IB DP Further Mathematics HL: Past Years Question Bank with Solution Paper -3

IB DP PhysicsIB DP MathsIB DP ChemistryIB DP Biology

IB DP Maths HL - Paper 2 - All Topics

Paper 1

SL

  • Time: 45 minutes (30 marks)
  • 30 multiple – choice questions (core)
  • No marks deducted from incorrect answers
  • NO CALCULATOR ALLOWED
  • Data booklet provided
  • 20% weight

HL

  • Time: 60 minutes (40 marks)
  • 40 multiple – choice questions (core & AHL)
  • No marks deducted from incorrect answers
  • NO CALCULATOR ALLOWED
  • Data booklet provided
  • 20% weight

Topic 1 – Core: Algebra

  • Topic 1.1 :
    • Arithmetic sequences and series; sum of finite arithmetic series; geometric sequences and series; sum of finite and infinite geometric series,
    • Sigma notation,
    • Applications. 
  • Topic 1.2 :
    • Exponents and logarithms,
    • Laws of exponents; laws of logarithms,
    • Change of base 
  • Topic 1.3 :
    • Counting principles, including permutations and combinations
    • The binomial theorem: expansion of \({\left( {a + b} \right)^n}\), \(n \in N\) 
  • Topic 1.4 :
    • Proof by mathematical induction. 
  • Topic 1.5 :
    •  Complex numbers: the number \({\text{i}} = \sqrt { – 1} \) ; the terms real part, imaginary part, conjugate, modulus and argument.
    • Cartesian form \(z = a + {\text{i}}b\) .
    • Sums, products and quotients of complex numbers.
  • Topic 1.6 :
    • Modulus–argument (polar) form \(z = r\left( {\cos \theta + {\text{i}}\sin \theta } \right) = r{\text{cis}}\theta = r{e^{{\text{i}}\theta }}\).
    • The complex plane.
    • Sums, products and quotients of complex numbers.
  • Topic 1.7 :
    • Powers of complex numbers: de Moivre’s theorem.
    •  \(n\)th roots of a complex number.
  • Topic 1.8 :
    • Conjugate roots of polynomial equations with real coefficients.
  • Topic 1.9 :
    • 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 – Core: Functions and equations

  • Topic 2.1
    • Concept of function \(f:x \mapsto f\left( x \right)\) : domain, range; image (value)
    • Odd and even functions.
    • Composite functions \(f \circ g\)
    • Identity function.
    • One-to-one and many-to-one functions.
    • Inverse function \({f^{ – 1}}\), including domain restriction. Self-inverse functions.
  • Topic 2.2
    • The graph of a function; its equation \(y = f\left( x \right)\) .
    • Investigation of key features of graphs, such as maximum and minimum values, intercepts, horizontal and vertical asymptotes and symmetry, and consideration of domain and range.
    • The graphs of the functions \(y = \left| {f\left( x \right)} \right|\) and \(y = f\left( {\left| x \right|} \right)\)
    • The graph of \(\frac{1}{{f\left( x \right)}}\) given the graph of \(y = f(x)\) .
  • Topic 2.3
    • Transformations of graphs: translations; stretches; reflections in the axes.
    • The graph of the inverse function as a reflection in \(y = x\).
  • Topic 2.4
    • The rational function \(x \mapsto \frac{{ax + b}}{{cx + d}}\) and its graph.
    • The function \(x \mapsto {a^x}\) , \(a > 0\) , and its graph.
    • The function \(x \mapsto {\log _a}x\) , \(x > 0\) , and its graph
  • Topic 2.5
    • Polynomial functions and their graphs.
    • The factor and remainder theorems.
    • The fundamental theorem of algebra.
  • Topic 2.6
    • Solving quadratic equations using the quadratic formula.
    • Use of the discriminant \(\Delta = {b^2} – 4ac\) to determine the nature of the roots.
    • Solving polynomial equations both graphically and algebraically.
    • Sum and product of the roots of polynomial equations.
    • Solution of \({a^x} = b\) using logarithms.
    • Use of technology to solve a variety of equations, including those where there is no appropriate analytic approach.
  • Topic 2.7
    • Solutions of \(g\left( x \right) \geqslant f\left( x \right)\) .
    • Graphical or algebraic methods, for simple polynomials up to degree 3.
    • Use of technology for these and other functions.

Topic 3 – Core: Circular functions and trigonometry

  • Topic 3.1 :
    • The circle: radian measure of angles.
    • Length of an arc; area of a sector. 
  • Topic 3.2 :
    • Definition of \(\cos \theta \) , \(\sin \theta \) and \(\tan \theta \) in terms of the unit circle.
    • Exact values of \(\sin\), \(\cos\) and \(\tan\) of \(0\), \(\frac{\pi }{6}\), \(\frac{\pi }{4}\), \(\frac{\pi }{3}\), \(\frac{\pi }{2}\) and their multiples.
    • Definition of the reciprocal trigonometric ratios \(\sec \theta \) , \(\csc \theta \) and \(\cot \theta \) .
    • Pythagorean identities: \({\cos ^2}\theta + {\sin ^2}\theta = 1\) ; \(1 + {\tan ^2}\theta = {\sec ^2}\theta \) ; \(1 + {\cot ^2}\theta = {\csc ^2}\theta \) .
  • Topic 3.3 :
    • Compound angle identities.
    • Double angle identities
  • Topic 3.4 :
    • Composite functions of the form \(f(x) = a\sin (b(x + c)) + d\) .
    • Applications.
  • Topic 3.5 :
    • The inverse functions \(x \mapsto \arcsin x\) , \(x \mapsto \arccos x\) , \(x \mapsto \arctan x\) ; their domains and ranges; their graphs.
  • Topic 3.6 :
    • Algebraic and graphical methods of solving trigonometric equations in a finite interval, including the use of trigonometric identities and factorization.
  • Topic 3.7 :
    • The cosine rule.
    • The sine rule including the ambiguous case.
    • Area of a triangle as \(\frac{1}{2}ab\sin C\) .
    • Applications.

Topic 4 – Core: Vectors

  • Topic 4.1
    • Concept of a vector.
    • Representation of vectors using directed line segments.
    • Unit vectors; base vectors \(i\), \(j\), \(k\).
    • Components of a vector: \(v = \left( {\begin{array}{*{20}{c}} {{v_1}} \\ {{v_2}} \\ {{v_3}} \end{array}} \right) = {v_1}i + {v_2}j + {v_3}k\) .
    • Algebraic and geometric approaches to the sum and difference of two vectors.
    • Algebraic and geometric approaches to the zero vector \(0\), the vector \( – v\) .
    • Algebraic and geometric approaches to multiplication by a scalar, \(kv\) .
    • Algebraic and geometric approaches to magnitude of a vector, \(\left| v \right|\) .
    • Algebraic and geometric approaches to position vectors \(\overrightarrow {OA} = a\) .
    • \(\overrightarrow {AB} = b – a\) .
  • Topic 4.2
    • 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 4.3
    • Vector equation of a line in two and three dimensions: \(r = a + \lambda b\) .
    • Simple applications to kinematics.
    • The angle between two lines.
  • Topic 4.4
    • Coincident, parallel, intersecting and skew lines; distinguishing between these cases.
    • Points of intersection.
  • Topic 4.5
    • 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 4.6
    • 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 4.7
    • Intersections of: a line with a plane; two planes; three planes.
    • Angle between: a line and a plane; two planes.

Topic 5 – Core: Statistics and probability

  • Topic 5.1 :
    • Concepts of population, sample, random sample and frequency distribution of discrete and continuous data.
    • Grouped data: mid-interval values, interval width, upper and lower interval boundaries.
    • Mean, variance, standard deviation.
  • Topic 5.2 :
    • Concepts of trial, outcome, equally likely outcomes, sample space (U) and event.
    • The probability of an event \(A\) is \(P\left( A \right) = \frac{{n\left( A \right)}}{{n\left( U \right)}}\)
    • The complementary events \(A\) and \({A’}\) (not \(A\)).
    • Use of Venn diagrams, tree diagrams, counting principles and tables of outcomes to solve problems.
  • Topic 5.3 :
    • Combined events; the formula for \(P\left( {A \cup B} \right)\) .
    • Mutually exclusive events.
  • Topic 5.4 :
    • Conditional probability; the definition \(P\left( {\left. A \right|P} \right) = \frac{{P\left( {A\mathop \cap \nolimits B} \right)}}{{P\left( B \right)}}\) .
    • Independent events; the definition \(P\left( {\left. A \right|B} \right) = P\left( A \right) = P\left( {\left. A \right|B’} \right)\) .
    • Use of Bayes’ theorem for a maximum of three events.
  • Topic 5.5 :
    • Concept of discrete and continuous random variables and their probability distributions.
    • Definition and use of probability density functions.
    • Expected value (mean), mode, median, variance and standard deviation.
    • Applications.
  • Topic 5.6 :
    • Binomial distribution, its mean and variance.
    • Poisson distribution, its mean and variance.
  • Topic 5.7 :
    • Normal distribution.
    • Properties of the normal distribution.
    • Standardization of normal variables.

Topic 6 – Core: Calculus

  • Topic 6.1
    • Informal ideas of limit, continuity and convergence.
    • Definition of derivative from first principles as \(f’\left( x \right) = \mathop {\lim }\limits_{h \to 0} {\frac{{f\left( {x + h} \right) – f\left( x \right)}}{h}} \).
    • The derivative interpreted as a gradient function and as a rate of change.
    • Finding equations of tangents and normals.
    • Identifying increasing and decreasing functions.
    • The second derivative.
    • Higher derivatives.
  • Topic 6.2
    • Derivatives of \({x^n}\) , \(\sin x\) , \(\cos x\) , \(\tan x\) , \({{\text{e}}^x}\) and \\(\ln x\) .
    • Differentiation of sums and multiples of functions.
    • The product and quotient rules.
    • The chain rule for composite functions.
    • Related rates of change.
    • Implicit differentiation.
    • Derivatives of \(\sec x\) , \(\csc x\) , \(\cot x\) , \({a^x}\) , \({\log _a}x\) , \(\arcsin x\) , \(\arccos x\) and \(\arctan x\) .
  • Topic 6.3
    • Local maximum and minimum values.
    • Optimization problems.
    • Points of inflexion with zero and non-zero gradients.
    • Graphical behaviour of functions, including the relationship between the graphs of \(f\) , \({f’}\) and \({f”}\) .
  • Topic 6.4
    • Indefinite integration as anti-differentiation.
    • Indefinite integral of \({x^n}\) , \(\sin x\) , \(\cos x\) and \({{\text{e}}^x}\) .
    • Other indefinite integrals using the results from 6.2.
    • The composites of any of these with a linear function.
  • Topic 6.5
    • Anti-differentiation with a boundary condition to determine the constant of integration.
    • Definite integrals.
    • Area of the region enclosed by a curve and the \(x\)-axis or \(y\)-axis in a given interval; areas of regions enclosed by curves.
    • Volumes of revolution about the \(x\)-axis or \(y\)-axis.
  • Topic 6.6
    • Kinematic problems involving displacement \(s\), velocity \(v\) and acceleration \(a\).
    • Total distance travelled.
  • Topic 6.7
    • Integration by substitution.
    • Integration by parts.

Topic 7 – Option: Statistics and probability

  • Topic 7.1
    • Cumulative distribution functions for both discrete and continuous distributions.
    • Geometric distribution.
    • Negative binomial distribution.
    • Probability generating functions for discrete random variables.
    • Using probability generating functions to find mean, variance and the distribution of the sum of \(n\) independent random variables.
  • Topic 7.2
    • Linear transformation of a single random variable.
    • Mean of linear combinations of \(n\) random variables.
    • Variance of linear combinations of \(n\) independent random variables.
    • Expectation of the product of independent random variables.
  • Topic 7.3
    • Unbiased estimators and estimates.
    • Comparison of unbiased estimators based on variances.
    • \({\bar X}\) as an unbiased estimator for \(\mu \) .
    • \({S^2}\) as an unbiased estimator for \({\sigma ^2}\) .
  • Topic 7.4
    • A linear combination of independent normal random variables is normally distributed. In particular, \(X{\text{ ~ }}N\left( {\mu ,{\sigma ^2}} \right) \Rightarrow \bar X{\text{ ~ }}N\left( {\mu ,\frac{{{\sigma ^2}}}{n}} \right)\) .
    • The central limit theorem.
  • Topic 7.5
    • Confidence intervals for the mean of a normal population.
  • Topic 7.6
    • Null and alternative hypotheses, \({H_0}\) and \({H_1}\) .
    • Significance level.
    • Critical regions, critical values, \(p\)-values, one-tailed and two-tailed tests.
    • Type I and II errors, including calculations of their probabilities.
    • Testing hypotheses for the mean of a normal population.
  • Topic 7.7
    • Introduction to bivariate distributions.
    • Covariance and (population) product moment correlation coefficient \(\rho \).
    • Proof that \(\rho = 0\) in the case of independence and \( \pm 1\) in the case of a linear relationship between \(X\) and \(Y\).
    • Definition of the (sample) product moment correlation coefficient \(R\) in terms of n paired observations on \(X\) and \(Y\).
    • Its application to the estimation of \(\rho \).Informal interpretation of \(r\), the observed value of \(R\). Scatter diagrams.
    • Topics based on the assumption of bivariate normality: use of the \(t\)-statistic to test the null hypothesis \(\rho = 0\) .
    • Topics based on the assumption of bivariate normality: knowledge of the facts that the regression of \(X\) on \(Y\) (\({E\left. {\left( X \right)} \right|Y = y}\)) and \(Y\) on \(X\) (\({E\left. {\left( Y \right)} \right|X = x}\)) are linear.
    • Topics based on the assumption of bivariate normality: least-squares estimates of these regression lines (proof not required).
    • Topics based on the assumption of bivariate normality: the use of these regression lines to predict the value of one of the variables given the value of the other.

Topic 8 – Option: Sets, relations and groups

  • Topic 8.1
    • Finite and infinite sets.
    • Subsets.
    • Operations on sets: union; intersection; complement; set difference; symmetric difference.
    • De Morgan’s laws: distributive, associative and commutative laws (for union and intersection).
  • Topic 8.2
    • Ordered pairs: the Cartesian product of two sets.
    • Relations: equivalence relations; equivalence classes.
  • Topic 8.3
    • Functions: injections; surjections; bijections.
    • Composition of functions and inverse functions.
  • Topic 8.4
    • Binary operations.
    • Operation tables (Cayley tables).
  • Topic 8.5
    • Binary operations: associative, distributive and commutative properties.
  • Topic 8.6
    • The identity element \(e\).
    • The inverse \({a^{ – 1}}\) of an element \(a\).Proof that left-cancellation and right-cancellation by an element a hold, provided that a has an inverse.
    • Proofs of the uniqueness of the identity and inverse elements.
  • Topic 8.7
    • The definition of a group \(\left\{ {G, * } \right\}\) .
    • The operation table of a group is a Latin square, but the converse is false.
    • Abelian groups.
  • Topic 8.8
    • Example of groups: \(\mathbb{R}\), \(\mathbb{Q}\), \(\mathbb{Z}\) and \(\mathbb{C}\) under addition.
    • Example of groups: integers under addition modulo \(n\).
    • Example of groups: non-zero integers under multiplication, modulo \(p\), where \(p\) is prime.
    • Symmetries of plane figures, including equilateral triangles and rectangles.
    • Invertible functions under composition of functions.
  • Topic 8.9
    • The order of a group.
    • The order of a group element.
    • Cyclic groups.
    • Generators.
    • Proof that all cyclic groups are Abelian.
  • Topic 8.10
    • Permutations under composition of permutations.
    • Cycle notation for permutations.
    • Result that every permutation can be written as a composition of disjoint cycles.
    • The order of a combination of cycles.
  • Topic 8.11
    • Subgroups, proper subgroups.
    • Use and proof of subgroup tests.
    • Definition and examples of left and right cosets of a subgroup of a group.
    • Lagrange’s theorem.
    • Use and proof of the result that the order of a finite group is divisible by the order of any element. (Corollary to Lagrange’s theorem.)
  • Topic 8.12
    • Definition of a group homomorphism.
    • Definition of the kernel of a homomorphism.
    • Proof that the kernel and range of a homomorphism are subgroups.
    • Proof of homomorphism properties for identities and inverses.
    • Isomorphism of groups.
    • The order of an element is unchanged by an isomorphism.

Topic 9 – Option: Calculus

  • Topic 9.1
    • Infinite sequences of real numbers and their convergence or divergence.
  • Topic 9.2
    • Convergence of infinite series.
    • Tests for convergence: comparison test; limit comparison test; ratio test; integral test.
    • The \(p\)-series, \(\mathop \sum \nolimits \frac{1}{{{n^p}}}\) .Series that converge absolutely.
    • Series that converge conditionally.
    • Alternating series.
    • Power series: radius of convergence and interval of convergence. Determination of the radius of convergence by the ratio test.
  • Topic 9.3
    • Continuity and differentiability of a function at a point.
    • Continuous functions and differentiable functions.
  • Topic 9.4
    • The integral as a limit of a sum; lower and upper Riemann sums.
    • Fundamental theorem of calculus.
    • Improper integrals of the type \(\int\limits_a^\infty {f\left( x \right){\text{d}}} x\) .
  • Topic 9.5
    • First-order differential equations.
    • Geometric interpretation using slope fields, including identification of isoclines.
    • Numerical solution of \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = f\left( {x,y} \right)\) using Euler’s method.
    • Variables separable.
    • Homogeneous differential equation \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = f\left( {\frac{y}{x}} \right)\) using the substitution \(y = vx\) .Solution of \(y’ + P\left( x \right)y = Q\left( x \right)\), using the integrating factor.
  • Topic 9.6
    • Rolle’s theorem.
    • Mean value theorem.
    • Taylor polynomials; the Lagrange form of the error term.
    • Maclaurin series for \({{\text{e}}^x}\) , \(\\sin x\) , \(\cos x\) , \(\ln (1 + x)\) , \({(1 + x)^p}\) , \(P \in \mathbb{Q}\) .
    • Use of substitution, products, integration and differentiation to obtain other series.
    • Taylor series developed from differential equations.
  • Topic 9.7
    • The evaluation of limits of the form \(\mathop {\lim }\limits_{x \to a} \frac{{f\left( x \right)}}{{g\left( x \right)}}\) and \(\mathop {\lim }\limits_{x \to \infty } \frac{{f\left( x \right)}}{{g\left( x \right)}}\) .
    • Using l’Hôpital’s rule or the Taylor series.

Topic 10 – Option: Discrete mathematics

  • Topic 10.1
    • Strong induction.
    • Pigeon-hole principle.
  • Topic 10.2
    • \(\left. a \right|b \Rightarrow b = na\) for some \(n \in \mathbb{Z}\) .
    • The theorem \(\left. a \right|b\) and \(a\left| {c \Rightarrow a} \right|\left( {bx \pm cy} \right)\) where \(x,y \in \mathbb{Z}\) .
    • Division and Euclidean algorithms.
    • The greatest common divisor, gcd(\(a\),\(b\)), and the least common multiple, lcm(\(a\),\(b\)), of integers \(a\) and \(b\).
    • Prime numbers; relatively prime numbers and the fundamental theorem of arithmetic.
  • Topic 10.3
    • Linear Diophantine equations \(ax + by = c\) .
  • Topic 10.4
    • Modular arithmetic.
    • The solution of linear congruences.
    • Solution of simultaneous linear congruences (Chinese remainder theorem).
  • Topic 10.5
    • Representation of integers in different bases.
  • Topic 10.6
    • Fermat’s little theorem.
  • Topic 10.7
    • Graphs, vertices, edges, faces.
    • Adjacent vertices, adjacent edges.
    • Degree of a vertex, degree sequence.
    • Handshaking lemma.
    • Simple graphs; connected graphs; complete graphs; bipartite graphs; planar graphs; trees; weighted graphs, including tabular representation.
    • Subgraphs; complements of graphs
    • .Euler’s relation: \(v – e + f = 2\) ; theorems for planar graphs including \(e \leqslant 3v – 6\) , \(e \leqslant 2v – 4\) , leading to the results that \({\kappa _5}\) and \({\kappa _{3,3}}\) are not planar.
  • Topic 10.8
    • Walks, trails, paths, circuits, cycles.
    • Eulerian trails and circuits.
    • Hamiltonian paths and cycles.
  • Topic 10.9
    • Graph algorithms: Kruskal’s; Dijkstra’s.
  • Topic 10.10
    • Chinese postman problem.
    • Travelling salesman problem.
    • Nearest-neighbour algorithm for determining an upper bound.
    • Deleted vertex algorithm for determining a lower bound.
  • Topic 10.11
    • Recurrence relations.
    • Initial conditions, recursive definition of a sequence.
    • Solution of first- and second-degree linear homogeneous recurrence relations with constant coefficients.
    • The first-degree linear recurrence relation \({u_n} = a{u_{n – 1}} + b\) .
    • Modelling with recurrence relations.
Scroll to Top