AQA MATHEMATICS Paper 2

A-level MATHEMATICS Paper 2 Tuesday 11 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 2 7357/2 Section A Answer all questions in the spaces provided. 1 One of the equations below is the equation of a circle. Identify this equation. [1 mark] box Tick () one box. (x + 1)2 – (y + 2)2 = –36 (x + 1)2 – (y + 2)2 = 36 (x + 1)2 + (y + 2)2 = –36 (x + 1)2 + (y + 2)2 = 36 2 The graph of y = f (x) intersects the x‑axis at (–3, 0), (0, 0) and (2, 0) as shown in the diagram below. box y A – 3 The shaded region A has an area of 189 The shaded region B has an area of 64 2 Find the value of ∫–3 f (x) dx Circle your answer. 2 x B [1 mark] –253 –125 125 253 Turn over for the next question Turn over U 3 Solve the inequality (1 – x)(x – 4) 0 [1 mark] box Tick () one box. {x : x 1} {x : x 4} {x : x 1} ∩ {x : x 4} {x : x 1} {x : x ≥ 4} {x : x 1} ∩ {x : x ≥ 4} 4 Use logarithms to solve the equation 5x–2 = 71570 box Give your answer to two decimal places. [3 marks] Turn over for the next question Turn over U 5 Given that find dy dx y = x3 sin x [3 marks] box 6 It is given that and that θ is obtuse. (2 sin θ + 3 cos θ)2 + (6 sin θ – cos θ)2 = 30 box Find the exact value of sin θ. Fully justify your answer. [6 marks] Turn over U (07) 7 On the first day of each month, Kate pays £50 into a savings account. Interest is paid on the total amount in the account on the last day of each month. The interest rate is 0.2% At the end of the nth month, the total amount of money in Kate’s savings account is £Tn Kate correctly calculates T1 and T2 as shown below: T1 = 50 × 1.002 = 50.10 T2 = (T1 + 50) × 1.002 = ((50 × 1.002) + 50) × 1.002 = 50 × 1.0022 + 50 × 1.002 ≈ 100.30 box 7 (a) Show that T3 is given by T3 = 50 × 1.0023 + 50 × 1.0022 + 50 × 1.002 [1 mark] 7 (b) Kate uses her method to correctly calculate how much money she can expect to have in her savings account at the end of 10 years. 7 (b) (i) Find the amount of money Kate expects to have in her savings account at the end of 10 years. [3 marks] (0 8) box 7 (b) (ii) The amount of money in Kate’s savings account at the end of 10 years may not be the amount she has correctly calculated. Explain why. [1 mark] Turn over for the next question Turn over U 8 A zookeeper models the median mass of infant monkeys born at their zoo, up to the age of 2 years, by the formula box y = a + b log10 x where y is the median mass in kilograms, x is age in months and a and b are constants. The zookeeper uses the data shown below to determine the values of a and b. Age in months (x) 3 24 Median mass (y) 6.4 12 8 (a) The zookeeper uses the data for monkeys aged 3 months to write the correct equation 6.4 = a + b log10 3 8 (a) (i) Use the data for monkeys aged 24 months to write a second equation. [1 mark] 8 (a) (ii) Show that b = 5.6 log10 8 [3 marks] box 8 (a) (iii) Find the value of a. Give your answer to two decimal places. [1 mark] Question 8 continues on the next question Turn over U 8 (b) Use a suitable value for x to determine whether the model can be used to predict the median mass of monkeys less than one week old. box [2 marks] Turn over for the next question DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED box Turn over U 9 (a) (i) Find the binomial expansion of (1 + 3x)–1 up to and including the term in x2 [2 marks] box 9 (a) (ii) Show that the first three terms in the binomial expansion of 1 2 – 3x form a geometric sequence and state the common ratio. [5 marks] 9 (b) It is given that 36x ≡ P + Q (1 + 3x)(2 – 3x) where P and Q are integers. Find the value of P and the value of Q (2 – 3x) (1 + 3x) [3 marks] Turn over U 9 (c) (i) Using your answers to parts (a) and (b), find the binomial expansion of 12x (1 + 3x)(2 – 3x) up to and including the term in x2 [2 marks] 9 (c) (ii) Find the range of values of x for which the binomial expansion of 12x (1 + 3x)(2 – 3x) box is valid. [1 mark] Turn over for the next question