surname names
Number Number
Afternoon
🟐 🟐
Advanced Subsidiary
Further Mathematics options
21: Further Pure Mathematics 1
(Part of options A, B, C and D)
Candidates may use any calculator allowed by Pearson regulations.
Calculators must not have the facility for symbolic algebra manipulation,
differentiation and integration, or have retrievable mathematical
formulae stored in them.
Instructions
•• Use black ink or ball-point pen.
If pencil is used for diagrams/sketches/graphs it must be dark (HB or B).
• Fill in the boxes at the top of this page with your name,
• Answer
centre number and candidate number.
all questions and ensure that your answers to parts of questions are
• Answer
clearly labelled.
the questions in the spaces provided
• You
– there may be more space than you need.
should show sufficient working to make your methods clear.
• Inexact answers should be given to three significant figures unless otherwise stated.
Answers without working may not gain full credit.
Information
•• AThebooklet ‘Mathematical Formulae and Statistical Tables’ is provided.
• The
total mark for this part of the examination is 40. There are 5 questions.
– usemarks
this asfor each as
a guide question
to how are
muchshown
timeintobrackets
spend on each question.
• Read each question carefully before you start to answer it.
Advice
•• Try to answer every question.
Check your answers if you have time at the end.
• 2025 Pearson Edexcel As level further Mathematics Question
Paper Option 21 June+ Mark scheme
,1. In this question you must show all stages of your working.
Solutions based entirely on calculator technology are not acceptable.
DO NOT WRITE IN THIS AREA
The surface temperature of the water in a lake during a particular year is modelled by
the equation
15 27
S = 12 – cos x° – sin x° (I)
2 10
where S is the temperature in degrees Celsius and x is the number of days after the start
of the year.
(a) Use the model to write down the surface temperature of the water in the lake at the
start of the year.
(1)
Using the substitution t = tan
x
2
(b) show that equation ( I ) can be rewritten as
DO NOT WRITE IN THIS AREA
At 2 + Bt + C
S=
10 (1 + t 2 )
where A, B and C are integers to be determined.
(3)
(c) Hence determine, according to the model, the number of days after the start of the
year when the surface temperature of the water in the lake is 10 °C for the second
time that year. Give your answer to the nearest day.
(5)
DO NOT WRITE IN THIS AREA
2
■■■■
, DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA
■■■■
Question 1 continued
3
Turn over
, DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA
■■■■
Question 1 continued
4
Number Number
Afternoon
🟐 🟐
Advanced Subsidiary
Further Mathematics options
21: Further Pure Mathematics 1
(Part of options A, B, C and D)
Candidates may use any calculator allowed by Pearson regulations.
Calculators must not have the facility for symbolic algebra manipulation,
differentiation and integration, or have retrievable mathematical
formulae stored in them.
Instructions
•• Use black ink or ball-point pen.
If pencil is used for diagrams/sketches/graphs it must be dark (HB or B).
• Fill in the boxes at the top of this page with your name,
• Answer
centre number and candidate number.
all questions and ensure that your answers to parts of questions are
• Answer
clearly labelled.
the questions in the spaces provided
• You
– there may be more space than you need.
should show sufficient working to make your methods clear.
• Inexact answers should be given to three significant figures unless otherwise stated.
Answers without working may not gain full credit.
Information
•• AThebooklet ‘Mathematical Formulae and Statistical Tables’ is provided.
• The
total mark for this part of the examination is 40. There are 5 questions.
– usemarks
this asfor each as
a guide question
to how are
muchshown
timeintobrackets
spend on each question.
• Read each question carefully before you start to answer it.
Advice
•• Try to answer every question.
Check your answers if you have time at the end.
• 2025 Pearson Edexcel As level further Mathematics Question
Paper Option 21 June+ Mark scheme
,1. In this question you must show all stages of your working.
Solutions based entirely on calculator technology are not acceptable.
DO NOT WRITE IN THIS AREA
The surface temperature of the water in a lake during a particular year is modelled by
the equation
15 27
S = 12 – cos x° – sin x° (I)
2 10
where S is the temperature in degrees Celsius and x is the number of days after the start
of the year.
(a) Use the model to write down the surface temperature of the water in the lake at the
start of the year.
(1)
Using the substitution t = tan
x
2
(b) show that equation ( I ) can be rewritten as
DO NOT WRITE IN THIS AREA
At 2 + Bt + C
S=
10 (1 + t 2 )
where A, B and C are integers to be determined.
(3)
(c) Hence determine, according to the model, the number of days after the start of the
year when the surface temperature of the water in the lake is 10 °C for the second
time that year. Give your answer to the nearest day.
(5)
DO NOT WRITE IN THIS AREA
2
■■■■
, DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA
■■■■
Question 1 continued
3
Turn over
, DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA
■■■■
Question 1 continued
4
