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Mathematics HL - Nov 2005 - P2

# Mathematics HL - Nov 2005 - P2 - 3 hours IB DIPLOMA...

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IB DIPLOMA PROGRAMME PROGRAMME DU DIPLÔME DU BI PROGRAMA DEL DIPLOMA DEL BI N05/5/MATHL/HP2/ENG/TZ0/XX MATHEMATICS HIGHER LEVEL PAPER 2 Friday 4 November 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. 8805-7202 11 pages 3 hours 88057202

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N05/5/MATHL/HP2/ENG/TZ0/XX 8805-7202 – 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: 12] (a) Given that x x x a x bx c x 2 2 2 1 1 1 1 ( )( ) ( ) ( ) + + + + + + , calculate the value of a, of b and of c . [5 marks] (b) (i) Hence, find I x x x x = + + 2 2 1 1 ( )( ) d . (ii) If I = π 4 when x = 1 , calculate the value of the constant of integration giving your answer in the form p q r + ln where p q r , , ¡ . [7 marks] 2. [Maximum mark: 16] (i) (a) Let M = 1 3 4 5 1 1 1 k k . Find det M . [2 marks] (b) Find the values of k for which the following system of equations does not have a unique solution. + = − + + = + = x ky z x y z x y kz 3 1 4 5 2 1 [3 marks] (ii) The plane π contains the line x y z = = 1 2 1 3 5 6 and the point (1, 2, 3). (a) Show that the equation of π is 6 2 3 7 x y z + = − . [7 marks] (b) Calculate the distance of the plane π from the origin. [4 marks]
N05/5/MATHL/HP2/ENG/TZ0/XX 8805-7202 – 3 – Turn over 3. [Maximum mark: 17] (i) In a game a player pays an entrance fee of \$ n . He then selects one number from 1, 2, 3, 4, 5, 6 and rolls three standard dice. If his chosen number appears on all three dice he wins four times his entrance fee. If his number appears on exactly two of the dice he wins three times the entrance fee. If his number appears on exactly one die he wins twice the entrance fee. If his number does not appear on any of the dice he wins nothing.

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