Edexcel (I)GCSE
Mathematics
Higher Tier
Geometry – Trigonometry and Pythagoras’
Theorem
Total Marks: 144
You must have:
Ruler
Calculator
Instructions:
Use black ink or ball-point pen.
Answer All questions.
Answer the questions in the spaces provided there may be more space than you need
Show all the steps in any calculations and state the units.
Information:
The total mark for this paper is 144
The marks for each question are shown in brackets use this as a guide as to how much time to spend
on each question.
Advice:
Read each question carefully before you start to answer it.
Keep an eye on the time.
Write your answers neatly and in good English.
Try to answer every question.
Check your answers if you have time at the end.
1
18
C
Diagram NOT
accurately drawn
47 m
51°
32°
A
D
B
Triangle ABC is right-angled at B.
Angle BAC = 32°
AC = 47 m.
D is the point on AB such that angle BDC = 51°
Calculate the length of BD.
Give your answer correct to 3 significant figures.
....... . . . . . . . . . . . . . . . . . . . . . . . . .
1 is 5 marks)
(Total for Question 18
m
11
2
L
Diagram NOT
accurately drawn
9.3cm
42º
M
N
Calculate the length of LM.
Give your answer correct to 3 significant figures.
............................... . . . . . . . . . . . . . . . . . . . . . .
cm
(Total for Question 11
2 is 3 marks)
12 (i) Solve the inequality
2x + 13 . 6
.............................. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(ii) n is a negative integer.
Write down all the values of n which satisfy 2n + 13 . 6
.............................. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(Total for Question 12 is 4 marks)
22
4
Diagram NOT
accurately drawn
H
G
E
F
3cm
A
D
C
7cm
5cm
B
The diagram shows a cuboid ABCDEFGH.
AB = 5cm
BC = 7cm
AE = 3cm
(a) Calculate the length of AG.
Give your answer correct to 3 significant figures.
........................................ . . . . . . . . . . . . . . . . . . . . . .
cm
(3)
(b) Calculate the size of the angle between AG and the plane ABCD.
Give your answer correct to 1 decimal place.
................................... . . . . . . . . . . . . . . . . . . . . . . . . . . .
(2)
(Total for Question 22
4 is 5 marks)
ƕ
9
Diagram NOT
accurately drawn
Q
5
3.4 cm
P
R
5.8 cm
Triangle PQR has a right angle at Q.
PQ = 3.4 cm and PR = 5.8 cm.
(a) Work out the size of angle QRP.
Give your answer correct to 1 decimal place.
.................................... . . . . . . . . . . . . . . . . . . . . . . . . . .
°
(3)
The length 5.8 cm, of PR, is correct to 2 significant figures.
(b) (i) Write down the upper bound of the length of PR.
......................................... . . . . . . . . . . . . . . . . . . . . .
cm
........................................ . . . . . . . . . . . . . . . . . . . . . .
cm
(ii) Write down the lower bound of the length of PR.
(2)
(Total for Question 95 is 5 marks)
18
6
8 cm
Diagram NOT
accurately drawn
4 cm
x°
6 cm
Calculate the value of x.
Give your answer correct to 1 decimal place.
x = ...........................................
6 is 3 marks)
(Total for Question 18
Do NOT write in this space.
47
x cm
3.7 cm
Diagram NOT
accurately drawn
5.6 cm
Work out the value of x.
Give your answer correct to 3 significant figures.
x = ............................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(Total for Question 74 is 3 marks)
5 Three positive whole numbers have a mean of 4 and a range of 7
Find the three positive whole numbers.
...............................
...............................
...............................
(Total for Question 5 is 2 marks)
8
C
8
38°
7.9 cm
ABC is a triangle.
AC"#
Angle B = 90°
Angle C = 38°
Diagram NOT
accurately drawn
A
B
(a) Calculate the length of BC.
Give your answer correct to 3 significant figures.
..............................................................
cm
(3)
(b) The size of angle C is 38°, correct to 2 significant figures.
(i) Write down the lower bound of the size of angle C.
°
..............................................................
(ii) Write down the upper bound of the size of angle C.
°
..............................................................
(2)
(Total for Question 88 is 5 marks)
11
9 A washing line is attached at points A and B on two vertical posts standing on horizontal
ground.
Point A is 2.1 metres above the ground on one post.
Point B$#&!
The horizontal distance between the two posts is 6 metres.
Diagram NOT
accurately drawn
A
B
2.1 m
1.7 m
6m
Calculate the distance AB.
Give your answer correct to 3 significant figures.
...................................... . . . . . . . . . . . . . . . . . . . . . . . .
m
9 is 4 marks)
(Total for Question 11
12 Make h the subject of the formula A = 2π r (r + h)
h = .............................. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(Total for Question 12 is 2 marks)
1610
9 cm
7 cm
Diagram NOT
accurately drawn
x°
13 cm
Calculate the value of x.
Give your answer correct to 1 decimal place.
x = .............................. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(Total for Question 10
16 is 3 marks)
4 x 2 − 25
17 Simplify fully
6 x 2 + 13 x − 5
............................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(Total for Question 17 is 3 marks)
21
11 The diagram shows a pyramid with a horizontal rectangular base PQRS.
PQ = 16 cm.
QR = 10 cm.
M is the midpoint of the line PR.
The vertex, T, is vertically above M.
MT = 15cm.
Diagram NOT
accurately drawn
T
15 cm
S
R
10 cm
M
P
Q
16 cm
Calculate the size of the angle between TP and the base PQRS.
Give your answer correct to 1 decimal place.
°
................................... . . . . . . . . . . . . . . . . . . . . . . . . . . .
11 is 4 marks)
(Total for Question 21
TOTAL FOR PAPER IS 100 MARKS
912
Diagram NOT
accurately drawn
8.6 cm
x cm
6.9 cm
Work out the value of x.
Give your answer correct to 3 significant figures.
x = ................................
(Total for Question 12
9 is 3 marks)
10 Solve 3x
x
Show clear algebraic working.
x = ................................
(Total for Question 10 is 3 marks)
11 Jack, Kate and Lila share some money in the ratios 5 : 9 : 6
In total, Jack and Kate receive £56
Work out the amount of money Lila receives.
£ ...............................
(Total for Question 11 is 3 marks)
13
25
A
16.5 m
Diagram NOT
accurately drawn
D
10°
59°
B
C
The diagram shows a vertical flagpole in Chennai, India.
The point A is at the top of the flagpole.
The point B is at the foot of the flagpole.
There is a platform at the point D on the flagpole.
B and C are points on horizontal ground.
AD = 16.5 m
The angle of elevation of A from C is 69°
The angle of elevation of D from C is 59°
Calculate the height, AB, of the flagpole.
Give your answer correct to 3 significant figures.
..................................... . . . . . . . . . . . . . . . . . . . . . . . . .
13 is 6 marks)
(Total for Question 25
Turn over for Question 26.
m
Diagram NOT
accurately drawn
1014
B'
x°
8.3 cm
Work out the value of x.
Give your answer correct to 1 decimal place.
x =........ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14 is 3 marks)
(Total for Question 10
11 ?"@''?
@?"B>
........ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(2)
5 ?";''<$
;<"B>
........ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(2)
(Total for Question 11 is 4 marks)
18
15
N
S
2.9 km
62°
Diagram NOT
accurately drawn
P
36°
Q
P and Q are two points on a coast.
P is due North of Q.
A ship is at the point S.
PS = 2.9 km.
The bearing of the ship from P is 062°
The bearing of the ship from Q is 036°
Calculate the distance QS.
Give your answer correct to 3 significant figures.
................... . . . . . . . . . . . . . . . . . . . . . . . .
(Total for Question 15
18 is 3 marks)
km
1116
6.8 m
Diagram NOT
accurately drawn
xm
52°
Calculate the value of x.
Give your answer correct to 3 significant figures.
x = ........................ . .. . .. . .. . .. . .. . . .. . .. . .. . .. . .
(Total for Question 16
11 is 3 marks)
12 (a) Write as an ordinary number
(i) 4.2 × 106
......................... . .. . .. . .. . .. . . .. . .. . .. . .. . . ..
(ii) 3.82 × 10–4
........................ . .. . . .. . .. . .. . .. . .. . . .. . .. . .. .
(2)
(b) Here are three numbers written in standard form.
Arrange these numbers in order of size.
Start with the smallest number.
5.6 × 10–7
8.6 × 10–9
5.64 × 10–8
............................................................................................................ . .. . .. . .. . .. . . .. . .. . .. . .. . . ..
(2)
(Total for Question 12 is 4 marks)
17
24
The diagram shows a cube ABCDEFGH.
The sides of the cube are of length 5 cm.
Calculate the size of the angle between the diagonal AH and the base EFGH.
Give your answer correct to 1 decimal place.
A
B
Diagram NOT
accurately drawn
D
C
5 cm
F
G
5 cm
E
5 cm
H
.......................... .. . . .. . .. . .. . .. . .. . . .. . .. . ..
(Total for Question 17
24 is 4 marks)
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FP
7
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FP
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0
$
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*LYH \RXU DQVZHU FRUUHFW WR GHFLPDO SODFH
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q
16
20
x cm
5.2 cm
Diagram NOT
accurately drawn
117q
6.7 cm
Calculate the value of x.
Give your answer correct to 3 significant figures.
x = ..............................
(Total for Question 20
16 is 3 marks)
17 y is directly proportional to x3
When x = 10, y = 250
(a) Find a formula for y in terms of x.
........................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(3)
(b) Calculate the value of x when y = 54
x = ..............................
(2)
(Total for Question 17 is 5 marks)
722
Diagram NOT
accurately drawn
48.6 cm
64.8 cm
A TV screen is rectangular.
The width of the rectangle is 64.8 cm and the height is 48.6 cm.
The length of a diagonal of the rectangle gives the ‘size’ of the TV screen.
(a) Calculate the ‘size’ of the TV screen.
..... . . .. . .. . .. . .. . .. . . .. . .
cm
(3)
4
m
8c
c
102
m
Diagram NOT
accurately drawn
38.4 cm
The diagram shows two rectangular TV screens.
The rectangles are similar.
The ‘size’ of the smaller screen is 48 cm.
The width of the smaller screen is 38.4 cm.
The ‘size’ of the larger screen is 102 cm.
(b) Calculate the width of the larger TV screen.
..... . . .. . .. . .. . .. . .. . . .. . .
(2)
(Total for Question22
7 is 5 marks)
cm
17
23
C
Diagram NOT
accurately drawn
8.3 cm
A
4.7 cm
7.5 cm
B
D
The diagram shows triangle ABC.
D is the point on AB, such that CD is perpendicular to AB.
AC = 8.3 cm.
AD = 4.7 cm.
BD = 7.5 cm.
Calculate the size of angle ABC.
Give your answer correct to 1 decimal place.
°
. .. . .. . . .. . .. . .. . .. . . .. . .. .
(Total for Question23
17 is 4 marks)
17
24 A circular clock face, centre O, has a minute hand OA and an hour hand OB.
OA = 10 cm.
OB = 7 cm.
Calculate the length of AB when the hands show 5 o’clock.
Give your answer correct to 3 significant figures.
Diagram NOT
accurately drawn
A
10 cm
O
7 cm
B
...... .. . .. . . .. . .. . .. . .. . . ..
(Total for Question 24
17 is 4 marks)
cm
25 A pyramid has a horizontal square base ABCD with sides of length 230 metres.
23
M is the midpoint of AC.
The vertex, T, is vertically above M.
The slant edges of the pyramid are of length 218 metres.
Calculate the height, MT, of the pyramid.
Give your answer correct to 3 significant figures.
T
Diagram NOT
accurately drawn
218 m
D
C
230 m
M
A
230 m
B
..... .. . . .. . .. . .. . .. . .. . . .. .
(Total for Question 25
23 is 5 marks)
TOTAL FOR PAPER IS 100 MARKS
m
22
27 The diagram shows a metal plate.
B
Diagram NOT
accurately drawn
65°
A
8 cm
35°
O
C
The metal plate is made from a sector OAB of a circle, centre O, and a triangle OCB.
Angle AOB = 65° Angle OCB = 35°
OA = OB = 8 cm.
AOC is a straight line.
(a) Calculate the length of BC.
Give your answer correct to 3 significant figures.
............... . . . . . . . . . . . . . . . . . . . . . . .
cm
(3)
(b) Calculate the total area of the metal plate.
Give your answer correct to 3 significant figures.
................. . . . . . . . . . . . . . . . . . . . . .
(4)
27 is 7 marks)
(Total for Question 22
cm2
18
E
28
H
F
16 cm
Diagram NOT
accurately drawn
D
G
6 cm
C
A
8 cm
B
The diagram shows a cuboid ABCDEFGH.
AB = 8 cm, AF = 6 cm and FC = 16 cm.
(a) Find the length of BC.
Give your answer correct to 3 significant figures.
BC = ................... . .. . .. . . .. . .. . .. . .. . . . cm
(3)
(b) Find the size of the angle between the line FC and the plane ABGF.
Give your answer correct to 1 decimal place.
............. .. . .. . .. . .. . . .. . .. . .. . .. . . ..
(2)
(Total for Question28
18 is 5 marks)
°
23
30
E
Diagram NOT
accurately drawn
12 cm
A
F
D
9 cm
M
B
15 cm
C
ABCDEF is a triangular prism.
AB = 9 cm, BC = 15 cm and AE = 12 cm.
Angle ABC = 90°
M is the midpoint of CD.
Calculate the size of the angle between AM and the plane BCDF.
Give your answer correct to 1 decimal place.
. . .. . .. . .. . . .. . .. . .. . .. . . .. .
30 is 5 marks)
(Total for Question 23
°
19
31
Diagram NOT
DFFXUDWHO\GUDZQ
North
ƒ
A
6.4 km
B
3.8 km
C
ABDQGC are 3 villages.
BLVNPGXHHDVWRIA.
C is 3.8 km from ARQDEHDULQJRIƒ
Calculate the bearing of B from C.
*LYH\RXUDQVZHUFRUUHFWWRWKHQHDUHVWGHJUHH
Show your working clearly.
. .. . .. . .. . . .. . .. . .. . .. . .. . . .
(Total for Question 31
19 is 6 marks)
°
8
32
B
Diagram NOT
accurately drawn
15 cm
10 cm
A
D
12.5 cm
C
ABC is a triangle.
The point D lies on AC.
Angle BDC = 90q
BD = 10 cm, AB = 15 cm and DC = 12.5 cm.
(a) Calculate the length of AD.
Give your answer correct to 3 significant figures.
......................... . .. . .. . . .. . .. . .. . .. . .
cm
(3)
(b) Calculate the size of angle BCD.
Give your answer correct to 1 decimal place.
.................. . . .. . . .. . .. . .. . .. . . .. . .. . .. .
(3)
(Total for Question 32
8 is 6 marks)
q
22
33
C
Diagram NOT
accurately drawn
2 7 cm
(2x – 1) cm
60q
A
(2x + 1) cm
B
The diagram shows a triangle ABC.
AB = (2x + 1) cm, AC = (2x – 1) cm and BC = 2 7 cm.
Angle BAC = 60q
Work out the value of x.
Show clear algebraic working.
x = .............. .. . .. . . .. . .. . .. . .. . . .. . .. . .. . ..
(Total for Question 33
22 is 3 marks)
TOTAL FOR PAPER IS 100 MARKS
11
34
North
B
Diagram NOT
accurately drawn
80 km
North
A
35 km
Town B is 35 km east and 80 km north of town A.
Work out the bearing of A from B.
Give your answer correct to the nearest degree.
............. . .. . .. . .. . .. . . .. . .. . .. . .. . . .
(Total for Question 34
11 is 4 marks)
Do NOT write in this space.
°
35 Here is triangle LMN, where angle LMN is an obtuse angle.
23
M
8.5 cm
47°
13.8 cm
N
Diagram NOT
accurately drawn
L
ML 13.8 cm
MN 8.5 cm
Angle MNL 47°
Work out the area of triangle LMN.
Give your answer correct to 3 significant figures.
.................... . . .. . .. . .. . .. . . .. . .. .
(Total for Question 35
23 is 6 marks)
cm2