Official June 2024 AQA AS FURTHER MATHEMATICS 7366/1 Paper 1 Merged Question Paper + Mark Scheme Ace your Mocks!!! G/LM/Jun24/G4001/V5 7366/1 (JUN247366101) AS FURTHER MATHEMATICS Paper 1 Monday 13 May 2024 Afternoon Time allowed: 1 hour 30 minutes Materials l You must have the AQA Formulae and statistical tables booklet for A‑level Mathematics and A‑level Further Mathematics. 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 require 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. For Examiner’s Use Question Mark 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 TOTAL Please write clearly in block capitals. Centre number Candidate number Surname _________________________________________________________________________ Forename(s) _________________________________________________________________________ Candidate signature _________________________________________________________________________ I declare this is my own work. 2 Do not write outside the box (02) G/Jun24/7366/1 Answer all questions in the spaces provided. 1 Express cosh2 x in terms of sinh x Circle your answer. [1 mark] 1 + sinh2 x 1 – sinh2 x sinh2 x – 1 –1 – sinh2 x 2 The function f is defined by f(x) = 2x + 3 0 ≤ x ≤ 5 The region R is enclosed by y = f(x), x = 5, the x-axis and the y-axis. The region R is rotated through 2π radians about the x-axis. Give an expression for the volume of the solid formed. Tick () one box. [1 mark] π ∫ 5 0 (2x + 3) dx π∫ 5 0 (2x + 3)2dx 2π ∫ 5 0 (2x + 3)dx 2π ∫ 5 0 (2x + 3)2dx 3 Do not write outside the box (03) G/Jun24/7366/1 Turn over U 3 The matrix A is such that det(A) = 2 Determine the value of det(A–1) Circle your answer. [1 mark] –2 –1 2 1 2 2 4 The line L has vector equation r = [ 4 ] –7 0 + λ [ –9 ] 1 3 Give the equation of L in Cartesian form. Tick () one box. [1 mark] x + 4 –9 = y – 7 1 = z 3 x – 4 –9 = y + 7 1 = z 3 x + 9 4 = y – 1 –7 , z = 3 x – 9 4 = y + 1 –7 , z = 3 4 Do not write outside the box (04) G/Jun24/7366/1 5 The vectors a and b are given by a = 3i + 4j – 2k and b = 2i – j – 5k 5 (a) Calculate a.b [1 mark] ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ 5 (b) Calculate |a| and |b| [2 marks] ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ |a| = ____________________________ |b| = ____________________________ 5 (c) Calculate the acute angle between a and b Give your answer to the nearest degree. [2 marks] ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ 5 Do not write outside the box (05) G/Jun24/7366/1 Turn over U 6 (a) On the axes below, sketch the graph of y = cosh x Indicate the value of any intercept of the curve with the axes. [2 marks] x y O 6 (b) Solve the equation cosh x = 2 Give your answers to three significant figures. [2 marks] ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ 6 (06) G/Jun24/7366/1 Do not write outside the box There are no questions printed on this page DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED 7 Do not write outside the box (07) G/Jun24/7366/1 Turn over U 7 The function f is defined by f(x) = 1 √x 4 ≤ x ≤ 7 Find the mean value of f over the interval 4 ≤ x ≤ 7 Give your answer in exact form. [3 marks] ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ 8 Do not write outside the box (08) G/Jun24/7366/1 8 (a) The complex number z is given by z = x + iy where x, y ∈ ℝ 8 (a) (i) Write down the complex conjugate z* in terms of x and y [1 mark] ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ 8 (a) (ii) Hence prove that zz* is real for all z ∈ ℂ [2 marks] ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ 9 Do not write outside the box (09) G/Jun24/7366/1 Turn over U 8 (b) The complex number w satisfies the equation 3w + 10i = 2w* + 5 8 (b) (i) Find w [3 marks] ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ 8 (b) (ii) Calculate the value of w2(w*)2 [1 mark] ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ 10 Do not write outside the box G/Jun24/7366/1 (10) 9 (a) Show that, for all positive integers r, r + 1 r + 2 – r r + 1 = 1 (r + 1)(r + 2) [1 mark] ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ 9 (b) Hence, using the method of differences, show that n ∑r=1 1 (r + 1)(r + 2) = n an + b where a and b are integers to be determined. [3 marks] ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ 11 Do not write outside the box G/Jun24/7366/1 Turn over U (11) ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ 9 (c) Hence find the exact value of 2000 ∑r=1001 1 (r + 1)(r + 2) [3 marks] ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ 12 Do not write outside the box G/Jun24/7366/1 (12) 10 The curve C has equation y = 2x – 10 3x –