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GCE JAN 2011 : A2, C4 : Core Mathematics 4

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ADVANCED General Certificate of Education January 2011 Mathematics assessing Module C4: Core Mathematics 4 AMC41 Assessment Unit C4 [AMC41] FRIDAY 28 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 111531 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 6136 Answer all eight questions. Show clearly the full development of your answers. Answers should be given to three significant figures unless otherwise stated. 1 (a) (i) Write 1 ( x + 1)( x 1) in partial fractions. [4] (ii) Hence find [4] x cos x dx 1 dx ( x + 1)( x 1) [5] (b) Use integration by parts to find 2 The vector equation of the line PQ is r = (2i + 3j) + (i j 2k). The line PR has direction vector (4i + 5j + k) Find the angle between the lines PQ and PR. 3 Solve the equation [7] cos x = 2 sin (x + 60 ) for 180 x 180 6136 [7] 2 [Turn over 4 A curve is given parametrically as x = cot q y = cosec q (i) Show that the cartesian equation of the curve is y2 = 1 + x2 (ii) Hence find the exact gradients of the tangents to the curve at the points where x = 1 5 [2] [7] (i) Show that tan x sec4 x tan x sec2 x + tan3 x sec2 x [3] (ii) Hence, and by using the substitution u = tan x, or otherwise, find 4 0 6 tan x sec 4 x dx Solve the differential equation (1 + x 2 ) dy = x(1 + y ) dx to find y in terms of x, given that x = 0 when y = 0 6136 [7] 3 [11] [Turn over 7 The bowl of a glass can be modelled by the rotation of the curve x y = e6 between x = 0 and x = 9 cm, as shown in Fig. 1 below, through 2p radians about the x-axis. x y = e6 y 0 x 9 Fig. 1 Find the maximum volume that the glass can hold. 8 [7] The function f is defined as 2 x f : x sin f or x 2 (i) Write down the inverse function f 1 and state its domain and range. [4] The function g is defined as g : x |x| for x (ii) Find the composite function gf, stating its range. [4] (iii) Hence sketch the graph of y = gf(x) 111531 6136 4 [3] [Turn over

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Additional Info : Gce Mathematics January 2011 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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