IB DP Further Mathematics HL: Past Years Question Bank with Solution Paper – 1

Paper 1

HL

  • Time: 150 minutes (150 marks)
  • No marks deducted from incorrect answers
  • A graphic display calculator is required for this paper.
  • A clean copy of the mathematics HL and further mathematics HL formula booklet is
    required for this paper

Topic 1 – Linear Algebra

Topic 2 – Geometry

  • Topic 2.1
    • Similar and congruent triangles.
    • Euclid’s theorem for proportional segments in a right-angled triangle.
  • Topic 2.2
    • Centres of a triangle: orthocentre, incentre, circumcentre and centroid.
  • Topic 2.3
    • Circle geometry.
    • Tangents; arcs, chords and secants.
    • In a cyclic quadrilateral, opposite angles are supplementary, and the converse.
  • Topic 2.4
  • Topic 2.5
  • Topic 2.6
    • Conic sections.
    • The parabola, ellipse and hyperbola, including rectangular hyperbola.
    • Focus–directrix definitions.
    • Tangents and normals.
  • Topic 2.7
    • Parametric equations.
    • Parametric differentiation.
    • Tangents and normals.
    • The standard parametric equations of the circle, parabola, ellipse, rectangular hyperbola, hyperbola.
  • Topic 2.8
    • The general conic \(a{x^2} + 2bxy + c{y^2} + dx + ey + f = 0\) and the quadratic form \({x^{\text{T}}}Ax = a{x^2} + 2bxy + c{y^2}\) .
    • Diagonalizing the matrix \(A\) with the rotation matrix \(P\) and reducing the general conic to standard form.

Topic 3 – Statistics and probability

  • Topic 3.1

    • 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 3.2
    • 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 3.3
    • Unbiased estimators and estimates.
    • Comparison of unbiased estimators based on variances.\(\overline X \) as an unbiased estimator for \(\mu \) .\({S^2}\) as an unbiased estimator for \({\sigma ^2}\) .
  • Topic 3.4
    • 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 3.5
    • Confidence intervals for the mean of a normal population.
  • Topic 3.6
  • Topic 3.7
    • 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 ρ.
    • Informal interpretation of \(r\), the observed value of \(R\).
    • Scatter diagrams.
    • The following topics are based on the assumption of bivariate normality.
    • Use of the \(t\)-statistic to test the null hypothesis \(\rho = 0\) .Knowledge of the facts that the regression of \(X\) on \(Y\) (\({\text{E}}(X)|Y = y\)) and \(Y\) on \(X\) (\({\text{E}}(Y)|X = x\)) are linear.
    • Least-squares estimates of these regression lines (proof not required).The use of these regression lines to predict the value of one of the variables given the value of the other.

Topic 4 – Sets, relations and groups

  • Topic 4.1
    • 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 4.2
    • Ordered pairs: the Cartesian product of two sets.
    • Relations: equivalence relations; equivalence classes.4.3Functions: injections; surjections; bijections.
    • Composition of functions and inverse functions.4.4Binary operations.
    • Operation tables (Cayley tables).
  • Topic 4.5
    • Binary operations: associative, distributive and commutative properties.
  • Topic 4.6
    • 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 4.7
  • Topic 4.8
    • 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.
    • Example of groups: symmetries of plane figures, including equilateral triangles and rectangles.
    • Example of groups: invertible functions under composition of functions.
  • Topic 4.9
    • The order of a group.
    • The order of a group element.
    • Cyclic groups.
    • Generators.
    • Proof that all cyclic groups are Abelian.
  • Topic 4.10
    • 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 4.11
    • 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 4.12
    • 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 5 – Calculus

  • Topic 5.1
    • Infinite sequences of real numbers and their convergence or divergence.
  • Topic 5.2
    • 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 5.3
    • Continuity and differentiability of a function at a point.
    • Continuous functions and differentiable functions.
  • Topic 5.4
    • 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 5.5
    • 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 5.6
    • 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 5.7
    • The evaluation of limits of the form \(\mathop {\lim }\limits_{x \to a} \frac{{f(x)}}{{g(x)}}\) and \(\mathop {\lim }\limits_{x \to \infty } \frac{{f(x)}}{{g(x)}}\) .
    • Using l’Hôpital’s rule or the Taylor series.

Topic 6 – Discrete mathematics

  • Topic 6.1
    • Strong induction.
    • Pigeon-hole principle.
  • Topic 6.2
    • \(\left. a \right|b \Rightarrow b = na\) for some \(n \in \mathbb{Z}\) .
    • The theorem \(\left. a \right|b\) and \(\left. a \right|c \Rightarrow \left. a \right|(bx \pm cy)\) 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 6.3
    • Linear Diophantine equations \(ax + by = c\) .
  • Topic 6.4
    • Modular arithmetic.
    • The solution of linear congruences.
    • Solution of simultaneous linear congruences (Chinese remainder theorem).
  • Topic 6.5
    • Representation of integers in different bases.
  • Topic 6.6
    • Fermat’s little theorem.
  • Topic 6.7
    • 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 6.8
    • Walks, trails, paths, circuits, cycles.Eulerian trails and circuits.
    • Hamiltonian paths and cycles.
  • Topic 6.9
    • Graph algorithms: Kruskal’s; Dijkstra’s.
  • Topic 6.10
    • Chinese postman problem.
    • Travelling salesman problem.
    • Nearest neighbour algorithm for determining an upper bound.
    • Deleted vertex algorithm for determining a lower bound.
  • Topic 6.11
    • 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.

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