AQA A-level MATHEMATICS 7357/2 Paper 2 Question Paper + Mark scheme [MERGED] June 2022 PB/Jun22/E7 7357/2 (JUN227357201) A-level MATHEMATICS Paper 2 Time allowed: 2 hours Materials

Document Content and Description Below

AQA A-level MATHEMATICS 7357/2 Paper 2 Question Paper + Mark scheme [MERGED] June 2022 PB/Jun22/E7 7357/2 (JUN227357201) A-level MATHEMATICS Paper 2 Time allowed: 2 hours Materials l You ... must have the AQA Formulae for A‑level Mathematics booklet. l You should have a graphical or scientific calculator that meets the requirements of the specification. Instructions l Use black ink or black ball-point pen. Pencil should only be used for drawing. l Fill in the boxes at the top of this page. l Answer all questions. l You must answer each question in the space provided for that question. If you need extra space for your answer(s), use the lined pages at the end of this book. Write the question number against your answer(s). l Do not write outside the box around each page or on blank pages. l Show all necessary working; otherwise marks for method may be lost. l Do all rough work in this book. Cross through any work that you do not want to be marked. Information l The marks for questions are shown in brackets. l The maximum mark for this paper is 100. Advice l Unless stated otherwise, you may quote formulae, without proof, from the booklet. l You do not necessarily need to use all the space provided. Please write clearly in block capitals. Centre number Candidate number Surname ________________________________________________________________________ Forename(s) ________________________________________________________________________ Candidate signature ________________________________________________________________________ For Examiner’s Use Question Mark 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 TOTAL I declare this is my own work. 2 Section A Answer all questions in the spaces provided. 1 A circle has centre (4, 5) and radius 6 Find the equation of the circle. Tick (3) one box. [1 mark] (x  4)2 þ (y þ 5)2 ¼ 6 (x þ 4)2 þ (y  5)2 ¼ 6 (x  4)2 þ (y þ 5)2 ¼ 36 (x þ 4)2 þ (y  5)2 ¼ 36 2 State the value of lim h!0 sin (p þ h)  sin p h Circle your answer. [1 mark] cos h 10 1 Jun22/7357/2 Do not write outside the box (02) 3 3 The function f is concave and is represented by one of the graphs below. Identify the graph which represents f. Tick (3) one box. [1 mark] x y O x y O x y O x y O Do not write outside the box Jun22/7357/2 Turn over s (03) 4 4 8.7 cm 6.1 cm A B C 38° The diagram shows a triangle ABC. AB is the shortest side. The lengths of AC and BC are 6.1 cm and 8.7 cm respectively. The size of angle ABC is 38 Find the size of the largest angle. Give your answer to the nearest degree. [3 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Do not write outside the box Jun22/7357/2 (04) DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED 5 Turn over for the next question Do not write outside the box Jun22/7357/2 Turn over s (05) 6 5 The binomial expansion of (2 þ 5x) 4 is given by (2 þ 5x) 4 ¼ A þ 160x þ Bx2 þ 1000x3 þ 625x4 5 (a) Find the value of A and the value of B. [2 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 5 (b) Show that (2 þ 5x) 4  (2  5x) 4 ¼ Cx þ Dx3 where C and D are constants to be found. [2 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Do not write outside the box Jun22/7357/2 (06) 7 5 (c) Hence, or otherwise, find ð (2 þ 5x) 4  (2  5x) 4  dx [2 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Turn over for the next question Do not write outside the box Jun22/7357/2 Turn over s (07) 8 6 (a) Asif notices that 242 ¼ 576 and 2 þ 4 ¼ 6 gives the last digit of 576 He checks two more examples: 272 ¼ 729 292 ¼ 841 2 þ 7 ¼ 9 2 þ 9 ¼ 11 Last digit 9 Last digit 1 Asif concludes that he can find the last digit of any square number greater than 100 by adding the digits of the number being squared. Give a counter example to show that Asif’s conclusion is not correct. [2 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 6 (b) Claire tells Asif that he should look only at the last digit of the number being squared. 272 ¼ 729 242 ¼ 576 72 ¼ 49 42 ¼ 16 Last digit 9 Last digit 6 Using Claire’s method determine the last digit of 234567892 [1 mark] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Do not write outside the box Jun22/7357/2 (08) 9 6 (c) Given Claire’s method is correct, use proof by exhaustion to show that no square number has a last digit of 8 [2 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Turn over for the next question Do not write outside the box Jun22/7357/2 Turn over s (09) 10 7 The curve y ¼ 15  x2 and the isosceles triangle OPQ are shown on the diagram below. q x y O P Q Vertices P and Q lie on the curve such that Q lies vertically above some point (q, 0) The line PQ is parallel to the x-axis. 7 (a) Show that the area, A, of the triangle OPQ is given by A ¼ 15q  q3 for 0 < q < c where c is a constant to be found. [3 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Do not write outside the box Jun22/7357/2 (10) 11 7 (b) Find the exact maximum area of triangle OPQ. Fully justify your answer. [6 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Turn over for the next question Do not write outside the box Jun22/7357/2 Turn over s (11) 12 8 (a) Sketch the graph of y ¼ 1 x2 [2 marks] x y Do not write outside the box Jun22/7357/2 (12) 13 8 (b) The graph of y ¼ 1 x2 can be transformed onto the graph of y ¼ 9 x2 using a stretch in one direction. Beth thinks the stretch should be in the y-direction. Paul thinks the stretch should be in the x-direction. State, giving reasons for your answer, whether Beth is correct, Paul is correct, both are correct or neither is correct. [3 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Turn over for the next question Do not write outside the box Jun22/7357/2 Turn over s (13) 14 9 Given that log2 x3  log2 y2 ¼ 9 show that x ¼ Ay p where A is an integer and p is a rational number. [4 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Do not write outside the box Jun22/7357/2 (14) DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED 15 Turn over for the next question Do not write outside the box Jun22/7357/2 Turn over s (15) 16 10 A gardener has a greenhouse containing 900 tomato plants. The gardener notices that some of the tomato plants are damaged by insects. Initially there are 25 damaged tomato plants. The number of tomato plants damaged by insects is increasing by 32% each day. 10 (a) The total number of plants damaged by insects, x, is modelled by x ¼ A  Bt where A and B are constants and t is the number of days after the gardener first noticed the damaged plants. 10 (a) (i) Use this model to find the total number of plants damaged by insects 5 days after the gardener noticed the damaged plants. [3 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 10 (a) (ii) Explain why this model is not realistic in the long term. [2 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Do not write outside the box Jun22/7357/2 (16) 17 10 (b) A refined model assumes the rate of increase of the number of plants damaged by insects is given by dx dt ¼ x(900  x) 2700 10 (b) (i) Show that ð A x þ B 900  x  dx ¼ ð dt where A and B are positive integers to be found. [3 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Question 10 continues on the next page Do not write outside the box Jun22/7357/2 Turn over s (17) 18 10 (b) (ii) Hence, find t in terms of x. [5 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 10 (b) (iii) Hence, find the number of days it takes from when the damage is first noticed until half of the plants are damaged by the insects. [2 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Do not write outside the box Jun22/7357/2 (18) 19 Section B Answer all questions in the spaces provided. 11 A moon vehicle has a mass of 212 kg and a length of 3 metres. On the moon the vehicle has a weight of 345 N Calculate a value for acceleration due to gravity on the moon. Circle your answer. [1 mark] 0.614 m s2 1.63 m s2 1.84 m s2 4.89 m s2 12 A car is travelling along a straight horizontal road with initial velocity u m s1 The car begins to accelerate at a constant rate a m s2 for 5 seconds, to reach a final velocity of 4u m s1 Express a in terms of u. Circle your answer. [1 mark] a ¼ 0:2u a ¼ 0:4u a ¼ 0:6u a ¼ 0:8u Turn over for the next question Do not write outside the box Jun22/7357/2 Turn over s (19) 20 13 In this question use g = 9:8ms2 A ball is projected from a point on horizontal ground with an initial velocity of 7 m s1 at an angle y above the horizontal. The ball reaches a maximum vertical height of h metres above the ground. 13 (a) Show that h ¼ 2:5 sin2 y [3 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 13 (b) Hence, given that 0  y  60, find the maximum value of h. [2 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Do not write outside the box Jun22/7357/2 (20) 21 13 (c) Nisha claims that the larger the size of the ball, the greater the maximum vertical height will be. State whether Nisha is correct, giving a reason for your answer. [1 mark] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Turn over for the next question Do not write outside the box Jun22/7357/2 Turn over s (21) 22 14 A £2 coin has a diameter of 28 mm and a mass of 12 grams. A uniform rod AB of length 160 mm and a fixed load of mass m grams are used to check that a £2 coin has the correct mass. The rod rests with its midpoint on a support. A £2 coin is placed face down on the rod with part of its curved edge directly above A. The fixed load is hung by a light inextensible string from a point directly below the other end of the rod at B, as shown in the diagram. A B m 14 (a) Given that the rod is horizontal and rests in equilibrium, find m. [3 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 14 (b) State an assumption you have made about the £2 coin to answer part (a). [1 mark] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Do not write outside the box Jun22/7357/2 (22) 23 15 A car is moving in a straight line along a horizontal road. The graph below shows how the car’s velocity v m s1 changes with time, t seconds. 0 –1 –2 –3 –4 1 5 t 2 3 4 v 10 15 Over the period 0  t  15 the car has a total displacement of 7 metres. Initially the car has velocity 0 m s1 Find the next time when the velocity of the car is 0 m s1 [4 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Do not write outside the box Jun22/7357/2 Turn over s (23) 24 16 Two particles, P and Q, move in the same horizontal plane. Particle P is initially at rest at the point with position vector (4i þ 5j) metres and moves with constant acceleration (3i  4j)ms2 Particle Q moves in a straight line, passing through the points with position vectors (i  j) metres and (10i þ cj) metres. P and Q are moving along parallel paths. 16 (a) Show that c ¼ 13 [4 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 16 (b) (i) Find an expression for the position vector of P at time t seconds. [1 mark] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Do not write outside the box Jun22/7357/2 (24) 25 16 (b) (ii) Hence, prove that the paths of P and Q are not collinear. [3 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Turn over for the next question Do not write outside the box Jun22/7357/2 Turn over s (25) 26 17 A particle is moving such that its position vector, r metres, at time t seconds, is given by r ¼ et cos t i þ et sin t j Show that the magnitude of the acceleration of the particle, a m s2, is given by a ¼ 2et Fully justify your answer. [7 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Do not write outside the box Jun22/7357/2 (26) DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED 27 Turn over for the next question Do not write outside the box Jun22/7357/2 Turn over s (27) 28 18 An object, O, of mass m kilograms is hanging from a ceiling by two light, inelastic strings of different lengths. The shorter string, of length 0.8 metres, is fixed to the ceiling at A. The longer string, of length 1.2 metres, is fixed to the ceiling at B. This object hangs 0.6 metres directly below the ceiling as shown in the diagram. A Ceiling O 0.6 m B 18 (a) Show that the tension in the shorter string is over 30% more than the tension in the longer string. [4 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Do not write outside the box Jun22/7357/2 (28) 29 18 (b) The tension in the longer string is known to be 2g newtons. Find the value of m [Show More]

Last updated: 10 months ago

Preview 1 out of 66 pages