AQA AS MATHEMATICS 7356/2 Paper 2 Question Paper + Mark scheme [MERGED] June 2022 PB/Jun22/E6 7356/2 (JUN) AS MATHEMATICS Paper 2 Time allowed: 1 hour 30 minutes Materials

AQA AS MATHEMATICS 7356/2 Paper 2 Question Paper + Mark scheme [MERGED] June 2022 PB/Jun22/E6 7356/2 (JUN) AS MATHEMATICS Paper 2 Time allowed: 1 hour 30 minutes Materials l You must have the AQA Formulae for A‑level Mathematics booklet. l You should have a graphical or scientific calculator that meets the requirements of the specification. Instructions l Use black ink or black ball-point pen. Pencil should only be used for drawing. l Fill in the boxes at the top of this page. l Answer all questions. l 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). l Do not write outside the box around each page or on blank pages. l Show all necessary working; otherwise marks for method may be lost. l Do all rough work in this book. Cross through any work that you do not want to be marked. Information l The marks for questions are shown in brackets. l The maximum mark for this paper is 80. Advice l Unless stated otherwise, you may quote formulae, without proof, from the booklet. l You do not necessarily need to use all the space provided. Please write clearly in block capitals. Centre number Candidate number Surname ________________________________________________________________________ Forename(s) ________________________________________________________________________ Candidate signature ________________________________________________________________________ For Examiner’s Use Question Mark 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 TOTAL I declare this is my own work. 2 Section A Answer all questions in the spaces provided. 1 Find ð 12x3 dx Circle your answer. [1 mark] 36x2 þ c 3x4 þ c 3x2 þ c 36x4 þ c 2 Given that cos (y  20) ¼ cos 60 which one of the following is a possible value for y? Circle your answer. [1 mark] 40 140 280 320 Jun22/7356/2 Do not write outside the box (02) 3 3 A curve has equation y ¼ k ffiffiffi x p where k is a constant. Find d2y dx2 at the point (4, 2k) on the curve, giving your answer as an expression in terms of k. [5 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Turn over for the next question Do not write outside the box Jun22/7356/2 Turn over s (03) 4 4 The equation 9x2 þ 4x þ p2 ¼ 0 has no real solutions for x. Find the set of possible values of p. Fully justify your answer. [4 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Do not write outside the box Jun22/7356/2 (04) 5 5 Kaya is investigating the function f (x) ¼ 2x3  7x2  12x þ 45 Kaya makes two statements. Statement 1: f (3) ¼ 0 Statement 2: this shows that (x þ 3) must be a factor of f (x). 5 (a) State, with a reason, whether each of Kaya’s statements is correct. [2 marks] Statement 1: ________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Statement 2: ________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 5 (b) Fully factorise f (x). [3 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Do not write outside the box Jun22/7356/2 Turn over s (05) 6 6 An on-line science website states: ‘To find a dog’s equivalent human age in years, multiply the natural logarithm of the dog’s age in years by 16 then add 31.’ 6 (a) Calculate the equivalent age to the nearest human year of a dog aged 5 years. [1 mark] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 6 (b) A dog’s equivalent age in human years is 40 years. Find the dog’s actual age to the nearest month. [3 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 6 (c) Explain why the behaviour of the natural logarithm for values close to zero means that the formula given on the website cannot be true for very young dogs. [2 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Do not write outside the box Jun22/7356/2 (06) 7 7 The expression 3  pn 2 þ pn can be written in the form a þ b pn , where a and b and n are rational but pn is irrational. Find expressions for a and b in terms of n. [4 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Turn over for the next question Do not write outside the box Jun22/7356/2 Turn over s (07) 8 8 Triangle ABC has sides of length (m  n) , m and (m þ n) where 0 < 2n < m Angle A is the largest angle in the triangle. 8 (a) (i) Explain why angle A must be opposite the side of length (m þ n). [1 mark] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 8 (a) (ii) Using the cosine rule, show that cosA ¼ m  4n 2(m  n) [3 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Do not write outside the box Jun22/7356/2 (08) 9 8 (b) You are given that BC is the diameter of a circle, and A lies on the circumference of the circle. The value of m is 8 Calculate the value of n