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  • 2008-ASLM&S

    HONG KONG EXAM]NATIONS AND ASSESSMENT AUTHORITY

    HONG KONG ADVANCED LEVEL EXAMINATION 2OO8

    II

    'ls

    tt'4

    MATHEMATICS AND STATISTICS AS-LEVEL

    8.30 am -

    11.30 am (3 hours)

    This paper must be answered in English

    l. This paper consists of Section A and Section B.

    2. Answer ALL questions in Section A, using the AL(E) answer book.

    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.

    t42008-AS-M & S-l

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  • 4. A and B are two events. A' and B' are the complementary events of A and B respectively.l9rSuppose P(A)=i , P(lvB)=i, PQqIB)=i and P(B)=* ,where 0
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  • 8. Abiologist studied the population of fruit fly I under limited food supply. Let r be the number ofdayssincethebeginningoftheexperimentand NO bethenumberoffruitfly I attime /. Thebiologist modelled the rate of change of the number of fruit tly A by

    N'(') = 'o ,- ('> o) 'I + he-^'where h and k are positive constants.

    ,rl

    $tIi+

    (a) (i) Express ,r[jl -

    r-] u, a linear tunction of r .' LN'(t) 'l

    (ii) It is given that the intercepts on the vertical axis and the horizontal axis ofthe graph ofthe linear function in (i) are 1.5 and 7.6 respectively. Find the values of h ard k.

    (4 marks)

    (b) Take h = 4.5 , k = 0.2 and assume that N(0) = JQ .

    (i) Let v= h+ekt,find I .'dt

    Hence, or otherwise, find N(r) .

    (iD The population of fruit fly B can be modelled by

    M(r)= x(t*L"-k'\*t\"/ -'[- t- ) " 'where 6 is a constant. It is known that M(20) = N(20) .

    (l) Find the value of 6.(2) Thebiologistclaimsthatthenumberoffruitfly I willbesmallerthanthatoffruit ;

    [Hint: Consider the difference between the rates of change of the two populations.](1l marks)

    2008-AS-M & S-5 l8

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  • 10. Assume that the number of visitors arriving at each corurter in an immigration hall is independentand follows a Plisson distrlbution with a rneanpl-3.9 visitols per minute. A counter is classified asbusy if at least 4 visitors arriving at it in one minute.

    ":'

    (a) Find the probability that a counter is busy in a certain minute.(3 marks)

    (b) An officer checks 4 counters in a certain minute.counter is found.

    Find the probability that at least one busy

    (2 marks)

    If 10 counters are open, find the probability that more thanminute.

    7 ofthem are busy in a certain

    (3 marks)

    (d) Suppose l0 counters are open and one of them is randomly selected. Find the probability thatmore than 7 of them are busy and the randomly selected counter is not busy in a certainminute.

    (3 marks)

    (e) The immigration hall is called congested if more than 90% of the open counters are busy in aminute. Suppose 15 counters in the hall are open. A senior officer checks the counters in acertain minute. It is given that more than 7 of the first 10 checked counters are busy. Findthe probability that the hall is congested.

    (4 marks)

    (c)

    1

    l

    2008-AS-M & S-7 20

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  • -I

    12. offrcials of the Food Safety centre of a city inspect the imported "choy s*" !y selecting 40samples of "Choy i.,." to. each lorry and testing for an unregistered insecticide' A lorry of;Crr'"v Sum" is .iurrin.a as risky if more than 2 samples show positive results in the test'

    Farm,4 supplies"ChoySum"tothecity. Pastdataindicatedthat l% oftheFarm l "ChoySum"showed positive ,"*ti, in the test. on a certain day, "choy Sum" supplied by Farm A istransportld by a number of lorries to the city'

    (a) Find the probability that a lorry of "Choy Sum" is risky'

    END OFPAPER

    (3 marks)

    (b) Find the probability that the 5ft lorry is the first lorry transportitg rislE "Choy tu-',, marks)

    (c) If t lorries of "choy Sum" are-insp_ected, find the least value of t such that the probability offinding at least o.t. iotry of risky "Choy Sum" is greater than 0'05 (3 marks)

    (d) Farm B also supplies "choy Sum" to the ciry' It is known that l'5% of the Farm B "choysum,, showed positive resulis in the test. on a certain day, "choy Sum" supplied by Farm

    IandFarmBistransportedbySand12lorriesrespectivelytothecity.

    (i) Find the probability that a lorry of "Choy Sum" supplied by Farm B is rislE'(ii) Find the probability that exactly 2 of these 20 lorries of "choy Sum" are risfty'(iii) It is given that exactly 2 of these 20 lorries of "choy Sum" are rrsty' Find the' proUaUility that these i lorries transport "Choy Sum" from Farm 'B ' (7 marks)

    2008-AS-M & S-9,,)

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