Cambridge International AS & A Level
* 6 0 8 1 9 6 6 3 5 3 *
FURTHER MATHEMATICS 9231/32
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) 316268/2
© UCLES 2023 [Turn over
, 2
1 One end of a light elastic string, of natural length a and modulus of elasticity 3mg, is attached to a
fixed point O. The other end of the string is attached to a particle P of mass m. The string hangs with P
vertically below O. The particle P is pulled vertically downwards so that the extension of the string is
2a. The particle P is then released from rest.
(a) Find the speed of P when it is at a distance 34 a below O. [3]
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
(b) Find the initial acceleration of P when it is released from rest. [2]
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
© UCLES 2023 9231/32/M/J/23
, 3
2
P
u
a
i
A particle P of mass m is moving with speed u on a fixed smooth horizontal surface. It collides at an
angle a with a fixed smooth vertical barrier. After the collision, P moves at an angle i with the barrier,
where tan i = 12 (see diagram). The coefficient of restitution between P and the barrier is e. The particle
P loses 20% of its kinetic energy as a result of the collision.
Find the value of e. [5]
....................................................................................................................................................................
....................................................................................................................................................................
....................................................................................................................................................................
....................................................................................................................................................................
....................................................................................................................................................................
....................................................................................................................................................................
....................................................................................................................................................................
....................................................................................................................................................................
....................................................................................................................................................................
....................................................................................................................................................................
....................................................................................................................................................................
....................................................................................................................................................................
....................................................................................................................................................................
....................................................................................................................................................................
....................................................................................................................................................................
© UCLES 2023 9231/32/M/J/23 [Turn over
, 4
3 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 i 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 u. It then starts to move along a circular path
in a vertical plane. The string goes slack when P is at B, where angle AOB is 90° and the speed of
P is 45 ag .
(a) Find the value of sin i . [2]
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
(b) Find, in terms of m and g, the tension in the string when P is at A. [5]
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
© UCLES 2023 9231/32/M/J/23
* 6 0 8 1 9 6 6 3 5 3 *
FURTHER MATHEMATICS 9231/32
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) 316268/2
© UCLES 2023 [Turn over
, 2
1 One end of a light elastic string, of natural length a and modulus of elasticity 3mg, is attached to a
fixed point O. The other end of the string is attached to a particle P of mass m. The string hangs with P
vertically below O. The particle P is pulled vertically downwards so that the extension of the string is
2a. The particle P is then released from rest.
(a) Find the speed of P when it is at a distance 34 a below O. [3]
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
(b) Find the initial acceleration of P when it is released from rest. [2]
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
© UCLES 2023 9231/32/M/J/23
, 3
2
P
u
a
i
A particle P of mass m is moving with speed u on a fixed smooth horizontal surface. It collides at an
angle a with a fixed smooth vertical barrier. After the collision, P moves at an angle i with the barrier,
where tan i = 12 (see diagram). The coefficient of restitution between P and the barrier is e. The particle
P loses 20% of its kinetic energy as a result of the collision.
Find the value of e. [5]
....................................................................................................................................................................
....................................................................................................................................................................
....................................................................................................................................................................
....................................................................................................................................................................
....................................................................................................................................................................
....................................................................................................................................................................
....................................................................................................................................................................
....................................................................................................................................................................
....................................................................................................................................................................
....................................................................................................................................................................
....................................................................................................................................................................
....................................................................................................................................................................
....................................................................................................................................................................
....................................................................................................................................................................
....................................................................................................................................................................
© UCLES 2023 9231/32/M/J/23 [Turn over
, 4
3 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 i 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 u. It then starts to move along a circular path
in a vertical plane. The string goes slack when P is at B, where angle AOB is 90° and the speed of
P is 45 ag .
(a) Find the value of sin i . [2]
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
(b) Find, in terms of m and g, the tension in the string when P is at A. [5]
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
© UCLES 2023 9231/32/M/J/23
