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OCR A-level Further Maths A questions by topic
Original topic questions for OCR A-level Further Maths A (H245), set out under the sections of OCR's Further Mathematics A specification from Proof to Further Calculus, in the order it prints them. Every question has a worked solution, and every set comes as a printable PDF.
Other specifications: Edexcel A-level Maths · Edexcel A-level Further Maths · AQA A-level Maths · AQA A-level Further Maths · OCR A-level Maths A · OCR MEI A-level Maths · every topic, by unit
4 Pure Core
4.01 Proof45 questions3 topics
- Proof by induction: sums and series15 questions · 64 marks · 4.01a, 4.01b · PDF, 74 KB
- Induction: divisibility and matrices15 questions · 71 marks · 4.01a, 4.01b · PDF, 55 KB
- Leibniz's theorem and the Weierstrass substitution15 questions · 60 marks · 4.01b · PDF, 75 KB
4.02 Complex Numbers136 questions9 topics
- Complex arithmetic and the Argand diagram15 questions · 56 marks · 4.02a, 4.02b, 4.02c, 4.02e and 5 more · PDF, 55 KB
- Modulus, argument and loci16 questions · 67 marks · 4.02a, 4.02b, 4.02c, 4.02e and 3 more · PDF, 79 KB
- Further loci and regions in the Argand diagram15 questions · 64 marks · 4.02c, 4.02o, 4.02p · PDF, 55 KB
- Transformations of the complex plane15 questions · 66 marks · 4.02c · PDF, 56 KB
- De Moivre's theorem and trigonometric identities15 questions · 60 marks · 4.02d, 4.02m, 4.02n, 4.02q · PDF, 77 KB
- Roots of unity and complex roots15 questions · 69 marks · 4.02d, 4.02h, 4.02r, 4.02s · PDF, 77 KB
- Roots of polynomials15 questions · 68 marks · 4.02g, 4.02j · PDF, 77 KB
- Quadratic functions15 questions · 56 marks · 4.02p · PDF, 54 KB
- Simultaneous equations and inequalities15 questions · 60 marks · 4.02p · PDF, 53 KB
4.03 Matrices91 questions6 topics
- Matrix algebra and transformations16 questions · 65 marks · 4.03a, 4.03b, 4.03c, 4.03d and 6 more · PDF, 82 KB
- Determinants and inverses15 questions · 53 marks · 4.03a, 4.03h, 4.03i, 4.03j and 7 more · PDF, 74 KB
- Induction: divisibility and matrices15 questions · 71 marks · 4.03b · PDF, 55 KB
- Groups and their axioms15 questions · 62 marks · 4.03c · PDF, 55 KB
- Systems of equations and invariance15 questions · 64 marks · 4.03g, 4.03r, 4.03s, 4.03t · PDF, 76 KB
- Lines and planes in three dimensions15 questions · 61 marks · 4.03t · PDF, 75 KB
4.04 Further Vectors90 questions6 topics
- Straight lines15 questions · 56 marks · 4.04a · PDF, 54 KB
- Vectors in two dimensions15 questions · 56 marks · 4.04a · PDF, 74 KB
- Lines and planes in three dimensions15 questions · 61 marks · 4.04a, 4.04b, 4.04c, 4.04e · PDF, 75 KB
- Vectors in three dimensions15 questions · 57 marks · 4.04c · PDF, 73 KB
- Intersections, angles and distances15 questions · 58 marks · 4.04c, 4.04d, 4.04f, 4.04h and 2 more · PDF, 77 KB
- The vector product and the scalar triple product15 questions · 56 marks · 4.04g, 4.04h, 4.04i · PDF, 78 KB
4.05 Further Algebra60 questions4 topics
- Roots of polynomials15 questions · 68 marks · 4.05a, 4.05b · PDF, 77 KB
- Polynomials and the factor theorem15 questions · 56 marks · 4.05c · PDF, 54 KB
- Partial fractions15 questions · 60 marks · 4.05c · PDF, 52 KB
- Calculus with inverse trigonometric functions15 questions · 59 marks · 4.05c · PDF, 74 KB
4.06 Series75 questions5 topics
- Sequences and sigma notation15 questions · 57 marks · 4.06a · PDF, 74 KB
- Proof by induction: sums and series15 questions · 64 marks · 4.06a · PDF, 74 KB
- Summing series and the method of differences15 questions · 67 marks · 4.06a, 4.06b · PDF, 77 KB
- Partial fractions15 questions · 60 marks · 4.06b · PDF, 52 KB
- Geometric series15 questions · 59 marks · 4.06b · PDF, 55 KB
4.07 Hyperbolic Functions30 questions2 topics
- Hyperbolic functions and identities15 questions · 60 marks · 4.07a, 4.07b, 4.07c, 4.07e and 1 more · PDF, 74 KB
- Calculus with hyperbolic functions15 questions · 62 marks · 4.07d · PDF, 53 KB
4.07 Hyperbolic Functions, 4 Pure Core, on the specification map
4.08 Further Calculus121 questions8 topics
- Maclaurin series15 questions · 60 marks · 4.08a, 4.08b · PDF, 73 KB
- The general binomial expansion15 questions · 70 marks · 4.08b · PDF, 56 KB
- Mean values and improper integrals15 questions · 60 marks · 4.08c, 4.08e · PDF, 75 KB
- Volumes of revolution16 questions · 65 marks · 4.08d · PDF, 81 KB
- Integrating rational functions15 questions · 62 marks · 4.08f · PDF, 73 KB
- Calculus with inverse trigonometric functions15 questions · 59 marks · 4.08f, 4.08g, 4.08h · PDF, 74 KB
- Hyperbolic functions and identities15 questions · 60 marks · 4.08g · PDF, 74 KB
- Calculus with hyperbolic functions15 questions · 62 marks · 4.08g, 4.08h · PDF, 53 KB
4.08 Further Calculus, 4 Pure Core, on the specification map
4.09 Polar Coordinates31 questions2 topics
- Polar curves16 questions · 66 marks · 4.09a, 4.09b · PDF, 93 KB
- Areas with polar coordinates15 questions · 65 marks · 4.09c · PDF, 76 KB
4.09 Polar Coordinates, 4 Pure Core, on the specification map
4.10 Differential Equations136 questions9 topics
- Solving differential equations12 questions · 56 marks · 4.10a · PDF, 51 KB
- First order equations and integrating factors15 questions · 68 marks · 4.10a, 4.10c · PDF, 56 KB
- Second order equations18 questions · 85 marks · 4.10a, 4.10d, 4.10e · PDF, 78 KB
- Rates of change and building differential equations15 questions · 59 marks · 4.10b · PDF, 81 KB
- Kinematics with variable acceleration15 questions · 66 marks · 4.10b · PDF, 80 KB
- Modelling with differential equations15 questions · 68 marks · 4.10b, 4.10f, 4.10g, 4.10h · PDF, 78 KB
- Newton's laws with a variable force15 questions · 65 marks · 4.10b · PDF, 80 KB
- Compound angles and the harmonic form15 questions · 62 marks · 4.10f · PDF, 75 KB
- Simple harmonic motion16 questions · 78 marks · 4.10f · PDF, 84 KB
4.10 Differential Equations, 4 Pure Core, on the specification map
5 Statistics
5.01 Probability45 questions3 topics
- Probability and Venn diagrams15 questions · 56 marks · 5.01a · PDF, 75 KB
- Conditional probability15 questions · 52 marks · 5.01a · PDF, 75 KB
- Combinatorics15 questions · 56 marks · 5.01a, 5.01b · PDF, 54 KB
5.02 Discrete Random Variables92 questions6 topics
- The binomial distribution15 questions · 52 marks · 5.02a, 5.02d, 5.02e · PDF, 54 KB
- Discrete random variables and expectation15 questions · 57 marks · 5.02a, 5.02b, 5.02c, 5.02d and 2 more · PDF, 76 KB
- Measures of location and spread16 questions · 59 marks · 5.02c · PDF, 82 KB
- The Poisson distribution15 questions · 65 marks · 5.02d, 5.02i, 5.02j, 5.02k and 3 more · PDF, 77 KB
- Goodness-of-fit tests16 questions · 67 marks · 5.02e · PDF, 85 KB
- Geometric and negative binomial distributions15 questions · 57 marks · 5.02f, 5.02g, 5.02h · PDF, 58 KB
5.02 Discrete Random Variables, 5 Statistics, on the specification map
5.03 Continuous Random Variables75 questions5 topics
- The normal distribution15 questions · 56 marks · 5.03a · PDF, 75 KB
- Continuous random variables: density and distribution functions15 questions · 61 marks · 5.03a, 5.03b, 5.03e, 5.03f and 1 more · PDF, 78 KB
- Mean, variance and skewness of continuous variables15 questions · 63 marks · 5.03a, 5.03c, 5.03d, 5.03f and 1 more · PDF, 78 KB
- The continuous uniform distribution15 questions · 67 marks · 5.03a · PDF, 56 KB
- Discrete random variables and expectation15 questions · 57 marks · 5.03d · PDF, 76 KB
5.03 Continuous Random Variables, 5 Statistics, on the specification map
5.04 Linear Combinations of Random Variables30 questions2 topics
- Discrete random variables and expectation15 questions · 57 marks · 5.04a · PDF, 76 KB
- Combinations of normal random variables15 questions · 68 marks · 5.04a, 5.04b · PDF, 80 KB
5.04 Linear Combinations of Random Variables, 5 Statistics, on the specification map
5.05 Hypothesis Tests and Confidence Intervals60 questions4 topics
- Hypothesis testing: correlation and the normal15 questions · 61 marks · 5.05a, 5.05c · PDF, 82 KB
- The Central Limit Theorem15 questions · 60 marks · 5.05a, 5.05c · PDF, 77 KB
- Estimators, standard error and confidence intervals15 questions · 57 marks · 5.05a, 5.05b, 5.05d · PDF, 79 KB
- Comparing two normal means15 questions · 59 marks · 5.05c, 5.05d · PDF, 79 KB
5.05 Hypothesis Tests and Confidence Intervals, 5 Statistics, on the specification map
5.06 Chi-squared Tests32 questions2 topics
- Goodness-of-fit tests16 questions · 67 marks · 5.06a, 5.06b, 5.06c, 5.06d · PDF, 85 KB
- Contingency tables16 questions · 62 marks · 5.06a · PDF, 83 KB
5.06 Chi-squared Tests, 5 Statistics, on the specification map
5.07 Non-parametric tests91 questions6 topics
- Goodness-of-fit tests16 questions · 67 marks · 5.07a · PDF, 85 KB
- Type I and Type II errors, test size and power15 questions · 59 marks · 5.07a · PDF, 79 KB
- Testing a correlation coefficient15 questions · 63 marks · 5.07a · PDF, 79 KB
- Non-parametric tests15 questions · 67 marks · 5.07a, 5.07b, 5.07c, 5.07d and 2 more · PDF, 61 KB
- Comparing two normal means15 questions · 59 marks · 5.07d · PDF, 79 KB
- Confidence intervals and tests with the t-distribution15 questions · 66 marks · 5.07d · PDF, 80 KB
5.07 Non-parametric tests, 5 Statistics, on the specification map
5.08 Correlation78 questions5 topics
- Correlation and regression16 questions · 57 marks · 5.08a, 5.08c, 5.08g · PDF, 61 KB
- Correlation coefficients: product moment and Spearman16 questions · 64 marks · 5.08a, 5.08b, 5.08c, 5.08e and 1 more · PDF, 84 KB
- Hypothesis testing with the binomial16 questions · 67 marks · 5.08d · PDF, 61 KB
- Hypothesis testing: correlation and the normal15 questions · 61 marks · 5.08d · PDF, 82 KB
- Testing a correlation coefficient15 questions · 63 marks · 5.08d, 5.08f, 5.08g · PDF, 79 KB
5.09 Linear Regression63 questions4 topics
- Correlation and regression16 questions · 57 marks · 5.09a, 5.09c, 5.09e · PDF, 61 KB
- Least squares regression and residuals15 questions · 60 marks · 5.09a, 5.09b, 5.09c, 5.09d and 1 more · PDF, 79 KB
- Measures of location and spread16 questions · 59 marks · 5.09d · PDF, 82 KB
- Correlation coefficients: product moment and Spearman16 questions · 64 marks · 5.09d · PDF, 84 KB
5.09 Linear Regression, 5 Statistics, on the specification map
6 Mechanics
6.01 Dimensional Analysis15 questions1 topic
- Modelling, quantities and units15 questions · 51 marks · 6.01a, 6.01b, 6.01c, 6.01d and 1 more · PDF, 76 KB
6.01 Dimensional Analysis, 6 Mechanics, on the specification map
6.02 Work, Energy and Power136 questions9 topics
- Work, energy and power15 questions · 65 marks · 6.02a, 6.02b, 6.02c, 6.02d and 6 more · PDF, 79 KB
- Friction and inclined planes16 questions · 73 marks · 6.02b, 6.02l · PDF, 84 KB
- Statics of a particle15 questions · 68 marks · 6.02b · PDF, 77 KB
- Vectors in three dimensions15 questions · 57 marks · 6.02c, 6.02f, 6.02m · PDF, 73 KB
- Newton's laws with a variable force15 questions · 65 marks · 6.02c · PDF, 80 KB
- Impulse and momentum as vectors15 questions · 60 marks · 6.02f · PDF, 79 KB
- Hooke's law and elastic strings15 questions · 60 marks · 6.02g · PDF, 80 KB
- Elastic potential energy15 questions · 64 marks · 6.02h, 6.02j · PDF, 80 KB
- Oscillations on strings and springs15 questions · 65 marks · 6.02j · PDF, 80 KB
6.02 Work, Energy and Power, 6 Mechanics, on the specification map
6.03 Impulse and Momentum76 questions5 topics
- Momentum and impulse16 questions · 72 marks · 6.03a, 6.03b, 6.03e, 6.03f and 1 more · PDF, 107 KB
- Impulse and momentum as vectors15 questions · 60 marks · 6.03c, 6.03d, 6.03g · PDF, 79 KB
- Direct impact and Newton's law of restitution15 questions · 65 marks · 6.03f, 6.03i, 6.03j, 6.03k · PDF, 80 KB
- Successive impacts and impacts with a wall15 questions · 64 marks · 6.03f, 6.03k · PDF, 79 KB
- Oblique impact and impact with a smooth surface15 questions · 62 marks · 6.03g, 6.03l · PDF, 81 KB
6.03 Impulse and Momentum, 6 Mechanics, on the specification map
6.04 Centre of Mass91 questions6 topics
- Modelling, quantities and units15 questions · 51 marks · 6.04a · PDF, 76 KB
- Centre of mass of a discrete distribution15 questions · 59 marks · 6.04a, 6.04c · PDF, 56 KB
- Centres of mass of plane figures and frameworks15 questions · 58 marks · 6.04a, 6.04b, 6.04c, 6.04e · PDF, 59 KB
- Centres of mass by integration15 questions · 72 marks · 6.04b, 6.04d · PDF, 57 KB
- Moments16 questions · 69 marks · 6.04e · PDF, 83 KB
- Equilibrium, suspension, toppling and sliding15 questions · 57 marks · 6.04e · PDF, 59 KB
6.05 Motion in a Circle45 questions3 topics
- Angular speed and horizontal circular motion15 questions · 69 marks · 6.05a, 6.05b, 6.05c · PDF, 81 KB
- Motion in a vertical circle15 questions · 66 marks · 6.05d, 6.05e, 6.05f · PDF, 80 KB
- Projectiles15 questions · 59 marks · 6.05f · PDF, 77 KB
6.05 Motion in a Circle, 6 Mechanics, on the specification map
6.06 Further Dynamics and Kinematics57 questions4 topics
- Solving differential equations12 questions · 56 marks · 6.06a · PDF, 51 KB
- First order equations and integrating factors15 questions · 68 marks · 6.06a · PDF, 56 KB
- Newton's laws with a variable force15 questions · 65 marks · 6.06a · PDF, 80 KB
- Further kinematics: acceleration as a function of x, t or v15 questions · 59 marks · 6.06a · PDF, 78 KB
6.06 Further Dynamics and Kinematics, 6 Mechanics, on the specification map
7 Discrete Mathematics
7.01 Mathematical Preliminaries90 questions6 topics
- Algorithms, sorting and bin packing15 questions · 67 marks · 7.01a · PDF, 58 KB
- Simultaneous equations and inequalities15 questions · 60 marks · 7.01b · PDF, 53 KB
- Probability and Venn diagrams15 questions · 56 marks · 7.01b, 7.01k, 7.01l · PDF, 75 KB
- Combinatorics15 questions · 56 marks · 7.01b, 7.01c, 7.01d, 7.01e and 8 more · PDF, 54 KB
- The binomial expansion15 questions · 55 marks · 7.01f · PDF, 54 KB
- Graphs: order, Eulerian paths and planarity15 questions · 57 marks · 7.01j · PDF, 57 KB
7.01 Mathematical Preliminaries, 7 Discrete Mathematics, on the specification map
7.02 Graphs and Networks124 questions8 topics
- Graphs: order, Eulerian paths and planarity15 questions · 57 marks · 7.02a, 7.02b, 7.02c, 7.02d and 11 more · PDF, 57 KB
- Shortest paths: Dijkstra and Floyd16 questions · 76 marks · 7.02a, 7.02c, 7.02p, 7.02q · PDF, 62 KB
- Minimum spanning trees: Prim and Kruskal16 questions · 60 marks · 7.02b, 7.02k, 7.02p, 7.02q and 1 more · PDF, 67 KB
- Route inspection15 questions · 66 marks · 7.02c, 7.02g, 7.02r · PDF, 57 KB
- The travelling salesman problem15 questions · 71 marks · 7.02d, 7.02h, 7.02r · PDF, 57 KB
- Critical path analysis16 questions · 66 marks · 7.02k, 7.02p, 7.02r · PDF, 59 KB
- Flows in networks: cuts and capacity16 questions · 66 marks · 7.02k, 7.02p, 7.02r · PDF, 64 KB
- Maximum flow and the labelling procedure15 questions · 59 marks · 7.02k · PDF, 59 KB
7.02 Graphs and Networks, 7 Discrete Mathematics, on the specification map
7.03 Algorithms76 questions5 topics
- Algorithms, sorting and bin packing15 questions · 67 marks · 7.03a, 7.03b, 7.03c, 7.03d and 8 more · PDF, 58 KB
- Shortest paths: Dijkstra and Floyd16 questions · 76 marks · 7.03a · PDF, 62 KB
- The travelling salesman problem15 questions · 71 marks · 7.03a, 7.03e · PDF, 57 KB
- Summing series and the method of differences15 questions · 67 marks · 7.03d · PDF, 77 KB
- Further packing and the knapsack problem15 questions · 67 marks · 7.03m · PDF, 60 KB
7.03 Algorithms, 7 Discrete Mathematics, on the specification map
7.04 Network Algorithms77 questions5 topics
- Shortest paths: Dijkstra and Floyd16 questions · 76 marks · 7.04a, 7.04e, 7.04f · PDF, 62 KB
- Minimum spanning trees: Prim and Kruskal16 questions · 60 marks · 7.04b, 7.04c, 7.04f · PDF, 67 KB
- The travelling salesman problem15 questions · 71 marks · 7.04c, 7.04d, 7.04f · PDF, 57 KB
- Route inspection15 questions · 66 marks · 7.04e, 7.04f · PDF, 57 KB
- Float, Gantt charts and scheduling15 questions · 58 marks · 7.04f · PDF, 57 KB
7.04 Network Algorithms, 7 Discrete Mathematics, on the specification map
7.05 Decision Making in Project Management31 questions2 topics
- Critical path analysis16 questions · 66 marks · 7.05a, 7.05b, 7.05d · PDF, 59 KB
- Float, Gantt charts and scheduling15 questions · 58 marks · 7.05c, 7.05d, 7.05e · PDF, 57 KB
7.05 Decision Making in Project Management, 7 Discrete Mathematics, on the specification map
7.06 Graphical Linear Programming31 questions2 topics
- Linear programming: formulation and graphical solution16 questions · 65 marks · 7.06a, 7.06b, 7.06c, 7.06d and 2 more · PDF, 63 KB
- The Simplex algorithm15 questions · 68 marks · 7.06b · PDF, 58 KB
7.06 Graphical Linear Programming, 7 Discrete Mathematics, on the specification map
7.07 The Simplex Algorithm47 questions3 topics
- Linear programming: formulation and graphical solution16 questions · 65 marks · 7.07a, 7.07e · PDF, 63 KB
- The Simplex algorithm15 questions · 68 marks · 7.07a, 7.07b, 7.07c, 7.07d and 2 more · PDF, 58 KB
- Transportation problems and the north-west corner method16 questions · 70 marks · 7.07d · PDF, 63 KB
7.07 The Simplex Algorithm, 7 Discrete Mathematics, on the specification map
7.08 Game Theory45 questions3 topics
- Game theory: play safe and stable solutions15 questions · 65 marks · 7.08a, 7.08b, 7.08c, 7.08d and 1 more · PDF, 58 KB
- Mixed strategies in game theory15 questions · 70 marks · 7.08e, 7.08f · PDF, 57 KB
- The Simplex algorithm15 questions · 68 marks · 7.08f · PDF, 58 KB
7.08 Game Theory, 7 Discrete Mathematics, on the specification map
8 Additional Pure Mathematics
8.01 Sequences and Series108 questions7 topics
- Sequences and sigma notation15 questions · 57 marks · 8.01a, 8.01c, 8.01d · PDF, 74 KB
- Recurrence relations15 questions · 77 marks · 8.01a, 8.01b, 8.01c, 8.01d and 5 more · PDF, 77 KB
- Proof by induction: sums and series15 questions · 64 marks · 8.01b · PDF, 74 KB
- Induction: divisibility and matrices15 questions · 71 marks · 8.01b · PDF, 55 KB
- Geometric series15 questions · 59 marks · 8.01d · PDF, 55 KB
- First order equations and integrating factors15 questions · 68 marks · 8.01d · PDF, 56 KB
- Second order equations18 questions · 85 marks · 8.01f, 8.01g · PDF, 78 KB
8.01 Sequences and Series, 8 Additional Pure Mathematics, on the specification map
8.02 Number Theory75 questions5 topics
- The Euclidean algorithm and Bézout's identity15 questions · 62 marks · 8.02a, 8.02d, 8.02f, 8.02h and 3 more · PDF, 55 KB
- Modular arithmetic and Fermat's little theorem15 questions · 60 marks · 8.02b, 8.02c, 8.02e, 8.02f and 8 more · PDF, 74 KB
- Groups and their axioms15 questions · 62 marks · 8.02e, 8.02n · PDF, 55 KB
- Disproof and proof by contradiction15 questions · 51 marks · 8.02i · PDF, 51 KB
- The binomial expansion15 questions · 55 marks · 8.02o · PDF, 54 KB
8.02 Number Theory, 8 Additional Pure Mathematics, on the specification map
8.03 Groups30 questions2 topics
- Groups and their axioms15 questions · 62 marks · 8.03a, 8.03b, 8.03c, 8.03d and 6 more · PDF, 55 KB
- Subgroups, Lagrange's theorem and isomorphism15 questions · 60 marks · 8.03e, 8.03f, 8.03h, 8.03i and 4 more · PDF, 55 KB
8.03 Groups, 8 Additional Pure Mathematics, on the specification map
8.04 Further Vectors15 questions1 topic
- The vector product and the scalar triple product15 questions · 56 marks · 8.04a, 8.04b, 8.04c, 8.04d and 1 more · PDF, 78 KB
8.04 Further Vectors, 8 Additional Pure Mathematics, on the specification map
8.05 Surfaces and Partial Differentiation15 questions1 topic
- Functions of two variables15 questions · 64 marks · 8.05a, 8.05b, 8.05c, 8.05d and 3 more · PDF, 55 KB
8.05 Surfaces and Partial Differentiation, 8 Additional Pure Mathematics, on the specification map
8.06 Further Calculus30 questions2 topics
- Reduction formulae15 questions · 69 marks · 8.06a · PDF, 55 KB
- Arc length and surface area15 questions · 64 marks · 8.06b · PDF, 75 KB
8.06 Further Calculus, 8 Additional Pure Mathematics, on the specification map
A topic is listed under every section its lesson teaches, so some appear more than once. The codes beside each one are the specification references it covers in that section.
All questions, worked solutions and marking guidance are original InkMaths material. They are not copied from awarding-body papers or mark schemes. The codes show how InkMaths allocates the stated total; an awarding body may credit a valid alternative method differently.
The sections and codes are OCR's. Check the current specification when a mark depends on it. The H245 specification map lists every statement, with the lesson that teaches it.