FURTHER MATHEMATICS Paper 2

FURTHER MATHEMATICS Paper 2 Monday 5 June 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/E4 7367/2 Answer all questions in the spaces provided. outside the box 1 Given that y ¼ sin x þ sinh x , find d2y d x2 þ y Circle your answer. 2 sin x —2 sin x 2 sinh x —2 sinh x 2 Which one of the expressions below is not equal to zero? Circle your answer. [1 mark] [1 mark] lim (x2 e—x ) lim (x5 ln x) lim ex lim (x3ex) x !1 x ! 0 x !1 x5 x ! 0 1 1 1 3 The determinant A ¼ 2 0 2 3 2 1 Which one of the determinants below has a value which is not equal to the value of A? outside the box Tick (3) one box. 3 1 3 2 0 2 . 3 2 1 . . 1 0 2 . . 1 2 1 . . 1 0 1 . . 3 2 1 . . 3 2 1 . . 2 0 2 . 4 It is given that f (x) ¼ cosh—1 (x — 3) [1 mark] Which of the sets listed below is the greatest possible domain of the function f ? Circle your answer. fx : x ≥ 4g fx : x ≥ 3g fx : x ≥ 1g fx : x ≥ 0g [1 mark] Turn over 5 Josh and Zoe are solving the following mathematics problem: outside the box Josh says that to solve this problem you must first carry out the transformation on C1 to find C2, and then find the asymptotes of C2 Zoe says that you will get the same answer if you first find the asymptotes of C1, and then carry out the transformation on these asymptotes to obtain the asymptotes of C2 Show that Zoe is correct. [5 marks] 6 (a) Express —5 — 5i in the form reiy , where —p y ≤ p [2 marks] outside the box 6 (b) The point on an Argand diagram that represents 5 5i is one of the vertices of an equilateral triangle whose centre is at the origin. Find the complex numbers represented by the other two vertices of the triangle. Give your answers in the form reiy, where —p y ≤ p [3 marks] Turn over 7 Show that n 1 r 3 r¼11 1 2 ¼ 4 (n þ an þ b)(n2 þ an þ c) outside the box where a, b and c are integers to be found. [3 marks] 8 A is a non-singular 2 × 2 matrix and AT 8 (a) Using the result is the transpose of A outside the box show that (AB)T ¼ BTAT (A—1)T ¼ (AT)—1 [3 marks] 8 (b) It is given that A ¼ " 4 5 , where k is a real constant. —1 k 8 (b) (i) Find (A—1)T, giving your answer in terms of k [2 marks] 8 (b) (ii) State the restriction on the possible values of k [1 mark] Turn over (07) 9 The complex number z is such that where k is a real number. z 1 þ i 1 — ki box 9 (a) Find the real part of z and the imaginary part of z, giving your answers in terms of k [2 marks] 9 (b) In the case where k ¼ pffi3ffiffi , use part (a) to show that cos 7p ¼ pffi2ffiffi — pffi6ffiffi 12 4 [5 marks] (0 ) box Turn over p 10 The region R on an Argand diagram satisfies both jz þ 2ij≤ 3 and — 6 p ≤ arg (z) ≤ 2 outside the box 10 (a) Sketch R on the Argand diagram below. Im 5 4 3 2 1 [3 marks] – 5 – 4 – 3 – 2 – 1 O – 1 – 2 – 3 – 4 – 5 1 2 3 4 5 Re 10 (b) Find the maximum value of z in the region R, giving your answer in exact form. [5 marks] box Turn over for the next question Turn over 11 The line l1 passes through the points A(6, 2, 7) and B(4, —3, 7) 11 (a) Find a Cartesian equation of l1 [2 marks] box 2 8 3 2 1 3 11 (b) The line l2 has vector equation r ¼ 64 9 75 þ m64 1 75 where c is a constant. 11 (b) (i) Explain how you know that the lines l1 and l2 are not perpendicular. [2 marks] 11 (b) (ii) The lines l1 and l2 both lie in the same plane. Find the value of c [5 marks] box Turn over for the next question Turn over