A-level Further Mathematics Specification for first teaching in 2017
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A-level Further Mathematics7367
Specification7367
7367
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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. |