Cambridge International AS & A Level
* 2 0 0 4 4 0 4 3 4 7 *
FURTHER MATHEMATICS 9231/21
Paper 2 Further Pure Mathematics 2 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 (LK) 327441
© UCLES 2023 [Turn over
, 2
BLANK PAGE
© UCLES 2023 9231/21/M/J/23
, 3
1 (a) Show that the system of equations
x + 2y + 3z = 1,
4x + 5y + 6z = 1,
7x + 8y + 9z = 1,
does not have a unique solution. [2]
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(b) Show that the system of equations in part (a) is consistent. Interpret this situation geometrically.
[3]
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© UCLES 2023 9231/21/M/J/23 [Turn over
, 4
2 Use the substitution z = x + y to find the solution of the differential equation
d y 1 + 3x + 3y
=
d x 3x + 3y - 1
for which y = 0 when x = 1. Give your answer in the form a ln (x + y) + b (x - y) + c = 0 , where a, b
and c are constants to be determined. [7]
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© UCLES 2023 9231/21/M/J/23
* 2 0 0 4 4 0 4 3 4 7 *
FURTHER MATHEMATICS 9231/21
Paper 2 Further Pure Mathematics 2 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 (LK) 327441
© UCLES 2023 [Turn over
, 2
BLANK PAGE
© UCLES 2023 9231/21/M/J/23
, 3
1 (a) Show that the system of equations
x + 2y + 3z = 1,
4x + 5y + 6z = 1,
7x + 8y + 9z = 1,
does not have a unique solution. [2]
............................................................................................................................................................
............................................................................................................................................................
............................................................................................................................................................
............................................................................................................................................................
............................................................................................................................................................
............................................................................................................................................................
............................................................................................................................................................
............................................................................................................................................................
(b) Show that the system of equations in part (a) is consistent. Interpret this situation geometrically.
[3]
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............................................................................................................................................................
© UCLES 2023 9231/21/M/J/23 [Turn over
, 4
2 Use the substitution z = x + y to find the solution of the differential equation
d y 1 + 3x + 3y
=
d x 3x + 3y - 1
for which y = 0 when x = 1. Give your answer in the form a ln (x + y) + b (x - y) + c = 0 , where a, b
and c are constants to be determined. [7]
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© UCLES 2023 9231/21/M/J/23
