Cambridge A Level Mathematics - Further 9231 — 2021 May/June Paper 3 · Variant 3

1 D A C 16a 3a O 6a B A uniform lamina ABCD consists of two isosceles triangles ABD and BCD. The diagonals of ABCD meet at the point O. The length of AO is 3a, the length of OC is 6a and the length of BD is 16a (see diagram). Find the distance of the centre of mass of the lamina from DB. [3] .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... ....................................................................................................................................................................

Mark scheme: 1 Area Centre of mass from DB ABD 24 2 a – a BCD 48 2 a 2a Combined 72 2 a x B1 All distances correct. ABCD can be split in other ways, for example ADC and ABC. Taking moments about DB: 72 2 2 2 24 48 2 = × − + × a x a a a a OR Taking moments about A: 72 2 2 2 24 2 48 5 = × + × a x a a a a OR Taking moments about G: ( ) ( ) 2 2 24 48 2 a x a a a x + = × − M1 Moments equation with masses in correct ratio. = x a A1 CWO Alternative method for question 1 ADC: distance of centre of mass from BD = 6 3 3 − = a a a ABC: distance of centre of mass from BD = 6 3 3 − = a a a B1 One calculation. Second calculation or statement about symmetry M1 = x a A1 3

More questions on Equilibrium of a rigid body

2 One end of a light elastic string of natural length 0.8 m and modulus of elasticity 36 N is attached to a fixed point O on a smooth plane. The plane is inclined at an angle a to the horizontal, where sin a = 35 . A particle P of mass 2 kg is attached to the other end of the string. The string lies along a line of greatest slope of the plane with the particle below the level of O. The particle is projected with speed 2 ms -1 directly down the plane from the position where OP is equal to the natural length of the string. Find the maximum extension of the string during the subsequent motion. [5] .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... ....................................................................................................................................................................

Mark scheme: 2 2 1 36 2 0.8 x × (= 2 22.5 ) x B1 EPE correct. Loss in GPE + loss in KE = gain in EPE 2 1 1 36 sin 2 2 2 0.8 x x mg m α + × = × *M1 Energy equation with only GPE, EPE and KE terms, allow sign errors, allow missing g for M1 only, weight must be resolved (allow sin or cos). All terms correct A1 2 6 1 9 5 5 4   + =     x x [leading to 2 45 24 4 0 − − = x x ] DM1 Simplify to 3-term quadratic and attempt to solve. ( )( ) 3 2 15 2 0 − + = x x 2 3 = x only A1 5 Question Answer Marks Guidance B1 May be embedded.

More questions on Hooke's law

3 R i B A Particles A and B, of masses 3m and m respectively, are connected by a light inextensible string of length a that passes through a fixed smooth ring R. Particle B hangs in equilibrium vertically below the ring. Particle A moves in horizontal circles on a smooth horizontal surface with speed 25 ga . The angle between AR and BR is i (see diagram). The normal reaction between A and the surface is 125 mg. (a) Find cosi. [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (b) Find, in terms of a, the distance of B below the ring. [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................

Mark scheme: 3(a) For B: = T mg For A: cos 3 θ + = R T mg M1 All 3 terms, allow sign errors, allow sin/cos mix. Use given R to obtain 3 cos 5 θ = A1 3 3(b) 2 3 sinθ = mv T r M1 May be seen in part (a), allow sin/cos mix. sinθ = r AR B1 Or equivalent. [Combine to give 3 4 = a AR , so] 1 4 = BR a A1 3 Question Answer Marks Guidance

More questions on Circular motion

4 1 2 u B O i a A u A particle 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 is initially held with the string taut at the point A, where OA makes an angle i with the downward vertical through O. The particle is then projected with speed u perpendicular to OA and begins to move upwards in part of a vertical circle. The string goes slack when the particle is at the point B where angle AOB is a right angle. The speed of the particle when it is at B is 12 u (see diagram). Find the tension in the string at A, giving your answer in terms of m and g. [8] .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... ....................................................................................................................................................................

Mark scheme: 4 ( ) 2 2 1 1 cos sin 2 2 2 θ θ   − = +     u mu m mg a a *M1 Energy equation, with 2 KE terms and a two-part GPE term, allow cos/sin mix. ( ) 2 3 2 cos sin 4 u ag θ θ = + A1 At B, tension in string is zero, so 2 2 sin u m mg a θ       = ( 2 4 sinθ = u ag ) B1 N2L Eliminate 2 u DM1 tan 2 θ = OE A1 At A, 2 cosθ − = mu T mg a B1 N2L 9 5 5 = T mg ( = 4.02 ) mg M1 A1 Substitute to find T. Alternative method for question 4 ( ) 2 2 1 1 cos sin 2 2 2 θ θ   − = +     u mu m mg a a *M1 Energy equation, with 2 KE terms and a two-part GPE term, allow cos/sin mix. ( ) 2 3 2 cos sin 4 u ag θ θ = + A1 Question Answer Marks Guidance 4 At B, tension in string is zero, so 2 2 sin u m mg a θ       = ( 2 4 sinθ = u ag ) B1 N2L Eliminate θ : 2 8 5 5 = ag u DM1 A1 At A, 2 cosθ − = mu T mg a B1 N2L 9 5 5 = T mg ( = 4.02 ) mg M1 A1 Substitute to find T. 8 Question Answer Marks Guidance

More questions on Circular motion

5 A particle P of mass m kg is projected vertically upwards from a point O, with speed 20ms -1 , and moves under gravity. There is a resistive force of magnitude 2mv N, where v ms -1 is the speed of P at time t s after projection. (a) Find an expression for v in terms of t, while P is moving upwards. [6] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ The displacement of P from O is x m at time t s. (b) Find an expression for x in terms of t, while P is moving upwards. [2] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (c) Find, correct to 3 significant figures, the greatest height above O reached by P. [2] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................

Mark scheme: 5(a) 2 = − − dv m mg mv dt B1 Use of SUVAT implies 0 marks. N2L, must include m . ( ) ( ) ln 5 2 + =− + v t A M1 Separate variables and integrate 3-term N2L, condone omission of constant. ( ) ln 5 2 + =−+ v t A A1 FT FT only sign error in N2L. 0, 20, ln25 = = = t v A M1 Use correct initial condition. 25 2 ln 5   =   +   t v , 2 25 5 = + t e v M1 Remove all logs. 2 25 5 − = − t v e A1 Alternative method for question 5(a) 2 = − − dv m mg mv dt B1 N2L, must include m . 2 + = − dv v g dt : Integrating factor = 2t e M1 ( ) 2 2 = − t t d ve ge dt , ( ) 2 2 2 = − + t t g ve e A M1 Integrate both sides, condone omission of constant. 2 2 A 2 = − + t t g ve e A1 FT FT only sign error in N2L. Question Answer Marks Guidance 5(a) 0, 20, 25 = = = t v A M1 Use correct initial condition. 2 2 2 25, 25 5 2 − = − + = − t t t g ve e v e A1 6 5(b) ( ) 2 25 5 2 − = − − + t x e t B M1 Use of SUVAT implies 0 marks. Integrate their expression from part (a). 25 0, 0, 2 = = = t x B ( ) 2 25 1 5 2 − = − − t x e t A1 FT FT only expressions of the form = + kt v Pe Q for P, Q non-zero. 2 5(c) Greatest height when v = 0, so t = 0.8047…or 1 ln 5 2 M1 Use of SUVAT in part (a) or part (b) implies 0 marks. Find value of t, may be embedded. x = 5.98 m A1 CWO 2

More questions on Linear motion under a variable force

6 A B i u m km Two uniform smooth spheres A and B of equal radii have masses m and km respectively. Sphere A is moving with speed u on a smooth horizontal surface when it collides with sphere B which is at rest. Immediately before the collision, A’s direction of motion makes an angle i with the line of centres (see diagram). The coefficient of restitution between the spheres is 1.3 4u cos i (a) Show that the speed of B after the collision is . [3] 3 ( 1 + k) ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ 70% of the total kinetic energy of the spheres is lost as a result of the collision. (b) Given that tan i = 13 , find the value of k. [6] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................

Mark scheme: 6(a) Let velocities of A and B along line of centres after collision be 1 v and 2 v . 1 2 cosθ + = mv kmv mu . 2 1 1 cos 3 θ − = v v u M1 Restitution, consistent signs, correct way up. Solve: ( ) 2 4 cos 3 1 θ = + u v k A1 AG shown convincingly. 3 6(b) ( ) ( ) 1 3 cos 3 1 θ − = + k u v k B1 Or equivalent, may be unsimplified. Use velocity of A with both components. B1 ( ) 2 2 1 sinθ + v u seen. ( ) ( ) 2 2 2 2 2 1 1 1 3 1 sin 2 2 10 2 kmv m v u mu θ + + = × M1 KE after = 30% KE before (all terms present). M0 if incorrect masses. Substitute from part (a) and for θ. M1 Eliminate trigonometric terms, must be KE equation, in terms of k only. ( ) ( ) 2 2 3 16 2 1 − + = + k k k , 2 6 7 0 − − = k k M1 Obtain simplified quadratic equation in k . 7 = k A1 6

More questions on Momentum

7 A particle P is projected with speed u at an angle i above the horizontal from a point O on a horizontal plane and moves freely under gravity. The horizontal and vertical displacements of P from O at a subsequent time t are denoted by x and y respectively. (a) Use the equation of the trajectory given in the List of formulae (MF19), together with the condition y = 0 , to establish an expression for the range R in terms of u, i and g. [2] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (b) Deduce an expression for the maximum height H, in terms of u, i and g. [2] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ 4HIt is given that R = . 3 (c) Show that i = 60° . [1] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ It is given also that u = 40 ms -1 . (d) Find, by differentiating the equation of the trajectory or otherwise, the set of values of x for which the direction of motion makes an angle of less than 45° with the horizontal. [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) y = 0 in trajectory equation: ( ) 2 2 2 tan 0 2 cos θ θ − = R R g u ( ) 2 2 sin cos θ θ = u R g only A1 Any equivalent single term expression, for example: 2 2 2 sin2 2 tan , sec θ θ θ u u g g , at least one intermediate line of working, not just quoting a result. SC B1 using SUVAT. 2 7(b) 2 sin cos θ θ = u x their g and substitute in trajectory equation. M1 Or use SUVAT. ( ) 2 2 sin 2 θ = u H g A1 Single term. 2 7(c) Use 4 3 = H R and simplify: tan 3 , 60 θ θ = = ° B1 AG 1 Question Answer Marks Guidance 7(d) ( ) 2 2 tan cos θ θ = − dy gx dx u M1 Differentiate with respect to x. ( ) 2 tan 1 4 cos θ θ − = ± x used M1 Use 1 = ± dy dx as limiting case. 3 1 = + x , 3 1 = − x A1 3 1 3 1 − < < + x A1 Strict inequality, exact values. Alternative method for question 7(d) 2 1 3 , 3 2 = − = − dy y x x x dx M1 Differentiate with respect to x. 1 = ± dy dx used M1 Use 1 = ± dy dx as limiting case. 3 1 = + x , 3 1 = − x A1 3 1 3 1 − < < + x A1 Strict inequality, exact values. Question Answer Marks Guidance 7(d) Alternative method for question 7(d) When moving at 45° to horizontal, =± x y v v M1 Used, both cases considered. 40 cosθ = xv , 40sin 10 θ = − yv t ( ) 1 30 10 10 = − t , ( ) 1 30 10 10 = + t M1 3 1 = + x , 3 1 = − x A1 3 1 3 1 − < < + x A1 Strict inequality, exact values. 4

More questions on Motion of a projectile