AQA AS MATHEMATICS Paper 2 MAY 2023

AQA AS MATHEMATICS Paper 2 Thursday 25 May 2023 Afternoon 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 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 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. 7356/2 Section A Answer all questions in the spaces provided. outside the box 1 Simplify log2 8a Circle your answer. a3 2a 3a 8a 2 It is given that sin y ¼ 5 and 90° y 180° Find the value of cos y Circle your answer. [1 mark] [1 mark] 3 3 3 3 — 4 — 5 5 4 3 (a) Find ð 2x3 þ 8 d x [3 marks] outside the box 3 (b) A curve has gradient function dy ¼ 2x3 þ 8 d x x2 The x-intercept of the curve is at the point (2, 0) Find the equation of the curve. [2 marks] Turn over 4 Find the exact solution of the equation ln (x þ 1) þ ln (x — 1) ¼ ln 15 — 2 ln7 Fully justify your answer. [5 marks] outside the box pffiffiffi pffiffiffi pffiffiffi pffiffiffi outside the Use these two expressions to show that tan 15° ¼ 2 — p3 Fully justify your answer. [4 marks] Turn over for the next question Turn over 6 A curve has equation A line has equation y ¼ 2x2 þ px þ 1 y ¼ 5x — 2 box Find the set of values of p for which the line intersects the curve at two distinct points. Give your answer in exact form. [5 marks] 7 The curve C has equation y ¼ f (x) C has a maximum point at P with coordinates (a, 2b) as shown in the diagram below. y box 2b P O a x 7 (a) C is mapped by a single transformation onto curve C1 with equation y ¼ f (x þ 2) State the coordinates of the maximum point on curve C1 [1 mark] 7 (b) C is mapped by a single transformation onto curve C2 with equation y ¼ 4f (x) State the coordinates of the maximum point on curve C2 [1 mark] 7 (c) C is mapped by a stretch in the x-direction onto curve C3 with equation y ¼ f (3x) State the scale factor of the stretch. [1 mark] Turn over (07) 8 Prove that the sum of the cubes of two consecutive odd numbers is always a multiple of 4. box [5 marks] Turn over for the next question box DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED Turn over 9 A craft artist is producing items and selling them in a local market. The selling price, £P, of an item is inversely proportional to the number of items produced, n 9 (a) When n ¼ 10 , P ¼ 24 240 box Show that P ¼ n [1 mark] 9 (b) The production cost, £C, of an item is inversely proportional to the square of the number of items produced, n When n ¼ 10 , C ¼ 60 Find the set of values of n for which P C [4 marks] 9 (c) Explain the significance to the craft artist of the range of values found in part (b). [1 mark] box Turn over for the next question Turn over 10 A piece of wire of length 66 cm is bent to form the