VCE Specialist Mathematics 2023 Exam 2

Every question from the 2023 VCE Specialist Mathematics Examination 2, the technology-active (CAS) paper and the first under the current study design. 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 2023 Examination 1 walkthrough.

How to use this page

• Questions are from the 2023 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 2023 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. In 2023 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.
• Section B (60 marks): 6 extended-response questions (10, 10, 10, 10, 11 and 9 marks).

Section A: Multiple choice

Q1

Contrapositive of "If my football team plays badly, then they are not training enough". Answer: C - "If they are training enough, then my football team does not play badly."

Q2

$y = \dfrac{x^3}{ax^2 + bx + c}$ has asymptotes $y = 2x + 1$ and $x = 1$. Find $a, b, c$. Answer: B - division gives $y = \tfrac{x}{a} - \tfrac{b}{a^2} + \ldots$, so $a = \tfrac12$ and $-4b = 1$ ($b = -\tfrac14$); the denominator must vanish at $x = 1$, so $c = -\tfrac14$.

Q3

When does $y = a + \sec(x)$ have two $x$-intercepts on $[-\pi, \pi]$? Answer: E - $\sec x = -a$ has two solutions when $-a > 1$ (two values near $\pm\tfrac{\pi}{2}$ inside) or when $-a \le -1$ (including $-a = -1$, giving $x = \pm\pi$), but only one ($x = 0$) when $-a = 1$. So $a < -1$ or $a \ge 1$.

Q4

$z = -(2a + 1) + 2ai$, $a \ne 0$. Find $\dfrac{4a}{1 + \bar z}$. Answer: B - $1 + \bar z = -2a - 2ai = -2a(1 + i)$, so the quotient is $\dfrac{-2}{1 + i} = -1 + i = \sqrt2\,\text{cis}\left(\tfrac{3\pi}{4}\right)$.

Q5

$\text{Re}(z), \text{Im}(z) > 0$, $|\bar z| = 4$ and $z^3$ is a negative real number. $z^2$ equals? Answer: E - $\arg z = \tfrac{\pi}{3}$, so $z^2 = 16\,\text{cis}\left(\tfrac{2\pi}{3}\right)$, and $-4\bar z = -16\,\text{cis}\left(-\tfrac{\pi}{3}\right) = 16\,\text{cis}\left(\tfrac{2\pi}{3}\right)$. (34% correct.)

Q6

Euler steps $y \leftarrow y + 0.5e^{xy}$, $x \leftarrow x + 0.5$ from $(0, 0)$. After how many iterations is 2.709 reached? Answer: C - $y = 0.5$, then $0.5 + 0.5e^{0.25} \approx 1.142$, then $1.142 + 0.5e^{1.142} \approx 2.709$: three iterations.

Q7

Following the direction field from $y = 2$ at $x = -1$, estimate $y$ at $x = 1.5$. Answer: D - the solution curve falls steeply while $x < 0$, flattens, and is close to $y = 1.0$ at $x = 1.5$.

Q8

A spa holds 8000 L; 20 L/min of mixed water is pumped out and 15 L/min of fresh water pumped in. The differential equation for $Q$? Answer: A - the volume is $8000 - 5t$, so $\dfrac{dQ}{dt} = -\dfrac{20Q}{8000 - 5t} = \dfrac{4Q}{t - 1600}$. (37% correct; 29% chose E, which has the wrong sign.)

Q9

Slope of the tangent to $x = \tfrac{6t}{t + 1}$, $y = \tfrac{-8}{t^2 + 4}$ at $t = 2$. Answer: D - $\dot x = \tfrac{6}{(t + 1)^2} = \tfrac23$ and $\dot y = \tfrac{16t}{(t^2 + 4)^2} = \tfrac12$, so $\tfrac{dy}{dx} = \tfrac34$.

Q10

$I_n = \int_0^1(1 - x)^ne^x\,dx$. Express $I_n$ in terms of $I_{n-1}$. Answer: A - by parts, $I_n = \left[(1 - x)^ne^x\right]_0^1 + n\int_0^1(1 - x)^{n-1}e^x\,dx = -1 + nI_{n-1}$. (33% correct.)

Q11

Surface area when $y = \cos^{-1}(x)$ from $\left(0, \tfrac{\pi}{2}\right)$ to $(1, 0)$ is rotated about the $y$-axis. Answer: E - with $x = \cos y$, $S = 2\pi\int_0^{\pi/2}\cos y\sqrt{1 + \sin^2y}\,dy$, and $u = \sin y$ gives $2\pi\int_0^1\sqrt{1 + u^2}\,du$.

Q12

A particle from rest has $a = 1 + v$. Velocity after $\log_e(e + 1)$ seconds? Answer: A - $\tfrac{dv}{dt} = 1 + v$ with $v(0) = 0$ gives $v = e^t - 1$, which is $e$ at $t = \log_e(e + 1)$.

Q13

A phone is dropped from a balloon rising at 2.5 m/s, 80 m up. Time to hit the ground? Answer: E - the phone starts with velocity $+2.5$: $-80 = 2.5t - 4.9t^2$ gives $t \approx 4.30$ s.

Q14

$\mathbf a = \mathbf i + \mathbf j$, $\mathbf b = \mathbf i - \mathbf j$, $\mathbf c = \mathbf i + 2\mathbf j + 3\mathbf k$ and unit $\mathbf n$ is perpendicular to $\mathbf a$ and $\mathbf b$. Find $|\mathbf c\cdot\mathbf n|$. Answer: B - $\mathbf n = \pm\mathbf k$, so $|\mathbf c\cdot\mathbf n| = 3$.

Q15

The sum of two unit vectors is a unit vector. Magnitude of their difference? Answer: D - $|\mathbf a + \mathbf b|^2 = 2 + 2\mathbf a\cdot\mathbf b = 1$ gives $\mathbf a\cdot\mathbf b = -\tfrac12$, so $|\mathbf a - \mathbf b|^2 = 2 + 1 = 3$ and the magnitude is $\sqrt3$. (18% correct.)

Q16

A ball has $\mathbf r_B(t) = 5t\mathbf i + 7t\mathbf j + (15t - 4.9t^2 + 1.5)\mathbf k$. Total vertical distance travelled before it lands? Answer: D - it rises from 1.5 m to about 13.0 m (11.5 m up) and falls 13.0 m to the ground: about $24.5$ m.

Q17

$\mathbf a = \alpha\mathbf i + \mathbf j - \mathbf k$, $\mathbf b = 3\mathbf i + \beta\mathbf j + 4\mathbf k$ and $\mathbf a\times\mathbf b = 2\mathbf i - 7\mathbf j + \gamma\mathbf k$. Answer: C - $\mathbf a\times\mathbf b = (4 + \beta)\mathbf i - (4\alpha + 3)\mathbf j + (\alpha\beta - 3)\mathbf k$, so $\beta = -2$, $\alpha = 1$, $\gamma = -5$.

Q18

For which $k$ are $2x - ky + 3z = 1$ and $2kx + 3y - 2z = 4$ perpendicular? Answer: C - the normals' dot product is $4k - 3k - 6 = k - 6 = 0$, so $k = 6$.

Q19

Invoices are $N(800, 200^2)$ dollars. Probability 16 invoices total more than $13 500? Answer: B - the total has mean 12 800 and standard deviation $200\sqrt{16} = 800$, so $\Pr(Z > 0.875) \approx 0.191$.

Q20

A 99% interval from $n = 100$ is $(10\,500, 15\,500)$. Find $\sigma$. Answer: A - the margin is 2500 $= 2.5758 \times \tfrac{\sigma}{10}$, so $\sigma \approx 9710$.

Section B: Extended response

Question 1 (10 marks)

A track from $O$ to $D$ follows $f(x) = -x(x + a)^2$ on $[0, 1]$ and $f(x) = e^{x - 1} - x + b$ on $(1, 2]$, meeting at $C(1, 0)$; $A$ is the minimum and $B$ a point of inflection on $OABC$.

a

Show that $a = -1$ and $b = 0$. (1 mark)

b

Verify that the two curves meet smoothly at $C$. (2 marks)

c. i

Find the coordinates of $A$. (1 mark)

ii

Find the coordinates of $B$. (1 mark)

The return path is $x = 2\cos t + 2$, $y = (e - 2)\sin t$, $t \in \left[\tfrac{\pi}{2}, \pi\right]$.

d

Find the Cartesian equation of this elliptical path. (2 marks)

e

Sketch the elliptical path from $D$ to $O$. (1 mark)

f. i

Write a definite integral in $t$ for the length of the elliptical path. (1 mark)

ii

Find this length to three decimal places. (1 mark)

Question 2 (10 marks)

Let $w = \text{cis}\left(\tfrac{2\pi}{7}\right)$.

a

Verify that $w$ is a root of $z^7 - 1 = 0$. (1 mark)

b

List the other roots in polar form. (1 mark)

c

Plot and label all the roots on an Argand diagram. (2 marks)

d. i

Sketch the ray from the real root through $\text{cis}\left(\tfrac{2\pi}{7}\right)$. (1 mark)

ii

Write its equation as $\text{Arg}(z - z_0) = \theta$. (1 mark)

e

Verify that $z^7 - 1 = (z - 1)(z^6 + z^5 + z^4 + z^3 + z^2 + z + 1)$. (1 mark)

f. i

Express $\text{cis}\left(\tfrac{2\pi}{7}\right) + \text{cis}\left(\tfrac{12\pi}{7}\right)$ as $A\cos(B\pi)$. (1 mark)

ii

Use De Moivre's theorem to show that $\cos\tfrac{2\pi}{7} + \cos\tfrac{4\pi}{7} + \cos\tfrac{6\pi}{7} = -\tfrac12$. (2 marks)

Question 3 (10 marks)

The curve $y^2 = x - 1$, $2 \le x \le 5$, is rotated about the $x$-axis.

a. i

Write a definite integral for the volume. (1 mark)

ii

Find the volume. (1 mark)

b. i

Express the curved surface area as $\pi\int_a^b\sqrt{Ax - B}\,dx$. (2 marks)

ii

Find the curved surface area to three decimal places. (1 mark)

The total surface area includes the two end discs; the "efficiency ratio" is total surface area divided by volume.

c. Find the efficiency ratio to two decimal places. (2 marks)

d. Another solid from $y^2 = x - 1$ on $2 \le x \le k$ has volume $24\pi$. Find its efficiency ratio to two decimal places. (3 marks)

Question 4 (10 marks)

200 fish are released into a pond; $\dfrac{dP}{dt} = P\left(1 - \dfrac{P}{1000}\right)$.

a. The equation can be written $\int\left(\dfrac{A}{P} + \dfrac{B}{1 - \frac{P}{1000}}\right)dP = \int dt$. Find $A$ and $B$. (1 mark)

b. With $P = \dfrac{1000}{1 + De^{-t}}$, find $D$. (1 mark)

In pond 2, $Q = \dfrac{1000}{1 + 9e^{-1.1t}}$ after $n$ fish are released.

c

Find $n$. (1 mark)

d

Find $Q$ when $t = 6$, to the nearest integer. (1 mark)

e. i

Given $\dfrac{dQ}{dt} = \dfrac{11}{10}Q\left(1 - \dfrac{Q}{1000}\right)$, express $\dfrac{d^2Q}{dt^2}$ in terms of $Q$. (1 mark)

ii

Find the population and the time (nearest year) when the growth rate is greatest. (2 marks)

f

Sketch $Q$ against $t$, labelling intercepts and asymptotes. (2 marks)

g

With 5.5% harvested each year, $\dfrac{dQ}{dt} = \dfrac{11}{10}Q\left(1 - \dfrac{Q}{1000}\right) - 0.055Q$. Find the maximum population the pond could support. (1 mark)

Question 5 (11 marks)

$A(1, 1, 2)$, $B(1, 2, 3)$ and $C(3, 2, 4)$ lie in a plane $\Pi$.

a. Find $\overrightarrow{AB}$ and $\overrightarrow{AC}$ and show that triangle $ABC$ has area 1.5. (2 marks)

b. Find the shortest distance from $B$ to the segment $AC$. (2 marks)

A plane $\psi$ has equation $2x - 2y - z = -18$.

c. Find the acute angle at which $\mathbf r(t) = 3\mathbf i + 2\mathbf j + 4\mathbf k + t(\mathbf i - 2\mathbf j + 2\mathbf k)$ meets $\psi$, to the nearest degree. (2 marks)

A line $L$ through the origin is normal to $\psi$ and meets it at $D$.

d

Write $L$ in parametric form. (1 mark)

e

Find the shortest distance from the origin to $\psi$. (2 marks)

f

Find $D$. (2 marks)

Question 6 (9 marks)

Adult male koala mass is normal with $\sigma = 1$ kg; a sample of 20 has mean 11.39 kg.

a

Find a 95% confidence interval for the population mean, to two decimal places. (1 mark)

b

Of 60 such intervals, how many would be expected to contain the true mean? (1 mark)

c

How many koalas should be sampled to reduce the width of the 95% interval by 60%? (1 mark)

The mean is thought to be 12 kg; a sample of 40 has mean 11.6 kg and a one-tailed test is proposed.

d

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

e. i

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

ii

Draw a conclusion at the 1% level, with a reason. (1 mark)

f

Find the critical sample mean to three decimal places. (1 mark)

g

If the true mean is 11.4 kg, find the probability of a type II error, to three decimal places. (1 mark)

h

Label the critical mean on the given sampling distributions and shade the type II error region. (1 mark)