MATHEMATICS Paper 1 Tuesday 6 June 2023

MATHEMATICS Paper 1 Tuesday 6 June 2023 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).  Do not write outside the box around each page or on blank pages.  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. PB/KL/Jun23/E7 7357/1 Answer all questions in the spaces provided. 1 Find the coefficient of x7 in the expansion of (2x — 3)7 Circle your answer. —2187 —128 2 128 [1 mark] box dy 2 Given that y ¼ 2x find d x Circle your answer. [1 mark] dy d x ¼ 5x 2 dy d x ¼ 6x 2 dy x4 d x ¼ 2 dy 3 d x ¼ 6x 3 The curve with equation y ln x is transformed by a stretch parallel to the x-axis with scale factor 2 Find the equation of the transformed curve. box Circle your answer. y ¼ 1 ln x y ¼ 2 ln x y ¼ ln x y ¼ ln 2x [1 mark] 4 Given that y is a small angle, find an approximation for cos 2y Circle your answer. y2 2 [1 mark] 2 2 1 — 2 2 — 2y 1 — 2y 1 — y Turn over for the next question Turn over 5 5 The graph of y ¼ ex — 1 is shown in the diagram below. y box O 1 4 x The trapezium rule with 6 ordinates (5 strips) is to be used to find an approximation for the shaded area. The values required to obtain this approximation are shown in the table below. x 1 1.6 2.2 2.8 3.4 4 y 2.90988 1.26485 0.62305 0.32374 0.17263 0.09329 5 (a) Use the trapezium rule with 6 ordinates (5 strips) to find an approximate value for the shaded area. Give your answer to four decimal places. [3 marks] 5 (b) Using your answer to part (a) deduce an estimate for 4 20 ex — 1 dx [1 mark] box Turn over for the next question Turn over 6 Show that the equation has exactly one solution. Fully justify your answer. 2 log10 x ¼ log10 4 þ log10 (x þ 8) [5 marks] box 7 (a) Given that n is a positive integer, express box 7 7 3 þ 5pn — 5pn — 3 as a single fraction not involving surds. [3 marks] 7 (b) Hence, deduce that 7 7 3 þ 5pn — 5pn — 3 is a rational number for all positive integer values of n [1 mark] Turn over (07) 8 Show that p 2 (x sin 4x) d x 0 p ¼— 8 [6 marks] box (0 ) Turn over for the next question box DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED