13+ Scholarship Examinations 2015
MATHEMATICS I
45 minutes (plus reading time)
Use the reading time wisely; gain an overview of the paper and start to think of how you will answer the
questions.
Do as many questions as you can (clearly numbered) on the lined paper provided. Clearly name each sheet
used. You are encouraged to attempt these questions in order.
The questions are not of equal length or mark allocation. Make sure you avoid spending too much time on any
one question; don’t get bogged down! Move on quickly if you get stuck. The paper is quite long; you are not
necessarily expected to finish everything.
Some of the later questions are more difficult, but not necessarily longer. Some questions are designed to test
your ability to work with unfamiliar ideas, or familiar ones with a twist. Don’t give up!
You are expected to use a calculator where appropriate, but you must show full and clear working, diagrams
and arguments wherever you can. Marks will be awarded for method as well as answers: merely writing down
an answer might score very few marks.
Complete solutions are preferable to fragments. You can sometimes, however, manage to complete later parts
of questions, even if you have failed to answer the earlier sections.
This paper has eleven questions.
1
Last month the UK athlete Mo Farah broke the indoor two-mile world record; he won his
race in 8 minutes 3.40 seconds. The previous record time was 8 minutes 4.35 seconds.
(a) Work out the percentage change in the record time, writing down all the figures on
your calculator.
(b) What was Mo’s average time per 100 m? [You are given that there are 1609 m in a
mile.]
(c) Comment on your answer to (b), and say for how long you think you could keep up
with Mo.
2
3
Hawking starts solving a set of ten hard scholarship problems on the morning of
1st March. He solves one-third of a problem every morning and one-fifth of a
problem every evening. On what day of the month does he finish all ten problems
(he takes no days off)?
Solve the following for x:
(a)
(b)
(c)
4
5
3(2𝑥 − 7) = −105
9𝑥
5
3
9
4
20
+ =1
1
2 − 8(5 − 𝑥) = 2𝑥 + (𝑥 − 7)
3
Dr Palmer has a pile of fifteen scholarship papers to mark. She finds that when she marks
all of them except Hetherington's, the mean mark so far is 80%.
Once she has marked the last test the mean rises to 81%. What is Hetherington’s mark?
China's Tianhe-2 is currently ranked the world's fastest supercomputer with a record speed
of 33.86 petaflops. This is approximately 34 million billion calculations per second.
(a) The BBC claims this is equivalent to 4702000 calculations per second being made by
every single person on the planet.
What does this suggest is the approximate current world population?
(b) The Colossus computer at Bletchley Park in World War II could do 5000 calculations
per second.
If Tianhe-2 can complete a code-breaking task in one second, estimate how long Colossus
would take to do the same task. Give your answer in appropriate units.
6
Mr Bateman decides to make some new business cards. The style rules he must follow are:
 Business cards are 85mm by 55mm
 Margins (blank) must be 7mm on all sides
Mr Fermat likewise makes some new letter stationery. The style rules he must follow are:
 A4 paper size (see below) is 297mm by 210mm
 Margins (including the head and foot of the page) must be:
o Left 25mm
o Right 20mm
o Top and bottom 35mm
Showing your full working carefully, decide which of Bateman’s card or Fermat’s letter has
the greater proportion of its area in the margins.
7
Solve the simultaneous equations
5𝑥 + 3𝑦 = 41
4𝑦 − 29𝑥 = 126
8
Nigel Farage, leader of the United Kingdom Independence Party (UKIP), has a certain
number of supporters.
After a programme on UKIP airs on TV, he loses 15% of his supporters.
After a second programme about UKIP is shown on TV he loses 15% of his remaining
support.
He now has 578000 supporters.
How many did he have originally?
9
Pansy does a large shop at her local supermarket and buys seventy items of food.
(In the below, the statements do not mean e.g. organic only).






She buys 41 items labelled organic
She buys 39 items labelled Fairtrade
She buys 31 items labelled free-range
In fact, 15 are labelled organic free-range
Also, 16 are labelled organic Fairtrade
Further, 17 are labelled Fairtrade free-range.
She buys nothing without at least one label.
How many of the items are actually all three of organic, free-range and Fairtrade?
10
Dave has 89 members in his team and Ed has 11 in his.
Some members of Dave’s team join Ed’s instead.
At this point Dave still has three times as many in in his team as Ed has.
How many people moved teams? Show all your working.
11
Makka Pakka has a trolley (the Og-Pog) in which he keeps four items: sponge, soap, orange
trumpet, and his Uff-Uff (a set of bellows).
(a) In how many ways can he choose two items from the four in his trolley?
He also has two separate sets of six stones; stones in each pile are all of different sizes.
Makka Pakka chooses two from the first set of six stones. He stacks them on the ground in
order of size (with the larger on the ground).
(b) Explain, without listing the possibilities, why he has fifteen ways of choosing the two
stones.
He then does the same with three stones from the second set making a separate, new, pile.
(c) In how many ways can he do this?
(d) What is the combined number of ways in which he chooses two items from the
trolley, and makes one pile of two stones and one of three stones?
END OF PAPER
THE KING’S SCHOOL, CANTERBURY
SCHOLARSHIP ENTRANCE EXAMINATION
March 2014
MATHEMATICS 1
Time: 45 minutes (plus reading time)
Use the reading time wisely; gain an overview of the paper and start to think of how you will answer the questions.
Do as many questions as you can (clearly numbered) on the lined paper provided. Clearly name each sheet used. You
are encouraged to attempt these questions in order.
The questions are not of equal length or mark allocation. Make sure you avoid spending too much time on any one
question; don’t get bogged down! Move on quickly if you get stuck. The paper is quite long; you are not necessarily
expected to finish everything.
Some of the later questions are more difficult, but not necessarily longer. Some questions are designed to test your
ability to work with unfamiliar ideas, or familiar ones with a twist. Don’t give up!
You are expected to use a calculator where appropriate, but you must show full and clear working, diagrams and
arguments wherever you can. Marks will be awarded for method as well as answers: merely writing down an answer
might score very few marks.
Complete solutions are preferable to fragments. You can sometimes, however, manage to complete later parts of
questions, even if you have failed to answer the earlier sections.
This paper has nine questions.
1
Here is a soup carton I bought last week.
The list of ingredients on the packet shows the percentage of basil in the pesto (59%) and the percentage
of pesto in the soup (1.8%).
What mass of basil is there in the soup if the carton contains 600g soup in total?
2
Below is given the names and sizes of the different champagne bottles available. If we buy one of each of
the below, how many basic bottles is this equivalent to?
Bouteille (basic bottle)
Magnum
Jeroboam
Rehoboam
Methuselah
Salmanazar
Balthazar
Nebuchadnezzar
Midas
3
Solve the following for x:
(a)
(b)
4
750 ml
1.5 litres
3 litres
4.5 litres
6 litres
9 litres
12 litres
15 litres
30 litres
(
)
(
)
(
)
In ancient Greece a surveyor might want to know the area, A, of a piece of land after measuring its three
sides (a, b, c in this question).
Here is a formula attributed to Hero of Alexandria (45 AD):
√ (
)(
where s is the semi-perimeter of the triangle i.e.
What is the area of a triangle of sides 7, 6 and 3 metres?
)(
)
s = ½ (a + b + c)
5
(a) Last term we encountered the date 11/12/13 which is made up of consecutive numbers in
ascending order. When is the next date with this property?
(b) Last term we also had the “odd day” 9/11/13 with the day/month/year being consecutive odd
numbers in ascending order. How many more odd days will there be this century?
6
Last year the Indian space agency launched a mission to Mars, which will cost £45 million for the 485
million mile journey.
The UK government’s plan for the High Speed 2 railway link (HS2) is scheduled to cost £32 billion for a
railway line stretching 119 miles.
(a) How many such Mars missions could be completed for the same amount of money as the
predicted cost of HS2?
(b) If HS2 was built for the same cost per mile as the Indian Mars mission, how much would it cost?
7
Pansy buys a petrol/electric hybrid car. She drives on battery power for 40 miles and then on petrol for
the rest of the journey, using petrol at 0.02 gallons per mile.
The petrol consumption for the whole journey was 55 miles per gallon.
How long was her journey?
8
(a)
Solve the simultaneous equations:
(b) Solve the simultaneous equations, giving all values for p and q :
TURN OVER FOR THE LAST QUESTION
9
The ancient Babylonian tablet YBC 7289 (second millennium BC), shown below, contains an intriguing
piece of mathematical working.
The cuneiform (ancient alphabet) markings have been deciphered as an calculation of the hypotenuse of
the triangle shown in the tablet, namely:
[the Babylonians used powers of sixty rather than ten in their number system]
The exact answer is supposed to be
√2.
(a) Use your calculator to work out the percentage error in the Babylonians’ calculation.
(b) Comment on the accuracy of the Babylonians’ approximation to √2.
END OF PAPER