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

Paper 3

HL

  • Time: 60 minutes  [55 Maximum marks].
  • 2 questions only
  • No marks deducted from incorrect answers
  • Answer all the questions
  • A graphic display calculator is required for this paper
  • Only HL syllabus for Maths AA

Topic 1: Number and algebra– AHL content

  • Topic : AHL 1.10
    • Counting principles, including permutations and combinations.
    • Extension of the binomial theorem to fractional and negative indices, ie (a + b)n , n ∈ ℚ.
  • Topic : AHL 1.11
    • Partial fractions
  • Topic : AHL 1.12
    • Complex numbers: the number i, where i2 = − 1.
    • Cartesian form z = a + bi; the terms real part, imaginary part, conjugate, modulus and argument.
    • The complex plane.
  • Topic : AHL 1.13
    • Modulus–argument (polar) form \(z = r\left( {\cos \theta + {\text{i}}\sin \theta } \right) = r{\text{cis}}\theta = r{e^{{\text{i}}\theta }}\).
    • Euler form: z = re.
    • Sums, products and quotients in Cartesian, polar or Euler forms and their geometric interpretation.
  • Topic : AHL 1.14
    • Complex conjugate roots of quadratic and polynomial equations with real coefficients.
    • De Moivre’s theorem and its extension to rational exponents. 
    • Powers and roots of complex numbers.
  • Topic : AHL 1.15
    • Proof by mathematical induction.
    • Proof by contradiction.
    • Use of a counterexample to show that a statement is not always true.
  • Topic : AHL 1.16
    • Solutions of systems of linear equations (a maximum of three equations in three unknowns), including cases where there is a unique solution, an infinity of solutions or no solution.

Topic 2: FunctionsAHL content

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.

Topic 5: Calculus-AHL content

IBDP Mathematics – Paper 3 – Old Syllabus

Topic 7 – Option: Statistics and probability

Topic 8 – Option: Sets, relations and groups

Topic 9 – Option: Calculus

Topic 10 – Option: Discrete mathematics

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