ATARMAxxing
← All official papers

NSW · HSC · Practice by topic

HSC Mathematics Advanced Questions by Topic

Browse 30 selected questions from the 2023 and 2024 HSC exams. Choose a skill, open the question’s page in NESA’s PDF and check your work with the marking guidelines in NESA’s marking guide.

These papers were set under the 2017 syllabus. The first HSC exam for the replacement Mathematics Advanced syllabus is in 2027. Check your syllabus before using these questions. NESA syllabus and implementation dates.

Functions

Graphing techniques

  • 2023 Q19 · 4 marks. Sketch a linear function and a quadratic function on the same axes and use the graphs to solve an inequality between them. Paper (PDF) · Marking guidelines
  • 2023 Q27 · 5 marks. Determine the parameters of an absolute-value function from three given points on its graph, then find the values of a line's gradient for which it intersects the graph exactly twice. Paper (PDF) · Marking guidelines

Trigonometric Functions

Trigonometric functions and graphs

  • 2023 Q16 · 4 marks. Calculate the perimeter of a composite shape combining a rectangle with a circular arc, using arc length formulas. Paper (PDF) · Marking guidelines
  • 2023 Q20 · 3 marks. Solve a trigonometric equation involving a phase-shifted sine function over a restricted domain in degrees. Paper (PDF) · Marking guidelines
  • 2023 Q22 · 3 marks. Use three-dimensional trigonometry to find an angle within a rectangular prism given a midpoint of one edge. Paper (PDF) · Marking guidelines

Calculus

Integral calculus

  • 2024 Q14 · 4 marks. Find the intersection points of a parabola and another quadratic curve, then calculate the area enclosed between them by integration. Paper (PDF) · Marking guidelines
  • 2024 Q15 · 3 marks. Given a rate of change of volume as a function of time, find the volume at the instant the rate becomes zero. Paper (PDF) · Marking guidelines
  • 2023 Q13 · 2 marks. Solve a differential equation involving an exponential growth rate to find an explicit function of time, given an initial condition. Paper (PDF) · Marking guidelines
  • 2023 Q17 · 2 marks. Evaluate an indefinite integral of a rational function using a substitution-style technique. Paper (PDF) · Marking guidelines
  • 2023 Q26 · 4 marks. Integrate a sinusoidal rate-of-change function to model the position of a swinging object over time, then count how many times it reaches its closest point within a time interval. Paper (PDF) · Marking guidelines
  • 2023 Q32 · 6 marks. Find the area enclosed between two exponential curves and the y-axis using integration, then determine the range of a parameter for which two shifted exponential curves intersect twice. Paper (PDF) · Marking guidelines

Differential calculus

  • 2023 Q14 · 3 marks. Find the equation of a tangent line to a cubic-composite curve at a given point using the chain rule. Paper (PDF) · Marking guidelines
  • 2023 Q24 · 5 marks. Derive a formula linking two variables for a garden-and-path optimisation problem, then find the dimension that minimises the path area using calculus, confirming it is a minimum. Paper (PDF) · Marking guidelines
  • 2023 Q28 · 4 marks. Given a tangent line at one point on a curve and the curve's gradient function, find the coordinates of a second point where the tangent is parallel. Paper (PDF) · Marking guidelines

The second derivative

  • 2024 Q19 · 5 marks. Sketch a quartic curve by finding all stationary points, classifying their nature, and locating points of inflection. Paper (PDF) · Marking guidelines
  • 2023 Q30 · 5 marks. Find and classify (by inspection, without further calculus) the stationary points of a product of an exponential and a trigonometric function over one period, then sketch the curve. Paper (PDF) · Marking guidelines

Exponential and Logarithmic Functions

Applications of exponential and logarithmic functions

  • 2024 Q13 · 3 marks. Compare two exponential population-growth/decay models from a graph, completing a table of initial values, percentage rates of change, and a predicted future value. Paper (PDF) · Marking guidelines
  • 2024 Q17 · 6 marks. Sketch the charging-curve behaviour of a capacitor's voltage over time, solve for an unknown decay constant given one data point, then find the instantaneous rate of voltage increase at a specified time. Paper (PDF) · Marking guidelines

Financial Mathematics

Modelling financial situations

  • 2023 Q15 · 5 marks. Use a future-value annuity factor table to calculate required periodic contributions and a final annuity balance under compounding. Paper (PDF) · Marking guidelines
  • 2023 Q21 · 3 marks. Find possible common ratios and first terms of a geometric sequence given its fourth and eighth terms. Paper (PDF) · Marking guidelines
  • 2023 Q25 · 6 marks. Derive a recurrence and closed-form expression for an account balance under monthly compounding with regular withdrawals, then find the maximum sustainable withdrawal for a target number of withdrawals. Paper (PDF) · Marking guidelines

Statistical Analysis

Random variables

  • 2023 Q12 · 3 marks. Verify the expected value of a discrete random variable from a probability distribution table, then calculate its standard deviation. Paper (PDF) · Marking guidelines
  • 2023 Q29 · 6 marks. For a continuous random variable with a given probability density function, find the mode, derive the cumulative distribution function, and prove an inequality between the mode and median without computing the median directly. Paper (PDF) · Marking guidelines

Descriptive statistics and bivariate data analysis

  • 2024 Q16 · 3 marks. Compare two datasets shown as parallel box plots in terms of skewness, central tendency, and spread. Paper (PDF) · Marking guidelines
  • 2023 Q18 · 6 marks. Plot the mean point and y-intercept of a least-squares regression line from summary statistics, state the regression equation, and identify a limitation of extrapolating the model. Paper (PDF) · Marking guidelines

The normal distribution

  • 2023 Q23 · 4 marks. Use a standard normal probability table and given mean/standard deviation to find an expected count exceeding a threshold weight in a normally distributed population. Paper (PDF) · Marking guidelines

Probability

Conditional probability and independence

  • 2024 Q18 · 3 marks. Find the probability of no successes in two independent attempts, then determine the minimum number of attempts needed for a cumulative success probability to exceed a threshold. Paper (PDF) · Marking guidelines
  • 2023 Q31 · 5 marks. Use given conditional probabilities for two events to test independence, then compute a marginal probability and the probability that at least one of several independent trials fails. Paper (PDF) · Marking guidelines

Sequences and series

Modelling financial situations