# IB DP Further Mathematics HL: Past Years Question Bank with Solution Paper -2

### Paper 2

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 1.1
• Topic 1.2
• Topic1.3
• Elementary row and column operations for matrices.
• Scaling, swapping and pivoting.
• Corresponding elementary matrices.
• Row reduced echelon form.
• Row space, column space and null space.
• Row rank and column rank and their equality.
• Topic 1.4
• Solutions of $$m$$ linear equations in $$n$$ unknowns: both augmented matrix method, leading to reduced row echelon form method, and inverse matrix method, when applicable.
• Topic 1.5
• The vector space $${\mathbb{R}^n}$$ .
• Linear combinations of vectors.
• Spanning set.
• Linear independence of vectors.
• Basis and dimension for a vector space.
• Subspaces.
• Topic 1.6
• Linear transformations: $$T(u + v) = T(u) + T(v)$$ , $$T(ku) = kT(u)$$ .
• Composition of linear transformations.
• Domain, range, codomain and kernel.
• Result and proof that the kernel is a subspace of the domain.
• Result and proof that the range is a subspace of the codomain.
• Rank-nullity theorem (proof not required).
• Topic 1.7
• Result that any linear transformation can be represented by a matrix, and the converse of this result.
• Result that the numbers of linearly independent rows and columns are equal, and this is the dimension of the range of the transformation (proof not required).
• Application of linear transformations to solutions of system of equations.
• Solution of $$A{\text{x}} = {\text{b}}$$ .
• Topic 1.8
• Geometric transformations represented by $$2 \times 2$$ matrices include general rotation, general reflection in $$y = (\tan \alpha )x$$ , stretches parallel to axes, shears parallel to axes, and projection onto $$y = (\tan \alpha )x$$.
• Compositions of the above transformations.
• Geometric interpretation of determinant.
• Topic 1.9
• Eigenvalues and eigenvectors of $$2 \times 2$$ matrices.
• Characteristic polynomial of $$2 \times 2$$ matrices.
• Diagonalization of $$2 \times 2$$ matrices (restricted to the case where there are distinct real eigenvalues).
• Applications to powers of $$2 \times 2$$ matrices.

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
• 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.