2
MJC/2007 JC2 Preliminary Examination/9740/02
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Section A: Pure Mathematics [40 marks]
1
The function f is defined by
2
f :
2,
0
x
x
x
x
.
(i)
Find the range of f.
[1]
(ii)
Sketch the graphs of
f( )
y
x
and
1
f
( )
y
x
on the same diagram. Label clearly
the coordinates of any intersection point(s).
[3]
The function g is defined by
g:
ln( ),
0
e
x
x
x
.
(iii)
Show that the composite function fg does not exist.
[1]
(iv)
The function fg exists if the domain of g is restricted to
e
k
x
. Find
K
, the
least value of
k
.
[2]
(v)
Find the range of fg, where fg is defined on [
K
, e].
[1]
2
Samuel owes the bank $15000 for his university fees which he intends to pay by monthly
instalments. He intends to pay $300 in the middle of every month. Interest is added to the
amount owed at the end of every month at a fixed rate of 0.5% of the outstanding
amount, after making the payment in the middle of the month.
(i)
Show that the amount he owes at the end of the second month after the interest
has been added is
$14545.87
.
[1]
(ii)
Show that, at the end of the
n
th
month, after the interest has been added, he will
still owe the bank
$ 60300
45300 1.005
n
.
[4]
(iii)
Calculate how many more months he will take to repay his debt, compared to the
case when no interest is imposed.
[3]
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3
MJC/2007 JC2 Preliminary Examination/9740/02
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3
A curve is defined parametrically by
2
x
,
2
ln 2
y
, where
0
.
Find
d
d
y
x
, expressing your answer in terms of
x
.
[2]
(i)
The tangent and normal to the curve at the point where
2
x
meet the
x
axis at
P
and
Q
respectively. Find the distance between
P
and
Q
in exact form.
[4]
(ii)
Determine the rate of change of
xy
when
1
3
and
x
increases at a constant rate
of 0.1 unit per second.
[4]
4
A line
l
has equation
4
3
0
2
5
1
r
,
.
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 Summer '07
 MOE
 Math, Normal Distribution, Standard Deviation, Antique shop, Preliminary Examination/9740/02

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