Total marks: 100 · Duration: 2 hours 40 minutes
Instructions to candidates
- This paper consists of TWO sections: Section I and Section II.
- Section I has SEVEN questions worth a total of 66 marks. Answer ALL questions in Section I.
- Section II has THREE questions worth a total of 34 marks. Answer ALL questions in Section II.
- In the January 2027 sitting this paper is "hybrid": you read the questions on screen and write your answers in the answer booklet. Show all working clearly in the booklet.
- A silent, non-programmable electronic calculator, mathematical instruments and graph paper may be used.
- Diagrams are NOT drawn to scale. Where a question describes a shape in words, sketch it first.
- Take π to be 22/7 unless the question says otherwise. Give non-exact answers correct to 3 significant figures unless told otherwise.
- In this paper a 2 × 2 matrix is written row by row, with a semicolon between the rows. For example, (3 2 ; 1 4) has first row 3, 2 and second row 1, 4. A column vector is written (x, y).
- The marks follow the syllabus allocation for Paper 02: Number and Consumer Arithmetic 9, Algebra 10, Measurement 9, Statistics 9, Relations, Functions and Graphs 20, Geometry and Trigonometry 21, Vectors and Matrices 12, Investigation 10.
Paper
Section I (66 marks)
Question 1
(a) Without using a calculator, and showing all your working,
(i) evaluate (2.7 × 0.45) ÷ 0.15 (2 marks)
(ii) write 0.15 as a fraction in its lowest terms. (1 mark)
(b) Mrs Joseph's electricity meter read 23 415 units last month and 23 795 units this month. The company charges
- a fixed charge of $18.00
- $0.42 per unit for the first 100 units
- $0.58 per unit for every unit after the first 100
VAT of 12.5% is then added to the whole bill.
(i) How many units did Mrs Joseph use? (1 mark)
(ii) Calculate her bill before VAT. (2 marks)
(iii) Calculate the total amount she pays, including VAT. (1 mark)
(c) A sum of $2500 earns $375 simple interest in 2½ years. Calculate the rate of interest per annum. (2 marks)
Total: 9 marks
Question 2
(a) Write as a single fraction in its simplest form
3x/4 − (x − 2)/3 (2 marks)
(b) Factorise completely
(i) 6p² − 9pq (1 mark)
(ii) x² − 5x − 14 (2 marks)
(c) Solve the equation 2x² + 3x − 9 = 0. (3 marks)
(d) y varies inversely as x, and y = 6 when x = 4. Find the value of y when x = 8. (2 marks)
Total: 10 marks
Question 3
A water trough on a farm is a prism. Its cross-section is a trapezium with parallel sides of 60 cm (across the top) and 40 cm (across the bottom), and the perpendicular distance between them (the depth) is 30 cm. The trough is 2 m long.
(a) Calculate the area of the cross-section, in cm². (2 marks)
(b) Calculate the volume of the trough
(i) in cm³ (1 mark)
(ii) in litres. (1 mark)
(c) The empty trough is filled from a hose at 12 litres per minute. How long does it take to fill? (1 mark)
(d) A circular flower bed has radius 3.5 m. A sector of the bed with an angle of 72° at the centre is planted with roses. Using π = 22/7, calculate
(i) the area of the sector (2 marks)
(ii) the length of the curved edge of the sector. (2 marks)
Total: 9 marks
Question 4
The masses, in kg, of 40 parcels at a post office are shown in the table.
| Mass (kg) | 1 – 5 | 6 – 10 | 11 – 15 | 16 – 20 | 21 – 25 |
|---|---|---|---|---|---|
| Number of parcels | 6 | 11 | 13 | 7 | 3 |
(a) State the modal class. (1 mark)
(b) Using the midpoint of each class, calculate an estimate of the mean mass of a parcel. (3 marks)
(c) State the upper class boundary of the class 6 – 10. (1 mark)
(d) A parcel is chosen at random. Find the probability that its mass is more than 15 kg. (2 marks)
(e) Explain why the value you found in (b) is only an estimate of the mean. (2 marks)
Total: 9 marks
Question 5
The functions f and g are defined by f(x) = 2x − 5 and g(x) = x² + 1.
(a) Evaluate f(4). (1 mark)
(b) Evaluate gf(4). (2 marks)
(c) Find f⁻¹(x). (2 marks)
(d) The straight line L passes through the points (−1, 4) and (3, −4).
(i) Calculate the gradient of L. (1 mark)
(ii) Find the equation of L in the form y = mx + c. (2 marks)
(e) Determine whether L is perpendicular to the line y = ½x + 7. Give a reason for your answer. (2 marks)
Total: 10 marks
Question 6
(a) In triangle PQR, PQ = 8 cm, QR = 11 cm and angle PQR = 50°.
(i) Calculate the area of triangle PQR. (2 marks)
(ii) Calculate the length of PR. (3 marks)
(b) Each interior angle of a regular polygon is 156°. Calculate the number of sides of the polygon. (2 marks)
(c) From a point on level ground 40 m from the foot of a vertical tower, the angle of elevation of the top of the tower is 32°. Calculate the height of the tower. (2 marks)
Total: 9 marks
Question 7 — Investigation
A patio is made from square tiles. Design n has a square of n by n white tiles in the middle, completely surrounded by a single border of grey tiles.
- Design 1 has 1 white tile and 8 grey tiles.
- Design 2 has 4 white tiles and 12 grey tiles.
- Design 3 has 9 white tiles and 16 grey tiles.
(a) Copy and complete the table.
| Design (n) | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| White tiles | 1 | 4 | 9 | ||
| Grey tiles | 8 | 12 | 16 |
(2 marks)
(b) Write an expression, in terms of n, for the number of grey tiles in Design n. (2 marks)
(c) Write an expression, in terms of n, for the TOTAL number of tiles in Design n. (1 mark)
(d) Which design uses exactly 52 grey tiles? (2 marks)
(e) Explain why no design uses exactly 50 grey tiles. (1 mark)
(f) Find the smallest design that has more white tiles than grey tiles. Show your reasoning. (2 marks)
Total: 10 marks
Section II (34 marks)
Question 8 — Relations, Functions and Graphs
The table shows some values of y = x² − 2x − 3.
| x | −2 | −1 | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|---|---|
| y | 5 | −3 | 0 |
(a) Copy and complete the table. (2 marks)
(b) Using a scale of 2 cm to 1 unit on the x-axis and 1 cm to 1 unit on the y-axis, draw the graph of y = x² − 2x − 3 for −2 ≤ x ≤ 4. (2 marks)
(c) Use your graph to state
(i) the values of x where the graph crosses the x-axis (1 mark)
(ii) the coordinates of the minimum point (1 mark)
(iii) the equation of the axis of symmetry. (1 mark)
(d) By drawing a suitable straight line on your graph, solve the equation x² − 2x − 3 = 2. (2 marks)
(e) State the range of values of x for which x² − 2x − 3 < 0. (1 mark)
Total: 10 marks
Question 9 — Geometry and Trigonometry
(a) A, B, C and D are points on a circle with centre O. B lies on the major arc AC and D lies on the minor arc AC, so that ABCD is a cyclic quadrilateral. Angle AOC (the angle at the centre on the same side as D) = 124°.
(i) Calculate angle ABC, giving a reason. (2 marks)
(ii) Calculate angle ADC, giving a reason. (2 marks)
(b) Triangle ABC has vertices A(1, 1), B(3, 1) and C(1, 4).
(i) Triangle ABC is enlarged with centre the origin and scale factor 2 to give triangle A′B′C′. State the coordinates of A′, B′ and C′. (2 marks)
(ii) Triangle ABC is rotated 90° anticlockwise about the origin to give triangle A″B″C″. State the coordinates of A″, B″ and C″. (3 marks)
(iii) The area of triangle ABC is 3 square units. State the area of triangle A′B′C′. (1 mark)
(c) A ship leaves a harbour and sails 8 km on a bearing of 060°. Calculate how far
(i) north of the harbour (1 mark)
(ii) east of the harbour (1 mark)
the ship is now.
Total: 12 marks
Question 10 — Vectors and Matrices
(a) M = (3 2 ; 1 4).
(i) Find the determinant of M. (1 mark)
(ii) Find M⁻¹, the inverse of M. (2 marks)
(iii) Use M⁻¹ to solve the simultaneous equations
3x + 2y = 7 x + 4y = 9 (3 marks)
(b) Relative to an origin O, the position vectors of P and Q are p = (2, 3) and q = (8, −1).
(i) Express the vector PQ as a column vector. (1 mark)
(ii) Calculate |PQ|, giving your answer correct to 3 significant figures. (2 marks)
(iii) M is the midpoint of PQ. Find the position vector of M. (2 marks)
(iv) Show that OM is parallel to the vector (10, 2). (1 mark)
Total: 12 marks