Mathematics HL - May 2005 - TZ2 - P2

Mathematics HL - May 2005 - TZ2 - P2 - IB DIPLOMA PROGRAMME...

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Unformatted text preview: IB DIPLOMA PROGRAMME PROGRAMME DU DIPLÔME DU BI PROGRAMA DEL DIPLOMA DEL BI M05/5/MATHL/HP2/ENG/TZ2/XX MATHEMATICS HIGHER LEVEL PAPER 2 Wednesday 4 May 2005 (morning) INSTRUCTIONS TO CANDIDATES ¡ Do not open this examination paper until instructed to do so. ¡ Answer all fve 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 signifcant fgures. 2205-7208 11 pages 3 hours 22057208 M05/5/MATHL/HP2/ENG/TZ2/XX 2205-7208 – 2 – 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 Fnd a solution, you should sketch these as part of your answer. SECTION A Answer all fve questions from this section. 1. [Maximum mark: 14] (a) Let R be a rotation through k degrees, centre (0, 0). R maps the point (5, 10) onto the point (–2, 11). Find the matrix R . [6 marks] (b) A transformation T is represented by the matrix 1 2 1 1 − − . Describe the geometric effect of applying transformation T four times in succession. [2 marks] (c) The matrix Q represents rotation R followed by transformation T . Show that Q = − − 2 1 1 4 0 2 . . . [1 mark] (d) Under transformation Q , ¡nd (i) the set of points which are mapped onto themselves; (ii) the image of the line y x = − . [5 marks] M05/5/MATHL/HP2/ENG/TZ2/XX 2205-7208 – 3 – Turn over 2. [Maximum mark: 13] The function f is deFned by f x x p px ( ) ( ), = + ∈ e where 1 R . (a) (i) Show that ′ = ( ) f x p x px ( ) ( e + 1) + 1 . (ii) Let f x n ( ) ( ) denote the result of differentiating f x ( ) with respect to x , n times. Use mathematical induction to prove that f x p p x n n n px ( ) ( ) ( = ( ) − 1 e + 1) + , n ∈ Z + . [7 marks] (b) When p = 3 , there is a minimum point and a point of in¡exion on the graph of f . ¢ind the exact value of the x-coordinate of (i) the minimum point; (ii) the point of in¡exion. [4 marks] (c) Let p = 1 2 . Let R be the region enclosed by the curve, the x-axis and the lines x = –2 and x = 2. ¢ind the area of R . [2 marks] 3. [Maximum mark: 12] (a) The plane π 1 has equation r = 2 1 1 + λ − 2 1 8 + µ 1 3 9 − − ....
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This note was uploaded on 05/06/2010 for the course MATH 1102 taught by Professor Chang during the Spring '10 term at Savannah State.

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Mathematics HL - May 2005 - TZ2 - P2 - IB DIPLOMA PROGRAMME...

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