AQA A-level MATHEMATICS FINAL Paper 3 JUNE 2022

A-level MATHEMATICS Paper 3 JUNE 2022 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. (JUN) PB/Jun22/E7 7357/3 Section A Answer all questions in the spaces provided. 1 State the range of values of x for which the binomial expansion of rffi1ffiffiffi—ffiffiffiffiffixffiffi box is valid. Circle your answer. 1 jxj < 4 jxj < 1 jxj < 2 jxj < 4 [1 mark] 2 The shaded region, shown in the diagram below, is defined by x2 — 7x þ 7 ≤ y ≤ 7 — 2x box y O 5 x Identify which of the following gives the area of the shaded region. Tick (3) one box. ð (7 — 2x) d x — ð (x2 — 7x þ 7) d x [1 mark] 5 (x2 0 ð5 — 5x) d x 2 (5x x 0 ð5 2 ) d x Turn over for the next question Turn over 3 The function f is defined by Solve the equation Circle your answer. f (x) ¼ 2x þ 1 f (x) ¼ f —1ðx) [1 mark] box x ¼ —1 x ¼ 0 x ¼ 1 x ¼ 2 4 Find ð x2 1 þ x2 d x [2 marks] 5 (a) Sketch the graph of for 0° ≤ x ≤360° y y ¼ sin 2x box O 90° 180° 270° 360° x [2 marks] 5 (b) The equation sin 2x ¼ A has exactly two solutions for 0° ≤ x ≤360° State the possible values of A. [1 mark] Turn over 6 A design for a surfboard is shown in Figure 1. Figure 1 box width length The curve of the top half of the surfboard can be modelled by the parametric equations x ¼ —2t 2 y ¼ 9t — 0:7t 2 for 0 ≤ t ≤ 9:5 as shown in Figure 2, where x and y are measured in centimetres. Figure 2 y O x 6 (a) Find the length of the surfboard. [2 marks] dy 6 (b) (i) Find an expression for d x in terms of t. [3 marks] box 6 (b) (ii) Hence, show that the width of the surfboard is approximately one third of its length. [4 marks] Turn over (07) 7 A planet takes T days to complete one orbit of the Sun. T is known to be related to the planet’s average distance d, in millions of kilometres, from the Sun. A graph of log10 T against log10 d is shown with data for Mercury and Uranus labelled. log10 T box Uranus (3.46, 4.49) Mercury (1.76, 1.94) log10 d 7 (a) (i) Find the equation of the straight line in the form log10 T ¼ a þ b log10 d where a and b are constants to be found. [3 marks] 7 (a) (ii) Show that T ¼ K d n box where K and n are constants to be found. [2 marks] 7 (b) Neptune takes approximately 60 000 days to complete one orbit of the Sun. Use your answer to 7(a)(ii) to find an estimate for the average distance of Neptune from the Sun. [2 marks] Turn over for the next question Turn over 8 Water is poured into an empty cone at a constant rate of 8 cm3/s outside the box After t seconds the depth of the water in the inverted cone is h cm, as shown in the diagram below. h When the depth of the water in the inverted cone is h cm, the volume, V cm3, is given by 8 (a) Show that when t ¼ 3 ph3 V 12 dV ¼ 6 p3 ffi6ffiffipffiffiffi [4 marks] box 8 (b) Hence, find the rate at which the depth is increasing when t ¼ 3 Give your answer to three significant figures. [3 marks] Turn over