Official June 2024 AQA A-level FURTHER MATHEMATICS 7367/3M Paper 3 Mechanics Merged Question Paper + Mark Scheme Ace your Mocks!!! G/LM/Jun24/G4006/V7 7367/3M (JUN2473673M01) A-level FURTHER MATHEMATICS Paper 3 Mechanics Friday 7 June 2024 Afternoon Time allowed: 2 hours Materials l You must have the AQA Formulae and statistical tables booklet for A‑level Mathematics and A‑level Further Mathematics. l You should have a graphical or scientific calculator that meets the requirements of the specification. l You must ensure you have the other optional Question Paper/Answer Book for which you are entered (either Discrete or Statistics). You will have 2 hours to complete both papers. Instructions l Use black ink or black ball‑point pen. Pencil should only be used for drawing. l Fill in the boxes at the top of this page. l Answer all questions. l You must answer each question in the space provided for that question. If you require extra space for your answer(s), use the lined pages at the end of this book. Write the question number against your answer(s). l Do not write outside the box around each page or on blank pages. l Show all necessary working; otherwise marks for method may be lost. l Do all rough work in this book. Cross through any work that you do not want to be marked. Information l The marks for questions are shown in brackets. l The maximum mark for this paper is 50. Advice l Unless stated otherwise, you may quote formulae, without proof, from the booklet. l You do not necessarily need to use all the space provided. For Examiner’s Use Question Mark 1 2 3 4 5 6 7 8 9 TOTAL Please write clearly in block capitals. Centre number Candidate number Surname _________________________________________________________________________ Forename(s) _________________________________________________________________________ Candidate signature _________________________________________________________________________ I declare this is my own work. 2 Do not write outside the box (02) G/Jun24/7367/3M Answer all questions in the spaces provided. 1 A particle moves in a circular path so that at time t seconds its position vector, r metres, is given by r = 4sin(2t)i + 4cos(2t)j Find the velocity of the particle, in m s–1, when t = 0 Circle your answer. [1 mark] 8i –8j 8j 8i – 8j 2 As a particle moves along a straight horizontal line, it is subjected to a force F newtons that acts in the direction of motion of the particle. At time t seconds, F = t 5 Calculate the magnitude of the impulse on the particle between t = 0 and t = 3 Circle your answer. [1 mark] 0.3 N s 0.6 N s 0.9 N s 1.8 N s 3 Do not write outside the box (03) G/Jun24/7367/3M Turn over U 3 A conical pendulum consists of a light string and a particle of mass m kg The conical pendulum completes horizontal circles with radius r metres and angular speed ω radians per second. The string makes an angle θ with the downward vertical. The tension in the string is T newtons. The conical pendulum and the forces acting on the particle are shown in the diagram. mg T θ Which one of the following statements is correct? Tick () one box. [1 mark] T cosθ = mrω2 T sinθ = mrω2 T cosθ = mω2 r T sinθ = mω2 r 4 Do not write outside the box (04) G/Jun24/7367/3M 4 A particle of mass 3 kg is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point on a smooth horizontal surface. The particle is set into motion so that it moves with a constant speed 4 m s–1 in a circular path with radius 0.8 metres on the horizontal surface. 4 (a) Find the acceleration of the particle. [2 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 4 (b) Find the tension in the string. [1 mark] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 4 (c) Show that the angular speed of the particle is 48 revolutions per minute correct to two significant figures. [2 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 5 Do not write outside the box (05) G/Jun24/7367/3M Turn over U 5 When a sphere of radius r metres is falling at v m s–1 it experiences an air resistance force F newtons. The force is to be modelled as F = krαvβ where k is a constant with units kg m–2 5 (a) State the dimensions of F [1 mark] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 5 (b) Use dimensional analysis to find the value of α and the value of β [3 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Turn over U 6 Do not write outside the box (06) G/Jun24/7367/3M 6 In this question use g = 9.8 m s–2 A light elastic string has natural length 3 metres and modulus of elasticity 18 newtons. One end of the elastic string is attached to a particle of mass 0.25 kg The other end of the elastic string is attached to a fixed point O The particle is released from rest at a point A, which is 4.5 metres vertically below O 6 (a) Calculate the elastic potential energy of the string when the particle is at A [2 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 6 (b) The point B is 3 metres vertically below O Calculate the gravitational potential energy gained by the particle as it moves from A to B [2 marks]