VCE Specialist Mathematics 2025 Exam 2

Every question from the 2025 VCE Specialist Mathematics 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 Specialist Mathematics Examination 2, copyright Victorian Curriculum and Assessment Authority (VCAA). Each question is summarised briefly here; open the official examination PDF for the full wording, diagrams and answer options.
• Answers are original ExamExplained working. Every multiple-choice answer matches the key in the 2025 Specialist Mathematics 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 Specialist Mathematics examinations page.

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. Take $g = 9.8 \text{ m s}^{-2}$.

• Section A (20 marks): 20 multiple-choice questions. Aim for about 25 minutes.
• Section B (60 marks): 6 extended-response questions of 10 marks each.

Section A: Multiple choice

Q1

Contrapositive of "If I have a tiger, then I have a cat". Answer: C - swap and negate both parts: "If I do not have a cat, then I do not have a tiger".

Q2

Which function is a counter-example to "if $f''(0) = 0$ then $f$ has a point of inflection at $x = 0$"? Answer: D - $f(x) = x^4 - x$ has $f''(x) = 12x^2$, which is 0 at $x = 0$ but does not change sign, so there is no inflection there. (48% correct; 40% chose C.)

Q3

$y = \dfrac{x^2 + a}{bx + c}$ has asymptote $y = -\tfrac12x + \tfrac14$ and $y$-intercept $-2$. Find $a, b, c$. Answer: A - dividing gives $y = \tfrac{x}{b} - \tfrac{c}{b^2} + \ldots$, so $\tfrac1b = -\tfrac12$ ($b = -2$) and $-\tfrac{c}{4} = \tfrac14$ ($c = -1$); then $\tfrac{a}{c} = -2$ gives $a = 2$.

Q4

An algorithm adds $\pi\big(f(\text{left})\big)^2$ for left $= 1, 2, 3$ with $f(x) = \sqrt{x + 1}$. What is printed? Answer: B - $\pi(2 + 3 + 4) = 9\pi$.

Q5

$z^3 + az^2 + bz - 52 = 0$ has root $2 - 3i$. Find $ab$. Answer: A - the conjugate is also a root, giving the factor $z^2 - 4z + 13$; the third factor is $z - 4$ (since $13 \times (-4) = -52$). Expanding gives $a = -8$ and $b = 29$, so $ab = -232$.

Q6

If $|z| = 1$ and $z \ne 1$, find $\text{Re}\left(\dfrac{1}{1 - z}\right)$. Answer: C - with $z = \cos\theta + i\sin\theta$, $\dfrac{1}{1 - z} = \dfrac{1 - \bar z}{|1 - z|^2}$ and $|1 - z|^2 = 2 - 2\cos\theta = 2(1 - \cos\theta)$, so the real part is $\dfrac{1 - \cos\theta}{2(1 - \cos\theta)} = \dfrac12$.

Q7

Use $u = \cos\theta$ to rewrite $\tfrac12\int_0^{\pi/2}\frac{\sin 2\theta}{1 + \cos\theta}\,d\theta$. Answer: D - $\sin 2\theta = 2\sin\theta\cos\theta$ and $du = -\sin\theta\,d\theta$, giving $\int_0^1\frac{u}{1 + u}\,du = \int_0^1\left(1 - \frac{1}{1 + u}\right)du$.

Q8

Which differential equation matches the direction field? Answer: A - the slopes are symmetric about the $y$-axis (so $x$ appears squared), and on the positive $y$-axis the slope is negative: $\dfrac{dy}{dx} = x^2 - y$.

Q9

Surface area when $x = kt$, $y = e^{kt}$, $a \le t \le b$, is rotated about the $x$-axis. Answer: C - $2\pi\int_a^b e^{kt}\sqrt{k^2 + k^2e^{2kt}}\,dt$; with $u = e^{kt}$, $du = ke^{kt}\,dt$, this is $2\pi\int_{e^{ka}}^{e^{kb}}\sqrt{1 + u^2}\,du$.

Q10

The region under $y = 3\cos^{-1}(x)$, $0 \le y \le a$, beside $x = 0$ is rotated about the $y$-axis, with volume $\dfrac{\pi(4\pi + 3\sqrt3)}{8}$. Find $a$. Answer: D - $V = \pi\int_0^a\cos^2\!\left(\tfrac y3\right)dy = \tfrac{\pi}{2}\left(a + \tfrac32\sin\tfrac{2a}{3}\right)$, which equals the given volume at $a = \pi$.

Q11

$\dfrac{dy}{dx} = x^2y^3$ with $y(1) = 3$. Domain of $y$? Answer: B - separating gives $y^2 = \dfrac{9}{7 - 6x^3}$, which needs $7 - 6x^3 > 0$: $x < \left(\tfrac76\right)^{1/3}$.

Q12

Constant acceleration, velocity $u$ at $A$ and $v$ at $B$. Velocity at the midpoint of $AB$? Answer: D - $v_m^2 = u^2 + 2a\tfrac{s}{2}$ and $v^2 = u^2 + 2as$ give $v_m = \sqrt{\dfrac{u^2 + v^2}{2}}$. (33% correct.)

Q13

A ball is thrown up at 20 m/s from 50 m and lands on a tray 1 m above the ground. Time taken? Answer: B - taking up as positive, $-49 = 20t - 4.9t^2$, so $t \approx 5.80$ s.

Q14

Non-zero $\mathbf{a}, \mathbf{b}$ with $\mathbf{a}\cdot\mathbf{b} = |\mathbf{a}\times\mathbf{b}|$. Angle between them? Answer: B - $|\mathbf a||\mathbf b|\cos\theta = |\mathbf a||\mathbf b|\sin\theta$, so $\tan\theta = 1$ and $\theta = \tfrac{\pi}{4}$.

Q15

Planes $2x + 2y + z = 2$ and $ax + 4z = 1$ meet at angle $\cos^{-1}\tfrac23$. Equation for $a$? Answer: A - using normals, $\dfrac{2a + 4}{3\sqrt{a^2 + 16}} = \dfrac23$, which simplifies to $a + 2 = \sqrt{a^2 + 16}$.

Q16

$\mathbf{r}(t) = ne^{-2t}\mathbf{i} - t^2\mathbf{j}$; for which $n$ are acceleration and velocity perpendicular at $t = \tfrac12$? Answer: C - $\mathbf{v}\cdot\mathbf{a} = -8n^2e^{-4t} + 4t = 0$ at $t = \tfrac12$ gives $n^2 = \tfrac{e^2}{4}$, so $n = \tfrac{e}{2}$.

Q17

A particle from rest has $\mathbf{a}(t) = 4\cos(2t)\mathbf{i} + 10\sin(2t)\mathbf{j} - 6e^{-2t}\mathbf{k}$. Find $\mathbf{v}(t)$. Answer: C - integrate and choose constants so $\mathbf{v}(0) = \mathbf 0$: $2\sin(2t)\mathbf{i} - 5\big(\cos(2t) - 1\big)\mathbf{j} + 3\big(e^{-2t} - 1\big)\mathbf{k}$.

Q18

Two lines meet at $(4, 3, t)$. Find $r$, $s$, $t$. Answer: D - matching coordinates gives $\lambda = 2$, $\mu = 3$, and so $r = 5$, $s = 8$, $t = 5$.

Q19

Area of the triangle cut from the axes by $x + y + z = a$. Answer: B - the intercepts are $a$ along each axis, forming an equilateral triangle of side $a\sqrt2$, with area $\tfrac{\sqrt3}{4}(2a^2) = \tfrac{\sqrt3a^2}{2}$.

Q20

$P \sim N(-2, 2^2)$, $Q \sim N(3, 3^2)$, $R \sim N(5, 6^2)$ independent. Find $\Pr(3P + 2Q - R > 25)$. Answer: A - the mean is $-6 + 6 - 5 = -5$ and the variance is $36 + 36 + 36 = 108$, so $\Pr\!\left(Z > \tfrac{30}{6\sqrt3}\right) = \Pr\!\left(Z > \tfrac{5\sqrt3}{3}\right)$.

Section B: Extended response

Question 1 (10 marks)

a

Sketch $y = \dfrac{3x}{x^3 + x + 2}$, labelling asymptotes, the turning point, and the point of inflection (to one decimal place). (3 marks)

b

The region bounded by this graph, the coordinate axes and $x = 2$ is rotated about the $x$-axis. i. Write a definite integral for the volume. (1 mark)

ii

Find the volume to two decimal places. (1 mark)

c

Find the vertical asymptotes of $y = \dfrac{3x}{x^3 - 5x + 2}$. (1 mark)

d

For the family $y = \dfrac{3x}{x^3 + ax + 2}$: i. if the graph has a stationary point $P$, find its $y$-coordinate in terms of $a$. (1 mark)

ii

For one value of $a$ there is no stationary point. Find the vertical asymptotes in this case. (1 mark)

iii

Find $a$ if the graph has a point of inflection at $x = 2$. (2 marks)

Question 2 (10 marks)

a

Sketch $\{z : z\bar z = 4\}$ on the Argand plane. (1 mark)

b. i

Show that $\{z : |z - 2i| = |z - \sqrt3 - i|\}$ is the line $y = \sqrt3x$. (2 marks)

ii

Sketch this line on the same diagram. (1 mark)

c. i

Find the intersection points of the curves in parts a and b.i, in the form $a + bi$. (2 marks)

ii

Label them on the diagram. (1 mark)

d

A ray starts at $P\left(\tfrac32, \tfrac{3\sqrt3}{2}\right)$ and passes through $Q(3, 0)$, with equation $\text{Arg}(z - z_0) = \theta$. Find $z_0$ and $\theta$. (1 mark)

e

Find the area of the minor segment cut from $|z| = 3$ by the chord $PQ$, in the form $c\pi + d$. (2 marks)

Question 3 (10 marks)

A 3000 L tank initially holds 5 kg of salt. Brine at 0.1 kg/L flows in at 20 L/min and the well-mixed solution drains at 20 L/min. $Q$ kg is the salt at time $t$ min.

a

By considering concentration, explain whether the quantity of salt increases. (1 mark)

b

Show that $\dfrac{dQ}{dt} = \dfrac{300 - Q}{150}$. (1 mark)

c

Use Euler's method with step 15 minutes to estimate $Q(30)$ to two decimal places. (2 marks)

d

Use calculus to solve for $Q$ in terms of $t$. (3 marks)

e

What does $Q$ approach as $t \to \infty$? (1 mark)

f

Find the time for the salt to reach 100 kg. (1 mark)

g

At that moment the drain is closed (inflow continues). How many minutes until the concentration reaches $\tfrac{1}{20}$ kg/L? (1 mark)

Question 4 (10 marks)

A particle has position $\mathbf{r}(t) = \left(5\cos t - 4\cos\tfrac{5t}{2}\right)\mathbf{i} + \left(5\sin t - 4\sin\tfrac{5t}{2}\right)\mathbf{j}$, $t \ge 0$.

a

Give the starting point. (1 mark)

b

Draw an arrow at $(9, 0)$ showing the direction of motion. (1 mark)

c

Find when the particle first returns to its starting point. (1 mark)

d

Show that its speed is $\sqrt{125 - 100\cos\left(\tfrac{3t}{2}\right)}$. (3 marks)

e

Find the maximum speed. (1 mark)

f

Trace the path for $t \in [0, \pi]$. (1 mark)

g

Find the length of that path, to one decimal place. (2 marks)

Question 5 (10 marks)

Planes $\Pi_1: 2x + 9z = 8$, $\Pi_2: 3x + 6y + 5z = 7$, $\Pi_3: x + 9y - 3z = 7$.

a

Find the point of intersection of the three planes. (1 mark)

b. i

Find a direction vector for the line of intersection of $\Pi_2$ and $\Pi_3$. (2 marks)

ii

Give parametric equations for this line. (1 mark)

c

Find the shortest distance from $(1, 1, 2)$ to $\Pi_3$. (2 marks)

d

$\Psi$ is the family of planes $6x + 27z = m$, $m \in N$. i. Show that $\Pi_1$ is parallel to every member of $\Psi$. (1 mark)

ii

Find all $m$ for which the distance between $\Pi_1$ and $6x + 27z = m$ is $\dfrac{23}{3\sqrt{85}}$. (3 marks)

Question 6 (10 marks)

Water drunk per day, $V$ mL, is normal with mean 1000 and standard deviation 80.

a. i. Give the mean and standard deviation of the sample mean for samples of 25 students. (1 mark)

ii. Find the probability that such a sample mean exceeds 970 mL, to four decimal places. (1 mark)

Wasser bottle volumes have $\sigma = 5$ mL; a sample of 30 bottles has mean 750 mL.

b

Find a 95% confidence interval for the mean, to one decimal place. (1 mark)

c

Of 300 independent such 95% intervals, how many would be expected to contain the true mean? (1 mark)

d

Find the minimum sample size so that the sample mean is within 1 mL of the true mean at the 95% level. (1 mark)

Apa bottles are normal with mean 750 mL and $\sigma = 5$ mL. After a service, a sample of 50 has mean 748 mL; the company claims the mean is now less than 750 mL. A one-tailed test at the 1% level is proposed.

e

State $H_0$ and $H_1$. (1 mark)

f. i

Find the $p$ value to four decimal places. (1 mark)

ii

Is the claim correct? Explain using the $p$ value. (1 mark)

g

Find the critical sample mean for $n = 50$ at the 1% level, to three decimal places. (1 mark)

h

If the true mean is actually 747.5 mL, find the probability of concluding the mean has not been reduced, to three decimal places. (1 mark)