# 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: Functions– AHL content

• Topic: AHL 2.12
• Topic: AHL 2.13
• Rational functions of the form
• $$f(x)=(\frac{ax + b}{cx^2 + dx + e}),and \; f(x)=\frac{ax^2 + bx + c}{dx + e}$$
• Topic: AHL 2.14
• Odd and even functions.
• Finding the inverse function, $${f^{ – 1}}$$, including domain restriction. Self-inverse functions.
• Topic: AHL 2.15
• Topic: AHL 2.16
• The graphs of the functions, $$y = \left| {f\left( x \right)} \right|$$ and $$y = f\left( {\left| x \right|} \right)$$,
• $$\frac{1}{{f\left( x \right)}}$$ , y = f(ax + b), y = [f(x)]2

### 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

• Topic: AHL 5.12
• Topic: AHL 5.13
• Topic: AHL 5.14
• Implicit differentiation.
• Related rates of change.  Optimization problems.
• Topic: AHL 5.15
• Derivatives of $$\tan x$$, $$\sec x$$ , cosec x , $$\cot x$$ , $${a^x}$$ , $${\log _a}x$$ , $$\arcsin x$$ , $$\arccos x$$ and $$\arctan x$$ .
• Indefinite integrals of the derivatives of any of the above functions. The composites of any of these with a linear function.
• Use of partial fractions to rearrange the integrand.
• Topic: AHL 5.16
• Integration by substitution.
• Integration by parts.
• Repeated integration by parts.
• Topic: AHL 5.17
• Area of the region enclosed by a curve and the y-axis in a given interval.
• Volumes of revolution about the $$x$$-axis or $$y$$-axis.
• Topic: AHL 5.18
• Topic: AHL 5.19

### IBDP Mathematics – Paper 3 – Old Syllabus

Topic 9 – Option: Calculus

Topic 10 – Option: Discrete mathematics