Cambridge A Level Mathematics - Further 9231 — 2024 May/June Paper 3 · Variant 2

1 u A B i m 5m Two smooth uniform spheres A and B of equal radii have masses m and 5m respectively. Sphere A is moving on a smooth horizontal surface with speed u when it collides with sphere B which is at rest on the surface. Immediately before the collision, A’s direction of motion makes an angle of i with the line of centres. After the collision, the kinetic energies of A and B are equal. The coefficient of restitution between the spheres is 1.2 Find the value of tani. [6] .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... ....................................................................................................................................................................

Mark scheme: 1 Along line of centres, PCLM: 5 B A mv mv mu NEL: 1 2 cos B A v v u    M1 Signs consistent with PCLM equation. cos , cos 4 4     B A u u v v A1 Perpendicular to line of centres: speed of A is sin u B1   2 2 2 1 1 cos sin 5 cos 2 4 2 4                           u u m u m M1 Equate final kinetic energies, 3 terms, correct masses.   2 4 5 cos   , 2 1 2 5 cos , tan     A1 6

More questions on Momentum

2 The points A and B are at the same horizontal level a distance 4a apart. The ends of a light elastic string, of natural length 4a and modulus of elasticity m, are attached to A and B. A particle P of mass m is attached to the midpoint of the string. The system is in equilibrium with P at a distance 3a2 below M, the midpoint of AB. (a) Find m in terms of m and g. [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ The particle P is pulled down vertically and released from rest at a distance 8a3 below M. (b) Find, in terms of a and g, the speed of P as it passes through M in the subsequent motion. [4] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................

Mark scheme: 2(a) T mg Hooke’s law: 5 2 2 2 4             a T a a B1 Equate and use 3 5 cos :   10 3 mg   A1 3 2(b) EPE loss = 2 1 20 4 2 4 3        a a a B1 80 27 mga Energy equation: 2 1 8 80 ' ' 2 3 27   a mv mg mga M1 A1 All 3 terms required, dimensionally correct, their . 4 3 3  ga v A1 Any equivalent form. 4

More questions on Hooke's law

3 At time t = 0 seconds, a particle P is projected with speed u m s -1 at an angle 60° above the horizontal from a point O. In the subsequent motion P moves freely under gravity. The direction of motion of P when t = 5 is perpendicular to its direction of motion when t = 15 . Find the value of u. [5] .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... ....................................................................................................................................................................

Mark scheme: 3 u u B1 If  is direction of velocity at  sin60 5 5, tan cos60      u g t u M1* Accept equivalent for 15  t . For perpendicular directions, sin60 5 sin60 15 1 cos60 cos60       u g u g u u M1dep Multiply two expressions involving relevant velocities and equate to – 1. Simplify: 3 1 2 2 2 2 4 4 75 10 3 0, 100 3 7500 0 u g ug u u u        M1 Simplify to quadratic in u (may see g ). 5 3  u g A1 OE. Accept 50 3 or 86.6. 5

More questions on Motion of a projectile

4 P Q a O a A ring of weight W, with radius a and centre O, is at rest on a rough surface that is inclined to the horizontal at an angle a where tan a = 12 . The plane of the ring is perpendicular to the inclined surface and parallel to a line of greatest slope of the surface. The point P on the circumference of the ring is such that OP is parallel to the surface. A light inextensible string is attached to P and to the point Q, which is on the surface, such that PQ is horizontal (see diagram). The points O, P and Q are in the same vertical plane. The system is in limiting equilibrium and the coefficient of friction between the ring and the surface is n. (a) Find, in terms of W, the tension in the string PQ. [4] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (b) Find the value of n. [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................

Mark scheme: 4(a) Frictional force F and normal reaction R at point of contact of ring with plane. Resolve parallel to plane: cos sin     F T W M1 Only allow cos/sin errors for T and W components, sign errors. Accept equations for vertical and horizontal (both needed). Moments about O: sin  Fa Ta B1 Combine and substitute for : M1 Expression for T in terms of W . 1 3 T W  A1 CAO Alternative solution for question 4(a) Moments about point where ring touches plane: sin cos sin      Ta Ta Wa M1 A1 Only allow cos/sin errors, sign errors. Must be dimensionally correct. Rearrange and substitute for : M1 Expression for T in terms of W . 1 3 T W  A1 CAO 4 4(b) Resolve perpendicular to plane: sin cos     R T W M1 Only allow cos/sin errors for T and W components, sign errors. Use   F R and combine to reach an equation in  only. M1 From part (a), cos sin F T W    or sin  F T . 1 7   A1 3

More questions on Equilibrium of a rigid body

5 Two particles A and B of masses m and km respectively are connected by a light inextensible string of length a. The particles are placed on a rough horizontal circular turntable with the string taut and lying along a radius of the turntable. Particle A is at a distance a from the centre of the turntable and particle B is at a distance 2a from the centre of the turntable. The coefficient of friction between each particle and the turntable is 1.5 2 g When the turntable is made to rotate with angular speed , the system is in limiting equilibrium. 5 a (a) Find the tension in the string, in terms of m and g. [4] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (b) Find the value of k. [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................

Mark scheme: 5(a) For A: 2     A F T m a M1 Only allow sign errors. 1 5 A F mg mg    B1 Accept with g replaced by 10. Combine: 1 4 5 25 T mg mg   M1 To reach an equation in T and mg only. Accept with g replaced by 10. 1 25 T mg  A1 CAO 4 5(b) For B: 2 2     B F T km a M1 Only allow sign errors. 1 5 B F kmg kmg    and combine to find k M1 To reach an equation in k only. 1 3 k  A1 3

More questions on Circular motion

6 A particle P of mass 2 kg moving on a horizontal straight line has displacement x m from a fixed point O on the line and velocity v m s -1 at time t s. The only horizontal force acting on P has magnitude 1 2 -t 10 ( 2v - 1) e N and acts towards O. When t = 0 , x = 1 and v = 3 . (a) Find an expression for v in terms of t. [5] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (b) Find an expression for x in terms of t. [4] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................

Mark scheme: 6(a)   2 d 1 2 2 1 e d 10    t v v t so   2 d 1 e d 20 2 1    t v t v   e 2 1     t p q A v *M1 Separate variables and attempt to integrate both sides. Where p and q are constants.   1 1 e 2 2 1 20      t A v A1 AEF 3 0, 3, 20          t v A DM1 Substituting the boundary condition and obtain a value. 1 5e 2 3e 1    t t v *M1 A1 Find v in terms of t . AEF. 5 6(b) Integrate:   ln(re )      t x pt q s B *M1   1 5ln(3e 1) 2 3     t x t B A1 AEF 5 0, 1, 1 ln2 3    t x B DM1 Substituting the boundary condition and obtain a value. 1 5 (3e 1) 1 ln 2 3 2    t x t A1 AEF 4

More questions on Differential equations

7 A smooth sphere with centre O and of radius a is fixed to a horizontal plane. A particle P of mass m is projected horizontally from the highest point of the sphere with speed u, so that it begins to move along the surface of the sphere. The particle P loses contact with the sphere at the point Q on the sphere, where OQ makes an angle i with the upward vertical through O. u 2 + 2ag (a) Show that cos i = . [4] 3ag ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ It is given that cos i = 56 . (b) Find, in terms of a and g, an expression for the vertical component of the velocity of P just before it hits the horizontal plane to which the sphere is fixed. [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (c) Find an expression for the time taken by P to fall from Q to the plane. Give your answer in the a form k , stating the value of k correct to 3 significant figures. [2] g ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ Additional page If you use the following page to complete the answer to any question, the question number must be clearly shown. ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................

Mark scheme: 7(a) Energy: 1 1 2 2 2 2 1 cos mu mv mga     *M1 m must be present, dimensionally correct, no missing terms. Allow sin instead of cos . Allow sign errors. N2L: 2 cosmv mg a B1 No reaction when P loses contact. Eliminate 2 v DM1 2 2 cos 3 u ag ag    A1 AG 4 7(b) Vertical component of velocity of P when it leaves the sphere: sin  v 55 216          ag *B1 Must not come from u .     2 2 sin 2 1 cos       V v g a DM1 Use of 2 2 ' 2 .   v u as Allow sin for cos. Allow sign errors. 847 216  ag V A1 AEF 3 Question Answer Marks Guidance 7(c) 1 847 55 216 216           ag ag t g M1   1 847 55 1.48 6 6   a a g g A1 2

More questions on Motion of a projectile