The Haberdashers' Aske's Boys' School 2017 Paper 1

IndependentBoys

30 questions

Original PDFWorked solutions in progress

Questions

1
Add: $88 + 37$
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2
Subtract: $96 - 47$
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3
Multiply: $58 \times 7$
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4
Divide: $78 \div 6$
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5
Write down two numbers that add to $14$ and multiply to $48$ .
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6
You are given the number $2864$ .

You are allowed to swap the position of any two digits.

For example, swapping the positions of $8$ and $6$ in $2864$ gives $2684$ .

Or, swapping the positions of $2$ and $4$ gives $4862$ .
1. 6.a
  What is the largest possible number you can make using one swap?
2. 6.b
  What is the smallest possible number you can make using one swap?
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7
Work out the following: $21 - 5 \times 2 + 6$
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8
What number is:
1. 8.a
  Six less than $-10$
2. 8.b
  Twelve more than $-8$
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9
Alison, Bethany and Catherine are three sisters. They are $6$ years old, $7$ years old and $12$ years old.

Bethany is older than Alison. Catherine's age is a prime number.

What is each girl's age?
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10
Draw the hour and minute hands on this clock to show the time $22 \rtqMathsTimeSeparator 30$ .
An analogue clock has numerals 1 to 12, minute ticks and a central hub. No hands are shown so the learner can draw 22:30.
What is the reflex angle between the two hands on this clock?
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11
The sums below were correct before someone rubbed out the brackets. Write down the correct sums, including the brackets.
1. 11.a
  $8 - 5 + 2 = 1$
2. 11.b
  $12 - 11 + 2 - 1 = 0$
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12
Geoff counted the number of lorries he saw on his journey to school each day.

The results for Monday, Tuesday, Wednesday and Thursday are shown in the pictogram.

Monday
Tuesday
Wednesday
Thursday
Friday

Key: Four lorries
1. 12.a
  How many lorries did Geoff see on his journey on Tuesday?
2. 12.b
  How many lorries did Geoff see on his journey on Wednesday?
3. 12.c
  On Friday Geoff saw six lorries on his journey into school.
  
  Complete the pictogram for Friday.
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13
Here is a map of Secret Island.
A signed square coordinate grid runs from negative 9 to positive 9 with one-unit ticks and grid lines. An irregular unfilled outline labelled Secret Island surrounds the origin, which is marked with an open circle. Lighthouse is at (-5, 7), Sunken treasure at (7, 8), Pirate ship at (-6, -8), and Whirlpool at (8, -5).
1. 13.a
  What are the co-ordinates of the Sunken Treasure?
2. 13.b
  What are the co-ordinates of the Whirlpool?
3. 13.c
  There is a crocodile at $(-7, 6)$ .
  
  Mark the crocodile on the map. Label the crocodile with the word “CROCODILE”.
4. 13.d
  Which of the features shown on the map is the crocodile closest to?
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14
What is the name of each of these three regular polygons?
1. 14.a
  A plain five-sided regular polygon is shown for identification.
2. 14.b
  A plain six-sided regular polygon is shown for identification.
3. 14.c
  A plain nine-sided regular polygon is shown for identification.
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15
From Year $10$ all pupils must join the Combined Cadet Force (“CCF”) or opt for School Community Service (“SCS”). Pupils can not opt for both CCF and SCS.

Rachel asked $150$ Year $9$ pupils whether they wanted to opt for CCF or SCS next year.

$60\%$ of the pupils said that they wanted to opt for CCF.

$\dfrac{1}{6}$ of the pupils said that they wanted to opt for SCS.

The rest of the pupils said that they hadn't yet decided.

How many pupils hadn't yet decided whether to do CCF or carry out SCS next year?
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16
Sam thinks of a number.

He multiplies that number by $5$ .

Then he subtracts $12$ .

Then he divides by $3$ .

Finally he adds $17$ .

His answer is $38$ .

What number did Sam originally think of?
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17
A bus can carry $52$ passengers.

How many buses will be needed to transport $993$ people to a sports day?
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18
In the American state of Kentucky sales tax is charged at $6\%$ .
1. 18.a
  Anila wants to buy a pair of jeans in Kentucky.
  
  They are priced at $\text{\rtqMathsSymbolDollar}72$ before allowing for the sales tax.
  
  How much does Anila pay for the pair of jeans?
2. 18.b
  Peter buys a bag of sweets in Kentucky.
  
  He pays $\text{\rtqMathsSymbolDollar}4.77$ including sales tax.
  
  What was the price of sweets before sales tax was added?
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19
I put the individual twelve letters of the word “HABERDASHERS” into an empty bag. I draw out a letter at random from the bag.

What is the probability that I draw out:
1. 19.a
  The letter “B”?
2. 19.b
  The letter “T”?
3. 19.c
  A vowel?
4. 19.d
  A letter that is also in the word “ASKES”?
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20
The sails of a windmill complete one full turn every $40 \rtqMathsUnit{seconds}$ .
1. 20.a
  How long does it take the sails to turn through a right angle?
2. 20.b
  How many turns do the sails make in fifty-six minutes?
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21
A class has thirty pupils.

Eight pupils are left-handed.

Sixteen pupils are girls.

Of the sixteen girls, thirteen are right handed.
1. 21.a
  Enter this information into the table below. Then complete the rest of the table.
  
  Left handed Right handed Total
  Boys
  Girls
  Total
2. 21.b
  How many boys in the class are right-handed?
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22
Stuart's patio is $5 \rtqMathsUnit{metres}$ long and $5 \rtqMathsUnit{metres}$ wide.

Stuart wants to cover the patio with paving stones. Each paving stone is $50 \rtqMathsUnit{centimetres}$ long and $50 \rtqMathsUnit{centimetres}$ wide.
1. 22.a
  How many paving stones does Stuart need to buy?
2. 22.b
  Stuart has $\rtqMathsSymbolPound 200$ left after buying the paving stones. Stuart wants to put a picket fence around three sides of his patio.
  
  Each panel of picket fencing is $1 \rtqMathsUnit{metre}$ long and costs $\rtqMathsSymbolPound 10.50$ . Stuart will also have to pay a delivery charge of $\rtqMathsSymbolPound 5$ regardless of the number of panels of picket fencing he purchases.
  
  How much money does Stuart have left after buying the picket fencing?
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23
Elizabeth writes down:

one multiple of $13$ , and

two different factors of $77$ .

Elizabeth adds up her three numbers. Her answer is greater than fifty but less than sixty.

What three numbers could Elizabeth have written down?
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24
1. 24.a
  A cube has side lengths of $5 \rtqMathsUnit{cm}$ .
  
  What is the volume of the cube?
2. 24.b
  A second cube has a volume of $27 \rtqMathsUnit{cm}^3$ .
  
  What is the total surface area of the second cube?
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25
1. 25.a
  Rinesh started painting his house two years ago. During that year, Rinesh painted one-third of his house.
  
  Last year, Rinesh painted another five-twelfths of his house.
  
  What fraction of his house does Rinesh need to paint this year, in order to finish completely painting his house?
2. 25.b
  Bijal started weeding her garden two weeks ago. During that week, she weeded one-fifth of her garden.
  
  Last week, Bijal weeded two and a half times as much of her garden as she weeded two weeks ago.
  
  What fraction of her garden does Bijal need to weed this week, in order to finish completely weeding her garden?
3. 25.c
  Nina is painting garden gnomes.
  
  She has four-fifths of a litre of paint.
  
  Each garden gnome needs one-twentieth of a litre of paint.
  
  How many garden gnomes can she paint?
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Monday
Tuesday
Wednesday
Thursday
Friday

Here is a portion of the Monday to Friday bus timetable for the $724$ bus between St. Albans and Heathrow.

	Hsch	HNsch	Hsch	HNsch	Hsch	HNsch	Hsch
St. Albans, Railway Station	$0555$	$0603$	$0641$	$0715$	$0745$	$0808$	$0909$
St. Albans, St Peter's Street	$0600$	$0608$	$0646$	$0721$	$0751$	$0814$	$0915$
Garston Bus Garage	$0625$	$0633$	$0716$	$0748$	$0831$	$0845$	$0944$
Watford Junction Railway Station	$0637$	$0645$	$0733$	$0803$	$0853$	$0901$	$1002$
Watford, Town Hall	$0640$	$0648$	$0737$	$0807$	$0857$	$0905$	$1006$
Rickmansworth Railway Station	$0650$	$0658$	$0754$	$0817$	$0909$	$0915$	$1016$
Denham, Station Parade	$0709$	$0713$	$0820$	$0832$	$0925$	$0930$	$1031$
Uxbridge, Belmont Road	$0717$	$0721$	$0833$	$0840$	$0936$	$0938$	$1039$
Heathrow Airport	$0737$	$0737$	$0857$	$0857$	$0954$	$0954$	$1055$

	Hsch	HNsch	Hsch	HNsch	Hsch	HNsch	Hsch
Heathrow Airport	$0610$	$0615$	$0700$	$0710$	$0755$	$0810$	$0915$
Uxbridge, Belmont Road	$0626$	$0631$	$0717$	$0727$	$0817$	$0827$	$0932$
Denham, Station Parade	$0634$	$0639$	$0726$	$0736$	$0828$	$0838$	$0941$
Rickmansworth Railway Station	$0650$	$0655$	$0743$	$0753$	$0850$	$0855$	$0957$
Watford, Clarendon Road	$0704$	$0709$	$0804$	$0814$	$0911$	$0911$	$1011$
Watford Junction Railway Station	$0707$	$0712$	$0807$	$0817$	$0914$	$0914$	$1014$
Garston Bus Garage	$0722$	$0727$	$0823$	$0833$	$0929$	$0929$	$1029$
St. Albans, St Peter's Street	$0745$	$0750$	$0850$	$0900$	$0952$	$0952$	$1052$
St. Albans, Railway Station	$0756$	$0801$	$0901$	$0911$	$1003$	$1003$	$1103$

Notes:

Hsch – Hertfordshire schooldays only
HNsch – Hertfordshire school holidays only

26.a
Andy caught a bus at Denham at $6 \rtqMathsTimeSeparator 39 \rtqMathsTimeAm$ .

What time does he arrive at Watford Junction Railway Station?
26.b
Ben wants to travel from Watford Town Hall to Uxbridge by bus.

If he catches a bus at Watford Town Hall at $8 \rtqMathsTimeSeparator 57 \rtqMathsTimeAm$ how long will his journey take?
26.c
Charlie lives in Uxbridge and works $10 \rtqMathsUnit{minutes}$ walk away from Garston Bus Garage. He needs to arrive at work by $9 \rtqMathsTimeSeparator 30 \rtqMathsTimeAm$ .

What time does he need to be at the bus stop in Uxbridge to get to work on time:
1. 26.c.i
  if it is a Hertfordshire school day?
2. 26.c.ii
  if it is a Hertfordshire school holiday?
26.d
Tom has arranged to meet Nic at St. Peters Street in St. Albans at $8 \rtqMathsTimeSeparator 30 \rtqMathsTimeAm$ .

Tom lives in Rickmansworth and is planning to travel by bus. Tom thinks that it is a Hertfordshire school holiday and arrives at the bus stop at Rickmansworth just in time to catch the bus. Unfortunately, it is a Hertfordshire school day.

How late does Tom arrive for his meeting with Nic?

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27
One angle of an isosceles triangle is $80 \degree$ .

What are the other angles?

There are two possible solutions to this question.
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28
1. 28.a
  What is the area of this triangle?
  The horizontal base is labelled 5 cm. A separate vertical double-headed arrow on the right spans the base and apex levels and is labelled 4 cm.
2. 28.b
  What is the area of this triangle?
  The upper vertex is marked as a right angle. The left and right sloping sides are labelled 6 cm and 8 cm, and the horizontal lower side 10 cm. A separate unlabelled vertical double-headed arrow on the right spans the base and apex levels.
3. 28.c
  By thinking of two different ways to work out the area of the triangle, calculate the length of arrow.
Worked solution
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29
Here is a map of the roads in Askeshire.

The route from Catsworth to Dogsville via Alysford is written “ $\text{C} \rtqMathsSpaceQuarterEm \rtqMathsSymbolRightArrow \rtqMathsSpaceQuarterEm \text{A} \rtqMathsSpaceQuarterEm \rtqMathsSymbolRightArrow \rtqMathsSpaceQuarterEm \text{D}$ ”.
Alysford A, Brickton B, Catsworth C, Dogsville D, Edgwood E, Foxton F, Goldsworth G and Habsville Airport H form a road network. The labelled roads are A-B 30 miles, A-C 50, A-D 40, B-D 20, B-E 50, C-G 40, D-E 20, D-F 20, D-G 10, E-F 10, E-H 30, F-G 30, F-H 50 and G-H 70 miles.
1. 29.a
  There has been a robbery in Alysford. The thieves are planning to go to Habsville Airport to flee the country with their loot.
  
  The police want to go from Alysford to Habsville Airport by the shortest possible route.
  
  What route should the police take?
2. 29.b
  The thieves also want to go from Alysford to Habsville Airport by the shortest possible route.
  
  However, the thieves also want to make sure that they don't travel on any of the roads being used by the police.
  
  What route should the thieves take?
Worked solution
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For each of the prime numbers in the table below:

Find the remainder when the prime number is divided by $4$ , and
Find whether the prime number can be expressed as the sum of two square numbers.

If the prime number can be expressed as the sum of two square numbers, state the two square numbers. If the prime number cannot be expressed as the sum of two square numbers state “Not possible”.

Complete the table below. The first two rows have been completed for you.

Prime number	The remainder when the prime number is divided by $4$	Can the prime number be expressed as the sum of two square numbers?
$3$	$3$	Not possible
$5$	$1$	$1^2 +2^2$
$7$
$11$
$13$
$17$

30.a
What is the link between whether a prime number can be expressed as the sum of two square numbers and the remainder when that prime number is divided by $4$ ? If you are not sure you can extend the table above and see what happens for other prime numbers.

30.b

The first proof that this pattern is true for all prime numbers was claimed by Pierre de Fermat and is called Fermat's Christmas Theorem (because Fermat's claim was dated December $25, 1640$ ).

Circle “YES” for each of the following prime numbers that can be expressed as the sum of two square numbers.

Circle “NO” for each of the following prime numbers that cannot be expressed as the sum of two square numbers.

You do not need to find the two square numbers.

Prime number	Can the prime number be expressed as the sum of two square numbers?
$58 \rtqMathsDigitGroupSeparator 031$	YES / NO
$58 \rtqMathsDigitGroupSeparator 043$	YES / NO
$58 \rtqMathsDigitGroupSeparator 049$	YES / NO
$58 \rtqMathsDigitGroupSeparator 057$	YES / NO

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Monday
Tuesday
Wednesday
Thursday
Friday

Monday

Tuesday

Wednesday

Thursday

Friday

Left handed

Right handed

Total

Boys

Girls

Total

Hsch

HNsch

Hsch

HNsch

Hsch

HNsch

Hsch

St. Albans, Railway Station

0555

0603

0641

0715

0745

0808

0909

St. Albans, St Peter's Street

0600

0608

0646

0721

0751

0814

0915

Garston Bus Garage

0625

0633

0716

0748

0831

0845

0944

Watford Junction Railway Station

0637

0645

0733

0803

0853

0901

1002

Watford, Town Hall

0640

0648

0737

0807

0857

0905

1006

Rickmansworth Railway Station

0650

0658

0754

0817

0909

0915

1016

Denham, Station Parade

0709

0713

0820

0832

0925

0930

1031

Uxbridge, Belmont Road

0717

0721

0833

0840

0936

0938

1039

Heathrow Airport

0737

0737

0857

0857

0954

0954

1055

Hsch

HNsch

Hsch

HNsch

Hsch

HNsch

Hsch

Heathrow Airport

0610

0615

0700

0710

0755

0810

0915

Uxbridge, Belmont Road

0626

0631

0717

0727

0817

0827

0932

Denham, Station Parade

0634

0639

0726

0736

0828

0838

0941

Rickmansworth Railway Station

0650

0655

0743

0753

0850

0855

0957

Watford, Clarendon Road

0704

0709

0804

0814

0911

0911

1011

Watford Junction Railway Station

0707

0712

0807

0817

0914

0914

1014

Garston Bus Garage

0722

0727

0823

0833

0929

0929

1029

St. Albans, St Peter's Street

0745

0750

0850

0900

0952

0952

1052

St. Albans, Railway Station

0756

0801

0901

0911

1003

1003

1103

Prime
number

The remainder when the
prime number is divided by

4

Can the prime number be expressed
as the sum of two square numbers?

3

3

Not possible

5

1

1^2 +2^2

7

11

13

17

Prime number

Can the prime number be expressed
as the sum of two square numbers?

58 \rtqMathsDigitGroupSeparator 031

YES / NO

58 \rtqMathsDigitGroupSeparator 043

YES / NO

58 \rtqMathsDigitGroupSeparator 049

YES / NO

58 \rtqMathsDigitGroupSeparator 057

YES / NO

The Haberdashers' Aske's Boys' School 2017 · Paper 1

Questions

The Haberdashers' Aske's Boys' School 2017 · Paper 1

Questions

The Haberdashers' Aske's Boys' School 2017 Paper 1

The Haberdashers' Aske's Boys' School 2017 Paper 1