ABC Education · 11+ Practice

Week 60 · Maths Practice Paper

5 questions · 43 marks · Suggested time: 40 minutes.

Question 1

[13 marks]

Priya adds odd numbers together and writes down her results as follows: 1 = 1 = 1² 1 + 3 = 4 = 2² 1 + 3 + 5 = 9 = 3²

(a)(i)

Write down the next three lines of this pattern.

(a)(ii)

Using this pattern, write down the line which contains 289 at the centre.

(b)(i)

Priya then adds different odd numbers and puts her results in a table again: 1 = 1 = 1³ 3 + 5 = 8 = 2³ 7 + 9 + 11 = 27 = 3³ Write down the next three lines of this pattern.

(b)(ii)

Using this pattern, how many numbers do you need to add together in the line with: ... = 1000 = ...

(c)

Using your answers from parts (a) and (b), find three numbers A, B and C such that A − B = C and A² − B² = C³.

Question 2

[9 marks]

The symbol φ represents a mathematical operation. The rule for φ is 'add the two numbers and then multiply their sum by the second number'. For example, 2 φ 6 = (2 + 6) × 6 = 8 × 6 = 48. Work out:

(a)

2 φ 6 =

(b)

12 φ 3 =

(c)

14 φ 12 =

(d)

If 6 φ m = 91, what positive number must m be? Show your working.

(e)

If p φ p = 72, what number must p be? Show your working.

(f)

Explain why x φ y is not the same as y φ x.

Question 3

[6 marks]

A road network connects villages P, Q, R, S, T and U. A route travels along the straight connections listed below (distances in miles): P–Q: 10, P–R: 6, Q–R: 4, Q–S: 12, R–S: 8, R–T: 15, S–T: 5, S–U: 9, T–U: 7.

(a)

What is the shortest route between P and T, and how long is it?

(b)

What is the shortest route between U and Q, and how long is it?

(c)

What is the shortest route between P and U, and how long is it?

Question 4

[6 marks]

Priya is calculating 32 × 26. She constructs this number pattern: 32 × 26 16 × 52 8 × 104 ___ × 208 Fill in the missing numbers.

(a)

Complete the pattern, and explain how it is made.

(b)

Using a similar method, work out 52 × 34.

(c)

Adapt this method to work out 1080 ÷ 24 (halve both numbers repeatedly until you reach an easy division).

Question 5

[9 marks]

Tilly decides to count using a clock. She starts counting from 1 in the normal way: 1, 2, 3, 4... But when she gets to 12, the count goes back to 1, so she counts: ...10, 11, 12, 1, 2, 3... So, for example, using this counting method, 4 + 9 = 1 and 10 + 5 = 3. Similarly, 3 × 5 = 3 and 2 × 13 = 2.

(a)(i)

5 + 6 =

(a)(ii)

8 + 9 =

(a)(iii)

7 + 11 =

(a)(iv)

4 × 7 =

(a)(v)

6 × 9 =

(b)

Using this counting method, can you find two different positive numbers n and m such that n² = m²?

(c)

Using this method, can you find two different numbers p and q such that p³ = q³?