IB DP Maths HL: Past Years Question Bank with Solution Paper – 3

Topic 3: Geometry and trigonometry-AHL content

• Topic : AHL 3.9
  
  • Definition of the reciprocal trigonometric ratios secθ, cosecθ and cotθ.
  • Pythagorean identities:
    
    • 1 + tan²θ = sec²θ
    • 1 + cot2θ = cosec²θ
  • The inverse functions f(x) = arcsinx, f(x) = arccosx, f(x) = arctanx; their domains and ranges; their graphs.
• Topic : AHL 3.10
  
  • Compound angle identities.
  • Double angle identity for tan.
• Topic : AHL 3.11
  
  • Relationships between trigonometric functions and the symmetry properties of their graphs.
• Topic : AHL 3.12
  
  • Concept of a vector; position vectors; displacement vectors.
  • Representation of vectors using directed line segments.
  • Base vectors i, j, k.
  • Components of a vector: \(v = \left( {\begin{array}{*{20}{c}} {{v_1}} \\ {{v_2}} \\ {{v_3}} \end{array}} \right) = {v_1}i + {v_2}j + {v_3}k\) .
  • Algebraic and geometric approaches to the following:
    
    • sum and difference of two vectors.
    • the zero vector \(0\), the vector \( – v\) .
    • multiplication by a scalar, \(kv\) , parallel vectors
    • magnitude of a vector, \(\left| v \right|\) .unit vectors=\(\frac{\vec{v}}{\left | \vec{v} \right |}\)
    • position vectors \(\overrightarrow {OA} = a\) .
    • displacement vector \(\overrightarrow {AB} = b – a\) .
  • Proofs of geometrical properties using vectors.
• Topic : AHL 3.13
  
  • The definition of the scalar product of two vectors.
  • Properties of the scalar product: \({\boldsymbol{v}} \cdot {\boldsymbol{w}} = {\boldsymbol{w}} \cdot {\boldsymbol{v}}\) ; \({\boldsymbol{u}} \cdot \left( {{\mathbf{v}} + {\boldsymbol{w}}} \right) = {\boldsymbol{u}} \cdot {\boldsymbol{v}} + {\boldsymbol{u}} \cdot {\boldsymbol{w}}\) ; \(\left( {k{\boldsymbol{v}}} \right) \cdot {\boldsymbol{w}} = k\left( {{\boldsymbol{v}} \cdot {\boldsymbol{w}}} \right)\) ; \({\boldsymbol{v}} \cdot {\boldsymbol{v}} = {\left| {\boldsymbol{v}} \right|^2}\) .
  • The angle between two vectors.
  • Perpendicular vectors; parallel vectors.
• Topic : AHL 3.14
  
  • Vector equation of a line in two and three dimensions: \(r = a + \lambda b\) .
  • The angle between two lines.
  • Simple applications to kinematics.
• Topic : AHL 3.15
  
  • Coincident, parallel, intersecting and skew lines; distinguishing between these cases.
  • Points of intersection.
• Topic : AHL 3.16
  
  • The definition of the vector product of two vectors.
  • Properties of the vector product: \({\text{v}} \times {\text{w}} = – {\text{w}} \times {\text{v}}\) ; \({\text{u}} \times ({\text{v}} + {\text{w}}) = {\text{u}} \times {\text{v}} + {\text{u}} \times {\text{w}}\) ; \((k{\text{v}}) \times {\text{w}} = k({\text{v}} + {\text{w}})\) ; \({\text{v}} \times {\text{v}} = 0\) .
  • Geometric interpretation of \({\text{v}} \times {\text{w}}\) .
• Topic : AHL 3.17
  
  • Vector equation of a plane \(r = a + \lambda b + \mu c\) .
  • Use of normal vector to obtain the form \(r \cdot n = a \cdot n\) .
  • Cartesian equation of a plane \(ax + by + cz = d\) .
• Topic : AHL 3.18
  
  • Intersections of: a line with a plane; two planes; three planes.
  • Angle between: a line and a plane; two planes.