### 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

### **New IBDP Mathematics: applications and interpretation HL Paper 1- Syllabus**

**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 n
^{th}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.

- Arithmetic sequences and series. Use of the formulae for the n
- 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

- Financial applications of geometric sequences and series:
- 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

- Use technology to solve:

**Topic 1: Number and algebra****– AH**L content

- Topic : AHL 1.9
- Laws of logarithms.
- log
_{a}xy = log_{a}x + log_{a}y - loga\(\frac{x}{y}\)=logax – log
_{a}y - log
_{a}x^{m}= mlog_{a}x for a, x, y > 0

- log

- Laws of logarithms.
- 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 i
^{2}= − 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 ax
^{2}+bx+c=0, a≠0, with real coefficients where b^{2 }– 4ac<0.

- Complex numbers: the number i, where i
- 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
- Definition of a matrix: the terms element, row, column and order for m×n matrices.
- Algebra of matrices: equality; addition; subtraction; multiplication by a scalar for m×n matrices.
- Multiplication of matrices.
- Identity and zero matrices
- Awareness that a system of linear equations can be written in the form Ax=b.
- Solution of the systems of equations using inverse matrix.

- Topic : AHL 1.15

### Topic 2: Functions**– **SL content

- Topic: SL 2.1
- Different forms of the equation of a straight line.
- Gradient; intercepts.
- Lines with gradients m
_{1}and m_{2} - Parallel lines m
_{1}= m_{2}. - Perpendicular lines m
_{1}× m_{2}= − 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.

- Quadratic models.
- f(x)=ax
^{2}+bx+c ; a≠0. Axis of symmetry, vertex, zeros and roots, intercepts on the x-axis and y -axis.

- f(x)=ax
- Exponential growth and decay models.
- f(x)=ka
^{x}+c - f(x)=ka
^{-x}+c (for a>0) - f(x)=ke
^{rx}+c

- f(x)=ka
- Equation of a horizontal asymptote.
- Direct/inverse variation:
- f(x)=ax
^{n}, n∈ℤ - The y-axis as a vertical asymptote when n<0.

- f(x)=ax
- Cubic models:
- f(x)=ax
^{3}+bx^{2}+cx+d.

- f(x)=ax
- 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

- (f ∘ f

- Composite functions.
- 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

- Modelling skills:
- Topic: SL 2.7
- Solution of quadratic equations and inequalities. The quadratic formula.
- 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.

- Transformations of graphs.
- 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:
- cos
^{2}θ+sin^{2}θ=1

- cos
- 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.

- Vector applications to kinematics.
- 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.

- Definition and calculation of the scalar product of two vectors.
- Topic : AHL 3.14
- Topic : AHL 3.15
- Adjacency matrices.
- Walks.
- Number of k -length walks (or less than k -length walks) between two vertices.

- Weighted adjacency tables.
- Construction of the transition matrix for a strongly-connected, undirected or directed graph.

- Adjacency matrices.
- Topic : AHL 3.16
- Tree and cycle algorithms with undirected graphs.Walks, trails, paths, circuits, cycles.
- Chinese postman problem and algorithm for solution, to determine the shortest route around a weighted graph with up to four odd vertices, going along each edge at least once.
- Travelling salesman problem to determine the Hamiltonian cycle of least weight in a weighted complete graph.

- Tree and cycle algorithms with undirected graphs.Walks, trails, paths, circuits, cycles.

**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
- Linear correlation of bivariate data. Pearson’s product-moment correlation coefficient, r.
- Scatter diagrams; lines of best fit, by eye, passing through the mean point.
- Equation of the regression line of y on x.
- Use of the equation of the regression line for prediction purposes.
- Interpret the meaning of the parameters, a and b, in a linear regression y = ax + b.

- 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, r
_{s}. - 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.

- Spearman’s rank correlation coefficient, r
- 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 χ
^{2}table and justifying the choice of categorisation.- Choosing an appropriate number of degrees of freedom when estimating parameters from data when carrying out the χ
^{2}goodness of fit test.

- Choosing an appropriate number of degrees of freedom when estimating parameters from data when carrying out the χ
- Definition of reliability and validity.
- Reliability tests.
- Validity tests.

- Design of valid data collection methods, such as surveys and questionnaires.
- Topic: AHL 4.13
- Non-linear regression.
- Evaluation of least squares regression curves using technology.
- Sum of square residuals (SS
_{res}) as a measure of fit for a model. - The coefficient of determination (R
^{2}).- Evaluation of R
^{2}using technology.

- Evaluation of R

- 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.

- Poisson distribution, its mean and variance.
- Topic: AHL 4.18
- Critical values and critical regions. Test for population mean for normal distribution.
- Test for proportion using binomial distribution.
- Test for population mean using Poisson distribution.
- Use of technology to test the hypothesis that the population product moment correlation coefficient (ρ) is 0 for bivariate normal distributions.
- Type I and II errors including calculations of their probabilities.

- 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) = ax
^{n}is f ′(x) = anx^{n−1}, n ∈ ℤ - The derivative of functions of the form f(x) = ax
^{n}+ bx^{n−1}. . . . where all exponents are integers.

- Derivative of f(x) = ax
- Topic SL 5.4
- Topic: SL 5.5
- Introduction to integration as anti-differentiation of functions of the form f(x) = ax
^{n}+ bx^{n−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.

- Introduction to integration as anti-differentiation of functions of the form f(x) = ax
- 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.

- Values of x where the gradient of a curve is zero.
- 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, e
^{x}, lnx, x^{n}where n∈ℚ. - The chain rule, product rule and quotient rules.
- Related rates of change.

- The derivatives of sin x, cos x, tan x, e
- 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 x
^{n}where n∈ℚ, including n=-1 , sin x, cos x, \(\frac{1}{cos^2x}\) and e^{x}. - Integration by inspection, or substitution of the form ∫f(g(x))g′(x)dx.

- Definite and indefinite integration of x
- 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.