YEAR 9 (13+) SCHOLARSHIP
May 2016
for entry in September 2016
MATHEMATICS
Name:____________________________________________________
School:____________________________________________________
Time allowed: 1 hour
Equipment needed: Pen, pencil, lined paper and eraser.
Information for candidates:
1. Calculators are NOT allowed.
2. Write your name and school on this page.
3. Write your answers on the separate paper provided. Please put your
name on all the sheets of paper you use.
4. There are 14 questions in this paper. Attempt all of them.
5. Show your working out for all questions.
6. Leave your answers in their simplest form.
7. The marks for each question or part question are indicated in square brackets [ ].
There are 50 marks on the paper.
1.
If 𝑛𝑛 is an odd number, what kind of a number, odd, even or either, must m be in each of
the following in order for the expression to be even?
a)
b)
c)
2.
4𝑛𝑛 + 2𝑚𝑚2
[2 marks]
(𝑛𝑛 + 𝑚𝑚)(𝑛𝑛 − 𝑚𝑚)
[2 marks]
[2 marks]
2𝑛𝑛2 − 3𝑚𝑚
Each of the circles shown in the diagram below has a radius of 1. Calculate the total area of
the shaded parts. Leave your answer in terms of 𝜋𝜋.
[3 marks]
3.
Ian, Barbara and Martin share £50. The ratio of the amount Ian received to the total
amount shared by Barbara and Martin was 3:2. Barbara received £4 more than Martin.
How much did each receive?
[4 marks]
4.
How much smaller is the area of a 60cm by 40cm rectangle than that of a square with the
same perimeter?
[3 marks]
5.
One pump used on its own fills a swimming pool in 2 hours. A second pump used on its
own fills the same pool in 3 hours. A third pump can be used to empty all the water from
the same pool in 90 minutes.
How long would it take to fill the pool if all three pumps were in use at the same time?
[4 marks]
6.
Biscuits occupy
7
8
th of the volume of their packet. The remaining
1
8
th is unused. It is
decided to re-design the packet so that the amount of unused space is reduced by 40%.
What fraction of the packet will the biscuits now take up?
[2 marks]
1
7.
A can of cola has a label on it saying “20% extra free”.
a)
If the can holds 960ml, what did the original can hold?
b)
The original can cost 45p. If the company prices the new can at 60p, does this
represent an increase in the price of cola? You must explain your answer.
[4 marks]
8.
5 numbers have a mean 4, median 3, mode 3. List all the possible sets of positive integers
that there could be.
[4 marks]
9.
Regular pentagons of side length 1cm are joined together to make a pattern as shown
below. What is the largest number of pentagons that can be joined together like this to
have a perimeter less than 1000cm?
[3 marks]
10.
S is 25% of 60. 60 is 80% of U. 80 is M% of 25. What is the value of the S + U + M?
[3 marks]
11.
The diagram, which is not drawn accurately, shows a parallelogram inside a triangle. The
marked lengths are equal. What is the value of 𝑥𝑥?
[3 marks]
2
12.
Given that 𝐴𝐴𝐴𝐴 = 𝐴𝐴𝐴𝐴 and 𝑧𝑧 < 90. State whether each of the following are true or false:
a)
b)
c)
𝑤𝑤 = 𝑥𝑥
𝑥𝑥 + 𝑦𝑦 + 𝑧𝑧 = 180
𝑧𝑧 = 2𝑥𝑥
[3 marks]
13.
Two trains travel towards each other on the same track, starting 200 miles apart. One train
travels at 40 miles per hour; the other travels at 60 miles per hour.
A fast bird flies at a speed of 90 miles per hour. It starts its flight at the same location as the
faster train. When it reaches the slower train it turns around, flying in the other direction
at the same speed. When it reaches the faster train again, it turns around. And so on.
When the trains collide, how far will the bird have flown?
14.
[2 marks]
In this question X is the number with prime factorisation 24 × 11. A square number is a
member of the sequence 1, 4, 9, 16,… and a cube number is a member of the sequence 1, 8,
27, 64,…
a)
Give the square root of 11X as a product of primes.
[2 marks]
b)
What is the smallest number which, when multiplied by X, gives a cube number?
[2 marks]
c)
Find a number by which X must be multiplied to make both a square and a cube
number.
[2 marks]
Total: [50 marks]
3