A-level MATHEMATICS Paper 3 Time allowed: 2 hou

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A-level MATHEMATICS Paper 3 Time allowed: 2 hours Materials l You must have the AQA Formulae for A‑level Mathematics booklet. l You should have a graphical or scientific calculator that meets ... the requirements of the specification. Instructions l Use black ink or black ball-point pen. Pencil should only be used for drawing. l Fill in the boxes at the top of this page. l Answer all questions. l 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). l Do not write outside the box around each page or on blank pages. l Show all necessary working; otherwise marks for method may be lost. l Do all rough work in this book. Cross through any work that you do not want to be marked. Information l The marks for questions are shown in brackets. l The maximum mark for this paper is 100. Advice l Unless stated otherwise, you may quote formulae, without proof, from the booklet. l You do not necessarily need to use all the space provided. Please write clearly in block capitals. Centre number Candidate number Surname ________________________________________________________________________ Forename(s) ________________________________________________________________________ Candidate signature ________________________________________________________________________ For Examiner’s Use Question Mark 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 TOTAL I declare this is my own work. 2 Section A Answer all questions in the spaces provided. 1 State the range of values of x for which the binomial expansion of ffiffiffiffiffiffiffiffiffiffiffi 1  x 4 r is valid. Circle your answer. [1 mark] jxj < 1 4 jxj < 1 jxj < 2 jxj < 4 Jun22/7357/3 Do not write outside the box (02) 3 2 The shaded region, shown in the diagram below, is defined by x2  7x þ 7  y  7  2x O 5 x y Identify which of the following gives the area of the shaded region. Tick (3) one box. [1 mark] ð (7  2x) dx  ð (x2  7x þ 7) dx ð5 0 (x2  5x) dx ð5 0 (5x  x2) dx ð5 0 (x2  9x þ 14) dx Turn over for the next question Do not write outside the box Jun22/7357/3 Turn over s (03) 4 3 The function f is defined by f (x) ¼ 2x þ 1 Solve the equation f (x) ¼ f 1ðx) Circle your answer. [1 mark] x ¼ 1 x ¼ 0 x ¼ 1 x ¼ 2 4 Find ð x2 þ x 1 2   dx [2 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Do not write outside the box Jun22/7357/3 (04) 5 5 (a) Sketch the graph of y ¼ sin 2x for 0  x 360 O x y 90° 180° 270° 360° [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] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Do not write outside the box Jun22/7357/3 Turn over s (05) 6 6 A design for a surfboard is shown in Figure 1. Figure 1 length width The curve of the top half of the surfboard can be modelled by the parametric equations x ¼ 2t 2 y ¼ 9t  0:7t2 for 0  t  9:5 as shown in Figure 2, where x and y are measured in centimetres. Figure 2 O y x 6 (a) Find the length of the surfboard. [2 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Do not write outside the box Jun22/7357/3 (06) 7 6 (b) (i) Find an expression for dy dx in terms of t. [3 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 6 (b) (ii) Hence, show that the width of the surfboard is approximately one third of its length. [4 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Do not write outside the box Jun22/7357/3 Turn over s (07) 8 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 log10 d Uranus (3.46, 4.49) Mercury (1.76, 1.94) 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] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Do not write outside the box Jun22/7357/3 (08) 9 7 (a) (ii) Show that T ¼ K d n 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 Do not write outside the box Jun22/7357/3 Turn over s (09) 10 8 Water is poured into an empty cone at a constant rate of 8 cm3/s 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, Vcm3, is given by V ¼ ph3 12 8 (a) Show that when t ¼ 3 dV dh ¼ 6 ffiffiffiffiffiffi 6p p3 [4 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Do not write outside the box Jun22/7357/3 (10) 11 _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Do not write outside the box Jun22/7357/3 Turn over s (11) 12 9 Assume that a and b are integers such that a2  4b  2 ¼ 0 9 (a) Prove that a is even. [2 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 9 (b) Hence, prove that 2b þ 1 is even and explain why this is a contradiction. [3 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Do not write outside the box Jun22/7357/3 (12) 13 9 (c) Explain what can be deduced about the solutions of the equation a2  4b  2 ¼ 0 [1 mark] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Turn over for the next question Do not write outside the box Jun22/7357/3 Turn over s (13) 14 10 The function f is defined by f (x) ¼ x2 þ 10 2x þ 5 where f has its maximum possible domain. The curve y ¼ f (x) intersects the line y ¼ x at the points P and Q as shown below. x y = f (x) y = x O Q P y 10 (a) State the value of x which is not in the domain of f. [1 mark] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Do not write outside the box Jun22/7357/3 (14) 15 10 (b) Explain how you know that the function f is many-to-one. [2 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 10 (c) (i) Show that the x-coordinates of P and Q satisfy the equation x2 þ 5x  10 ¼ 0 [2 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 10 (c) (ii) Hence, find the exact x-coordinate of P and the exact x-coordinate of Q. [1 mark] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Do not write outside the box Jun22/7357/3 Turn over s (15) 16 10 (d) Show that P and Q are stationary points of the curve. Fully justify your answer. [5 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 10 (e) Using set notation, state the range of f [Show More]

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