AQA MATHEMATICS Paper 1

A-level MATHEMATICS Paper 1 Tuesday 4 June 2024 Afternoon Time allowed: 2 hours Materials  You must have the AQA Formulae for A‑level Mathematics booklet.  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 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).  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. (JUN) PAPER 1 7357/1 Answer all questions in the spaces provided. 1 Find the coefficient of x in the expansion of (4x3 – 5x2 + 3x – 2)(x5 + 4x + 1) Circle your answer. [1 mark] box –5 –2 7 11 2 The function f is defined by f (x) = ex + 1 for x ℝ Find an expression for f –1(x) Tick () one box. [1 mark] box f –1(x) = ln (x – 1) f –1(x) = ln (x) – 1 f –1(x) = 1 ex + 1 f –1(x) = x – 1 Turn over for the next question Turn over U 3 The expression 12x2 + 3x + 7 3x – 5 box can be written as State the value of A Circle your answer. Ax + B + C [1 mark] 3 4 7 9 4 One of the diagrams below shows the graph of y = arccos x Identify the graph of y = arccos x Tick () one box. [1 mark] box y π 2 –1 O x y π 2 O 1 x Turn over for the next question Turn over U 5 Solve the equation box for 0° x 360° sin2 x = 1 [3 marks] 6 Use the chain rule to find dy d x when y = (x3 + 5x)7 [2 marks] box Turn over for the next question Turn over U (07) 7 Show that box 3 + √ 8n 1 + √ 2n can be written as 4n – 3 + √ 2n 2n – 1 where n is a positive integer. [4 marks] (08) 8 (a) Find the first three terms, in ascending powers of x, in the expansion of (2 + k x)5 box where k is a positive constant. [3 marks] 8 (b) Hence, given that the coefficient of x is four times the coefficient of x2, find the value of k [2 marks] Turn over U 9 (a) Show that, for small values of θ measured in radians cos 4θ + 2 sin 3θ – tan 2θ ≈ A + Bθ + Cθ 2 where A, B and C are constants to be found. [3 marks] box 9 (b) Use your answer to part (a) to find an approximation for cos 0.28 + 2 sin 0.21 – tan 0.14 Give your answer to three decimal places. [2 marks] box Turn over for the next question Turn over U 10 (a) An arithmetic sequence has 300 terms. The first term of the sequence is –7 and the last term is 32 Find the sum of the 300 terms. [2 marks] box 10 (b) A school holds a raffle at its summer fair. There are nine prizes. The total value of the prizes is £1260 The values of the prizes form an arithmetic sequence. The top prize has the highest value, and the bottom prize has the least value. The value of the top prize is six times the value of the bottom prize. Find the value of the top prize. [4 marks] box Turn over U 11 It is given that box f (x) = x(x – a)(x – 6) where 0 a 6 11 (a) Sketch the graph of y = f (x) on the axes below. [3 marks] y O x 11 (b) Sketch the graph of y = f (–2x) on the axes below. [2 marks] box y O x Turn over for the next question Turn over U 12 The terms, un, of a periodic sequence are defined by u1 = 3 and un+1 = – 6 box 12 (a) Find u2, u3 and u4 [2 marks] 12 (b) State the period of the sequence. [1 mark] 101 12 (c) Find the value of ∑un n =1 [2 marks] box Turn over for the next question Turn over U (17)