Further Mathematics AS/F1/D17 AS PAPER 1

Transcription

1 Surname Other Names Candidate Signature Centre Number Candidate Number Examiner Comments Total Marks Further Mathematics AS PAPER 1 CM December Mock Exam (AQA Version) Time allowed: 1 hour and 30 minutes Instructions to candidates: In the boxes above, write your centre number, candidate number, your surname, other names and signature. Answer ALL of the questions. You must write your answer for each question in the spaces provided. You may use a calculator. Information to candidates: Full marks may only be obtained for answers to ALL of the questions. The marks for individual questions and parts of the questions are shown in square brackets. There are 14 questions in this question paper. The total mark for this paper is 80. Advice to candidates: You should ensure your answers to parts of the question are clearly labelled. You should show sufficient working to make your workings clear to the Examiner. Answers without working may not gain full credit. AS/F1/D crashmaths Ltd.

2 2 Answer all questions in the spaces provided. 1 The matrix A is represented by A = 2 2 k. k k 2 The determinant of A is negative. Which of the following options is correct? Circle your answer. [1 mark] Option The matrix A is... Range of values of k A B C D singular non-singular singular non-singular 2 < k < 2 2 < k < 2 k < 2, k > 2 k < 2, k > The points A and B have positions vectors and respectively, relative to a 2 3 fixed origin O. 1 4 Which of the following vectors is a unit vector perpendicular to the line AB? Circle your answer. [1 mark]

3 3 3 The roots of the quadratic x 2 4x + 2 = 0 are α and β. Find the value of α 3 + β 3. Circle your answer. [1 mark] Turn over for the next question Turn over

4 4 4 (a) The complex number z is defined such that z = 1 i 3 1+ i 3 ( ) θ π,π Express z in the form r cosθ + isinθ, where. [5 marks]

5 5 4 (b) (i) The function f is defined such that ( ) = 3x 4 + ax 3 + bx 2 + cx 104 f x where a, b and c are real constants. Two of the solutions to the equation f(x) = 0 are 4 and 3 + 2i. Factorise f(x) fully into a product of real factors. [4 marks] 4 (b) (ii) Find the values of the constants a, b and c. [2 marks] Turn over

6 6 5 The function g is defined such that g(x) = tanh x. 5 (a) On the axes below, sketch the curve with equation y = g(x). [2 marks] 5 (b) The inverse of g(x) is denoted by g 1 (x). Prove that g 1 (x) = 1 1+ x ln 2 1 x, 1< x < 1 [3 marks]

7 7 ( ) ln3 = 0 5 (c) Find the solution to the equation 2g 1 x 1 and explain why it does not contradict that g 1 (x) is only defined for 1< x < 1. [3 marks] ( ) = 36 6 Use algebra to show that the curve with polar equation r sin 2 θ is a hyperbola with asymptotes y = ± kx, where k is a constant to be found. [4 marks] Turn over

8 8 7 Figure 1 shows two semi-circles, C 1 and C 2, centred at the origin. y R C C x Figure 1 The region R, shown shaded in the diagram, is bounded by C 1 and C 2. The region R is rotated 360 degrees about the x axis to form a solid S. Using integration, find the volume of the solid S. [6 marks]

9 9 8 The matrix M represents a reflection in the line y = x 3 followed by an anti-clockwise rotation of 30 degrees about the origin. Find M. [4 marks] Turn over

10 10 9 The function f is defined such that f(x) = x p+2, where p > 2. The mean value of f '(x) over 1, 2 is Find the mean value of f(x) over 1, 2. Give your answer as an exact value in its simplest form. [6 marks]

11 11 BLANK PAGE Turn over

12 12 10 A sensor is able to detect motion within a distance of 10 m from its position. Edward walks from the point A to the point B. The sensor is modelled as a particle fixed at the origin O. Edward is modelled as a particle moving in a straight line from A to B. Relative to O, the point A has coordinates ( 12, 9, 4) and the point B has coordinates (10, 1, 18), where the unit of distance is metres. 10 (a) Using the model, determine whether the sensor will detect the motion of Edward. [7 marks]

13 13 10 (a) [Extra space] 10 (b) Suggest one way in which the model in part (a) could be refined. [1 mark] Turn over

14 14 11 (i) Using the principle of mathematical induction, prove that n k=1 x k = ( ) 1 x x 1 xn, n! +, x 1 [5 marks]

15 15 11 (ii) Given that f(n) = 10 n 1, n! +, 11 (ii) (a) show that f(2) is divisible by 11. [1 mark] 11 (ii) (b) Assuming that f(2k) is divisible by 11, show that f(2k + 2) is divisible by 11. [3 marks] 11 (ii) (c) Explain what has been proved about f in parts (a) and (b). [1 mark] Turn over

16 16 12 (a) On an Argand diagram, shade the set of points given by { z! : z < 2} z! : 0 argz k, π 2 < k < π [5 marks] 12 (b) The complex number z = 1 + ai satisifies both z 2 and arg z = k, where a is a constant. Find the exact value of k. [3 marks]

17 17 13 The curve C has the equation y = x2 10x 3, where a is a constant. x 2 + 2x (a) State the equation of the horizontal asymptote to C. [1 mark] 13 (b) Show that the curve C has no vertical asymptotes. [2 marks] Turn over

18 18 14 (a) Using standard results for series, show that n r=1 ( 3r 2 11r n 19) = n an 2 + bn + c ( ) where a, b and c are integers to be found. [4 marks] 14 (b) (i) The function f is defined such that n ( ) = ( 3r 2 11r n 19) f n r=0 where n is a positive integer. Find the value of n for which f has its minimum value. [4 marks]

19 19 14 (b) (i) [Extra space] 14 (b) (ii) Hence, or otherwise, write down the minimum value of f. [1 mark] END OF QUESTIONS Copyright 2017 crashmaths Ltd.