AS & A Level Further Mathematics (9231) — October–November 2023, Paper 32

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 - conical pendulum (circular motion under gravity)

One end of a light inextensible string of length a is attached to a fixed point O. The other end of the string is attached to a particle of mass m. The string is taut and makes an…

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Question 2 · 6 marks · Further Mechanics - motion in a straight line with variable force (v dv/dx)

A particle P of mass 0.5 kg moves in a straight line. At time t s the velocity of P is v m s^-1 and its displacement from a fixed point O on the line is x m. The only forces…

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Question 3 · 7 marks · Further Mechanics - centre of mass of a composite lamina (removed portion)

A uniform lamina is in the form of an isosceles triangle ABC in which AC = 2a and angle ABC = 90 degrees. The point D on AB is such that the ratio DB : AB = 1 : k. The point E on…

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Question 4 · 7 marks · Further Mechanics - oblique impact of two smooth spheres (line of centres)

Two smooth vertical walls meet at right angles. The smooth sphere A, with mass m, is at rest on a smooth horizontal surface and is at a distance d from each wall. An identical…

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Question 5 · 8 marks · Further Mechanics - motion in a vertical circle (energy and radial Newton's second law)

A bead of mass m moves on a smooth circular wire, with centre O and radius a, in a vertical plane. The bead has speed vA when it is at the point A where OA makes an angle alpha…

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Question 6 · 9 marks · Further Mechanics - projectile motion (Cartesian equation of the 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 7 · 9 marks · Further Mechanics - energy conservation with an elastic spring (rod and spring system)

A particle P of mass m is attached to one end of a light rod of length 3a. The other end of the rod is able to pivot smoothly about the fixed point A. The particle is also…

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