Mechanics
Chapters: Contents Preface 1. Nature of Mathematics 2. Using Graphs 3. Functions 4. Graph Transforms 5. Solving Problems 6. Extending Algebra 7. Straight Lines 8. Rates of Change 9. Powers 10. Trigonometry 11. Growth and Decay 12. Integration 13. Sequences and Series 14. Further Calculus 15. Further Trigonometry 16. Data Analysis 17. Cartesian Geometry 18. Even more Calculus 19. Iteration 20. Probability 21. Mathematical Modelling Appendix Answers P Index	Chapters: Contents Preface 1. Probability 2. Data Collection 3. Descriptive Statistics 4. Discrete Probaility Distributions 5. Binomial Distribution 6. Poisson Distributions 7. Continuous Probability Distributions 8. The Normal Distribution 9. Estimation 10. Hypothesis Testing 11. Chi-Squared 12. Correlation and Regression Answers P Index	Chapters: Contents Preface 1. Modelling and Mechanics 2. One-Dimensional Motion 3. Vectors 1 4. Vectors 2 5. Projectiles 6. Work and Energy 7. Circular Motion 8. Simple Harmonic Motion 9. Physical Structures Answers P Index
Further Pure Mathematics	Further Statistics	Discrete Mathematics
Chapters: Contents Preface 1. Trigonometry 2. Hyperbolic Functions 3. Complex Numbers 4. Vectors 5. Vector Geometry 6. Sequences and Series 7. Properties of Curves 8. Further Calculus 9. Matrices and Transformations Answers P Index	Chapters: Contents Preface 1. Continuous Probability Distributions 2. Estimation 3. Hypothesis Testing: One Sample Tests 4. Hypothesis Testing: Two Sample Tests 5. Goodness of Fit Tests 6. Experimental Design 7. Analysis of Variance (Anova) 8. Statistical Process Control 9. Acceptance Sampling Answers P Index	Chapters: Contents Preface 1. Graphs 2. Travel Problems 3. Enumeration 4. Inequalities 5. Linear Progamming 6. Planar Graphs 7. Network Flows 8. Codes in Everyday Use 9. Theory of Codes 10. Logic 11. Boolean Algebra 12. Critical Path Analysis 13. Scheduling 14. Difference Equations 1 15. Difference Equations 2 Appendix Answers P Index

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