Maths: Full Practice Paper (Additional, Level 3)
Additional Mathematics (still often called Level 3, its name under the older three-level Common Entrance Maths system) is a further paper sat only by candidates entered for it by their prep school, usually the more mathematically able and those heading to more academically selective senior schools.
Method
- Expect harder content than the Core papers from the first question onwards, so revise Pythagoras' theorem, right-angled trigonometry, direct and inverse proportion, standard form and quadratic expressions specifically, not just the Core topics.
- For a worded problem, identify which area of maths it needs before starting to calculate, since Additional Mathematics questions often disguise a Pythagoras, proportion or algebra question inside a real-life context.
- Even though a calculator is allowed, still write out every algebraic or trigonometric step, since method marks are awarded for the working, not just for reaching the calculator's final display.
- For proportion questions, find the constant (the multiplier for direct proportion, or the product for inverse proportion) explicitly and write it down before using it, rather than trying to scale directly from one given pair of values to another.
- For quadratics, check whether the question wants a solution by factorising, or by another method, and always check a solution by substituting it back into the original equation.
- Round only your final answer, and only to the accuracy the question asks for, keeping full calculator accuracy in every step in between.
- Manage time by mark value: a question near the end worth 5 or 6 marks deserves several minutes of focused working, so avoid over-checking an early 2-mark question at the expense of a later multi-part one.
Worked example
A ladder 6.5 m long leans against a vertical wall, with its foot 2.5 m from the base of the wall. Work out how high up the wall the ladder reaches.
- Recognise this as a right-angled triangle problem, with the ladder as the hypotenuse: use Pythagoras' theorem, a^2 + b^2 = c^2.
- Substitute the known values: height^2 + 2.5^2 = 6.5^2.
- Calculate: height^2 = 42.25 - 6.25 = 36.
- Square root both sides: height = 6.
- So the ladder reaches 6 m up the wall.
Practice questions
Type your answer and press Check to be marked straight away, or reveal the answer and mark yourself.
Q1A right-angled triangle has two shorter sides of length 9 cm and 12 cm. Work out the length of the hypotenuse.Show answer
Answer: 15 cm (9^2 + 12^2 = 81 + 144 = 225, square root of 225 = 15)
Q2In a right-angled triangle, the angle at A is 40 degrees and the side opposite A is 8 cm. Use trigonometry to find the length of the hypotenuse, giving your answer to 1 decimal place.Show answer
Answer: 12.4 cm (hypotenuse = 8 / sin(40 degrees) = 12.4457..., rounds to 12.4)
Q3y is directly proportional to x. When x = 5, y = 30. Work out the value of y when x = 8.Show answer
Answer: 48 (the multiplier is 30 / 5 = 6, so y = 6 x 8 = 48)
Q4p is inversely proportional to q. When q = 4, p = 15. Work out the value of p when q = 10.Show answer
Answer: 6 (the constant is 4 x 15 = 60, so p = 60 / 10 = 6)
Q5Write 0.00042 in standard form.Show answer
Answer: 4.2 x 10^-4
Q6Solve x^2 + 3x - 10 = 0.Show answer
Answer: x = 2 or x = -5 (factorises to (x + 5)(x - 2) = 0)
Q7The three angles of a triangle are in the ratio 2 : 3 : 4. Work out the size of the largest angle.Show answer
Answer: 80 degrees (parts sum to 9, one part = 180 / 9 = 20, largest = 4 x 20 = 80)
Exam-style questions
Written in the style of a 13+ Common Entrance exam paper, with a full mark scheme.
A wheelchair access ramp rises 0.9 m over a horizontal distance of 5.4 m. Work out the angle the ramp makes with the horizontal, giving your answer to 1 decimal place.
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It takes 6 identical pumps 10 hours to empty a flooded basement, all pumps working at the same constant rate. (a) Work out how long it would take 4 of these pumps to empty the same basement. (b) Using 4 pumps, after 6 hours a fifth identical pump joins them for the rest of the job. Work out the total time now needed to empty the basement, from the very start.
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Free printable worksheet
Want more practice on paper? Download the maths: full practice paper (additional, level 3) worksheet pack - 9 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
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