IBDP Maths Applications and Interpretation: IB Style Question Bank HL Paper-3

Topic 3: Geometry and trigonometry-AHL content

• Topic : AHL 3.7
  
  • The definition of a radian and conversion between degrees and radians.
  • Using radians to calculate area of sector, length of arc.
• Topic : AHL 3.8
  
  • The definitions of cosθ and sinθ in terms of the unit circle.
  • The Pythagorean identity:
    
    • cos²θ+sin²θ=1
  • Definition of tanθ as \(\frac{sin\theta }{cos\theta }\)
  • Extension of the sine rule to the ambiguous case.
  • Graphical methods of solving trigonometric equations in a finite interval.
• Topic : AHL 3.9
  
  • Geometric transformations of points in two dimensions using matrices: reflections, horizontal and vertical stretches, enlargements, translations and rotations.
  • Compositions of the above transformations.
  • Geometric interpretation of the determinant of a transformation matrix.
• Topic : AHL 3.10
  
  • Concept of a vector; position vectors; displacement vectors.
  • Representation of vectors using directed line segments.
  • Unit vectors ; Base vectors i, j, k.
  • Components of a vector: \(v = \left( {\begin{array}{*{20}{c}} {{v_1}} \\ {{v_2}} \\ {{v_3}} \end{array}} \right) = {v_1}i + {v_2}j + {v_3}k\) .
  • The zero vector 0, the vector -v.
  • Position vectors \(\vec{{OA}}=a\)
  • Rescaling and normalizing vectors.
• Topic : AHL 3.11
  
  • Vector equation of a line in two and three dimensions: \(r = a + \lambda b\) .
• Topic : AHL 3.12
  
  • Vector applications to kinematics.
    
    • Modelling linear motion with constant velocity in two and three dimensions.
  • Motion with variable velocity in two dimensions.
• Topic : AHL 3.13
  
  • Definition and calculation of the scalar product of two vectors.
    
    • The angle between two vectors; the acute angle between two lines.
  • Definition and calculation of the vector product of two vectors.
  • Geometric interpretation of |v×w|.
  • Components of vectors.
• Topic : AHL 3.14
  
  • Graph theory: Graphs, vertices, edges, adjacent vertices, adjacent edges. Degree of a vertex.
  • Simple graphs; complete graphs; weighted graphs
  • Directed graphs; in degree and out degree of a directed graph.
  • Subgraphs; trees.
• Topic : AHL 3.15
  
  • Adjacency matrices.
    
    • Walks.
    • Number of k -length walks (or less than k -length walks) between two vertices.
  • Weighted adjacency tables.
    
    • Construction of the transition matrix for a strongly-connected, undirected or directed graph.
• Topic : AHL 3.16
  
  • Tree and cycle algorithms with undirected graphs.Walks, trails, paths, circuits, cycles.
    
    • Eulerian trails and circuits.
    • Hamiltonian paths and cycles.
    • Minimum spanning tree (MST) graph algorithms:
      
      • Kruskal’s and Prim’s algorithms for finding minimum spanning trees.
  • Chinese postman problem and algorithm for solution, to determine the shortest route around a weighted graph with up to four odd vertices, going along each edge at least once.
  • Travelling salesman problem to determine the Hamiltonian cycle of least weight in a weighted complete graph.
    
    • Nearest neighbour algorithm for determining an upper bound for the travelling salesman problem.
    • Deleted vertex algorithm for determining a lower bound for the travelling salesman problem.