HSC Extension One Maths Papers

Explore free HSC extension one maths questions organised into exam style topic tests with worked solutions. Unlike the other subjects on this site, these tests will be of varying lengths as each topic has a different amount of content and I didn’t want to write tests that ask the same fundamental question multiple times. Each test includes 5 multiple choice questions and 1-3 15 mark short answer segments based on the amount of content in each topic. For any questions or concerns about these papers, feel free to contact me and I’ll reply as soon as possible.

The HSC extension one maths syllabus includes a range of topics which can make it confusing to remember what to study, below is a skills checklist based on the syllabus with everything you should understand going into the final exam. Note that the year 11 section is based on the 2024 syllabus while the year 12 section is based on the 2017 syllabus and will be updated to reflect the 2024 syllabus at the beginning of term 4 when this syllabus is taught to year 12.

Graphical Relationships:

  • Graph the reciprocal of an algebraic function, or of a graph

  • Graph trigonometric functions in radians and degrees

  • Graph absolutes of functions and functions of absolutes

  • Graph the addition and subtraction of functions which may be given in algebraic or graphical form

Inequalities:

  • Solve cubic inequalities where the cubic is expressed as a product of linear factors

  • Solve rational inequalities

  • Solve absolute value inequalities

Inverse Functions:

  • Understand which functions can have an inverse and know the horizontal line test

  • Recognise that a function’s inverse is its reflection over the line y=x

  • Determine the inverse of a given function

  • Graph a function’s inverse

  • Restrict the domain of a function such that it has an inverse

Parametric Form of a Function or Relation:

  • Express linear and quadratic functions and circles in parametric form

  • Convert between cartesian and parametric form

Language and Graphs of Polynomials:

  • Understand the definition of a polynomial

  • Understand terms such as leading term, degree and leading coefficient

  • Understand how the leading coefficient and degree determine behaviours as x approaches positive and negative infinity

  • Understand the multiplicity of roots

  • Find the zeros of polynomials expressed as a product of linear factors

Remainder and Factor Theorems:

  • Divide polynomials by other polynomials

  • Determine the remainder of polynomial division

  • Determine when x-a is a factor of a polynomial

  • Use the factor theorem to factorise polynomials

Sums and Products of Zeroes of Polynomials:

  • Derive the coefficients of a polynomials up to degree 4 from their roots

  • Use these relationships to solve problems involving zeroes of polynomials up to degree 4

Trigonometry in Three Dimensions:

  • Interpret information about 3 dimensional spaces from written or visual prompts

  • Use trigonometry to solve problems in three dimensions with a diagram

Further Trigonometric Identities:

  • Derive and use the sum and difference expansions for the trigonometric functions sin(x), cos(x) and tan(x)

  • Derive and use the double angle formulas for sin(x), cos(x) and tan(x)

  • Use these formulas to solve problems and prove results

Further Trigonometric Equations:

  • Solve trigonometric equations involving factorisation and/or substitution of trigonometric identities

  • Understand the relationship acos(x)+bsin(x)=Rsin(x+α) or Rcos(x+α)

  • Use this relationship to solve equations

  • Interpret solutions of trigonometric equations graphically

Permutations and Combinations:

  • Understand factorial notation

  • Understand and use the multiplication principle

  • Understand permutation notation and how it relates to the multiplication principle

  • Solve problems involving permutations with restrictions on the placement of certain objects

  • Understand why the number of ways to arrange n objects in a circle is (n-1)!

  • Understand the concept of a combination where order is no longer important

  • Solve problems involving permutations, combinations or both

The Binomial Theorem:

  • Understand the link between the coefficients of binomial expansions and Pascal’s triangle

  • Understand the link between binomial coefficients and combinations

  • Derive the binomial theorem

  • Use the binomial theorem to expand and simplify expressions

  • Use the binomial theorem to determine specific coefficients in an expansion

  • Prove identities using the binomial theorem

Proof by Mathematical Induction:

  • Understand the steps of an inductive proof

  • Use induction to prove results for sums and for divisibility

  • Identify false proofs by induction

  • Recognise what induction can be used to prove and what it cannot

Introduction to Vectors:

  • Understand the concept of a vector

  • Understand the difference between a position vector and a displacement vector

  • Use vector notation

  • Perform basic vector operations such as addition, subtraction and multiplication by a scalar

  • Understand the parallelogram and triangle laws for vector addition and subtraction

Further Operations with Vectors:

  • Define and calculate the magnitude of a vector

  • Understand that the magnitude of the difference of two vectors is the distance between them

  • Convert a non-zero vector into a unit vector by dividing by its length

  • Define and use the dot product of two vectors

  • Use the dot product to solve problems

  • Determine of two vectors are parallel or perpendicular

  • Use vector projections

  • Use vectors to prove geometric results

Projectile Motion:

  • Understand the assumptions made in projectile motion calculations

  • Use vectors to model the motion of a projectile

  • Derive the horizontal and vertical equations of motion for a given particle

  • Apply calculus to the equations of motion to solve projectile motion questions

Trigonometric Functions:

  • Understand the compound angle formulas

  • Understand the t-formulae

  • Use the compound angle or t-formulae to solve equations

  • Prove trigonometric identities

Further Calculus Skills:

  • Use integration by substitution where the substitution is given

  • Solve problems involving the integrals of squared trigonometric functions

  • Find derivatives using the relationship dy/dx=1/(dx/dy)

  • Solve problems involving the derivatives of inverse trigonometric functions

Further Area and Volumes of Solids of Revolution:

  • Calculate the area of regions between curves

  • Sketch the graph of a solid of revolution

  • Calculate the volume of a solid of revolution

Differential Equations:

  • Recognise that the solutions to differential equations are functions and that they may not be unique

  • Sketch the graph of a particular solution given a direction field and initial conditions

  • Solve simple first order differential equations

  • Understand the logistic equation

Bernoulli and Binomial Distributions:

  • Use Bernoulli random variables to solve problems

  • Understand the formulas for the mean and variance

  • Understand the concept of Bernoulli trials

  • Use the binomial distribution to solve problems

  • Understand notations involved with the binomial distribution

  • Understand when it is suitable to use the binomial distribution

Normal Approximation for the Sample Proportion:

  • Understand the concept of the sample proportion as a random variable whose value varies between samples

  • Understand and use the normal approximation to the distribution of the sample proportion and its limitations