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GCE JUN 2006 : A2, C4 : Core Mathematics 4

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ADVANCED General Certificate of Education 2006 Mathematics assessing Module C4: Core Mathematics 4 AMC41 Assessment Unit C4 [AMC41] THURSDAY 8 JUNE, 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 log e z AMC4S6 1558 Answer all eight questions. Show clearly the full development of your answers. Answers should be given to three significant figures unless otherwise stated. 1 The vertices of triangle ABC have position vectors 2i j + 3k, 2i + j k and i + 5j 2k respectively. Verify that the triangle is right-angled at B. [7] 2 Find the equation of the normal to the curve x2 4xy + y2 = 13 at the point (2, 1). 3 Sketch the curve y = sec x for 4 [9] , indicating relevant values on the axes. Find x 4xe 2x dx 5 [4] [5] (i) Starting with the compound angle formula for cos(A + B), show that 2 cos2 1 + cos 2 [3] (ii) Using the substitution x = 2 sin , or otherwise, show that 1 2 x 2 dx = 0 AMC4S6 1558 2 2+ 4 [12] [Turn over 6 Solve the differential equation 1 dy x y = 2 dx x 1 given that y = 4 when x = 2 7 [9] The designers of a bowl use an area rotated through 2 radians around the x-axis as the basis for their design. The area used is between the curve y = 4 x + 2, the x-axis and the lines x = 0 and x = a, as shown in Fig. 1 below. y (cm) y=4 x+2 0 a x (cm) Fig. 1 (i) Find an expression for the capacity of the bowl in terms of a. [7] The specification requires the capacity of the bowl to be 1000 cm3 (ii) Find the value of a correct to one decimal place. AMC4S6 1558 3 [7] [Turn over 8 (i) Given that 2 sin x + 2 3 cos x = r sin(x + ) 3 find r. [2] The function f is defined as f : x 2 sin x + 2 3 cos x 0 2 x (ii) State the range of f. [1] (iii) Using a sketch of f, or otherwise, explain why f does not have an inverse. [2] The function g is defined as g : x 2 sin x + 2 3 cos x where 0 a b a x b 2 The range of g is the same as the range of f and the inverse function g 1 exists. (iv) Write down the values of a and b. [2] (v) Find the inverse function g 1 [5] S 4/05 4000 302507(1) [Turn over

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Additional Info : Gce Mathematics June 2006 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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