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GCE JUN 2007 : A2, C4 : Core Mathematics 4

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ADVANCED General Certificate of Education 2007 Assessment Unit C4 assessing Module C4: Core Mathematics 4 AMC41 Mathematics [AMC41] MONDAY 11 JUNE, AFTERNOON 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. 1 - 23/3/06GG 2 - 19/7/06EA 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 AMC4S7 2169 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 points A, B and C have position vectors i + 2j + k, 3i + 4j + k and 3i + 2j + 3k respectively. (i) Find the vectors AB, BC and AC. Hence show that the triangle ABC is equilateral. (ii) Find a vector equation of the line through B parallel to AC. 2 [6] [3] (i) Sketch the graph of y = sin x for 0 [2] x 360 (ii) Hence sketch the graph of y = cosec x for 0 3 x 360 [3] A curve is defined by 3x2 + xy 2y2 = 0 (i) Show that dy y + 6 x = dx 4 y x [6] (ii) Find the equation of the normal to the curve at the point (2, 3). 4 (i) Write 3 cos + 4 sin in the form r cos( ), where r is real and 0 [4] 90 [4] 1 - 23/3/06GG 2 - 19/7/06EA (ii) Hence solve the equation 3 cos + 4 sin = 2 for 0 AMC4S7 2169 360 [5] 2 [Turn over 5 (a) Prove that cos 2 A cos A + 1 cot A sin 2 A sin A [5] (b) By expressing tan 2A in terms of tan A, find the exact value of tan 221 2 6 [6] The voltage V of the battery in a smoke alarm can be modelled by dV = kV dt where t is the number of days after installation and k is a positive constant. The initial voltage is 9 volts and after 30 days the voltage drops to 4.5 volts. (i) Solve this differential equation. [7] The smoke alarm does not function properly if the battery s voltage drops below 1.5 volts. (ii) Find after how many days the battery should be changed. 7 [2] The functions f and g are defined as 1 +2 x 1 g: x x+2 f:x x ,x > 0 x ,x > 0 (i) State the range of f and the range of g. [2] (ii) Find the composite function gf stating its domain. [4] 1 - 23/3/06GG 2 - 19/7/06EA f and g can be written as composite functions where f(x) = uv(x) and g(x) = vu(x) for all positive x. (iii) Find the functions u and v. AMC4S7 2169 [3] 3 [Turn over 8 (a) Find 16 x ln x dx 3 [7] (b) Find (sin x + cos x) dx 2 [6] 1 - 23/3/06GG 2 - 19/7/06EA THIS IS THE END OF THE QUESTION PAPER S 4/05 0000 7-042-1 [Turn over

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Additional Info : Gce Mathematics June 2007 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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