A-level MATHEMATICS Paper 3 Tuesday 20 June 2023

MATHEMATICS Paper 3 Tuesday 20 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 7357/3 Section A Answer all questions in the spaces provided. 1 The graph of y ¼ f (x) is shown below. y box O x One of the four equations listed below is the equation of the graph y ¼ f (x) Identify which one is the correct equation of the graph. Tick (3) one box. y ¼ jx þ 2jþ 3 y ¼ jx þ 2j— 3 y ¼ jx — 2jþ 3 y ¼ jx — 2j— 3 [1 mark] 2 The trapezium rule is used to estimate the area of the shaded region in each of the graphs below. Identify the graph for which the trapezium rule produces an overestimate. box Tick (3) one box. y [1 mark] O x y O x y O x y O x Turn over 3 A curve with equation y ¼ f (x) passes through the point (3, 7) Given that f ’(3) ¼ 0 find the equation of the normal to the curve at (3, 7) Circle your answer. y ¼ 7 x y ¼ 0 x ¼ 3 x ¼ 7 [1 mark] outside the box 4 Express 5 — p3 x x2 in the form where p and q are constants. 5xp — xq [2 marks] 5 A curve has equation y ¼ 3e2x outside the box Find the gradient of the curve at the point where y ¼ 10 [3 marks] Turn over for the next question Turn over 6 (a) Sketch the curve with equation where a 0 y ¼ x2(2x þ a) y [3 marks] box O x 6 (b) The polynomial p(x) is given by p(x) ¼ x2(2x þ a) þ 36 6 (b) (i) It is given that x þ 3 is a factor of p(x) Use the factor theorem to show a ¼ 2 [2 marks] 6 (b) (ii) State the transformation which maps the curve with equation y ¼ x2(2x þ 2) box onto the curve with equation y ¼ x2(2x þ 2) þ 36 [2 marks] 6 (b) (iii) The polynomial x2(2x þ 2) þ 36 can be written as (x þ 3)(2x2 þ bx þ c) Without finding the values of b and c, use your answers to parts (a) and (b)(ii) to explain why b2 8c [2 marks] Turn over for the next question Turn over (07) 7 A new design for a company logo is to be made from two sectors of a circle, ORP and OQS, and a rhombus OSTR, as shown in the diagram below. T box R S θ P O Q The points P, O and Q lie on a straight line and the angle ROS is y radians. A large copy of the logo, with PQ ¼ 5 metres, is to be put on a wall. 7 (a) Show that the area of the logo, A square metres, is given by A ¼ 25 (p — y þ 2 sin y) [4 marks] (0 ) 7 (b) (i) Show that the maximum value of A occurs when y p 3 Fully justify your answer. [6 marks] outside the box Question 7 continues on the next page Turn over 7 (b) (ii) Find the exact maximum value of A [2 marks] box 7 (c) Without further calculation, state how your answers to parts (b)(i) and (b)(ii) would change if PQ were increased to 10 metres. [2 marks] Turn over for the next question box DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED Turn over 8 Use the substitution u ¼ x5 þ 2 to show that outside the box ð1 x9 1 (x5 þ 2)3 d x ¼ 180 [7 marks] box Turn over for the next question Turn over 9 A water slide is the shape of a curve PQ as shown in Figure 1 below. box Figure 1 P Q O The curve can be modelled by the parametric equations x ¼ t 1 — t þ 4:8 where 0:2 ≤ t ≤ 3 y ¼ t þ 2 The horizontal distance from O is x metres. The vertical distance above the point O at ground level is y metres. P is the point where t ¼