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

### 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
• 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.
• 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.
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