Total marks: 100 · Duration: 2 hours · Tier: Extended
Instructions to candidates
- Answer all questions.
- Use a black or dark blue pen. You may use an HB pencil for any diagrams or graphs.
- Write your answer to each question in the space provided. Show all necessary working clearly.
- You should use a calculator where appropriate.
- Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless the question says otherwise.
- For π, use either your calculator value or 3.142.
- Diagrams are not drawn to scale. Where a question describes a shape in words, you may find it helpful to sketch it.
Information
- The total mark for this paper is 100.
- The number of marks for each question or part question is shown in brackets [ ].
Paper
1 Ana, Ben and Chen share $480 in the ratio 3 : 4 : 5.
(a) Show that Ben receives $160. [2]
(b) Ana spends 35% of her share. Work out how much she spends. [1]
(c) Chen invests $200 at a rate of 2.5% per year compound interest. Work out the amount of interest he earns in 6 years. Give your answer correct to the nearest cent. [2]
(d) In a sale, the price of a jacket is reduced by 18%. The sale price is $36.08. Work out the original price. [2]
(e) Ben changes his $160 into euros. The exchange rate is €1 = $1.08. Work out how many euros he receives. [2]
2 (a) Expand and simplify (2x − 3)(x + 5). [2]
(b) Factorise 4x² − 25. [1]
(c) Solve 3/(x − 1) − 2/(x + 2) = 1. Give your answers correct to 2 decimal places. [4]
(d) V = (4/3)πr³. Make r the subject of the formula. [2]
3 The masses, m grams, of 200 apples are recorded in the table.
| Mass (m grams) | 80 < m ≤ 100 | 100 < m ≤ 120 | 120 < m ≤ 140 | 140 < m ≤ 160 | 160 < m ≤ 200 |
|---|---|---|---|---|---|
| Frequency | 18 | 46 | 64 | 48 | 24 |
(a) Calculate an estimate of the mean mass. [4]
(b) Complete the cumulative frequency table.
| Mass (m grams) | m ≤ 100 | m ≤ 120 | m ≤ 140 | m ≤ 160 | m ≤ 200 |
|---|---|---|---|---|---|
| Cumulative frequency |
[2]
(c) Assuming the masses are evenly spread within each interval, estimate the median mass. [1]
(d) Estimate the number of apples with a mass greater than 150 g. [1]
(e) A histogram is drawn to show the information in the first table. Work out the frequency density of the bar for 80 < m ≤ 100 and of the bar for 160 < m ≤ 200. [2]
4 ABCD is a quadrilateral.
AB = 7.2 cm, BC = 9.5 cm and angle ABC = 112°. CD = 6.4 cm and angle ACD = 48°.
(a) Calculate the length of the diagonal AC. [3]
(b) Calculate angle BAC. [3]
(c) Calculate the area of triangle ACD. [2]
(d) Calculate the shortest distance from D to the line AC. [2]
5 f(x) = 2x + 3, g(x) = x² − 1, h(x) = 3ˣ
(a) Find f(−4). [1]
(b) Find gf(x), giving your answer in the form ax² + bx + c. [2]
(c) Find f⁻¹(x). [2]
(d) Find h⁻¹(81). [1]
(e) Solve g(x) = f(x). Give your answers correct to 2 decimal places. [3]
6 A solid metal cone has base radius 6 cm and perpendicular height 8 cm.
[The curved surface area of a cone of radius r and slant height l is πrl. The volume of a cone of radius r and height h is (1/3)πr²h.]
(a) Show that the slant height of the cone is 10 cm. [1]
(b) Calculate the curved surface area of the cone. [2]
(c) Calculate the volume of the cone. [2]
(d) The cone is melted down and recast as a solid cylinder of radius 4 cm, with no metal wasted. Calculate the height of the cylinder. [2]
(e) A larger cone is mathematically similar to the first cone. Its volume is 8 times the volume of the first cone.
(i) Find the height of the larger cone. [1]
(ii) Calculate the curved surface area of the larger cone. [2]
7 A bag contains 5 red, 3 green and 2 blue counters. Two counters are taken at random, without replacement.
(a) Find the probability that both counters are red. [2]
(b) Find the probability that both counters are the same colour. [3]
(c) Find the probability that at least one counter is blue. [2]
(d) The counters are replaced. Counters are now taken at random, one at a time without replacement, until a blue counter is taken. Find the probability that the first blue counter is the third counter taken. [2]
8 (a) Complete the table of values for y = x³ − 3x + 1.
| x | −2 | −1.5 | −1 | −0.5 | 0 | 0.5 | 1 | 1.5 | 2 |
|---|---|---|---|---|---|---|---|---|---|
| y | −1 | 3 | 1 | −1 | 3 |
[2]
(b) On a grid with −2 ≤ x ≤ 2 and −2 ≤ y ≤ 4, draw the graph of y = x³ − 3x + 1. [3]
(c) Use your graph to solve the equation x³ − 3x + 1 = 0. [2]
(d) By drawing a suitable straight line on your grid, solve the equation x³ − 5x + 2 = 0 for −2 ≤ x ≤ 2. [2]
9 Triangle T has vertices (1, 1), (3, 1) and (1, 2).
(a) Reflect triangle T in the line y = −x. Write down the coordinates of the vertices of the image. [2]
(b) Rotate triangle T through 90° anticlockwise about the origin. Write down the coordinates of the vertices of the image. [2]
(c) Triangle U has vertices (−2, −2), (−6, −2) and (−2, −4). Describe fully the single transformation that maps triangle T onto triangle U. [3]
10 (a) Find the nth term of the sequence 7, 12, 19, 28, 39, … [3]
(b) The nth term of a geometric sequence is 3 × 2ⁿ⁻¹. Write down the first three terms. [2]
(c) Find the smallest value of n for which the nth term of this geometric sequence is greater than 10 000. [2]
(d) A car costs $25 000. Its value decreases by 14% each year. Find the number of complete years after which its value is first less than $10 000. [2]
11 A curve has equation y = x³ − 6x² + 9x + 2.
(a) Find dy/dx. [2]
(b) Find the coordinates of the two turning points of the curve. [4]
(c) Find the gradient of the curve at the point where x = 4. [1]
(d) Determine whether each turning point is a maximum or a minimum. Show how you decide. [2]
END OF QUESTIONS — TOTAL: 100 MARKS