mathhl_p1_m06_tz0

mathhl_p1_m06_tz0 - 2 hours IB DIPLOMA PROGRAMME PROGRAMME...

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Unformatted text preview: 2 hours IB DIPLOMA PROGRAMME PROGRAMME DU DIPLÔME DU BI PROGRAMA DEL DIPLOMA DEL BI M06/5/MATHL/HP1/ENG/TZ0/XX 22067204 MATHEMATICS HIGHER LEVEL PAPER 1 Wednesday 3 May 2006 (afternoon) 0 0 Candidate session number INSTRUCTIONS TO CANDIDATES Ÿ Ÿ Ÿ Ÿ Write your session number in the boxes above. Do not open this examination paper until instructed to do so. Answer all the questions in the spaces provided. Unless otherwise stated in the question, all numerical answers must be given exactly or correct to three significant figures. 2206-7204 17 pages 0117 –2– M06/5/MATHL/HP1/ENG/TZ0/XX Full marks are not necessarily awarded for a correct answer with no working. Answers must be supported by working and/or explanations. In particular, 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. Where an answer is incorrect, some marks may be given for a correct method, provided this is shown by written working. You are therefore advised to show all working. Working may be continued below the lines, if necessary. 1. In an arithmetic sequence the second term is 7 and the sum of the first five terms is 50. Find the common difference of this arithmetic sequence. ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... 2206-7204 0217 –3– 2. π π Let z1 = r cos + isin and z2 = 1 + 3 i . 4 4 (a) (b) Write z2 in modulus-argument form. Find the value of r if z1 z23 = 2 . M06/5/MATHL/HP1/ENG/TZ0/XX ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... 2206-7204 Turn over 0317 –4– 3. M06/5/MATHL/HP1/ENG/TZ0/XX 2 The graph of y = 2 x 2 + 4 x + 7 is translated using the vector . Find the equation of the −1 2 translated graph, giving your answer in the form y = ax + bx + c . ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... 4. Let f ( x) = 3 x 2 − x + 4 . Find the values of m for which the line y = mx + 1 is a tangent to the graph of f . ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... 2206-7204 0417 –5– 5. M06/5/MATHL/HP1/ENG/TZ0/XX The polynomial P ( x) = 2 x3 + ax 2 − 4 x + b is divisible by ( x − 1) and by ( x + 3) . Find the value of a and of b. ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... 2206-7204 Turn over 0517 –6– 6. M06/5/MATHL/HP1/ENG/TZ0/XX The following is the cumulative frequency diagram for the heights of 30 plants given in centimetres. (a) Use the diagram to estimate the median height. ......................................................................... ......................................................................... (b) Complete the following frequency table. Height (h) 0≤h<5 5 ≤ h < 10 10 ≤ h < 15 15 ≤ h < 20 20 ≤ h < 25 (c) Hence estimate the mean height. Frequency 4 9 ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... 2206-7204 0617 –7– 7. M06/5/MATHL/HP1/ENG/TZ0/XX $ In the obtuse-angled triangle ABC, AC = 10.9 cm , BC = 8.71 cm and BA C = 50o . Not to scale Find the area of triangle ABC. ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... 2206-7204 Turn over 0717 –8– 8. M06/5/MATHL/HP1/ENG/TZ0/XX The weights in grams of bread loaves sold at a supermarket are normally distributed with mean 200 g . The weights of 88 % of the loaves are less than 220 g . Find the standard deviation. ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... 9. Solve ln ( x + 3) = 1 . Give your answers in exact form. ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... 2206-7204 0817 –9– 10. M06/5/MATHL/HP1/ENG/TZ0/XX 5 Let f ( x) = 20.5 x and g ( x) = 3− 0.5 x + . Let R be the region completely enclosed by the 3 graphs of f and g, and the y-axis. Find the area of R. ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... 2206-7204 Turn over 0917 – 10 – 2 2 −1 Let a = 1 , b = p and c = − 4 . 3 0 6 (a) (b) Find a × b . Find the value of p, given that a × b is parallel to c. M06/5/MATHL/HP1/ENG/TZ0/XX 11. ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... 12. Find ∫ e 2 x sin x dx . ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... 2206-7204 1017 – 11 – 13. M06/5/MATHL/HP1/ENG/TZ0/XX 1 1 7 Let A and B be events such that P ( A) = , P ( B | A) = and P ( A ∪ B) = . 5 4 10 (a) (b) (c) Find P ( A ∩ B) . Find P (B) . Show that A and B are not independent. ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... 2206-7204 Turn over 1117 – 12 – 14. Let f ( x) = cos3 (4 x + 1) , 0 ≤ x ≤ 1 . (a) (b) Find f ′ ( x) . Find the exact values of the three roots of f ′ ( x) = 0 . M06/5/MATHL/HP1/ENG/TZ0/XX ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... 2206-7204 1217 – 13 – 15. M06/5/MATHL/HP1/ENG/TZ0/XX 1 Let f be the function f ( x) = x arccos x + x for −1 ≤ x ≤ 1 and g the function 2 g ( x) = cos 2 x for −1 ≤ x ≤ 1 . (a) On the grid below, sketch the graph of f and of g . (b) (c) Write down the solution of the equation f ( x) = g ( x) . Write down the range of g . ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... Turn over 1317 2206-7204 – 14 – 16. M06/5/MATHL/HP1/ENG/TZ0/XX The number of car accidents occurring per day on a highway follows a Poisson distribution with mean 1.5 . (a) (b) Find the probability that more than two accidents will occur on a given Monday. Given that at least one accident occurs on another day, find the probability that more than two accidents occur on that day. ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... 2206-7204 1417 – 15 – 17. M06/5/MATHL/HP1/ENG/TZ0/XX 2 6 h 3 Let A = and B = , where h and k are integers. Given that det A = det B k −1 −3 7 and that det AB = 256h , (a) (b) show that h satisfies the equation 49h 2 − 130h + 81 = 0 ; hence find the value of k. ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... 2206-7204 Turn over 1517 – 16 – 18. Given that 3x+ y = x3 + 3 y , find dy . dx M06/5/MATHL/HP1/ENG/TZ0/XX ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... 19. There are 10 seats in a row in a waiting room. There are six people in the room. (a) (b) In how many different ways can they be seated? In the group of six people, there are three sisters who must sit next to each other. In how many different ways can the group be seated? ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... 2206-7204 1617 – 17 – 20. M06/5/MATHL/HP1/ENG/TZ0/XX Each of the diagrams below shows the graph of a function f . Sketch on the given axes the graph of (a) f (− x) ; (b) 1 . f ( x) 2206-7204 1717 ...
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This note was uploaded on 06/22/2010 for the course COMCC 05123 taught by Professor Mcgee during the Spring '10 term at York County CC.

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