Name
Class
………………………………………………………………
………………………………………………………………
TIME ALLOWED
1 hour 30 minutes
INSTRUCTIONS TO CANDIDATES
 Answer all the questions.
 Read each question carefully. Make sure you know what you have to
do before starting your answer.
 You are permitted to use a calculator in this paper.
 Do all rough work in this book.
INFORMATION FOR CANDIDATES
 The number of marks is given in brackets [ ] at the end of each
question or part question on the Question Paper.
 You are reminded of the need for clear presentation in your
answers.
 The total number of marks for this paper is 80.
© The PiXL Club Limited 2017
This resource is strictly for the use of member schools for as long as they remain members of The
PiXL Club. It may not be copied, sold nor transferred to a third party or used by the school after
membership ceases. Until such time it may be freely used within the member school. All opinions
and contributions are those of the authors. The contents of this resource are not connected with nor
endorsed by any other company, organisation or institution.
1
Out of
Mark
Question
Pre Public Examination
GCSE Mathematics (Edexcel style)
March 2017
Higher Tier
Paper 3H
Worked Solutions
1
3
2
4
3
2
4
4
5
2
6
3
7
3
8
4
9
6
10
2
11
2
12
2
13
6
14
4
15
3
16
2
17
5
18
3
19
4
20
5
21
7
22
4
Total
80
Answer ALL questions.
Write your answers in the spaces provided.
You must write down all the stages in your working.
Question 1.
Trains to Slough leave Reading Station every 26 minutes.
Trains to Southall leave the same train station every 10 minutes.
A train to Slough and a train to Southall both leave the train station at 9:30 a.m.
When will a train to Slough and a train to Southall next leave the train station at the same time?
26, 52, 78, 104, 130
10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, 130
P1
9.30 + 2hrs 10mins
130 minutes = 2 hour and 10 minutes
A1
11:40am
A1
(Total 3 marks)
Question 2.
Timothy, Jane and Sarah share £275.
The ratio of the amount of money Timothy gets to the amount of money Sarah gets is 1 : 5
Timothy gets £96 less than Sarah gets.
What percentage of the £275 does Jane get?
Give your answer to 3sf.
5–1=4
4 shares = £96
P1
1 share = 96 ÷ 4 = £24
5 shares = 24 x 5 = £120
M1
275 – (24 + 120) = 131
M1
(131/275) x 100
47.6% A1
(Total 4 marks)
2
Question 3.
Gurdeep thinks that (y + 7)2 = y2 + 49 for all values of y.
Is Gurdeep right?
You must show how you get your answer.
(y + 7) (y + 7) = y2 + 7y + 7y + 49
M1
y2 + 14y + 49
C1
No Gurdeep is not right
(Total 2 marks)
Question 4.
The table gives information about the speeds of 85 motorbikes on a road.
Speed (s) (km/h)
Frequency
30  s < 40
9
35
9x35=315
40  s < 50
25
45
25x25=1125
50  s < 60
34
55
34x55=1870
60  s < 70
17
65
17x65=1105
M1
M1
Work out an estimate for the mean speed.
315 + 1125 + 1870 + 1105 = 4415
4415 ÷85
M1
51.9km/h
A1
(Total 4 marks)
3
Question 5.
The Rail Company increased all prices of train tickets by 2 %.
The cost of a return ticket from Birmingham to Banbury increased by £1.20.
Work out the price of the ticket before the increase.
2% = £1.20
M1
100% = 1.2 x 50
£60 A1
(Total 2 marks)
Question 6.
(a) Write 476 00000 in standard form.
4.76 x 107
B1
(1)
(b) Work out (2.6 × 10−5) × (3.71 × 10−2)
Give your answer as an ordinary number correct to 3 significant figures.
9.646 x 10-7
M1
= 0.0000009646
0.000000965 A1
(2)
(Total 3 marks)
4
Question 7.
A, B, C and D are four vertices of a regular hexagon.
Angle BCY = 90°.
Work out the size of angle DCY.
You must show how you get your answer.
Sum of interior angle of a hexagon (6 – 2) × 180 = 720
One interior angle = 720 ÷ 6 = 120 P1
Angle BCD = 120, angle BCY = 90.
Angles around a point is 360
Angle DCY = 360 – (120 + 90) M1
150° A1
(Total 3 marks)
5
Question 8.
There are 5 banana smoothies and 4 apple smoothies in a box.
Mostafa takes ‘at random’ one smoothie from the box.
He writes down its flavour, and puts it back in the box.
Mostafa then takes ‘at random’ a second smoothie from the box.
(a) Complete the probability tree diagram.
B2
(2)
(b) Work out the probability that both smoothies are apple flavour.
𝟒
𝟗
𝟒
× = M1
𝟗
𝟏𝟔
𝟖𝟏
A1
(2)
(Total 4 marks)
6
Question 9.
The diagram shows the floor plan of Olivia’s dining room.
Olivia is going to cover the floor with wooden floorboards.
The floorboards are sold in packs.
A pack of floorboards costs £14.60 and will cover 2.25 m2 of floor.
(a) Work out the cost of buying enough floorboards to cover the dining room.
Area of rectangle A = 4.5 × 2 = 9m2
P1
Area of rectangle B = 3 × 2 = 6m2
Total area = 9 + 6 = 15m2
𝟏𝟓
𝟐.𝟐𝟓
= 𝟔. 𝟔̇
M1
M1
So Olivia will need 7 packs
Cost of 7 packs: 14.60 × 7 =
M1
£102.20 A1
(5)
Olivia finds out that a pack of floorboards will cover more than 2.25m2 of floor.
(b) Explain how this might affect the number of packs she needs to buy.
She might need less than 7 packs C1
(1)
(Total 6 marks)
7
Question 10.
ABC is a right-angled triangle.
Work out the size of angle BAC.
Give your answer correct to 3 significant figures.
Tan-1(8÷7)
M1
48.8° A1
(Total 2 marks)
8
Question 11.
Here are six graphs.
Write down the letter of the graph that could have the equation
(i) y = x2 - 4
E B1
(ii) y = 3x
C B1
(Total 2 marks)
Question 12.
Simplify
7y4w × 2y5w7
7 × 2 = 14 y4 × y5 = y9 w × w7 = w8
M1
14y9w8 A1
(Total 2 marks)
9
Question 13.
Martha recorded the times in seconds, some girls and boys took to find a word in the dictionary.
The box plot below shows the distribution of the times it took the girls to find the word in the dictionary.
(a) Work out the interquartile range.
Upper Quartile =35, Lower quartile = 12
Interquartile range = 35 – 12 = 23
23 seconds B1
(1)
Here are the times, in seconds, that 15 boys took to find the same word in the dictionary.
5 9
11
14
15
20
22
25
27
27
28
30
32
35
44
(b) On the grid below, draw a box plot for this information.
𝟏𝟓 + 𝟏
𝟏𝟓 + 𝟏
𝟏𝟓 + 𝟏
Median is the 𝟐 value = 8th value. LQ is the 𝟒 value = 4th value. UQ is the 𝟑 (
Median = 25, Lower Quartile = 14 and Upper Quartile = 30
𝟐
) = 12th value
B1 at least 2 correctly plotted values including box or whiskers/tails or 5 correct values and no
whiskers/tails
B1 at least 2 correctly plotted values including box and whiskers/tails
B1 fully correct box plot
(3)
(c) Compare the distributions of the girls' times and the boys' times.
The girls had a smaller median time of 21 seconds (boys 25 seconds). The boys had a smaller
interquartile range of 16 compared to the girls 23 seconds. The smaller interquartile range means that
the boys had a more consistent timing of finding the word in the dictionary. C2
(2)
(Total 6 marks)
10
Question 14.
The diagram shows a piece of wood cut in the shape of a triangular prism.
The piece of wood is 5 cm by 12 cm by 2.3 m.
The mass of the piece of wood is 6.5 kg.
The piece of wood will float in sea water if the density of the wood is less than the density of the sea water.
In a large pool, 1 litre of sea water has a mass of 980 g.
Will the piece of wood float in this pool?
You must show how you get your answer.
2.3 m = 230 cm
1 litre = 1000 cm3
6.5 kg = 6500 g
Volume of the piece of wood =
𝒎𝒂𝒔𝒔
𝟓 ×𝟏𝟐
Density of the wood = 𝒗𝒐𝒍𝒖𝒎𝒆 =
Density of sea water =
𝟗𝟖𝟎
𝟏𝟎𝟎𝟎
𝟐
× 230 = 6900 cm3
𝟔𝟓𝟎𝟎
𝟔𝟗𝟎𝟎
= 0.942 g/cm3
= 0.98 g/cm3
P1
M1
M1
Yes it will float C1
(Total 4 marks)
11
Question 15.
David invested £2800 in a savings account.
He was paid x % per annum compound interest.
The interest is paid into the account at the end of each year.
At the end of 3 years, the amount of money in the savings account is £3241.35.
Work out the value of x.
2800 × r3 = 3241.35 M1
r3 =
𝟑𝟐𝟒𝟏.𝟑𝟓
𝟑
𝟐𝟖𝟎𝟎
r= √
𝟑𝟐𝟒𝟏.𝟑𝟓
𝟐𝟖𝟎𝟎
M1
r = 1.05
1.05 – 1 = 0.05
So x = 5% A1
(Total 3 marks)
Question 16.
The diagram shows two similar solids, A and B.
The total surface area of the solid B is 240 cm2.
Work out the total surface area of solid A.
Linear scale factor = 8 ÷ 4 = 2
Area scale factor = 22 = 4 M1
Total surface area of solid A is 240 ÷ 4
60 cm2 A1
(Total 2 marks)
12
Question 17.
𝑛=
r = 3.74
s = 8.123
√𝑟
𝑠
correct to 2 decimal places.
correct to 3 decimal places.
By considering bounds, work out the value of n to a suitable degree of accuracy.
Give a reason for your answer.
Upper Bound of r = 3.745
Lower Bound of r = 3.735
Upper Bound of s = 8.1235
Lower Bound of s = 8.1225 P1
√𝟑.𝟕𝟒𝟓
Upper Bound of n =
𝟖.𝟏𝟐𝟐𝟓
= 0.23825
M1
√𝟑.𝟕𝟑𝟓
Lower Bound of n =
= 0.23790
𝟖.𝟏𝟐𝟑𝟓
M1
0.23825 and 0.23790 seen A1
Both upper and lower bound round to 0.238 C1
(Total 5 marks)
13
Question 18.
The diagram shows a solid metal cylinder.
The cylinder has base radius 2x cm and height h cm.
The metal cylinder is melted.
All the metal is then used to make 162 spheres.
Each sphere has a radius of
1
3
𝑥 cm.
Find an expression, in its simplest form, for h in terms of x.
Volume of cylinder = 162 × volume of sphere
𝟒
𝟏
𝝅 × (𝟐𝒙)𝟐 × 𝒉 = 162 × 𝟑 × 𝝅 × (𝟑 𝒙)𝟑 P1
𝟒
𝟑
𝟏
𝟑
𝟏𝟔𝟐 × × 𝝅 × ( 𝒙)𝟑
h=
𝝅× (𝟐𝒙)𝟐
𝟒
𝟑
𝟏𝟔𝟐 × ×
h=
h=
𝟒𝒙𝟐
𝟏 𝟑
𝒙
𝟐𝟕
M1
𝟖𝒙𝟑
𝟒𝒙𝟐
h = 2x A1
(Total 3 marks)
14
Question 19.
Make n the subject of
𝑔=
g(3 + n) = 6 – 7n
6 − 7𝑛
3+𝑛
M1
3g + gn = 6 – 7n
gn + 7n = 6 – 3g M1
n(g + 7) = 6 – 3g M1
n=
𝟔 − 𝟑𝒈
𝒈+𝟕
A1
(Total 4 marks)
15
Question 20.
The diagram shows a pyramid with base ABC.
CD is perpendicular to both CA and CB.
Angle CBD = 39°
BC = 18 cm.
Angle ADB = 45°
Angle DBA = 55°
Calculate the size of the angle between the line AD and the plane ABC.
Give your answer correct to 1 decimal place.
The angle AD makes with the plane ABC is angle DAC
𝑪𝑫
First find CD using tan:
tan 39 = 𝟏𝟖 P1
Then find BD using cos:
cos 39 = 𝑩𝑫
Use sine rule to find AD;
𝑨𝑫
𝐬𝐢𝐧 𝑩
𝑩𝑫
= 𝐬𝐢𝐧 𝑨
AD =
𝟐𝟑.𝟏𝟔𝟐
𝐬𝐢𝐧 𝟖𝟎
𝑨𝑫
𝐬𝐢𝐧 𝟓𝟓
𝟏𝟖
CD = tan 39 × 18 = 14.576
BD = 18 ÷ cos 39 = 23.162 M1
Angle BAD (angle A) = 180 – (55 + 45) = 80
=
𝟐𝟑.𝟏𝟔𝟐
𝐬𝐢𝐧 𝟖𝟎
M1
× 𝐬𝐢𝐧 𝟓𝟓 = 19.26589…
𝑪𝑫
𝟏𝟒.𝟓𝟕𝟔
Now use sine to find angle DAC: sin DAC = 𝑨𝑫 = 𝟏𝟗.𝟐𝟔𝟕 M1
𝟏𝟒.𝟓𝟕𝟔
Angle DAC = 𝐬𝐢𝐧−𝟏 ( 𝟏𝟗.𝟐𝟔𝟕) = 49.1589
49.2° A1
(Total 5 marks)
16
Question 21.
For all values of x
f(x) = 2x + 5 and
(a) Find
g(x) = x2 − 25
g(−6)
(−6)2 – 25
36 – 25 = 11
11 B1
(1)
gf(x) = 4x2 + 20x
(b) Show that
(2x + 5)2 – 25 B1
(2x + 5)(2x + 5) – 25
4x2 + 10x +10x + 25 – 25
4x2 + 20x C1
(2)
(c) Solve
gf(x) = g(7)
4x2 + 20x = (7)2 – 25 P1
4x2 + 20x = 49 – 25
4x2 + 20x = 24
4x2 + 20x – 24 = 0 M1
Divide all through by 4, factorise and solve
x2 + 5x – 6 = 0 M1
(x + 6)( x – 1) = 0
x = – 6 or x = 1
A1
(4)
(Total 7 marks)
17
Question 22.
P, Q and R are points on the circumference of a circle centre O.
Prove that angle QOR is twice the size of angle QPR stating your reasons.
First draw a radius from the centre to point P at the circumfernce.
This makes 2 isoseles triangles. P1
Angle y =
𝟏𝟖𝟎−𝒛
𝟐
Angle w =
y+w=
𝟏𝟖𝟎−𝒙
𝟐
𝟏𝟖𝟎−𝒛
𝟐
+
𝒛
M1
𝟏𝟖𝟎−𝒙
𝟐
=
𝟑𝟔𝟎− 𝒛 − 𝒙
𝟐
𝒙
y + w = 180 - 𝟐 - 𝟐
a + x + z = 360
a = 360 – x – z M1
Angle a (QOR) is twice the size of angle y + w (QPR) C1
(Total 4 marks)
TOTAL FOR PAPER: 80 MARKS
18