Cambridge A Level Mathematics - Further 9231 — 2021 Oct/Nov Paper 3 · Variant 1

1 One end of a light elastic string, of natural length a and modulus of elasticity 3mg, is attached to a fixed point O on a smooth horizontal plane. A particle P of mass m is attached to the other end of the string and moves in a horizontal circle with centre O. The speed of P is 43 ga . Find the extension of the string. [4] .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... ....................................................................................................................................................................

Mark scheme: 1 3 = mgx T a CAO. ( ) 4 3 = + mga T a x B1 2 2 9 9 4 0 + − = x ax a leading to ( )( ) 3 3 4 0 − + = x a x a M1 1 3 = x a A1 4 Must include logs. Condone missing modulus.

More questions on Circular motion

2 A particle P of mass m kg moves along a horizontal straight line with acceleration ams -2 given by v ( 1 - 2 t 2 ) a = , t where v ms -1 is the velocity of P at time t s. (a) Find an expression for v in terms of t and an arbitrary constant. [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (b) Given that a = 5 when t = 1, find an expression, in terms of m and t, for the horizontal force acting on P at time t. [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................

Mark scheme: 2(a) Separate variables and integrate: 2 1 2   − =     dv t dt v t so 2 ln ln = − + v t t c M1 A1 2 − = t v Ate , 2 − −= t v Ate , 2 − = − t v Ate A1 CAO. 3 Question Answer Marks Guidance 2(b) ( ) ( ) 2 2 2 2 1 2 1 2 − − − − = = − − t t Ate t a Ae t t M1 Substituting their answer to part (a) into given formula ( ) 1, 5 5 t a A e = = = M1 Use initial condition. Force = ( ) 2 1 2 5 2 1 − − t me t A1 Use N2L, correct work only. Alternative method for question 2(b) ( ) 2 1 2 − = v t a t substitute 1, 5 t a = = so 5 = − v M1 Use initial condition. Use N2L, correct work only. Substituting in their answer to part (a) so ( ) 5 = A e M1 Force = ( ) 2 1 2 5 2 1 − − t me t A1 3

More questions on Differential equations

3 A light elastic string has natural length a and modulus of elasticity 12mg . One end of the string is attached to a fixed point O. The other end of the string is attached to a particle of mass m. The particle hangs in equilibrium vertically below O. The particle is pulled vertically down and released from rest with the extension of the string equal to e, where e 2 13 a . In the subsequent motion the particle has speed 2ga when it has ascended a distance 13 a. Find e in terms of a. [6] .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... ....................................................................................................................................................................

Mark scheme: 3 Loss in EPE = 2 2 1 12 1 12 2 2 3   × − × × −     mge mg a e a a ( ) 2 6 3   = −     mg e a B1 Either term correct. Gain in KE = 2 1 2 mv and Gain in GPE = 3 mga B1 Gain in KE + Gain in GPE = Loss in EPE M1 KE, GPE and at least one EPE term. ( ) 2 1 2 6 2 3 3 + = − mga mg mv e a A1 All terms correct. Simplify to a linear equation in e. M1 1 2 = e a A1 6

More questions on Hooke's law

4 D 3a C h F 3a h A E B A uniform lamina AECF is formed by removing two identical triangles BCE and CDF from a square lamina ABCD. The square has side 3a and EB = DF = h (see diagram). (a) Find the distance of the centre of mass of the lamina AECF from AD and from AB, giving your answers in terms of a and h. [5] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ The lamina AECF is placed vertically on its edge AE on a horizontal plane. (b) Find, in terms of a, the set of values of h for which the lamina remains in equilibrium. [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................

Mark scheme: 4(a) Area Centre of mass from AD Square 2 9a 3 2 a CDF 3 2 ah a BEC 3 2 ah 1 3 3 − a h Resulting AEFC 2 9 3 − a ah x M1 Attempt at moments with three terms. Taking moments about AD: ( ) 2 9 3 − a ah = x 2 3 3 3 1 9 3 2 2 2 3 a a ah a ah a h         × − × − × −                 A1 A1 Two terms correct. All correct. ( ) 2 2 27 12 6 3 − + = − a ah h x a h 9 6 −   =     a h A1 AEF = y x B1 By symmetry or equal to their x . 5 4(b) For equilibrium, x ⩽ 3a – h 27a2 – 12ah + h2 ⩽ 6(3a – h)2 B1 Accept strict inequality. 27a2– 24ah + 5h2 ⩾ 0 M1 Homogeneous 3-term quadratic inequality. h ⩽ 9 5 a A1 CAO. 3

More questions on Equilibrium of a rigid body

5 A particle P is projected from a point O on a horizontal plane and moves freely under gravity. Its initial speed is ums -1 and its angle of projection is sin -1 ( 45 ) above the horizontal. At time 8 s after projection, P is at the point A. At time 32 s after projection, P is at the point B. The direction of motion of P at B is perpendicular to its direction of motion at A. Find the value of u. [7] .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... ....................................................................................................................................................................

Mark scheme: 5 At A: sin 8 cos θ θ ↑ − → u g u M1 Both. sin 8 tan cos θ α θ − = u g u A1 At B: sin 32 cos θ θ ↑ − → u g u M1 Both. sin 32 tan cos θ β θ − = u g u A1 sin 8 sin 32 1 cos cos θ θ θ θ − − × = − u g u g u u B1 Perpendicular directions, so tan tan 1 α β × = −. 2 320 25600 0 − + = u u M1 Simplify to a quadratic in u. 160 = u A1 7

More questions on Motion of a projectile

6 A particle P, of mass m, is attached to one end of a light inextensible string of length a. The other end of the string is attached to a fixed point O. The particle P moves in complete vertical circles about O with the string taut. The points A and B are on the path of P with AB a diameter of the circle. OA makes an angle i with the downward vertical through O and OB makes an angle i with the upward vertical through O. The speed of P when it is at A is 5ag . The ratio of the tension in the string when P is at A to the tension in the string when P is at B is 9 : 5. (a) Find the value of cosi. [6] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (b) Find, in terms of a and g, the greatest speed of P during its motion. [2] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................

Mark scheme: 6(a) At A 5 cosθ − = × A ag T mg m a B1 N2L At B 2 cosθ + = × B v T mg m a B1 N2L 2 1 1 5 2cos 2 2 θ × − = × m ag mv mga M1 Energy equation with correct number of terms. 2 5 4 cosθ = − v ag ga A1 Accept multiplied by m and/or divided by a. Use ratio of tensions = 9 : 5 M1 Use ratio and simplify to an expression in cosθ . 2 cos 5 θ = A1 CAO 6 6(b) Greatest speed at lowest point ( ) 2 1 1 5 1 cos 2 2 θ − × + = × − m ag mV mga M1 Energy equation including lowest point, correct number of terms. 31 5 = ag V A1 FT Ft their cosθ from part (a). 2

More questions on Circular motion

7 A B c b a u P C The smooth vertical walls AB and CB are at right angles to each other. A particle P is moving with speed u on a smooth horizontal floor and strikes the wall CB at an angle a. It rebounds at an angle b to the wall CB. The particle then strikes the wall AB and rebounds at an angle c to that wall (see diagram). The coefficient of restitution between each wall and P is e. (a) Show that tan b = e tan a . [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (b) Express c in terms of a and explain what this result means about the final direction of motion of P. [4] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ As a result of the two impacts the particle loses 89 of its initial kinetic energy. (c) Given that a + b = 90° , find the value of e and the value of tana. [4] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ Additional Page If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown. .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. ..................................................................................................................................................................

Mark scheme: 7(a) α β = M1 sin sin α β = eu v M1 Divide: tan tan β α = e A1 AG. Must see divide OE. 3 7(b) ( ) sin cos sin β γ α = = v w eu M1 ( ) cos sin cos β γ α = = ev w eu M1 Divide: tan 1/ tan γ α = : 90 γ α = ° − *A1 After second rebound, direction of motion is parallel to initial path. DB1 4 7(c) Final KE = ( ) ( ) ( ) 2 2 1 sin cos 2 α α + m eu eu 2 2 1 2   =     me u M1 Energy expression in terms of u. So 2 2 2 1 1 1 2 9 2 = × me u mu giving 1 3 = e A1 Part (a) gives ( ) tan 90 tan α α − = e M1 So tan 3 α = A1 4

More questions on Momentum