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Pearson Edexcel GCSE (9-1) Mathematics Higher Tier Paper 2 (Calculator) November 2021
Mark Scheme Legend
- M: Method mark (awarded for a correct method or partial method)
- P: Process mark (awarded for a correct process as part of a problem solving question)
- A: Accuracy mark (awarded after a correct method; full marks implied if correct answer seen)
- C: Communication mark (awarded for a fully correct statement with no contradiction)
- B: Unconditional accuracy mark (no method needed)
- oe: Or Equivalent
- ft: Follow Through
Table of Contents
- Question 1 (Inequalities)
- Question 2 (HCF and LCM)
- Question 3 (Travel Graphs)
- Question 4 (Quadratic Graphs)
- Question 5 (Perimeter of Composite Shapes)
- Question 6 (Trigonometry)
- Question 7 (Density and Volume)
- Question 8 (Cumulative Frequency Error)
- Question 9 (Box Plots)
- Question 10 (Compound Interest)
- Question 11 (Coordinate Geometry)
- Question 12 (Expanding Brackets)
- Question 13 (Combinations)
- Question 14 (Circle Theorems)
- Question 15 (Cosine Rule and Area)
- Question 16 (Iteration)
- Question 17 (Histograms)
- Question 18 (Trigonometric Graphs)
- Question 19 (Similarity and Spheres)
- Question 20 (Conditional Probability)
- Question 21 (Error Bounds)
Question 1 (3 marks)
(a) Write down the inequality shown on this number line.
(b) On the number line below, show the inequality \( -3 \le y < 4 \)
Question 2 (4 marks)
(a) Find the Highest Common Factor (HCF) of 60 and 84.
(b) Find the Lowest Common Multiple (LCM) of 24 and 40.
Question 3 (5 marks)
Sam drives his car on a journey.
Here is the travel graph for the first 15 minutes of his journey.
(a) Work out Sam’s speed, in km/h, for the first 15 minutes of his journey.
At 10:15 Sam stops for 10 minutes and then drives for 20 minutes at a speed of 75 km/h.
(b) On the grid, complete the travel graph for Sam’s journey.
Question 4 (6 marks)
(a) Complete the table of values for \( y = x^2 – 2x + 2 \)
| x | -2 | -1 | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|---|---|
| y | 10 | 2 | 5 |
(b) On the grid, draw the graph of \( y = x^2 – 2x + 2 \) for values of x from -2 to 4.
(c) Use your graph to find estimates of the solutions of the equation \( x^2 – 2x + 2 = 4 \)
Question 5 (4 marks)
Here is a right-angled triangle.
The shaded shape below is made from two of these triangles.
Question 6 (4 marks)
(a) \( ABC \) is a right-angled triangle.
Work out the length of \( BC \).
Give your answer correct to 1 decimal place.
(b) \( PQR \) is a right-angled triangle.
Work out the size of the angle marked \( x \).
Give your answer correct to 1 decimal place.
Question 7 (3 marks)
Liquid A has a density of \( 1.8 \text{ g/cm}^3 \)
Liquid B has a density of \( 1.2 \text{ g/cm}^3 \)
\( 80 \text{ cm}^3 \) of liquid A is mixed with \( 40 \text{ cm}^3 \) of liquid B to make \( 120 \text{ cm}^3 \) of liquid C.
Work out the density of liquid C.
Question 8 (1 mark)
The grouped frequency table gives information about the time, in minutes, taken by 50 people to solve a puzzle.
| Time (t minutes) | Frequency |
|---|---|
| \( 0 < t \le 10 \) | 5 |
| \( 10 < t \le 20 \) | 8 |
| \( 20 < t \le 30 \) | 12 |
| \( 30 < t \le 40 \) | 15 |
| \( 40 < t \le 50 \) | 7 |
| \( 50 < t \le 60 \) | 3 |
Brian was asked to draw a cumulative frequency table for this information.
This is the table that Brian drew.
| Time (t minutes) | Cumulative frequency |
|---|---|
| \( 0 < t \le 10 \) | 5 |
| \( 10 < t \le 20 \) | 13 |
| \( 20 < t \le 30 \) | 25 |
| \( 30 < t \le 40 \) | 40 |
| \( 40 < t \le 50 \) | 47 |
| \( 50 < t \le 60 \) | 50 |
Write down one thing that is wrong with this cumulative frequency table.
Question 9 (4 marks)
The box plot shows information about the length of time, in minutes, some people waited to see a doctor at a hospital on Monday.
(a) Work out the interquartile range of the information in the box plot.
Becky says, “50% of the people waited for at least 2 hours.”
(b) Is Becky correct? Explain why.
The table gives information about the length of time, in minutes, some people waited to see a doctor at the same hospital on Tuesday.
| Waiting time (minutes) |
|---|
| Shortest time: 20 |
| Lower quartile: 50 |
| Median: 100 |
| Upper quartile: 140 |
| Longest time: 210 |
Becky was asked to compare the distribution of the lengths of times people waited on Monday with the distribution of the lengths of times people waited on Tuesday.
She wrote, “People had to wait longer on Tuesday than on Monday.”
(c) Give one reason why Becky may be wrong.
Question 10 (4 marks)
Louise invests £\(x\) in Better Investments for 3 years.
Sadiq invests £\(x\) in County Bank for 3 years.
Compound Interest
2.5% per annum
Compound Interest
2% per annum for the first two years
3.5% per annum for each extra year
At the end of the 3 years, the value of Louise’s investment is £344605.
Work out the value of Sadiq’s investment at the end of the 3 years.
Question 11 (5 marks)
Here is a sketch of the line L.
The points \( P(-6, 0) \) and \( Q(0, 3) \) are points on the line L.
The point \( R \) is such that \( PQR \) is a straight line and \( PQ : QR = 2 : 3 \).
(a) Find the coordinates of \( R \).
(b) Find an equation of the line that is perpendicular to L and passes through \( Q \).
Question 13 (2 marks)
In a school there are 16 teachers and 220 students.
Of these students 120 are girls and 100 are boys.
One teacher, one girl and one boy are going to be chosen to represent the school.
Work out the number of different ways there are to choose one teacher, one girl and one boy.
Question 14 (4 marks)
A, B, C and D are four points on a circle.
SBT is a tangent to the circle.
Angle ABD = 20°
the size of angle BAD : the size of angle BCD = 3 : 1
Find the size of angle SBA.
Give a reason for each stage of your working.
Question 15 (5 marks)
Here is triangle ABC.
(a) Find the length of \( BC \).
Give your answer correct to 3 significant figures.
(b) Find the area of triangle \( ABC \).
Give your answer correct to 3 significant figures.
Question 16 (4 marks)
(a) Use the iteration formula \( x_{n+1} = \sqrt[3]{10 – 2x_n} \) to find the values of \( x_1 \), \( x_2 \) and \( x_3 \).
Start with \( x_0 = 2 \).
The values of \( x_1 \), \( x_2 \) and \( x_3 \) found in part (a) are estimates of the solution of an equation of the form \( x^3 + ax + b = 0 \) where \( a \) and \( b \) are integers.
(b) Find the value of \( a \) and the value of \( b \).
Question 17 (4 marks)
Some people took part in the first round of a competition.
The histogram gives information about the scores of these people in the first round.
20% of the people got a score high enough for them to qualify for the second round.
Work out an estimate for the score needed to qualify for the second round.
Question 18 (4 marks)
Here is a graph of \( y = \sin x^\circ \) for \( 0 \le x \le 360 \).
(a) Using this graph, find estimates of all four solutions of \( \sin x^\circ = 0.6 \) for \( 0 \le x \le 720 \).
The graph of \( y = \sin x^\circ \) is reflected in the x-axis.
(b) Write down an equation of the reflected graph.
Here is a graph of \( y = f(x) \).
(c) On the grid, draw the graph of \( y = f(x – 2) \).
Question 19 (3 marks)
A, B and C are three spheres.
- The volume of sphere A is \( 125 \text{ cm}^3 \).
- The volume of sphere B is \( 27 \text{ cm}^3 \).
- The ratio of the radius of sphere B to the radius of sphere C is \( 1 : 2 \).
Work out the ratio of the surface area of sphere A to the surface area of sphere C.
Question 20 (4 marks)
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.
Question 21 (4 marks)
The time period, \( T \) seconds, of a simple pendulum of length \( l \) cm is given by the formula
\[ T = 2\pi \sqrt{\frac{l}{g}} \]Katie uses a simple pendulum in an experiment to find an estimate for the value of \( g \).
Here are her results.
- \( l = 52.0 \) correct to 3 significant figures.
- \( T = 1.45 \) correct to 3 significant figures.
Work out the upper bound and the lower bound for the value of \( g \).
Use \( \pi = 3.142 \)
You must show all your working.