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Mathematics HL - May 2005 - TZ2 - P2

# Mathematics HL - May 2005 - TZ2 - P2 - 3 hours IB DIPLOMA...

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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 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. 2205-7208 11 pages 3 hours 22057208

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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 find a solution, you should sketch these as part of your answer. SECTION A Answer all five 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 , find (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 defined 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 inflexion on the graph of f . Find the exact value of the x -coordinate of (i) the minimum point; (ii) the point of inflexion. [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. Find 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 . The plane π 2 has the equation r = 2 0 1 + s 1 2 1 + t 1 1 1 .

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