Mathematics HL - Nov 2002 - P2

Mathematics HL - Nov 2002 - P2 - c 3 hours IB DIPLOMA...

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MATHEMATICS HIGHER LEVEL PAPER 2 Monday 11 November 2002 (morning) 3 hours N02/510/H(2) c IB DIPLOMA PROGRAMME PROGRAMME DU DIPL&ME DU BI PROGRAMA DEL DIPLOMA DEL BI 882-237 11 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. ! Write the make and model of your calculator on the front cover of your answer booklets e.g. Casio fx-9750G , Sharp EL-9600, Texas Instruments TI-85.
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Please start each question on a new page. You are advised to show all working, where possible. Solutions found from a graphic display calculator should be supported by suitable working. For example, if graphs are used to find a solution, you should sketch these as part of your answer. Incorrect answers with no working will normally receive no marks. SECTION A Answer all five questions from this section. 1. [Maximum mark: 16] (i) A sequence is defined by . { } n u 01 1 1 1, 2, 3 2 where nn n uuu u u n +− == = + Z [3 marks] (a) Find . 234 ,, (b) (i) Express in terms of n . n u [3 marks] (ii) Verify that your answer to part (b)(i) satisfies the equation . 11 32 n uu u + =− (ii) The matrix M is defined as . 21 10  =   M [3 marks] (a) Find . 23 4 ,a n d MM M (b) (i) State a conjecture for , i.e. express in terms of n M n M . ,where + Z [7 marks] (ii) Prove this conjecture using mathematical induction. & 2 & N02/510/H(2) 882-237
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2. [Maximum mark: 11] Consider the complex number . 23 4 cos isin cos 4 cos 24 z ππ  −+  43 3  = 24 (a) (i) Find the modulus of z . [4 marks] (ii) Find the argument of z , giving your answer in radians. [2 marks] (b) Using De Moivre&s theorem, show that z is a cube root of one, i.e. . 3 1 z = [5 marks] (c) Simplify , expressing your answer in the form a + b i, 2 (1 2 ) (2 ) z z ++ where a and b are exact real numbers. [2 marks] 3. [Maximum mark: 14] (a) On the same axes sketch the graphs of the functions , where ()a n d () f xg x , 2 () 4 ( 1 ),f o r 2 4 fx x x =−− −≤≤ . () l n ( 3 ) 2 ,f o r 3 5 gx x x =+ (b) (i) Write down the equation of any vertical asymptotes. [3 marks] (ii) State the x -intercept and y -intercept of . () g x [2 marks] (c) Find the values of x for which . f x = (d) Let A be the region where . n d 0 fx gx x ≥≥ (i) On your graph shade the region A . (ii) Write down an integral that represents the area of A . [4 marks] (iii) Evaluate this integral. [3 marks] (e) In the region A find the maximum vertical distance between .
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Mathematics HL - Nov 2002 - P2 - c 3 hours IB DIPLOMA...

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