AQA AS MATHEMATICS Paper 1 MAY 2023

AQA AS MATHEMATICS Paper 1 Thursday 18 May 2023 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. PB/KL/Jun23/E4 7356/1 Section A Answer all questions in the spaces provided. 1 At a point P on a curve, the gradient of the tangent to the curve is 10 State the gradient of the normal to the curve at P Circle your answer. —10 —0.1 0.1 10 [1 mark] box 2 Identify the expression below which is equivalent to 2x —3 Circle your answer. [1 mark] 8x3 125 125x3 8 125 8x3 8 125x3 3 The coefficient of x2 in the binomial expansion of (1 ax)6 is 20 3 outside the box Find the two possible values of a [3 marks] Turn over for the next question Turn over 4 It is given that 5 cos2 y — 4 sin2 y ¼ 0 outside the box 4 (a) Find the possible values of tan y, giving your answers in exact form. [3 marks] 4 (b) Hence, or otherwise, solve the equation 5 cos2 y — 4 sin2 y ¼ 0 giving all solutions of y to the nearest 0.1° in the interval 0° ≤ y ≤ 360° [2 marks] 5 (a) Given that y ¼ xpffixffi , find d x [2 marks] outside the box 5 (b) The line, L, has equation 6x — 2y þ 5 ¼ 0 L is a tangent to the curve with equation y ¼ xpx þ k Find the value of k [5 marks] Turn over 6 (a) The curve C1 has equation y ¼ 2x2 — 20x þ 42 outside the box Express the equation of C1 in the form y ¼ a(x — b)2 þ c where a, b and c are integers. [3 marks] 6 (b) Write down the coordinates of the minimum point of C1 [1 mark] 6 (c) The curve C1 is mapped onto the curve C2 by a stretch in the y-direction. The minimum point of C2 is at (5, —4) Find the equation of C2 [2 marks] 7 Points P and Q lie on the curve with equation y ¼ x4 The x-coordinate of P is x The x-coordinate of Q is x þ h 7 (a) Expand (x þ h)4 [2 marks] outside the box 7 (b) Hence, find an expression, in terms of x and h, for the gradient of the line PQ [1 mark] 7 (c) Explain how to use the answer from part (b) to obtain the gradient function of y x4 [2 marks] Turn over (07) 8 (a) Show that ð a 12 pffiffiffi box 8 (b) The curve 12 ¼ 6 — pffixffi , the line x ¼ 1 and the line x ¼ a are shown in the box diagram below. The shaded region R1 is bounded by the curve, the line x ¼ 1 and the x-axis. The shaded region R2 is bounded by the curve, the line x ¼ a and the x-axis. y 1 R2 O R1 a x It is given that the areas of R1 and R2 are equal. Find the value of a Fully justify your answer. [4 marks] Turn over 9 A continuous curve has equation y ¼ f (x) The curve passes through the points A(2, 1), B(4, 5) and C(6, 1) It is given that f ’(4) ¼ 0 Jasmin made two statements about the nature of the curve y ¼ f (x) at the point B: Statement 1: There is a turning point at B Statement 2: There is a maximum point at B 9 (a) Draw a sketch of the curve y f (x) such that Statement 1 is correct and Statement 2 is correct. box [1 mark] y 8 7 6 5 B 4 3 2 1 A C 0 0 1 2 3 4 5 6 7 8 x 9 (b) Draw a sketch of the curve y f (x) such that Statement 1 is correct and Statement 2 is not correct. box [1 mark] y 8 7 6 5 B 4 3 2 1 A C 0 0 1 2 3 4 5 6 7 8 x 9 (c) Draw a sketch of the curve y f (x) such that Statement 1 is not correct and Statement 2 is not correct. [1 mark] y 8 7 6 5 B 4 3 2 1 A C 0 0 1 2 3 4 5 6 7 8 x Turn over 10 Charlie buys a car for £18 000 on 1 January 2016. The value of the car decreases exponentially. The car has a value of £12 000 on 1 January 2018. 10 (a) Charlie says: * because the car has lost £6000 after two years, after another two years it will be worth £6000. Charlie’s friend Kaya says: * because the car has lost one third of its value after two years, after another two years it will be worth £8000. box Explain whose statement is correct, justifying the value they have stated. [2 marks]