§-Past paper

NSWMaths Standard 22021

HSC Maths Standard 2 2021

Walkthrough of the 2021 HSC Mathematics Standard 2 exam: what it assessed, how to pace it, original exam-style worked questions on tax, energy costs, bearings, critical paths, the normal distribution and regression, and common errors from NESA's 2021 marking feedback.

Marks: 100
Time: 150 min
Authority: NESA
Updated: 2026-09-28

Open the official PDF Official marking notes

What this paper assessed

The 2021 HSC Mathematics Standard 2 paper sampled every strand of the course, with a strong emphasis on applying skills in everyday contexts rather than abstract manipulation. Reading the paper and NESA's mapping grid, the main areas were:

Financial mathematics: declining-balance and straight-line depreciation, compound interest with monthly compounding compared against a simple interest rate, income tax and the Medicare levy from a tax table, dividend yield, and annuities using both present value and future value interest factor tables (including a two-stage problem where an annuity is left to grow at compound interest).
Measurement: perimeter and volume (including a hemispherical tank), scale drawings converted to real distances and speeds, fuel consumption and electricity costs as rates, time zones from longitude, and the trapezoidal rule for an irregular area.
Non-right-angled trigonometry: right-angled ratios inside a semicircle, the sine rule including the ambiguous (obtuse) case, the area formula $A = \tfrac{1}{2}ab\sin C$ rearranged to find an angle, the cosine rule in a compass radial survey, and true bearings.
Networks: degree sums, minimum spanning trees, shortest paths, and critical path analysis with earliest and latest starting times and float.
Statistics: stem-and-leaf plots and cumulative frequency, outliers from a five-number summary, least-squares regression lines read from a graph, interpreting gradient and correlation, probability of dependent events (selection without replacement) on a tree diagram, and the normal distribution using z-scores, a probability table and the empirical rule.
Algebra: forming and solving linear equations (including one with a fraction), inverse variation, exponential models, simultaneous linear equations solved graphically, and a quadratic revenue model with an axis of symmetry.

Structure and timing

The paper cover gives 10 minutes reading time and 2 hours and 30 minutes working time for 100 marks. A NESA reference sheet is provided and approved calculators may be used.

Section	Questions	Marks	Suggested time on the paper
Section I (multiple choice)	1 to 15	15	about 25 minutes
Section II (written, two booklets)	16 to 41	85	about 2 hours 5 minutes

Section II is split across two answer booklets: Booklet 1 holds Questions 16 to 33 (55 marks) and Booklet 2 holds Questions 34 to 41 (30 marks).

Pacing arithmetic: $150 \div 100 = 1.5$ minutes per mark overall. Section II works out at $125 \div 85 \approx 1.47$ minutes per mark, so a 4-mark question deserves about 6 minutes and a 5-mark question about 7 to 8 minutes. A practical split is:

Section I: 20 to 25 minutes. Most items are one-step, so bank the time.
Booklet 1 (55 marks): about 75 minutes.
Booklet 2 (30 marks): about 45 minutes. The later normal distribution and annuity questions are multi-step, so do not arrive with less than 40 minutes left.
Check: use any spare minutes to confirm units, rounding instructions and that each answer is realistic.

Worked practice questions (exam-style)

These are original questions written in the style of the 2021 paper and covering the same syllabus areas. They are not NESA questions.

Worked example

Question 1 (4 marks): Priya wants to know her total tax payable for the year and whether she will get a refund. Use the details and the tax table below (the 2024-25 Australian resident rates), with a Medicare levy of 2% of taxable income.

Gross salary: $67 500
Bank interest earned: $1200
Deduction for work travel: $1350
Deduction for professional memberships: $650
PAYG tax withheld by her employer: $12 500

Taxable income ($)	Tax on this income
0 to 18 200	Nil
18 201 to 45 000	16 cents for each dollar over 18 200
45 001 to 135 000	4288 plus 30 cents for each dollar over 45 000
135 001 to 190 000	31 288 plus 37 cents for each dollar over 135 000
190 001 and over	51 638 plus 45 cents for each dollar over 190 000

Step 1: Taxable income

All income counts, then deductions come off:

\text{Taxable income} = 67\,500 + 1200 - 1350 - 650 = 66\,700

Step 2: Income tax from the correct bracket

$66\,700$ lies in the third bracket:

\text{Tax} = 4288 + 0.30 \times (66\,700 - 45\,000) = 4288 + 6510 = 10\,798

Step 3: Medicare levy on taxable income

\text{Levy} = 0.02 \times 66\,700 = 1334

Step 4: Total and comparison with PAYG

\text{Total tax payable} = 10\,798 + 1334 = 12\,132

\text{Refund} = 12\,500 - 12\,132 = 368

Reasonableness check: the total tax is roughly 18% of her taxable income, which is sensible for this bracket.

Final answer: total tax payable is 12 132 dollars, so Priya receives a refund of 368 dollars.

Worked example

Question 2 (4 marks): A 2.4 kW heater runs for 5 hours a day for a 90-day winter. Electricity costs 32 cents per kWh. (a) Find the cost of running the heater for one winter. (b) A 1.8 kW model gives similar warmth but costs $180 more to buy. How many full winters will it take for the cheaper-to-run model to have paid off its extra purchase price?

Step 1: Energy used by the 2.4 kW heater

Energy is power multiplied by time, with power already in kilowatts:

E = 2.4 \times 5 \times 90 = 1080 \text{ kWh}

Step 2: Cost for one winter

\text{Cost} = 1080 \times 0.32 = 345.60

Step 3: Saving per winter from the 1.8 kW model

The difference in power is $2.4 - 1.8 = 0.6$ kW, so

\text{Saving} = 0.6 \times 5 \times 90 \times 0.32 = 86.40 \text{ per winter}

Step 4: Break-even

\frac{180}{86.40} \approx 2.08 \text{ winters}

After 2 winters the saving is only 172.80 dollars, which is still short of the 180-dollar price difference, so a third winter is needed.

Final answer: (a) $345.60 per winter; (b) the extra cost is recovered during the 3rd winter, so 3 full winters.

Worked example

Question 3 (4 marks): A kayaker paddles 12 km from A to B on a true bearing of 040°, then 8 km from B to C on a true bearing of 150°. Find the distance AC to one decimal place and the true bearing of C from A to the nearest degree.

Step 1: Angle at B

At B, the back-bearing to A is $040^\circ + 180^\circ = 220^\circ$ . The angle between the direction to A ( $220^\circ$ ) and the direction to C ( $150^\circ$ ) is

\angle ABC = 220^\circ - 150^\circ = 70^\circ

Step 2: Cosine rule for AC

AC^2 = 12^2 + 8^2 - 2 \times 12 \times 8 \times \cos 70^\circ \approx 142.33

AC \approx 11.93 \text{ km}

Step 3: Angle at A by the sine rule

\frac{\sin \angle BAC}{8} = \frac{\sin 70^\circ}{11.93} \quad\Rightarrow\quad \angle BAC \approx 39.06^\circ

This is acute, as it must be because it faces the shortest side.

Step 4: Bearing of C from A

C lies clockwise from the line AB, so add the angle to the bearing of B:

040^\circ + 39.06^\circ \approx 079^\circ

Final answer: AC is about 11.9 km and C is on a true bearing of 079° from A.

Worked example

Question 4 (4 marks): A small renovation has seven tasks. A (4 days) and B (6 days) can start immediately. C (3 days) and D (5 days) follow A. E (7 days) follows both B and C. F (2 days) follows D. G (4 days) follows both E and F. (a) Find the minimum completion time and the critical path. (b) Find the float of tasks B and D.

Step 1: Earliest starting times (forward scan)

A and B: EST $= 0$ .
C and D: EST $= 4$ .
E: EST $= \max(0 + 6,\ 4 + 3) = 7$ .
F: EST $= 4 + 5 = 9$ .
G: EST $= \max(7 + 7,\ 9 + 2) = 14$ .

Finish $= 14 + 4 = 18$ days.

Step 2: Latest starting times (backward scan)

G: LST $= 18 - 4 = 14$ .
E: LST $= 14 - 7 = 7$ ; F: LST $= 14 - 2 = 12$ .
D: LST $= 12 - 5 = 7$ ; C: LST $= 7 - 3 = 4$ ; B: LST $= 7 - 6 = 1$ .
A: LST $= \min(4, 7) - 4 = 0$ .

Step 3: Critical path and floats

Tasks with $\text{LST} - \text{EST} = 0$ are A, C, E and G. Check: $4 + 3 + 7 + 4 = 18$ .

\text{Float of B} = 1 - 0 = 1 \text{ day}, \qquad \text{Float of D} = 7 - 4 = 3 \text{ days}

Final answer: (a) 18 days along A, C, E, G; (b) B has 1 day of float and D has 3 days of float.

Worked example

Question 5 (4 marks): The heights of a batch of 400 tomato seedlings are normally distributed with a mean of 42 mm and a standard deviation of 6 mm. (a) Find the z-score of a seedling 51 mm tall. (b) What percentage of seedlings are between 30 mm and 48 mm tall? (c) How many seedlings are expected to be taller than 54 mm?

Step 1: z-score

z = \frac{x - \mu}{\sigma} = \frac{51 - 42}{6} = 1.5

Step 2: Convert the interval to z-scores

z_{30} = \frac{30 - 42}{6} = -2, \qquad z_{48} = \frac{48 - 42}{6} = 1

By the empirical rule, 95% lies within 2 standard deviations, so 47.5% is between $z = -2$ and $0$ . Also 68% lies within 1 standard deviation, so 34% is between $0$ and $z = 1$ .

47.5\% + 34\% = 81.5\%

Step 3: Tail above 54 mm

$z_{54} = \dfrac{54 - 42}{6} = 2$ . The area above $z = 2$ is $\dfrac{100\% - 95\%}{2} = 2.5\%$ .

0.025 \times 400 = 10

Final answer: (a) $z = 1.5$ ; (b) 81.5%; (c) about 10 seedlings.

Worked example

Question 6 (4 marks): A tutor records the weekly hours of practice, $x$ , and the test score out of 100, $y$ , for a class. The least-squares regression line passes through $(0, 38)$ and $(10, 83)$ , and the correlation coefficient is $r = 0.82$ . Practice hours in the data range from 1 to 12. (a) Find the equation of the line. (b) Interpret the gradient. (c) Predict the score for 6 hours of practice. (d) Explain why using the line for 25 hours is not reliable.

Step 1: Gradient and intercept

m = \frac{83 - 38}{10 - 0} = 4.5, \qquad \text{vertical intercept} = 38

y = 38 + 4.5x

Step 2: Interpret the gradient in context

For each extra hour of weekly practice, the test score is predicted to rise by about 4.5 marks.

Step 3: Interpolation

y = 38 + 4.5 \times 6 = 65

Six hours lies inside the data range of 1 to 12 hours, and $r = 0.82$ indicates a strong positive linear relationship, so this estimate is reasonable.

Step 4: Extrapolation

y = 38 + 4.5 \times 25 = 150.5

This is impossible for a test out of 100, and 25 hours is far outside the data range, so the linear pattern cannot be assumed to continue.

Final answer: (a) $y = 38 + 4.5x$ ; (b) about 4.5 more marks per extra hour; (c) 65 marks; (d) 25 hours is extrapolation well beyond the data and gives an impossible score of 150.5.

Common errors students made

NESA's 2021 marking feedback for Mathematics Standard 2 noted these recurring problems (paraphrased):

Formula adaptation and setting out. In the volume question, some students did not adjust the given sphere formula to the shape shown, and working was often written across the page without operation signs. The feedback asked for the formula, the substitution and the result to be set out down the page.
Outliers. Some students drew a box plot they did not need, confused range with interquartile range, or did not use the course definition of an outlier (more than 1.5 IQR beyond a quartile).
Rates and early rounding. In the fuel cost question some divided by the price per litre instead of multiplying, and rounding part-way through cost accuracy. Energy questions exposed weak conversions between watts and kilowatts and between units of time.
Time zones. Students were reminded that only the longitude difference matters, that dividing by 15 converts degrees to hours, and that when a question asks for both the time and the day, both must be stated.
Tax. Common slips were not subtracting deductions to get taxable income, not using taxable income for both the tax and the Medicare levy, mixing up cents and dollars, and not checking that the answer was realistic.
Networks. Students confused a spanning tree with the minimum spanning tree, assumed the shortest path must lie on the minimum spanning tree, and confused the minimum time to complete a project (the longest path) with the shortest path through the network.
Interest. Errors included writing 6% incorrectly as a decimal, not matching the rate and number of periods to monthly compounding, using the future value instead of just the interest in the simple interest formula, and guessing and checking instead of rearranging.
Regression and correlation. Some wrote a regression "equation" without an equals sign, mixed up the gradient and the constant, or gave geographical explanations rather than mathematical ones. The feedback stressed that of two negative correlation coefficients, the one closer to $-1$ is stronger.
Trigonometry. Students used right-angled methods or Pythagoras in triangles that only looked right-angled, did not go on to the obtuse angle in the ambiguous case of the sine rule, and copied the area formula with cosine instead of sine.
Normal distribution. Students struggled to turn a z-score into a probability using a table, to handle "greater than" regions, and to use percentages such as 16%, 84% and 97.5% that come from combining the 68/95/99.7 rule. Some read $1.05\mu$ as $1.05 = \mu$ instead of $1.05 \times \mu$ .
Annuity tables. Some students recalculated compound interest when a present value factor table was provided, did not see that a two-stage savings problem needs a table value then a compound interest step, and did not check whether their answer was reasonable.

How to use this paper

Download the official 2021 paper from the link above and sit it in exam conditions: 10 minutes reading, then 2 hours 30 minutes working, aiming for about 25 minutes on Section I.
Mark your attempt with the official marking guidelines at https://www.nsw.gov.au/sites/default/files/noindex/2025-05/2021-hsc-mathematics-standard-2-mg.pdf, noting where you lost the second or third mark of a multi-mark question rather than just the final answer.
Read NESA's 2021 marking feedback (linked above) question by question and match each of your errors to one of the patterns listed on this page.
Rework the six practice questions here without looking at the solutions, then redo any official question you dropped marks on a few days later.
Before your exam, make a one-page checklist from your own errors (units, rounding, taxable income, ambiguous case, "time and day") and read it during reading time.

Use this paper well

Sit the paper under exam conditions (150 minutes, 100 marks).
Mark yourself against the official NESA marking notes.
Compare against the Maths Standard 2 hub to find the syllabus dot points this paper tested.

What this paper assessed

Structure and timing

Worked practice questions (exam-style)

Question 1 (4 marks): Priya wants to know her total tax payable for the year and whether she will get a refund. Use the details and the tax table below (the 2024-25 Australian resident rates), with a Medicare levy of 2% of taxable income.

Step 1: Taxable income

Step 2: Income tax from the correct bracket

Step 3: Medicare levy on taxable income

Step 4: Total and comparison with PAYG

Step 1: Energy used by the 2.4 kW heater

Step 2: Cost for one winter

Step 3: Saving per winter from the 1.8 kW model

Step 4: Break-even

Question 3 (4 marks): A kayaker paddles 12 km from A to B on a true bearing of 040°, then 8 km from B to C on a true bearing of 150°. Find the distance AC to one decimal place and the true bearing of C from A to the nearest degree.

Step 1: Angle at B

Step 2: Cosine rule for AC

Step 3: Angle at A by the sine rule

Step 4: Bearing of C from A

Step 1: Earliest starting times (forward scan)

Step 2: Latest starting times (backward scan)

Step 3: Critical path and floats

Step 1: z-score

Step 2: Convert the interval to z-scores

Step 3: Tail above 54 mm

Step 1: Gradient and intercept

Step 2: Interpret the gradient in context

Step 3: Interpolation

Step 4: Extrapolation

Common errors students made

How to use this paper

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