AS & A Level Further Mathematics (9231) — May–June 2025, Paper 33

7 questions · every question answerable online with instant point-by-point AI marking against the official mark scheme.

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Question 1 · 3 marks · Further Mechanics: motion in a horizontal circle (conical pendulum)

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 moves in a…

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Question 2 · 8 marks · Further Mechanics: equilibrium and energy with elastic strings (Hooke's law, elastic PE)

A particle P of mass m is attached to one end of a light elastic string of natural length a and modulus of elasticity mg. The other end of the string is attached to a fixed point…

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Question 3 · 8 marks · Further Mechanics: motion with resistance proportional to v^2 (using v dv/dx)

A ball of mass m kg is projected vertically upwards with initial speed U m s^-1 and moves under gravity. At time t s after projection, the ball has travelled a distance x m and…

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Question 4 · 4 marks · Further Mechanics: centre of mass of a composite lamina; condition for toppling

An object consists of a uniform lamina with a particle attached. The uniform lamina ABCEFD of mass m is formed from a rectangle ABCD and an isosceles triangle CEF, where F is the…

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Question 5 · 8 marks · Further Mechanics: motion in a vertical circle (loss of contact, energy + N2L radially)

A hollow cylinder of radius r is fixed with its axis horizontal. Points A, B and O are in the same vertical plane perpendicular to the axis of the cylinder, with A and B on the…

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Question 6 · 8 marks · Further Mechanics: oblique impact of two smooth spheres (momentum + restitution along the line of centres)

Two identical uniform smooth spheres A and B, each with mass m, are moving on a horizontal surface with speeds 2u and u respectively when they collide. Immediately before the…

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Question 7 · 11 marks · Further Mechanics: projectile motion (velocity components)

A particle P is projected from a point O with speed U at an angle 45 degrees above the horizontal and moves freely under gravity. State the vertical and horizontal components of…

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