Cambridge International AS & A Level
* 1 3 7 0 4 5 3 0 4 4 *
FURTHER MATHEMATICS 9231/33
Paper 3 Further Mechanics May/June 2023
1 hour 30 minutes
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.
● Where a numerical value for the acceleration due to gravity (g) is needed, use 10 m s–2.
INFORMATION
● The total mark for this paper is 50.
● 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/SG) 316276/1
© UCLES 2023 [Turn over
, 2
1 A particle P of mass m is attached to one end of a light inextensible string of length a. The other end
of the string is attached to a fixed point O. The particle P is held at the point A, where OA makes an
angle a with the downward vertical through O, and with the string taut. The particle P is projected
perpendicular to OA in an upwards direction with speed 3ag . It then starts to move along a circular
path in a vertical plane. The string goes slack when P is at B, where OB makes an angle i with the
upward vertical.
Given that cos a = 45 , find the value of cos i . [4]
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© UCLES 2023 9231/33/M/J/23
, 3
2 One end of a light elastic string, of natural length a and modulus of elasticity mmg , is attached to a
fixed point O. The string lies on a smooth horizontal surface. A particle P of mass m is attached to the
other end of the string. The particle P is projected in the direction OP. When the length of the string is
4 5 1
3 a , the speed of P is 2ag . When the length of the string is 3 a , the speed of P is 2 2ag .
Find the value of m. [4]
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© UCLES 2023 9231/33/M/J/23 [Turn over
* 1 3 7 0 4 5 3 0 4 4 *
FURTHER MATHEMATICS 9231/33
Paper 3 Further Mechanics May/June 2023
1 hour 30 minutes
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.
● Where a numerical value for the acceleration due to gravity (g) is needed, use 10 m s–2.
INFORMATION
● The total mark for this paper is 50.
● 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/SG) 316276/1
© UCLES 2023 [Turn over
, 2
1 A particle P of mass m is attached to one end of a light inextensible string of length a. The other end
of the string is attached to a fixed point O. The particle P is held at the point A, where OA makes an
angle a with the downward vertical through O, and with the string taut. The particle P is projected
perpendicular to OA in an upwards direction with speed 3ag . It then starts to move along a circular
path in a vertical plane. The string goes slack when P is at B, where OB makes an angle i with the
upward vertical.
Given that cos a = 45 , find the value of cos i . [4]
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© UCLES 2023 9231/33/M/J/23
, 3
2 One end of a light elastic string, of natural length a and modulus of elasticity mmg , is attached to a
fixed point O. The string lies on a smooth horizontal surface. A particle P of mass m is attached to the
other end of the string. The particle P is projected in the direction OP. When the length of the string is
4 5 1
3 a , the speed of P is 2ag . When the length of the string is 3 a , the speed of P is 2 2ag .
Find the value of m. [4]
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© UCLES 2023 9231/33/M/J/23 [Turn over
