Samyak Math AI HL Tracker – Sprout Education

Operations with numbers in the form $a \times 10^k$

Arithmetic sequences and series

Geometric sequences and series

Financial applications of geometric sequences and series (compound interest, annual depreciation)

Laws of exponents with integer exponents

Approximation: decimal places, significant figures, upper and lower bounds, percentage errors, estimation

Amortization and annuities using technology

Use technology to solve systems of linear equations and polynomial equations

Laws of logarithms

Simplifying expressions involving rational exponents

The sum of infinite geometric sequences

Complex numbers: $i$ such that $i^2 = -1$, Cartesian form $z = a + bi$, calculations

Complex numbers as solutions to quadratic equations ($b^2 – 4ac < 0$)

Modulus-argument (polar) form: $z = r(\cos\theta + i\sin\theta)$, exponential form: $z = re^{i\theta}$

Conversion between forms, products, quotients, integer powers of complex numbers

Geometric interpretation of complex numbers

Matrices: definition, algebra, multiplication

Determinants and inverses of $n \times n$ matrices (by hand for $2 \times 2$)

Solution of systems of equations using inverse matrix

Eigenvalues and eigenvectors, diagonalization of $2 \times 2$ matrices

Applications to powers of $2 \times 2$ matrices

Different forms of the equation of a straight line; gradient; intercepts

Concept of a function, domain, range and graph; function notation

Informal concept that an inverse function; inverse function as a reflection in $y=x$

The graph of a function $y=f(x)$; creating sketches

Using technology to graph functions

Determine key features of graphs (max/min, intercepts, symmetry, asymptotes)

Finding the point of intersection of two curves/lines using technology

Modelling with linear, quadratic, exponential, direct/inverse variation, cubic, sinusoidal functions

Modelling skills: use the modelling process, develop and fit the model, use the model

Composite functions; notation $f \circ g(x) = f(g(x))$

Inverse function $f^{-1}$, including domain restriction

Transformations of graphs (translations, reflections, stretches)

Modelling with natural logarithmic models $f(x)=a+b \ln x$

Modelling with sinusoidal models $f(x)=a\sin(b(x-c))+d$

Modelling with logistic models $f(x)=\frac{L}{1+Ce^{-kx}}$

Piecewise models

Scaling very large or small numbers using logarithms

Linearizing data using logarithms to determine exponential or power relationship

Interpretation of log-log and semi-log graphs

Distance between two points in three-dimensional space, and their midpoint

Volume and surface area of three-dimensional solids

Size of an angle between two intersecting lines or between a line and a plane

Use of sine, cosine and tangent ratios in right-angled triangles

The sine rule, cosine rule, area of a triangle

Applications of right and non-right angled trigonometry (Pythagoras’ theorem, angles of elevation/depression, bearings)

The circle: length of an arc; area of a sector

Equations of perpendicular bisectors

Voronoi diagrams: sites, vertices, edges, cells, addition of a site

Nearest neighbour interpolation; applications of “toxic waste dump” problem

Definition of a radian and conversion between degrees and radians

Using radians to calculate area of sector, length of arc

Definitions of $\cos\theta$ and $\sin\theta$ in terms of the unit circle

Pythagorean identity: $\cos^2\theta + \sin^2\theta = 1$; definition of $\tan\theta$

Extension of the sine rule to the ambiguous case

Graphical methods of solving trigonometric equations

Geometric transformations of points in two dimensions using matrices

Compositions of transformations

Geometric interpretation of the determinant of a transformation matrix

Concept of a vector and a scalar; representation of vectors

Unit vectors; base vectors $i, j, k$; components of a vector; column representation

Zero vector and vector $-v$

Position vectors $\vec{OA} = a$; rescaling and normalizing vectors

Vector equation of a line in two and three dimensions: $r = a + \lambda b$

Vector applications to kinematics; modelling linear motion with constant velocity

Motion with variable velocity in two dimensions

Scalar product of two vectors; angle between two vectors

Vector product of two vectors; geometric interpretation

Components of vectors

Graph theory: Graphs, vertices, edges, adjacent vertices, degree of a vertex

Simple graphs; complete graphs; weighted graphs

Directed graphs; in-degree and out-degree; subgraphs; trees

Adjacency matrices; walks; number of $k$-length walks

Weighted adjacency tables; construction of transition matrix

Tree and cycle algorithms with undirected graphs (walks, trails, paths, circuits, cycles)

Eulerian trails and circuits; Hamiltonian paths and cycles

Minimum spanning tree (MST) graph algorithms (Kruskal’s and Prim’s)

Chinese postman problem and algorithm for solution

Travelling salesman problem (Hamiltonian cycle, nearest neighbour, deleted vertex)

Concepts of population, sample, random sample, discrete and continuous data

Reliability of data sources and bias in sampling

Interpretation of outliers

Sampling techniques and their effectiveness

Presentation of data (discrete and continuous): frequency distributions (tables), histograms

Cumulative frequency; cumulative frequency graphs; use to find median, quartiles, percentiles, range, IQR

Production and understanding of box and whisker diagrams

Measures of central tendency (mean, median and mode); estimation of mean from grouped data

Measures of dispersion (interquartile range, standard deviation and variance)

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 for prediction

Interpret the meaning of the parameters, $a$ and $b$, in a linear regression $y = ax+b$

Concepts of trial, outcome, equally likely outcomes, relative frequency, sample space, event

Probability of an event $A$ is $P(A) = \frac{n(A)}{n(U)}$; complementary events $A$ and $A’$

Expected number of occurrences

Use of Venn diagrams, tree diagrams, sample space diagrams, tables of outcomes

Combined events: $P(A \cup B) = P(A) + P(B) – P(A \cap B)$; mutually exclusive events

Conditional probability: $P(A|B) = \frac{P(A \cap B)}{P(B)}$; independent events

Concept of discrete random variables and their probability distributions

Expected value (mean), $E(X)$ for discrete data

Applications of discrete random variables

Binomial distribution; mean and variance

The normal distribution and curve; properties of the normal distribution

Diagrammatic representation of normal distribution

Normal probability calculations; inverse normal calculations

Spearman’s rank correlation coefficient, $r_s$

Awareness of appropriateness and limitations of Pearson’s and Spearman’s coefficients

Modelling and finding structure in seemingly random events

Different probability distributions provide a representation of relationship between theory and reality

Statistical literacy involves identifying reliability and validity of samples and whole populations

Systematic approach to hypothesis testing allows statistical inferences to be tested for validity

Representation of probabilities using transition matrices to predict long-term behaviour

Design of valid data collection methods (surveys, questionnaires)

Selecting relevant variables and appropriate data to analyse

Categorizing numerical data in a $\chi^2$ table and justifying the choice of categorization

Choosing appropriate number of degrees of freedom

Definition of reliability and validity; reliability and validity tests

Non-linear regression; evaluation of least squares regression curves using technology

Sum of square residuals ($SS_{res}$) as a measure of fit

The coefficient of determination ($R^2$); evaluation of $R^2$ using technology

Linear transformation of a single random variable; expected value and variance

Expected value and variance of linear combinations of random variables

$\overline{x}$ as an unbiased estimate of $\mu$; $s_{n-1}^2$ as an unbiased estimate of $\sigma^2$

Linear combination of independent normal random variables is normally distributed

Central limit theorem

Confidence intervals for the mean of a normal population

Poisson distribution, its mean and variance

Sum of two independent Poisson distributions

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

Type I and II errors including calculations of their probabilities

Transition matrices; powers of transition matrices

Regular Markov chains; initial state probability matrices

Calculation of steady state and long-term probabilities

Introduction to the concept of a limit

Derivative interpreted as gradient function and as rate of change

Forms of notation: $\frac{dy}{dx}, \frac{dV}{dr}, \frac{ds}{dt}$, $f'(x)$

Informal understanding of the gradient of a curve as a limit

Increasing and decreasing functions

Graphical interpretation of $f'(x)>0, f'(x)=0, f'(x)<0$

Derivative of $f(x)=ax^n$ is $f'(x)=anx^{n-1}$, $n \in \mathbb{Z}$

Derivative of functions of the form $f(x)=ax^n+bx^{n-1}+…$

Tangents and normals at a given point, and their equations

Introduction to integration as anti-differentiation

Anti-differentiation with a boundary condition to determine the constant term

Definite integrals using technology; area of a region enclosed by a curve and the x-axis

Values of $x$ where the gradient of a curve is zero; solution of $f'(x)=0$

Local maximum and minimum points

Optimization problems in context

Approximating areas using the trapezoidal rule

The derivatives of $\sin x, \cos x, \tan x, e^x, \ln x, x^n$ where $n \in \mathbb{Q}$

The chain rule, product rule and quotient rules

Related rates of change

The second derivative; forms of notation $\frac{d^2y}{dx^2}$, $f”(x)$

Use of second derivative test to distinguish between max/min

Integration by inspection or substitution of the form $\int f(g(x))g'(x)dx$

Area of the region enclosed by a curve and the x or y-axes

Volumes of revolution about the x-axis or y-axis

Kinematic problems involving displacement $s$, velocity $v$, acceleration $a$

Motion with variable velocity in two dimensions

Setting up a model/differential equation from a context

Solving by separation of variables

Numerical solution of the coupled system $\frac{dx}{dt}=f_1(x,y,t)$ and $\frac{dy}{dt}=f_2(x,y,t)$

Phase portrait for the solutions of coupled differential equations

Qualitative analysis of future paths for distinct, real, complex and imaginary eigenvalues

Sketching trajectories and using phase portraits to identify key features

Euler’s method for finding the approximate solution to first order differential equations

Numerical solution of $\frac{dy}{dx}=f(x,y)$

Solutions of $\frac{d^2x}{dt^2}=f(x,\frac{dx}{dt})$ by Euler’s method

Solutions of $\frac{d^2x}{dt^2}+a\frac{dx}{dt}+b=0$ can be investigated using phase portrait method

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