2024_AQA A-Level Mathematics Paper 3 (Merged Question Paper and Marking Scheme) Thursday 20 June 2024

2024_AQA A-Level Mathematics Paper 3 (Merged Question Paper and Marking Scheme) Thursday 20 June 2024 Please write clearly in block capitals. Centre number Surname Forename(s) Candidate signature Candidate number I declare this is my own work. A-level MATHEMATICS Paper 3 Thursday 20 June 2024 Materials Afternoon  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 Time allowed: 2 hours  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. For Examiner’s Use Question Mark 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 TOTAL A-Level Mathematics: Paper 3 (Thursday 20 June 2024): Exam Preparation area This paper typically focuses on Applied Mathematics, which includes Mechanics and Statistics. It builds upon the foundations covered in Paper 1 and Paper 2, but focuses more on real-world applications of mathematics. 1. Mechanics:  Kinematics: o Motion in one and two dimensions. Understanding displacement, velocity, and acceleration, and solving problems using the equations of motion. o Motion under constant acceleration, including free fall and projectile motion.  Forces: o Newton’s Laws of Motion: Solving problems involving forces, friction, tension, and normal force. o Equilibrium: Analyzing forces acting on an object at rest and solving static equilibrium problems. o Circular Motion: Understanding centripetal force and angular velocity for objects moving in a circular path.  Work, Energy, and Power: o Applying the work-energy principle to problems involving the transfer of energy (e.g., work done by a force, gravitational potential energy, kinetic energy). o Conservation of Energy: Understanding how energy is conserved in mechanical systems and solving related problems.  Momentum: o Conservation of momentum in collisions, both elastic and inelastic. o Solving problems related to impulse and change in momentum. 2. Statistics:  Probability: o Understanding and applying basic probability concepts, including conditional probability and the use of Venn diagrams and tree diagrams. o Bayes’ Theorem: Solving problems involving conditional probability and revising probabilities based on new information.  Distributions: o Binomial Distribution: Solving problems related to trials with two outcomes (success/failure). o Normal Distribution: Working with standard normal distributions, Z-scores, and using the properties of normal distribution to solve problems (e.g., finding percentiles, probabilities). o Other possible distributions such as Poisson Distribution (for counting events in a fixed interval of time/space).  Statistical Inference: o Hypothesis Testing: Performing significance tests for means or proportions, interpreting p values, and making decisions based on the results (e.g., z-tests and t-tests). o Confidence Intervals: Constructing and interpreting confidence intervals for population parameters.  Bivariate Data: o Correlation and Regression: Analyzing the relationship between two variables, interpreting the correlation coefficient, and fitting a line to data using linear regression. 3. Key Focus Areas:  Mechanics: Strengthen your understanding of forces, energy, momentum, and motion, as these topics are frequently tested.  Statistics: Ensure you are comfortable with various probability distributions and statistical inference techniques, as they form a core part of Paper 3. 7357/3 G/LM/Jun24/G4005/E6 2 Do not write outside the Section A Answer all questions in the spaces provided. 1 Each of the series below shows the first four terms of a geometric series. Identify the only one of these geometric series that is convergent. Tick () one box. 0.1 + 0.2 + 0.4 + 0.8 + … 1 – 1 + 1 – 1 + … 128 – 64 + 32 – 16 + … 1 + 2 + 4 + 8 + … box [1 mark] G/Jun24/7357/3 3 Do not write outside the G/Jun24/7357/3 2 The quadratic equation box 4x2 + bx + 9 = 0 has one repeated real root. Find b Circle your answer. [1 mark] b = 0 b = ±12 b = ±13 b = ±36 Turn over for the next question Turn over U 3 4 Do not write outside the One of the graphs shown below cannot have an equation of the form y = ax Identify this graph. Tick () one box. y where a > 0 O x y O x box [1 mark] G/Jun24/7357/3 5 Do not write outside the G/Jun24/7357/3 4 A curve has equation y = x4 + 2x Find an expression for dy dx [2 marks] box Turn over for the next question Turn over U 5 6 Do not write outside the The diagram below shows a sector of a circle OAB. The chord AB divides the sector into a triangle and a shaded segment. Angle AOB is π radians. 6 The radius of the sector is 18 cm. Show that the area of the shaded segment is k (π – 3) cm2 where k is an integer to be found. box [3 marks] G/Jun24/7357/3 7 Do not write outside the G/Jun24/7357/3 Turn over for the next question DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED box Turn over U 6 (a) 8 Do not write outside the Find ( ∫ 6x2 – 5 d x √x ) box [3 marks] G/Jun24/7357/3 9 Do not write outside the G/Jun24/7357/3 6 (b) The gradient of a curve is given by dy = 6x2 – 5 box dx √x The curve passes through the point (4, 90). Find the equation of the curve. [2 marks] Turn over for the next question Turn over U 7 10 Do not write outside the The graphs with equations are shown in the diagram below. The graphs intersect at the points A and B 7 (a) On the diagram above, shade and label the region, R, that is satisfied by the inequalities 0 ≤ y ≤ 2 + 3x – 2x2 and x + y ≥ 1 box y = 2 + 3x – 2x2 and x + y = 1 [2 marks] G/Jun24/7357/3 11 Do not write outside the G/Jun24/7357/3 box 7 (b) Find the exact coordinates of A [3 marks] Turn over for the next question Turn over U 8 12 Do not write outside the The temperature θ °C of an oven t minutes after it is switched on can be modelled by the equation θ = 20 (11 – 10e–kt) where k is a positive constant. Initially the oven is at room temperature. The maximum temperature of the oven is T °C The temperature predicted by the model is shown in the graph below. 8 (a) Find the room temperature. [2 marks] G/Jun24/7357/3 box 13 Do not write outside the G/Jun24/7357/3 box 8 (b) Find the value of T [2 marks] Question 8 continues on the next page Turn over U 14 Do not write outside the G/Jun24/7357/3 8 (c) The oven reaches a temperature of 86 °C one minute after it is switched on. 8 (c) (i) Find the value of k. [2 marks] box 8 (c) (ii) Find the time it takes for the temperature of the oven to be within 1°C of its maximum. [2 marks] 15 Do not write outside the G/Jun24/7357/3 Turn over for the next question DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED box Turn over U 9 16 Do not write outside the Figure 1 below shows a circle. P O x The centre of the circle is P and the circle intersects the y‑ axis at Q as shown in Figure 1. The equation of the circle is 9 (a) x2 + y2 = 12y – 8x – 27 Express the equation of the circle in the form (x – a)2 + (y – b)2 = k where a, b and k are constants to be found. box Figure 1 y Q [3 marks] G/Jun24/7357/3 17 Do not write outside the G/Jun24/7357/3 box 9 (b) State the coordinates of P [1 mark] 9 (c) Find the y‑coordinate of Q [2 marks] Question 9 continues on the next page Turn over U 9 (d) 18 Do not write outside the The line segment QR is a tangent to the circle as shown in Figure 2 below. Figure 2 The point R has coordinates (9, –3). Find the angle QPR Give your answer in radians to three significant figures. box [3 marks] G/Jun24/7357/3 19 Do not write outside the G/Jun24/7357/3 10 It is given that f (x) = 5x3 + x box Use differentiation from first principles to prove that f ' (x) = 15x2 + 1 [5 marks] Turn over for the next ques