Stpm Trial 2009 Kl

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STPM TRIAL 2009

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AN PElAJAt(;\i\I ',VIV, rAH I--'tK~t;rl.U I UI\N rliU,i LA tIJMPUR

.lAMA I AN I"l:.UWI\KAN V':ILA Y.4.H PERSEKUTUA

JABA rAN PElAJARAN W1LAYAH PERSEKUI UA AN PHCo JAnAN W 'LAYAH PER!->EKU'TUAN " TAN PElAJ,'\RAN W ,LAYAH PERSEKUTUAN KUAli. LUMPUR JABATAN PELAJARAN W'lLAYAH PERSEKUTUAJ> PAPER 2 TAN PELAJARAN \VILAYAH PERSEKUTUAN KUALA LU f·/,PUR JABATAN PELAJARAN WtLAYAH PERSEKUTUM iAN PEL,\ J"'V\ N 'NltAYAH PERSEKUTUAN KUALA LUMPUR JABATAN PELAJARAN WtLAYAH PERSEKUTUAJI. " V .... L" u ... ,."- ,,,". . " " ' ,, rAN PEU~JARAN V>lIU,YAH PERSEKUrUAN KUALA LuMPUR .lA6ATAN PELAJARAN WILAY,\H PERSEKUT' ' " ., .•• , < . . .. . ,., • • ~ • • ~ ' '''' ' 1 PElA.JAR.i\N WILA YAH PERSEKUTUAN KUALA LUMPUR

MATHEMATICS T

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Three hours

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JABATAN PELAJARAN WILAYAH PERSEKUTUAN KUALA LUMPUR SIJIL TINGGI PERSEKOLAHAN MALAYSIA

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Instructions to cand id ates:

DO NOT OPEN THIS QUESTION PAPER UNTI L YOU ARE TOLD TO DO SO. Answer a ll questions. All necessary working should be shown clearly. Non-exact numerical answers may be given correct to three Significant fig ures, or one decimal place in the case ofangles in degrees, unless a diffe rent level 0) accuracy is specified in rhe question. Mathematical tables, a

/iSI

of mathematical formlllae and graph paper are pro vided.

This question pape r consists of3 printed pa ges and I blank pa ge.

STPM 95412 *Th is qu estio n paper is CONF IDENT IAL until the exam inati on is ove r.

I Turn oyer CONF IIH~ NT I AL*

2

CONFlDENTIAL· I

Show that the homogenous equation x(y - x) dy = y(x + y) can be transfonned into dx . t lle sub' . Y"" vx. x -dv = 2. by usmg Stltut lOD [4 marks1 dx v - I

2

Forces (4i + 3j )N, (3 i + 8j)N and ( -6; - 5j )N act at a po int. Calculate the mal:oitude o f the

resultant force and the angle between the resul tant force-and the unit vector j .

}5 marks ]

. ,t.,L-k -51 . , G ( ,lL , t ) 1\

\..1.

, In the fi gure above, AS is the diameter of circle O. DE is a tangent at 0 which is perpendicular to BE. AD and BE produced meet at C. Prove that AS = Be. [7 marks1

::--:-. ,~c , I .1 ... - ') "\....

4

tan ) x - 3 tan x

Show that tan ] x = -c:--,"--.,-3tan 2 x- l Hence, by taking

( =

.....

.;. A. t\ ..,

Y.

tan x, find all the roots of the equation

I ' - JI 'Ji - ] 1+Ji = O.

5

6

The rate of increase in temperature of a thennometer is proportional to the difference between the room temperature, Xo and the temperature of the thcmometer, x. Write a diITerential equation to express thi s informati on. A thennometer with a reading of lODe is brought into a room with room temperature 3 iDe . The thennometer reads 24 DC after f minutes. Find the time taken for the reading to be 30DC. . 't. - \b'~L'I'L ' [10 marks1

'-~ BCD is a paralle logram . E and F are points on the diagonal AC such that AE - rc. ~ove that BFDE is a parallelogram.

15 marks]

(b) A ship P sails at a spced 40 km h' l along the directi on N45 D E. Anothe r ship Q sa ils at a speed of 65 kmlf' along the direction S60oE. Find the re lative vc!ocity of P with respect to Q. Initiall y Q is exactly 100 km north of P. Find the shortest distance between P and Q.

f8 marksJ 7

A discrete random variable X takes integer values probabilities gi ven by P( X = x) :o 2 J 2 5 6

betwcc~ 0~j ~\nelusive with

~ . Find the ex pectation of X.

(4 marks)

95412 ~Th is

question paper is CONFIDENTIAL lmtil the examination is over,

CONFIDENTIAL·

3

/"ctlf'/FIDENTlAL'

8

The letters in the word 'ADA!1; are randomly arranged in a lim!. Find the probabi lity that the arrangements (a) begin and end with 'A'

1

~ .,~

( b ) have separated consonants " .'to ~

f5 marksJ

A box containSl fcri\.mangoes.~f the mangoes are of vari ety A and are all good. The remai ning ma;~ are of varicly-B and of these, th ree are good and three are not good. Events X and Yare defin~~s fo llows: X: A random selection ~angoes which containSI9fvatiety A a~variety B.

t'

It

Y: A random selection of 4 mangoes which contains exactly 3 good mangoes. Find P( X), P( Y) and P( X n Y). State, with a rcason, whether the events X and Yare independent . "f.

10

IS·,

'{

The times spent in visiting a particular bookstore

by(~

and ~are

~

arks] to be

<\SS U

independent and are nonna lly di stributed with means'ii{ minu~nd 25 minutes and standard dev iat ions 2.2 minutes and 0.8 minute respective ly. Find the probabihty that

-

c

(a) the time spent by Ali is longer than that spent by Ravi in a particular visi t,

[5 marks]

'/<.-1 (b) the tota l time spem by Al i in 3 separate visits exceeds twice the ti me spent by Rav i in a si nglevi~ it. 't\1"-"1."t'lj ~ "2'( [4 marks]

,

II

Souvenir c ups man!ifactured by a certai n company are pac ked in boxes of(lOO. On the average o ne cup irvSO cups is defective. A box which contain s,) or more defec~ps will be rejected. UsingVsuitable approximation, find the probab ility that a given box wi ll be rejected. A de li very van carries ~uch boxes. The number o f boxes. out ol 20, that wi ll be rejected is denoted bV~nn i ne P( Y:5 ~) and fin d the expected number of rejected boxes. lJI marks1

12

The we ight, to the nearest kg, of 90 new born babies are summarized in the table below.

W~4'!' (~) 1.0 - 1.4 1.5 - 1.9

N lImber of Babies 8

2.0 - 2.4 2.5 - 2.9

12 18 24

3.0 -

20

]A

3.5 - 3.9

-

8,

(a) Draw the frequen cy cu rve for the data above. State the mode for the weight o f the (4 marks] babies. ( b) Plot the cumulative frequency graph for the data above. Esti mate the median and [7 marks ) the se nti.i.t:.'.:~r.tilc range of weight for the babies. (c) Find the probabi lity that a baby chosen at random has a weight excet!ding 3.5 kg.

[2 marks ) (d) If 10% o f the babies with the lowest weight are transferred to the intensive care unit, estimate the nHlximum weight orlhe babies transferred there. [2 marksJ 95412 ' This question paper is CON FIDEN riAL until thc

c~aminali on

is O\cr.

CONFIDENT IAL '

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