AQA MATHEMATICS Paper 2

AS MATHEMATICS Paper 2 Thursday 23 May 2024 Afternoon Time allowed: 1 hour 30 minutes 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 80. 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) 7356/2 box 3 It is given that 3 loga x = loga 72 – 2 loga 3 box Solve the equation to find the value of x Fully justify your answer. [4 marks] Turn over for the next question Turn over U 4 Curve C has equation y = 8 sin x 4 (a) Curve C is transformed onto curve C1 by a translation of vector [ 0 ] box Find the equation of C1 [1 mark] 4 (b) Curve C is transformed onto curve C2 by a stretch of scale factor 4 in the y direction. Find the equation of C2 [1 mark] 4 (c) Curve C is transformed onto curve C3 by a stretch of scale factor 2 in the x direction. Find the equation of C3 [1 mark] 5 A student suggests that for any positive integer n the value of the expression 4n2 + 3 is always a prime number. Prove that the student’s statement is false by finding a counter example. Fully justify your answer. [3 marks] box Turn over for the next question Turn over U 6 In the expansion of (3 + ax)n, where a and n are integers, the coefficient of x2 is 4860 box 6 (a) Show that 3n a2 n (n – 1) = 87480 [3 marks] 6 (b) The constant term in the expansion is 729 The coefficient of x in the expansion is negative. 6 (b) (i) Verify that n = 6 [1 mark] 6 (b) (ii) Find the value of a [3 marks] box Turn over for the next question Turn over U (07) 7 Point A has coordinates (4, 1) and point B has coordinates (– 8, 5) 7 (a) Find the equation of the perpendicular bisector of AB [5 marks] box (0 8) 7 (b) A circle passes through the points A and B A diameter of the circle lies along the x‑axis. Find the equation of the circle. [4 marks] box Turn over for the next question Turn over U 8 Prove that the graph of the curve with equation y = x3 + 15x – 18 box has no stationary points. [5 marks] Turn over for the next question DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED box Turn over U 9 A curve has equation where a and b are constants. y = x – a√x + b box The curve intersects the line y = 2 at points with coordinates (1, 2) and (9, 2), as shown in the diagram below. y 2 (1, 2) (9, 2) O x 9 (a) Show that a has the value 4 and find the value of b [3 marks] 9 (b) On the diagram, the region enclosed between the curve and the line y = 2 is shaded. Show that the area of this shaded region is 16 3 box Fully justify your answer. [6 marks] Turn over for the next question Turn over U