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Form 3 Mathematics Practice Paper: Algebra, Equations, Mensuration and Statistics
Form 3 Mathematics – 12 questions – 100 marks – 75 minutes
Form 3 Mathematics practice paper: 12 questions, 100 marks in 75 minutes on algebra, equations, percentages, mensuration, statistics and probability, with full answers.
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Paper details Form 3 Mathematics paper: 12 questions, 100 marks in 75 minutes on algebra, equations, mensuration and statistics, with full worked answers.
Section A: Algebra and Equations
Answer all parts and show all working.
| # | Question | Answer and marking notes |
|---|---|---|
| 1 | Let x and y be positive numbers. Do all of the following. (a) Simplify (3x²y)(4xy³). (b) Expand (x+3)(x−5). (c) Factorise x²−9. (d) Factorise 2x^2 – 8. | (a) 12x³y⁴ (3 marks). (b) x²−2x−15 (3 marks). (c) (x+3)(x−3) (2 marks). (d) 2(x^2 – 4) = 2(x + 2)(x – 2) (2 marks). |
| 2 | In each equation x is unknown; solve it and check. Solve these equations. (a) 5(x−2) = 3x+4. (b) (3x−1)/2 = 4. (c) Solve 4(x + 1) = 2x + 10. | (a) 5x−10 = 3x+4, 2x = 14, x = 7 (4 marks). (b) 3x−1 = 8, 3x = 9, x = 3 (4 marks). (c) 4x + 4 = 2x + 10, so 2x = 6 and x = 3 (3 marks). Check: both sides equal 16 (1 mark). |
| 3 | Two integers x and y satisfy the equations below. Solve the simultaneous equations x+y = 12 and 2x−y = 3, and check the answer. (b) Two numbers have a sum of 12 and a product of 35; find them. | Adding: 3x = 15, x = 5 (3 marks); substituting into x+y = 12 gives y = 7 (2 marks). Check: 2(5)−7 = 3, correct (4 marks). (b) The numbers are 5 and 7 (4 marks). |
| 4 | A book is priced at $240 and the shop adjusts the price twice. A book costs $240. Its price is raised by 15% and then reduced by 20%. (a) Find the price after the rise. (b) Find the final price. (c) Find the percentage change from the original price. (d) The final price is then reduced by a further 10%; find the new price. | (a) 240 x 1.15 = $276 (3 marks). (b) 276 x 0.8 = $220.8 (3 marks). (c) (220.8−240)/240 x 100% = −8%, that is 8% below the original price (3 marks). (d) 220.8 x 0.9 = $198.72 (3 marks). |
Section B: Mensuration, Shape and Space
Answer all parts and give units with each answer.
| # | Question | Answer and marking notes |
|---|---|---|
| 5 | A city map uses a scale of 1:25000. A map has a scale of 1:25000. (a) Two places are 4.8 cm apart on the map; find the actual distance in km. (b) How far apart on the map are two places 3 km apart in reality? (c) What actual area in km^2 does 1 cm^2 on the map represent? | (a) 4.8 x 25000 = 120000 cm = 1.2 km (4 marks). (b) 3 km = 300000 cm, 300000 / 25000 = 12 cm (4 marks). (c) 1 cm represents 0.25 km, so the area is 0.25^2 = 0.0625 km^2 (4 marks). |
| 6 | The following questions are about right-angled triangles. In a right-angled triangle the two legs are 6 cm and 8 cm. (a) Find the length of the hypotenuse. (b) In another right-angled triangle the hypotenuse is 13 cm and one leg is 5 cm; find the other leg. (c) Is a triangle with sides 9 cm, 12 cm and 15 cm right-angled? Explain. | (a) sqrt(36+64) = sqrt(100) = 10 cm (5 marks). (b) sqrt(169−25) = sqrt(144) = 12 cm (4 marks). (c) 9^2 + 12^2 = 81 + 144 = 225 = 15^2, so it is right-angled (4 marks). |
| 7 | A circle has a radius of 7 cm. A circle has radius 7 cm. Take pi = 22/7. (a) Find the area. (b) Find the circumference. (c) If the radius is doubled, by what factor does the area increase? | (a) (22/7) x 49 = 154 cm² (4 marks). (b) 2 x (22/7) x 7 = 44 cm (4 marks). (c) Area is proportional to the square of the radius, so it becomes 4 times as large (4 marks). |
| 8 | A solid cylindrical container has radius 3 cm and height 10 cm. A cylinder has radius 3 cm and height 10 cm. Take pi = 3.14. (a) Find the volume. (b) Find the total surface area. (c) If the height is halved and the radius is unchanged, what fraction of the volume remains? | (a) 3.14 x 9 x 10 = 282.6 cm³ (5 marks). (b) 2 x 3.14 x 9 + 2 x 3.14 x 3 x 10 = 56.52 + 188.4 = 244.92 cm² (4 marks). (c) Volume is proportional to height, so half the volume remains (4 marks). |
Section C: Statistics and Probability
Answer all parts; give probabilities as fractions.
| # | Question | Answer and marking notes |
|---|---|---|
| 9 | Below are the marks of six students in a test. The data are 3, 5, 5, 7, 9, 10. (a) Find the mean. (b) Find the median. (c) Find the mode. (d) Find the range of the data. | (a) (3+5+5+7+9+10)/6 = 39/6 = 6.5 (3 marks). (b) The middle values are 5 and 7, so the median is 6 (3 marks). (c) The mode is 5 (2 marks). (d) Range = 10 – 3 = 7 (3 marks). |
| 10 | The balls in a bag are identical except in colour. A bag contains 3 red, 5 blue and 2 green balls. One ball is drawn at random. (a) Find the probability of drawing red. (b) Find the probability that it is not blue. (c) Find the probability of drawing a red or a green ball. | (a) 3/10 (4 marks). (b) (3+2)/10 = 1/2 (4 marks). (c) (3 + 2)/10 = 1/2 (4 marks). |
| 11 | Use the laws of indices to do the following. Simplify (a) 2³ x 2⁴, giving the answer as a power and as a number. (b) (a²)³. (c) 5⁰. (d) Simplify (2x^3)^2. | (a) 2⁷ = 128 (3 marks). (b) a⁶ (3 marks). (c) 1 (2 marks). (d) 4x^6 (3 marks). |
| 12 | Solve the inequality below. Solve the inequality 3x + 5 less than 20 and state the largest integer solution. (b) If x is a positive integer, list all possible values of x. | 3x less than 15, so x less than 5 (4 marks); the largest integer solution is 4 (4 marks). (b) x can be 1, 2, 3 or 4 (4 marks). |
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