VCE Vault

VCE Mathematical Methods past exams 2023–2025

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

2025 Exam 1

  • Q1a (1 mark, class average 88%): Differentiate $x^2\cos(x)$ with the product rule. Lesson: Differentiation: Rules and Techniques
  • Q1b (2 marks, class average 75%): Gradient of the tangent to $y = 6\sqrt{x+1} + 5$ at $x = 8$. Lesson: Differentiation: Rules and Techniques
  • Q2 (2 marks, class average 60%): Find $g(x)$ from $g'(x) = \dfrac{1}{2x+3}$ and $g(1) = 0$. Lesson: Antidifferentiation
  • Q3a (1 mark, class average 81%): Range of $f(x) = 2\cos(2x) + 1$ over $[0, 2\pi]$. Lesson: Circular (Trigonometric) Functions
  • Q3b (3 marks, class average 67%): Solve $2\cos(2x) + 1 = 0$ for $x$ in $[0, 2\pi]$. Lesson: Circular (Trigonometric) Functions, Trigonometric Identities and Equation Solving
  • Q3c (2 marks, class average 50%): Sketch $y = 2\cos(2x) + 1$ for $x$ from $\frac{\pi}{2}$ to $\frac{3\pi}{2}$, labelling endpoints. Lesson: Circular (Trigonometric) Functions
  • Q4a (2 marks, class average 65%): Use the probabilities summing to 1 to show $k = 10$ or $15$. Lesson: Discrete Random Variables
  • Q4b.i (1 mark, class average 75%): $\Pr(X > 1)$ for the distribution with $k = 15$. Lesson: Discrete Random Variables
  • Q4b.ii (1 mark, class average 71%): Expected value of the distribution with $k = 15$. Lesson: Discrete Random Variables
  • Q5a (2 marks, class average 75%): Solve $e^{2x} - 8e^x + 7 = 0$. Lesson: Exponential and Logarithmic Functions
  • Q5b (2 marks, class average 50%): Largest $a$ so $e^{2x} - 8e^x + 7$ on $(-\infty, a]$ has an inverse. Lesson: Composite and Inverse Functions, Applications of Differentiation
  • Q6a (1 mark, class average 73%): Variance of a binomial variable with $n = 6$, $p = \frac{1}{4}$. Lesson: Discrete Random Variables
  • Q6b (2 marks, class average 50%): $\Pr(X \ge 5)$ for the same binomial, written as $\frac{a}{2^b}$. Lesson: Discrete Random Variables
  • Q7a (1 mark, class average 90%): Check that $x = 5$ is a root of a cubic. Lesson: Polynomial Functions
  • Q7b (2 marks, class average 80%): Write the cubic as $(x + d)^2(x - 5)$. Lesson: Polynomial Functions
  • Q7c (1 mark, class average 76%): Axis intercepts of the cubic. Lesson: Polynomial Functions
  • Q7d.i (1 mark, class average 44%): Stationary point of inflection of $f(x) \times (x + 2)$. Lesson: Polynomial Functions
  • Q7d.ii (1 mark, class average 27%): Where the quartic $(x + 2)^3(x - 5)$ is non-negative. Lesson: Polynomial Functions
  • Q8a (3 marks, class average 47%): Find $k$ with $\Pr(X > k) = \frac{9}{16}$ for a straight-line pdf. Lesson: Continuous Random Variables and the Normal Distribution
  • Q8b (2 marks, class average 40%): Integral of $m f(x) + n$ over the pdf's domain. Lesson: Continuous Random Variables and the Normal Distribution
  • Q9a (3 marks, class average 33%): With $w = -3$, the four solutions of $\frac{9}{(x-1)^2} = (x+3)^2$. Lesson: Solving Equations Analytically and Numerically
  • Q9b.i (2 marks, class average 40%): Minimum point of $y = (x - 1)(x - w)$ in terms of $w$. Lesson: Polynomial Functions
  • Q9b.ii (2 marks, class average 5%): Positive $w$ giving exactly three solutions to $f(x) = g(x)$. Lesson: Solving Equations Analytically and Numerically

2025 Exam 2

  • MC 1 (1 mark, class average 91%): Which circular function has range $[6, 12]$. Lesson: Circular (Trigonometric) Functions
  • MC 2 (1 mark, class average 51%): Asymptotes of $y = 2\tan\left(\pi\left(x + \frac{1}{2}\right)\right)$. Lesson: Circular (Trigonometric) Functions
  • MC 3 (1 mark, class average 71%): Graph of $y = f(-x) + 2$ from a given graph. Lesson: Functions, Domain and Transformations
  • MC 4 (1 mark, class average 55%): Values of $k$ for which a pair of linear equations has no solution. Lesson: Solving Equations Analytically and Numerically
  • MC 5 (1 mark, class average 76%): Which set of ordered pairs has an inverse function. Lesson: Composite and Inverse Functions
  • MC 6 (1 mark, class average 50%): When does the trapezium rule overestimate an area. Lesson: Pseudocode and Numerical Methods — Bisection, Newton’s Method and the Trapezium Rule
  • MC 7 (1 mark, class average 72%): Trace a while loop that subtracts until $n$ is no longer above $k$. Lesson: Pseudocode and Numerical Methods — Bisection, Newton’s Method and the Trapezium Rule
  • MC 8 (1 mark, class average 66%): Sample size from a 95% confidence interval $(0.248, 0.552)$. Lesson: Confidence Intervals for a Proportion
  • MC 9 (1 mark, class average 57%): Conditional probability that a slow-commute student walked. Lesson: Probability — Conditional, Independence and Tree Diagrams
  • MC 10 (1 mark, class average 57%): When is $f(\sin x) > 0$ for a quadratic $f$. Lesson: Composite and Inverse Functions, Circular (Trigonometric) Functions
  • MC 11 (1 mark, class average 75%): Greatest average rate of change on a share price chart. Lesson: Differentiation: Rules and Techniques
  • MC 12 (1 mark, class average 58%): Mean and standard deviation of a normal variable from two probabilities. Lesson: Advanced Normal Distribution Problems
  • MC 13 (1 mark, class average 45%): Graph of $g(f(x))$ when $f$ is a decreasing line through the origin. Lesson: Composite and Inverse Functions, Functions, Domain and Transformations
  • MC 14 (1 mark, class average 66%): Find $k$ for a pdf made of $k\sin(x)$ then $k\cos(x)$. Lesson: Continuous Random Variables and the Normal Distribution
  • MC 15 (1 mark, class average 44%): Image of the point $(1, 3)$ under $y = 1 - g(2x - 3)$. Lesson: Functions, Domain and Transformations
  • MC 16 (1 mark, class average 18%): Sign conditions on $a$ and $b$ when $a\log_e(bx)$ has derivative range $(0, \infty)$. Lesson: Differentiation: Rules and Techniques, Exponential and Logarithmic Functions
  • MC 17 (1 mark, class average 38%): Which graph fits $\int_1^2 f > \int_1^3 f$. Lesson: Definite Integrals and Area
  • MC 18 (1 mark, class average 50%): Which two probability mass functions have the same mean. Lesson: Discrete Random Variables
  • MC 19 (1 mark, class average 14%): Value of $c$ when the shortest distance from $y = x + c$ to a log curve is $\sqrt{2}$. Lesson: Applications of Differentiation
  • MC 20 (1 mark, class average 36%): Which sequence of transformations does not turn $a^x$ into $a^{2x+2}$. Lesson: Functions, Domain and Transformations, Exponential and Logarithmic Functions
  • Q1a (2 marks, class average 95%): Both stationary points of $g(x) = 4x^3 - 3x^4$. Lesson: Applications of Differentiation
  • Q1b (2 marks, class average 80%): Sketch the quartic, labelling stationary points and intercepts. Lesson: Applications of Differentiation, Polynomial Functions
  • Q1c (2 marks, class average 65%): Gradient table showing a stationary point of inflection. Lesson: Applications of Differentiation
  • Q1d (2 marks, class average 75%): Average value of $g$ on $[0, 2]$. Lesson: Average Value and Applications of Integration
  • Q1e (3 marks, class average 40%): Three transformations taking $g$ to a function with given key points. Lesson: Functions, Domain and Transformations
  • Q1f (2 marks, class average 50%): Show $\Pr(X \ge 3) = g(p)$ for $X \sim \text{Bi}(4, p)$. Lesson: Discrete Random Variables
  • Q2a (3 marks, class average 50%): Show $A = 2^{\frac{18}{7}}$ and $k = \frac{3}{14}\log_e(2)$ from two intersection points. Lesson: Exponential and Logarithmic Functions
  • Q2b (1 mark, class average 55%): Write $g(x) = Ae^{kx}$ as $A \times 2^{bx}$. Lesson: Exponential and Logarithmic Functions
  • Q2c (2 marks, class average 80%): Area between the line and the exponential, to two decimal places. Lesson: Definite Integrals and Area
  • Q2d.i (1 mark, class average 71%): Derivative of $h(x) = f(x) - g(x)$. Lesson: Differentiation: Rules and Techniques
  • Q2d.ii (1 mark, class average 56%): Maximum value of $h$ on $[-12, 2]$, to two decimal places. Lesson: Applications of Differentiation
  • Q2e (2 marks, class average 70%): Where $y = g^{-1}(x)$ meets $y = 2(x - 7)$. Lesson: Composite and Inverse Functions
  • Q2f.i (2 marks, class average 50%): Show an antiderivative of $f$ can't pass through both points. Lesson: Antidifferentiation
  • Q2f.ii (2 marks, class average 35%): Find the dilation factor $m$ and constant $c$ so $mF$ passes through both points. Lesson: Functions, Domain and Transformations, Antidifferentiation
  • Q3a.i (1 mark, class average 87%): Mean commute time from a polynomial pdf on $[29, 59]$. Lesson: Continuous Random Variables and the Normal Distribution
  • Q3a.ii (2 marks, class average 75%): Exact standard deviation of the commute time. Lesson: Continuous Random Variables and the Normal Distribution
  • Q3b.i (1 mark, class average 81%): Integral for the probability of a trip longer than 47 minutes. Lesson: Continuous Random Variables and the Normal Distribution
  • Q3b.ii (2 marks, class average 65%): Probability of being late at least once in five days. Lesson: Discrete Random Variables
  • Q3b.iii (2 marks, class average 50%): $\Pr(0.4 \le \hat{P} \le 0.6)$ for the proportion of late days in a week. Lesson: Sampling and Sample Proportions
  • Q3b.iv (2 marks, class average 25%): Integer number of minutes $k$ that makes the weekly chance of lateness 0.2. Lesson: Continuous Random Variables and the Normal Distribution, Discrete Random Variables
  • Q3c.i (1 mark, class average 79%): Normal wait time: probability of under 3.5 minutes. Lesson: Continuous Random Variables and the Normal Distribution
  • Q3c.ii (1 mark, class average 48%): Find $\sigma$ so that only 2% of waits exceed 3.5 minutes. Lesson: Advanced Normal Distribution Problems
  • Q3d (2 marks, class average 50%): Distribution of the number of red lights out of three independent lights. Lesson: Discrete Random Variables, Probability — Conditional, Independence and Tree Diagrams
  • Q4a (1 mark, class average 93%): Evaluate $\sin(x) + 1$ at $x = \frac{2\pi}{3}$. Lesson: Circular (Trigonometric) Functions
  • Q4b (1 mark, class average 78%): Solve $\sin(x) + 1 = \frac{3}{2}$ on $\left[0, \frac{5\pi}{2}\right]$. Lesson: Circular (Trigonometric) Functions, Trigonometric Identities and Equation Solving
  • Q4c (2 marks, class average 30%): Shift $k$ and largest $a$ with $f(x + k) = f(x)$ on $[0, a]$. Lesson: Circular (Trigonometric) Functions
  • Q4d (1 mark, class average 70%): Tangent to $y = \sin(x) + 1$ at $x = \frac{2\pi}{3}$. Lesson: Applications of Differentiation
  • Q4e.i (1 mark, class average 65%): Two iterations of Newton's method from $x_0 = \frac{2\pi}{3}$. Lesson: Pseudocode and Numerical Methods — Bisection, Newton’s Method and the Trapezium Rule
  • Q4e.ii (1 mark, class average 28%): Draw the tangent at $x = x_1$ on the graph. Lesson: Pseudocode and Numerical Methods — Bisection, Newton’s Method and the Trapezium Rule
  • Q4f.i (2 marks, class average 55%): Show the tangent at $x = p$ is $t(x) = \cos(p)(x - p) + \sin(p) + 1$. Lesson: Applications of Differentiation
  • Q4f.ii (2 marks, class average 25%): Exact minimum and maximum of the tangent's $y$-intercept as $p$ varies. Lesson: Applications of Differentiation
  • Q4f.iii (2 marks, class average 20%): Values of $p$ where the tangent's only $x$-intercept is the curve's $x$-intercept. Lesson: Applications of Differentiation, Solving Equations Analytically and Numerically
  • Q4g.i (2 marks, class average 70%): Show $c = 1$ and $d = 1$ when a cubic matches $f$ and $f'$ at $x = 0$. Lesson: Differentiation: Rules and Techniques
  • Q4g.ii (2 marks, class average 35%): Area between $f$ and the matching cubic on $[0, 2\pi]$, two decimal places. Lesson: Definite Integrals and Area
  • Q4g.iii (2 marks, class average 25%): Find $b$ and $r$ where $bx^2 + x + 1$ touches $\sin(x) + 1$. Lesson: Applications of Differentiation, Solving Equations Analytically and Numerically

2024 Exam 1

  • Q1a (1 mark, class average 82%): Differentiate $e^x\cos(3x)$. Lesson: Differentiation: Rules and Techniques
  • Q1b (2 marks, class average 60%): $f'(3)$ for $f(x) = \log_e(x^3 - 3x + 2)$. Lesson: Differentiation: Rules and Techniques, Exponential and Logarithmic Functions
  • Q2 (3 marks, class average 57%): Value of $k$ for which two linear equations have no solution. Lesson: Solving Equations Analytically and Numerically
  • Q3a (3 marks, class average 70%): Sketch $y = \frac{1}{(x + 3)^2} - 2$ with asymptotes and intercepts. Lesson: Other Function Families: Modulus, Reciprocal and Piecewise, Functions, Domain and Transformations
  • Q3b (2 marks, class average 40%): Area between the curve, the $x$-axis, the $y$-axis and $x = -2$. Lesson: Definite Integrals and Area
  • Q4a (1 mark, class average 59%): Standard deviation of $X \sim \text{Bi}\left(4, \frac{9}{10}\right)$. Lesson: Discrete Random Variables
  • Q4b (2 marks, class average 50%): $\Pr(X < 2)$ for the same binomial. Lesson: Discrete Random Variables
  • Q5a (1 mark, class average 95%): Population after four years from $h(t) = \frac{3000}{t + 1}$. Lesson: Other Function Families: Modulus, Reciprocal and Piecewise
  • Q5b (2 marks, class average 20%): Two transformations giving a model that starts the same but falls at half the rate. Lesson: Functions, Domain and Transformations
  • Q5c.i (2 marks, class average 65%): Approximate 95% confidence interval from 60 out of 100, using $z = 2$. Lesson: Confidence Intervals for a Proportion
  • Q5c.ii (1 mark, class average 33%): Smallest sample size giving standard deviation $\frac{\sqrt{2}}{50}$ for the proportion. Lesson: Sampling and Sample Proportions
  • Q6 (4 marks, class average 38%): Solve $2\log_3(x - 4) + \log_3(x) = 2$. Lesson: Exponential and Logarithmic Functions
  • Q7a (3 marks, class average 50%): Trapezium rule with step $\frac{\pi}{3}$ for the area under $x\sin(x)$ on $[0, \pi]$. Lesson: Pseudocode and Numerical Methods — Bisection, Newton’s Method and the Trapezium Rule
  • Q7b.i (1 mark, class average 84%): Derivative of $x\sin(x)$. Lesson: Differentiation: Rules and Techniques
  • Q7b.ii (1 mark, class average 20%): Range of $f'(x)$ over $\left[\frac{\pi}{2}, \frac{2\pi}{3}\right]$. Lesson: Applications of Differentiation
  • Q7b.iii (1 mark, class average 12%): Use the range of $f'$ to verify a stationary point in the interval. Lesson: Applications of Differentiation
  • Q7c (3 marks, class average 37%): Sketch $y = f'(x)$ on $[-\pi, \pi]$. Lesson: Applications of Differentiation
  • Q8a (1 mark, class average 61%): $y$-intercept of $g(x) = \sqrt[3]{x - k} + m$. Lesson: Functions, Domain and Transformations
  • Q8b (2 marks, class average 40%): Gradient of the cube root graph at its $y$-intercept, in terms of $k$. Lesson: Differentiation: Rules and Techniques
  • Q8c (1 mark, class average 47%): Express $k$ in terms of $m$ when the graph passes through the origin. Lesson: Functions, Domain and Transformations
  • Q8d (3 marks, class average 17%): Another point with the same gradient as $P$, in terms of $m$. Lesson: Applications of Differentiation

2024 Exam 2

  • MC 1 (1 mark, class average 74%): Asymptotes of $y = \log_e(x + 1) - 3$. Lesson: Exponential and Logarithmic Functions
  • MC 2 (1 mark, class average 81%): Find $g(2)$ from $g'(x) = x^3 - x$ and $g(0) = 5$. Lesson: Antidifferentiation
  • MC 3 (1 mark, class average 67%): Conditional probability from a discrete table with parameter $k$. Lesson: Discrete Random Variables, Probability — Conditional, Independence and Tree Diagrams
  • MC 4 (1 mark, class average 70%): Combine two definite integrals to find $\int_b^c 2f(x)\,dx$. Lesson: Definite Integrals and Area
  • MC 5 (1 mark, class average 52%): Range of $e^{-f(x)}$ where $f(x) = x^2 - 4x$ on $(1, \infty)$. Lesson: Composite and Inverse Functions
  • MC 6 (1 mark, class average 50%): Rule and domain of the inverse of $\frac{2x + 1}{3 - x}$. Lesson: Composite and Inverse Functions
  • MC 7 (1 mark, class average 68%): Rolls of a die needed for a better than 0.95 chance of at least one six. Lesson: Discrete Random Variables
  • MC 8 (1 mark, class average 65%): Which point must be an $x$-intercept of $f(x) - g(x)$ from a table of values. Lesson: Functions, Domain and Transformations
  • MC 9 (1 mark, class average 66%): Probability a student came by limousine given they had a photo taken. Lesson: Probability — Conditional, Independence and Tree Diagrams
  • MC 10 (1 mark, class average 38%): What must be true if $f'(2) < 0$ and $f'(4) > 0$. Lesson: Applications of Differentiation, Composite and Inverse Functions
  • MC 11 (1 mark, class average 40%): Chance exactly two of three chosen students sit in the first row. Lesson: Probability — Conditional, Independence and Tree Diagrams
  • MC 12 (1 mark, class average 47%): Graph of $y = f(2x + 1)$ from the graph of $f$. Lesson: Functions, Domain and Transformations
  • MC 13 (1 mark, class average 45%): Local minimum after a dilation and a translation of $\frac{x}{2} + \frac{2}{x}$. Lesson: Functions, Domain and Transformations, Applications of Differentiation
  • MC 14 (1 mark, class average 53%): $\Pr(X < 0.5)$ for a pdf made of two line segments. Lesson: Continuous Random Variables and the Normal Distribution
  • MC 15 (1 mark, class average 53%): Points of inflection of $y = 2 - \tan\left(\pi\left(x - \frac{1}{4}\right)\right)$. Lesson: Circular (Trigonometric) Functions
  • MC 16 (1 mark, class average 36%): Gradient of $\sqrt{f(x)}$ at $x = 4$ from $f(4)$ and $f'(4)$. Lesson: Differentiation: Rules and Techniques
  • MC 17 (1 mark, class average 27%): Output of an algorithm that searches for roots of a cubic. Lesson: Pseudocode and Numerical Methods — Bisection, Newton’s Method and the Trapezium Rule, Polynomial Functions
  • MC 18 (1 mark, class average 58%): Value of $x$ that maximises the area of a trapezium with slanted sides 10. Lesson: Applications of Differentiation
  • MC 19 (1 mark, class average 57%): $\Pr(X < 12)$ for a normal variable with symmetric tail probabilities. Lesson: Advanced Normal Distribution Problems
  • MC 20 (1 mark, class average 41%): Integral over $[2, 6]$ of a period-2 function with average value $k$ on $[0, 2]$. Lesson: Average Value and Applications of Integration
  • Q1a (1 mark, class average 85%): Roots of $(x + 1)(x + a)(x - 2)(x - 2a)$. Lesson: Polynomial Functions
  • Q1b.i (2 marks, class average 40%): Values of $a$ giving exactly three $x$-intercepts. Lesson: Polynomial Functions
  • Q1b.ii (1 mark, class average 31%): Values of $a$ giving exactly four $x$-intercepts. Lesson: Polynomial Functions
  • Q1c.i (1 mark, class average 95%): Derivative of $(x + 1)^2(x - 2)^2$. Lesson: Differentiation: Rules and Techniques
  • Q1c.ii (1 mark, class average 76%): Local maximum of $(x + 1)^2(x - 2)^2$. Lesson: Applications of Differentiation
  • Q1c.iii (1 mark, class average 71%): Where $g'(x) > 0$. Lesson: Applications of Differentiation
  • Q1c.iv (2 marks, class average 60%): Where the tangents at $x = \frac{1 \pm \sqrt{3}}{2}$ meet. Lesson: Applications of Differentiation
  • Q1d.i (1 mark, class average 37%): Translations moving the local maximum of $h$ onto that of $g$. Lesson: Functions, Domain and Transformations
  • Q1d.ii (2 marks, class average 25%): A dilation and translations putting both minimums of $h$ on those of $g$. Lesson: Functions, Domain and Transformations, Applications of Differentiation
  • Q2a (2 marks, class average 70%): Derivative of a two-piece heating model as a hybrid function. Lesson: Differentiation: Rules and Techniques, Other Function Families: Modulus, Reciprocal and Piecewise
  • Q2b (1 mark, class average 65%): Average rate of temperature change from $t = 0$ to $t = \frac{1}{2}$. Lesson: Differentiation: Rules and Techniques
  • Q2c.i (1 mark, class average 95%): Derivative of $22 - 10e^{-6t}$. Lesson: Differentiation: Rules and Techniques
  • Q2c.ii (1 mark, class average 85%): When the second model's rate of change is 10. Lesson: Exponential and Logarithmic Functions
  • Q2d (1 mark, class average 63%): When the two temperature models agree, for $t$ between 0 and 1. Lesson: Solving Equations Analytically and Numerically
  • Q2e (1 mark, class average 29%): When the two models differ the most, for $t$ between 0 and 1. Lesson: Applications of Differentiation
  • Q2f.i (1 mark, class average 55%): Use continuity at $t = 0.4$ to show $A = 1.2e^4$. Lesson: Other Function Families: Modulus, Reciprocal and Piecewise
  • Q2f.ii (1 mark, class average 37%): Time to use 0.5 kilowatt hours. Lesson: Definite Integrals and Area
  • Q2f.iii (2 marks, class average 35%): Time to use 1 kilowatt hour, two decimal places. Lesson: Definite Integrals and Area
  • Q3a.i (3 marks, class average 57%): Cubic coefficients from a local minimum and a local maximum. Lesson: Applications of Differentiation, Polynomial Functions
  • Q3a.ii (2 marks, class average 30%): Translation values $h$ and $k$ moving the cubic's maximum to $(23, 4750)$. Lesson: Functions, Domain and Transformations
  • Q3b.i (2 marks, class average 75%): Continue the sales model's graph to $t = 36$ and label the endpoint. Lesson: Circular (Trigonometric) Functions
  • Q3b.ii (1 mark, class average 59%): How much the sales model rises every 12 months. Lesson: Circular (Trigonometric) Functions
  • Q3b.iii (1 mark, class average 77%): Derivative of the sales model. Lesson: Differentiation: Rules and Techniques
  • Q3b.iv (2 marks, class average 30%): Maximum rate of change of sales and when it happens. Lesson: Applications of Differentiation
  • Q4a (1 mark, class average 79%): Integral for the chance a piece of luggage is over 23 kg. Lesson: Continuous Random Variables and the Normal Distribution
  • Q4b.i (1 mark, class average 85%): Mean luggage mass. Lesson: Continuous Random Variables and the Normal Distribution
  • Q4b.ii (2 marks, class average 75%): Standard deviation of luggage mass. Lesson: Continuous Random Variables and the Normal Distribution
  • Q4b.iii (2 marks, class average 60%): Probability a bag is heavy, given it's above the mean. Lesson: Continuous Random Variables and the Normal Distribution, Probability — Conditional, Independence and Tree Diagrams
  • Q4c.i (1 mark, class average 36%): Show the chance of two heavy bags is 0.027. Lesson: Probability — Conditional, Independence and Tree Diagrams
  • Q4c.ii (2 marks, class average 15%): Distribution of the number of heavy bags per traveller. Lesson: Discrete Random Variables, Probability — Conditional, Independence and Tree Diagrams
  • Q4d.i (2 marks, class average 45%): $\Pr(\hat{P} > 0.2)$ for samples of 35 bags. Lesson: Sampling and Sample Proportions
  • Q4d.ii (2 marks, class average 25%): Chance $\hat{P}$ is within one standard deviation of its mean, exactly. Lesson: Sampling and Sample Proportions
  • Q4e.i (1 mark, class average 56%): Approximate 90% confidence interval from 10 heavy bags out of 50. Lesson: Confidence Intervals for a Proportion
  • Q4e.ii (1 mark, class average 8%): Fewest and most heavy bags giving a wider 90% interval from 50 bags. Lesson: Confidence Intervals for a Proportion
  • Q5a.i (1 mark, class average 69%): Local maximum of $\sin(2\sin(x))$ in $\left[0, \frac{\pi}{2}\right]$. Lesson: Composite and Inverse Functions, Applications of Differentiation
  • Q5a.ii (1 mark, class average 63%): Range of $\sin(2\sin(x))$ on $[0, 2\pi]$. Lesson: Composite and Inverse Functions
  • Q5b.i (1 mark, class average 82%): Derivative of $\sin(\sin(2x))$. Lesson: Differentiation: Rules and Techniques
  • Q5b.ii (2 marks, class average 40%): Show $\cos(\sin(2x)) = 0$ has no real solutions. Lesson: Circular (Trigonometric) Functions
  • Q5b.iii (1 mark, class average 47%): Stationary points of $\sin(\sin(2x))$ on $[0, 2\pi]$. Lesson: Applications of Differentiation
  • Q5b.iv (1 mark, class average 34%): Range of $\sin(\sin(2x))$. Lesson: Composite and Inverse Functions
  • Q5c.i (1 mark, class average 32%): Single integral for the area between the two compositions on $[0, 2\pi]$. Lesson: Definite Integrals and Area
  • Q5c.ii (1 mark, class average 45%): Value of that area to two decimal places. Lesson: Definite Integrals and Area
  • Q5d (2 marks, class average 15%): Where $\sin(x)$ restricted to $(0, 2\pi)$ composed with $\sin(2x)$ is defined. Lesson: Composite and Inverse Functions

2023 Exam 1

  • Q1a (2 marks, class average 65%): Differentiate and simplify $\frac{x^2 - x}{e^x}$. Lesson: Differentiation: Rules and Techniques
  • Q1b (2 marks, class average 80%): $f'\left(\frac{\pi}{4}\right)$ for $f(x) = \sin(x)e^{2x}$. Lesson: Differentiation: Rules and Techniques
  • Q2 (3 marks, class average 67%): Solve $e^{2x} - 12 = 4e^x$. Lesson: Exponential and Logarithmic Functions
  • Q3a (3 marks, class average 80%): Sketch $y = 2 - \frac{3}{x - 1}$ with asymptotes and intercepts. Lesson: Other Function Families: Modulus, Reciprocal and Piecewise, Functions, Domain and Transformations
  • Q3b (1 mark, class average 38%): Values of $x$ where $2 - \frac{3}{x - 1} \le 1$. Lesson: Other Function Families: Modulus, Reciprocal and Piecewise
  • Q4 (2 marks, class average 55%): Two trapeziums to estimate the area under $x + \frac{1}{x}$ from 1 to 3. Lesson: Pseudocode and Numerical Methods — Bisection, Newton’s Method and the Trapezium Rule
  • Q5a (1 mark, class average 66%): Evaluate $\int_0^{\frac{\pi}{3}}\sin(x)\,dx$. Lesson: Definite Integrals and Area
  • Q5b (3 marks, class average 60%): Values of $k$ where a cosine integral up to $\frac{\pi}{2}$ equals that sine integral. Lesson: Definite Integrals and Area, Trigonometric Identities and Equation Solving
  • Q6a (1 mark, class average 52%): Sample proportion behind the interval $(0.04, 0.16)$. Lesson: Confidence Intervals for a Proportion
  • Q6b (2 marks, class average 35%): Sample size for that interval, using $z = 2$. Lesson: Confidence Intervals for a Proportion
  • Q6c (1 mark, class average 23%): Effect on the interval's width of a sample four times bigger. Lesson: Confidence Intervals for a Proportion
  • Q7a (1 mark, class average 90%): Range of $x^2 - 2x$ on $(-\infty, 1]$. Lesson: Polynomial Functions
  • Q7b (2 marks, class average 55%): Sketch the inverse function on the same axes. Lesson: Composite and Inverse Functions
  • Q7c (2 marks, class average 50%): Rule and domain of the inverse. Lesson: Composite and Inverse Functions
  • Q7d (2 marks, class average 25%): Total area enclosed by $f$, its inverse and $y = -x$. Lesson: Definite Integrals and Area, Composite and Inverse Functions
  • Q8a (1 mark, class average 43%): Show $k = \frac{1}{64}$ for a queueing-time pdf. Lesson: Continuous Random Variables and the Normal Distribution
  • Q8b (2 marks, class average 40%): Expected queueing time. Lesson: Continuous Random Variables and the Normal Distribution
  • Q8c (3 marks, class average 23%): Chance of queueing over two minutes, given you've already queued one. Lesson: Continuous Random Variables and the Normal Distribution, Probability — Conditional, Independence and Tree Diagrams
  • Q9a (1 mark, class average 90%): Verify the constants in two walking-track functions. Lesson: Polynomial Functions
  • Q9b (2 marks, class average 50%): Show both tracks have a turning point at the same point $P$. Lesson: Applications of Differentiation
  • Q9c (3 marks, class average 27%): Largest area of a triangle under track 2. Lesson: Applications of Differentiation

2023 Exam 2

  • MC 1 (1 mark, class average 77%): Amplitude and period of $-\frac{1}{2}\sin(3x + 2\pi)$. Lesson: Circular (Trigonometric) Functions
  • MC 2 (1 mark, class average 53%): Axis of symmetry of $y = ax^2 + 2bx + c$. Lesson: Polynomial Functions
  • MC 3 (1 mark, class average 47%): Domain of a sum of two functions. Lesson: Functions, Domain and Transformations
  • MC 4 (1 mark, class average 55%): Values of $k$ giving infinitely many solutions to a linear system. Lesson: Solving Equations Analytically and Numerically
  • MC 5 (1 mark, class average 62%): Which power of $x$ has a horizontal tangent at the origin. Lesson: Differentiation: Rules and Techniques, Other Function Families: Modulus, Reciprocal and Piecewise
  • MC 6 (1 mark, class average 49%): Combine integrals over $[3, 10]$ and $[7, 10]$ to get $\int_7^3 f(x)\,dx$. Lesson: Definite Integrals and Area
  • MC 7 (1 mark, class average 60%): Domain of the derivative of $\log_e\left(\sqrt{1 - x}\right)$. Lesson: Composite and Inverse Functions, Differentiation: Rules and Techniques
  • MC 8 (1 mark, class average 49%): Chance of at least one green ball in 8 draws with replacement. Lesson: Discrete Random Variables
  • MC 9 (1 mark, class average 42%): Value of $a$ that makes a tan-then-sine hybrid smooth at $2\pi$. Lesson: Other Function Families: Modulus, Reciprocal and Piecewise, Differentiation: Rules and Techniques
  • MC 10 (1 mark, class average 60%): Find $k$ with $\Pr(X < k) = 0.35$ for a two-piece pdf. Lesson: Continuous Random Variables and the Normal Distribution
  • MC 11 (1 mark, class average 22%): Gradient of $f(x)g(x)$ at $x = -2$ from given values. Lesson: Differentiation: Rules and Techniques
  • MC 12 (1 mark, class average 29%): Largest possible mean of a distribution with parameter $k$. Lesson: Discrete Random Variables
  • MC 13 (1 mark, class average 52%): Result of three Newton's method steps in an algorithm. Lesson: Pseudocode and Numerical Methods — Bisection, Newton’s Method and the Trapezium Rule
  • MC 14 (1 mark, class average 29%): Number of tangents to a quartic through its positive $x$-intercept. Lesson: Applications of Differentiation
  • MC 15 (1 mark, class average 65%): Pick the graphs of two normal curves with different means and spreads. Lesson: Continuous Random Variables and the Normal Distribution
  • MC 16 (1 mark, class average 69%): Find $g$ when $e^{x-1} \times g(x) = e^{(x-1)^2}$. Lesson: Exponential and Logarithmic Functions
  • MC 17 (1 mark, class average 28%): Volume of a cylinder made from a cut metal sheet, in terms of $x$ and $y$. Lesson: Applications of Differentiation
  • MC 18 (1 mark, class average 29%): Number of local minima of $\sin(ax)$ on $[-a\pi, a\pi]$. Lesson: Circular (Trigonometric) Functions
  • MC 19 (1 mark, class average 32%): Values of $k$ giving one positive and one negative root of a quadratic. Lesson: Polynomial Functions
  • MC 20 (1 mark, class average 30%): Largest interval where both $f \circ g$ and $g \circ f$ exist. Lesson: Composite and Inverse Functions
  • Q1a (1 mark, class average 91%): Axis intercepts of $x(x - 2)(x + 1)$. Lesson: Polynomial Functions
  • Q1b (2 marks, class average 80%): Stationary points of the cubic $x(x - 2)(x + 1)$. Lesson: Applications of Differentiation
  • Q1c.i (1 mark, class average 86%): Where the cubic meets the line $y = x - 2$. Lesson: Polynomial Functions
  • Q1c.ii (2 marks, class average 70%): Integral expression for the area between the cubic and the line. Lesson: Definite Integrals and Area
  • Q1c.iii (1 mark, class average 58%): Value of that area to two decimal places. Lesson: Definite Integrals and Area
  • Q1d (4 marks, class average 25%): Possible $a$ and $b$ when a vertical shift of the cubic equals $(x - a)(x - b)^2$. Lesson: Polynomial Functions, Applications of Differentiation
  • Q2a (2 marks, class average 75%): Show $b = \frac{\pi}{15}$ and $c = 75$ for an observation wheel model. Lesson: Circular (Trigonometric) Functions
  • Q2b (2 marks, class average 45%): Average height from the bottom of the wheel to the side point $B$. Lesson: Average Value and Applications of Integration
  • Q2c (1 mark, class average 55%): Average rate of change of height from $A$ to $B$. Lesson: Differentiation: Rules and Techniques
  • Q2d.i (1 mark, class average 40%): Height while stopped, and the speed factor after restarting. Lesson: Circular (Trigonometric) Functions, Other Function Families: Modulus, Reciprocal and Piecewise
  • Q2d.ii (2 marks, class average 20%): All values of $n$ that make the restart join on correctly. Lesson: Circular (Trigonometric) Functions
  • Q2d.iii (3 marks, class average 40%): Sketch the piecewise height model with endpoints. Lesson: Circular (Trigonometric) Functions, Other Function Families: Modulus, Reciprocal and Piecewise
  • Q3a (1 mark, class average 75%): Limit of $2^x + 5$ as $x \to -\infty$. Lesson: Exponential and Logarithmic Functions
  • Q3b (1 mark, class average 85%): Write the derivative of $2^x + 5$ as $k \times 2^x$. Lesson: Differentiation: Rules and Techniques
  • Q3c.i (1 mark, class average 52%): Tangent to $2^x + 5$ at $x = a$. Lesson: Applications of Differentiation
  • Q3c.ii (2 marks, class average 30%): Tangent to $2^x + 5$ through the origin, three decimal places. Lesson: Applications of Differentiation
  • Q3d (1 mark, class average 58%): Point of inflection of $2^x - x^2$. Lesson: Applications of Differentiation
  • Q3e (1 mark, class average 35%): Largest interval where $2^x - x^2$ is strictly decreasing. Lesson: Applications of Differentiation
  • Q3f (2 marks, class average 60%): Three Newton's method estimates for an $x$-intercept of $2^x - x^2$. Lesson: Pseudocode and Numerical Methods — Bisection, Newton’s Method and the Trapezium Rule
  • Q3g (1 mark, class average 21%): Why a root of $\ln(2)2^x - 2x = 0$ is a bad starting point for Newton's method. Lesson: Pseudocode and Numerical Methods — Bisection, Newton’s Method and the Trapezium Rule
  • Q3h (2 marks, class average 10%): Positive $n$ for which $n^x - x^n$ has a local minimum on the $x$-axis. Lesson: Applications of Differentiation, Exponential and Logarithmic Functions
  • Q4a (1 mark, class average 79%): $\Pr(D > 6.8)$ for tennis ball diameters $N(6.7, 0.1^2)$. Lesson: Continuous Random Variables and the Normal Distribution
  • Q4b (1 mark, class average 59%): Diameter that is bigger than 90% of balls. Lesson: Advanced Normal Distribution Problems
  • Q4c (1 mark, class average 77%): Chance a ball fits through a 6.95 cm opening. Lesson: Continuous Random Variables and the Normal Distribution
  • Q4d (2 marks, class average 65%): Chance at least 3 of 4 balls fit. Lesson: Discrete Random Variables
  • Q4e (2 marks, class average 60%): Chance a ball is grade A, given it fits through the opening. Lesson: Advanced Normal Distribution Problems, Probability — Conditional, Independence and Tree Diagrams
  • Q4f (2 marks, class average 35%): Standard deviation needed for over 99% of balls to be grade A. Lesson: Advanced Normal Distribution Problems
  • Q4g (2 marks, class average 25%): Confidence level of an interval from 32 balls. Lesson: Confidence Intervals for a Proportion
  • Q4h (1 mark, class average 57%): Chance a grade A serve is faster than 50 metres per second. Lesson: Continuous Random Variables and the Normal Distribution
  • Q4i (1 mark, class average 55%): Exact mean serving speed for grade A balls. Lesson: Continuous Random Variables and the Normal Distribution
  • Q4j (2 marks, class average 10%): Dilation constants mapping one serve-speed pdf to another. Lesson: Continuous Random Variables and the Normal Distribution, Functions, Domain and Transformations
  • Q5a (2 marks, class average 60%): Finish a sequence of transformations from $e^x + e^{-x}$ to $\frac{1}{2}f(2 - x)$. Lesson: Functions, Domain and Transformations
  • Q5b (2 marks, class average 50%): Domain and range of the inverse of the increasing branch. Lesson: Composite and Inverse Functions
  • Q5c.i (1 mark, class average 55%): Points $P$ and $Q$ where $g$, its inverse branches and $y = x$ meet. Lesson: Composite and Inverse Functions, Solving Equations Analytically and Numerically
  • Q5c.ii (2 marks, class average 30%): Area enclosed by $g$ and the two inverse branches. Lesson: Definite Integrals and Area
  • Q5d (1 mark, class average 13%): Integer $n$ so the turning point of $\frac{1}{k}f(k - x)$ lies on $y = 2x^n$. Lesson: Applications of Differentiation
  • Q5e (1 mark, class average 4%): Smallest $k$ for which $h$ meets the inverse of $h_1$. Lesson: Composite and Inverse Functions
  • Q5f (2 marks, class average 15%): Area between $h$ and the inverse of $h_1$ when $k = 5$. Lesson: Definite Integrals and Area