1
PAGE 16
INNOVA JUNIOR COLLEGE
JC 2 PRELIMINARY EXAMINATION 2
in preparation for General Certificate of Education Advanced
Level
Higher 1
CANDIDATE
NAME
Civics Group
INDEX NUMBER
Mathematics
Paper 1
Additional materials: Answer Papers, List of Formulae (MF15),
Cover Page.
8863
10 September 2008
3 hours
This document consists of 8 printed pages and NO blank
page.Section A: Pure Mathematics [40 marks]1 Given that , show that
it can be simplified to y = x.[1]Hence solve the simultaneous
equations
giving the answers in exact form.[4]
2As part of an Art Project, a student designed a letterbox in
the form of a prism. The cross-section forms a pentagon with two
vertical sides of equal height, y cm, and two slant edges of equal
length, 5x cm. The remaining faces are rectangles. A rectangular
sheet of cardboard, ABCD, of area 1500 cm2, is folded to make the
surface ABCD of the prism as shown in the diagram. The front, back
and bottom surfaces are made of another material.If the length of
the letterbox is 30 cm and the width is 6x cm,
(i) show that the volume, V , enclosed by the letterbox is given
by
V = ,
[3]
(ii)determine the maximum value of V.
[4]
3 (i)Show that the equation of the tangent to the curve y = at
the point
where is . Hence determine the coordinates of thepoint where
this tangent cuts the y-axis.
[5]
(ii)Find .
[1](iii) Find the exact area of the region bounded by the curve
y = , the tangent to the curve at the point where and the
y-axis.
[3]4 (a) Sketch the graph of , giving the equations of any
asymptotes and
the coordinates of any points of intersection with the axes.
[3]
Hence, by sketching a suitable graph on the same diagram, solve
the inequality
.
[3]
(b) The diagram below shows the graph of . The curve passes
through the origin O and has a minimum point . The asymptotes of
the graph are x = 1 and y = 0. Sketch the graph of .
[3]
5The functions f and g are defined by
,
. (i)The function f has an inverse if its domain is restricted
to . State the least value of k. Find an expression for f(x)
corresponding to this domain.[3]
(ii)Sketch the graph of y = g(x). Show that the composite
function fg exists and find an expression for fg(x). State the
range of fg.
[5]
(iii)Solve the equation ff(x) = 4.
[2]Section B: Statistics [60 marks]
6 In a factory of 1500 employees, there are 150 management
staff, 300 administrative staff and 1050 production workers. A
survey is to be taken to check the factorys working conditions. A
sample of 30 employees is needed for the survey. The factory uses a
list with all the employees names arranged in alphabetical order.
(i) One method of obtaining this sample is to select every fiftieth
name on the name list. Identify the sampling method used here. Give
a reason why the sample obtained may not be a good representation
of the employees. [2](ii)Describe an alternative sampling method
which would be better in this case. [2]7The life of fluorescent
tubes made by a particular manufacturer has mean 758 hours and
standard deviation 12 hours.
(i)Find the probability that the mean life of a random sample of
100 fluorescent tubes is greater than 756 hours. State, giving a
reason, whether it is necessary to assume that the life of the
fluorescent tubes has a normal distribution.[3]
(ii)A large random sample, n, of fluorescent tubes is randomly
selected. Find the least value of n such that the probability the
mean life of the fluorescent tubes exceeds 762 hours is at most
0.01.
[4]8The volume of perfume in the bottles of perfume made by a
particular manufacturer has mean ml. A random sample of 50 bottles
is taken and the volume of the perfume in each bottle, x ml, is
measured. The data are summarized by
and
(i)Find unbiased estimates of the population mean and
variance.[2]
(ii)The manufacturer claims that. Test, at the 4% significance
level, whether the manufacturers claim overstates the value of
.
[3](iii)The null hypothesis is being tested. Find the range of
possible values of for which is rejected in favour ofat the 6%
level of significance.
[3]
9 A manufacturer of soft drinks launched a new drink. The table
shows the weekly advertising expenditure (x) and weekly sales (y)
during the launch period.Advertising, $x (in
thousands)1.603.050.452.000.752.65
Weekly sales, $y (in thousands)240450130324200365
(a) (i)Find the equation of the regression line of y on x in the
form , giving the values of a and b correct to 1 decimal
place.[1](ii) Interpret the values of and in terms of the amount
spent on advertising and the weekly sales. [2]
(b)Find the equation of the regression line of on . [1]
(c)Find the product moment correlation coefficient of the data,
and say what it leads you to expect about the scatter diagram for
the data.
[2](d)Some time later, the manufacturer launched another new
drink. The manufacturer wants a weekly sale of $ for this new
drink. Using the regression line of y on x, estimate the amount of
money the manufacturer has to spend on advertising. Comment on the
reliability of the estimate. [2]
10A bag contains 3 black balls and 7 red balls. A player draws
balls at random from the bag, one by one without replacement,
continuing until he gets a black ball. The tree diagram below
illustrates the possibilities for the first 2 draws.
1st Draw
(i)Draw the tree diagram until the fifth draw, showing the
probability for each branch.
[2]
(ii)Show that the probability the player gets his first black
ball on the fourth draw is .
[1]
(iii)Find the probability that the player needs at least 4 draws
to get his first black ball.
[4]
(iv)Given that the player needs at least 4 draws to get his
first black ball, find the
probability that he needs fewer than 6 draws to get his first
black ball.[3]
11The random variable has a normal distribution with mean 15 and
standard deviation 3. The independent random variable has a normal
distribution with mean and standard deviation .
(i)Find the value of the constant such that . [3] (ii)Given ,
show that the value of is 32.7, correct to
three significant figures.
[5]
(iii)Two independent observations of are denoted by and .
Find
.[3]
12(a)A bag contains red and green balls. An experiment is
conducted where n balls
are drawn one by one with replacement. The colour of the ball
drawn is noted before it is placed back in the bag. The random
variable X denotes the number of red balls drawn. When the
experiment was repeated a large number of times, it was found that
the mean number of red balls drawn is 10.8 and the variance is
4.32. Find
[5]
(b) Market research has shown that 4 out of 7 clients of a bank
make use of its internet banking services.
(i)A random sample of N clients is selected. Find the smallest
value of N required so that the probability the sample contains at
least one client who makes use of the banks internet banking
services is greater than 99.9 %.[3]
(ii)A random sample of 210 clients is chosen. Using a suitable
approximation, find the probability that more than 100 clients use
the banks internet banking services.[4]End of PaperInnova Junior
College 2008 Prelim 2 Exam H1 Maths
QnSolutions
1 (
( (
Let w be , so
Solving, or (rejected) ( and
2
For maximum volume,
Need to show that Volume is maximum when ,
Since , V is maximum when .
3y = =
and
The coordinates are
3ii(*)
3iii
=
=
4
Asymptotes:
Coordinates of points of intersection with axes: and
To satisfy the inequality,
5i
Since ,
,
ii
ii, therefore fg exists.
,
Range of =
iii
or ( rejected)
or
6(i)Systematic random samplingBiased as may not select enough
management staff.
(ii)
Stratified Random Sampling
StrataNo. of employeesManagement Staff(150/1500) x 30 = 3Admin
Staff(300/1500) x 30 = 6Workers(1050/1500) x 30 = 21 The employees
selected from each strata are to be chosen randomly.
7(i)Let X be the random variable denoting the life of a
fluorescent tube (in hours).
Since is large, by Central Limit Theorem
approx
(3 s.f.)
No. Since is large, Central Limit Theorem applies.
(ii)Let be the random variable denoting the mean life of n
fluorescent tubes.
approx by CLT since n is large
when
when
Hence the least value of n is 49 fluorescent tubes.
Alternatively: (Algebraic Method)
Let be the random variable denoting the mean life of n
fluorescent tubes.
approx by CLT since n is large
(
( (
Least value of .
8iLet be the random variable volume of contents in each
bottle.
Unbiased estimate for population mean
Unbiased estimate for population variance
iiTo test
against
Perform a 1-tail test at 4 % level of significance.
Under Ho, since is large, by CLT,
Using GC
Then p-value
Since p-value Reject .
We conclude that there is sufficient evidence at the 4 % level
of significance that .
Alternatively
Critical Region: Reject if
Test statistic:
Since , reject .
iiiTo test
against
Perform a 2-tail test at 6 % level of significance.
is rejected
9ai
aiiThe weekly sale of the soft drink is $89300 when no money is
spent on advertising.
There is an increase of $111.8 in weekly sales for every dollar
invested in advertising.
b
c
Since is close to 1, the points representing the data on a
scatter diagram would be close to a line with a positive
gradient.
dSub into
Then .
The manufacturer has to spend $3680 on advertising.
Since is outside the range [130,450], the estimate is an
extrapolation, it is not reliable.
10(i)
(ii)P(obtain 1st black ball in 4th draw) = =
(iii)P(needs at least 4 draws)
= 1 [ P(1 draw) + P(2 draws) + P(3 draws) ]
=
= =
(iv)P(needs fewer than 6 draws | needs at least 4 draws )
=
=
= =
11(i)Given:
and
(3 s.f.)
(ii)Consider :
(
( ( (3 s.f.) (shown)
(iii)Consider :
EMBED Equation.DSMT4 (3 s.f.)
12aLet X be the r.v. number of red balls. .
( , ,
bi (
(
A sample size of at least 9 clients is needed.
iiLet Y be the r.v. number of clients using internet banking out
of 210 clients.
= 0.997 (3 s.f.)
B
EMBED Equation.DSMT4
B
5x
C
EMBED Equation.DSMT4
R
1st Draw
READ THESE INSTRUCTIONS FIRST
Do not open this booklet until you are told to do so.
Write your name, civics group and index number in the spaces at
the top of this page.
Write in dark blue or black pen on both sides of the paper. You
may use a soft pencil for any diagrams or graphs.
Do not use staples, paper clips, highlighters, glue or
correction fluid.
Answer all the questions.
Give non-exact numerical answers correct to 3 significant
figures, or 1 decimal place in the case of angles in degrees,
unless a different level of accuracy is specified in the
question.
You are expected to use a graphic calculator.
Unsupported answers from a graphic calculator are allowed unless
a question specifically states otherwise.
Where unsupported answers from a graphic calculator are not
allowed in a question, you are required to present the mathematical
steps using mathematical notations and not calculator commands.
You are reminded of the need for clear presentation in your
answers.
The number of marks is given in brackets [ ] at the end of each
question or part question.
At the end of the examination, fasten all your work securely
together.
Innova Junior College[Turn over
5x
D
EMBED Equation.DSMT4
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[Turn over
[Turn over
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Red
Black
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R
B
2nd Draw
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R
[Turn over
O
x
y
B
1
y
A
y
30
6x
2nd Draw
Black
Red
3rd Draw
R
B
R
EMBED Equation.DSMT4
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4th Draw
B
R
EMBED Equation.DSMT4
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5th Draw
IJC/2008/JC2 8863/M/08
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