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Secondary 3 Mathematics Practice Paper: Quadratic Equations, Functions and Graphs, Circle Properties, Trigonometric Ratios, Probability and Statistics
Mathematics – Secondary 3 – Practice Paper
S3 mathematics practice paper: quadratic equations, quadratic graphs, circle properties, trigonometric ratios, probability and statistics. Full answers included.
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Paper details
Section A: Quadratic Equations (18 marks)
Answer all four parts. Show your working in each part and simplify all answers.
| # | Question | Answer and marking notes |
|---|---|---|
| 1 | (a) Solve 2x^2 – 7x + 3 = 0 by factorisation. | Answer: 2x^2 – 7x + 3 = (2x – 1)(x – 3) = 0, so x = 0.5 or x = 3. (Marking: correct factorisation 2 marks, each root 1 mark) |
| 2 | (b) The equation 3x^2 + kx + 12 = 0 has a repeated root. Find the two possible values of k. | Answer: repeated root requires k^2 = 4 x 3 x 12 = 144, so k = 12 or k = -12. (Marking: condition 2 marks, k squared 1 mark, both values 1 mark) |
| 3 | (c) Find the sum and the product of the roots of 2x^2 – 7x + 3 = 0. | Answer: sum of roots = 7 / 2 = 3.5, product of roots = 3 / 2 = 1.5. (Marking: sum 2 marks, product 2 marks) |
| 4 | (d) Solve x^2 + 4x – 5 = 0 by completing the square. | Answer: x^2 + 4x – 5 = (x + 2)^2 – 9 = 0, so (x + 2)^2 = 9 and x = 1 or x = -5. (Marking: completing the square 3 marks, taking square root 2 marks, both roots 1 mark) |
Section B: Functions and Graphs (16 marks)
A quadratic function is given by y = x^2 – 6x + 5.
| # | Question | Answer and marking notes |
|---|---|---|
| 5 | (a) Find the coordinates of the vertex of the graph. | Answer: y = (x – 3)^2 – 4, so the vertex is (3, -4). (Marking: completing the square 2 marks, vertex 2 marks) |
| 6 | (b) Find all intercepts of the graph with the two axes. | Answer: the y-intercept is (0, 5); the x-intercepts are (1, 0) and (5, 0). (Marking: y-intercept 1 mark, each x-intercept 1 mark, factorisation 1 mark) |
| 7 | (c) State the equation of the axis of symmetry and the minimum value of the function. | Answer: the axis of symmetry is x = 3 and the minimum value is -4. (Marking: axis 2 marks, minimum value 2 marks) |
| 8 | (d) Describe how the graph of y = 2(x – 1)^2 + 3 is obtained from the graph of y = 2x^2, and state its vertex. | Answer: shift the graph of y = 2x^2 one unit to the right and three units up; the vertex is (1, 3). (Marking: right shift 1 mark, upward shift 1 mark, vertex 2 marks) |
Section C: Circle Properties (18 marks)
In each part the circle has centre O. Give units in every answer.
| # | Question | Answer and marking notes |
|---|---|---|
| 9 | (a) An arc subtends an angle of 128 degrees at the centre. Find the angle it subtends at the circumference. | Answer: 128 / 2 = 64 degrees (the angle at the centre is twice the angle at the circumference). (Marking: correct relation 2 marks, value 2 marks) |
| 10 | (b) In a cyclic quadrilateral ABCD, angle A = (3x + 10) degrees and angle C = (2x + 20) degrees. Find x and angle A. | Answer: (3x + 10) + (2x + 20) = 180, so 5x = 150, x = 30 and angle A = 100 degrees. (Marking: opposite angles sum to 180 2 marks, x 1 mark, angle A 1 mark) |
| 11 | (c) A circle of radius 9 cm has a tangent from an external point P of length 12 cm. Find the distance OP. | Answer: the tangent is perpendicular to the radius, so OP^2 = 9^2 + 12^2 = 225 and OP = 15 cm. (Marking: right angle 2 marks, Pythagoras 2 marks, answer 1 mark) |
| 12 | (d) A sector has radius 6 cm and angle 120 degrees. Find its arc length and area, in terms of pi and to one decimal place. | Answer: arc length = 4 pi, about 12.6 cm; area = 12 pi, about 37.7 cm squared. (Marking: arc length 2 marks, area 3 marks) |
Section D: Trigonometric Ratios (18 marks)
Give numerical answers to three significant figures.
| # | Question | Answer and marking notes |
|---|---|---|
| 13 | (a) In a right-angled triangle the side opposite angle theta is 8 and the hypotenuse is 17. Find sin theta, cos theta and tan theta. | Answer: the adjacent side is 15; sin theta = 8 / 17 = 0.471, cos theta = 15 / 17 = 0.882, tan theta = 8 / 15 = 0.533. (Marking: adjacent side 1 mark, each ratio 1 mark) |
| 14 | (b) A person 1.6 m tall stands 30 m from a building. The angle of elevation of the top of the building is 40 degrees. Find the height of the building. | Answer: 1.6 + 30 x tan 40 degrees, about 26.8 m. (Marking: tangent relation 2 marks, 30 tan 40 2 marks, adding eye height 1 mark) |
| 15 | (c) A ladder 5 m long has its foot 3 m from a wall. Find the height reached and the angle the ladder makes with the ground. | Answer: height = 4 m; angle = inverse cos of 3 / 5, about 53.1 degrees. (Marking: height 2 marks, angle 2 marks) |
| 16 | (d) In triangle ABC, angle B = 90 degrees, AB = 7 cm and angle A = 32 degrees. Find BC. | Answer: BC = 7 x tan 32 degrees, about 4.4 cm. (Marking: choosing tangent 3 marks, value 2 marks) |
Section E: Probability (15 marks)
A bag contains 4 red, 6 blue and 5 green balls, identical except in colour.
| # | Question | Answer and marking notes |
|---|---|---|
| 17 | (a) One ball is drawn at random. Find the probability that it is red. | Answer: there are 15 balls, so P(red) = 4 / 15 = 0.267. (Marking: total 1 mark, probability 2 marks) |
| 18 | (b) Two balls are drawn without replacement. Find the probability that both are red. | Answer: P = (4 / 15) x (3 / 14) = 2 / 35 = 0.057. (Marking: second draw over 14 2 marks, multiplication 1 mark, answer 1 mark) |
| 19 | (c) Two fair dice are thrown. Find the probability that the total is 9. | Answer: there are 4 favourable outcomes, so P = 4 / 36 = 1 / 9 = 0.111. (Marking: listing outcomes 2 marks, probability 2 marks) |
| 20 | (d) Three fair coins are tossed. Find the probability of getting at least one head. | Answer: P = 1 – 1/8 = 7/8 = 0.875. (Marking: complementary event 2 marks, value 2 marks) |
Section F: Statistics (15 marks)
The test scores of seven students are 12, 15, 15, 18, 20, 22 and 25.
| # | Question | Answer and marking notes |
|---|---|---|
| 21 | (a) Find the mean, correct to two decimal places. | Answer: the total is 127, so the mean is 127 / 7 = 18.14. (Marking: total 2 marks, dividing by 7 2 marks) |
| 22 | (b) Find the median and the mode. | Answer: the median is 18 (the fourth value) and the mode is 15 (it appears twice). (Marking: median 2 marks, mode 2 marks) |
| 23 | (c) Find the range and the interquartile range. | Answer: range = 25 – 12 = 13; Q1 = 15 and Q3 = 22, so the interquartile range is 7. (Marking: range 2 marks, interquartile range 2 marks) |
| 24 | (d) If 3 marks are added to every score, what is the new mean? If every score is doubled, what is the new mean? | Answer: adding 3 gives 21.14; doubling gives 36.29. (Marking: 1.5 marks each, correct working accepted) |
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