AQA A LEVEL MATHEMATICS PAPER 1

AQA A LEVEL MATHEMATICS PAPER 1 JUNE 2020 A-level MATHEMATICS Paper 1 Wednesday 3 June 2020 Afternoon Time allowed: 2 hours 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 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).  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 100. 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/Jun20/E7 7357/1 Answer all questions in the spaces provided. 1 The first three terms, in ascending powers of x, of the binomial expansion of 1 (9 þ 2 x) 2 are given by box 1 x x2 (9 þ 2 x) 2 ≈ a þ — where a is a constant. 1 (a) State the range of values of x for which this expansion is valid. Circle your answer. [1 mark] 2 2 9 jxj < 9 jxj < 3 jxj < 1 jxj < 2 1 (b) Find the value of a. Circle your answer. [1 mark] 1 2 3 9 2 A student is searching for a solution to the equation f (x) ¼ 0 He correctly evaluates box f (—1) ¼ —1 and f (1) ¼ 1 and concludes that there must be a root between —1 and 1 due to the change of sign. Select the function f (x) for which the conclusion is incorrect. Circle your answer. f (x) 1 x f (x) x f (x) x3 f (x) 2x þ 1 x þ 2 [1 mark] 3 The diagram shows a sector OAB of a circle with centre O and radius 2 A B 2 θ O The angle AOB is y radians and the perimeter of the sector is 6 Find the value of y Circle your answer. 1 pffi3ffiffi [1 mark] 2 3 Turn over for the next question Turn over 4 (a) Sketch the graph of y ¼ 4 — j2x — 6j y box O x [3 marks] 4 (b) Solve the inequality 4 — j2x — 6j > 2 [2 marks] 5 Prove that, for integer values of n such that 0 ≤ n < 4 2nþ2 > 3n [2 marks] box Turn over for the next question Turn over 6 Four students, Tom, Josh, Floella and Georgia are attempting to complete the indefinite integral box 1 dx for x > 0 x Each of the students’ solutions is shown below: Tom ð 1 dx ¼ ln x Josh ð 1 dx ¼ k ln x Floella ð 1 dx ¼ ln Ax Georgia ð 1 dx ¼ ln x þ c 6 (a) (i) Explain what is wrong with Tom’s answer. [1 mark] 6 (a) (ii) Explain what is wrong with Josh’s answer. [1 mark] 6 (b) Explain why Floella and Georgia’s answers are equivalent. [2 marks] 7 Consecutive terms of a sequence are related by unþ1 ¼ 3 — (un)2 box 7 (a) In the case that u 2 1 7 (a) (i) Find u 3 [2 marks] 7 (a) (ii) Find u 50 [1 mark] 7 (b) State a different value for u 1 which gives the same value for u 50 as found in part (a)(ii). [1 mark] Turn over for the next question Turn over (07) 8 Mike, an amateur astronomer who lives in the South of England, wants to know how the number of hours of darkness changes through the year. On various days between February and September he records the length of time, H hours, of darkness along with t, the number of days after 1 January. His results are shown in Figure 1 below. box H 16 14 12 10 8 6 4 2 0 0 50 100 150 Figure 1 200 250 300 350 t Mike models this data using the equation H ¼ 3:87 sin 2p(t þ 101:75) þ 11:7 8 (a) Find the minimum number of hours of darkness predicted by Mike’s model. Give your answer to the nearest minute. [2 marks] (0 ) 8 (b) Find the maximum number of consecutive days where the number of hours of darkness predicted by Mike’s model exceeds 14 box [3 marks] Question 8 continues on the next page Turn over 8 (c) Mike’s friend Sofia, who lives in Spain, also records the number of hours of darkness on various days throughout the year. Her results are shown in Figure 2 below. box Figure 2 H 16 14 12 10 8 6 4 2 0 0 50 100 150 200 250 300 350 t Sofia attempts to model her data by refining Mike’s model. She decides to increase the 3.87 value, leaving everything else unchanged. Explain whether Sofia’s refinement is appropriate. [2 marks] Turn over for the next question box DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED Turn over 9 Chloe is attempting to write numerators. 2x2 x (x þ 1)(x þ 2)2 as partial fractions, with constant box Her incorrect attempt is shown below. 2x2 þ x A B Step 1 (x þ 1)(x þ 2)2 Ξ x þ 1 þ (x þ 2)2 Step 2 2x2 þ x Ξ A(x þ 2)2 þ B(x þ 1) Step 3 Let x ¼ —1 ) A ¼ 1 Let x ¼ —2 ) B ¼ —6 2x2 þ x 1 6 Answer (x þ 1)(x þ 2)2 Ξ x þ 1 — (x þ 2)2 9 (a) (i) By using a counter example, show that the answer obtained by Chloe cannot be correct. [2 marks] 9 (a) (ii) Explain her mistake in Step 1. [1 mark] 9 (b) Write 2x2 x (x þ 1)(x þ 2)2 as partial fractions, with constant numerators. [4 marks] box Turn over 10 (a) An arithmetic series is given by box 20 (4r þ 1) r¼5 10 (a) (i) Write down the first term of the series. [1 mark] 10 (a) (ii) Write down the common difference of the series. [1 mark] 10 (a) (iii) Find the number of terms of the series. [1 mark] 10 (b) A different arithmetic series is given by box where b and c are constants. The sum of this series is 7735 10 (b) (i) Show that 55b þ c ¼ 85 100 (br þ c) r¼10 [4 marks] Turn over 10 (b) (ii) The 40th term of the series is 4 times the 2nd term. Find the values of b and c. [4 marks] box 11 The region R enclosed by the lines x ¼ 1, x ¼ 6, y ¼ 0 and the curve y ¼ ln (8 — x) box is shown shaded in Figure 3 below. Figure 3 y R 0 1 2 3 4 5 6 7 x All distances are measured in centimetres. 11 (a) Use a single trapezium to find an approximate value of the area of the shaded region, giving your answer in cm2 to two decimal places. [2 marks] Question 11 continues on the next page Turn over (17) 11 (b) Shape B is made from four copies of region R as shown in Figure 4 below. Figure 4 box Shape B is cut from metal of thickness 2 mm The metal has a density of 10.5 g/cm3 Use the trapezium rule with six ordinates to calculate an approximate value of the mass of Shape B. Give your answer to the nearest gram. [5 marks] (1 ) box 11 (c) Without further calculation, give one reason why the mass found in part (b) may be: 11 (c) (i) an underestimate. [1 mark] 11 (c) (ii) an overestimate. [1 mark] Turn over for the next question Turn over 12 A curve C has equation where A is a constant. x3 sin y þ cos y ¼ Ax box C passes through the point P pffi3ffiffi, p 12 (a) Show that A ¼ 2 [2 marks] dy 2 — 3x2 sin y 12 (b) (i) Show that d x ¼ x3 cos y — sin y [5 marks] box 12 (b) (ii) Hence, find the gradient of the curve at P. [2 marks] 12 (b) (iii) The tangent to C at P intersects the x-axis at Q. Find the exact x-coordinate of Q. [4 marks] Turn over 13 The function f is defined by f (x) 2x þ 3 x — 2 x 2 R, x 2 box 13 (a) (i) Find f —1 [3 marks] 13 (a) (ii) Write down an expression for ff (x). [1 mark] 13 (b) The function g is defined by g(x) ¼ 2 x2 5x 2 x 2 R, 0 ≤ x ≤ 4 box 13 (b) (i) Find the range of g. [3 marks] 13 (b) (ii) Determine whether g has an inverse. Fully justify your answer. [2 marks] Turn over 13 (c) Show that gf (x) ¼ 48 þ 29x — 2 x2 2 x2 — 8x þ 8 [4 marks] box 13 (d) It can be shown that fg is given by fg(x) ¼ 4x2 — 10x þ 6 2 x2 — 5x — 4 box with domain fx 2 R : 0 ≤ x ≤ 4, x ag Find the value of a. Fully justify your answer. [2 marks] Turn over for the next question Turn over 14 The function f is defined by f (x) ¼ 3xpffixffi — 1 where x ≥ 0 14 (a) f (x) ¼ 0 has a single solution at the point x ¼ a By considering a suitable change of sign, show that a lies between 0 and 1 [2 marks] box 14 (b) (i) Show that 0 3xð1 þ x ln 9) f (x) ¼ 2 x pffiffiffi [3 marks] 14 (b) (ii) Use the Newton–Raphson method with x1 ¼ 1 to find x3 , an approximation for a. Give your answer to five decimal places. box [2 marks] 14 (b) (iii) Explain why the Newton–Raphson method fails to find a with x1 ¼ 0 [2 marks] Turn over (27) x 15 The region enclosed between the curves y ex, y 6 e2 and the line x 0 is shown shaded in the diagram below. y y = e x outside the box x y = 6 – e 2 O x Show that the exact area of the shaded region is 6 ln4 — 5 Fully justify your answer. [10 marks] (2 ) box END OF QUESTIONS There are no questions printed on this page box DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED box box Copyright information For confidentiality purposes, all acknowledgements of third-party copyright material are published in a separate booklet. This booklet is published after each live examination series and is available for free download from . 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. Copyright ª 2020 AQA and its licensors. All rights reserved. (32) Jun20/7357/1 (206A7357/1)