VCE Specialist Mathematics 2022 Exam 2

Every published question from the 2022 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 2022 Examination 1 walkthrough.

How to use this page

• Questions are from the 2022 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 2022 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.
• Redacted questions. VCAA has redacted multiple-choice Questions 4 and 19 and Section B Question 6f from the published paper and report, following the Independent Review into its examination-setting policies, processes and procedures. They are not covered here.
• Study design. This was the last Examination 2 set on the previous Specialist Mathematics study design (2016 to 2022); the current one began in 2023. Most of the paper is still on the course. The exceptions, flagged where they appear, are the mechanics questions. Momentum is not in the current study design, so multiple-choice Question 17 and Question 5b are outside it. Forces and Newton's second law are covered only in Specialist Mathematics Unit 2 (vectors applied to statics and motion under a constant force), which Units 3 and 4 assume but do not list as a topic, so multiple-choice Questions 15, 16 and 20 and Questions 5a and 5e are partly outside it; the formula sheet also no longer lists the Mechanics formulas (momentum and the equation of motion) that it gave in 2022. Anything not flagged is still examinable.

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. In 2022 the multiple-choice questions had five options (A to E). 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 (11, 9, 10, 11, 10 and 9 marks).

According to the report, students did well at finding asymptotes and sketching rational functions, working with velocity and speed from vector functions of time, resolving forces and solving equations of motion, related rates, hypothesis testing and using CAS. Weaker areas were answering every part of a multi-part question (the ray in 2c, the point in 3b.ii and the time in 4b.ii were often left out), how a function involving inverse tangent behaves in the limit, and random variables that are functions of other random variables.

Section A: Multiple choice

Q1

For $\tfrac12 \le x \le 3$, which line matches $y = |2x - 1| - |x - 3|$? Answer: B - on this interval $2x - 1 \ge 0$ and $x - 3 \le 0$, so $y = (2x - 1) - (3 - x) = 3x - 4$.

Q2

Simplify $1 - \dfrac{4\sin^2(x)}{\tan^2(x) + 1}$. Answer: E - $\tan^2x + 1 = \sec^2x$, so the expression is $1 - 4\sin^2x\cos^2x = 1 - \sin^2(2x) = \cos^2(2x)$.

Q3

What must the graph of $y = \dfrac{x^2 + 2x + c}{x^2 - 4}$ always have? Answer: E - the degrees match, so $y = 1$ is always a horizontal asymptote, but a factor can cancel: with $c = 0$ the function reduces to $1 + \tfrac{2}{x - 2}$, leaving only $x = 2$. So there is at least one vertical asymptote. (38% correct; 41% chose A, which fails when a factor cancels.)

Q4

Redacted by VCAA from the published paper and report; not covered here.

Q5

If $\text{Arg}(z - i) = \tfrac{3\pi}{4}$, which is true? Answer: A - this is a ray from $(0, 1)$ heading up and to the left at gradient $-1$, with the endpoint excluded: $y = 1 - x$, $x < 0$.

Q6

Which graph meets the circle $|z - 5| = 2$ in two points? Answer: E - $|z - 5 - 5i| = 4$ is a circle centred at $(5, 5)$ of radius 4; the centres are 5 apart and $4 - 2 < 5 < 4 + 2$, so the circles cross twice. Options B and C touch the circle once, D misses it, and the ray in A only reaches its excluded endpoint $(3, 0)$.

Q7

Rewrite $\displaystyle\int_0^{\log_e 2}\frac{1}{1 + e^x}\,dx$ using $u = 1 + e^x$. Answer: D - $du = (u - 1)\,dx$ and the terminals become $u = 2$ and $u = 3$, giving $\displaystyle\int_2^3\frac{du}{u(u - 1)} = \int_2^3\left(\frac{1}{u - 1} - \frac{1}{u}\right)du$.

Q8

Which differential equation matches the direction field? Answer: C - the slopes are negative in the first and third quadrants and positive in the second and fourth, and they are steep near the $x$-axis: $\dfrac{dy}{dx} = -\dfrac{2x}{y}$ gives slope $-2$ at $(1, 1)$, while option B would give only $-\tfrac12$.

Q9

Euler's method on $\dfrac{dy}{dx} = 2x^2$ with $x_0 = 1$, $y_0 = 2$ gives $y_2 = 2.976$. Find $h$. Answer: B - $y_1 = 2 + 2h$ and $y_2 = y_1 + 2h(1 + h)^2$; with $h = 0.2$ this is $2.4 + 0.576 = 2.976$.

Q10

When is the tangent to $5x^2y - 3xy + y^2 = 10$ at $(1, m)$ negatively sloped? Answer: E - at $x = 1$ the curve gives $m^2 + 2m = 10$, so $m = -1 \pm \sqrt{11}$. Implicit differentiation gives $\tfrac{dy}{dx} = \tfrac{-7m}{2m + 2}$ at $x = 1$, which is about $-2.44$ and $-4.56$ for the two values: both negative. (21% correct; 38% chose A.)

Q11

$\mathbf a = 2\mathbf i - 3\mathbf j + p\mathbf k$, $\mathbf b = \mathbf i + 2\mathbf j - q\mathbf k$ and $\mathbf c = -3\mathbf i + 2\mathbf j + 5\mathbf k$ are linearly dependent. Then? Answer: A - the determinant of the three vectors is $8p - 5q + 35$, which must be zero, so $8p = 5q - 35$.

Q12

When is $\mathbf u(x) = -\text{cosec}(x)\mathbf i + \sqrt3\mathbf j$ perpendicular to $\mathbf v(x) = \cos(x)\mathbf i + \mathbf j$? Answer: A - $\mathbf u\cdot\mathbf v = -\cot x + \sqrt3 = 0$ gives $\tan x = \tfrac{1}{\sqrt3}$, so $x = \tfrac{\pi}{6}$ or $\tfrac{7\pi}{6}$.

Q13

$\ddot{\mathbf r}(t) = \sin(t)\mathbf i + 2\cos(t)\mathbf j$ and $\dot{\mathbf r}(0) = 2\mathbf i + \mathbf j$. Find $\dot{\mathbf r}(t)$. Answer: B - antidifferentiating gives $(-\cos t + c_1)\mathbf i + (2\sin t + c_2)\mathbf j$; the initial velocity forces $c_1 = 3$ and $c_2 = 1$.

Q14

Constant acceleration, 7 m/s at $A$ and 17 m/s at $B$. Velocity at the midpoint of $AB$? Answer: D - $17^2 = 7^2 + 2as$ gives $as = 120$, and at the midpoint $v^2 = 7^2 + 2a\cdot\tfrac{s}{2} = 49 + 120 = 169$, so $v = 13$. (28% correct; 49% chose C, the average of the two velocities, which applies at the time midpoint, not the distance midpoint.)

Q15

A mass on a smooth incline is acted on by $\mathbf N$, $\mathbf T$ (at angle $\alpha$ to the slope) and $\mathbf W$, and accelerates at $\mathbf a$. What is necessarily true? Answer: E - Newton's second law in vector form, $\mathbf N + \mathbf W + \mathbf T = m\mathbf a$, holds whatever the direction of motion. Option C assumes the object accelerates down the slope, but it could be accelerating up it. (26% correct; 49% chose C.)

Study design: partly outside the current course. Resolving forces and Newton's second law on an inclined plane are covered only in Specialist Mathematics Unit 2 (vectors applied to statics and motion under a constant force), which Units 3 and 4 assume.

Q16. Forces of 5 N, 7 N and 10 N keep a particle in equilibrium. Which equation gives the angle $\theta$ between the 5 N and 7 N forces? Answer: A - the 5 N and 7 N forces must add to a resultant of magnitude 10, so $|\mathbf F_5 + \mathbf F_7|^2 = 25 + 49 + 2 \times 5 \times 7\cos\theta = 100$. In the triangle of forces the interior angle is $\pi - \theta$, which is why the cosine rule there carries a minus sign. (17% correct; 63% chose B.)

Study design: partly outside the current course. Equilibrium of coplanar forces is covered only in Specialist Mathematics Unit 2 (vectors applied to statics and motion under a constant force), which Units 3 and 4 assume.

Q17. A 7 kg particle with constant acceleration starts at 3 m/s and covers 30 m in 6 s. Change in momentum? Answer: C - $30 = 3(6) + \tfrac12a(36)$ gives $a = \tfrac23$, so the final velocity is $3 + 4 = 7$ m/s and the change in momentum is $7(7 - 3) = 28$ kg m/s.

Study design: outside the current course, which does not include momentum.

Q18

Travel times are $N(30, 2.5^2)$ and independent. Probability that two consecutive times differ by more than 6 minutes? Answer: B - the difference $D = T_1 - T_2$ has mean 0 and variance $2 \times 2.5^2 = 12.5$, so $1 - \Pr(-6 < D < 6) \approx 0.0897$. (42% correct.)

Q19

Redacted by VCAA from the published paper and report; not covered here.

Q20

A 4 kg mass hangs on one side of a pulley and two 2 kg masses on the other; the actual masses are normal with means 1.980 and 3.940 and standard deviations 0.015 and 0.002. Probability the 4 kg mass moves up? Answer: C - it rises when the two 2 kg masses together outweigh it. $X = M_1 + M_2 - M_4$ has mean $0.02$ and standard deviation $\sqrt{2(0.015)^2 + 0.002^2} \approx 0.0213$, so $\Pr(X > 0) \approx 0.826$. (41% correct; 24% chose B.)

Study design: partly outside the current course. The probability step (a linear combination of independent normal variables) is on the course, but deciding the direction of motion of a connected-particle pulley system relies on forces and Newton's second law, covered only in Specialist Mathematics Unit 2 (vectors applied to statics and motion under a constant force), which Units 3 and 4 assume.

Section B: Extended response

Question 1 (11 marks)

The family $f(x) = \dfrac{x^2}{x - k}$, $k \in R \setminus \{0\}$.

a

State the equations of the two asymptotes when $k = 1$. (2 marks)

b

Sketch $y = f(x)$ for $k = 1$ on the grid (about $-4.5 \le x \le 4.5$, $-8 \le y \le 8$), labelling turning points with coordinates and asymptotes with equations. (3 marks)

c. i

Find the asymptotes in terms of $k$. (1 mark)

ii

Find, in terms of $k$, the distance between the two turning points. (2 marks)

Now let $h(x) = x + 3$ and $g(x) = \left|\dfrac{x^2}{x - 1}\right|$. The region bounded by their graphs is rotated about the $x$-axis.

d. i. Write a definite integral for the volume of the solid. (2 marks)

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

Question 2 (9 marks)

Let $u = a + i$ and $v = b - \sqrt{2}\,i$, where $a, b \in R$.

a. i

Given $uv = \left(\sqrt{2} + \sqrt{6}\right) + \left(\sqrt{2} - \sqrt{6}\right)i$, show that $a^2 + \left(1 - \sqrt{3}\right)a - \sqrt{3} = 0$. (2 marks)

ii

One solution is $a = \sqrt{3}$, $b = \sqrt{2}$. Find the other pair of values. (1 mark)

b

Plot and label $u = \sqrt{3} + i$ and $v = \sqrt{2} - \sqrt{2}\,i$ on the polar-grid Argand diagram. (2 marks)

c

The ray $\text{Arg}(z) = \theta$ passes through the midpoint of the interval joining $u$ and $v$. Find $\theta$ in radians and draw the ray on the diagram. (2 marks)

d

The interval joining $u$ and $v$ divides the circle $|z| = 2$ into a major and a minor segment. Find the area of the minor segment, correct to two decimal places. (2 marks)

Question 3 (10 marks)

A particle moves in a straight line with $\dfrac{dx}{dt} = \dfrac{2e^{-x}}{1 + 4t^2}$, where $x$ metres is its distance from $O$ at time $t$ seconds and $x = 0$ when $t = 0$.

a. i

Write the differential equation in the form $\displaystyle\int g(x)\,dx = \int f(t)\,dt$. (1 mark)

ii

Hence show that $x = \log_e\left(\tan^{-1}(2t) + 1\right)$. (2 marks)

b. i

State the equation of the horizontal asymptote of this graph. (1 mark)

ii

Sketch the graph and the asymptote for $0 \le t \le 10$, and plot and label the point where $t = 10$, with $x$ correct to two decimal places. (2 marks)

c

Find the speed when $t = 3$, in m/s correct to two decimal places. (1 mark)

A second particle passes through $O$ two seconds after the first, and its distance is $x = \log_e\left(\tan^{-1}(3t - 6) + 1\right)$, with $t$ measured from when the first particle passed $O$.

d. Verify that the particles are the same distance from $O$ when $t = 6$. (1 mark)

e. Find the ratio of the first particle's speed to the second's at that moment, as $\dfrac{a}{b}$ in simplest form, where $a$ and $b$ are positive integers. (2 marks)

Question 4 (11 marks)

A minigolf ball is blown off course by a strong wind. It passes through the origin $O$ at $t = 0$ and the hole is at $(0, 7)$, 7 m from $O$ in the forward ($\mathbf j$) direction. Its path is $\mathbf r(t) = \dfrac{1}{2}\sin\left(\dfrac{\pi t}{4}\right)\mathbf i + 2t\,\mathbf j$ for $t \in [0, 5]$, in metres, where $\mathbf i$ points to the right. At $O$ the path makes an angle of $\theta$ degrees with the forward direction.

a

Find $\theta$, correct to one decimal place. (2 marks)

b. i

Find the speed of the ball as it passes through $O$, in m/s correct to two decimal places. (2 marks)

ii

Find the minimum speed of the ball and the time at which it occurs. (2 marks)

c

Find the minimum distance from the ball to the hole, in metres correct to three decimal places. (3 marks)

d

How far does the ball travel in the first four seconds after passing through $O$? Answer in metres correct to three decimal places. (2 marks)

Question 5 (10 marks)

An object of mass 5 kg moves to the right along a horizontal surface. Forces of $20\sqrt{3}$ N (at $30^\circ$ above the horizontal) and $10\sqrt{2}$ N (at a further $15^\circ$ above that force, so $45^\circ$ to the horizontal) act on it, with a horizontal resistance of $5k$ N, $k \in R$. At point $O$ its speed is 0.5 m s$^{-1}$.

a. Show that the acceleration of the object is $(8 - k)$ m s$^{-2}$. (2 marks)

After 5 seconds the object reaches point $P$ with speed 2 m s$^{-1}$.

b

Find the change in momentum, in kg m s$^{-1}$, from $t = 0$ to $t = 5$. (1 mark)

c

Show that $k = 7.7$. (2 marks)

d

Find the distance, in metres, from $O$ to $P$. (2 marks)

e

At $P$ the two slanting forces stop acting and the object moves up a plane $PQ$, inclined at $\theta^\circ$ to the horizontal, at a reduced speed of 1.95 m s$^{-1}$. A resistance of 38.5 N acts parallel to the plane, and the object comes to rest 0.2 m up the plane from $P$. Find $\theta$, correct to one decimal place. (3 marks)

Study design: part b is outside the current course, which does not include momentum. Parts a and e, which resolve forces and apply Newton's second law (on a horizontal surface and on an inclined plane with resistance), are partly outside: the current study design covers forces only in Specialist Mathematics Unit 2 (vectors applied to statics and motion under a constant force), which Units 3 and 4 assume but do not list as a topic. Parts c and d use only the constant acceleration formulas with the acceleration from part a, which is on the current course.

Question 6 (9 marks)

A supplier claims the mass of aluminium in each empty can is normally distributed with mean 15 g and standard deviation 0.25 g. A random sample of 64 cans has mean mass 14.94 g. The company conducts a one-tailed test at the 5% level of significance, taking the standard deviation as 0.25 g.

a

Write down suitable hypotheses $H_0$ and $H_1$. (1 mark)

b

Find the $p$ value for the test, correct to three decimal places. (1 mark)

c

Does the sample mean support the supplier's claim at the 5% level for a one-tailed test? Justify your answer. (1 mark)

d

What is the smallest value of the sample mean of 64 cans for which $H_0$ is not rejected? Answer correct to two decimal places. (1 mark)

Filled cans have masses normally distributed with mean 406 g and standard deviation 5 g.

e. Find the probability that the masses of two randomly selected filled cans differ by no more than 3 g, correct to three decimal places. (2 marks)

Part f was redacted by VCAA from the paper and the report (following the Independent Review into VCAA's examination-setting policies, processes and procedures), so it is not covered here.