AQA AS MATHEMATICS 7356/1 Paper 1 Question Paper + Mark scheme [MERGED] June 2022 PB/Jun22/E5 7356/1 (JUN) AS MATHEMATICS Paper 1 Time allowed: 1 hour 30 minutes Materials

AQA AS MATHEMATICS 7356/1 Paper 1 Question Paper + Mark scheme [MERGED] June 2022 PB/Jun22/E5 7356/1 (JUN) AS MATHEMATICS Paper 1 Time allowed: 1 hour 30 minutes 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 80. 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 TOTAL I declare this is my own work. 2 Section A Answer all questions in the spaces provided. 1 Express as a single logarithm log10 2  log10 x Circle your answer. [1 mark] log10 (2 þ x) log10 (2  x) log10 (2x) log10 2 x   Jun22/7356/1 Do not write outside the box (02) 3 2 The graph of the function y ¼ cos 1 2 x for 0  x  360 is one of the graphs shown below. Identify the correct graph. Tick (3) one box. [1 mark] 2 360° –2 O y x 360° – 1 2 1 2 O y x 1 360° –1 O y x 1 360° –1 O y x Do not write outside the box Jun22/7356/1 Turn over s (03) 4 3 Find the coefficient of the x3 term in the expansion of 3x þ 1 2  4 [3 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Do not write outside the box Jun22/7356/1 (04) 5 4 Find all the solutions of the equation cos2 y ¼ 10 sin y þ 4 for 0 < y < 360, giving your answers to the nearest degree. Fully justify your answer. [5 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Turn over for the next question Do not write outside the box Jun22/7356/1 Turn over s (05) 6 5 Express 3x3 þ 5x2  27x þ 10 in the form (x  2)(ax2 þ bx þ c) , where a, b and c are integers. [3 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Do not write outside the box Jun22/7356/1 (06) DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED 7 Turn over for the next question Do not write outside the box Jun22/7356/1 Turn over s (07) 8 6 AB is a diameter of a circle where A is (1, 4) and B is (7, 2) 6 (a) Find the coordinates of the midpoint of AB. [1 mark] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 6 (b) Show that the equation of the circle may be written as x2 þ y2  8x  2y ¼ 1 [4 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Do not write outside the box Jun22/7356/1 (08) 9 6 (c) The circle has centre C and crosses the x-axis at points D and E. Find the exact area of triangle DEC. [4 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Do not write outside the box Jun22/7356/1 Turn over s (09) 10 7 A curve has equation y ¼ a2  x2 , where a > 0 The area enclosed between the curve and the x-axis is 36 units. Find the value of a. Fully justify your answer. [6 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Do not write outside the box Jun22/7356/1 (10) 11 _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Turn over for the next question Do not write outside the box Jun22/7356/1 Turn over s (11) 12 8 A curve has equation y ¼ x3  6x þ 9 x 8 (a) Show that the x coordinates of the stationary points of the curve satisfy the equation x4  2x2  3 ¼ 0 [3 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 8 (b) Deduce that the curve has exactly two stationary points. [3 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Do not write outside the box Jun22/7356/1 (12) 13 8 (c) Find the coordinates and nature of the two stationary points. Fully justify your answer. [4 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 8 (d) Write down the equation of a line which is a tangent to the curve in two places. [1 mark] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Turn over for the next question Do not write outside the box Jun22/7356/1 Turn over s (13) 14 9 Integers m and n are both odd. Prove that m2 þ n2 is a multiple of 2 but not a multiple of 4 [5 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Do not write outside the box Jun22/7356/1 (14) DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED 15 Turn over for the next question Do not write outside the box Jun22/7356/1 Turn over s (15) 16 10 Curve C has equation y ¼ ffiffiffi 2 p x2 10 (a) Find an equation of the tangent to C at the point 2, ffiffiffi 2 p 4 ! [4 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 10 (b) Show that the tangent to C at the point 2, ffiffiffi 2 p 4 ! is also a normal to the curve at a different point. (2, ) 2 4 O x y [5 marks] Do not write outside the box Jun22/7356/1 (16) 17 _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Turn over for Section B Do not write outside the box Jun22/7356/1 Turn over s (17) 18 Section B Answer all questions in the spaces provided. 11 A car, initially at rest, moves with constant acceleration along a straight horizontal road. One of the graphs below shows how the car’s velocity, v m s1, changes over time, t seconds. Identify the correct graph. Tick (3) one box. [1 mark] t v O t v O t v O t v O Do not write outside the box Jun22/7356/1 (18) 19 12 A horizontal force of 30 N causes a crate to travel with an acceleration of 2 m s2, in a straight line, on a smooth horizontal surface