AQA 7356 AS June 2019 Paper 1 Pure and Mechanics

AQA 7356 AS June 2019 Paper 1 Pure and Mechanics AS MATHEMATICS Paper 1 Wednesday 15 May 2019 Morning Time allowed: 1 hour 30 minutes Materials  You must have the AQA Formulae for A‑level Mathematics booklet.  You should have a graphical or scientific calculator that meets the requirements of the specification. Instructions  Use black ink or black ball‑point pen. Pencil should only be used for drawing.  Fill in the boxes at the top of this page.  Answer all questions.  You must answer each question in the space provided for that question. If you require extra space, use an AQA supplementary answer book; do not use the space provided for a different question.  Show all necessary working; otherwise marks for method may be lost.  Do all rough work in this book. Cross through any work that you do not want to be marked. Information  The marks for questions are shown in brackets.  The maximum mark for this paper is 80. Advice  Unless stated otherwise, you may quote formulae, without proof, from the booklet.  You do not necessarily need to use all the space provided. (JUN) PB/Jun19/E4 7356/1 box 3 It is given that (x þ 1) and (x — 3) are two factors of f (x), where f (x) ¼ px3 — 3x2 — 8x þ q 3 (a) Find the values of p and q. [3 marks] outside the box 3 (b) Fully factorise f (x). [2 marks] Turn over 4 Show that integers. p6 pffi3ffiffi — p2ffiffiffi can be expressed in the form mpffinffiffi þ npffimffiffiffi , where m and n are outside the box Fully justify your answer. [4 marks] 5 (a) Sketch the curve y ¼ g(x) where g(x) ¼ (x þ 2)(x — 1)2 [3 marks] box 5 (b) Hence, solve g(x) ≤ 0 [2 marks] Turn over 1 6 (a) (i) Show that cos y ¼ 2 is one solution of the equation 6 sin2 y þ 5 cos y ¼ 7 [2 marks] box 6 (a) (ii) Find all the values of y that solve the equation 6 sin2 y þ 5 cos y ¼ 7 for 0° ≤ y ≤ 360° Give your answers to the nearest degree. [2 marks] 6 (b) Hence, find all the solutions of the equation 6 sin2 2y þ 5 cos 2y ¼ 7 box for 0° ≤ y ≤ 360° Give your answers to the nearest degree. [2 marks] Turn over for the next question Turn over (07) 7 Given that y 2 R, prove that (2 þ 3y)4 þ (2 — 3y)4 ≥ 32 box Fully justify your answer. [6 marks] (0 ) 8 Prove that the curve with equation y ¼ 2x5 þ 5x4 þ 10x3 — 8 has only one stationary point, stating its coordinates. [6 marks] box Turn over for the next question Turn over 9 A curve cuts the x-axis at (2, 0) and has gradient function box dy 24 d x ¼ x3 9 (a) Find the equation of the curve. [4 marks] 9 (b) Show that the perpendicular bisector of the line joining A( 2, 8) to B( 6, 4) is the normal to the curve at (2, 0) box [6 marks] Turn over 10 On 18 March 2019 there were 12 hours of daylight in Inverness. On 16 June 2019, 90 days later, there will be 18 hours of daylight in Inverness. Jude decides to model the number of hours of daylight in Inverness, N , by the formula box N ¼ A þ B sin t ° where t is the number of days after 18 March 2019. 10 (a) (i) State the value that Jude should use for A. [1 mark] 10 (a) (ii) State the value that Jude should use for B. [1 mark] 10 (a) (iii) Using Jude’s model, calculate the number of hours of daylight in Inverness on 15 May 2019, 58 days after 18 March 2019. [1 mark] 10 (a) (iv) Using Jude’s model, find how many days during 2019 will have at least 17.4 hours of daylight in Inverness. box [4 marks] 10 (a) (v) Explain why Jude’s model will become inaccurate for 2020 and future years. [1 mark] 10 (b) Anisa decides to model the number of hours of daylight in Inverness with the formula N ¼ A þ B sin 360 t ° Explain why Anisa’s model is better than Jude’s model. [1 mark] Turn over box Turn over for the next question box DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED Turn over 13 A car, starting from rest, is driven along a horizontal track. The velocity of the car, v m s—1, at time t seconds, is modelled by the equation v ¼ 0:48t 2 — 0:024 t 3 for 0 ≤ t ≤ 15 13 (a) Find the distance the car travels during the first 10 seconds of its journey. box [3 marks] 13 (b) Find the maximum speed of the car. Give your answer to three significant figures. [4 marks] box 13 (c) Deduce the range of values of t for which the car is modelled as decelerating. [2 marks] Turn over for the next question Turn over (17) 14 Two particles, A and B, lie at rest on a smooth horizontal plane. A has position vector rA ¼ (13i — 22j) metres B has position vector rB ¼ (3i þ 2j) metres 14 (a) Calculate the distance between A and B. [2 marks] box 14 (b) Three forces, F1, F2 and F3 are applied to particle A, where F1 ¼ (—2i þ 4j) newtons F2 ¼ (6i — 10j) newtons Given that A remains at rest, explain why F3 ¼ (—4i þ 6j) newtons [1 mark] (1 ) 14 (c) A force of (5i 12j) newtons, is applied to B, so that B moves, from rest, in a straight line towards A. box B has a mass of 0.8 kg 14 (c) (i) Show that the acceleration of B towards A is 16.25 m s—2 [2 marks] 14 (c) (ii) Hence, find the time taken for B to reach A. Give your answer to two significant figures. [2 marks] Turn over 15 A tractor and its driver have a combined mass of m kilograms. The tractor is towing a trailer of mass 4m kilograms in a straight line along a horizontal road. The tractor and trailer are connected by a horizontal tow bar, modelled as a light rigid rod. A driving force of 11 080 N and a total resistance force of 160 N act on the tractor. A total resistance force of 600 N acts on the trailer. The tractor and the trailer have an acceleration of 0.8 m s—2 box 15 (a) Find m. [3 marks] 15 (b) Find the tension in the tow bar. [2 marks] 15 (c) At the instant the speed of the tractor reaches 18 km h—1 the tow bar breaks. The total resistance force acting on the trailer remains constant. Starting from the instant the tow bar breaks, calculate the time taken until the speed of the trailer reduces to 9 km h—1 box [4 marks] END OF QUESTIONS There are no questions printed on this page box DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED There are no questions printed on this page box DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED There are no questions printed on this page box DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED Copyright information For confidentiality purposes, from the November 2015 examination series, acknowledgements of third-party copyright material are published in a separate booklet rather than including them on the examination paper or support materials. This booklet is published after each examination series and is available for free download from after the live examination series. Permission to reproduce all copyright material has been applied for. In some cases, efforts to contact copyright-holders may have been unsuccessful and AQA will be happy to rectify any omissions of acknowledgements. If you have any queries please contact the Copyright Team, AQA, Stag Hill House, Guildford, GU2 7XJ. Copyright ª 2019 AQA and its licensors. All rights reserved. (24) Jun19/7356/1 (196A7356/1)