VCE Vault

VCE Specialist Mathematics past exams 2023–2025

Every question from the last three VCE Specialist Mathematics exams, matched to the lesson that teaches it, with class averages from the examiners' reports and worked solutions. VCE Specialist Mathematics (Year 12) lessons.

2025 Exam 1

  • Q1 (4 marks, class average 73%): Tangent at $(4, -2)$ to the curve $xe^{-2y} + y^2e^x = 8e^4$. Lesson: Implicit Differentiation and Related Rates
  • Q2 (3 marks, class average 73%): Point where two lines in 3D meet. Lesson: Lines and Planes in 3D
  • Q3a (1 mark, class average 71%): Integrate $\frac{t}{\sqrt{t^2 + k}}$ to show the displacement formula. Lesson: Kinematics with Calculus
  • Q3b (2 marks, class average 70%): Initial acceleration when $v = \frac{t}{\sqrt{t^2 + k}}$. Lesson: Kinematics with Calculus
  • Q3c (2 marks, class average 40%): Find $k$ when a second particle at $s = t$ leads by 1 m after 3 s. Lesson: Kinematics with Calculus
  • Q4a (3 marks, class average 50%): Use partial fractions to show a waiting-time pdf has mean $\frac{1}{2}$. Lesson: Integration Techniques — Substitution and Partial Fractions
  • Q4b (2 marks, class average 60%): Chance a mean of 25 waiting times is between 0.44 and 0.5 hours. Lesson: Linear Combinations of Random Variables and Sample Means
  • Q5a (1 mark, class average 94%): Show $c = -4$ for two particles to collide at $t = 1$. Lesson: Vector Calculus — Kinematics in Two Dimensions
  • Q5b (2 marks, class average 50%): Values of $a$ when the velocities at collision are perpendicular. Lesson: Vector Calculus — Kinematics in Two Dimensions
  • Q5c (1 mark, class average 25%): Find $a$ and $b$ when the accelerations also have equal size. Lesson: Vector Calculus — Kinematics in Two Dimensions
  • Q6 (4 marks, class average 63%): Volume when $\sqrt{\frac{\arctan(x)}{1 + x^2}}$ is rotated about the $x$-axis, from 1 to $\sqrt{3}$. Lesson: Volumes of Revolution and Arc Length, Integration Techniques — Substitution and Partial Fractions
  • Q7 (4 marks, class average 65%): Prove the formula for $2^2 + 3^2 + \ldots + (n + 1)^2$ by induction. Lesson: Proof by Mathematical Induction
  • Q8a (1 mark, class average 84%): Plot $1 + 2i$ and its conjugate on an Argand diagram. Lesson: Complex Numbers — Introduction and the Complex Plane
  • Q8b (2 marks, class average 70%): Quadratic factor of $z^4 + 6z^2 + 25$ from the root $1 + 2i$. Lesson: Complex Numbers — Introduction and the Complex Plane
  • Q8c (2 marks, class average 55%): Remaining roots of $z^4 + 6z^2 + 25$. Lesson: Complex Numbers — Introduction and the Complex Plane
  • Q9a (2 marks, class average 60%): Rewrite $\frac{x^3 + x^2 - 2x}{1 - x^2}$ as $-x - 1 + \frac{1}{x + 1}$. Lesson: Rational Functions — Asymptotes, Sketching and Partial Fractions
  • Q9b (1 mark, class average 46%): Value of $k$ that makes the function continuous at $x = 1$. Lesson: Rational Functions — Asymptotes, Sketching and Partial Fractions
  • Q9c (3 marks, class average 33%): Sketch the rational function with its asymptotes. Lesson: Rational Functions — Asymptotes, Sketching and Partial Fractions

2025 Exam 2

  • MC 1 (1 mark, class average 93%): Identify the contrapositive of a conditional statement. Lesson: Proof Techniques — Contradiction and Contrapositive
  • MC 2 (1 mark, class average 48%): Find an example where a zero second derivative does not imply an inflection. Lesson: Implicit Differentiation and Related Rates
  • MC 3 (1 mark, class average 68%): Use an oblique asymptote and an intercept to find constants in a rational function. Lesson: Rational Functions — Asymptotes, Sketching and Partial Fractions
  • MC 4 (1 mark, class average 72%): Evaluate a left-endpoint algorithm for a volume approximation. Lesson: Volumes of Revolution and Arc Length
  • MC 5 (1 mark, class average 69%): Find polynomial coefficients using a known complex root. Lesson: Complex Numbers — Introduction and the Complex Plane
  • MC 6 (1 mark, class average 70%): Calculate the real part of a reciprocal involving a unit complex number. Lesson: Complex Numbers — Introduction and the Complex Plane
  • MC 7 (1 mark, class average 79%): Substitute a cosine expression into a trigonometric integral. Lesson: Integration Techniques — Substitution and Partial Fractions
  • MC 8 (1 mark, class average 80%): Match a direction field to a differential equation. Lesson: Differential Equations — Separation of Variables
  • MC 9 (1 mark, class average 49%): Set up a surface-of-revolution integral for a parametric curve. Lesson: Volumes of Revolution and Arc Length
  • MC 10 (1 mark, class average 69%): Find a rotational boundary from a given volume. Lesson: Volumes of Revolution and Arc Length
  • MC 11 (1 mark, class average 55%): Determine the valid interval of a separable differential-equation solution. Lesson: Differential Equations — Separation of Variables
  • MC 12 (1 mark, class average 33%): Find speed halfway through a displacement under constant acceleration. Lesson: Kinematics with Calculus
  • MC 13 (1 mark, class average 55%): Find the time for a projectile to reach a lower platform. Lesson: Kinematics with Calculus
  • MC 14 (1 mark, class average 48%): Find the angle between vectors from dot- and cross-product magnitudes. Lesson: Vector Proofs and Applications
  • MC 15 (1 mark, class average 52%): Relate the angle between planes to their normal vectors. Lesson: Lines and Planes in 3D
  • MC 16 (1 mark, class average 67%): Make velocity perpendicular to acceleration at a specified time. Lesson: Vector Calculus — Kinematics in Two Dimensions
  • MC 17 (1 mark, class average 52%): Integrate three-dimensional acceleration with an initial rest condition. Lesson: Vector Calculus — Kinematics in Two Dimensions
  • MC 18 (1 mark, class average 49%): Solve for parameters at the intersection of two lines in space. Lesson: Lines and Planes in 3D
  • MC 19 (1 mark, class average 59%): Calculate the area of a triangle intercepted by a plane. Lesson: Vector Proofs and Applications, Lines and Planes in 3D
  • MC 20 (1 mark, class average 54%): Find a tail probability for a linear combination of independent normal variables. Lesson: Linear Combinations of Random Variables and Sample Means
  • Q1a (3 marks, class average 70%): Sketch a rational curve with its turning point, inflection and asymptotes. Lesson: Rational Functions — Asymptotes, Sketching and Partial Fractions, Implicit Differentiation and Related Rates
  • Q1b.i (1 mark, class average 90%): Set up the volume integral for rotation of the positive rational curve. Lesson: Volumes of Revolution and Arc Length
  • Q1b.ii (1 mark, class average 81%): Evaluate the volume integral to two decimal places. Lesson: Volumes of Revolution and Arc Length
  • Q1c (1 mark, class average 74%): Find vertical asymptotes from the real roots of a cubic denominator. Lesson: Rational Functions — Asymptotes, Sketching and Partial Fractions
  • Q1d.i (1 mark, class average 84%): Find the stationary-point height for a family of rational functions. Lesson: Rational Functions — Asymptotes, Sketching and Partial Fractions, Implicit Differentiation and Related Rates
  • Q1d.ii (1 mark, class average 46%): Find asymptotes when the usual stationary point is undefined. Lesson: Rational Functions — Asymptotes, Sketching and Partial Fractions
  • Q1d.iii (2 marks, class average 75%): Find the parameter that puts an inflection point at a specified input. Lesson: Implicit Differentiation and Related Rates, Rational Functions — Asymptotes, Sketching and Partial Fractions
  • Q2a (1 mark, class average 88%): Identify the circle described by a complex number with modulus 2. Lesson: Argand Diagrams and Polar Form
  • Q2b.i (2 marks, class average 81%): Derive the perpendicular bisector of two points represented by complex numbers. Lesson: Complex Number Geometry — Loci and Transformations, Argand Diagrams and Polar Form
  • Q2b.ii (1 mark, class average 78%): Add the complex-number locus line to an Argand diagram. Lesson: Argand Diagrams and Polar Form
  • Q2c.i (2 marks, class average 76%): Find where a circle of radius 2 meets the line with gradient square root 3. Lesson: Complex Number Geometry — Loci and Transformations, Argand Diagrams and Polar Form
  • Q2c.ii (1 mark, class average 74%): Label both intersection points on the Argand diagram. Lesson: Argand Diagrams and Polar Form
  • Q2d (1 mark, class average 51%): Express the ray from one complex point through another using argument. Lesson: Argand Diagrams and Polar Form
  • Q2e (2 marks, class average 69%): Find the smaller circular segment cut off by a chord of a radius-3 circle. Lesson: Argand Diagrams and Polar Form
  • Q3a (1 mark, class average 20%): Explain why salt initially accumulates in a well-mixed tank. Lesson: Differential Equations — Separation of Variables
  • Q3b (1 mark, class average 62%): Form a differential equation for salt entering and leaving a tank. Lesson: Differential Equations — Separation of Variables
  • Q3c (2 marks, class average 46%): Use two Euler steps to estimate the amount of dissolved salt. Lesson: Differential Equations — Separation of Variables
  • Q3d (3 marks, class average 75%): Solve the salt-mixing differential equation using separation. Lesson: Differential Equations — Separation of Variables
  • Q3e (1 mark, class average 79%): Find the limiting quantity of salt in the tank. Lesson: Differential Equations — Separation of Variables
  • Q3f (1 mark, class average 61%): Find when the tank first contains 100 kilograms of salt. Lesson: Differential Equations — Separation of Variables
  • Q3g (1 mark, class average 18%): Find the extra time to reach a target concentration after closing the drain. Lesson: Differential Equations — Separation of Variables
  • Q4a (1 mark, class average 93%): Find the starting point of a particle moving along a trigonometric path. Lesson: Vector Calculus — Kinematics in Two Dimensions
  • Q4b (1 mark, class average 63%): Determine the particle's direction through a given point on its path. Lesson: Vector Calculus — Kinematics in Two Dimensions
  • Q4c (1 mark, class average 53%): Find the first complete repeat of a two-frequency particle trajectory. Lesson: Vector Calculus — Kinematics in Two Dimensions
  • Q4d (3 marks, class average 58%): Derive the particle's speed by differentiating its position vector. Lesson: Vector Calculus — Kinematics in Two Dimensions, Kinematics with Calculus
  • Q4e (1 mark, class average 73%): Find the greatest possible speed of a moving particle. Lesson: Vector Calculus — Kinematics in Two Dimensions
  • Q4f (1 mark, class average 66%): Trace the first part of the parametric path with its endpoint. Lesson: Vector Calculus — Kinematics in Two Dimensions
  • Q4g (2 marks, class average 68%): Calculate a particle's travel distance from its speed function. Lesson: Vector Calculus — Kinematics in Two Dimensions, Volumes of Revolution and Arc Length
  • Q5a (1 mark, class average 74%): Locate the common point of three planes. Lesson: Lines and Planes in 3D
  • Q5b.i (2 marks, class average 62%): Find a direction vector parallel to the intersection of two planes. Lesson: Lines and Planes in 3D
  • Q5b.ii (1 mark, class average 51%): Write parametric equations for the common line of two planes. Lesson: Lines and Planes in 3D
  • Q5c (2 marks, class average 71%): Find the shortest distance from a point to a plane. Lesson: Lines and Planes in 3D
  • Q5d.i (1 mark, class average 68%): Explain why every plane in a family is parallel to a given plane. Lesson: Lines and Planes in 3D
  • Q5d.ii (3 marks, class average 45%): Find plane constants producing a prescribed separation. Lesson: Lines and Planes in 3D
  • Q6a.i (1 mark, class average 86%): Find the mean and spread of a sample average. Lesson: Linear Combinations of Random Variables and Sample Means
  • Q6a.ii (1 mark, class average 86%): Calculate the probability that a sample average exceeds a threshold. Lesson: Linear Combinations of Random Variables and Sample Means
  • Q6b (1 mark, class average 88%): Calculate a confidence interval for a population mean. Lesson: Confidence Intervals and Hypothesis Testing for a Mean
  • Q6c (1 mark, class average 77%): Estimate the number of confidence intervals covering the true mean. Lesson: Confidence Intervals and Hypothesis Testing for a Mean
  • Q6d (1 mark, class average 53%): Choose the smallest sample size for a specified error margin. Lesson: Confidence Intervals and Hypothesis Testing for a Mean
  • Q6e (1 mark, class average 91%): State hypotheses for a one-sided population-mean test. Lesson: Confidence Intervals and Hypothesis Testing for a Mean
  • Q6f.i (1 mark, class average 85%): Calculate a one-tailed p-value for a sample mean. Lesson: Confidence Intervals and Hypothesis Testing for a Mean
  • Q6f.ii (1 mark, class average 68%): Decide whether a one-tailed test supports a reduction claim. Lesson: Confidence Intervals and Hypothesis Testing for a Mean
  • Q6g (1 mark, class average 63%): Find the critical sample-mean threshold for a one-tailed test. Lesson: Confidence Intervals and Hypothesis Testing for a Mean
  • Q6h (1 mark, class average 54%): Find the chance of missing a real decrease in mean bottle volume. Lesson: Confidence Intervals and Hypothesis Testing for a Mean

2024 Exam 1

  • Q1a (1 mark, class average 73%): Show $3z + 2i$ is a factor of a complex cubic. Lesson: Complex Numbers — Introduction and the Complex Plane
  • Q1b (2 marks, class average 60%): Solve the complex cubic in Cartesian form. Lesson: Complex Numbers — Introduction and the Complex Plane
  • Q1c (1 mark, class average 49%): Plot the three roots on an Argand diagram. Lesson: Argand Diagrams and Polar Form
  • Q2 (3 marks, class average 83%): Direct proof that $2x^2 - 3x - 7$ is even when $x$ is odd. Lesson: Proof Techniques — Contradiction and Contrapositive
  • Q3a (1 mark, class average 63%): Show $\frac{(x-1)^2}{(x+1)^2} = 1 - \frac{4}{x+1} + \frac{4}{(x+1)^2}$. Lesson: Rational Functions — Asymptotes, Sketching and Partial Fractions
  • Q3b (2 marks, class average 80%): Turning point of $\frac{(x-1)^2}{(x+1)^2}$. Lesson: Rational Functions — Asymptotes, Sketching and Partial Fractions
  • Q3c (3 marks, class average 33%): Sketch the graph with asymptotes and intercepts. Lesson: Rational Functions — Asymptotes, Sketching and Partial Fractions
  • Q4a (2 marks, class average 70%): Angle between two given vectors in 3D. Lesson: Vector Proofs and Applications
  • Q4b (2 marks, class average 75%): Values of $n$ where a dot product equals the size of a cross product. Lesson: Vector Proofs and Applications
  • Q5 (3 marks, class average 70%): Show $k$ satisfies a cubic, from a volume of revolution of $\frac{7\pi}{2}$. Lesson: Volumes of Revolution and Arc Length
  • Q6a (1 mark, class average 79%): Mean and variance of the total time across three stages. Lesson: Linear Combinations of Random Variables and Sample Means
  • Q6b (2 marks, class average 45%): Variance of the total cost, at a different hourly rate for each stage. Lesson: Linear Combinations of Random Variables and Sample Means
  • Q6c (2 marks, class average 50%): Chance Stage 2 takes less time than Stage 1. Lesson: Linear Combinations of Random Variables and Sample Means
  • Q7 (4 marks, class average 57%): Solve $x + 2y\sqrt{x^2 + 1}\frac{dy}{dx} = 0$ with $y(0) = -2$. Lesson: Differential Equations — Separation of Variables
  • Q8a (2 marks, class average 70%): Show $\frac{dy}{dx} = -\frac{y}{x}$ for $x^2y^2 + xy = 2$. Lesson: Implicit Differentiation and Related Rates
  • Q8b (2 marks, class average 50%): Points where the tangent to $x^2y^2 + xy = 2$ has slope $-1$. Lesson: Implicit Differentiation and Related Rates
  • Q9a (1 mark, class average 32%): Does a car trip the speed detector at $x = 0$?. Lesson: Kinematics with Calculus
  • Q9b (3 marks, class average 53%): Acceleration at $x = 12$, in the form $\frac{k}{\pi}$. Lesson: Kinematics with Calculus
  • Q10 (3 marks, class average 27%): Values of $m$ giving a shortest distance of $\frac{14}{\sqrt{35}}$ between skew lines. Lesson: Lines and Planes in 3D

2024 Exam 2

  • MC 1 (1 mark, class average 72%): Contrapositive of a statement about two integers. Lesson: Proof Techniques — Contradiction and Contrapositive
  • MC 2 (1 mark, class average 48%): Which statement is true for a rational function patched at $x = 2$. Lesson: Rational Functions — Asymptotes, Sketching and Partial Fractions
  • MC 3 (1 mark, class average 70%): Values of $h$ for which $\frac{x - h}{(x + 1)(x - 4)}$ has no turning points. Lesson: Rational Functions — Asymptotes, Sketching and Partial Fractions
  • MC 4 (1 mark, class average 27%): $\cos\left(\frac{x}{2}\right)$ from $\sin(x) = a$ with $x$ in the fourth quadrant. Lesson: Reciprocal and Inverse Circular Functions — Compound and Double Angle Formulas
  • MC 5 (1 mark, class average 70%): Transformation taking $1 + \sqrt{3}i$ to $-\bar{z}$. Lesson: Argand Diagrams and Polar Form
  • MC 6 (1 mark, class average 77%): Value of $k$ making $z^2 + 4iz + 3$ purely imaginary for $z = 3 + ki$. Lesson: Complex Numbers — Introduction and the Complex Plane
  • MC 7 (1 mark, class average 72%): Separating a differential equation with $\cos(x - y) - \cos(x + y)$. Lesson: Differential Equations — Separation of Variables
  • MC 8 (1 mark, class average 60%): Step size that gives a stated $y_3$ in Euler's method for $\frac{dy}{dx} = xy^2$. Lesson: Differential Equations — Separation of Variables
  • MC 9 (1 mark, class average 45%): Arc length of a cycloid-type curve over $[0, 2\pi]$. Lesson: Volumes of Revolution and Arc Length
  • MC 10 (1 mark, class average 74%): Surface area when a parametric line segment is rotated about the $y$-axis. Lesson: Volumes of Revolution and Arc Length
  • MC 11 (1 mark, class average 58%): Position after 150 s from a two-segment velocity–time graph. Lesson: Kinematics with Calculus
  • MC 12 (1 mark, class average 50%): Acceleration in terms of $k$ when $x = e^{(k-1)t}$ and $x = k + 1$. Lesson: Kinematics with Calculus
  • MC 13 (1 mark, class average 81%): Find positive $m$ from the angle between two vectors. Lesson: Vector Proofs and Applications
  • MC 14 (1 mark, class average 36%): Scalar resolute of $\underset{\sim}{s}$ on $\underset{\sim}{r}$ from the vector resolute the other way. Lesson: Vector Proofs and Applications
  • MC 15 (1 mark, class average 36%): Describe the path and motion for $\sin(t)\underset{\sim}{i} + \cos(2t)\underset{\sim}{j}$. Lesson: Vector Calculus — Kinematics in Two Dimensions
  • MC 16 (1 mark, class average 43%): How often one particle's velocity is perpendicular to another's position. Lesson: Vector Calculus — Kinematics in Two Dimensions
  • MC 17 (1 mark, class average 58%): Distance between two parallel lines. Lesson: Lines and Planes in 3D
  • MC 18 (1 mark, class average 68%): Where a line meets the plane $3x - 2y + 4z = 5$. Lesson: Lines and Planes in 3D
  • MC 19 (1 mark, class average 68%): Definition of a type II error. Lesson: Confidence Intervals and Hypothesis Testing for a Mean
  • MC 20 (1 mark, class average 48%): Chance four avocados give more than 570 g of flesh. Lesson: Linear Combinations of Random Variables and Sample Means
  • Q1a (3 marks, class average 67%): Sketch $\frac{x^4 - x^2 + 1}{1 - x^2}$ with asymptotes and stationary points. Lesson: Rational Functions — Asymptotes, Sketching and Partial Fractions
  • Q1b.i (2 marks, class average 70%): Integral in $y$ for the volume when the middle branch is rotated about the $y$-axis. Lesson: Volumes of Revolution and Arc Length
  • Q1b.ii (1 mark, class average 56%): Volume of that solid to one decimal place. Lesson: Volumes of Revolution and Arc Length
  • Q1c (1 mark, class average 72%): Value of $b$ that removes every asymptote of $\frac{x^4 + b}{1 - x^2}$. Lesson: Rational Functions — Asymptotes, Sketching and Partial Fractions
  • Q1d.i (1 mark, class average 27%): Values of $b$ giving exactly one stationary point. Lesson: Rational Functions — Asymptotes, Sketching and Partial Fractions
  • Q1d.ii (1 mark, class average 27%): Values of $b$ giving exactly three stationary points. Lesson: Rational Functions — Asymptotes, Sketching and Partial Fractions
  • Q1d.iii (1 mark, class average 25%): Values of $b$ giving exactly five stationary points. Lesson: Rational Functions — Asymptotes, Sketching and Partial Fractions
  • Q2a (2 marks, class average 75%): Line of points equidistant from $1 + 2i$ and $4$ as $y = mx + c$. Lesson: Complex Number Geometry — Loci and Transformations
  • Q2b (2 marks, class average 60%): Circle with the segment from $1 + 2i$ to $4$ as diameter. Lesson: Complex Number Geometry — Loci and Transformations
  • Q2c (2 marks, class average 85%): Sketch $|z - (1 + 2i)| = 2$ and label its imaginary-axis intercepts. Lesson: Complex Number Geometry — Loci and Transformations
  • Q2d.i (1 mark, class average 59%): Sketch the ray from $2 - i$ through $-2 + 3i$. Lesson: Complex Number Geometry — Loci and Transformations
  • Q2d.ii (1 mark, class average 56%): Equation of that ray in Arg form. Lesson: Complex Number Geometry — Loci and Transformations
  • Q2e (2 marks, class average 65%): Area of the minor segment the ray cuts from the circle. Lesson: Complex Number Geometry — Loci and Transformations
  • Q3a (1 mark, class average 76%): Maximum inflow rate of pollutant and when it happens. Lesson: Vector Proofs and Applications
  • Q3b (3 marks, class average 53%): Rate the radius of the pollutant disc grows at $t = 4$. Lesson: Implicit Differentiation and Related Rates
  • Q3c.i (1 mark, class average 61%): Rewrite $\int\frac{8t}{240 + 5t^4}\,dt$ with $u = \sqrt{5}t^2$. Lesson: Integration Techniques — Substitution and Partial Fractions
  • Q3c.ii (1 mark, class average 55%): Total pollutant after $t$ days in the form $\frac{1}{a\sqrt{b}}\arctan\left(\frac{t^c}{d\sqrt{b}}\right)$. Lesson: Integration Techniques — Substitution and Partial Fractions
  • Q3d (2 marks, class average 25%): Pond area the pollutant approaches in the long run. Lesson: Volumes of Revolution and Arc Length
  • Q3e (2 marks, class average 15%): Days after clean-up starts until the pond is clear. Lesson: Integration Techniques — Substitution and Partial Fractions
  • Q4a (1 mark, class average 78%): Cartesian equation of the path $3\sec(t)\underset{\sim}{i} + 2\tan(t)\underset{\sim}{j}$. Lesson: Vector Calculus — Kinematics in Two Dimensions, Reciprocal and Inverse Circular Functions — Compound and Double Angle Formulas
  • Q4b (2 marks, class average 50%): Sketch the yacht's path with endpoints and direction. Lesson: Vector Calculus — Kinematics in Two Dimensions
  • Q4c.i (1 mark, class average 23%): Square of the yacht's speed in terms of $\sec(t)$. Lesson: Vector Calculus — Kinematics in Two Dimensions
  • Q4c.ii (1 mark, class average 61%): Time of minimum speed. Lesson: Vector Calculus — Kinematics in Two Dimensions
  • Q4c.iii (1 mark, class average 48%): Minimum speed of the yacht. Lesson: Vector Calculus — Kinematics in Two Dimensions
  • Q4c.iv (1 mark, class average 54%): Position of the yacht at minimum speed. Lesson: Vector Calculus — Kinematics in Two Dimensions
  • Q4d.i (1 mark, class average 61%): Definite integral for the distance sailed. Lesson: Volumes of Revolution and Arc Length
  • Q4d.ii (1 mark, class average 57%): Distance sailed to one decimal place. Lesson: Volumes of Revolution and Arc Length
  • Q4e (2 marks, class average 30%): Shortest distance between a drone and the yacht while the yacht sails its path. Lesson: Vector Calculus — Kinematics in Two Dimensions, Lines and Planes in 3D
  • Q5a (1 mark, class average 55%): Vector equation of the line through $A(1, -2, 3)$ and $B(2, -5, -1)$. Lesson: Lines and Planes in 3D
  • Q5b (3 marks, class average 63%): Shortest distance from point $A$ to a given line, as $\frac{a\sqrt{b}}{c}$. Lesson: Lines and Planes in 3D
  • Q5c (3 marks, class average 80%): Cartesian equation of the plane through $A$, $B$ and $C(0, 2, -5)$. Lesson: Lines and Planes in 3D
  • Q5d.i (1 mark, class average 63%): Axis intercepts of $2x - 3y + 4z = 12$. Lesson: Lines and Planes in 3D
  • Q5d.ii (2 marks, class average 50%): Area of the triangle formed by those intercepts, as $m\sqrt{n}$. Lesson: Vector Proofs and Applications
  • Q6a (1 mark, class average 91%): Hypotheses for the one-sided test on the olive oil machine. Lesson: Confidence Intervals and Hypothesis Testing for a Mean
  • Q6b.i (1 mark, class average 85%): $p$ value for a sample mean of 997.5 from nine bottles. Lesson: Confidence Intervals and Hypothesis Testing for a Mean
  • Q6b.ii (1 mark, class average 75%): Should the machine be paused?. Lesson: Confidence Intervals and Hypothesis Testing for a Mean
  • Q6c (2 marks, class average 50%): Type II error probability if the true mean is 997 mL. Lesson: Confidence Intervals and Hypothesis Testing for a Mean
  • Q6d (1 mark, class average 58%): Lower and upper quality-control limits on the mean of nine bottles. Lesson: Confidence Intervals and Hypothesis Testing for a Mean
  • Q6e (1 mark, class average 82%): 95% confidence interval from 50 bottles with mean 1005 mL. Lesson: Confidence Intervals and Hypothesis Testing for a Mean
  • Q6f (1 mark, class average 51%): Expected number of 40 intervals that contain the true mean. Lesson: Confidence Intervals and Hypothesis Testing for a Mean
  • Q6g (1 mark, class average 46%): Minimum sample size for a 1 mL margin at 95% confidence. Lesson: Confidence Intervals and Hypothesis Testing for a Mean

2023 Exam 1

  • Q1a (1 mark, class average 87%): Show $\frac{x^2 + x - 6}{x - 1} = x + 2 - \frac{4}{x - 1}$. Lesson: Rational Functions — Asymptotes, Sketching and Partial Fractions
  • Q1b (3 marks, class average 63%): Sketch the rational function with its asymptotes. Lesson: Rational Functions — Asymptotes, Sketching and Partial Fractions
  • Q2 (3 marks, class average 50%): Find $b > 0$ when $\arg\big((b - i)^3\big) = -\frac{\pi}{2}$. Lesson: De Moivre's Theorem and Roots of Unity, Complex Numbers — Introduction and the Complex Plane
  • Q3a (2 marks, class average 45%): Acceleration at $x = 2$ when $v = \frac{3x + 2}{2x - 1}$. Lesson: Kinematics with Calculus
  • Q3b (1 mark, class average 50%): Limiting velocity as $x$ grows large. Lesson: Kinematics with Calculus
  • Q4 (3 marks, class average 60%): Gradient of $x\arcsin(y^2) = \pi$ at $\left(6, \frac{1}{\sqrt{2}}\right)$. Lesson: Implicit Differentiation and Related Rates
  • Q5 (3 marks, class average 67%): Evaluate $\int_1^2 x^2\log_e(x)\,dx$. Lesson: Techniques of Integration
  • Q6a (2 marks, class average 85%): Mean and standard deviation of a total commute made of three stages. Lesson: Linear Combinations of Random Variables and Sample Means
  • Q6b (2 marks, class average 45%): Standardised bounds for an average waiting time over 12 days. Lesson: Linear Combinations of Random Variables and Sample Means
  • Q7 (4 marks, class average 55%): Surface area when a parametric curve is rotated about the $x$-axis. Lesson: Volumes of Revolution and Arc Length
  • Q8 (4 marks, class average 53%): Induction proof of a formula for the $n$th derivative of $xe^{2x}$. Lesson: Proof by Mathematical Induction
  • Q9a (1 mark, class average 91%): Coordinates of the plane's $y$-intercept. Lesson: Lines and Planes in 3D
  • Q9b (1 mark, class average 91%): Show the vectors from $A$ to $B$ and from $A$ to $D$. Lesson: Vector Proofs and Applications
  • Q9c (2 marks, class average 70%): Cartesian equation of the plane through $A$, $B$ and $D$. Lesson: Lines and Planes in 3D
  • Q9d (1 mark, class average 62%): Find $a$ so that $C(a, -1, 5)$ is in the plane. Lesson: Lines and Planes in 3D
  • Q9e (1 mark, class average 57%): Area of the parallelogram with sides $\overrightarrow{AB}$ and $\overrightarrow{AD}$. Lesson: Vector Proofs and Applications
  • Q10a (1 mark, class average 80%): Write $5 - 6\sin^2(t)$ as $\alpha + \beta\cos(2t)$. Lesson: Reciprocal and Inverse Circular Functions — Compound and Double Angle Formulas
  • Q10b (2 marks, class average 70%): Show the particle moves on $(x - 2)^2 + (y - 1)^2 = 9$. Lesson: Vector Calculus — Kinematics in Two Dimensions, Reciprocal and Inverse Circular Functions — Compound and Double Angle Formulas
  • Q10c (1 mark, class average 36%): Find $a$ when the distance travelled from $t = 0$ to $t = a$ is $\frac{3\pi}{4}$. Lesson: Vector Calculus — Kinematics in Two Dimensions
  • Q10d (2 marks, class average 30%): All times when the position vector is perpendicular to the velocity. Lesson: Vector Calculus — Kinematics in Two Dimensions

2023 Exam 2

  • MC 1 (1 mark, class average 85%): Contrapositive of a statement about a football team. Lesson: Proof Techniques — Contradiction and Contrapositive
  • MC 2 (1 mark, class average 59%): Constants in $\frac{x^3}{ax^2 + bx + c}$ from its asymptotes. Lesson: Rational Functions — Asymptotes, Sketching and Partial Fractions
  • MC 3 (1 mark, class average 72%): Values of $a$ for which $a + \sec(x)$ has two $x$-intercepts on $[-\pi, \pi]$. Lesson: Reciprocal and Inverse Circular Functions — Compound and Double Angle Formulas
  • MC 4 (1 mark, class average 62%): Polar form of $\frac{4a}{1 + \bar{z}}$ for a given $z$. Lesson: Argand Diagrams and Polar Form
  • MC 5 (1 mark, class average 34%): Which expression equals $z^2$ given $|\bar{z}| = 4$ and $\arg(z^3) = -\pi$. Lesson: De Moivre's Theorem and Roots of Unity
  • MC 6 (1 mark, class average 69%): Iterations of an Euler's method algorithm before 2.709 is printed. Lesson: Differential Equations — Separation of Variables
  • MC 7 (1 mark, class average 68%): Follow a direction field from $(-1, 2)$ to find $y$ at $x = 1.5$. Lesson: Differential Equations — Separation of Variables
  • MC 8 (1 mark, class average 37%): Differential equation for chemical in a spa where more leaves than enters. Lesson: Differential Equations — Separation of Variables
  • MC 9 (1 mark, class average 65%): Slope of a parametric path at $t = 2$. Lesson: Vector Calculus — Kinematics in Two Dimensions
  • MC 10 (1 mark, class average 33%): Reduction formula for $\int_0^1 (1 - x)^ne^x\,dx$. Lesson: Techniques of Integration
  • MC 11 (1 mark, class average 46%): Surface area when part of $y = \cos^{-1}(x)$ is rotated about the $y$-axis. Lesson: Volumes of Revolution and Arc Length
  • MC 12 (1 mark, class average 54%): Velocity after $\log_e(e + 1)$ seconds when $a = 1 + v$. Lesson: Kinematics with Calculus
  • MC 13 (1 mark, class average 45%): Time for a phone dropped from a rising balloon to reach the ground. Lesson: Kinematics with Calculus
  • MC 14 (1 mark, class average 48%): $|\underset{\sim}{c} \cdot \underset{\sim}{n}|$ for a unit vector perpendicular to two given vectors. Lesson: Vector Proofs and Applications
  • MC 15 (1 mark, class average 18%): Size of the difference of two unit vectors whose sum is a unit vector. Lesson: Vector Proofs and Applications
  • MC 16 (1 mark, class average 32%): Total vertical distance a thrown ball travels before landing. Lesson: Vector Calculus — Kinematics in Two Dimensions, Kinematics with Calculus
  • MC 17 (1 mark, class average 73%): Find $\alpha$, $\beta$ and $\gamma$ from a cross product. Lesson: Vector Proofs and Applications
  • MC 18 (1 mark, class average 74%): Value of $k$ making two planes perpendicular. Lesson: Lines and Planes in 3D
  • MC 19 (1 mark, class average 67%): Chance the total of 16 invoices is over 13 500 dollars. Lesson: Linear Combinations of Random Variables and Sample Means
  • MC 20 (1 mark, class average 63%): Population standard deviation from a 99% confidence interval. Lesson: Confidence Intervals and Hypothesis Testing for a Mean
  • Q1a (1 mark, class average 78%): Show the constants in a two-piece walking-track function. Lesson: Differential Equations — Separation of Variables
  • Q1b (2 marks, class average 70%): Verify the two pieces join smoothly at $C$. Lesson: Differential Equations — Separation of Variables
  • Q1c.i (1 mark, class average 92%): Coordinates of the minimum point $A$. Lesson: Vector Proofs and Applications
  • Q1c.ii (1 mark, class average 91%): Coordinates of the inflection point $B$. Lesson: Vector Proofs and Applications
  • Q1d (2 marks, class average 85%): Cartesian equation of the elliptical return path. Lesson: Vector Calculus — Kinematics in Two Dimensions, Reciprocal and Inverse Circular Functions — Compound and Double Angle Formulas
  • Q1e (1 mark, class average 22%): Sketch the elliptical path from $D$ to $O$. Lesson: Vector Calculus — Kinematics in Two Dimensions
  • Q1f.i (1 mark, class average 66%): Integral for the length of the elliptical path. Lesson: Volumes of Revolution and Arc Length
  • Q1f.ii (1 mark, class average 66%): Length of the path to three decimal places. Lesson: Volumes of Revolution and Arc Length
  • Q2a (1 mark, class average 63%): Verify $\operatorname{cis}\left(\frac{2\pi}{7}\right)$ is a root of $z^7 - 1 = 0$. Lesson: De Moivre's Theorem and Roots of Unity
  • Q2b (1 mark, class average 61%): The other roots of $z^7 = 1$ in polar form. Lesson: De Moivre's Theorem and Roots of Unity
  • Q2c (2 marks, class average 60%): Plot all seven roots on an Argand diagram. Lesson: De Moivre's Theorem and Roots of Unity
  • Q2d.i (1 mark, class average 42%): Sketch the ray from the real root through $\operatorname{cis}\left(\frac{2\pi}{7}\right)$. Lesson: Complex Number Geometry — Loci and Transformations
  • Q2d.ii (1 mark, class average 18%): Equation of that ray in Arg form. Lesson: Complex Number Geometry — Loci and Transformations
  • Q2e (1 mark, class average 54%): Show $z^7 - 1$ factorises with $z^6 + z^5 + \ldots + 1$. Lesson: De Moivre's Theorem and Roots of Unity
  • Q2f.i (1 mark, class average 47%): Write $\operatorname{cis}\left(\frac{2\pi}{7}\right) + \operatorname{cis}\left(\frac{12\pi}{7}\right)$ as $A\cos(B\pi)$. Lesson: De Moivre's Theorem and Roots of Unity
  • Q2f.ii (2 marks, class average 10%): Use the roots to show $\cos\frac{2\pi}{7} + \cos\frac{4\pi}{7} + \cos\frac{6\pi}{7} = -\frac{1}{2}$. Lesson: De Moivre's Theorem and Roots of Unity
  • Q3a.i (1 mark, class average 89%): Integral for the volume when $y^2 = x - 1$ is rotated, $2 \le x \le 5$. Lesson: Volumes of Revolution and Arc Length
  • Q3a.ii (1 mark, class average 85%): Value of that volume. Lesson: Volumes of Revolution and Arc Length
  • Q3b.i (2 marks, class average 75%): Curved surface area as $\pi\int_a^b\sqrt{Ax - B}\,dx$. Lesson: Volumes of Revolution and Arc Length
  • Q3b.ii (1 mark, class average 55%): Curved surface area to three decimal places. Lesson: Volumes of Revolution and Arc Length
  • Q3c (2 marks, class average 50%): Efficiency ratio (total surface area over volume) of the solid. Lesson: Volumes of Revolution and Arc Length
  • Q3d (3 marks, class average 50%): Efficiency ratio of a longer solid with volume $24\pi$. Lesson: Volumes of Revolution and Arc Length
  • Q4a (1 mark, class average 48%): Partial fractions for the logistic equation. Lesson: Differential Equations — Separation of Variables, Integration Techniques — Substitution and Partial Fractions
  • Q4b (1 mark, class average 57%): Constant $D$ when the pond starts with 200 fish. Lesson: Differential Equations — Separation of Variables
  • Q4c (1 mark, class average 79%): Number of fish released into pond 2. Lesson: Differential Equations — Separation of Variables
  • Q4d (1 mark, class average 85%): Pond 2 population after 6 years. Lesson: Differential Equations — Separation of Variables
  • Q4e.i (1 mark, class average 21%): $\frac{d^2Q}{dt^2}$ in terms of $Q$. Lesson: Differential Equations — Separation of Variables, Implicit Differentiation and Related Rates
  • Q4e.ii (2 marks, class average 65%): Population and time when pond 2 grows fastest. Lesson: Differential Equations — Separation of Variables
  • Q4f (2 marks, class average 55%): Sketch the logistic curve for pond 2. Lesson: Differential Equations — Separation of Variables
  • Q4g (1 mark, class average 40%): Largest population pond 2 can support with harvesting. Lesson: Differential Equations — Separation of Variables
  • Q5a (2 marks, class average 80%): Show triangle $ABC$ has area 1.5. Lesson: Vector Proofs and Applications
  • Q5b (2 marks, class average 35%): Shortest distance from $B$ to the segment $AC$. Lesson: Vector Proofs and Applications
  • Q5c (2 marks, class average 60%): Acute angle between a line and the plane $2x - 2y - z = -18$. Lesson: Lines and Planes in 3D
  • Q5d (1 mark, class average 52%): Parametric equations of the line through the origin normal to the plane. Lesson: Lines and Planes in 3D
  • Q5e (2 marks, class average 60%): Shortest distance from the origin to the plane. Lesson: Lines and Planes in 3D
  • Q5f (2 marks, class average 40%): Where the normal line through the origin meets the plane. Lesson: Lines and Planes in 3D
  • Q6a (1 mark, class average 84%): 95% confidence interval for koala mass from 20 koalas. Lesson: Confidence Intervals and Hypothesis Testing for a Mean
  • Q6b (1 mark, class average 58%): Expected number of 60 intervals containing the true mean. Lesson: Confidence Intervals and Hypothesis Testing for a Mean
  • Q6c (1 mark, class average 28%): Sample size to cut the interval's width by 60%. Lesson: Confidence Intervals and Hypothesis Testing for a Mean
  • Q6d (1 mark, class average 88%): Hypotheses for testing whether koala mass is below 12 kg. Lesson: Confidence Intervals and Hypothesis Testing for a Mean
  • Q6e.i (1 mark, class average 80%): $p$ value for a mean of 11.6 kg from 40 koalas. Lesson: Confidence Intervals and Hypothesis Testing for a Mean
  • Q6e.ii (1 mark, class average 78%): Conclusion of the koala test. Lesson: Confidence Intervals and Hypothesis Testing for a Mean
  • Q6f (1 mark, class average 60%): Critical sample mean at the 1% level. Lesson: Confidence Intervals and Hypothesis Testing for a Mean
  • Q6g (1 mark, class average 39%): Type II error probability if the true mean is 11.4 kg. Lesson: Confidence Intervals and Hypothesis Testing for a Mean
  • Q6h (1 mark): Label the critical mean and shade the type II error region. Lesson: Confidence Intervals and Hypothesis Testing for a Mean