AQA AS FURTHER MATHEMATICS Paper 1 MAY 2023

AQA AS FURTHER MATHEMATICS Paper 1 Monday 15 May 2023 Afternoon Time allowed: 1 hour 30 minutes Materials  You must have the AQA Formulae and statistical tables booklet for A-level Mathematics and A-level Further Mathematics.  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 require 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) G/LM/Jun23/E7 7366/1 Answer all questions in the spaces provided. 1 Which expression below is equivalent to tanh x ? box Circle your answer. sinh x cosh x sinh x cosh x cosh x sinh x [1 mark] sinh x + cosh x 2 The two vectors a and b are such that a.b = 0 State the angle between the vectors a and b Circle your answer. [1 mark] 0° 45° 90° 180° 3 The matrices A and B are given by A = 3 1 0 5 B = 0 4 7 1 box Calculate AB Circle your answer. 3 5 7 6 0 20 21 12 0 4 7 13 0 5 35 5 [1 mark] 4 The roots of the equation 5x3 + 2x2 – 3x + p = 0 are α, β and γ Given that p is a constant, state the value of αβ + βγ + γα Circle your answer. [1 mark] – 3 – 2 2 3 5 5 5 5 Turn over ► 5 The function f is defined by f (x) = 3x2 1 ≤ x ≤ 5 box 5 (a) Find the mean value of f [2 marks] 5 (b) The function g is defined by g (x) = f (x) + c 1 ≤ x ≤ 5 The mean value of g is 40 Calculate the value of the constant c [2 marks] 6 (a) Find and simplify the first five terms in the Maclaurin series for e2 x [2 marks] box 6 (b) Hence, or otherwise, write down the first five terms in the Maclaurin series for e–2 x [1 mark] 6 (c) Hence, or otherwise, show that the Maclaurin series for cosh (2x) is a + bx 2 + cx 4 + . . . where a, b and c are rational numbers to be determined. [3 marks] Turn over ► 7 (a) Show that, for all integers r, 1 – 1 = 2 box 2r – 1 2r + 1 (2r – 1) (2r + 1) [1 mark] 7 (b) Hence, using the method of differences, show that n 1 (2r – 1) (2r + 1) = an bn + c where a, b and c are integers to be determined. [4 marks] box 7 (c) Hence, or otherwise, evaluate 1 + 1 + 1 + . . . + 1 1 × 3 3 × 5 5 × 7 99 × 101 [2 marks] Turn over ► (07) 8 Abdoallah wants to write the complex number –1 + i√ 3 in the form r (cos θ + i sin θ) where r ≥ 0 and – π < θ ≤ π box Here is his method: r = √ (–1)2 + (√ 3)2 = √ 1 + 3 = √ 4 = 2 tan θ = √ 3 ⇒ tan θ = –√ 3 ⇒ θ = tan–1(–√ 3 ) ⇒ θ = – π 3 –1 + i√ 3 = 2 (cos (– π ) + i sin (– π )) There is an error in Abdoallah’s method. 8 (a) Show that Abdoallah’s answer is wrong by writing 2 (cos (– π ) + i sin (– π )) 3 3 in the form x + iy Simplify your answer. [1 mark] 8 (b) Explain the error in Abdoallah’s method. [1 mark] box 8 (c) Express –1 + i√ 3 in the form r(cos θ + i sin θ) [1 mark] 8 (d) Write down the complex conjugate of –1 + i√ 3 [1 mark] Turn over ► 9 The matrix M represents the transformation T and is given by box = 3p + 1 p + 2 12 p2 – 3 9 (a) In the case when p = 0 show that the image of the point (4, 5) under T is the point (64, –7) [2 marks] 9 (b) In the case when p = –2 find the gradient of the line of invariant points under T [3 marks]