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GCE JAN 2010 : A2, C4 : Core Mathematics 4

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ADVANCED General Certificate of Education January 2010 Mathematics assessing Module C4: Core Mathematics 4 AMC41 Assessment Unit C4 [AMC41] FRIDAY 29 JANUARY, MORNING TIME 1 hour 30 minutes. INSTRUCTIONS TO CANDIDATES Write your Centre Number and Candidate Number on the Answer Booklet provided. Answer all eight questions. Show clearly the full development of your answers. Answers should be given to three significant figures unless otherwise stated. You are permitted to use a graphic or scientific calculator in this paper. INFORMATION FOR CANDIDATES The total mark for this paper is 75 Figures in brackets printed down the right-hand side of pages indicate the marks awarded to each question or part question. A copy of the Mathematical Formulae and Tables booklet is provided. Throughout the paper the logarithmic notation used is ln z where it is noted that ln z loge z 5188 Answer all eight questions. Show clearly the full development of your answers. Answers should be given to three significant figures unless otherwise stated. 1 Relative to a fixed origin O, point A has position vector 2i + 3j + k and point C has position vector 4i + 5j + 2k (i) Find a vector equation of the line AC. [4] The points OABC are the vertices of a parallelogram. (ii) Find the position vector OB. (iii) Hence find the acute angle between the diagonals OB and AC. 2 [2] [5] Let x3 g(x) = 4x and 0 2 2 x 5 0 x 2 2 x3 h(x) = 4x + 1 x x 5 (a) Which of g or h is a function? Give a reason for your answer. [2] (b) A function f is defined as f(x) = 4 x 2 x (i) Sketch the graph of y = f(x). [2] (ii) Hence state the range of f(x). [1] (iii) Write down two functions a(x) and b(x) such that f(x) is equal to the composite function ab(x). State the domains of the two functions. [3] 5188 2 [Turn over 3 (i) Rewrite (8 sin + 6 cos ) in the form R sin ( + ) where R is an integer and 0 2 [3] (ii) Hence state the maximum and minimum values of 8 sin + 6 cos [2] (iii) A mass is suspended from the end of a spring, as shown in Fig. 1 below. > P d > Fig. 1 The mass is oscillating. After t seconds the distance d (cm) between the fixed point P and the mass is given by d = 15 + 8 sin 2t + 6 cos 2t Find the time at which the mass is first at its lowest point. 5188 3 [4] [Turn over 4 (i) Differentiate x 3 3x 2y + 2y 2 = 3 implicitly with respect to x. [5] (ii) Hence find the equation of the tangent to the curve x 3 3x 2y + 2y 2 = 3 at the point (1, 2). [3] dx (sin ) d = x4 2 5 2 Solve the differential equation dx (sin ) d = x4 2 2 given that x = 3 when = 4 [7] 4 5188 4 [Turn over 6 A trophy is to be made in the shape of a rugby ball. It can be modelled by the volume generated when the area between the curve y = sin x and the x-axis, between x = 0 and x = , is rotated through 2 radians about the x-axis, as shown in Fig. 2 below. y y = sin x x Fig. 2 Find the exact volume of the trophy. 7 [9] (a) Sketch the graph of for 180 y = cot x x 180 [2] (b) Prove the identity 1 + cot 2 cot sin 2 8 (a) Find 2x 4 ln 3x dx [7] [6] (b) Use partial fractions to find 5188 x+9 dx 3 2x x 2 5 [8] [Turn over THIS IS THE END OF THE QUESTION PAPER 5188 6 [Turn over 110531

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Additional Info : Gce Mathematics January 2010 Assessment Unit C4 Module C4 : Core Mathematics 4
Tags : General Certificate of Education, A Level and AS Level, uk, council for the curriculum examinations and assessment, gce exam papers, gce a level and as level exam papers , gce past questions and answer, gce past question papers, ccea gce past papers, gce ccea past papers

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