Created by T. Madas. Candidates may use any calculator allowed by the regulations of this examination.

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1 IYGB GCE Mathematics MP Advanced Level Practice Paper N Difficulty Rating: 3.550/.68 Time: hours Candidates may use any calculator allowed by the regulations of this eamination. Information for Candidates This practice paper follows closely the Pearson Edecel Syllabus, suitable for first assessment Summer 08. The standard booklet Mathematical Formulae and Statistical Tables may be used. Full marks may be obtained for answers to ALL questions. The marks for the parts of questions are shown in round brackets, e.g. (). There are 3 questions in this question paper. The total mark for this paper is 00. Advice to Candidates You must ensure that your answers to parts of questions are clearly labelled. You must show sufficient working to make your methods clear to the Eaminer. Answers without working may not gain full credit. Non eact answers should be given to an appropriate degree of accuracy. The eaminer may refuse to mark any parts of questions if deemed not to be legible.

2 Question (**+) y P y = + sin O The figure above shows the graph of the curve with equation y = + sin, R. The point P lies on the curve where π =. 3 Show that the area of the finite region bounded by the curve, the y ais and the straight line segment OP is eactly ( π π ) ( 7) Question (**) Relative to a fied origin O, the points A, B and C have respective position vectors 3i + k, i + 4j+ k and 5i + 4j. Calculate the size of the angle ABC and hence find the area of the triangle ABC. ( 6)

3 Question 3 (***+) y A(,4) O B( 4,0) y = 4 The figure above shows part of the graph of the curve with equation y = 4, R. The graph meets the coordinate aes at the origin O and B ( 4,0), and has a stationary point at A (,4). Sketch on separate diagrams, indicating the new coordinates of the points A and B, the graph of a) y = 4. ( ) b) y = 4. ( ) c) y = 4. ( )

4 Question 4 (***) A cubic equation has the following equation. 3 + = 4, R. a) Show that the above equation has a root α, which lies between 0 and. ( ) b) Show further that the above equation can be written as =. 4 ( ) An iterative formula, based on the rearrangement of part (b), is to be used to find α. c) Starting with = 0., find to 4 decimal places, the value of, 3 and 4. The diagram below is used to show the convergence of these iterations. ( ) y y = y = 4 O d) Draw, on a copy of this diagram, a staircase or cobweb pattern showing how the iterations of part (c) converge to α, marking the position of,, 3 and 4. ( )

5 Question 5 (***) By using the substitution u =, or otherwise, find. 4 d ( 6) Question 6 (***+) The function f is given by f 3 : +, R,. a) By sketching the graph of f, or otherwise, state its range. b) Determine an epression for f ( ) c) Find the domain and range of f ( ), the inverse of f.. ( ) ( 3) ( ) Question 7 (***+) Two variables and y are related by y = π ( 4 4 ). The variable y is changing with time t, at the constant rate of 0., in suitable units. Find the rate at which is changing with respect to t, when =. ( 6)

6 Question 8 (***+) The curve C has equation f ( ) = ( a), R, a, where a is a non zero constant. Given that f ( a) =, determine the value of a. ( 8) Question 9 (***+) Solve the differential equation dy ( + ) = y, d > 0, y > 0 subject to the condition y = at =, giving the answer in the form y f ( ) =. ( ) Question 0 (****+) Solve the following trigonometric equation tan 4 tan = 0, 0 < 360. ( 7) Question (****) The st rd th, 3 and term of an arithmetic progression are the first three terms of a geometric progression. It is further given that the sum of the first 3 terms of the arithmetic progression is 60. Find, in any order, the common ratio of the geometric progression and the first term and common difference of the arithmetic progression. ( 8)

7 Question (***+) A curve C has implicit equation ( y) 6 0 y + =, y,, y 0. Find the coordinates of the stationary point of C. ( 9) Question 3 (****+) A curve is given parametrically by the equations t =, + t t y =, t R. + t The point P, lies on this curve. Show that an equation of the tangent at the point P is given by + y =. ( 3)