VCE Specialist Mathematics 2021 Exam 2

Every question from the 2021 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 2021 Examination 1 walkthrough.

How to use this page

• Questions are from the 2021 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 2021 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.
• Percentages after multiple-choice answers are the share of students who chose the correct option. In Section B, the From the report notes summarise what the assessors said, with the share of students who gained full marks.
• Study design. This paper was 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 as partly outside where they appear, are the questions about forces: Section A Questions 14 (Newton's second law), 15 (forces in equilibrium) and 16 (an object sliding down a smooth slope), and all of Section B Question 5 (connected masses on a rough inclined plane). The current 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, and the formula sheet no longer lists the Mechanics formulas (momentum and the equation of motion) that it gave in 2021. Everything else, including Question 4 (the stunt car, which is kinematics and vector calculus with no forces), 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 2021 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 (10, 9, 10, 11, 10 and 10 marks).

The report named these strengths: sketching in the complex plane, setting up a volume-of-revolution integral, using the chain rule for related rates, drawing a force diagram and CAS use. The weaknesses were answering every part of a multi-part question (1c, 3b.ii and 6d were singled out), using the right angle for the gradient of a line, justifying an answer when asked to, and noticing when the conditions change part-way through a mechanics problem. Three parts (3b.i, 4a and 4d) ask you to show a given result, so every step must be set out.

Section A: Multiple choice

Q1

How many asymptotes does the graph of $f(x) = \dfrac{1}{\sec(3x) + \frac32}$ have on $\left[-\tfrac{\pi}{6}, \pi\right]$? Answer: B - $f(x) = \dfrac{2\cos(3x)}{2 + 3\cos(3x)}$ has a vertical asymptote wherever $\cos(3x) = -\tfrac23$. For $3x \in \left[-\tfrac{\pi}{2}, 3\pi\right]$ this happens three times ($3x \approx 2.30$, $3.98$ and $8.58$). Where $\cos(3x) = 0$ the rule is undefined but $f(x) \to 0$, so those points are holes, not asymptotes. (66% correct; 20% chose C.)

Q2

Implied domain of $f(x) = \cos^{-1}\big(\log_e(bx)\big)$, $b > 0$. Answer: E - need $-1 \le \log_e(bx) \le 1$, so $e^{-1} \le bx \le e$ and $x \in \left[\tfrac{1}{be}, \tfrac{e}{b}\right]$. (76% correct.)

Q3

Local maxima of $y = \dfrac{1}{\big(\cos(ax) + 1\big)^2 + 3}$, $a \ne 0$. Answer: E - $y$ is largest when the denominator is smallest, which is 3 when $\cos(ax) = -1$, that is $ax = (1 + 2k)\pi$: the points $\left(\tfrac{\pi(1 + 2k)}{a}, \tfrac13\right)$, $k \in Z$. (51% correct.)

Q4

For $\text{Im}(z) > 0$, find $\text{Arg}\left(\dfrac{z\bar z}{z - \bar z}\right)$. Answer: A - with $z = a + bi$ and $b > 0$, $z\bar z = a^2 + b^2$ and $z - \bar z = 2bi$, so the quotient is $\dfrac{a^2 + b^2}{2bi} = -\dfrac{a^2 + b^2}{2b}i$, on the negative imaginary axis: $-\tfrac{\pi}{2}$. (35% correct; 29% chose D, the wrong sign.)

Q5

Maximum of $|z|$ on the circle $\left|z - 2 - \sqrt3i\right| = 1$. Answer: D - the farthest point from the origin lies on the line through the centre: $\left|2 + \sqrt3i\right| + 1 = \sqrt{4 + 3} + 1 = \sqrt7 + 1$. (32% correct; 42% chose A, $\sqrt3 + 1$.)

Q6

$z \ne 0$ and $z^2 \in R$. Possible values of $\arg(z)$? Answer: A - if $\arg(z) = \theta$, then $z^2 = |z|^2\text{cis}(2\theta)$ is real exactly when $2\theta = k\pi$, so $\theta = \tfrac{k\pi}{2}$, $k \in Z$: real and purely imaginary $z$ both work. (23% correct; 30% chose C, which leaves out the real values of $z$.)

Q7

$x = 5\cos(2t) + 1$, $y = 5\sin(2t) - 1$. Shortest distance along the graph from $A(6, -1)$ to $B(1, 4)$. Answer: E - the graph is the circle with centre $(1, -1)$ and radius 5. $A$ is due east of the centre and $B$ due north, a quarter-turn apart, so the shorter arc is $\tfrac14 \times 2\pi \times 5 = \tfrac{5\pi}{2}$. (39% correct; 23% chose D.)

Q8

Euler's method, step 0.1, for $\dfrac{dy}{dx} = y\sin(x)$ with $y = 2$ at $x = 1$. Estimate $y$ at $x = 1.2$. Answer: C - $y_1 = 2 + 0.1 \times 2\sin(1) \approx 2.1683$, then $y_2 = 2.1683 + 0.1 \times 2.1683\sin(1.1) \approx 2.362$. (72% correct.)

Q9

Which $f'(x)$ gives a graph of $f$ with no points of inflection? Answer: B - for $f'(x) = 2(x - 3)^3 + 5$, $f''(x) = 6(x - 3)^2 \ge 0$ never changes sign. In A, C and D, $f''$ is linear and changes sign at $x = 3$; in E, $f''(x) = 3(x - 3)^2 - 12$ changes sign at $x = 1$ and $x = 5$. (38% correct; 22% chose E.)

Q10

Which differential equation has the given direction field? Answer: D - $\dfrac{dy}{dx} = y - 2x$. The slopes are zero along the line $y = 2x$ (for example at $(0.5, 1)$ and $(-0.5, -1)$), negative on the positive $x$-axis and positive on the negative $x$-axis. Only B and D are zero along $y = 2x$, and B has the opposite signs. (68% correct.)

Q11

With $\mathbf i$ east and $\mathbf j$ north, hikers walk 5 km in the direction south $30^\circ$ west, then 10 km north. Their position vector? Answer: B - the first leg is $5\left(-\sin 30^\circ\,\mathbf i - \cos 30^\circ\,\mathbf j\right) = -\tfrac52\mathbf i - \tfrac{5\sqrt3}{2}\mathbf j$; adding $10\mathbf j$ gives $-\tfrac52\mathbf i + \left(10 - \tfrac{5\sqrt3}{2}\right)\mathbf j$. (69% correct.)

Q12

$\mathbf a = x\mathbf i + \mathbf j$, $\mathbf b = \mathbf i - \mathbf j$, $\mathbf c = \mathbf i + x\mathbf j$; $\theta$ is the angle between $\mathbf a$ and $\mathbf b$, and $\phi$ the angle between $\mathbf b$ and $\mathbf c$. Find $\cos(\theta)\cos(\phi)$. Answer: D - $\cos\theta = \dfrac{x - 1}{\sqrt2\sqrt{x^2 + 1}}$ and $\cos\phi = \dfrac{1 - x}{\sqrt2\sqrt{x^2 + 1}}$, so the product is $-\dfrac{(x - 1)^2}{2(1 + x^2)}$. (60% correct.)

Q13

The scalar resolute of $\mathbf a$ in the direction of $\mathbf b = -\sqrt3\,\mathbf i$ is $-4$. The vector resolute? Answer: E - multiply the scalar resolute by the unit vector $\hat{\mathbf b} = -\mathbf i$: $-4(-\mathbf i) = 4\mathbf i$. (46% correct; 22% chose A, $-4\mathbf i$.)

Q14

A 5 kg body moves in a straight line with $v = 3 + 2x$. Find the net force $F$ when $x = 2$. Answer: D - $a = v\dfrac{dv}{dx} = (3 + 2x)(2) = 14$ at $x = 2$, so $F = ma = 5 \times 14 = 70$ N. (56% correct.) Study design: partly outside. Finding $a = v\frac{dv}{dx}$ is on the course, but $F = ma$ is not listed in Units 3 and 4: 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, and the formula sheet no longer lists the Mechanics formulas (momentum and the equation of motion) that it gave in 2021.

Q15

A stationary body is acted on by 8 N to the right, $F_1$ to the left, $F_2$ at $60^\circ$ above the horizontal (on the 8 N side) and 6 N at $30^\circ$ below the horizontal (on the $F_1$ side). Find $F_1$. Answer: A - vertically, $F_2\sin 60^\circ = 6\sin 30^\circ$, so $F_2 = 2\sqrt3$. Horizontally, $F_1 = 8 + F_2\cos 60^\circ - 6\cos 30^\circ = 8 + \sqrt3 - 3\sqrt3 = 8 - 2\sqrt3$. (53% correct.) Study design: partly outside. Resolving vectors into components is on the course, but forces in equilibrium (statics) are now Specialist Mathematics Unit 2 content, which Units 3 and 4 assume but do not list as a topic.

Q16

An object slides down a smooth slope inclined at $\theta^\circ$ ($0^\circ < \theta^\circ < 45^\circ$) with acceleration $a$. Its acceleration when the angle is doubled? Answer: B - on a smooth slope $a = g\sin\theta$, so $\cos\theta = \dfrac{\sqrt{g^2 - a^2}}{g}$ and $g\sin 2\theta = 2g\sin\theta\cos\theta = \dfrac{2a}{g}\sqrt{g^2 - a^2}$. (48% correct.) Study design: partly outside. The result $a = g\sin\theta$ comes from resolving the forces on the object, and the current study design covers forces only in Specialist Mathematics Unit 2 (motion under a constant force), which Units 3 and 4 assume. The trigonometry is still needed.

Q17

Fill volumes are normal with mean 1.26 L and standard deviation 0.01 L. Probability the mean of six bottles is at least 1.25 L? Answer: E - $\bar X$ has mean 1.26 and standard deviation $\tfrac{0.01}{\sqrt6}$, so $\Pr(\bar X \ge 1.25) = \Pr\left(Z \ge -\sqrt6\right) \approx 0.9928$. (50% correct; 20% chose B, 0.8413, which is the probability for a single bottle.)

Q18

A 95% confidence interval from a sample of 100 fish is $(70.2, 75.8)$ g. The sample mean divided by the population standard deviation? Answer: C - $\bar x = 73$ and the margin is $2.8 = 1.96 \times \tfrac{\sigma}{10}$, so $\sigma \approx 14.29$ and $\tfrac{73}{14.29} \approx 5.1$. (65% correct.)

Q19

Scores with mean 25 and variance 36 are scaled by $S = mX + n$ ($m > 0$) to mean 30 and variance 49, with $S$ rounded to the nearest integer. What does 32 scale to? Answer: D - $m^2 = \tfrac{49}{36}$ gives $m = \tfrac76$, and $30 = \tfrac76 \times 25 + n$ gives $n = \tfrac56$. Then $S = \tfrac76 \times 32 + \tfrac56 \approx 38.2$, which rounds to 38. (51% correct; 24% chose C.)

Q20

Two independent machines each take a normally distributed time with mean 30 s and standard deviation 5 s to make a cup of coffee. Probability their times differ by less than 3 seconds? Answer: C - $T_1 - T_2$ has mean 0 and variance $25 + 25 = 50$, so $\Pr\left(|T_1 - T_2| < 3\right) = \Pr\left(|Z| < \tfrac{3}{\sqrt{50}}\right) \approx 0.329$. (43% correct.)

Section B: Extended response

Question 1 (10 marks)

Let $f(x) = \dfrac{(2x - 3)(x + 5)}{(x - 1)(x + 2)}$.

a

Express $f(x)$ in the form $A + \dfrac{Bx + C}{(x - 1)(x + 2)}$. (1 mark)

b

State the equations of the asymptotes of the graph of $f$. (2 marks)

c

Sketch the graph of $f$, labelling the asymptotes, the maximum turning point and the point of inflection (coordinates to two decimal places), and the axis intercepts. (3 marks)

Let $g_k(x) = \dfrac{(2x - 3)(x + 5)}{(x - k)(x + 2)}$, where $k$ is a real constant.

d. i. For what values of $k$ does the graph of $g_k$ have two asymptotes? (2 marks)

ii. Given that the graph of $g_k$ has more than two asymptotes, for what values of $k$ does it have no stationary points? (2 marks)

Question 2 (9 marks)

The polynomial $p(z) = z^3 + \alpha z^2 + \beta z + \gamma$, with $\alpha, \beta, \gamma \in R$, can also be written $p(z) = (z - z_1)(z - z_2)(z - z_3)$, where $z_1 \in R$ and $z_2, z_3 \in C$.

a. i. State the relationship between $z_2$ and $z_3$. (1 mark)

ii. Find $\alpha$, $\beta$ and $\gamma$, given that $p(2) = -13$, $|z_2 + z_3| = 0$ and $|z_2 - z_3| = 6$. (3 marks)

Let $z_4 = \sqrt3 + i$.

b. Sketch the ray $\text{Arg}(z - z_4) = \dfrac{5\pi}{6}$ on an Argand diagram. (2 marks)

The ray meets the circle $|z - 3i| = 1$, dividing it into a major and a minor segment.

c. i. Sketch the circle $|z - 3i| = 1$ on the same diagram. (1 mark)

ii. Find the area of the minor segment. (2 marks)

Question 3 (10 marks)

A thin-walled vessel is formed by rotating the graph of $y = x^3 - 8$ about the $y$-axis for $0 \le y \le H$ (lengths in centimetres).

a. i. Write a definite integral in terms of $y$ and $H$ for the volume of the vessel. (1 mark)

ii. Hence find the volume in terms of $H$. (1 mark)

Water leaks out through a crack in the base at $\dfrac{dV}{dt} = -4\sqrt h$ cm³ per minute, where $h$ cm is the depth of water and $V$ cm³ the volume of water after $t$ minutes.

b. i

Show that $\dfrac{dh}{dt} = \dfrac{-4\sqrt h}{\pi(h + 8)^{2/3}}$. (2 marks)

ii

Find the maximum rate at which the depth decreases, in cm per minute to two decimal places, and the depth at which it occurs. (2 marks)

iii

A vessel with $H = 50$ is initially full and still leaking at $4\sqrt h$ cm³ per minute. Find the maximum rate at which water can be added without the vessel overflowing. (1 mark)

c

With $H = 50$ and the vessel initially full, extra water is poured in at $40\sqrt2$ cm³ per minute once the depth has dropped to 25 cm, while the leak continues. How long does the vessel take to refill completely from a depth of 25 cm, in minutes to one decimal place? (3 marks)

Question 4 (11 marks)

A stunt car accelerates from rest at $A$ along a horizontal track, is launched from the ramp $BO$ and lands on a second section of track, inclined at $10^\circ$ to the horizontal (sloping down), at or beyond $C(16, 4)$. Air resistance is negligible, $O$ is the origin and distances are in metres. At $O$ the car has speed $u$ m s$^{-1}$ at an angle $\theta$ to the horizontal, and its rear wheels then follow $\mathbf r(t) = ut\cos(\theta)\,\mathbf i + \left(ut\sin(\theta) - \tfrac12gt^2\right)\mathbf j$.

a

Show that the path of the rear wheels is $y = x\tan(\theta) - \dfrac{4.9x^2}{u^2\cos^2(\theta)}$. (1 mark)

b

If $\theta = 30^\circ$, find the minimum speed at $O$ for the rear wheels to land at or beyond $C$, to two decimal places. (2 marks)

c

For what values of $\theta$ (to the nearest degree) and $u$ (to one decimal place) does the path join the second section of track smoothly at $C$? (3 marks)

On the horizontal section $AB$, the acceleration after the car has travelled $s$ metres from $A$ is $a = \dfrac{60}{v}$, where $v$ is its speed.

d. Show that $v = (180s)^{1/3}$. (2 marks)

e. The car reaches 20 m s$^{-1}$ at $B$. If the stunt is called off, the car brakes at 9 m s$^{-2}$ and must stop at or before $B$; $W$ is the furthest point along $AB$ at which the stunt can be called off. How far is $W$ from $B$, to one decimal place? (3 marks)

Question 5 (10 marks)

A mass of $m_1$ kg on a plane inclined at $30^\circ$ is connected by a light inextensible string, over a frictionless pulley at the top of the incline, to a mass of $m_2$ kg hanging vertically.

a. If the plane is smooth, find the relationship between $m_1$ and $m_2$ for $m_1$ to move down the plane at constant speed. (2 marks)

The plane is now rough. With $N$ the magnitude of the normal force on $m_1$, a resistance force of magnitude $\lambda N$ opposes the motion of $m_1$.

b

The mass $m_1$ moves up the plane.

i

Mark and label all the forces acting on $m_1$. (1 mark)

ii

Taking up the plane as positive, find the acceleration of $m_1$ in terms of $m_1$, $m_2$ and $\lambda$. (2 marks)

Later, $m_2$ hits the ground at 4.5 m s$^{-1}$ and the string becomes slack, with $m_1$ at the point $P$ on the plane, 2 m from the pulley. Take $\lambda = 0.1$.

c. How far from $P$ does $m_1$ travel before it starts to slide back down the plane, to two decimal places? (2 marks)

d. Find the time from when the string becomes slack until $m_1$ returns to $P$, to the nearest tenth of a second. (3 marks)

Study design: partly outside. This question is about forces: resolving the weight on an inclined plane, the tension in a string over a pulley, a resistance force $\lambda N$ and Newton's second law. 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; pulleys, connected masses and resistance forces are not named in it, and the formula sheet no longer lists the Mechanics formulas (momentum and the equation of motion) that it gave in 2021. Parts c and d finish with constant-acceleration kinematics, which is still on the course.

Question 6 (10 marks)

A lift has a maximum load of 1000 kg. The masses of employees are normally distributed with mean 75 kg and standard deviation 8 kg, and $n$ employees are about to use the lift.

a. What is the maximum value of $n$ for there to be less than a 1% chance of the lift exceeding the maximum load? (2 marks)

The times to dispense hot drinks are independent and normally distributed with mean 2 minutes and standard deviation 0.5 minutes.

b. At 8.52 am Clare is fourth in the queue, and her meeting starts at 9.00 am. The waiting time between drinks is negligible and she needs 0.5 minutes to reach the meeting room. Find the probability, to four decimal places, that she is on time. (2 marks)

Daily sales of chocolate bars are normally distributed with mean 60 000 and standard deviation 5000. After an advertising campaign, the mean daily sales over 14 randomly selected days is 63 500. Clare performs a one-sided test at the 1% level of significance.

c. i

Write down suitable null and alternative hypotheses. (1 mark)

ii

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

iii

Giving a reason, state whether there is any evidence that the campaign succeeded. (1 mark)

d

Find the range of values of the mean daily sales over another 14 days that would lead to the null hypothesis being rejected at the 1% level, to the nearest integer. (1 mark)

e

The mean daily sales is now in fact 63 000. For a test at the 5% level, find the probability that the null hypothesis would be incorrectly accepted, based on another 14 days and a standard deviation of 5000, to three decimal places. (2 marks)