Non-Calculator Practice Questions
May/June 2017
GCSE Mathematics (OCR style)
Higher Tier
Paper 5
Name
………………………………………………………………
Class
………………………………………………………………
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 NOT 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 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.
Page 1 of 47
Number Operations and Integers
1
(a) Complete the table with suitable values. Do not use the same value more than once.
Square
number
Cube number
Prime number
Factor of 60
Even number
Multiple of 5
[3]
(b) Explain why a prime number can never also be a square number.
………………………………………………………………………………………………………………………
………………………………………………………………………………………………………………………
………………………………………………………………………………………………………………………
[1]
2
(a) Write the number 420 as a product of its prime factors
................................................. [2]
(b) The number 600 can be written in the form 2𝑥 × 3𝑦 × 5 𝑧 . Find the values 𝑥, 𝑦 and 𝑧.
................................................. [1]
(c) Find the highest common factor of 420 and 600.
................................................. [2]
Page 2 of 47
3
Theo thinks of a number greater than zero, doubles it, adds three and then squares it. He gets 64. What
was the number he first thought of?
................................................. [2]
Fractions, Decimals & Percentages
4
(a) Which fraction has the greatest value
3
15
or
4
24
?
................................................. [2]
(b) For each pair of numbers decide which is greater.
(i)
1
9
or
0.09
................................................. [1]
(ii) 25% or
0.3
................................................. [1]
(iii) 15% or
1
5
................................................. [1]
Page 3 of 47
5
Maria is getting a new carpet fitted in her living room. The room is 2
3
1
4
2
3
m wide and 3
1
4
m long.
m
2
2
3
m
(a) Calculate the total area of the carpet Maria needs to buy.
Give your answer as a mixed number.
………………………………….... m2 [4]
(b) She is also going to buy a skirting board which will go around the full perimeter of the living room.
Calculate the length of skirting board that Maria needs.
Give your answer as a mixed number.
................................................. m [3]
Page 4 of 47
6
Convert the recurring decimal 0.16̇ into a fraction
................................................. [3]
Indices & Surds
7
Write each of these as a single power of 2.
(a) (24)3
................................................. [1]
(b)
1
8
................................................. [2]
1
(c) 4
√163
................................................. [3]
Page 5 of 47
8
(a) The distance to the sun is approximately 150 million km away. Write this number in standard form.
................................................. [1]
(b) The thickness of a piece of paper is approximately 0.05 mm. Write this number is standard form.
................................................. [1]
(c) Using these values, calculate how many pieces of paper it would take to reach the sun.
Give your answer in standard form.
................................................. [3]
9
Simplify these expressions.
(a)
√20
................................................. [1]
(b)
1
√3+1
................................................. [3]
Page 6 of 47
10
Andy has designed his new kitchen to have carpeted and tiled sections.
The darkened area represents where he will be laying the tiles.
Calculate, in m2, how much carpet Andy will need for this room. (All measurements are in metres).
................................................. [4]
Approximation & Estimation
11
(a) Marco won a 60m sprint race.
His time was measured as 7.1 seconds, rounded to 1 decimal place.
Complete the error interval for Marco’s time, s.
(a) ………… ≤ 𝑠 < ………… [2]
(b) Justin came second in the same race.
His time was measured as 7.23 seconds, truncated to 2 decimal places.
Complete the error interval for Justin’s time, t.
(b) ………… ≤ 𝑡 < ………… [2]
Page 7 of 47
Ratio, Proportion & Rates of Change
12
Write 2 minutes to 45 seconds as a ratio in the form 1 : n.
1 : ............ [2]
13
(a) y is directly proportional to the square of x.
x = 3 when y = 54.
Work out y when x = 2.
(a) y = …………………… [2]
(b) y is inversely proportional to x.
y = 6 when x = 10.
Find the value of x when y = 15.
(b) x = …………………… [2]
Page 8 of 47
Algebra
14
(a) If n is a positive integer, explain why the expression 2n + 1 is always an odd number.
…………………………………………………………………………………………………………………
…………………………………………………………………………………………………………………
…………………………………………………………………………………………………………………
[1]
(b) Use algebra to prove that the product of two odd numbers is always odd.
[2]
15
Simplify the following.
(a)
9𝑎2 𝑏5
(
27𝑎−3 𝑏3
)
………………………………… [2]
1
(b)
𝑝2 𝑞 3 × 𝑝3 𝑞 −1
………………………………… [2]
Page 9 of 47
16
(a) Expand and simplify.
(x + 2)(2x – 1)2
………………………………… [3]
(b) Factorise completely
(i) x2 – 36
………………………………… [1]
(ii) 2x2 – 7x – 15
………………………………… [3]
(c)
(i) Express x2 – 4x – 10 in the form (x + a)2 + b.
………………………………… [2]
(ii) Hence, write down the minimum point of y = x2 – 4x – 10.
………………………………… [2]
17
Write as a single fraction in its simplest form.
(i)
𝑥
3
+𝑦
………………………………… [1]
(ii)
3𝑥
𝑥−4
−
𝑥
𝑥+3
………………………………… [3]
18
A circle touches two sides of a right-angled triangle at points P and Q.
Page 10 of 47
The centre of the circle, O, is on the hypotenuse of the triangle.
The circle had radius r and the triangle has sides of length x and y.
(a) Explain why OPBQ is a square.
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
[2]
(b) Write expressions for the area of triangles OAP, OQC and ABC.
OAP: …………………………
OQC: …………………………
ABC: …………………………
[3]
(c) Hence, show that
1
𝑟
1
1
𝑥
𝑦
= + .
………………………………… [5]
Page 11 of 47
19
Rearrange this formula to make u the subject.
v2 = u2 + 2as
u = ………………………………… [3]
20
Work out the perimeter of the semi-circle with radius 8 cm.
Give your answer in terms of π.
………………………………… cm [3]
21
1
Given that s = ut + at2 , work out the value of s when u = 3, t = 4, a = –5.
2
s = ………………… [3]
22
James is 35 years old.
Three years ago, he was four times as old as his son was then.
How old is his son now?
…………………………… [3]
Page 12 of 47
23
Solve the following equations:
(a) 7x – 5 = 3x + 9
…………………………… [2]
(b)
𝑥+3
5
−
𝑥−2
3
=4
…………………………… [4]
10
(c)
𝑥+3
2
− =1
𝑥
…………………………… [6]
Page 13 of 47
24
A rectangle has a width of x cm. Its length is 5 cm longer than the width.
(a) Write down an expression for the area of the rectangle.
……………………………cm2 [1]
(b) Given that the area is 84 cm2, work out the length and width of the rectangle.
length = …………………………… cm
width = …………………………… cm
[4]
25
Solve the quadratic inequality
2x2 + 9x < 5
…………………………………
[4]
Page 14 of 47
26
3 rulers and 4 pens cost £1.02.
2 rulers and 5 pens cost 96 pence.
Calculate the cost of each ruler and each pen.
Ruler = …………………………………
Pen = …………………………………
[3]
27
Solve simultaneously.
y – x = 10
x2 + y2 = 58
x = ……., y = …….
x = ……., y = …….
[6]
Page 15 of 47
28
(a) Complete the table of values for the equation y = x2 – 2x from -2 ≤ x ≤ 3
x
-2
y
8
-1
0
1
2
3
-1
[2]
(b) Plot the graph on the axes below.
[2]
(c) Use your graph to find out the solutions to the equation
x2 – 2x = 1.5
x = ………., x = ………. [2]
(d) By drawing an additional line, use the graph to solve the equation x2 – 3x + 1 = 0.
x = ……., x = …….
[4]
Page 16 of 47
29
Solve 6𝑥 − 7 ≤ 5
Represent your answer on the number line below.
[3]
30
Explain why for any value of x both function machines give the same value of y.
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
[1]
31
Function 1 is given by y = x + 2.
Function 2 is given by y = 2x – 3.
Show that the composite function, ‘Function 1 followed by Function 2’ is different to the function
‘Function 2 followed by Function 1’.
[4]
Page 17 of 47
32
A function is given by y = 5x – 3.
Work out the inverse of this function.
………………………………… [2]
33
Here is a linear sequence
8, 11, 14, 17, …, …
(a) Write down the next two terms.
………………… and ……………… [1]
th
(b) Find an expression for the n term.
………………………………… [2]
(c) David says that 63 is in the sequence. Is he correct?
Give reasons for your answer.
………………………………… [2]
34
(a) Write down the first three terms in the sequence n2 + 3n.
…………, …………, ………… [2]
th
(b) Find an expression for the n term of the sequence
6, 13, 24, 39, 58, …
………………………………… [3]
Page 18 of 47
35
Find the equation of the line perpendicular to 𝑦 = 3𝑥 − 2 that passes through the point (3, 9).
………………………………… [3]
36
Match the graphs with the corresponding equations.
Equation
y =(x + 2)2
y = 1 – 2x
y = 2x
y=1–
2
𝑥
Graph
[3]
Page 19 of 47
37
Use shading to identify the region, R, which satisfies the following three inequalities.
x≥0
1
y≥ x
2
x + 2y ≤ 4
[4]
38
Sketch the graph of 𝑦 = 𝑠𝑖𝑛𝑥 for the values 0 ≤ 𝑥 ≤ 360
[2]
Page 20 of 47
39
Write down the equation of this circle.
……………………………. [1]
40
The graph below shows a sketch of y = x3.
On the same pair of axes, sketch the graph of y = x3 – 8, indicating any points where the graph
intersects with the x and y axes.
……………………………. [3]
Page 21 of 47
41
The conversion graph below can be used to convert between pounds (GBP) and Euros.
(a) Write down the exchange rate from pounds to Euros shown by the graph.
£1 = ………………. Euros [1]
(b) Using this exchange rate, convert £150 into Euros.
………………. Euros [2]
Page 22 of 47
42
The velocity-time graph below shows the motion of a car, travelling in a straight line at v metres per
second, in the 20 seconds after it starts from rest.
(a) Calculate the acceleration of the car in the first 5 seconds of the journey.
……………………. [1]
(b) Was the car’s change of speed greatest between 5 and 10 seconds, or between 15 and 20
seconds?
Explain your answer.
[2]
(c) Between 10 and 15 seconds, how far did the car travel?
……………………. [2]
Page 23 of 47
Basic Geometry
43 (a) Construct the perpendicular bisector of the line below.
[2]
(b) Construct the bisector of the angle shown.
[2]
Page 24 of 47
44
Sam is building a new house and needs to supply water to the house.
He can take a water pipe from the main water supply at any point. He wants to save money so wants it
to be the shortest distance.
Construct the pipeline to his house.
[3]
Page 25 of 47
45
Jane wants to buy a new house.
She wants to live within 10km of Aldington and within 8km of Broughton.
She also wants to live closer to Crick than Aldington.
Shade the region that Jane should look to buy a house.
Scale: 2km=1cm.
[4]
Page 26 of 47
46
P,Q,R and S are points on a circle.
PR is parallel to QS.
PR is equal to PQ.
Angle PSQ=30°.
PS is a diameter.
Find angle PRQ giving reasons.
Angle PRQ is …………° because ………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
[4]
Page 27 of 47
47
AB and DE are parallel. AE and BD are straight lines
(a) Explain why angle CDE is 46°.
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………… [1]
(b) Calculate the size of angle ACB. Show your working and give reasons.
Angle ACB is …………° because ………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………… [2]
Page 28 of 47
48
ABCDE is a regular pentagon. DEFGHI is a regular hexagon.
(a) Calculate the size of the angle DEF.
DEF = …………° [2]
(b) Hence, or otherwise, calculate the size of angle AEF
AEF = …………° [2]
Page 29 of 47
49
(a) Describe the single transformation that maps shape A onto shape B.
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
[2]
(b) Describe the single transformation that maps shape B onto shape C.
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
[3]
(c) Translate shape C [
5
]. Label this image D.
0
[2]
(d) Describe the single transformation that maps shape A onto shape D.
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
[2]
Page 30 of 47
50
Enlarge the shape below by scale factor -0.5 using the origin as the centre of enlargement.
[2]
51
Point E is the midpoint of line AB and CD.
Prove that triangle ABE and CDE are congruent.
………………………………………………………………………………………………………………………
………………………………………………………………………………………………………………………
………………………………………………………………………………………………………………………
………………………………………………………………………………………………………………………
[2]
Page 31 of 47
52
State if the triangles below are similar.
Show how you decide.
………………………………………………………………………………………………………………………
………………………………………………………………………………………………………………………
………………………………………………………………………………………………………………………
………………………………………………………………………………………………………………………
[2]
53
A titanium bar measures 5cm by 10cm by 4cm. Titanium has a density of 4.506 g/cm³.
Calculate the mass of the titanium bar.
State the units of your answer
…………………… …………
[4]
Page 32 of 47
54
Triangle ABC is shown in the diagram below.
D is the midpoint of AB.
E is the midpoint of AC.
⃗⃗⃗⃗⃗
𝐴𝐷
= a and ⃗⃗⃗⃗⃗
𝐴𝐸 = b.
Find the following vectors. Give your answer in terms of a and b.
(a)
⃗⃗⃗⃗⃗
𝐷𝐸 .
…………………… [1]
(b)
⃗⃗⃗⃗⃗
𝐵𝐶 .
…………………… [1]
The point F lies on BC such that BF:FC = 2:1.
(c) Find ⃗⃗⃗⃗⃗
𝐷𝐹 .
…………………… [3]
55
A 40g tin of peas requires a label that is 20cm2.
A larger tin of peas, that is mathematically similar, requires a label that is 80cm2.
How many more grams are in the larger tin?
…………………… g [4]
Page 33 of 47
56
Calculate the area of the following sector, leaving your answer in terms of π.
…………………… m2 [2]
57
Triangle ABC is shown below.
State which type of triangle this is.
Show how you decide.
……………………………… triangle [3]
Page 34 of 47
58
Triangle OPQ is shown below.
OPQ is a triangle
R is the midpoint of OP.
S is the midpoint of PQ.
⃗⃗⃗⃗⃗ = p and ⃗⃗⃗⃗⃗⃗
𝑂𝑃
𝑂𝑄= q.
(a) Find ⃗⃗⃗⃗⃗
𝑂𝑆 in terms of p and q.
…………………… [1]
(b) Prove that the line RS is parallel to the line OQ.
…………………… [2]
Page 35 of 47
59
(a) Give a reason why angle BCD is 105°.
……………………………………………………………………………………………………………………
………………………………………………………………………………………………………………… [1]
(b) Calculate the value of angle BAX showing all of your working.
BAX = ………………° [5]
Page 36 of 47
60
A container is made in the shape of a cylinder joined to a hemisphere at both ends. The diameter of the
shape is 6 cm and the height of the cylinder is 10 cm.
Calculate the volume of the container in terms of 𝜋.
[The volume V of a sphere with radius r is 𝑉
4
= 𝜋𝑟 3 ]
3
................................................. [5]
61
The radius of the base of a cone is x cm and its height is h cm.
The radius of a sphere is 2x cm.
The volume of the cone and the volume of the sphere are equal.
Express h in terms of x in its simplest form.
4
1
[The volume V of a sphere with radius r is 𝑉 = 3 𝜋𝑟 3 and the volume of a cone is 3 𝜋𝑟 2 ℎ]
................................................. [4]
Page 37 of 47
62
The diagram shows triangle ABC.
The point D lies on BC such that CD = 12 cm.
AC = 13 cm, angle ABD = 60° and angle ADB = 90°.
Calculate the exact value of the length of AB in its simplest form.
................................................. [5]
63
(a) Complete this table of exact values.
[2]
(b) Triangles ABC and DEF are shown below.
By working out the exact area of each triangle, state which one has the larger area.
(b) ................................................. [5]
Page 38 of 47
Probability
64
The button in the picture below has a pattern on one side and is plain on the other.
Chris, Jim and Gary each drop a number of these identical buttons and count how many times the
buttons land with the pattern side facing up. This table shows some of their results.
(a) Chris says:
1
“There are only two sides to the button, so the probability of a button landing ‘pattern up’ is .”
2
Criticise Chris’s statement.
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………… [1]
(b) Gary’s results give the best estimate of a button landing ‘pattern up’.
Explain why.
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………… [1]
(c) Gary drops three further buttons.
Estimate the probability that all three buttons land ‘pattern up’.
(c) ................................................. [2]
Page 39 of 47
65
Phil uses two fair spinners in a game.
5
3
4
7
9
6
4
8
4
Spinner 1
2
Spinner 2
He spins both spinners and adds the two results together.
(a) Phil works out the probability of getting a total of 11.
His workings are shown below.
Phil has made a number of mistakes.
Describe two of these.
(i) ………………………………………………………………………………………………………………
………………………………………………………………………………………………………… [1]
(ii) ………………………………………………………………………………………………………………
………………………………………………………………………………………………………… [1]
(b) Zahra uses the same two spinners in a different game.
She finds the difference between the results when she spins the spinners.
Find the probability the difference will be 1.
[3]
Page 40 of 47
66
A class of 35 students are asked if they like cereal or fruit for breakfast.
21 students said they like fruit. 20 students said they like cereal. 4 students said they don’t like either.
(a) Complete the Venn diagram.
[3]
One student is chosen at random. What is the probability that this student:
(b) only likes fruit?
................................. [2]
(c) likes fruit, given that they like cereal.
................................. [2]
Page 41 of 47
67
Samir sometimes gets a bus to football training.
When he doesn’t get a bus, he walks.
The probability that he gets a bus to training is 0.6.
The probability that he walks home from training is 0.2.
(a) Complete the tree diagram.
[2]
(b) Find the probability that Samir takes a bus to training and then walks home.
................................. [2]
(c) Find the probability that Samir doesn’t get a bus on a particular day.
................................. [2]
Page 42 of 47
68
A bag contains n marbles.
There is 1 blue marble and the rest are red.
Two marbles are taken from the bag at random.
(a) Show the probability that the marbles chosen are the same colour is
𝑛−2
𝑛
[2]
(b) The probability that both marbles are red is greater than 99%.
Work out the least possible value of n.
................................. [3]
Page 43 of 47
Statistics
69
The table below shows information about the heights of 60 students.
(a) On the grid below, draw a cumulative frequency graph for the information in the table.
[3]
(b) Find an estimate for:
(i) the median.
................................. [1]
(ii) the interquartile range.
................................. [2]
Page 44 of 47
70
The table shows the weight of 815 parcels handled by a sorting office in one day.
(a) Draw a fully labelled histogram to show the weights of the parcels.
[4]
(b) Estimate the number of parcels that weighed more than 2500 grams.
................................. [2]
Page 45 of 47
71
Jodie measures the lengths of 120 snakes. The lengths were recorded and displayed in the boxplot
below.
(a) Use the boxplot to fill in the values in this summary table:
[3]
(b) Calculate the interquartile range.
................................. [2]
(c) One of the snakes measured at 42cm was actually 44cm, what affect would this change have on the
median? Explain your answer.
Decrease
□
Stay the same
□
Increase
□
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
…………………………………………………………………………………………………………… [1]
Page 46 of 47
72
The scatter diagram below shows the test data from a science test and maths test for a class of
students.
(a) One of the results is an outlier. Write down the scores this student achieved for maths and science.
Science ………… Maths …………[2]
(b) Another student sits the science test and scores 40 marks. Estimate the score they are likely to
achieve in maths.
................................. [2]
(c) One final student sits the maths test and scores 70. Estimate the score they are likely to achieve in
science.
................................. [1]
(d) Which of these two estimates is likely to be most accurate?
Explain your reasoning.
………………… because ………………………………………………………………………………………
…………………………………………….…………………………………………….………………………...
…………………………………………….…………………………………………….………………………...
[3]
(e) Which subject has the most consistent results? Explain your reasoning.
………………… because ………………………………………………………………………………………
…………………………………………….…………………………………………….………………………...
[2]
Page 47 of 47