MATHEMATICS Paper 2 Tuesday 13 June 2023

MATHEMATICS Paper 2 Tuesday 13 June 2023 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. PB/KL/Jun23/E6 7357/2 Section A Answer all questions in the spaces provided. 1 The graph of y ax2 bx c has roots x 2 and x 5 , as shown in the diagram below. box y O 2 5 x State the set of values of x which satisfy ax2 þ bx þ c 0 Tick (3) one box. fx : x 2g ¨ fx : x 5g fx : 0 x 2g ˙fx : x 5g fx : 2 x 5g fx : 2 x 5g [1 mark] 2 It is given that ð 3 6 f (x) d x 20 and 0 6 f (x) d x ¼ —10 box Circle your answer. —30 —10 10 30 [1 mark] 3 A circle has equation Find the radius of the circle. Circle your answer. (x — 5)2 þ ( y — 13)2 ¼ 16 [1 mark] 4 12 16 256 Turn over for the next question Turn over 4 A curve has equation x pffiffiffi box dy 4 (a) Find an expression for d x y ¼ 8 þ 4 x [3 marks] 4 (b) The point P with coordinates (4, 10) lies on the curve. Find an equation of the tangent to the curve at the point P [2 marks] 4 (c) Show that the curve has no stationary points. [2 marks] box Turn over for the next question Turn over 5 Ziad is training to become a long-distance swimmer. He trains every day by swimming lengths at his local pool. The length of the pool is 25 metres. Each day he increases the number of lengths that he swims by four. On his first day of training, Ziad swims 10 lengths of the pool. 5 (a) Write down an expression for the number of lengths Ziad will swim on his nth day of training. box [1 mark] 5 (b) (i) Ziad’s target is to be able to swim at least 3000 metres in one day. Determine the minimum number of days he will need to train to reach his target. [3 marks] 5 (b) (ii) Ziad’s coach claims that when he reaches his target he will have covered a total distance of over 50 000 metres. box Determine if Ziad’s coach is correct. [3 marks] Turn over for the next question Turn over (07) 6 Victoria, a market researcher, believes the average weekly value, £ V million, of online grocery sales in the UK has grown exponentially since 2009. Victoria models the incomplete data, shown in the table, using the formula V ¼ a × b N where N is the number of years since 2009 and a and b are constants. Year 2009 2010 2011 2012 2013 2014 2015 2016 Average Weekly Sales £ V million 56.4 74.5 86.9 97.7 109.3 141.9 6 (a) Victoria wishes to determine the values of a and b in her formula. To do this she plots a graph of log10 V against N and then draws a line of best fit as shown in the diagram below. log10 V 2.2 2.1 2.0 1.9 1.8 1.7 box 0 0 1 2 3 4 5 6 7 8 N The equation of Victoria’s line of best fit is log10 V ¼ 0:057 N þ 1:76 6 (a) (i) Use the equation of Victoria’s line of best fit to show that, correct to three significant figures, a = 57.5 [1 mark] (0 ) box 6 (a) (ii) Use the equation of Victoria’s line of best fit to find the value of b Give your answer to three significant figures. [1 mark] 6 (b) According to Victoria’s model, state the yearly percentage increase in the average weekly value of online grocery sales. [1 mark] 6 (c) (i) Use Victoria’s model to predict the average weekly value of online grocery sales in 2025. [2 marks] 6 (c) (ii) Explain why the prediction made in part (c)(i) may be unreliable. [1 mark] Turn over 7 The functions f and g are defined by f (x) ¼ pffi1ffiffi0ffiffiffiffi—ffiffiffiffiffi2ffiffiffixffi for x ≤ 5 box g(x) 1 x for x 0 The function h has maximum possible domain and is defined by h(x) ¼ gf (x) 7 (a) Find an expression for h(x) [1 mark] 7 (b) Find the domain of h [1 mark] 7 (c) Show that h—1(x) 5 1 2x2 [3 marks] Turn over for the next question box DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED Turn over 8 (a) Given that cos y 6¼ 1 , prove the identity 1 1 2 box 1 — cos y þ 1 þ cos y Ξ 2 cosec y [4 marks] 8 (b) Hence, find the set of values of A for which the equation 1 1 has real solutions. Fully justify your answer.