Kuching Stpm 2012-Mathst Paper 2(Q&A)

SMK ST.JOSEPH, KUCI{INGMATHEMATICS T 954/2

FINAL EXAMTNATION PAPBR2 -2012

NAME:CLASS: UPPER 6A18

Ansrvcr all questions

1. Prole the identitl' 2cosx(sin3x-sin;,:) = sin4x.l-lence, shorvthat sin540 -sinlBo =1.2

2. A person walks towards a lighthouse and observes the tip of the torver has an angle of elevation

of a . After walkingp km towards it on level road the angle of elevation is now BShorv that the height of the lighthouse is given as psinasin B cosec(B - a) .

3.

Two circles PQR and PQS intercept at points P and Q The tangents at R of the circle PQR and the

tangent S of the circle PQS meet at T on the PQ produced. Given that RQS is a straight line, prove

that

tsl

tsl

4 (a)

t3ll3ll2l

(a)(b)(c)

t5l

t5l(b)

TRPS is a cyclic quadrilateral,ITPR= ZTPS ,

TR = TS.

OAB is a triangle .Points A and B have position vectors a and 3b respectively

M is the midpoint of ' OA and K is a point on OB such that OK = : OB. P is theJ

point on MB such that MP = 2 MB. Find the value of I if A, P and K are collinear.

,"t irr\PQR is a triangle. L, M and N are thg nointllQ, 9| and RP respectively. if o is a point

in the plane of PQR, show that OP + OQ + OR = OL + OM + ON .

5. In time r, the velocity of a parlicle wl1ich mo1,es in a straight line is given by the equation?1

$ = i<1a,';' u'here K is a constant. If r,: V u,hen /:0 and v:8V when /: T, find v as adt

function in terms of r. Find the displacen-rent frorr time /: 0 to /: T.

6. (u) Event A and B are such thar p(A\ =1, pf elnl =l und P(A' n n') = | . rinaLIs( r \/r/ 1,,

\"1, \, _ 4

s.._ - \_ 6

(i) P(A t, B)(ii) P(B)

(b) A man writes 5 ietters, one each to A, B, C, D and E . Each letter is placed in a separate

envelope and sealed. He addresses the envelopes, at random, one each to A, B, C D and E.

(i) Find the probability that the letter to A is in the correct envelope and letter to B is inan incorrect envelope. 11]

(ii) Find the probability that the letter to A is in the correct envelope, given that letter to B isin an incorrect envelope. I2l

(iii) Find the probabiiity that both of the letters to A and B are in the incorrect envelopes.

l2l

7 . During weekdays , the number of people entering a bookshop per hour has a Poisson distribution

with mean 4. Independently the number of people leaving the bookshop per hour has a Poisson

distribution with mean 3.

(a) Find the probability that during a half-hour interval.

(i) no one will enter and no one will leave the bookshop' t2l(ii) at most one person will enter the bookshop, given that the total number of people

entering and leaving the bookshop is exactly 5. t3]

(b) The probability that a person buys a mathematics textbook from the bookshop is 0.07.

By using a suitable approximation, find the probability that in a random sample of 60

people, at most 5 of them will buy a mathematics textbook. t3]

8. The table below shows the number of students in a certain college

Construct a cumulative frequency table and sketch the ogive.

From tlre graph. find

(a) the rnedian(b) interquartilerange(c) the number of students with heights between 177 cm to 1 87cm.

t8l

t4l

tel

Heights(cm)

r50 -1 55 - 160 -165 - 170 -175 -1 80 -r 85 -190 >190

Number ofstudents

0 8 )z 78 181 122 52 24 3 0

g. At the Legoland Theme Park , the waiting time for the Dream Ride, has a normal distribution with

meal 50 minutes and varialce of 100 minutes. The u'aiting time for another ride, the Scar,v Ride

has a normal distribution rvith mean of 45 minutes and variance of 144 minutes. 1'he rvaitirlg

tir:res for the 2 rides are independent ofeach other.

Amanda and Brenda queued independently for the Dream Ride while Calvin queued for Scary

Ride. Calculate the probability that the sum of the waiting times of Anranda and Brenda is less

than twice the waiting times of Calvin.

Two in every five tourists rvho take Scary Ride purchase a souvenir photo at the end of the ride.

20 tourists existing from the Scary Ride are randomly chosen. Find the value of ft such that there is

a probability of more than 0.5 that at least /r of them purchased souvenir photos.

The amount spent by tourist who visit the Legoland souvenir has been found to have a mean of

RM 35.00 and standard deviation of RM 7.00. Find an approximate value of the probability that

the mean amount spent by a group of 60 tourists exceeds RM 36'50' L7)

10. The cumulative distribution function of time T, in minutes, a student waits for a bus every

morning is given by

0,

ltIJ

t <0,

0<r<5,F(0:

-!r!, - | ,',5</<15,2 5 150

l. f > 15.

(i) Find the probability density function ofT and sketch its graph' t5l

iill Calculate the mean waiting time of the siudent t4l

iiiil Find the probability that a student has to wait for a bus less than 10 minutes for

at least 3 out ofthl 5 days. t5l

l1 The probability that a pair of Mika sport shoes wear out after six months offusage is 0.2.

Using the suiiable approximation, find the probability that 75 to 100 pairs of Mika sport shoes

fromlhe 500 pairs oitutitu sport shoes that are chosen randomly wear out after six months off

usage. t5l

t2. The discrete random variable X has the probability function

Il ,r9-t)". x=5,7,8,p(x)= ll16 withnastheconstanl'

I O. otherwise.

(i) Determine the value of n .Hence, find the cumulative distribution function of Xand sketch its graph. tll

(ii) Sketch the graph for the probability function of X. l2l

iiil; catcutate the mean and variance of X. t4l

END OF QUESTION PAPER

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H BANTUA,^/ S?: JOSEPH

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SE,KOLAH MENENGAH BANTUATI S?iKUCTTING, SARAWAK

MATAPELAJARAN

JOSEPH

TIN(iK;\'['AN

-J a +)i i.t+

-t 3 +^t +32 ) =, (-2+it

J an ganlr is di

ruang rnr

i,+ = 4r+, t) c,

+)

--:i-J4i : A/vi rlip

= -!Z t-An6

7R,p..r Li 4 t ti c/tt-)-).J4K = ko +ok

,:Jc,n R f qn/

4, = d . (-ct lkt /,.'/e e9 jh.^ -!+jtr:r-l,=F- Ce/L,.rro*@.*orr"r) --L+2

/. (7K -- *,'f 6 r( erf /,r =su"n

"-!,',ica A/ fi /< a,<- c.//,

L.fTE =1RPS (ex+lJ:),r/cry b ,n, 4P: m +k-

foa-t t R,t,T ar< cwq h\ Gt-t;)g- +3r){ =r/h

Satr TRt's lw q- j-j a:-4_A 3)=

jt .S,ic< TRN iV q rgrl,t dauf @1 ,'16 6

, CqY l-t ,'4,t",',ai -ja"*-:;

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SEKOIAH MENENGAH BANTT]AII S?1 JOSEPHKUCHING, SARAWAK

NAMA MATAPELAJARAN TIN(;KAI AN

Jangantulis di

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NAMA:

SEKOLAH MENENGAH KEBANGSAAN ST JOSEPHKUCHING, SARAWAK

MAIAPELAJARAN: KELAS:

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R-+_t -r f'r I

c(tnq -- e @/a,t z(aWlt-u) = cltt4l *- /, v

v- -_ ( t,rv/s 13 -- 8T3tz

L2r'q " P (o,r) = P(b -t ?@J - P @na A

v'1 - Er'v /l{\\ rr<tt tv C)-t-E f l/'

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B - e+4'+ fl* tel/e. B i to4 e44

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1: t* a.ta /3 Qx- r) covreoA

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MATAPELAJARAN:

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: P(r=o aat [zt) + 2 ( t'/ dq Y'*) /-z?c", h /

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n &n&. 4k n. -

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=..v." Ct+f z+-+-+LS4z) f 4)f

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0,+ 4t fap4vQatf )r"1- K -. B (s, t/t) ,./

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A-r- o l"r%f/) (Lvs tc'7) {0,{ -f rc- ,z,t L+-r'

lu <7 ) : o,4,d(1 1utr .f\tf) (t'sd ) , o q46aa >0. 5 = o. / 6u1S -1, o. V o/*t + o.

61-u.ll k =(zt v/41

ln 'dl4rh |tz ,,'za tuna4

-tiu"t1\l /4 n At Ctl- ,2 [*,> ]e.s,,) =o.dtf (

ffiJHHINAMA:

SEKOLAH MENENGAH KEBANGSAAN ST JOSEPHKUCEING, SARAWAK.

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)tl7a d MBI

l-{-:>,. t 1 -l-d1*a \L

t+2\- 16'^)'..

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=t-tr(z<-L&J)-r2(>ruc/sr) / IL \ ,'/

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O. r>6 \e) nLaf c,i,i A(x: = Elr PA.n) = ! { late

/ x = fi .x C1-x2^. 4: r, 7, I ml

i/x-s--78 AO ) : !7' P(X=zc; :.

Pr=r, &r+l n ,Az\))^

J.

* lnl

Va6 = f(r) -Co.'lL

7/(x--x)tt / (nt ^ .q7/ -./6, ,/'l I

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$Cra) +Xrn)+b&=-4 / nl z 13y *t,/-{ q: Jn rc'

\t't 79 *tofcct4

Untuk peperiksaan sekolah sahaia. Gunakan ketlua-dua beiah muka

ffiiJHHI,NAMA:

SEKOLAH MENENGAH KEBANGSAAN ST IOSEPHKUCHING, SARAWAK.

MATAPBLAJARAN: KELAS:

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