Maths: Scholarship Challenge Paper
A 13+ Maths Scholarship paper is set by the individual senior school, or sometimes by a shared scholarship scheme used by a group of schools, rather than centrally by ISEB, and is sat by academically strong candidates competing for an academic scholarship at their chosen school.
Before you start
Make sure you're comfortable with these topics first:
Method
- Expect unfamiliar, multi-step questions that do not map onto a single memorised method: read the whole question at least twice before starting, to identify exactly what is given and what is being asked.
- Write down every step of your thinking, including ideas you try and reject, since method marks can be awarded for a sound approach even where the final answer is not reached.
- If a problem feels too general, try a smaller or simpler version first, such as testing a small number of items before generalising, since this often reveals the pattern or structure the full question needs.
- Revise beyond Core and Additional content, including algebraic proof, surds, harder circle theorems and more demanding trigonometry, since scholarship papers regularly draw on the top of the syllabus.
- Manage time deliberately: scholarship papers usually have fewer, longer questions than Core papers, so decide early which questions you can access fully and prioritise those.
- Present final answers exactly where possible, as a fraction, a surd or another exact value, rather than rounding early, since scholarship mark schemes often specifically reward an exact form.
- Since scholarship papers are set by individual schools, check any guidance issued by the specific senior school or scholarship scheme about format, calculator policy and syllabus coverage, since these vary in a way standard Common Entrance does not.
Worked example
Find three consecutive even numbers that add up to 96.
- Let the smallest of the three consecutive even numbers be n, so the three numbers are n, n + 2 and n + 4.
- Form an equation from their sum: n + (n + 2) + (n + 4) = 96.
- Simplify the left-hand side: 3n + 6 = 96.
- Solve for n: 3n = 90, so n = 30.
- State all three numbers: 30, 32 and 34 (check: 30 + 32 + 34 = 96).
Practice questions
Type your answer and press Check to be marked straight away, or reveal the answer and mark yourself.
Q1Simplify sqrt(50) - sqrt(8), giving your answer in the form k sqrt(2).Show answer
Answer: 3 sqrt(2) (sqrt(50) = 5 sqrt(2), sqrt(8) = 2 sqrt(2), so the difference is 3 sqrt(2))
Q2Prove that the sum of any three consecutive integers is always a multiple of 3.Show answer
Answer: Let the integers be n, n + 1 and n + 2. Their sum is n + (n + 1) + (n + 2) = 3n + 3 = 3(n + 1), which is 3 multiplied by a whole number, so it is always a multiple of 3
Q3Simplify (3x^2 y^3)^2 divided by 9x^3 y^4, giving your answer as simply as possible.Show answer
Answer: xy^2 ((3x^2y^3)^2 = 9x^4y^6, and 9x^4y^6 / 9x^3y^4 = xy^2)
Q4A two-digit number is 7 times the sum of its digits. The digit in the tens place is one more than the digit in the units place. Find the two-digit number.Show answer
Answer: 21 (digits 2 and 1: sum of digits = 3, and 7 x 3 = 21, which matches the number itself)
Q5Femi can complete a job alone in 2 hours, Grace can complete the same job alone in 3 hours, and Hassan can complete it alone in 6 hours. Working together at these constant rates, how long will it take them to complete the job?Show answer
Answer: 1 hour (combined rate = 1/2 + 1/3 + 1/6 = 3/6 + 2/6 + 1/6 = 1 whole job per hour)
Q6A triangle is inscribed in a circle so that one of its sides is a diameter of the circle. Explain, using a circle theorem, why the angle at the third vertex (opposite that side) must be 90 degrees.Show answer
Answer: This is the angle in a semicircle theorem: a diameter subtends an arc of 180 degrees at the centre of the circle, and the angle at the circumference subtending the same arc is always half the angle at the centre, so the angle at the third vertex is 180 / 2 = 90 degrees
Q7The product of two consecutive positive integers is 132. Find the two integers.Show answer
Answer: 11 and 12 (n(n+1) = 132 leads to n^2 + n - 132 = 0, which factorises to give n = 11, so the integers are 11 and 12)
Exam-style questions
Written in the style of a 13+ Common Entrance exam paper, with a full mark scheme.
Show that (n + 1)^2 - (n - 1)^2 = 4n for any integer n. Hence find two consecutive even numbers whose squares differ by 60.
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Two cyclists, Oliver and Freya, start 45 km apart and cycle towards each other along the same road, starting at the same time. Oliver cycles at 18 km/h and Freya cycles at 12 km/h. (a) Work out how long it takes for them to meet. (b) Work out how far from Oliver's starting point they meet.
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Free printable worksheet
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