,,llr", Ll'UCONFIDENTTAL*/SULIT*

951t2MathcrnaticsPapcr 2Thrce hours

PERSIDANGAN KEBANGSAAN PENGETUASEKOLAFI MENENGAI-I MALAYSIA (CAWANGAN IUELAKA)

PEPERIKS,{AN PERCUBAANSIJIL TINGGI PERSEKOLAHAN MALAYSIA 20IO

MATHEMATICS T (MATEMATIK T)PAPER 2 (KERTAS 2)Three hours (Tiga jarn)

Instructions to candidates :

Answer all clttestions. Ansu,ers may be written in either English or Malay.

All nece.ssary working shoulcl be shown clearly.

Non-exacl numericol ensvtcr. \ rnal ,be giveu corract t t - t lhree s/gtr i l icurt t . f igure.s, ot ' t ) , iet{ecimalplace in the case oJ-angles in degrees, unless a dtlferent level o.f'ac:cw'ecri.r speciJiatlin the question.

Mathemalical tables, a li.st of maftematicalformulqe and graph paper are provided.

Arahan kepada calon:

Jcrwab semua soalun. Jawrrpan boleh ditulis dalam hohasa Inggeris otau haha.saMelayu.

,Jemuu kerju yttng perlu hencluklah dituniukkun tlcngun jclas.

,Lut'upon berangku tak tcpat boleh diberiknn betul hinggu tiga angko bererti, otau scttutempot perpuluhan dalam kcs ,sudttl dalarn darjah, kecuali aras kejituan yang lain ditentuksndalam soalan.

SiJir matcmtttik, sanarai rumus'mtttematik, dan kcrtas graf dibekulkan

This question papcr consists of 6 printed pagcs(Kertrs soalan ini terdiri daripadl 6 hahman bercetak.)

STPlvt TRIAL (MELAKA) 954/2* This question paper is CONFIDENTIAL until the exarnination is over..*Kertas soalan ini SULIT sehingga peperiksaan kertas ini tamat.

[Turn over (Lihat sebelah)CONFIDENTIAL*

SULIT*

CONFIDENTIAL*

In the diagram below, ABED is a trapezium

Given that CED is a straight line, with AB :

and zCBE:60' .

Find

(i) the length of CD in terms of d,

( i i ) z CAD.

with right angles at E and D.

2d. BC : 2'13d, LBAD = 30"

------)

,,s e 9l and lN isr*-')[+2r l

12 marla)

13 marksl

13 markrl

12 marksl

13 marksl

13 marlcs)

Given thar ;,= ( t '.]., f-t)

( -2 l l2 )

perpendicular to the direction vector

Find

(D the value of s,

(ii) the position vector of N.

, f r=- i+2j .

(} Prove that sin 0 + sin 3d * sin 5d = 8sin3 I - sin d.. . , t l . )(-L'r?LcLu'\ lHence, determine the values of 0, where 0 < 0 < 2n, which satisfy the

filtslaK't 'equation sin 39: sin 5d

A hunter sees a rabbit 12 m away due north, running in the direction of 060]

rvith a speed of 2.5 ms-1 If both of thern move in straight lines and the

hunter can run at a maximum speed of 5.0 ms'', find

(a) the direction in which the hunter must run in order to catch the rabbit

as soon as possible, [3 marksf

(b) the time taken by the hunter to catch the rabbit. 14 markl

CONFIDENTIAL*

In a chemical process for the decomposition of substance A, the rate of

.dxreaction, A where x moles per liter decompose in time I, is propottional

to the concentrati on of A at time l. Given that the initial concentration of I

is v mole per liter, show that the differential equation which describes this

chemical reaction is given bY

where k is a oositive constant.

#-k@-x),Find the general solution of this differential equation.

AC, AD and CD are tangents to a circle at B, S and R respectively. PQ is a

straight line that passes through D and is parallel to AC. BSp and BRe are

also straight lines. Show that

(i) I SPD is an isosceles triangle,

( i i ) PD: DQ.

In a country, licence plirtes for cars display four digits follor,ved by

trvo ietters. Find the number of dift-erent possible licence plates that

can be obtained by using this system.

Three different letters are chosen randomly from the word cl,oRox.

Find the number of ways these tliree letters can be chosen.

l8 marksl

15 murksl

16 marksl

(a)

l1anrks)

954/2

(b)

[2 marks]

CONFIDENTIAL*

8 The ages, x years, of 20 people attending an evening class are given by

lx = B2B, Zr, _ 37764(i) Calculate the mean and standard deviation of the ages of this group

of people. 13 marksl

(ii) One person leaves the group and the mean age of the remaining 19

people is exactly 4[ years. Find the age of the person r.vho left and the

standard deviation of the remaining 19 people. [3 mark,cl

9 The Intenral Revenue Service (lRS) has estimated that 60 percent of their

audits find that there is either no change in the individual's tax or that the

individual actually paid too much.

Using a suitable approximate distribution.

(a) f ind the probabi l i t l ' that in a srrnple of 1000 audi ts.600 in. l i r i , ] l ra ls

wil l not or,ve addit ional taxes if the IRS esrimate is rnre. [- l nl,u.r. i ]

(b) hnd the probability that in a random sample of 1,500 IRS audits,

fewer than 800 will owe additional tax. 14 mark,rl

l0 Records show that the length, in mm, of a particular type of building block

can be modelled by a normal random variable with mean 450 mrn and

variance 9 mm2.

(a) Find the probability rhat a randomly chosen block rvill be

(i) longer than 455 mm, 13 marksl(ii) between 44-r mm and 452 mm. lz narksl

(b) Find the probability that, for three randonily chosen blocks, at least

one will be longer than 455 mm. 13 rtrctrksl(c) Determine the value of k such that 95 o/o of the blocks will have their

lengths in the range ( 450 t k) mm. 14 mark.sl

CONFTDENTIAL*

I I The random number generator on a calculator randomly generates numbers

between 0 and L Let X be a random variable representing the numbers

generated.

(a) Write the probability density function for X. ll mark;l

(b) Find the probability that a number generated is greater than 0.27

given that it is less than 0.90. 13 marlcsl

(c) The calculator simulation experiments use the uniform distribution in

determining random numbers. To simulate a certain probability

situation, the calculator is prograrnmed to generate random numbers

between 10 and 20.

Let Y represent the numbers generated in this way, with probabiliry

density function.

Find,

( i ) P(13.5<Y<18.5), l2marksl( i i ) P(Y>1S|Y>15), f2marks)

(iii) the cumulative distribution function of Y, F(y) and sketch its

graph. 15 marksl

11.f(y) = l;'

1'o < Y s 2o

t 0, otherwise

CONFTDENTTAL*

ln an agricultural experiment, the gains in mass, in kilogram, of 100 cows

during a certain period were recoded as follows:

Calculate the mode for the gains in mass of the 100 cows.

Plot a cumulative frequency curve for the above data.

Calculate the estimates of the mean and stanCard t ier iat l trn of thc

above data.

Estimate the percentage of tire gains in mass of corvs in the range of

one standard deviation from the mean mass.

END OF QUESTIO|{ PAPER

(a)

(b)

(c)

[2 marlc]

[4 marksl

[5 nrcr i i ]

13 morks)

(d)

Gain in mass, kg Frequency

5 -9 Aa

t0-14 t2

ts - 19 29

20 1,1L1 3L

25 -29 11l-)

10 - '14 ' 7

l rn

i1'his question paper is CONFIDENTIAL until the examination is over.