surname names
Number Number
Further Pure Mathematics
🟐 🟐
PAPER 1R
Instructions
•• Use black ink or ball-point pen.
Fill in the boxes at the top of this page with your name,
centre number and candidate number.
•• Answer all questions.
Without sufficient working, correct answers may be awarded no marks.
• Answer the questions
– there may in thethan
be more space spaces
youprovided
need.
• Anything you write on the formulae page will gain NO credit.
You must NOT write anything on the formulae page.
Information
•• The total mark for this paper is 100.
The marks for each question are shown in brackets
– use this as a guide as to how much time to spend on each question.
Advice
•• Read each question carefully before you start to answer it.
Check your answers if you have time at the end.
Pearson Edexcel International GCSE Further Pure Mathematics
(4PM1/01R) Question Paper And Mark Scheme 2025
, International GCSE in Further Pure Mathematics Formulae sheet
Mensuration
DO NOT WRITE IN THIS AREA
Surface area of sphere = 4πr2
Curved surface area of cone = πr slant height
4
Volume of sphere = πr3
3
Series
Arithmetic series
Sum to n terms, Sn =
n
2a + (n − 1)d
2
Geometric series
a(1 − rn )
Sum to n terms, Sn =
(1 − r)
a
Sum to infinity, S = r <1
DO NOT WRITE IN THIS AREA
1− r
Binomial series
n(n − 1) n(n − 1) (n − r + 1)
(1 + x)n = 1 + nx + x2 + + xr + for x < 1, n
2! r!
Calculus
Quotient rule (differentiation)
d f ( x) f' ( x)g( x) − f( x)g' ( x)
=
dx g( x) [g( x)]2
Trigonometry
Cosine rule
In triangle ABC: a2 = b2 + c2 – 2bc cos A
sin θ
tan θ =
DO NOT WRITE IN THIS AREA
cosθ
sin(A + B) = sin A cos B + cos A sin B sin(A – B) = sin A cos B – cos A sin B
cos(A + B) = cos A cos B – sin A sin B cos(A – B) = cos A cos B + sin A sin B
tan A + tan B tan A − tan B
tan( A + B) = tan( A − B) =
1 − tan A tan B 1 + tan A tan B
Logarithms
log x
loga x = b
logb a
2
■■■■
, Answer all ELEVEN questions.
Write your answers in the spaces provided.
DO NOT WRITE IN THIS AREA
You must write down all the stages in your working.
1 (8 − 6 ) w = 50
Without using a calculator, find the value of w
Give your answer in the form a + b 6 where a and b are integers and c is prime.
c
Show your working clearly.
(3)
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(Total for Question 1 is 3 marks)
3
■■■■ Turn over
, 2 An arithmetic series F has third term 8 and fifth term 20
Given that the sum to n terms of F is greater than 220
DO NOT WRITE IN THIS AREA
find the least value of n
(7)
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4
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Number Number
Further Pure Mathematics
🟐 🟐
PAPER 1R
Instructions
•• Use black ink or ball-point pen.
Fill in the boxes at the top of this page with your name,
centre number and candidate number.
•• Answer all questions.
Without sufficient working, correct answers may be awarded no marks.
• Answer the questions
– there may in thethan
be more space spaces
youprovided
need.
• Anything you write on the formulae page will gain NO credit.
You must NOT write anything on the formulae page.
Information
•• The total mark for this paper is 100.
The marks for each question are shown in brackets
– use this as a guide as to how much time to spend on each question.
Advice
•• Read each question carefully before you start to answer it.
Check your answers if you have time at the end.
Pearson Edexcel International GCSE Further Pure Mathematics
(4PM1/01R) Question Paper And Mark Scheme 2025
, International GCSE in Further Pure Mathematics Formulae sheet
Mensuration
DO NOT WRITE IN THIS AREA
Surface area of sphere = 4πr2
Curved surface area of cone = πr slant height
4
Volume of sphere = πr3
3
Series
Arithmetic series
Sum to n terms, Sn =
n
2a + (n − 1)d
2
Geometric series
a(1 − rn )
Sum to n terms, Sn =
(1 − r)
a
Sum to infinity, S = r <1
DO NOT WRITE IN THIS AREA
1− r
Binomial series
n(n − 1) n(n − 1) (n − r + 1)
(1 + x)n = 1 + nx + x2 + + xr + for x < 1, n
2! r!
Calculus
Quotient rule (differentiation)
d f ( x) f' ( x)g( x) − f( x)g' ( x)
=
dx g( x) [g( x)]2
Trigonometry
Cosine rule
In triangle ABC: a2 = b2 + c2 – 2bc cos A
sin θ
tan θ =
DO NOT WRITE IN THIS AREA
cosθ
sin(A + B) = sin A cos B + cos A sin B sin(A – B) = sin A cos B – cos A sin B
cos(A + B) = cos A cos B – sin A sin B cos(A – B) = cos A cos B + sin A sin B
tan A + tan B tan A − tan B
tan( A + B) = tan( A − B) =
1 − tan A tan B 1 + tan A tan B
Logarithms
log x
loga x = b
logb a
2
■■■■
, Answer all ELEVEN questions.
Write your answers in the spaces provided.
DO NOT WRITE IN THIS AREA
You must write down all the stages in your working.
1 (8 − 6 ) w = 50
Without using a calculator, find the value of w
Give your answer in the form a + b 6 where a and b are integers and c is prime.
c
Show your working clearly.
(3)
............................................................................................................................. .....................................................................................................................
............................................................................................................................. .....................................................................................................................
............................................................................................................................. ........................................................................................................... ..........
DO NOT WRITE IN THIS AREA
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(Total for Question 1 is 3 marks)
3
■■■■ Turn over
, 2 An arithmetic series F has third term 8 and fifth term 20
Given that the sum to n terms of F is greater than 220
DO NOT WRITE IN THIS AREA
find the least value of n
(7)
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