Maths: Scholarship Challenge Paper 2
Maths Scholarship Challenge Paper 2 is a second Common Entrance 13+ scholarship-level revision set, building on the first Challenge Paper (CE.M35) with non-routine questions drawn from geometry, spatial reasoning and logic problems used to separate the strongest mathematicians from the merely fluent.
Before you start
Make sure you're comfortable with these topics first:
Method
- For spatial and geometric scholarship questions, sketch the situation even when a diagram is given, adding your own labels for any lengths, angles or points you introduce during your solution.
- For 'how many ways' or logic-style questions, systematically list or organise possible cases, for example in a table, rather than trying to count them in your head, since scholarship mark schemes reward a visibly organised method.
- Look for symmetry or a repeating pattern in a spatial problem, since scholarship geometry questions are often designed so a clever observation shortens the working compared with checking every case by hand.
- Where a question asks you to 'explain' or 'show that' rather than simply 'calculate', write your reasoning in full sentences alongside the maths, since marks are available for the explanation itself.
- Keep exact values, such as surds, fractions or multiples of pi, through every stage of a multi-part geometry problem, since converting to a rounded decimal too early is a common source of an inexact final answer.
- If a question is split into parts (a), (b), (c), use the result of an earlier part in a later one wherever possible, since scholarship questions are frequently built to lead towards the final part.
- Where possible, practise scholarship-style papers from more than one school, since the amount of scaffolding and the amount of written explanation expected varies more between individual schools' scholarship papers than it does across standard Core Common Entrance.
Worked example
A square painting has side length 24 cm. It is surrounded by a frame of constant width w, so that the total area of painting plus frame is 900 cm^2. Work out the width of the frame.
- Let the frame have width w. Since the frame runs around all four sides, the outer square has side length 24 + 2w.
- Form an equation using the total area: (24 + 2w)^2 = 900.
- Take the square root of both sides, using the positive root since a length cannot be negative: 24 + 2w = 30.
- Solve for w: 2w = 6, so w = 3.
- So the frame is 3 cm wide.
Practice questions
Type your answer and press Check to be marked straight away, or reveal the answer and mark yourself.
Q1A cube with side length 4 cm is built from 64 unit cubes, each 1 cm x 1 cm x 1 cm. The outside of the large cube is painted red, then it is taken apart into the 64 unit cubes again. How many of the unit cubes have exactly two faces painted red?Show answer
Answer: 24 (these are the edge cubes: a cube has 12 edges, and on each edge there are n - 2 = 2 cubes with exactly two painted faces, so 12 x 2 = 24)
Q2At a small maths club meeting, every one of the 8 members shakes hands with every other member exactly once. Work out the total number of handshakes.Show answer
Answer: 28 (each of the 8 people shakes hands with 7 others, giving 8 x 7 = 56 handshakes counted twice, so 56 / 2 = 28)
Q3A floor is tiled with a repeating pattern of identical regular hexagonal tiles, with no gaps or overlaps. State the size of each interior angle of a regular hexagon, and explain why regular hexagons can tile a flat floor exactly.Show answer
Answer: 120 degrees (interior angle = (6 - 2) x 180 / 6 = 120); three hexagons meet at each point, and 3 x 120 = 360 degrees, exactly filling the angle around a point with no gaps or overlaps
Q4A dice has its six faces numbered 1 to 6, arranged so opposite faces always add up to 7. The face showing 2 is on top, and the face showing 3 is facing you. Work out which number is on the bottom face, and which number is on the face directly opposite the one facing you.Show answer
Answer: Bottom face = 5 (7 - 2, since it is opposite the top face showing 2); face opposite the one showing 3 = 4 (7 - 3)
Q5A rectangular field measuring 60 m by 40 m is to be divided into identical square plots, as large as possible, with no land left over. Work out the side length of each square plot, and the total number of plots.Show answer
Answer: 20 m side length (the highest common factor of 60 and 40), giving 3 x 2 = 6 plots in total
Q6A regular polygon has an exterior angle of 24 degrees. Work out how many sides the polygon has.Show answer
Answer: 15 sides (exterior angles of any polygon sum to 360 degrees, so 360 / 24 = 15)
Q7In a class of 30 pupils, 18 play football, 14 play tennis, and 7 play both football and tennis. How many pupils play neither football nor tennis?Show answer
Answer: 5 (18 + 14 - 7 = 25 play at least one sport, so 30 - 25 = 5 play neither)
Exam-style questions
Written in the style of a 13+ Common Entrance exam paper, with a full mark scheme.
A vertical flagpole casts a shadow 12 m long at the same time as a vertical metre stick, 1 m tall, casts a shadow 0.8 m long, with both shadows measured on level ground at the same moment. (a) Explain why the triangle formed by the flagpole and its shadow is similar to the triangle formed by the metre stick and its shadow. (b) Use this to work out the height of the flagpole.
Show mark scheme
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Isla has an unlimited supply of 20p coins and 50p coins. Using only these two types of coin, and at least one of each, she wants to make exactly 3.00 pounds. (a) Form an equation connecting the number of 20p coins (x) and the number of 50p coins (y) she could use. (b) List every possible combination of whole numbers of 20p and 50p coins that makes exactly 3.00 pounds.
Show mark scheme
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Nothing ticked yet - 6 available
Free printable worksheet
Want more practice on paper? Download the maths: scholarship challenge paper 2 worksheet pack - 11 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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