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Cambridge International AS & A LevelFURTHER MATHEMATICS 9231/03Paper 3 Further Mechanics For examination from 2020SPECIMEN PAPER 1 hour 30 minutesYou 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 [ ].

*0123456789*© UCLES 2017 [Turn overThis document has 14 pages.

Blank pages are indicated.


2 BLANK PAGE 9231/03/SP/20© UCLES 2017 1 A child’s toy consists of a uniform solid circular cone, of vertical height 3r and radius r, and a uniform solid hemisphere of radius r. The circular bases of the cone and the hemisphere are joined together so that they coincide.

The cone and the hemisphere are made of the same material.

3 Show that the centre of mass of the toy is at a distance 10 27 r from the vertex of the cone.
[4] .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... 9231/03/SP/20© UCLES 2017[Turn over 4 2 A light elastic string has natural length a and modulus of elasticity 24mg.
One end of the string is attached to a fixed point A.

The other end of the string is attached to a particle of mass 2m.

(a) Find, in terms of a, the extension of the string when the particle hangs freely in equilibrium [2] below A.
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Find, in terms of a, the distance of the particle below A when it first comes to instantaneous rest.
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............................................................................................................................................................ ............................................................................................................................................................ 9231/03/SP/20© UCLES 2017[Turn over 6 3 A particle P of mass m kg falls from rest under gravity.
There is a resistive force of magnitude mkv2 N, where v m s−1 is the speed of P after it has fallen a distance x m and k is a positive constant.

(a) By solving an appropriate differential equation, show that v2 = g k (1 – e–2kx ).
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It is now given that k = 0.01.

The speed of P when x becomes large approaches V m s–1.

(b) (i) Find V correct to 2 decimal places.
[1] .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... (ii) Hence find how far P has fallen when its speed is 2 1 V m s–1.
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Sphere B is at rest on a smooth horizontal surface.

Sphere A is moving on the surface with speed u at an angle of 30° to the line of centres of A and B when it collides with B (see diagram).

The coefficient of restitution between the spheres is e. (a) Show that the speed of B after the collision is 3 6 u(1 + e) and find the speed of A after the collision.
[6] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ 9231/03/SP/20© UCLES 2017 9 ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (b) Given that e = 3 1 , find the loss of kinetic energy as a result of the collision.
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The other end of the string is attached to a fixed point O.

10 (a) O 60° ω a P The particle P moves in a horizontal circle with a constant angular speed ω with the string inclined at 60° to the downward vertical through O (see diagram).

g2 Show that ω2 = a .
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It is then projected horizontally so that it begins to move in a vertical circle with centre O.

When the string makes an angle of 60° with the downward vertical through O, the angular speed of P is makes an angle θ with the upward vertical through O. g2 a .

The string first goes slack when OP Find the value of cos θ.
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The horizontal and vertical displacements of P from O at a subsequent time t are denoted by x and y respectively.

(a) Derive the equation of the trajectory of P in the form y = x tan α – 2 gx 2 2 u sec2 α.
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(b) The greatest height of P above the plane is denoted by H. When P is at a height of 4 3 H, it has travelled a horizontal distance d. Given that tan α = 2, find, in terms of H, the two possible values of d. 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............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ 9231/03/SP/20© UCLES 2017 14 Additional page If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ 9231/03/SP/20© UCLES 2017Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible.
Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity.Cambridge Assessment International Education is part of the Cambridge Assessment Group.

Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which itself is a department of the University of Cambridge.