For OCR
H
GENERAL CERTIFICATE OF SECONDARY EDUCATION
MATHEMATICS
Higher Paper 4E
Time: 2 hours
Candidates answer on the question paper.
Additional materials: Calculator
Geometrical instruments
Tracing paper (optional)
INSTRUCTIONS AND INFORMATION FOR CANDIDATES
•
•
•
•
•
Do not write on this question paper – use blank paper and the answer sheets provided.
Answer all the questions.
Read each question carefully.
Show all your working. Marks may be given for working even if you get the answer wrong.
The total number of marks for this paper is 100.
FORMULAE
Volume of prism = (area of cross-section) × length
1
Volume of cone = 3 πr2h
In any triangle ABC
Sine rule
a
b
c
=
=
sin A
sinB
sinC
Cosine rule a2 = b2 + c2 – 2bc cos A
1
Curved surface area of cone = πrl
The Quadratic Equation
The solutions of ax2 + bx + c = 0,
Area of triangle = 2 ab sin C
4
Volume of sphere = 3 πr3
where a ≠ 0, are given by
Surface area of sphere = 4πr2
x=
−b ± b −4 ac 
2a
2
Written by Shaun Armstrong
Only to be copied for use in the purchaser's school or college
OH4E short Page 1
© Churchill Maths Limited
1
Solder is an alloy made from tin and lead.
The ratio of the weight of tin to the weight of lead in solder is 5 : 3
Work out the weight of lead in 20 grams of solder.
[2]
2
R
Q
x°
P
y°
NOT TO SCALE
102°
41°
S
V
U
T
PQR and STUV are parallel lines.
(a)
Write down the value of x.
Give a reason for your answer.
[2]
(b)
Find the value of y.
[2]
3
The scatter graph on the answer sheets shows information about 8 companies.
For each company, it shows the percentage of employees doing unskilled jobs and the mean
weekly pay of all employees.
(a)
Describe the relationship between the mean weekly pay and the percentage of
employees doing unskilled jobs.
[1]
(b)
Draw a line of best fit on the diagram.
[1]
(c)
Use your line of best fit to estimate the mean weekly pay at another company where
59% of employees are doing unskilled jobs.
[1]
OH4E short Page 2
© Churchill Maths Limited
4
The diagram is a sketch of a prism.
The cross-section of the prism is a trapezium.
(a)
NOT TO SCALE
Sketch the plan of the prism.
[2]
(b)
Sketch a net for the prism.
[2]
5
This travel graph shows Dawn's journey when delivering a present to her sister's house,
40 km from her home.
40
35
30
Distance
from
home 25
(km)
20
15
10
5
0
1300 1310 1320 1330 1340 1350 1400 1410 1420 1430
Time
(a)
How long did Dawn spend at her sister's house?
[1]
(b)
Work out Dawn's average speed on her way home from her sister's house.
[2]
OH4E short Page 3
© Churchill Maths Limited
6
A mobile phone weighs 82 grams correct to the nearest gram.
(a)
Write down the least possible weight of the phone.
[1]
(b)
Write down the greatest possible weight of the phone.
[1]
7
(a)
(i)
Express 48 as the product of its prime factors.
[2]
(ii)
Express 180 as the product of its prime factors.
[2]
(b)
(i)
Find the Highest Common Factor (HCF) of 48 and 180.
[2]
(ii)
Find the Lowest Common Multiple (LCM) of 48 and 180.
Give your answer as the product of its prime factors.
[1]
8
(a)
Expand
y(y + 6)
[1]
(b)
Factorise fully
p2q + 4pq
[2]
(c)
Expand and simplify
(x – 3)2
[2]
(d)
C=
a−2
b
Find the value of b when C = 4 and a = 6.8
[3]
OH4E short Page 4
© Churchill Maths Limited
9
Chris records how long he spends in the supermarket each week.
His results are shown in the table.
(a)
Time (t minutes)
Frequency
20 ≤ t < 25
3
25 ≤ t < 30
5
30 ≤ t < 35
6
35 ≤ t < 40
3
40 ≤ t < 45
7
45 ≤ t < 50
2
Write down the modal class.
[1]
(b)
(i)
Work out an estimate for the mean time he spends in the supermarket.
[4]
(ii)
Explain why your answer is only an estimate of the mean.
[1]
10
(a)
(i)
Use your calculator to work out
6.19  2.382
 20.57
Write down all the figures on your calculator display.
[2]
(ii)
Write your answer to part (i) correct to 3 significant figures.
[1]
(b)
Work out
(4.8 × 104) × (2.3 × 107)
Give your answer in standard form correct to 2 significant figures.
[2]
OH4E short Page 5
© Churchill Maths Limited
11
NOT TO SCALE
30 cm
42 cm
One end of a rod is attached to a wall.
The other end of the rod is attached to a wire which keeps the rod horizontal.
The wire is also attached to the wall, 30 cm vertically above the rod.
The rod is 42 cm long.
(a)
Find the length of the wire.
Give your answer correct to 3 significant figures.
[3]
(b)
Find the angle which the wire makes with the rod.
Give your answer to a sensible degree of accuracy.
[3]
12
(a)
–3 ≤ n < 1
n is an integer.
Write down all the possible values of n.
[2]
The graph of y = 1 – x has been drawn on the grid on the answer sheets.
(b)
Draw the graph of y = 2x + 4 on the grid.
[3]
The point P satisfies all three of these inequalities.
y>0
y<1–x
y < 2x + 4
The coordinates of P are both integers.
(c)
Write down the coordinates of P.
[1]
OH4E short Page 6
© Churchill Maths Limited
13
Each morning, the probability that Ms. Toms buys a newspaper is 0.55
Each morning, the probability that Mr. Blake buys a newspaper is 0.4
Calculate the probability that exactly one of Ms. Toms and Mr. Blake will buy a newspaper
tomorrow morning.
State an assumption you have to make in order to carry out your calculation.
[5]
14
NOT TO SCALE
A child is playing with plastic shapes.
She uses 2 squares and 3 equilateral triangles to make a shape like a house.
The side length of each triangle is 6 cm.
The side length of each square is also 6 cm.
(a)
Find the perimeter of the whole shape.
[1]
(b)
Work out the area of the whole shape.
Give your answer in the form a + b  3 , where a and b are integers.
OH4E short Page 7
© Churchill Maths Limited
[5]
15
Part of a multiplication table is shown.
×
1
2
3
4
5
6
1
1
2
3
4
5
6
2
2
4
6
8
10
12
3
3
6
9
12
15
18
4
4
8
12
16
20
24
5
5
10
15
20
25
30
...
...
Rafiq is looking at the numbers in 2 by 2 squares drawn on the table.
(a)
This is one of the squares Rafiq looks at.
28
32
Fill the missing numbers in on the answer sheets.
[2]
(b)
Rafiq decides to use algebra to consider any 2 by 2 square like this:
ab
a(b + 1)
Fill the missing expressions in on the answer sheets.
[3]
OH4E short Page 8
© Churchill Maths Limited
16
2
2
2
2
2
2
NOT TO SCALE
2x + 3
2
2
6x – 1
The width of a rectangular piece of paper is (2x + 3) cm and its length is (6x – 1) cm.
Four squares of side length 2 cm are removed from the corners of the paper as shown.
The area of the remaining piece of paper is 65 cm2.
(a)
Show that
3x2 + 4x – 21 = 0
[4]
(b)
(i)
Solve the equation
3x2 + 4x – 21 = 0
[3]
(ii)
Hence find the length of the piece of paper.
Give your answer correct to 3 significant figures.
[1]
17
(a)
Complete the table of values on the answer sheets for y = x2 + 3x + 1
[2]
(b)
On the grid on the answer sheets, draw the graph of y = x2 + 3x + 1
[2]
(c)
Use your graph to estimate the minimum value of y.
[1]
(d)
By drawing an appropriate straight line on the grid, find estimates of the solutions to the
equation
x2 + x – 1 = 0
[4]
OH4E short Page 9
© Churchill Maths Limited
18
P
Q
NOT TO SCALE
9 cm
9 cm
50°
O
The diagram shows a sector OPQ of a circle, centre O.
OP = OQ = 9 cm.
Angle POQ = 50°.
(a)
Calculate the arc length PQ.
Give your answer correct to 3 significant figures.
[2]
(b)
Calculate the perimeter of the shaded segment.
[3]
19
6 cm
NOT TO SCALE
4 cm
The diagram shows a child's toy made from a cone and a hemisphere.
The diameter of the base of the cone is 4 cm and its slant height is 6 cm.
The diameter of the hemisphere is 4 cm.
(a)
Calculate the total surface area of the toy.
Give your answer in terms of π.
[3]
The volume of the toy is 40.45 cm3.
A larger version of the toy is similar in shape and has a surface area of 45π cm2.
(b)
Calculate the volume of this larger version of the toy.
Give your answer correct to 3 significant figures.
[3]
OH4E short Page 10
© Churchill Maths Limited