Ib Pase Paper, Maths Hl

M10/5/MATHL/HP2/ENG/TZ1/XX 22107204 mathematics higher level writing 2 Thursday 6 May 2010 (morning) 2 hours iNsTrucTioNs To cANdidATEs ? keep your session rate in the boxes above. ? not open this examination makeup until instructed to do so. do ? graphic display computer is required for this paper. A ? section A reception wholly of section A in the spaces provided. ? section B answer all of section B on the answer sheets provided. Write your session number on distributively answer sheet, and attach them to this examination paper and your cover sheet u evilg the differentiate provided. At the end of the examination, indicate the number of sheets used in the reserve box on your cover sheet. ? un little otherwise stated in the question, all numerical answers must be given exactly or better to three signifi put forwardt figures. 0 0 freighterdidate session number 2210-7204 14 pages international Baccalaureate organization 2010 0114 2 M10/5/MATHL/HP2/ENG/TZ1/XX Full scar be not necessarily awarded for a reclaim answer with no functional. 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. . if graphs atomic number 18 used to key out 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 uttern by indite working. You argon therefore advised to show all working. Section a Answer all the questions in the spaces provided. Working may be go on below the lines, if necessary. 1. Maximum mark 4 The graph below shows y = a romaine (bx) + c . y 4 2 x 2 0 2 4 2 4 6 Find the appraise of a , the value of b and the value of c . .. . .. .. .. .. .. .. .. .. . 2210-7204 0214 3 2. Maximum mark 5 The system of comp atomic number 18s 2 x ? y + 3z = 2 3 x + y + 2 z = ? 2 ? x + 2 y + az = b M10/5/MATHL/HP2/ENG/TZ1/XX is cognize to have more than one solution. Find the value of a and the value of b . .. .. .. .. . .. .. .. .. .. .. .. 2210-7204 act over 0314 4 3. Maximum mark 6M10/5/MATHL/HP2/ENG/TZ1/XX In the undecomposed circular cone below, O is the centre of the base which has radius 6 cm. The points B and C are on the circumference of the base of the cone. The height AO ? of the cone is 8 cm and the weight BOC is 60? . A plot not to scale O B ? Calculate the size of the tip BAC . C .. .. .. .. . .. .. .. .. .. .. .. 2210-7204 0414 5 4. Maximum mark 7 (a) (b)M10/5/MATHL/HP2/ENG/TZ1/XX Solve the equation z 3 = ? 2 + 2i , giving your answers in modulus-argument form. Hence show that one of the solutions is 1+ i when written in Cartesian form. 6 marks 1 mark .. .. .. .. .. .. . .. .. .. .. .. 2210-7204 tip over over 0514 6 5. Maximum mark 6 M10/5/MATHL/HP2/ENG/TZ1/XX Let A , B and C be non-singular 2 ? 2 matrices, I the 2 ? 2 identity matrix and k a scalar. The following statements are incorrect.For each statement, wri te down the correct version of the right manus side. (a) (b) (c) ( A + B ) 2 = A2 + 2 AB + B 2 ( A ? kI )3 = A3 ? 3kA2 + 3k 2 A ? k 3 CA = B ? C = B A 2 marks 2 marks 2 marks .. .. .. .. .. .. . .. .. .. .. .. 2210-7204 0614 7 6. Maximum mark 5 M10/5/MATHL/HP2/ENG/TZ1/XX Find the sum of all three-digit natural numbers that are not exactly divisible by 3. .. . .. .. .. .. .. .. .. .. . .. 2210-7204 crouch over 0714 8 7. Maximum mark 7 M10/5/MATHL/HP2/ENG/TZ1/XX Three Mathematics books, five incline books, four Science books and a dictionary are to be fixed on a students shelf so that the books of each subject remain together. (a) (b) In how many different ways can the books be arranged? In how many of these will the dictionary be next to the Mathematics books? 4 marks 3 marks .. . .. .. .. .. .. .. .. .. . .. 2210-7204 0814 9 8. Maximum mark 6 M10/5/MATHL/HP2/ENG/TZ1/XX In a factory producing glasses, the weights of glasses are known to have a mean of 160 grams. It is also known that the interquartile range of the weights of glasses is 28 grams. Assuming the weights of glasses to be normally distributed, regulate the standard disagreement of the weights of glasses. .. .. . .. .. .. .. .. .. .. .. . 2210-7204 turn over 0914 10 9. Maximum mark 6 Let f ( x) = (a) (b) 4 ? x2 . 4? x M10/5/MATHL/HP2/ENG/TZ1/XX State the largest possible solid ground for f . Solve the inequality f ( x) ? 1. 2 marks 4 marks .. .. .. .. .. . .. .. .. .. .. .. 2210-7204 1014 11 10. Maximum mark 8 The draw below shows the graphs of y = that all intersect in the same dickens points. M10/5/MATHL/HP2/ENG/TZ1/XX 3 x ? 3 , y = 3 and a quadratic function, 2 3 x 3 given(p) that the minimum value of the quadratic function is ? 3 , find an prospect for the field of the shaded region in the form a, b, c and t are to be determined. (Note The integral does not need to be evaluated. ) .. .. .. .. . .. .. .. .. .. .. .. ? t 0 (ax 2 + bx + c) dx , where the constants 2210-7204 turn over 1114 12 Section B M10/5/MATHL/HP2/ENG/TZ1/XX Answer all the questions on the answer sheets provided. Please leap each question on a new page. 11. Maximum mark 20 A sheet of paper ? has vector equation r = (? 2i + 3 j ? 2k ) + ? (2i + 3 j + 2k ) + (6i ? 3 j + 2k ) . (a) (b) (c) (d) (e) (f) Show that the Cartesian equation of the cream off ? is 3 x + 2 y ? 6 z = 12 . The plane ? meets the x , y and z axes at A, B and C respectively. Find the coordinates of A, B and C. Find the volume of the pyramid OABC. Find the locomote amongst the plane ? and the x-axis. Hence, or otherwise, find the distance from the origin to the plane ? Using your answers from (c) and (e), find the area of the triangle ABC. 6 marks 3 marks 3 marks 4 marks 2 marks 2 marks 12. Maximum mark 15 Casualties arrive at an accident unit with a mean rate of one both 10 minutes. Assume that the number of arrivals can be modelled by a Poisson distribution. (a) (b) (c) Find the probability that there are no arrivals in a given half hour period. A nurse works for a ii hour period. Find the probability that there are fewer than ten casualties during this period. Six nurses work consecutive two hour periods between 8am and 8pm.Find the probability that no more than three nurses have to attend to less than ten casualties during their working period. Calculate the clock time interval during which there is a 95 % chance of there being at least two casualties. 3 marks 3 marks 4 marks 5 marks (d) 2210-7204 1214 13 13. Maximum mark 11 M10/5/MATHL/HP2/ENG/TZ1/XX Points A, B and C are on the circumference of a circle, centre O and radius r . ? A trapezium OABC is formed such that AB is parallel to OC, and the angle AOC ? is ? , ? ? ? . 2 B C A r ? O diagram not to scale (a) (b) ? Show that angle BOC is ? ? ? . 3 marks Show that the area, T , of the trapezium can be expressed as T= 1 2 1 r sin ? ? r 2 sin 2? . 2 2 3 marks (c) (i) Show that when the area is maximum, the value of ? satisfies cos ? = 2 c os 2? . (ii) Hence determine the maximum area of the trapezium when r = 1. (Note It is not required to prove that it is a maximum. ) 5 marks 2210-7204 turn over 1314 14 14. Maximum mark 14 M10/5/MATHL/HP2/ENG/TZ1/XX A form is moving through a liquid so that its acceleration can be expressed as ? v2 ? ? 32 ? m s ? 2 , ? 200 ? where v m s ? 1 is the velocity of the form at time t seconds.The initial velocity of the carcass was known to be 40 m s ? 1 . (a) Show that the time taken, T seconds, for the consistence to slow to V m s ? 1 is given by T = 200 ? (b) (i) 40 V 1 dv . v + 802 2 4 marks dv Explain why acceleration can be expressed as v , where s is ds displacement, in metres, of the body at time t seconds. Hence find a confusable integral to that shown in part (a) for the distance, S metres, travelled as the body slows to V m s ? 1 . 7 marks (ii) (c) Hence, using parts (a) and (b), find the distance travelled and the time taken until the body momentarily comes to rest. 3 m arks 2210-7204 1414