# IBDP Maths Applications and Interpretation: IB Style Question Bank HL 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: Number and algebra– SL content

• Topic : SL 1.1
• Topic : SL 1.2
• Arithmetic sequences and series. Use of the formulae for the nth term and the sum of the first n terms of the sequence.
• Use of sigma notation for sums of arithmetic sequences.
• Applications.
• Analysis, interpretation and prediction where a model is not perfectly arithmetic in real life.
• approximate common differences.
• Topic : SL 1.3
• Geometric sequences and series Use of the formulae for the n th term and the sum of the first n terms of the sequence.
• Use of sigma notation for the sums of geometric sequences.
• Applications
• Examples include the spread of disease, salary increase and decrease and population growth
• Topic : SL 1.4
• Financial applications of geometric sequences and series:
• compound interest
• annual depreciation
• Topic : SL 1.5
• Laws of exponents with integer exponents.
• Introduction to logarithms with base 10 and e. Numerical evaluation of logarithms using technology.
• Topic : SL 1.6
• Approximation: decimal places, significant figures
• Upper and lower bounds of rounded numbers.
• Percentage errors.
• Estimation.
• Topic : SL 1.7
• Amortization and annuities using technology.
• Topic : SL 1.8
• Use technology to solve:
• Systems of linear equations in up to 3 variables
• Polynomial equations

### Topic 1: Number and algebra– AHL content

• Topic : AHL 1.9
• Laws of logarithms.
• logaxy = logax + logay
• loga$$\frac{x}{y}$$=logax – logay
• logaxm = mlogax for a, x, y > 0
• Topic : AHL 1.10
• Simplifying expressions, both numerically and algebraically, involving rational exponents.
• Topic : AHL 1.11
• The sum of infinite geometric sequences.
• Topic : AHL 1.12
• Complex numbers: the number i, where i2 = − 1.
• Cartesian form z = a + bi; the terms real part, imaginary part, conjugate, modulus and argument.
• The complex plane.
• Complex numbers as solutions to quadratic equations of the form ax2+bx+c=0, a≠0, with real coefficients where b2 – 4ac<0.
• Topic : AHL 1.13
• Modulus–argument (polar) form $$z = r\left( {\cos \theta + {\text{i}}\sin \theta } \right) = r{\text{cis}}\theta$$
• Exponential form:
• $$z =r{e^{{\text{i}}\theta }}$$.
• Conversion between Cartesian, polar and exponential forms, by hand and with technology.
• Calculate products, quotients and integer powers in polar or exponential forms.
• Adding sinusoidal functions with the same frequencies but different phase shift angles.
• Geometric interpretation of complex numbers.
• Topic : AHL 1.14
• Topic : AHL 1.15

### Topic 2: Functions– SL content

• Topic: SL 2.1
• Different forms of the equation of a straight line.
• Lines with gradients m1 and m2
• Parallel lines m1 = m2.
• Perpendicular lines m1 × m2 = − 1.
• Topic: SL 2.2
• Concept of a function, domain, range and graph. Function notation, for example f(x), v(t), C(n). The concept of a function as a mathematical model.
• Informal concept that an inverse function reverses or undoes the effect of a function. Inverse function as a reflection in the line y = x, and the notation f−1(x).
• Topic: SL 2.3
• Topic: SL 2.4
• Determine key features of graphs.
• Finding the point of intersection of two curves or lines using technology
• Topic: SL 2.5
• Modelling with the following functions:
• Linear models.
• f(x)=mx+c.
• f(x)=ax2+bx+c ; a≠0. Axis of symmetry, vertex, zeros and roots, intercepts on the x-axis and y -axis.
• Exponential growth and decay models.
• f(x)=kax+c
• f(x)=ka-x+c (for a>0)
• f(x)=kerx+c
• Equation of a horizontal asymptote.
• Direct/inverse variation:
• f(x)=axn, n∈ℤ
• The y-axis as a vertical asymptote when n<0.
• Cubic models:
• f(x)=ax3+bx2+cx+d.
• Sinusoidal models:
• f(x)=asin(bx)+d,
• f(x)=acos(bx)+d.
• Topic: SL 2.5
• Composite functions.
• (f ∘ g)(x) = f(g(x))
• Identity function.
• Finding the inverse function f−1(x)
• (f ∘ f−1)(x) = (f−1∘ f)(x) = x
• Topic: SL 2.6
• Modelling skills:
• Use the modelling process described in the “mathematical modelling” section to create, fit and use the theoretical models in section SL2.5 and their graphs.
• Develop and fit the model:
• Given a context recognize and choose an appropriate model and possible parameters.
• Determine a reasonable domain for a model.
• Find the parameters of a model.
• Test and reflect upon the model:
• Comment on the appropriateness and reasonableness of a model.
• Justify the choice of a particular model, based on the shape of the data, properties of the curve and/or on the context of the situation.
• Use the model:
• Reading, interpreting and making predictions based on the model
• Topic: SL 2.7
• The discriminant $$\Delta = {b^2} – 4ac$$ and the nature of the roots, that is, two distinct real roots, two equal real roots, no real roots.
• Topic: SL 2.8
• The reciprocal function f(x) = 1 x , x ≠ 0: its graph and self-inverse nature.
• The rational function $$f(x)=\frac{{ax + b}}{{cx + d}}$$ and its graph. Equations of vertical and horizontal asymptotes.

### Topic 2: Functions– AHL content

• Topic: AHL 2.7
• Composite functions in context.
• The notation (f∘g)(x)=f(g(x)).
• Inverse function f-1, including domain restriction.
• Finding an inverse function.
• Topic : AHL 2.8
• Transformations of graphs.
• Translations: y = f(x) + b; y = f(x − a).
• Reflections (in both axes): y = − f(x); y = f( − x).
• Vertical stretch with scale factor p: y= p f(x).
• Horizontal stretch with scale factor $$\frac{1}{q}$$: y = f(qx).
• Composite transformations.
• Topic : AHL 2.9
• In addition to the models covered in the SL content the AHL content extends this to include modelling with the following functions:
• Exponential models to calculate half-life.
• Natural logarithmic models:
• f(x)=a+blnx
• Sinusoidal models:
• f(x)=asin(b(x-c))+d
• Logistic models:
• $$f(x)=\frac{L}{1+Ce^{-kx}};L,C,k>0$$
• Piecewise models.
• Topic: AHL 2.10
• Scaling very large or small numbers using logarithms.
• Linearizing data using logarithms to determine if the data has an exponential or a power relationship using best-fit straight lines to determine parameters
• Interpretation of log-log and semi-log graphs.

### Topic 3: Geometry and trigonometry-SL content

• Topic : SL 3.1
• The distance between two points in three dimensional space, and their midpoint.
• Volume and surface area of three-dimensional solids including right-pyramid, right cone, sphere, hemisphere and combinations of these solids.
• The size of an angle between two intersecting lines or between a line and a plane.
• Topic SL 3.2
• Use of sine, cosine and tangent ratios to find the sides and angles of right-angled triangles.
• The sine rule including the ambiguous case.
• $$\frac{a}{sinA}=\frac{b}{sinB}=\frac{c}{sinC}$$
• The cosine rule.
• $$c^2 = a^2 +b^2-2abcosC;$$
• $$cosC =\frac{a^2+ b^2-c^2}{2ab}$$
• Area of a triangle as $$\frac{1}{2}ab\sin C$$ .
• Topic SL 3.3
• Topic SL 3.4
• Topic SL 3.5
• Equations of perpendicular bisectors.
• Topic SL 3.6
• Voronoi diagrams: sites, vertices, edges, cells.
• Addition of a site to an existing Voronoi diagram.
• Nearest neighbour interpolation.
• Applications of the “toxic waste dump” problem.

### Topic 3: Geometry and trigonometry-AHL content

• Topic : AHL 3.7
• The definition of a radian and conversion between degrees and radians.
• Using radians to calculate area of sector, length of arc.
• Topic : AHL 3.8
• The definitions of cosθ and sinθ in terms of the unit circle.
• The Pythagorean identity:
• cos2θ+sin2θ=1
• Definition of tanθ as $$\frac{sin\theta }{cos\theta }$$
• Extension of the sine rule to the ambiguous case.
• Graphical methods of solving trigonometric equations in a finite interval.
• Topic : AHL 3.9
• Topic : AHL 3.10
• Concept of a vector; position vectors; displacement vectors.
• Representation of vectors using directed line segments.
• Unit vectors ; Base vectors i, j, k.
• Components of a vector: $$v = \left( {\begin{array}{*{20}{c}} {{v_1}} \\ {{v_2}} \\ {{v_3}} \end{array}} \right) = {v_1}i + {v_2}j + {v_3}k$$ .
• The zero vector 0, the vector -v.
• Position vectors $$\vec{{OA}}=a$$
• Rescaling and normalizing vectors.
• Topic : AHL 3.11
• Vector equation of a line in two and three dimensions: $$r = a + \lambda b$$ .
• Topic : AHL 3.12
• Vector applications to kinematics.
• Modelling linear motion with constant velocity in two and three dimensions.
• Motion with variable velocity in two dimensions.
• Topic : AHL 3.13
• Definition and calculation of the scalar product of two vectors.
• The angle between two vectors; the acute angle between two lines.
• Definition and calculation of the vector product of two vectors.
• Geometric interpretation of |v×w|.
• Components of vectors.
• Topic : AHL 3.14
• Topic : AHL 3.15
• Walks.
• Number of k -length walks (or less than k -length walks) between two vertices.
• Construction of the transition matrix for a strongly-connected, undirected or directed graph.
• Topic : AHL 3.16
•

### Topic 4 : Statistics and probability-SL content

• Topic: SL  4.1
• Concepts of population, sample, random sample and frequency distribution of discrete and continuous data.
• Reliability of data sources and bias in sampling.
• Interpretation of outliers.
• Sampling techniques and their effectiveness
• Topic: SL  4.2
• Presentation of data (discrete and continuous): frequency distributions (tables).
• Histograms. Cumulative frequency; cumulative frequency graphs; use to find median, quartiles, percentiles, range and interquartile range (IQR).
• Production and understanding of box and whisker diagrams.
• Topic: SL  4.3
• Measures of central tendency (mean, median and mode).
• Estimation of mean from grouped data.
• Modal class.
• Measures of dispersion (interquartile range, standard deviation and variance).
• Effect of constant changes on the original data.
• Quartiles of discrete data.
• Topic: SL  4.4
• Topic: SL  4.5
• 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 $$A$$).
• Expected number of occurrences.
• Topic: SL  4.6
• Use of Venn diagrams, tree diagrams, sample space diagrams and tables of outcomes to calculate probabilities.
• Combined events: P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
• Mutually exclusive events: P(A ∩ B) = 0.
• 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 $$P\left( {\left. A \right|B} \right) = P\left( A \right) = P\left( {\left. A \right|B’} \right)$$ .
• Topic: SL  4.7
• Topic: SL  4.8
• Binomial distribution. Mean and variance of the binomial distribution.
• Topic: SL  4.9
• The normal distribution and curve.
• Properties of the normal distribution.
• Diagrammatic representation.
• Normal probability calculations.
• Inverse normal calculations
• Topic: SL  4.10
• Spearman’s rank correlation coefficient, rs.
• Awareness of the appropriateness and limitations of Pearson’s product moment correlation coefficient and Spearman’s rank correlation coefficient, and the effect of outliers on each.
• Topic: SL  4.11

### Topic 4 : Statistics and probability-AHL content

• Topic: AHL  4.12
• Design of valid data collection methods, such as surveys and questionnaires.
• Selecting relevant variables from many variables.
• Choosing relevant and appropriate data to analyse.
• Categorizing numerical data in a χ2 table and justifying the choice of categorisation.
• Choosing an appropriate number of degrees of freedom when estimating parameters from data when carrying out the χ2 goodness of fit test.
• Definition of reliability and validity.
• Reliability tests.
• Validity tests.
• Topic: AHL  4.13
• Non-linear regression.
• Evaluation of least squares regression curves using technology.
• Sum of square residuals (SSres) as a measure of fit for a model.
• The coefficient of determination (R2).
• Evaluation of R2 using technology.
• Topic: AHL  4.14
• Topic: AHL  4.15
• Topic: AHL  4.16
• Topic: AHL  4.17
• Poisson distribution, its mean and variance.
• Sum of two independent Poisson distributions has a Poisson distribution.
• Topic: AHL  4.18
• Topic: AHL  4.19
• Transition matrices. Powers of transition matrices.
• Regular Markov chains.
• Initial state probability matrices.
• Calculation of steady state and long-term probabilities by repeated multiplication of the transition matrix or by solving a system of linear equations.

### Topic 5: Calculus-SL content

• Topic SL 5.1
• Introduction to the concept of a limit.
• Derivative interpreted as gradient function and as rate of change.
• Topic SL 5.2
• Increasing and decreasing functions.
• Graphical interpretation of f ′(x) > 0, f ′(x) = 0, f ′(x) < 0.
• Topic SL 5.3
• Derivative of f(x) = axn is f ′(x) = anxn−1 , n ∈ ℤ
• The derivative of functions of the form f(x) = axn + bxn−1 . . . . where all exponents are integers.
• Topic SL 5.4
• Topic: SL 5.5
• Introduction to integration as anti-differentiation of functions of the form f(x) = axn + bxn−1 + …., where n ∈ ℤ, n ≠ − 1.
• Anti-differentiation with a boundary condition to determine the constant term.
• Definite integrals using technology.
• Area of a region enclosed by a curve y = f(x) and the x -axis, where f(x) > 0.
• Topic: SL 5.6
• Values of x where the gradient of a curve is zero.
• Solution of f′(x)=0. Local maximum and minimum points.
• Topic: SL 5.7
• Optimisation problems in context.
• Topic: SL 5.8
• Approximating areas using the trapezoidal rule.

### Topic 5: Calculus-AHL content

• Topic: AHL 5.9
• The derivatives of sin x, cos x, tan x, ex, lnx, xn where n∈ℚ.
• The chain rule, product rule and quotient rules.
• Related rates of change.
• Topic: AHL 5.10
• The second derivative.
• Use of second derivative test to distinguish between a maximum and a minimum point.
• Topic: AHL 5.11
• Definite and indefinite integration of xn where n∈ℚ, including n=-1 , sin x, cos x, $$\frac{1}{cos^2x}$$ and ex.
• Integration by inspection, or substitution of the form ∫f(g(x))g′(x)dx.
• Topic: AHL 5.12
• Area of the region enclosed by a curve and the x or y-axes in a given interval.
• Volumes of revolution about the x- axis or y- axis.
• Topic: AHL 5.13
• Kinematic problems involving displacement s, velocity v and acceleration a.
• Topic: AHL 5.14
• Setting up a model/differential equation from a context.
• Solving by separation of variables.
• Topic: AHL 5.15
• Slope fields and their diagrams.
• Topic: AHL 5.16
• Topic: AHL 5.17
• Phase portrait for the solutions of coupled differential equations of the form:
• $$\frac{dx}{dt}$$=ax+by
• $$\frac{dy}{dt}$$=cx+dy.
• Qualitative analysis of future paths for distinct, real, complex and imaginary eigenvalues.
• Sketching trajectories and using phase portraits to identify key features such as equilibrium points, stable populations and saddle points.
• Topic: AHL 5.18
• Solutions of $$\frac{d^2x}{dt^2}$$=f(x,$$\frac{dx}{dt}$$,t) by Euler’s method.