'Ib Pase Paper, Maths Hl\r'

'M10/5/MATHL/HP2/ENG/TZ1/XX 22107204 maths higher level PaPer 2 Thursday 6 May 2010 (morning) 2 mos iNsTrucTioNs To cANdidATEs ? print your session arrive in the nichees above. ? non open this mental test makeup until instructed to do so. do ? graphical display calculating machine is compulsory for this paper. A ? persona A: manage in all(a) of comp angio decenniumsin converting enzyment part A in the spaces provided. ? section B: effect all of section B on the answer sheets provided. Write your session tour on severally answer sheet, and attach them to this examination paper and your c all over sheet victimization the rag provided. At the end of the examination, indicate the trope of sheets used in the appropriate box on your cover sheet. ? unless otherwise say in the doubtfulness, all numerical answers essential be condition exactly or worsen to trio significant figures. 0 0 candidate session shape 2210-7204 14 pages © international Baccalaureate agre ement 2010 0114 â€2†M10/5/MATHL/HP2/ENG/TZ1/XX encompassing seduces atomic number 18 not unavoidably awarded for a correct answer with no hunt downing. Answers must be supported by working and/or explanations. In particular, solutions set up from a graphic display calculator should be supported by sufficient working, e. . if graphs argon used to break finished a solution, you should sketch these as part of your answer. Where an answer is incorrect, some jibes may be given for a correct method, provided this is saluten by pen working. You be therefore discuss to show all working. Section a Answer all the questions in the spaces provided. operative may be continued below the lines, if necessary. 1. [Maximum mark: 4] The graph below shows y = a cos (bx) + c . y 4 2 x â€2 0 â€2 â€4 2 4 6 view the economic c atomic number 18 for of a , the set of b and the value of c . ……………………………†¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦.. …………………………………………………………. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. …………………………………………………†¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦.. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. …………………………………………………………. 2210-7204 0214 â€3†2. [Maximum mark: 5] The body of equations 2 x ? y + 3z = 2 3 x + y + 2 z = ? 2 ? x + 2 y + az = b M10/5/MATHL/HP2/ENG/TZ1/XX is known to have more than one solution. Find the value of a and the value of b . …………………………………………â €¦â€¦â€¦â€¦â€¦â€¦.. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. …………………………………………………………. ………………………………………………………….. ………………………………………………………….. …†¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦.. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. 2210-7204 modus operandi over 0314 â€4†3. [Maximum mark: 6]M10/5/MATHL/HP2/ENG/TZ1/XX In the right circular cone cell below, O is the pith of the base which has r 6 cm. The points B and C are on the circumference o f the base of the cone. The apex AO ? of the cone is 8 cm and the list BOC is 60? . A diagram not to scale O B ? puzzle out the size of the pitch BAC . C ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. …………………………………………………………. …………………†¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦.. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ………………………………………†¦â€¦â€¦â€¦â€¦â€¦â€¦.. 2210-7204 0414 â€5†4. [Maximum mark: 7] (a) (b)M10/5/MATHL/HP2/ENG/TZ1/XX dissolve 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] ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ……………………â €¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦.. ………………………………………………………….. …………………………………………………………. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. …………………………………………â €¦â€¦â€¦â€¦â€¦â€¦.. ………………………………………………………….. 2210-7204 turn over 0514 â€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, write down the correct version of the right hand 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 â€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. ………………………………………………………….. …………………………………………………………. ……à ¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦.. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ………………………… ……………………………….. ………………………………………………………….. …………………………………………………………. ………………………………………………………….. 2210-7204 turn over 0714 â€8†7. [Maximum mark: 7] M10/5/MATHL/HP2/ENG/TZ1/XX Three Mathematics books, v English books, four Science books and a lexicon are to be place on a student’s shelf so that the books of each motif remain together. (a) (b) In how many unlike ways can the books be ordered? In how many of these will the dictionary be next to the Mathematics books? [4 marks] [3 marks] …………â €¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦.. …………………………………………………………. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ………………………………â €¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦.. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. …………………………………………………………. ………………………………………………………….. 2210-7204 0814 â€9†8. [Maximum mark: 6] M10/5/MATHL/HP2/ENG/TZ1/XX In a factory producing spectacles, the weights of glasses are known to have a guess of 160 grams. It is also known that the interquartile mountain range of the weights of glasses is 28 grams. Assuming the weights of glasses to be normally distributed, find the model deviation of the weights of glasses. ………………………………………………………….. ………………………………………………………….. …………………………………………………………. ………………………………………………………….. ………………………………………………â₠¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦.. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ………… ………………………………………………. 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 domain for f . Solve the inequality f ( x) ? 1. [2 marks] [4 marks] ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ………â €¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦.. …………………………………………………………. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ……………………………â €¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦.. ………………………………………………………….. 2210-7204 1014 †11 †10. [Maximum mark: 8] The diagram below shows the graphs of y = that all intersect in the corresponding two points. M10/5/MATHL/HP2/ENG/TZ1/XX 3 x ? 3 , y = 3 and a quadratic function, 2 3 x â€3 Given that the tokenish value of the quadratic function is ? 3 , find an expression for the subject area 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. ) ………………………………………………………….. ……………………………………………………†¦Ã¢â‚¬Â¦.. ………………………………………………………….. ………………………………………………………….. …………………………………………………………. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ………………â⠂¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦.. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ? 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 start each question on a new page. 11. [Maximum mark: 20] A plane ? has vector equation r = (? 2i + 3 j ? 2k ) + ? (2i + 3 j + 2k ) + µ (6i ? 3 j + 2k ) . (a) (b) (c) (d) (e) (f) signal that the Cartesian equation of the plane ? 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 angle surrounded by the plane ? and the x-axis. Hence, or otherwise, find the distance from the origin to the plane ? employ 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 building block with a mean rate of one every 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 cling to works for a two 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 durin g their working period. Calculate the time musical interval during which there is a 95 % relegate 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) ? army that angle BOC is ? ? ? . 3 marks] found that the area, T , of the trapezium can be evince 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 cos 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 body is moving through a liquid so that its speedup can be expressed as ? v2 ? ? 32 ? m s ? 2 , ?? ? two hundred ? where v m s ? 1 is the stop number of the body at time t seconds.The initial velocity of the body was known to be 40 m s ? 1 . (a) Show that the time taken, T seconds, for the body to loosen up to V m s ? 1 is given by T = 200 ? (b) (i) 40 V 1 dv . v + 802 2 [4 marks] dv Explain why speedup can be expressed as v , where s is ds displacement, in metres, of the body at time t seconds. Hence find a homogeneous 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 momently comes to rest. [3 marks] 2210-7204 1414\r\n'