Maths: Full Practice Paper (Additional, Level 3, Set 5)
The Additional, Level 3 paper is the hardest of the three 13+ Common Entrance Maths papers, sat by pupils in the top mathematical set; it uses a calculator throughout and, alongside the full Core specification, tests Additional content such as harder algebraic manipulation, surds, more advanced trigonometry, circle theorems and problems that combine several topics within one question.
Before you start
Make sure you're comfortable with these topics first:
Method
- Expect longer, multi-step questions rather than a series of short, single-skill questions, and read each one through fully before starting.
- Revise the Additional-only content specifically, including surds, harder simultaneous equations, circle theorems and trigonometry beyond right-angled triangles, since Level 3 papers draw on these more heavily than Core papers do.
- Keep exact values such as surds or fractions through the working of a question, only converting to a decimal at the very end if the question requires it.
- Show algebraic working line by line, including any factorising or expanding of brackets, since method marks are available even when a final simplification is missed.
- In multi-part questions, use the exact value found in an earlier part, not a rounded version of it, when it feeds into a later part.
- Manage your time carefully, since Level 3 papers typically have fewer, longer questions than Core papers, and decide early which questions you can access fully.
Worked example
Solve the simultaneous equations y = x + 1 and y = x^2 - 5.
- Since both expressions equal y, set them equal to each other: x + 1 = x^2 - 5.
- Rearrange into the form ax^2 + bx + c = 0: x^2 - x - 6 = 0.
- Factorise the quadratic: (x - 3)(x + 2) = 0.
- Solve for x: x = 3 or x = -2.
- Substitute each value of x back into y = x + 1 to find the matching y-values: when x = 3, y = 4; when x = -2, y = -1.
Practice questions
Type your answer and press Check to be marked straight away, or reveal the answer and mark yourself.
Q1Simplify sqrt(45) + sqrt(20), giving your answer in the form k sqrt(5).Show answer
Answer: 5 sqrt(5) (sqrt(45) = 3 sqrt(5), sqrt(20) = 2 sqrt(5), so the sum is 5 sqrt(5))
Q2Expand and simplify (x + 5)(x - 3).Show answer
Answer: x^2 + 2x - 15
Q3Factorise x^2 - 9x + 20.Show answer
Answer: (x - 4)(x - 5)
Q4A chord divides a circle so that the angle at the centre is 130 degrees. Work out the angle at the circumference subtended by the same chord, on the major arc.Show answer
Answer: 65 degrees (the angle at the centre is twice the angle at the circumference: 130 / 2 = 65)
Q5Solve x^2 = 7x - 12.Show answer
Answer: x = 3 or x = 4 (x^2 - 7x + 12 = 0 factorises to (x - 3)(x - 4) = 0)
Q6In a right-angled triangle, the hypotenuse is 13 cm and one other side is 5 cm. Work out the size of the angle opposite the 5 cm side, to 1 decimal place.Show answer
Answer: 22.6 degrees (sin(angle) = 5/13, so the angle = sin^-1(5/13) = 22.6 degrees to 1 dp)
Exam-style questions
Written in the style of a 13+ Common Entrance exam paper, with a full mark scheme.
(a) Simplify sqrt(72) - sqrt(8), giving your answer in the form k sqrt(2). (b) Hence, or otherwise, find the value of (sqrt(72) - sqrt(8))^2.
Show mark scheme
Tick each line you got. Your score builds from the marks on the scheme.
Nothing ticked yet - 5 available
A triangle ABC has AB = 8 cm, BC = 10 cm, and angle ABC = 60 degrees. (a) Use the cosine rule to work out the length of AC, giving your answer to 1 decimal place. (b) Work out the area of triangle ABC, to 1 decimal place.
Show mark scheme
Tick each line you got. Your score builds from the marks on the scheme.
Nothing ticked yet - 5 available
Free printable worksheet
Want more practice on paper? Download the maths: full practice paper (additional, level 3, set 5) 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.
Next topics
Not quite what you needed?
Tell us what is missing on maths: full practice paper (additional, level 3, set 5), or which topic to write up next. Every request is read, and we reply to every one.
Build a full practice pack.
This topic is one of hundreds in the library - pick the ones a student needs and generate a printable PDF in minutes.