A-level MATHEMATICS Paper 1 Time allowed: 2 hours Materials

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A-level MATHEMATICS Paper 1 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 TOTAL I declare this is my own work. 2 Answer all questions in the spaces provided. 1 A curve is defined by the parametric equations x ¼ cos y and y ¼ sin y where 0  y  2p Which of the options shown below is a Cartesian equation for this curve? Circle your answer. [1 mark] y x ¼ tan y x2 þ y2 ¼ 1 x2  y2 ¼ 1 x2y2 ¼ 1 2 A periodic sequence is defined by Un ¼ (1)n State the period of the sequence. Circle your answer. [1 mark] 10 1 2 3 The curve y ¼ log4 x is transformed by a stretch, scale factor 2, parallel to the y-axis. State the equation of the curve after it has been transformed. Circle your answer. [1 mark] y ¼ 1 2 log4 x y ¼ 2 log4 x y ¼ log4 2x y ¼ log8 x Jun22/7357/1 Do not write outside the box (02) DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED 3 Turn over for the next question Do not write outside the box Jun22/7357/1 Turn over s (03) 4 4 The graph of y ¼ f (x) where f (x) ¼ ax2 þ bx þ c is shown in Figure 1. Figure 1 x y Do not write outside the box Jun22/7357/1 (04) 5 Which of the following shows the graph of y ¼ f0 (x) ? Tick (3) one box. [1 mark] x y x y x y x y Do not write outside the box Jun22/7357/1 Turn over s (05) 6 5 Find an equation of the tangent to the curve y ¼ (x  2)4 at the point where x ¼ 0 [3 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Do not write outside the box Jun22/7357/1 (06) 7 6 (a) Find the first two terms, in ascending powers of x, of the binomial expansion of 1  x 2   1 2 [2 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 6 (b) Hence, for small values of x, show that sin 4x þ ffiffiffiffiffiffiffiffiffiffiffi cos x p  A þ Bx þ Cx2 where A, B and C are constants to be found. [4 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Do not write outside the box Jun22/7357/1 Turn over s (07) 8 7 Sketch the graph of y ¼ cot x  p 2  for 0  x  2p [3 marks] x y O  2 Do not write outside the box Jun22/7357/1 (08) DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED 9 Turn over for the next question Do not write outside the box Jun22/7357/1 Turn over s (09) 10 8 The lines L1 and L2 are parallel. L1 has equation 5x þ 3y ¼ 15 and L2 has equation 5x þ 3y ¼ 83 L1 intersects the y-axis at the point P. The point Q is the point on L2 closest to P, as shown in the diagram. x y L2 Q P L1 8 (a) (i) Find the coordinates of Q. [5 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Do not write outside the box Jun22/7357/1 (10) 11 _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 8 (a) (ii) Hence show that PQ ¼ k ffiffiffiffiffi 34 p , where k is an integer to be found. [2 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Do not write outside the box Jun22/7357/1 Turn over s (11) 12 8 (b) A circle, C, has centre (a, 17). L1 and L2 are both tangents to C. 8 (b) (i) Find a. [2 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 8 (b) (ii) Find the equation of C. [2 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Do not write outside the box Jun22/7357/1 (12) DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED 13 Turn over for the next question Do not write outside the box Jun22/7357/1 Turn over s (13) 14 9 The first three terms of an arithmetic sequence are given by 2x þ 5 5x þ 1 6x þ 7 9 (a) Show that x ¼ 5 is the only value which gives an arithmetic sequence. [3 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 9 (b) (i) Write down the value of the first term of the sequence. [1 mark] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 9 (b) (ii) Find the value of the common difference of the sequence. [1 mark] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Do not write outside the box Jun22/7357/1 (14) 15 9 (c) The sum of the first N terms of the arithmetic sequence is SN where SN < 100 000 SNþ1 > 100 000 Find the value of N. [4 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Do not write outside the box Jun22/7357/1 Turn over s (15) 16 10 The diagram shows a sector of a circle OAB. O θ C A B The point C lies on OB such that AC is perpendicular to OB. Angle AOB is y radians. 10 (a) Given the area of the triangle OAC is half the area of the sector OAB, show that y ¼ sin 2y [Show More]

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