A-level Further Mathematics Specification Specification for first teaching in 2017
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A-level Further Mathematics7367
3.5 Optional application 3 – discrete mathematics
3.5.1 DA: Graphs
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| DA1 | Understand and use the language of graphs including: vertex, edge, trail, cycle, connected, degree, subgraph, subdivision, multiple edge and loop. |
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| DA2 | Identify or prove properties of a graph including that a graph is Eulerian, semi-Eulerian or Hamiltonian. |
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| DA3 | Understand and use Euler’s formula for connected planar graphs. |
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| DA4 | Use Kuratowski’s Theorem to determine the planarity of graphs. |
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| DA5 | Understand and use complete graphs and bipartite graphs, including adjacency matrices and the complement of a graph. |
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| DA6 | Understand and use simple graphs, simple-connected graphs and trees. |
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| DA7 | Recognise and find isomorphism between graphs. |
3.5.2 DB: Networks
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| DB1 | Understand and use the language of networks including: node, arc and weight. |
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| DB2 | Solve network optimisation problems using spanning trees. |
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| DB3 | Solve route inspection problems. |
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| DB4 | Find and interpret upper bounds and lower bounds for the travelling salesperson problem. |
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| DB5 | Evaluate, modify and refine models which use networks. |
3.5.3 DC: Network flows
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| DC1 | Interpret flow problems represented by a network of directed arcs. |
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| DC2 | Find the value of a cut and understand its meaning. |
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| DC3 | Use and interpret the maximum flow-minimum cut theorem. |
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| DC4 | Introduce supersources and supersinks to a network. |
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| DC5 | Augment flows and determine the maximum flow in a network |
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| DC6 | Solve problems involving arcs with upper and lower capacities. |
3.5.4 DD: Linear programming
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| DD1 | Formulate constrained optimisation problems. |
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| DD2 | Solve constrained optimisation problems via graphical methods. |
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| DD3 | Use the Simplex algorithm for optimising (maximising and minimising) an objective function including the use of slack variables. |
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| DD4 | Interpret a Simplex tableau. |
3.5.5 DE: Critical path analysis
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| DE1 | Construct, represent and interpret a precedence (activity) network using activity-on-node. |
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| DE2 | Determine earliest and latest start and finish times for an activity network. |
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| DE3 | Identify critical activities, critical paths and the float of non-critical activities. |
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| DE4 | Refine models and understand the implications of possible changes in the context of critical path analysis. |
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| DE5 | Construct and interpret Gantt (cascade) diagrams and resource histograms. |
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| DE6 | Carry out resource levelling (using heuristic procedures) and solve problems where resources are restricted. |
3.5.6 DF: Game theory for zero-sum games
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| DF1 | Understand, interpret and construct pay-off matrices. |
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| DF2 | Find play-safe strategies and the value of the game. |
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| DF3 | Prove the existence or non-existence of a stable solution. |
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| DF4 | Identify and make use of dominated strategies. |
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| DF5 | Find optimal mixed strategies for a game including use of graphical methods. |
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| DF6 | Convert and solve higher order games to linear programming problems. |
3.5.7 DG: Binary operations
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| DG1 | Understand and use binary operations including use of modular arithmetic and matrix multiplication. |
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| DG2 | Understand, use and prove the commutativity of a binary operation. |
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| DG3 | Understand, use and prove the associativity of a binary operation. |
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| DG4 | Construct a Cayley table for a given set under a given binary operation. |
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| DG5 | Understand and prove the existence of an identity element for a given set under a given binary operation. |
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| DG6 | Find the inverse of an element belonging to a given set under a given binary operation. |
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| DG7 | Understand and use the language of groups including: order, period, subgroup, proper, trivial, non-trivial. |
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| DG8 | Understand and use the group axioms: closure, identity, inverses and associativity, including use of Cayley tables. |
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| DG9 | Recognise and use finite and infinite groups and their subgroups, including: groups of symmetries of regular polygons, cyclic groups and abelian groups. |
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| DG10 | Understand and use Lagrange’s theorem. |
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| DG11 | Identify and use the generators of a group. |
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| DG12 | Recognise and find isomorphism between groups of finite order. |