A-level MATHEMATICS BEST RATING Paper 2 JUNE 2022

A-level MATHEMATICS Paper 2 JUNE 2022 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) PB/Jun22/E7 7357/2 Section A Answer all questions in the spaces provided. 1 A circle has centre (4, —5) and radius 6 Find the equation of the circle. Tick (3) one box. (x — 4)2 þ (y þ 5)2 ¼ 6 (x þ 4)2 þ (y — 5)2 ¼ 6 (x — 4)2 þ (y þ 5)2 ¼ 36 (x þ 4)2 þ (y — 5)2 ¼ 36 [1 mark] box 2 State the value of Circle your answer. lim h ! 0 sin (p þ h) — sin p h [1 mark] cos h —1 0 1 3 The function f is concave and is represented by one of the graphs below. Identify the graph which represents f . Tick (3) one box. [1 mark] box y y O x O x y y O x O x Turn over 4 38° B A 8.7 cm 6.1 cm C box The diagram shows a triangle ABC. AB is the shortest side. The lengths of AC and BC are 6.1 cm and 8.7 cm respectively. The size of angle ABC is 38° Find the size of the largest angle. Give your answer to the nearest degree. [3 marks] Turn over for the next question box DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED Turn over 5 The binomial expansion of (2 þ 5x)4 is given by outside the box (2 þ 5x)4 ¼ A þ 160x þ Bx2 þ 1000x3 þ 625x4 5 (a) Find the value of A and the value of B. [2 marks] 5 (b) Show that (2 þ 5x)4 — (2 — 5x)4 ¼ Cx þ Dx3 where C and D are constants to be found. [2 marks] 5 (c) Hence, or otherwise, find ð (2 þ 5x)4 — (2 — 5x)4 dx [2 marks] outside the box Turn over for the next question Turn over (07) 6 (a) Asif notices that 242 ¼ 576 and 2 þ 4 ¼ 6 gives the last digit of 576 outside the box He checks two more examples: 272 ¼ 729 292 ¼ 841 2 þ 7 ¼ 9 2 þ 9 ¼ 11 Last digit 9 Last digit 1 Asif concludes that he can find the last digit of any square number greater than 100 by adding the digits of the number being squared. Give a counter example to show that Asif’s conclusion is not correct. [2 marks] 6 (b) Claire tells Asif that he should look only at the last digit of the number being squared. 272 ¼ 729 242 ¼ 576 72 ¼ 49 42 ¼ 16 Last digit 9 Last digit 6 Using Claire’s method determine the last digit of [1 mark] 6 (c) Given Claire’s method is correct, use proof by exhaustion to show that no square number has a last digit of 8 outside the box [2 marks] Turn over for the next question Turn over 7 The curve y 15 x2 below. and the isosceles triangle OPQ are shown on the diagram y outside the box P Q O q x Vertices P and Q lie on the curve such that Q lies vertically above some point (q, 0) The line PQ is parallel to the x-axis. 7 (a) Show that the area, A, of the triangle OPQ is given by A ¼ 15q — q3 for 0 q c where c is a constant to be found. [3 marks] 7 (b) Find the exact maximum area of triangle OPQ. Fully justify your answer. [6 marks] box Turn over for the next question Turn over