Cambridge A Level Mathematics - Further 9231 — 2022 May/June Paper 3 · Variant 2

1 A i 7.5 N A particle of weight 10 N is attached to one end of a light elastic string. The other end of the string is attached to a fixed point A on a horizontal ceiling. A horizontal force of 7.5 N acts on the particle. In the equilibrium position, the string makes an angle i with the ceiling (see diagram). The string has natural length 0.8 m and modulus of elasticity 50 N. (a) Find the tension in the string. [2] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (b) Find the vertical distance between the particle and the ceiling. [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................

Mark scheme: 1(a) B1   1 2 2 2 7.5 10 T   = 12.5 N B1 2 1(b) Hooke’s law: 50 0.8 x T  , 0.2 x  B1   10 0.8 sin 1 12.5 x    M1 4 Vertical distance 0.8 5   A1 3

More questions on Hooke's law

2 One end of a light inextensible string of length a is attached to a fixed point O. A particle of mass m is attached to the other end of the string. The particle is held at the point A with the string taut. The angle between OA and the downward vertical is equal to a, where cos a = 45 . The particle is projected from A, perpendicular to the string in an upwards direction, with a speed 3ga . It then moves along a circular path in a vertical plane. The string first goes slack when it makes an angle i with the upward vertical through O. Find the value of cosi. [5] .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... ....................................................................................................................................................................

Mark scheme: 2   2 1 1 3 cos cos 2 2 mv m ga mga        M1 A1 Energy equation.   2 cos m T mg v a    B1 N2L 4 cos 3 2 cos 2 5 ag ga ga ga       M1 Combine to find cos. 7 cos 15  A1 5

More questions on Circular motion

3 A particle P is moving in a horizontal straight line. Initially P is at the point O on the line and is moving -1 4000 -2 with velocity 25 m s . At time t s after passing through O, the acceleration of P is 3 m s in ( t5 + 4 ) the direction PO. The displacement of P from O at time t is x m. Find an expression for x in terms of t. [5] .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... ....................................................................................................................................................................

Mark scheme: 3   3 4000 5 4 dv dt t   ;     2 400 5 4 v A t    M1 A1 Integrate. Constant of integration needed for A1. 0, 25 25 25 0 t v A      M1 Find constant.   1 : 80 5 4 dx v x t B dt      0, 0 20 x t B    M1 Integrate and find constant. 80 100 20 5 4 5 4 t x t t             A1 5

More questions on Linear motion under a variable force

4 2a h a A B An object is composed of a hemispherical shell of radius 2a attached to a closed hollow circular cylinder of height h and base radius a. The hemispherical shell and the hollow cylinder are made of the same uniform material. The axes of symmetry of the shell and the cylinder coincide. AB is a diameter of the lower end of the cylinder (see diagram). (a) Find, in terms of a and h, an expression for the distance of the centre of mass of the object from AB. [4] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ The object is placed on a rough plane which is inclined to the horizontal at an angle i, where tan i = 23 . The object is in equilibrium with AB in contact with the plane and lying along a line of greatest slope of the plane. (b) Find the set of possible values of h, in terms of a. [4] .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... ....................................................................................................................................................................

Mark scheme: 4(a) Area Centre of mass from AB Cylinder 2 2π 2π ah a  1 2 h Shell   2 2π 2 a h a  M1 Moments equation, condone missing ends of cylinder. One expression on the RHS correct. Moments about AB          2 2 2 2 1 2π 2π 2π 2 2π 2 2π 2π 2 x ah a a a h a ah a h              A1 A1 One correct expression on RHS correct scores A1.   2 2 2 10 8 8 h a x h ah ah a        2 2 9 8 2 5 h ah a x h a     A1 4 4(b) tan a x  B1   2 2 9 8 3 2 5 2 h ah a x a h a      2 2 6 7 0 h ah a    M1 Form inequality and rearrange to quadratic, condone equation.    7 0 h a h a    M1 Attempt to solve, condone equation.   7a h a    A1 4

More questions on Equilibrium of a rigid body

5 C D i a i B 3a A A light inextensible string AB passes through two small holes C and D in a smooth horizontal table where AC = 3a and DB = a. A particle of mass m is attached at the end A and moves in a horizontal circle with angular velocity ~. A particle of mass 34 m is attached to the end B and moves in a horizontal circle with angular velocity k~. AC makes an angle i with the downward vertical and DB makes an angle i with the horizontal (see diagram). Find the value of k. [7] .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... ....................................................................................................................................................................

Mark scheme: 5 For A: 2 sin T mr    M1 N2L horizontal. 3 sin r a   B1 Correct expression for radius. 2 3 T m a   A1 Similarly, for B: 2 2 3 cos 4 T m r k     M1 N2L horizontal 2 2 3 4 T mak   A1 2 2 2 3 3 4 m a mak     M1 Equate expressions for T. 2 4, 2 k k   A1 Question Answer Marks Guidance 5 Alternative method for question 5 For A: 2 cos , sin T mg T mr      M1 N2L horizontal and vertical. 3 sin r a   B1 Correct expression for radius. 2 2 5 5 3 , 4 12 g T m a mg a        A1 Combine to obtain expression for 2 . Similarly, for B: 2 2 3 cos 4 T m r k     M1 N2L horizontal. 2 2 3 4 T mak   A1 2 5 3 5 4 4 12 g mg mak a   M1 Substitute for T and . 2 4, 2 k k   A1 7

More questions on Circular motion

6 Two uniform smooth spheres A and B of equal radii have masses m and km respectively. The two spheres are on a horizontal surface. Sphere A is travelling with speed u towards sphere B which is at rest. The spheres collide. Immediately before the collision, the direction of motion of A makes an angle a with the line of centres. The coefficient of restitution between the spheres is 1.2 3u cos a (a) Show that the speed of B after the collision is and find also an expression for the speed of 2 ( 1 + k) A along the line of centres after the collision, in terms of k, u and a. [4] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ After the collision, the kinetic energy of A is equal to the kinetic energy of B. (b) Given that tan a = 23 , find the possible values of k. [5] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................

Mark scheme: 6(a) Let v and w be speeds after collision: cos mv kmw mu    M1 Momentum along line of centres. 1 cos 2 w v u    M1 NEL consistent signs. Add to give   3 cos 2 1 u k   A1 AG Convincing working. Substitute back or re-solve:     2 cos 2 1 k u v k     A1 Accept without modulus sign. 4 Question Answer Marks Guidance 6(b)     2 2 2 cos ( sin ) 2 1 k u u k             B1 For speed of A (SOI). Equal KE after collision:       2 2 2 2 cos 1 3 cos 1 ( sin ) 2 2 1 2 2 1 k u u km m u k k                                         2 2 2 2 2 9 cos 4 1 sin 2 cos k k k            M1 Equate KEs. Use 2 tan : 3      2 2 16 1 2 9 4 4 81 k k k k k       M1 2 25 85 52 0 k k    leading to    5 4 5 13 0 k k    M1 Obtain quadratic and attempt to solve. 4 13 or 5 5 k  A1 5

More questions on Momentum

7 Particles P and Q are projected in the same vertical plane from a point O at the top of a cliff. The height of the cliff exceeds 50 m. Both particles move freely under gravity. Particle P is projected with a above the horizontal, where tan a = 3 . Particle Q is projected with speed speed 352 ms -1 at an angle 4 u m s -1 at an angle b above the horizontal, where tan b = 12 . Particle Q is projected one second after the projection of particle P. The particles collide T s after the projection of particle Q. (a) Write down expressions, in terms of T, for the horizontal displacements of P and Q from O when they collide and hence show that 4uT = 21 5 ( T + 1) . [4] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (b) Find the value of T. [4] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (c) Find the horizontal and vertical displacements of the particles from O when they collide. [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ 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) For Q: cos x u T For P:   35 cos 1 2 x T    B1 Collision, so   35 cos 1 cos 2 T u T     M1 Equate and attempt to rearrange.   35 3 2 1 2 5 5 T u T       4 21 5 1 uT T   A1 AG Shown convincingly. 4 7(b) Vertical motion to collision: For Q: 2 1 sin 2 y u T gT    For P:   2 35 1 sin 1 ( 1) 2 2 y T g T      M1 A1 M1 for both expressions, one correct. Equate:   2 2 1 1 35 4 1 1 ( 1) 2 2 5 2 5 u T gT T g T              2 2 1 21 14 1 2 1 1 2 4 T g T T T T       M1 Equate and attempt to solve 16 36 21 21, 1 5 5 T T T     3 T  A1 4 Question Answer Marks Guidance 7(c) 42 x  B1 24 y  M1 24 y  (or 24 m below O ) A1 Correct sign or in words. 3

More questions on Motion of a projectile