Sect ion A: Pure lV{athematics [40 marks]

3x3 +lLxz +lExoress ---------------- ln partlal tracttons.

' (x - I ) (3x ' + 1)t4]

13lHence find the integral f (x- l ) (3x2+l)3x3 +LZxz +l

The diagram belorv 5[6rvs the cross-section of a single track funnel entrance for

steam-engine in a model rrain system. The elliptical entrance of the tunne I can be

modelled by the equation

,2 +Ay2+By+C=0.

The highest point of the tunnel entrance is 7 cm dir,lctly above the origin at ground

level. The traclq mountcd on track bed PS, is cenbed about the origin A train

carrying containers which measured 2 cm wide may stack goods as high as 5-8 im'

Sotve for the constantsA, B and C, to 4 decimal places. Hence find the lenglh of the

widest part of the tunnel, to the nearest 0- l cm' t6I

4.3 cm

A nglo-Chinese Jnior CoIlege

tl2 l,4athemarics 97a0; 2OO8 JC 2 Preliminary Examination Paper2

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3 The rth term ofa sequence is given by u, = r2- ' where r : 1,2, 3, - . . .Write down the first 4 terms of the sequence-It is given that

Zr, :2- f (n) ,. r=l

rvhere f (n) is an expression in n.

Byevaluat i "g iu, forn=|,2,3and4,deducetheformulaforf(n)

With this formula of f (n), prove, by mathematical induction,

' t , r - ' - -z-r(n) t r l

4 Functions f, g and h are defined by

f :xHlr t -or+sl for xe R,2<x(6,

g:.rHh(r t ) for re lR., x;cO,

hrr* I

2-x

Anglo{hinese Juior College[{2 Mathematics 9740: 2008 JC 2 Preliminaq, Examinarion }aper 2

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for-xelR,x<2-

(i) Give a reason why f does not have an inverse- iu(ii) State the targestpossible domain of f in the form lo,b|, a,6e IR, flor whirn ihe

inverse function f exists. Hence, define f-t in similar form.(iii) Explain why the composite function hg does not exist.

function hr exist. Hence, state the range of hr.

( iv) Another funct ion r is def ined by r :xr+ h(r t ) for xe R, -k <x<k. r-*0.

Given that t e IR, state the maximum value of t for which the conrp,'site

t4lt l l

t3lI

The diagram above shows a simpte shetter designed by an engineer- It consists of

three vertical'columns., OE, AF and.BG with their bases on the horizlnul ground- The

point o is taken as the origin and the points E, F and G have position !'ec-to's given

respect ively by ":

6K f :8i + lOk and g: 7 ' + 4i + 4lL

(i) Given ttrai tfr! ptane .EFG forms the roof of the shelter, show that rhe cartesian

equat ion of the iane EFG is -2x +3y+42=24 I2l-

(ii) Find the angle befween the plane EFG and thahorizontal ground' t?l

tii i i The enginelr has to drill a hole on the roof of the shelter in order to install a [amp'

, , , ,, ' , ,, ,q,"

n the ground with position vector

(iv) M is a point that lies on the line segment E^F- The engineer decided to construct a

vertical partition that intersects plane EFG along the line segment GM- Given that the

partition has a cartesian equation a{ 2x+ y =d ' find the vector equation for line CM'

pl(*,) Find the shortest distance from I to the vertical wall partition' t3l

, Section B: Statistics [60 marks]

6 (a) The manager of a popular supermarket wishes to determine the effect of a new'store

Iayout on the customers- Cuitomers are coming out of the supermarket in i steady

stream-

Ai tf the manager has no information about the profile of his customers, explain

rvhether it iould be possibte to use stratified sarnpling, random sampling or

systematic sampling to choose the sample- Justiff yoor choice ' I2l

(ii) If the manager-decides to use the first 100 customers on that day as his sample'

give one reason why this sample may not be appropriate' tl l

(b) A s,irveyor is required io choose a person from a HDB estate' He decides to choose a

household at random and then randomly choose a member of that household' Explain

u,hether this method will produc" u p.rion chosen randomly from the estate' tzl

A nglo -Chi nese Junior Co fl ege

H2 Mathematics 9740: 2008 JC 2 Prcliminary E::amination Paper 2

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The mean l i fespan of dogs is normal ly distr ibuted and the local vet c laims that thismdan is 9.5 years. The v.et conducted a t-test at the 50% significance level using arandom sample of l0 dogs. He found that there is sufficient evidence that the meanlife time is not 9-5 years- State, giving a reason, whether the follorving statement is

(1) neeessary true; (2) necessarily false; or(3) neither necessarily true nor necessarily false:

There is suflicient evidence attheT-54/o level that there has been a change in the rn*lnof the distribution" I7l

A study of the number of male and lemale children in families of inhabitants on aplanet very far a\\'ay reveals that thc probabitiry that a baby' girl ' is bom is 0.995.500 babies are chosen at random. Use a suitable approximation to find the probabilitythat at most 495 of them are sirls. -

t3l

A particular monkey at a zoo is extremely intelligent- He is able to recognise shapes-In order to him from getting bored, his r gives him 5 circles, 5 squares andJ rlangles to ptay wrtn- AII me snapes nave olttereni sEes.

(i) [f the monkey tries to arrange the 5 circles and ihe 5 squares in a circular fashion andthat the circles and squares have to be alternated, find the number of dififerentpossible arrangements. [2]The monkey finds that he does not like circles, so he takes all the squares ancftriangles and tries to arrange them in a row such that no fwo triangles are placed nextto each other. Find the number of different possible arrangements. tzl

l0 Bertie found five strav kinens and brought them home. The-v are of fir,e diFferentcolours, namely, black, rvhite, orange, grey and brotvn. ln order to bring the kiften:s t,:the vet for an examination, Bertie prepared five cages, one for each kitten, with thecolour of the kinen indicated on the cage. As Bertie was in a hurry, he placed eachkinen in a cage at random, with one kinen to each cage.

(i) Find the probability that the orange kitten is in the correctly labelled cage given thatthe black kitten is in the wrong cage. I7l

(ii) Show that the probability that both the orange kinen and the black kinen are in cageswith labels not rnatching their colours is 0-65. i2]

(iii) If Bertie needs to make one hundred trips to the vet in the years to come rvith his fivecats, estimate the probability that both the orange cat and the black cat are placed inthe rvrong cages on sixty occasions using a suitable approximation. t3l

Anglo thinese J w i or CollegeH2 Ma*rematics 9740: 2OO8 JC 2 Prcliminary Lramination Paper 2

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I1 A fruit grower grolvs both red and green apptes which have masses that are normally

distributed. The mass of a randomly chosen red apple has mean 75g and standard

deviation I7.5g. The mass of a randomly chosen green apple has mean of 55g and

standard deviation I 0.59.Find the probability that the tolal mass of J randomly chosen green apples exceeds

fwice the mean mass of 3 randomly chosen red apples. I3l

A red apple is considered "undenveight" if it weighs less than 70g. Red apples are

packed into bugt of 10 for transportation to a supermarket. A bag is considered to

i-,uu. p^ssed the quality test if it contains tess than 2 "underweight" apples- Calculate

the piobability that in a randomly chosen batch of 20 bags of red apples, all the bags

fait the quality test. I4l

A marmalade manufacturer produces thousands ofjariof marmalade each week- The

mass, r grams, of marmalade in a randomly chosen jar has mean 25g' Following a

slight adjustment to the fil l ing machine, a random sampld 6f 50 jars is taken and the

muss of marmatade in the jars is measured. The masses are summarised by

lx=173o and I(:-20)t:1129' Correct to 3 significant fltgures'show that the

I2l

(i)

(i i)

t2

unbiased estimate of the population variance is l-45-

The manufacturer wishes to test whether there has been a change

disf-ribution atthe 5%o significance level.

(a) (i) Carry out the test above-

in the mean of the

I4l

t3

the test to be valid. trl

(b) Assurning that the mass of the mar-matade in a randomly chosen jar is normally

distributed, explain whether a z-test or t-test should be used when a random sample of

l0 jars is used. {21

In a certain country, the number of deaths due to old age in the northern and southern

parts is recorded separately- It is found that on the average, there are 2 deaths in the

northern parts in a day and 3 deaths in the southem parts every 3 days.

Find the probability that there are nine deaths recorded in the country over a period ofl? l

\ r l : r , 'c L- t

The number of deaths in the northern parts is recorded over periods of l0 days each'

Estimate the probability that the average number o[ deaths over 50 such periods is

befween 2l andl0 inclusive, explaining clearly how you arrive at your answer- t3l

Using a suitable approximation, find the teast value of n such that the probability that

there are at least 2j deaths in the southern parts exceeds l'/" over a period of2n days'

You may assume that n is an integer greater than 5' I7l

Anglolhinese Juior Col Ieg e

F[2 Malhematics 9740'.2008 JC 2 Preliminarl' Examinatioo Papcr 2

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la {a) State, giving a reason, whether each of the lollowing statements is true or ialse.(i) A set of bivariate data has a Iinear product moment correlation coefficient of +1.

This impties a'linear relarion holds for rhe whole popularion_ tl j(ii) The linear product moment correlation coefficient betlveen 2 random variables X

and i' is zero. This implies that the variables x and rare not correlared. t l l(b) The amount of a chemical, r (in grams), r,aries with tirne r (in minutes) for r,r'hich a

particular chemical reaction has taken place and is given by the forrnula x = 4,3 tu

,rvhere -ro and kare constants. The values of l may be considered to be exai:t v,,hi[ethe values ofx are subiect to experimentals ot -r are su tal error_

t (minute)' 0-2 0_4 0.6 0.8 t .0r {grams) 3.22 r ,63 0.89 0.4 t 436

The variabley is defined by y: ln r.(i) Using a least squares method, calculate the equation of the appropriate regression

line and hence give estimates of xn and,t- t3l(;i) From the equation estimate the decrease in7 when l increases by l- tl l

(i i i) Use a regression line to give the best estimate of t when x : 1.5. Explain yourchoice of the regression line used. I?l

imeasured. [t is given that the linear product moment correlation coefficientbefween u and y is 0-9 and the least squares regression line of u on ;, is f,u = 2-4y + l0 - Find the least squares regression line ofy on z. i3 l

Anglo t hincsc -t un ior Collc ge[{2 Mathemarics 9']t0:20O8 JC 2 Prcliminrrl, ExminariJn paper 2

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