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Mathematics HL - Nov 2004 - P2

Mathematics HL - Nov 2004 - P2 - c 3 hours IB DIPLOMA...

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MATHEMATICS HIGHER LEVEL PAPER 2 Thursday 4 November 2004 (morning) 3 hours N04/5/MATHL/HP2/ENG/TZ0/XX c IB DIPLOMA PROGRAMME PROGRAMME DU DIPL°ME DU BI PROGRAMA DEL DIPLOMA DEL BI 88047402 8804-7402 14 pages INSTRUCTIONS TO CANDIDATES ! Do not open this examination paper until instructed to do so. ! Answer all five questions from Section A and one question from Section B. ! Unless otherwise stated in the question, all numerical answers must be given exactly or to three significant figures. ! Indicate the make and model of your calculator in the appropriate box on your cover sheet .
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Please start each question on a new page. You are advised to show all working, where possible. Where an answer is wrong, some marks may be given for correct method, provided this is shown by written working. Solutions found from a graphic display calculator should be supported by suitable working, e.g. if graphs are used to find a solution, you should sketch these as part of your answer. SECTION A Answer all five questions from this section. 1. [Maximum mark: 11] Consider the complex number . cos isin z θ θ = + (a) Using De Moivre±s theorem show that [2 marks] . 1 2cos n n z n z θ + = (b) By expanding show that 4 1 z z + [4 marks] . 4 1 cos (cos4 4cos2 3) 8 θ θ θ = + + (c) Let . 4 0 ( ) cos d a g a θ θ = (i) Find . ( ) g a [5 marks] (ii) Solve . ( ) 1 g a = ² 2 ² N04/5/MATHL/HP2/ENG/TZ0/XX 8804-7402
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2. [Maximum mark: 16] A line has equation . 1 l 2 9 3 1 2 x y z + = = [1 mark] (a) Let M be a point on with parameter ° . Express the coordinates of M 1 l in terms of ° . (b) The line is parallel to and passes through . 2 l 1 l P(4, 0, 3) (i) Write down an equation for . 2 l [4 marks] (ii) Express in terms of ° . PM (c) The vector is perpendicular to . PM 1 l (i) Find the value of ° . [5 marks] (ii) Find the distance between . 1 2 and l l [4 marks] (d) The plane contains . Find an equation for , giving your π 1 2 and l l 1 π answer in the form . Ax By Cz D + + = [2 marks] (e) The plane has equation . Verify that is the line of 2 π 5 11 x y z = 1 l intersection of the planes . 1 2 and π π ² 3 ² N04/5/MATHL/HP2/ENG/TZ0/XX 8804-7402 Turn over
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3. [Maximum mark: 15] (a) Let be a rotation about the origin through angle A , and be a 1 T 2 T rotation about the origin through angle B . By considering the matrices , and the product , prove that 1 2 , T T 1 2 T T , sin( ) sin cos cos sin A B A B A B + = + [4 marks] . cos( ) cos cos sin sin A B A B A B + = [5 marks] (b) Hence prove that . 2 2tan tan 2 1 tan A A A = [6 marks] (c) The matrix represents a reflection in a line through the 5 12 13 13 12 5 13 13 origin. Using the result in part (b) , find the equation of the line.
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