Mathematics > Higher Education > Practice Exam Papers Pack contents: Two full sets of A-Level practice papers Answer book with mark s (All)

Practice Set 1 Paper 1: Pure Mathematics 1 Time allowed: 2 hours Instructions to candidates • Use black ink or ball-point pen. • A pencil may be used for diagrams, sketches and ... graphs. • Write your name and other details in the spaces provided above. • Answer all questions in the spaces provided. • Show clearly how you worked out your answers. • Round answers to 3 significant figures unless otherwise stated. Information for candidates • There are 16 questions in this paper. • There are 100 marks available for this paper. • The marks available are given in brackets at the end of each question. • You may get marks for method, even if your answer is incorrect. Advice to candidates • Work steadily through the paper and try to answer every question. • Don’t spend too long on one question. • If you have time at the end, go back and check your answers. Exam Set MEP71 Answer ALL the questions. Write your answers in the spaces provided. Leave blank 1 a) Given that f(x) = x3 – 4x2 – 3x + 7, find f '(x). (2) b) Hence find the values of x for which f(x) is a decreasing function, giving your answer in the form {x : x > a} {x : x < b} where a and b are real numbers to be found. (3) 2 A helicopter flies between 3 locations, A, B and C, which are positioned such that AB = 9 km, AC = 5 km and angle ABC = 24°. Find the possible values of angle ACB to 1 decimal place. (3) 3 a) Express 1 6 2 x 3 in the form axb where a and b are integers to be found. (2) Leave blank b) Hence find the x-coordinates of the points where the line y = 1 – 3x intersects the curve with equation 2 x 3 i + 6x – 18 = y – y. (4) 4 For each of the following, prove that the statement is false. a) The exterior angles of a regular n-sided polygon are always acute. (1) b) For n ℝ, n ≠ –1, n ≥ 0. (1) c) For n < 50, if n is an odd prime then one or both of n + 2 and n + 4 are prime. (1) 5 The elastic energy stored in a large industrial spring, E, in joules, is directly proportional to the square of how far it is extended, x, in metres. When the spring is extended by ( 2 – 1) m, it has (7 2 – 9) joules of elastic energy. Find the exact amount of elastic energy in the spring when it is extended by 2 m, giving your answer in joules in the form a + b 2, where a and b are integers. (5) Leave blank 6 The circle C has equation x2 – 8x + y2 + 4y – 29 = 0. The centre of C is at the point X. a) Find: i) the coordinates of the point X. (2) ii) the radius of the circle C. (1) A tangent from the point P(–16, 13) touches the circle at the point Y. b) Find the distance PY. (3) Leave blank 7 a) Express 4x2 + 8x + 3 as a single fraction in its simplest form. 4x 9 (2) b) Hence, or otherwise, solve the equation: log3(4x2 + 8x + 3) – log3(4x2 – 9) = 2, x > 1.5 (4) 8 Leave y blank x A B Figure 1 Figure 1 shows the graph of y = f(x) where x ℝ. The graph has stationary points A and B. a) State the nature of the stationary point A, justifying your answer with reference to the shape of the graph. (2) b) Explain why f(x) does not have an inverse function. (1) B is a minimum turning point with coordinates (p, q), where p and q are constants. c) Write down, in terms of p and q, the coordinates of the point B under these transformations: i) y = f(x – 1) (1) ii) y = 3f(2x) (1) d) Given that f(x) = 3x4 – 2x3 – 2: i) find the exact values of p and q. (4) ii) justify that B is a minimum turning point. (2) e) Sketch the graph of y = |f(–x)|. (2) Leave blank 9 Figure 2 shows how a manufacturer cuts pieces of cheese to sell. h cm Leave blank A B C Figure 2 h cm B Label A F C E Wax D A cylinder of cheese has a layer of wax with negligible thickness applied to its curved surface. The cylinder is sliced horizontally, h cm below the top face. The slice is then cut vertically along two radii, AB and CB, as shown above. Each piece, ABCDEF, has a triangular label, ABC, applied to the top face. For a particular piece, h = 4 cm, angle ABC = 1.02 radians and the area of the label is 85.9 cm2. Find the area of wax on this piece of cheese. (4) 10 a) A student is attempting to find a root, a, of the equation f(x) = 0. The student finds two values, a and b, such that f(a) < 0 and f(b) > 0 and deduces that a < a < b. Explain why the student’s deduction is not necessarily true. (1) b) The student then attempts to use the Newton-Raphson method to find the root a of the equation 2x3 + 9x = 125, using 5 as a first approximation for a. Their working is shown below. Leave blank i) Identify two errors that the student has made in their working. (2) ii) Find the correct final value of a, giving your answer to 3 decimal places. [Show More]

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