Cambridge International AS & A Level
* 0 8 3 4 5 5 7 7 1 1 *
FURTHER MATHEMATICS 9231/13
Paper 1 Further Pure Mathematics 1 May/June 2023
2 hours
You must answer on the question paper.
You will need: List of formulae (MF19)
INSTRUCTIONS
● Answer all questions.
● Use a black or dark blue pen. You may use an HB pencil for any diagrams or graphs.
● Write your name, centre number and candidate number in the boxes at the top of the page.
● Write your answer to each question in the space provided.
● Do not use an erasable pen or correction fluid.
● Do not write on any bar codes.
● If additional space is needed, you should use the lined page at the end of this booklet; the question
number or numbers must be clearly shown.
● You should use a calculator where appropriate.
● You must show all necessary working clearly; no marks will be given for unsupported answers from a
calculator.
● Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in
degrees, unless a different level of accuracy is specified in the question.
INFORMATION
● The total mark for this paper is 75.
● The number of marks for each question or part question is shown in brackets [ ].
This document has 16 pages. Any blank pages are indicated.
DC (CE) 311426/2
© UCLES 2023 [Turn over
, 2
BLANK PAGE
© UCLES 2023 9231/13/M/J/23
, 3
1 Prove by mathematical induction that, for all positive integers n, 5 3n + 32 n - 33 is divisible by 31. [6]
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© UCLES 2023 9231/13/M/J/23 [Turn over
, 4
2 (a) Use standard results from the list of formulae (MF19) to show that
n
/ (6r + 6r - 5) = an + bn + cn ,
2 3 2
r=1
where a, b and c are integers to be determined. [2]
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n
/ 6r r++6rr- 5 in terms of n.
2
(b) Use the method of differences to find 2 [4]
r=1
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© UCLES 2023 9231/13/M/J/23
* 0 8 3 4 5 5 7 7 1 1 *
FURTHER MATHEMATICS 9231/13
Paper 1 Further Pure Mathematics 1 May/June 2023
2 hours
You must answer on the question paper.
You will need: List of formulae (MF19)
INSTRUCTIONS
● Answer all questions.
● Use a black or dark blue pen. You may use an HB pencil for any diagrams or graphs.
● Write your name, centre number and candidate number in the boxes at the top of the page.
● Write your answer to each question in the space provided.
● Do not use an erasable pen or correction fluid.
● Do not write on any bar codes.
● If additional space is needed, you should use the lined page at the end of this booklet; the question
number or numbers must be clearly shown.
● You should use a calculator where appropriate.
● You must show all necessary working clearly; no marks will be given for unsupported answers from a
calculator.
● Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in
degrees, unless a different level of accuracy is specified in the question.
INFORMATION
● The total mark for this paper is 75.
● The number of marks for each question or part question is shown in brackets [ ].
This document has 16 pages. Any blank pages are indicated.
DC (CE) 311426/2
© UCLES 2023 [Turn over
, 2
BLANK PAGE
© UCLES 2023 9231/13/M/J/23
, 3
1 Prove by mathematical induction that, for all positive integers n, 5 3n + 32 n - 33 is divisible by 31. [6]
....................................................................................................................................................................
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....................................................................................................................................................................
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....................................................................................................................................................................
© UCLES 2023 9231/13/M/J/23 [Turn over
, 4
2 (a) Use standard results from the list of formulae (MF19) to show that
n
/ (6r + 6r - 5) = an + bn + cn ,
2 3 2
r=1
where a, b and c are integers to be determined. [2]
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............................................................................................................................................................
............................................................................................................................................................
............................................................................................................................................................
n
/ 6r r++6rr- 5 in terms of n.
2
(b) Use the method of differences to find 2 [4]
r=1
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© UCLES 2023 9231/13/M/J/23
