# IB DP Maths HL: Past Years Question Bank with Solution Paper – 1

### 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 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.