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PPE Shadow Paper 2
Question 1
Skill involved: 301b: Convert a large number to standard form/scientific notation.
Write 95 600 000 as a number in standard form.
(1 mark)
Question 2
Skill involved: 302b: Convert a small number in standard form/scientific notation to a normal
number.
Write 7.97 × 10−6
as an ordinary number.
(1 mark)
Question 3
Skill involved: 162c: Determine the Highest Common Factor (HCF) of 2 numbers by prime
factorising.
Find the highest common factor (HCF) of 96 and 120.
(2 marks)
Question 4
Skill involved: 162b: Determine the LCM of two numbers which are already prime factorised.
𝐴 = 32 × 54 × 7
𝐵 = 34 × 53 × 7 × 11
Find the lowest common multiple (LCM) of 𝐴 and 𝐵.
(2 marks)
Question 5
Skill involved: 315d: Draw a frequency polygon for grouped data.
The table shows information about the ages of people in a club.
Copy the graph into your homework books and draw a frequency polygon to represent the
data.
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(2 marks)
Question 6
Skill involved: 231b: Calculate the volume of a prism, given the area of the cross-section.
A pillar is in the shape of a hexagonal prism.
The area of the shaded cross-section is 960 cm 2 . The height of the pillar is 1.2 m.
Calculate the volume of the pillar.
(3 marks)
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Question 7
Skill involved: 243f: List elements using set notation.
Here is a Venn diagram.
Write down the numbers that are in the set 𝐴 ∩ 𝐵
Question 8
Skill involved: 243f: List elements using set notation.
The Venn diagram shows the numbers in the
universal set, and two sets 𝐴 and 𝐵.
List the members of the set 𝐴′.
(1 mark)
Question 9
Skill involved: E243: Union, intersection and complements of sets, including the universal and
empty sets
Here is a Venn diagram.
One of the numbers in the diagram is chosen at
random.
Find the probability that the number is in set 𝐴′
Question 10
Skill involved: 211e: Solve problems involving the cost of covering a fraction of a circle or the
quantity of materials needed.
Jessie needs to cover a wooden floor with varnish. The floor is in the
shape of a rectangle and a quarter circle. A tin of varnish:


covers 6 m 2
costs £5.41
Jessie has £25 to buy the tins of varnish she needs to cover this
wooden floor.
Is £25 enough to buy all the tins of varnish Jessie needs?
(6 marks)
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Question 11
Skill involved: 299b: Expand two brackets given in the form (𝐱 ± 𝐚)(𝐱 ± 𝐛)
Expand and simplify
(𝑦 − 2)(𝑦 − 5)
(2 marks)
Question 12
Skill involved: 340a: Solve a linear inequality with the unknown on both sides.
Solve the inequality 3𝑥 + 15 < 8𝑥 + 3
(3 marks)
Question 13
Skill involved: 328d: Solve problems involving mass, density and volume of a cuboid or prism
(excluding cylinders).
Here are two cubes, A and B.
Cube A has a mass of 81 g.
Cube B has a mass of 128 g.
Work out
the density of cube A : the density of cube B
Give your answer in the form 𝑎: 𝑏, where 𝑎 and 𝑏 are integers.
(3 marks)
Question 14
Skill involved: E310: Upper and lower bound of rounded values or identifying an error interval
Andrea is 165 cm tall, correct to the nearest cm.
Joel is 170 cm tall, correct to the nearest 10 cm.
Calculate the upper and lower bound for each person.
(3 marks)
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Question 15 – HIGHER ONLY FROM HERE
Skill involved: E160: Using a scientific calculator
Use your calculator to work out
√
sin25° +sin40°
cos25° −cos40°
Write down all the figures on your calculator display.
(2 marks)
Question 16
Skill involved: 361a: Determine time before a value reaches a given value given some known
compound percentage change.
On 1st January 2017, Samantha and Dyfan invested money into different savings accounts.
They did not make any further payments into their accounts or withdraw any money from
their accounts.
Dyfan invested £3000 in a savings account that paid interest at a rate of 1.02% every 3
months.
Interest is paid on the last day of each 3-month period.
Calculate the date when Dyfan will first have over £3600 in his account.
(4 marks)
Question 17
Skill involved: 297c: Enlarge/dilate a shape by a decimal or fractional scale factor greater than one.
1
Roy is going to enlarge triangle 𝑃𝑄𝑅 with centre 𝐶 and scale factor 1 2. He draws triangle
𝑋𝑌𝑍. Explain why Roy’s diagram is not correct.
(1 mark)
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Question 18
Skill involved: E322: Trigonometry to determine angles in a right-angled triangle
The diagram shows two right-angled triangles, joined together along a common side.
Angle 𝑆𝑃𝑄 = 90° , angle 𝑆𝑄𝑅 = 90° , angle 𝑆𝑄𝑃 = 38° ,𝑃𝑆 = 8 cm and 𝑄𝑅 = 15 cm.
Calculate the size of angle 𝑥.
(6 marks)
Question 19
Skill involved: 201h: Change the subject of a simple formula with a root of the subject.
Make 𝑔 the subject of the formula
𝑔+6
2
𝑇=√
(3 marks)
Question 20
Skill involved: 351a: Use the capture-recapture method to estimate the size of a population.
Shirley wants to find an estimate for the number of bees in her hive.
On Monday she catches 90 of the bees.
She puts a mark on each bee and returns them to her hive.
On Tuesday she catches 120 of the bees.
She finds that 20 of these bees have been marked.
Work out an estimate for the total number of bees in her hive.
(3 marks)
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Question 21
Skill involved: 464e: Determine an amount given a change in a ratio where two parts are changing.
The ratio of Marta’s hourly pay to Khalid’s hourly pay is 6: 5
Both Marta and Khalid get an increase of £1.50 in their hourly pay.
The ratio of Marta’s hourly pay to Khalid’s hourly pay after this increase is 13: 11
Work out the hourly pay before the increase for Marta and for Khalid.
(4 marks)
Question 22
Skill involved: 441c: Approximate a solution to an equation using (fixed point) iteration.
Use the iteration formula
𝑥𝑛+1 = 3√10 − 2𝑥𝑛
to find the values of 𝑥1 , 𝑥2 and 𝑥3 .
Start with 𝑥0 = 2.
(3 marks)
Question 23
Skill involved: 324g: Determine the surface area/volume of a similar solid when the scale factor of
surface area/volume is given in ratio form.
A and B are two similar cylindrical containers.
the surface area of container A : the surface area of container B = 4: 9
Tyler fills container A with water.
She then pours all the water into container B.
Tyler repeats this and stops when container B is full of water.
Work out the number of times that Tyler fills container A with water.
(4 marks)
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Question 24
Skill involved: 413a: Complete the square for quadratic expressions given in the form
𝐱 𝟐 + 𝐛𝐱 + 𝐜 where 𝐛 is even.
Write 𝑥2 + 6𝑥 − 7 in the form (𝑥 + 𝑎)2 + 𝑏 where 𝑎 and 𝑏 are integers.
(2 marks)
Question 25
Skill involved: 416a: Determine the minimum point of a quadratic graph when given in completed
the square form.
𝑦 = 𝑥2 + 12𝑥 + 24 can be written as 𝑦 = (𝑥 + 6)2 − 12.
Hence state the minimum value of 𝑦 = 𝑥2 + 12𝑥 + 24.
(1 mark)
Question 26
Skill involved: E474: Parallel vectors and straight line proofs using vectors
OAB is a triangle.
→
𝑂𝐴 = 2𝑎
→
𝑂𝐵 = 3𝑏
→
𝐴𝐵 = −2𝑎 + 3𝑏
P is the point on AB such that AP : PB = 2 : 3
→
→
To show that 𝑂𝑃 is parallel to the vector 𝑎 + 𝑏, write 𝑂𝑃 as a multiple of 𝑎 + 𝑏
(3 marks)
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Question 27
Skill involved: 401c: Use a single frequency to determine another for a given range on a histogram.
A policeman records the speed of the traffic on a busy road with a 30 mph speed limit.
He records the speeds of a sample of 450 cars. The histogram in Figure 2 represents the
results.
Calculate the number of cars that were exceeding the speed limit by at least 5 mph in the
sample.
(4 marks)
Question 28
Skill involved: E434: Solving simple trigonometric equations, e.g. 𝐬𝐢𝐧(𝐱) = 𝐤, using a graph
Here is a graph of 𝑦 = sin 𝑥° for 0 ≤ 𝑥 ≤ 360.
Using this graph, together with your calculator, find all four solutions of sin 𝑥° = 0.6 for 0
≤ 𝑥 ≤ 720. Round your answers to the nearest integer.
(2 marks)
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Question 29
Skill involved: 354c: Deal with multiple different chains of outcomes involving successive
dependent events.
In a village,
if it rains on one day, the probability that it will rain on the next day is 0.8
if it does not rain on one day, the probability that it will rain on the next day is 0.6
A weather forecaster says,
“There is a 70% chance that it will rain in the village on Monday.”
Work out an estimate for the probability that it will rain in the village on Wednesday.
(4 marks)
Question 30
Skill involved: 393h: Rationalise the denominator of a fraction where either the numerator or the
denominator is squared.
Simplify
24√𝑎
2
2
(√𝑎+3) −(√𝑎−3)
(3 marks)
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