VCE Math Methods 2025 Exam 2

Every question from the 2025 VCE Mathematical Methods Examination 2, the technology-active (CAS) paper. Multiple-choice answers come with a one-line reason; Section B solutions sit behind a Show worked solution toggle. For the no-calculator paper, see the 2025 Examination 1 walkthrough.

How to use this page

• Questions are from the 2025 VCE Mathematical Methods Examination 2, copyright Victorian Curriculum and Assessment Authority (VCAA). Each question is summarised briefly here; open the official examination PDF for the full wording, graphs and answer options.
• Answers are original ExamExplained working. Every multiple-choice answer matches the key in the 2025 Mathematical Methods Examination 2 external assessment report (Word document), and every Section B result was recomputed and compared with the report. Both files are listed on the VCAA Mathematical Methods examinations page.
• Where we say "CAS", any approved CAS calculator gives the value shown; the working shows what to enter.

Structure and timing

Examination 2 is 80 marks in 120 minutes (plus 15 minutes reading time), with a CAS calculator and one bound reference allowed. That is 1.5 minutes per mark.

• Section A (20 marks): 20 multiple-choice questions. Aim for about 25 minutes.
• Section B (60 marks): 4 extended-response questions (13, 14, 14 and 19 marks). Give exact answers unless told to round, and show the integral, equation or distribution you put into CAS.

Section A: Multiple choice

Q1

Which function has range $[6, 12]$? Answer: C - the centre is 9 and the amplitude is 3, so $f(x) = 9 - 3\cos(6x)$ runs from 6 to 12. (91% correct.)

Q2

Find all vertical asymptotes of $y = 2\tan\!\left(\pi\left(x + \tfrac12\right)\right)$. Answer: A - asymptotes occur where $\pi\left(x + \tfrac12\right) = \tfrac{\pi}{2} + k\pi$, so $x = k$, $k \in Z$ (the period is 1).

Q3

Given a graph of $y = f(x)$, choose the graph of $y = f(-x) + 2$. Answer: B - reflect in the $y$-axis, then shift up 2; for example the point $(0, 1)$ moves to $(0, 3)$ and $(4, 2)$ moves to $(-4, 4)$.

Q4

For which $k$ does the system $kx + 3y = k^2$, $2x + (2k+1)y = 6 - 2k$ have no real solutions? Answer: A - the determinant $k(2k+1) - 6 = (2k-3)(k+2)$ is zero at $k = \tfrac32$ or $k = -2$. At $k = \tfrac32$ the equations are multiples of each other (infinitely many solutions); at $k = -2$ they are parallel and inconsistent, so $k = -2$ only.

Q5

Which set of ordered pairs is a function with an inverse function? Answer: B - $\{(-1, 3), (2, 2), (3, 1)\}$ is one-to-one; each other set either repeats an $x$-value (not a function) or repeats a $y$-value (not one-to-one).

Q6

With two trapeziums on $[0, 1]$, for which function does the trapezium rule overestimate the area? Answer: B - the rule overestimates when the curve is concave up, and $x^3 + 1$ has $f''(x) = 6x \ge 0$ on $[0, 1]$; the other three are concave down there.

Q7

A loop starts with $n = 17$, $k = 5$ and repeatedly replaces $n$ by $n - k$ and prints it while $n > k$. What is printed? Answer: C - 12, 7, 2. After printing 2 the condition $2 > 5$ fails and the loop stops.

Q8

A 95% confidence interval for a proportion is $(0.248, 0.552)$. Find the sample size. Answer: C - $\hat p = 0.4$ and the margin is $0.152 = 1.96\sqrt{\tfrac{0.4 \times 0.6}{n}}$, which gives $n \approx 40$.

Q9

Of $m$ walkers, 20% take at least 30 minutes; of $n$ others, 40% do. Given a student took at least 30 minutes, find the probability they walked. Answer: A - $\dfrac{0.2m}{0.2m + 0.4n} = \dfrac{m}{m + 2n}$.

Q10

With $f(x) = 2x^2 + x - 1$ and $g(x) = \sin(x)$, when is $(f \circ g)(x) > 0$? Answer: C - $2\sin^2 x + \sin x - 1 = (2\sin x - 1)(\sin x + 1) > 0$ needs $\sin x > \tfrac12$ (the other branch $\sin x < -1$ is impossible), so $\tfrac12 < \sin(x) \le 1$.

Q11

From a 30-day share-price chart, which interval has the greatest average rate of change? Answer: D - day 14 to day 28 rises about $4.40 in 14 days (about $0.31 per day), steeper than the other three chords.

Q12

$\Pr(X > 200) = 0.325$ and $\Pr(180 < X < 200) = 0.589$ for a normal $X$. Find $\mu$ and $\sigma$. Answer: D - $\Pr(X < 180) = 0.086$, so $\tfrac{200 - \mu}{\sigma} = 0.4538$ and $\tfrac{180 - \mu}{\sigma} = -1.3658$. Subtracting gives $\sigma \approx 11$ and then $\mu \approx 195$.

Q13

Given the graphs of a line $y = f(x)$ through the origin with negative gradient and a quartic $y = g(x)$, which could be $y = (g \circ f)(x)$? Answer: C - $g(f(x)) = g(-ax)$ with $a > 0$ is $g$ reflected in the $y$-axis and dilated from the $y$-axis. Only C matches.

Q14

A pdf is $k\sin(x)$ on $\left[0, \tfrac{\pi}{4}\right)$ and $k\cos(x)$ on $\left[\tfrac{\pi}{4}, \tfrac{\pi}{2}\right]$. Find $k$. Answer: B - each piece integrates to $k\left(1 - \tfrac{\sqrt2}{2}\right)$, so $k(2 - \sqrt2) = 1$ and $k = \dfrac{1}{2 - \sqrt2}$.

Q15

$y = g(x)$ passes through $(1, 3)$. Which point must lie on $y = 1 - g(2x - 3)$? Answer: B - set $2x - 3 = 1$, so $x = 2$ and $y = 1 - 3 = -2$, giving $(2, -2)$.

Q16

$h(x) = a\log_e(bx)$ and $h'(x)$ has range $(0, \infty)$. What must be true? Answer: D - $h'(x) = \tfrac{a}{x}$. The domain is $x > 0$ when $b > 0$ and $x < 0$ when $b < 0$; for $\tfrac{a}{x}$ to be positive, $a$ must share the sign of $x$, which has the sign of $b$. So $ab > 0$.

Q17

Which graph satisfies $\int_1^2 f(x)\,dx > \int_1^3 f(x)\,dx$? Answer: A - the condition means $\int_2^3 f(x)\,dx < 0$, so the graph must lie mainly below the $x$-axis between 2 and 3. Only the parabola in A (negative between its roots 1 and 3) does.

Q18

Which pair of the four probability mass functions has the same mean? Answer: D - the means are 2.5 (I), 3 (II), 2 (III) and 3 (IV), so II and IV.

Q19

$A$ is on $y = x + c$, $B$ is on $y = \log_e(x - 1)$, and the minimum length of $AB$ is $\sqrt2$. Find $c$. Answer: D - the closest point on the curve has gradient 1: $\tfrac{1}{x-1} = 1$ gives $B(2, 0)$. Its distance to the line is $\tfrac{|2 + c|}{\sqrt2} = \sqrt2$, so $c = 0$ or $c = -4$; but $y = x - 4$ cuts the curve (distance 0), so $c = 0$.

Q20

For $a > 1$, which sequence of transformations does not map $f(x) = a^x$ to $g(x) = a^{2x+2}$? Answer: C - dilating by $a$ from the $x$-axis gives $a^{x+1}$, then by $\tfrac12$ from the $y$-axis gives $a^{2x+1}$, then 1 unit right gives $a^{2(x-1)+1} = a^{2x-1} \ne g(x)$. Options A, B and D all give $a^{2x+2}$.

From the report. The hardest multiple-choice items were Q19 (14% correct), Q16 (18% correct), Q20 (36% correct) and Q17 (38% correct). In Q16, 46% chose C ($a > 0$ and $b > 0$), which is sufficient but not necessary; in Q19 the most popular wrong answer was A.

Section B: Extended response

Question 1 (13 marks)

Let $g(x) = 4x^3 - 3x^4$, $x \in R$.

a

Find the coordinates of both stationary points of $g$. (2 marks)

b

Sketch $y = g(x)$, labelling stationary points and axial intercepts with coordinates. (2 marks)

c

Complete a gradient table of $x$ and $g'(x)$ values that shows $g$ has a stationary point of inflection. (2 marks)

d

Find the average value of $g$ between $x = 0$ and $x = 2$. (2 marks)

e

Give a sequence of three transformations mapping $g$ to a function $h$ with a stationary point of inflection at $(1, 0)$ and a local maximum at $(-1, 1)$. (3 marks)

f

For $X \sim \text{Bi}(4, p)$, show that $\Pr(X \ge 3) = g(p)$ for all $p \in [0, 1]$. (2 marks)

Question 2 (14 marks)

Let $f(x) = \tfrac{x}{2} + 7$ and $g(x) = Ae^{kx}$, whose graphs meet at $(-12, 1)$ and $(2, 8)$.

a

Write two simultaneous equations in $A$ and $k$ and solve them algebraically to show $A = 2^{\frac{18}{7}}$ and $k = \tfrac{3}{14}\log_e(2)$. (3 marks)

b

Find $b$ such that $g(x) = A \times 2^{bx}$. (1 mark)

c

Use a definite integral to find the area between the graphs for $x \in [-12, 2]$, to two decimal places. (2 marks)

d

Let $h(x) = f(x) - g(x)$. i. Write down $h'(x)$. (1 mark)

ii

Find the maximum value of $h$ on $[-12, 2]$, to two decimal places. (1 mark)

e

Find the points where $y = g^{-1}(x)$ meets $y = 2(x - 7)$. (2 marks)

f

$F$ is an antiderivative of $f$ through $(0, c)$. i. Show that $y = F(x)$ cannot pass through both $(-12, 1)$ and $(2, 8)$. (2 marks)

ii

If $y = F(x)$ is dilated by a factor of $m$ from the $x$-axis so that the image passes through both points, find $m$ and $c$. (2 marks)

Question 3 (14 marks)

A driver's travel time $T$ (minutes) has pdf $f(t) = \dfrac{1}{1\,215\,000}(t - 29)(59 - t)^3$ for $29 \le t \le 59$, and 0 elsewhere.

a. i

Find the mean travel time. (1 mark)

ii

Find the standard deviation of the travel time. (2 marks)

b

The driver is late if the trip takes longer than $k$ minutes; days are independent. i. For $k = 47$, write a definite integral showing that the probability of being late is 0.08704. (1 mark)

ii

For $k = 47$, find the probability of being late at least once in a five-day week, to four decimal places. (2 marks)

iii

For $k = 47$, with $\hat P$ the proportion of late days in a five-day week, find $\Pr(0.4 \le \hat P \le 0.6)$ to four decimal places. (2 marks)

iv

Find the integer $k$ for which the probability of being late at least once in a five-day week is 0.2, correct to one decimal place. (2 marks)

c

A traffic-light wait is normal with mean 2.5 minutes and standard deviation $\sigma$. i. If $\sigma = 0.6$, find the probability the wait is under 3.5 minutes, to two decimal places. (1 mark)

ii

Find $\sigma$ so that there is a 2% chance of a wait longer than 3.5 minutes, to two decimal places. (1 mark)

d

Lights $A$, $B$, $C$ are red with probabilities 0.2, 0.3 and 0.1, independently. Complete the probability distribution of $Y$, the number of red lights. (2 marks)

Question 4 (19 marks)

Let $f:\left[0, \tfrac{5\pi}{2}\right] \to R$, $f(x) = \sin(x) + 1$.

a

Evaluate $f\!\left(\tfrac{2\pi}{3}\right)$. (1 mark)

b

Find the exact values of $x$ for which $f(x) = \tfrac32$. (1 mark)

c

Real numbers $a$ and $k$ in $\left(0, \tfrac{5\pi}{2}\right)$ satisfy $f(x + k) = f(x)$ for all $x \in [0, a]$. Find $k$ and the largest possible $a$. (2 marks)

d

Find the equation of the tangent to $y = f(x)$ at $x = \tfrac{2\pi}{3}$ (point $A$). (1 mark)

e

Apply two iterations of Newton's method to $f$ with $x_0 = \tfrac{2\pi}{3}$. i. Give $x_2$ to one decimal place. (1 mark)

ii

Draw the tangent to $y = f(x)$ at $x = x_1$. (1 mark)

f

$y = t(x)$ is the tangent at $(p, f(p))$, $p \in \left(0, \tfrac{5\pi}{2}\right)$. i. Show that $t(x) = \cos(p)(x - p) + \sin(p) + 1$. (2 marks)

ii

Find the minimum and maximum possible $y$-intercepts of $y = t(x)$. (2 marks)

iii

Find $p$ (to two decimal places) for which $y = t(x)$ has a unique $x$-intercept equal to the $x$-intercept of $y = f(x)$. (2 marks)

g

$g(x) = ax^3 + bx^2 + cx + d$ on $\left[0, \tfrac{5\pi}{2}\right]$ with $g(0) = f(0)$ and $g'(0) = f'(0)$. i. Show $c = 1$ and $d = 1$. (2 marks)

ii

If also $g(2\pi) = f(2\pi)$ and $g'(2\pi) = f'(2\pi)$, find the area between the graphs for $x \in [0, 2\pi]$ to two decimal places. (2 marks)

iii

With $a = 0$, $c = 1$, $d = 1$, find $b$ and $r \in \left(0, \tfrac{5\pi}{2}\right)$ such that $g(r) = f(r)$ and $g'(r) = f'(r)$. (2 marks)

General advice from the 2025 report

• Show the set-up before using CAS. Write the integral, equation or binomial expression you entered; bare answers on multi-mark parts lost marks.
• Exact versus rounded. Read each part for "exact" or "correct to $n$ decimal places" and give exactly that; both an unrounded decimal where exact was needed and an exact value where a decimal was requested lost marks.
• Answer the actual question. A maximum value is not a coordinate pair, and an integer answer must be an integer.
• Use a ruler for tangents and lines on graphs.