Maths: Full Practice Paper (Additional, Level 3, Set 6)
This is another Additional, Level 3 paper, the hardest tier of the 13+ Common Entrance Maths papers, sat by pupils in the top mathematical set; it uses a calculator throughout and, on top of the full Core specification, tests Additional content such as surds, harder simultaneous and quadratic equations, circle theorems and trigonometry beyond right-angled triangles, with a fresh set of questions from the fifth Additional paper.
Before you start
Make sure you're comfortable with these topics first:
Method
- Read the whole of a multi-part question before starting, since a later part often depends on carrying forward an earlier result exactly, not a rounded version of it.
- For circle theorem questions, redraw the diagram larger and mark on every angle and length you are given or can work out, since a clear diagram often reveals which theorem applies.
- When solving simultaneous equations that include a quadratic, substitute to eliminate one variable and check your final quadratic is fully simplified before you factorise or use the quadratic formula.
- Simplify surds fully in your final answer, since an unsimplified surd is usually not accepted as a final answer on a mark scheme.
- For trigonometry questions beyond right-angled triangles, decide whether the sine rule, the cosine rule, or the area formula 0.5 x a x b x sin(C) is needed before you substitute any values.
- Where a question allows several correct methods, choose whichever you can complete most reliably under time pressure, rather than the most elegant approach.
Worked example
Solve the simultaneous equations y = 3x - 2 and y = x^2 - 2x - 8.
- Since both expressions equal y, set them equal to each other: 3x - 2 = x^2 - 2x - 8.
- Rearrange into the form ax^2 + bx + c = 0: x^2 - 5x - 6 = 0.
- Factorise the quadratic: (x - 6)(x + 1) = 0.
- Solve for x: x = 6 or x = -1.
- Substitute each value of x back into y = 3x - 2 to find the matching y-values: when x = 6, y = 16; when x = -1, y = -5.
Practice questions
Type your answer and press Check to be marked straight away, or reveal the answer and mark yourself.
Q1Simplify sqrt(75) - sqrt(12), giving your answer in the form k sqrt(3).Show answer
Answer: 3 sqrt(3) (sqrt(75) = 5 sqrt(3), sqrt(12) = 2 sqrt(3), so the difference is 3 sqrt(3))
Q2Expand and simplify (2x - 3)(x + 4).Show answer
Answer: 2x^2 + 5x - 12
Q3Factorise 2x^2 + 7x + 3.Show answer
Answer: (2x + 1)(x + 3)
Q4A chord subtends an angle of 76 degrees at the centre of a circle. Work out the angle it subtends at the circumference, on the major arc.Show answer
Answer: 38 degrees (half of 76)
Q5Solve x^2 + 3x - 18 = 0.Show answer
Answer: x = 3 or x = -6 (factorises to (x - 3)(x + 6) = 0)
Q6A triangle has sides 7 cm and 9 cm with an included angle of 50 degrees. Work out its area, to 1 decimal place.Show answer
Answer: 24.1 cm^2 (area = 0.5 x 7 x 9 x sin(50) = 31.5 x 0.766 = 24.1)
Exam-style questions
Written in the style of a 13+ Common Entrance exam paper, with a full mark scheme.
(a) Simplify sqrt(48) + sqrt(27), giving your answer in the form k sqrt(3). (b) Hence find the value of (sqrt(48) + sqrt(27))^2.
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In triangle PQR, PQ = 9 cm, angle P = 40 degrees and angle Q = 65 degrees. (a) Work out angle R. (b) Use the sine rule to work out the length of QR, to 1 decimal place.
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Free printable worksheet
Want more practice on paper? Download the maths: full practice paper (additional, level 3, set 6) 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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