Candidate surname Other names
Centre Number Candidate Number
Further Mathematics
🟐 🟐
Advanced
PAPER 3A: Further Pure Mathematics 1
Marks
Candidates may use any calculator permitted by Pearson regulations. Calculators
must not have the facility for symbolic algebraic 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,
•
centre number and candidate number.
Answer all questions and ensure that your answers to parts of questions are
•
clearly labelled.
Answer the questions in the spaces provided
•
– there may be more space than you need.
You should show sufficient working to make your methods clear.
•
Answers without working may not gain full credit.
Inexact answers should be given to three significant figures unless otherwise stated.
Information
••
A booklet ‘Mathematical Formulae and Statistical Tables’ is provided.
•
There are 9 questions in this question paper. The total mark for this paper is 75.
The
– usemarks
this asfora guide
each as
question
to howare shown
much timeintobrackets
spend on each question.
Advice
•
Read each question carefully before you start to answer it.
••
Try to answer every question.
Check your answers if you have time at the end.
EDEXCEL A LEVEL FURTHER MATHEMATICS (9FM0/3A) QUESTION
PAPER 3A AND MARK SCHEME SUMMER 2025
,1. (a) Given that
2
y = ecosec x
DO NOT WRITE IN THIS AREA
complete the table below with the value of y corresponding to x = 2, giving your
answer to 2 decimal places.
x 1.5 1.75 2 2.25 2.5
y 2.73 2.81 5.22 16.31
(1)
(b) Use Simpson’s rule, with all the values of y in the completed table, to estimate, to
one decimal place, the value of
2.5 2
cosec x
e dx
1.5
(3)
DO NOT WRITE IN THIS AREA
(c) Using your answer to part (b) and making your method clear, estimate the value of
2.5 2
cot x
e dx
1.5
(2)
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
(Total for Question 1 is 6 marks)
3
Turn over
,
2. The Taylor series expansion of f(x) about x = a is given by
f(x) = f(a) + (x – a)f ′(a) + (x − a)2 (x − a)r (r)
f ′′(a) + ... +
DO NOT WRITE IN THIS AREA
f (a) + ...
2! r!
Given that
d2 y dy
2
+ 3 − 2xy = 4 (I)
dx dx
(a) show that
d4 y dy d2 y d3 y
=a + bx +c
dx4 dx dx2 dx3
where a, b and c are integers to be determined.
(4)
dy
Hence, given that y = 1 and = 1 when x = 2
dx
DO NOT WRITE IN THIS AREA
(b) determine a Taylor series solution, in ascending powers of (x – 2), up to and
including the term in (x – 2)4, of the differential equation (I), giving each coefficient
in simplest form.
(4)
DO NOT WRITE IN THIS AREA
4
■■■■
Centre Number Candidate Number
Further Mathematics
🟐 🟐
Advanced
PAPER 3A: Further Pure Mathematics 1
Marks
Candidates may use any calculator permitted by Pearson regulations. Calculators
must not have the facility for symbolic algebraic 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,
•
centre number and candidate number.
Answer all questions and ensure that your answers to parts of questions are
•
clearly labelled.
Answer the questions in the spaces provided
•
– there may be more space than you need.
You should show sufficient working to make your methods clear.
•
Answers without working may not gain full credit.
Inexact answers should be given to three significant figures unless otherwise stated.
Information
••
A booklet ‘Mathematical Formulae and Statistical Tables’ is provided.
•
There are 9 questions in this question paper. The total mark for this paper is 75.
The
– usemarks
this asfora guide
each as
question
to howare shown
much timeintobrackets
spend on each question.
Advice
•
Read each question carefully before you start to answer it.
••
Try to answer every question.
Check your answers if you have time at the end.
EDEXCEL A LEVEL FURTHER MATHEMATICS (9FM0/3A) QUESTION
PAPER 3A AND MARK SCHEME SUMMER 2025
,1. (a) Given that
2
y = ecosec x
DO NOT WRITE IN THIS AREA
complete the table below with the value of y corresponding to x = 2, giving your
answer to 2 decimal places.
x 1.5 1.75 2 2.25 2.5
y 2.73 2.81 5.22 16.31
(1)
(b) Use Simpson’s rule, with all the values of y in the completed table, to estimate, to
one decimal place, the value of
2.5 2
cosec x
e dx
1.5
(3)
DO NOT WRITE IN THIS AREA
(c) Using your answer to part (b) and making your method clear, estimate the value of
2.5 2
cot x
e dx
1.5
(2)
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
(Total for Question 1 is 6 marks)
3
Turn over
,
2. The Taylor series expansion of f(x) about x = a is given by
f(x) = f(a) + (x – a)f ′(a) + (x − a)2 (x − a)r (r)
f ′′(a) + ... +
DO NOT WRITE IN THIS AREA
f (a) + ...
2! r!
Given that
d2 y dy
2
+ 3 − 2xy = 4 (I)
dx dx
(a) show that
d4 y dy d2 y d3 y
=a + bx +c
dx4 dx dx2 dx3
where a, b and c are integers to be determined.
(4)
dy
Hence, given that y = 1 and = 1 when x = 2
dx
DO NOT WRITE IN THIS AREA
(b) determine a Taylor series solution, in ascending powers of (x – 2), up to and
including the term in (x – 2)4, of the differential equation (I), giving each coefficient
in simplest form.
(4)
DO NOT WRITE IN THIS AREA
4
■■■■
