FURTHER MATHEMATICS Paper 1

FURTHER MATHEMATICS Paper 1 Thursday 25 May 2023 Afternoon Time allowed: 2 hours Materials  You must have the AQA Formulae and statistical tables booklet for A‑level Mathematics and A‑level Further Mathematics.  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 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).  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. PB/KL/Jun23/E6 7367/1 Answer all questions in the spaces provided. 1 Find the number of solutions of the equation tanh x ¼ cosh x Circle your answer. [1 mark] outside the box 0 1 2 3 2 The diagram below shows a locus on an Argand diagram. Im O 2 Re – 3 Which of the equations below represents the locus shown above? Circle your answer. [1 mark] jz — 2 þ 3ij¼ 2 jz þ 2 — 3ij¼ 2 jz — 2 þ 3ij¼ 4 jz þ 2 — 3ij¼ 4 1 2 3 The matrix A ¼ 0 1 represents a transformation. outside the box Which one of the points below is an invariant point under this transformation? Circle your answer. [1 mark] (1, 1) (0, 2) (3, 0) (2, 1) 4 The solution of a second order differential equation is f (t) The differential equation models heavy damping. Which one of the statements below could be true? Tick (3) one box. [1 mark] f (t) ¼ 2e—t cos (3t) þ 5e—t sin (3t) f (t) ¼ 3e—t þ 4te—t f (t) ¼ 7e—t þ 2e—2t f (t) ¼ 8e—t cos (3t — 0:1) Turn over for the next question Turn over 5 The function f is defined by f (r) ¼ 2r(r — 2) (r 2 Z) box 5 (a) Show that f (r þ 1) — f (r) ¼ r2 r [2 marks] 5 (b) Use the method of differences to show that box n r2 r ¼ 2nþ1(n — 1) þ 2 r¼1 [4 marks] Turn over for the next question Turn over 6 The matrix M is given by 2 a a —6 3 box M ¼ 6 0 10 0 7 where a is a real number. 10 4 9 14 —13 5 The vectors v1, v2, and v3 are eigenvectors of M The corresponding eigenvalues are l1, l2, and l3 respectively. 2 1 3 2 1 3 2 c 3 It is given that l2 ¼ 1 and v1 ¼ 6 0 7 , v2 ¼ 6 1 7 and v3 ¼ 64 0 75 , 6 (a) (i) Find the value of l1 [2 marks] 6 (a) (ii) Find the value of a [2 marks] 6 (b) Find the integer c and the value of l3 [4 marks] box 6 (c) Find matrices U, D and U—1, such that D is diagonal and M ¼ UDU—1 [3 marks] Turn over (07) 7 The function f is defined by f (x) ¼ .sin x þ . (0 ≤ x ≤ 2r) box Find the set of values of x 2. f (x) 1 2 Give your answer in set notation. [5 marks] (0 )