AS & A Level Further Mathematics (9231) — May–June 2025, Paper 34

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

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Question 1 · 5 marks · Further Mechanics: Hooke's law and elastic potential energy (vertical motion)

A light spring of natural length a and modulus of elasticity 20mg is placed so that it stands vertically on a horizontal plane. The lower end of the spring is fixed to the plane.…

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Question 2 · 7 marks · Further Mechanics: motion of a particle with variable resistance (Newton's second law, separation of variables)

A particle P of mass m kg moves along a horizontal straight line against a resistive force of magnitude 2mv^3 N, where v m s^-1 is the velocity of P at time t s. When t = 0, v =…

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Question 3 · 7 marks · Further Mechanics: projectiles (equation of trajectory)

A particle P is projected with speed u at an angle alpha above the horizontal from a point O on a horizontal plane and moves freely under gravity. The horizontal and vertical…

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Question 4 · 6 marks · Further Mechanics: circular motion (particle on the inner surface of a cone)

A hollow cone with a smooth inner surface is fixed with its vertex O downwards. The semi-vertical angle of the cone is alpha, where tan(alpha) = 3/4. A light inextensible string…

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Question 5 · 8 marks · Further Mechanics: centre of mass of a composite plane lamina

ABCD is a uniform square lamina of side 6a. Points E and F are on DC and AB respectively and are such that DE = FB = h. The quadrilateral BCEF is removed from the square lamina…

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Question 6 · 9 marks · Further Mechanics: oblique impact of two smooth spheres

Two uniform smooth spheres A and B of equal radii have masses 5m and 4m respectively. Sphere A is moving with speed u on a horizontal surface when it collides with sphere B which…

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Question 7 · 8 marks · Further Mechanics: motion in a vertical circle (tension and energy)

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…

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