AS & A Level Further Mathematics (9231) — May–June 2023, 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 · 4 marks · Further Mechanics: circular motion in a vertical plane (string goes slack; energy conservation and Newton's second law)

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…

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Question 2 · 4 marks · Further Mechanics: elastic strings — elastic potential energy and conservation of energy

One end of a light elastic string, of natural length a and modulus of elasticity lambda m g, is attached to a fixed point O. The string lies on a smooth horizontal surface. A…

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Question 3 · 7 marks · Further Mechanics: centre of mass of a composite lamina (by subtraction, taking moments)

A uniform lamina is in the form of a triangle ABC, with AC = 8a, BC = 6a and angle ACB = 90 degrees. The point D on AC is such that AD = 3a. The point E on CB is such that CE = x…

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Question 4 · 8 marks · Further Mechanics: oblique impact of two smooth spheres (momentum and restitution along the line of centres)

Two identical smooth uniform spheres A and B each have mass m. The two spheres are moving on a smooth horizontal surface when they collide with speeds u and 2u respectively.…

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Question 5 · 8 marks · Further Mechanics: conical pendulum with an elastic string (motion in a horizontal circle + Hooke's law)

One end of a light elastic string, of natural length 12a and modulus of elasticity k m g, is attached to a fixed point O. The other end of the string is attached to a particle of…

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Question 6 · 10 marks · Further Mechanics: motion in a straight line with resistance proportional to speed (variable force, dv/dt)

A particle of mass m kg falls vertically under gravity, from rest. At time t s, P has fallen x m and has velocity v m s^-1. The only forces acting on P are its weight and a…

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Question 7 · 9 marks · Further Mechanics: projectile motion — condition for collision of two moving bodies

The points O and P are on a horizontal plane, a distance 8 m apart. A ball is thrown from O with speed u m s^-1 at an angle theta above the horizontal, where tan theta = 4/3. At…

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