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AS & A Level FURTHER MATHEMATICS 9231/12 Paper 1 Further Pure Mathematics 1 May/June 2023 AUTHENTIC MARKING SCHEME ATTACHED
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FURTHER MATHEMATICS
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FURTHER MATHEMATICS
FURTHER MATHEMATICS 9231/12
Paper 1 Further Pure Mathematics 1 May/June 2023
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Cambridge International AS & A Level
* 5 5 7 2 6 8 0 2 0 5 *
FURTHER MATHEMATICS 9231/12
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 20 pages. Any blank pages are indicated.
DC (PQ) 311427/1
© UCLES 2023 [Turn over
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BLANK PAGE
© UCLES 2023 9231/12/M/J/23
, 3
3 0
1 Let A = e o.
1 1
(a) Prove by mathematical induction that, for all positive integers n,
2 # 3n 0
2A n = e n o. [5]
3 -1 2
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(b) Find, in terms of n, the inverse of A n . [2]
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© UCLES 2023 9231/12/M/J/23 [Turn over
, 4
2 The cubic equation x 3 + 4x 2 + 6x + 1 = 0 has roots a , b , c .
(a) Find the value of a 2 + b 2 + c 2 . [2]
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(b) Use standard results from the list of formulae (MF19) to show that
n
/`(a + r) 2
+ (b + r) 2 + (c + r) 2j = n (n 2 + an + b) ,
r=1
where a and b are constants to be determined. [6]
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© UCLES 2023 9231/12/M/J/23