ABC Education

ABC Education · 11+ Practice

Week 62 · Maths Practice Paper

5 questions · 50 marks · Suggested time: 40 minutes.

Question 1

[12 marks]

Using 1s and 2s, there is only one way to represent the value 1. However, there are two ways of making the value 2: 1+1 or 2. There are three ways of adding up an ordered combination of 1s and 2s to make 3: 1+1+1 or 2+1 or 1+2. Notice that order is important here!

(a)
How many ways can you add up an ordered combination of 1s and 2s to make 4?
(b)
How many ways can you add up an ordered combination of 1s and 2s to make 5?
(c)
By considering these results, prove that there are exactly 34 ways you can add up an ordered combination of 1s and 2s to make 8.

Question 2

[7 marks]

You are given that angles which form a Z-shape within parallel lines are equal.

(a)
Using a triangle with a line drawn through one vertex parallel to the opposite side, explain how you know the angles in a triangle sum to 180°.
(b)
A 5-pointed star shape (a pentagram) is formed by extending the sides of a regular pentagon until they meet. What is the sum of the five point-angles (the shaded angles at each tip of the star)?

Question 3

[8 marks]

A top cyclist rides 250 metres in 20 seconds.

(a)
What is her speed in metres per second?
(b)
To convert a speed from metres per second to miles per hour, multiply by 9 and divide by 4. Convert her speed to miles per hour.
(c)
How would you convert a speed from miles per hour into metres per second?
(d)
A new leaf blower blows at 90 miles per hour. How fast is that in metres per second?

Question 4

[9 marks]

Here is how you can solve problems like the example below: What's my number? When it is divided by 4 it leaves remainder 1, and when divided by 6 it leaves remainder 3. The numbers that leave remainder 1 when divided by 4 are 1, 5, 9, 13, ... The numbers that leave remainder 3 when divided by 6 are 3, 9, 15, 21, ... Looking for a number in both sequences, the smallest possible solution is 9, but the sequences also have 21, 33, ... in common. Use the approach described above to solve the questions below:

(a)
A box contains more than 50 but less than 60 tennis balls. When counted in fours there are 3 left over. When counted in sixes there are 3 left over. How many balls are in the box?
(b)
There are between 300 and 350 people at a concert. When they form groups of five, 1 person is left over. When they form groups of seven, 2 people are left over. When they form groups of nine, 7 people are left over. How many people are at the concert?

Question 5

[14 marks]

You are told that O_n is 'the value of the first n odd numbers multiplied together' and E_n is 'the value of the first n even numbers multiplied together'. For example, O_3 = 1×3×5 = 15, O_5 = 1×3×5×7×9 = 945, E_6 = 2×4×6×8×10×12 = 46,080.

(a)
Calculate O_7
(b)
What is the remainder when O_60 is divided by 21?
(c)
What is the remainder when O_80 is divided by 2?
(d)
What is the remainder when E_80 is divided by 5?
(e)
You are told that n! is 'the value of the first n counting numbers multiplied together' (1!=1, 2!=2, 3!=6, etc). Notice that E_n can also be written as 2ⁿ × n! (since each even number is 2 times a counting number). Use this idea to calculate 8!, given that E_8 = 10,321,920.
(f)
What is the remainder when 1! + 2! + 3! + 4! + 5! + 6! + 7! is divided by 6?