NSW · NESASyllabus
Maths Extension 1 syllabus, dot point by dot point
Every dot point in the NSW Maths Extension 1 syllabus, with a focused answer for each one. Click any dot point for a worked explainer, past exam questions, and links to related dot points. Generated by Claude Opus and reviewed by Better Tuition Academy tutors.
Calculus (ME-C1, C2, C3)
Module overview →- How do we model and solve problems involving exponential growth and decay using ?Model unrestricted growth and decay with and solve the resulting separable differential equation7 min answer →
- How do we use the substitution method to evaluate integrals that arise from the reverse chain rule?Apply integration by substitution to evaluate definite and indefinite integrals, including reverse chain rule cases9 min answer →
- Which integrals lead to inverse trigonometric antiderivatives, and how do we recognise them?Integrate functions whose antiderivative involves , or8 min answer →
- What are the derivatives and antiderivatives of the inverse trigonometric functions?Differentiate inverse trigonometric functions and integrate functions that involve them7 min answer →
- How do we model projectile motion in two dimensions, and what quantities (range, maximum height, time of flight) can we extract?Model projectile motion in two dimensions using parametric equations and find range, maximum height, time of flight and trajectory equation9 min answer →
- How do we link the rate of change of one quantity to the rate of change of another using the chain rule?Solve related-rates problems by linking two changing quantities via an equation and differentiating with respect to time7 min answer →
- How do we solve a first-order differential equation when the variables separate?Solve separable first-order differential equations of the form by separating variables and integrating both sides8 min answer →
- How do we compute the volume of a solid generated by rotating a region around an axis?Calculate volumes of revolution about the x-axis and y-axis using the disc method8 min answer →
Combinatorics (ME-A1)
Module overview →- How do we expand , and what does Pascal's triangle reveal about the coefficients?State and use the binomial theorem, identify general and specific terms, and relate it to Pascal's triangle9 min answer →
- How do we count the number of unordered selections (combinations) of objects from a larger set?Use the combination formula to count unordered selections, including with restrictions and complementary counting8 min answer →
- How do we count the number of ordered arrangements of a set of objects?Use the multiplication principle and the permutation formula to count ordered arrangements, including restrictions and repeated elements8 min answer →
- When can we guarantee that some box contains more than one object, and how do we apply this to counting and existence arguments?State and apply the pigeonhole principle in counting and existence problems6 min answer →
Functions (ME-F1, ME-F2)
Module overview →- How do parametric equations describe a curve, and how do we convert between parametric and Cartesian forms?Sketch curves given parametrically, eliminate the parameter to obtain Cartesian equations, and use parametric form for circles, parabolas and lines7 min answer →
- How do we divide one polynomial by another and use the remainder and factor theorems to find roots?Apply the division algorithm for polynomials and use the remainder and factor theorems to identify and verify factors and roots9 min answer →
- How does the multiplicity of a root affect the graph of a polynomial near that root?Sketch polynomial functions using leading-term behaviour, intercepts and the multiplicity of each root7 min answer →
- How do we solve polynomial and rational inequalities using sign analysis?Solve polynomial and rational inequalities by factoring and analysing the sign of each factor across critical values8 min answer →
- How do the coefficients of a polynomial relate to the sum and product of its roots?Use the relationships between roots and coefficients (Vieta's formulas) for polynomials of degree two, three and four8 min answer →
Proof (ME-P1)
Module overview →- How do we use mathematical induction to prove divisibility statements?Prove divisibility statements involving an integer parameter using mathematical induction8 min answer →
- How do we use induction to prove general statements involving a recursion, a formula or a property?Apply mathematical induction to prove general statements about a recursive sequence, a property of a formula, or a recursive procedure7 min answer →
- How do we use induction to prove an inequality holds for every positive integer?Prove inequalities involving an integer parameter using mathematical induction7 min answer →
- How do we use mathematical induction to prove identities involving sums of a sequence?Prove identities for sums of series using the principle of mathematical induction8 min answer →
Statistical Analysis (ME-S1)
Module overview →- What is a Bernoulli trial, and what are its mean and variance?Define a Bernoulli random variable, compute its mean and variance, and recognise scenarios that fit the model6 min answer →
- What is the binomial distribution, and what are its mean and variance?Define the binomial distribution , state its probability mass function, and find its mean and variance7 min answer →
- How do we calculate probabilities involving the binomial distribution, including ranges and complements?Compute exact probabilities for the binomial distribution including , , , and use complementary counting7 min answer →
- When and how do we use the normal distribution to approximate binomial probabilities?Use the normal approximation to approximate binomial probabilities for large7 min answer →
Trigonometric Functions (ME-T1, T2, T3)
Module overview →- How do we write as a single sinusoid, and what is this used for?Express in the form or and use this to solve equations and find extreme values7 min answer →
- How do we find every solution to a trigonometric equation, not just the principal one?Write general solutions to trigonometric equations using the period and the symmetries of , and7 min answer →
- What are the inverse trigonometric functions, and what are their domains, ranges and graphs?Define and sketch the inverse trigonometric functions , and , including their domains and ranges9 min answer →
- How do we convert between products and sums of trigonometric functions?Use the product-to-sum and sum-to-product identities to simplify trigonometric expressions and integrals7 min answer →
- How do we expand , and , and what are these identities used for?Use the sum and difference identities for sine, cosine and tangent to expand or simplify trigonometric expressions8 min answer →
- How does the t-substitution help simplify and solve trigonometric equations?Use the t-formula (Weierstrass substitution) to express , and as rational functions of7 min answer →
Vectors (ME-V1)
Module overview →- How do we use vectors to prove standard geometric results in the plane?Use vector methods to prove geometric properties, including parallelism, perpendicularity, midpoint and ratio division8 min answer →
- How do we represent a line in the plane using vector equations?Write parametric vector equations of lines and convert between vector and Cartesian forms7 min answer →
- What is the scalar (dot) product of two vectors, and what does it measure?Compute the scalar product of two vectors in component or geometric form and use it to find the angle between vectors and test orthogonality8 min answer →
- How do we add, subtract and scale two-dimensional vectors, and how do we find their magnitudes?Perform vector arithmetic with vectors in the plane, including component and column-vector notation, and find the magnitude and unit vector8 min answer →
- How do we find the projection of one vector onto another, and what does it mean geometrically?Compute the scalar and vector projection of one vector onto another and interpret it geometrically7 min answer →