2006_Question

2006_Question - DOB-ASL I 8: S HONG KONG EXAMINATIONS AND...

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Unformatted text preview: DOB-ASL I 8: S HONG KONG EXAMINATIONS AND ASSESSMENT AUTHORITY HONG KONG ADVANCED LEVEL EXAMINATION 2006 MATHEMATICS AND STATISTICS AS-LEVEL 8.30 am — 11.30 am (3 hours) This paper must be answered in English This paper consists of Section A and Section B. Answer ALL questions in Section A, using the AL(E) answer book. 2. 3. Answer any FOUR questions in Section B, using the AL(C) answer book. 4. Unless otherwise specified, all working must be clearly shown. 5. Unless otherwise specified, numerical answers should be either exact or given to 4 decimal places. ©i§i%%i€fifiijl=i¥% R¥Iflii§ Hong Kong Examinations and Assessment Authority All Rights Reserved 2006 2006—AS-M & S—l SECTION A (40 marks) Answer ALL questions in this section. Write your answers in the AL(E) answer book. —1 (a) The coefficients of x and x2 in the binomial expansion of [1+5] ’7 a —l 1 . . are ~— and -- respectlvely, where a IS a non-zero constant and 18 24a 11 is a positive integer. (i) Find the values of a and n . (ii) State the range of values of x for which the binomial expansion —l of [1 +1] ” is valid. a (b) Using the results of (a) , or otherwise, (i) write down the binomial expansion of (9 +x) 2 in ascending powers of x as far as the term in x2 ; (ii) state the range of values of x for which the binomial expansion 41 of (9+x) 2 is valid. (6 marks) After adding a chemical into a bottle of solution, the temperature S(t) of the surface ofthe bottle can be modelled by S(t)=2(t+1)ze_’”+15, where S(t) is measured in °C , t (2 0) is the time measured in seconds after the chemical has been added and xi is a positive constant. It is given that 8(9) 2 S(19) . (a) Find the exact value of xi. . (b) Will the temperature of the surface of the bottle get higher than 90°C ? Explain your answer. (6 marks) 2006—AS-M & 8—2 V 2 2006-AS-M & S—3 3 The rate of change of the amount of water in litres flowing into a tank can be modelled by 500 f :_ (I) (r+2)2e" where t (2 0) is the time measured in minutes. (a) Using the trapezoidal rule with 5 sub—intervals, estimate the total amount ofwater flowing into the tank from i=1 to t=ll . d2f(t) dt2 (b) Find . (c) Determine whether the estimate in (a) is an over-estimate or under-estimate. (7 marks) The stem—and—leaf diagram below shows the distribution of the numbers of books read by 24 students of a school in the first term: Stem {tens} Leaf g units) 6 7 4 223567889 3455789 0 UJNV—‘O oy—‘r—tUJ (a) Find the median and the interquartile range of the numbers of books read. (b) The librarian of the school ran a reading award scheme in the second term. The following table shows some statistics of the distribution of the numbers of books read by these 24 students in the second term: Minimum Lower quartile Median Upper quartile Maximum 8 26 35 41 46 (i) Draw two box—and-whisker diagrams of the same scale to compare the numbers ofbooks read by these students in the first term and in the second term. (ii) The librarian claims that not less than 50% of these students read at least 5 more books in the second term than that in the first term. Do you agree? Explain your answer. (7 marks) Go on to the next page 5. A and B are two events. Suppose that P(AflB):0.2 and P(AlB')=O.5 , where B’ is the complementary event of B . Let P(B) = b , where b <l . (a) Express P(AmB’) and P(A) in terms of b. (b) if A and B are independent events, find the value(s) of b . (7 marks) A teacher, Susan, built up a performance assessment scheme in which her 100 students were awarded merit points according to their test scores. She also used a normal distribution and a Poisson distribution to model the test scores and the merit points of the students respectively (see the following table). Expected number of students * Normal Poisson distribution distribution Merit Observed number points of students Test score (X) 50£X<150 0 20 14.65 l 24.66 150sX<250 41 44.00 34.52 250$X<350 28 33.45 24.17 350$X<450 9 6.38 11.28 X 2450 2 0.30 3.95 Sum = 100 Sum = 98.78 Sum : 98.58 hwai—A * Correct to 2 decimal places. (a) In the above table, why is the sum of the expected numbers of students under each distribution less than 100 ? (b) The absolute values of the differences between the observed numbers of students and the expected numbers of students are regarded as errors. The distribution with a smaller sum of errors will fit the observed data better. Which distribution, normal or Poisson, fits the observed data better? Explain your answer. (7 marks) 2006—AS-M & S—4 4 2006—AS-M & S—5 5 SECTEON B (60 marks) Answer any FOUR questions in this section. Each question carries 15 marks. Write your answers in the AL(C) answer book. Define f(x): a+bx 4 forall x¢4 . Let g(x)=f(—x) forall x¢—4 . —x Let C1 and C2 be the curves y=f(x) and y=g(x) respectively. 1‘ It is given that the y-intercept of C1 is 73 while the x-intercept of C2 is —2 . (a) Find the values of a and b . (2 marks) (b) (i) Find the equations of the vertical asymptote(s) and the horizontal asymptote(s) to C1 . (ii) Sketch C1 and indicate its asymptote(s) and its intercept(s). (5 marks) (c) On the diagram sketched in (b)(ii), sketch C2 and indicate its asymptote(s), its intercept(s) and the point(s) of intersection of the two curves. (4 marks) (d) Find the area enclosed by C1 , C2 and the straight line y = 9 . (4 marks) Go on to the next page An airline manager, Christine, notices that the weekly number of passengers of the airline is declining, so she starts a promotion plan to boost the weekly number of passengers. She models the rate of change of the weekly number of passengers by dx 301 — 90 ~=— :20 , dt [2—6z‘+ll ( ) where x is the weekly number of passengers recorded at the end of a week in thousands of passengers and t is the number of weeks elapsed since the start of the plan. Christine finds that at the start ofthe plan ( i.e. t = O ) , the weekly number of passengers is 40 thousand. (a) Let v=t2—6t+11,find %. Hence, or otherwise, express x in terms of t . (4 marks) (b) How many weeks after the start of the plan will the weekly number of passengers be the same as at the start of the plan? (2 marks) (c) Find the least weekly number of passengers after the start of the plan. Give your answer correct to the nearest thousand. (3 marks) (d) The week when the weekly number of passengers drops to the least is called the Recovery Week. (i) Find the change in the weekly number of passengers from the Recovery Week to its following week. Give your answer correct to the nearest thousand. (ii) Provethat(t+1)2—6(t+1)+11<3(12—6t+11) forall 1'. (iii) Christine’s assistant claims that after the Recovery Week, the change in the weekly number of passengers from a certain week to its following week will be greater than 25 thousand. Do you agree? Explain your answer. (6 marks) 2006—AS—M & S—6 6 2006—AS-M & S—7 7 After upgrading the production line of a cloth factory, two engineers, John and Mary, model the rate of change of the amount of cloth production in thousand metres respectively by f(z)=2522(t+10)3 and g(t)=28+keht2, where h and k are positive constants and t (2 0 ) is the time measured in months since the upgrading of the production line. (a) Using the substitution u =t+10 , or otherwise, find the total amount of cloth production from t = 0 to t = 3 under John’s model. (5 marks) (b) Express 1n (g0) —28) as a linear function of 22 . (1 mark) (c) Given that the slope and the intercept on the vertical axis of the graph of the linear function in (b) are measured to be 0.3 and 1.0 respectively, estimate the values of h and k correct to 1 decimal place. (2 marks) (d) Using the estimated values of h and k obtained in (c) correct to 1 decimal place, (i) expand g(t) in ascending powers of t as far as t6 , and hence estimate the total amount of cloth production from t=0 to t = 3 under Mary‘s model; (ii) determine whether the estimate in (d)(i) is an over—estimate or an under-estimate; (iii) determine whether the total amount of cloth production from t = 0 to t: 3 under Mary’s model is greater than that under John’s model. (7 marks) (30 on to the next page 10. A researcher models the number of cars entering a roundabout in five-second time intervals (FSTIs) by a Poisson distribution with a mean of 4.7 cars per FSTI, and the speed of a car entering the roundabout by a normal distribution with a mean of 42.8 km/h and a standard deviation of 12 km/h . A car is speeding if the speed of the car is over 50 km/h . (a) Find the probability that fewer than 6 cars enter the roundabout in a certain FSTI. (3 marks) (b) Find the probability that a car entering the roundabout is speeding. (2 marks) (c) Find the probability that the 6th car entering the roundabout is the lst Speeding car. >_ (3 marks) (d) The roundabout is hazardous in a certain FSTI if at least 4 cars enter the roundabout in that FSTI and more than 2 of them are speeding. (i) If exactly 4 cars enter the roundabout in a certain FSTI, find the probability that the roundabout is hazardous in that FSTI. (ii) Given that fewer than 6 cars enter the roundabout in a certain FSTI, find the probability that the roundabout is hazardous in that FSTI. (7 marks) 2006—AS-M & 8—8 8 ll. 2006—AS-_M & S—9 9 A manufacturer of brand E grape juice starts a marketing campaign by issuing points which can be exchanged for gifts. The number of points is shown on the back of the lid of each can of brand E grape juice. The probabilities for a customer to get a can of brand E grape juice with a 2—point lid and 5-point lid are 0.8 and 0.2 respectively. Atotal of 15 points or more can be exchanged for a packet of potato chips while a total of 20 points or more can be exchanged for a radio. (21) Find the probability that a customer can exchange for a packet of potato chips in buying 5 cans of brand E grapejuice. (3 marks) (b) A customer, Peter, buys 7 cans of brand E grapejuice. (i) Find the probability that only when the 7th can of brand E grape juice has been opened, Peter gets a 5—point lid. (ii) Find the probability that only when the 7th can of brand E grape juice has been opened, Peter can exchange for a radio. (iii) Given that Peter can exchange for a radio only when the 7th can ofbrand E grapejuice has been opened, find the probability that the 7th can of brand E grapejuice has a 5—point lid. (iv) Given that Peter cannot get a packet of potato chips after opening 5 cans of brand E grapejuice, find the probability that he can exchange for a radio only when the 7th can of brand E grapejuice has been opened. (12 marks) Go on to the next page Table: Area under the Standard Normal Curve 12. There are many plants in a greenhouse and all ofthem are ofthe same species, Assume that the numbers of infected leaves on the plants in the greenhouse are independent and the number of infected leaves on each plant follows a Poisson distribution with a mean of 2.6 . A plant with at least 4 infected leaves is classified as unhealthy. (a) Find the probability that a certain plant in the greenhouse is unhealthy. (3 marks) (b) A researcher, Teresa, inspects the plants one by one in the greenhouse. She finds that the Mth inspected plant is the first unhealthy plant. .0 (i) Find the probability that M is less than 5 . .1 .2 (ii) Given that M is less than 5 , find the probability that M is :2 greater than 2 . I 5 l.6 (iii) If Teresa inspects In plants in the greenhouse, find the least 1'7 value of In so that the probability of finding an unhealthy plant is greater than 0.95 . ' 7'0 (9 marks) 5:1 2.2 (c) It is given that there are 150 plants in the greenhouse. The number of 33 unhealthy plants in the greenhouse is recorded every Friday. Let N be "4 the number of unhealthy plants recorded on a Friday. Find the mean and the variance of N. g: (3 marks) END OF PAPER Note: An entry in the table is the proportion ofthe area under the entire curve which is between 2 = 0 and a positive value of z , Areas for negative values of z are obtained by symmetry, Ar ) f 1 :iz'dx z = e OVZn 2006-AS—M & 8—10 10 2006'AS'M & 5‘“ 11 ...
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This note was uploaded on 08/17/2011 for the course MATH 0001 taught by Professor Unknown during the Spring '11 term at CUHK.

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2006_Question - DOB-ASL I 8: S HONG KONG EXAMINATIONS AND...

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