SMK ST.MICHAEL, IPOH
STPM TRIAL EXAMINATION 2009
MATHEMATICS sn 9SQ/I,95411 PAPER 1 Thrte hours
Instructions: Answer all questions. All working should be shown clearly, Nonexact
numerical answers may he given COi ecllo threo significant figures, or ons decimal place in the case
of angles in degrees, unless a different level of accuracy is specified in the question.
n _ w .
=tn(n+IX
2n
+
I
), L;r' =tn'(n+I)'
,...1 r _1 . ,...\
ã
Trapezi
SMK
ST.MICHAEL,
IPOH
STPM
TRIAL
EXAMINATION 2009
MATHEMATICS
sn
9SQ/I,95411
PAPER
1
Thrtehours
Instructions:
Answer
all
questions.
All
neces~ary
working
should
be
shown
clearly,N
on
exact
numerical answers
may
hegiven
COi ecllo
threosignificant figures,
or
onsdecimal
place
in
thecase
of
angles
in
degrees, unless
a
different
level
of
accuracy
is
specified
in
the
question
.
n_
w.
~:;r=tn(n+l)
,~:;r'
=tn(n+IX
2n
+I
),
L;r'
=tn'(n+I)'
,...1
r
_1
.,...\
ã
TrapeziumRule:
ff(x)dr
=
thlY,
+
2(y,+
y,
+ ..... +
y_
,)
+
y,]
ã
1.
Express(i)
(3+.fi)'
in theform
a+bfi
(ii)in(2.,Je)

~I{;)I{~)
in
th~
form
c
+
Ind
Where
a,h,e
and
d
are
rational
numbers.
2.
By letting
z
= x
+Iy
,find
the complex number
z
which satisfiestheequation
z
+
2z'
=2
15
I
[3]
(3]
where
z· denotes
the
complex
conjugate
ofz
,
[5]
,
3.
Use
the
trapezium
rule toobtainan approximation
to
!xIn
xcaby
using
5
ordinates.
[6]
,
xl2x9
4.
Express
(2x
I)(x'
+
3)
,
in partial
fractions.
Hence, evaluate
J
,
x'
2x9
~
.,
(2x
l)(x
'
+
3) '
gtVlng
your
answer
correct
to
2
significant
figures.
[6]
5.
Find the equation·
of
the circle that touches the line
y
=
5x
+3
at
thepoint (2,7) and itscentre is
located on the line
x2y=19
.
.
[6]
{1
Xx'l,x
4
6.
Sketch the graph
ofthe
functionfdefined
by
f(x)
=.
x
l

5x+6,x
>4
.
.
Find
limf(x)
and
limf(x)
.
Hence,show
thatfis
notcontinuous atx=4.Determine
whetherfis
...
+4
~
...
...
continuous
at
x
=
o.
[9]
7.
Given that
y
= (2 +
3x)e
,prove
that
~;
+4:
+4y
=0
(6)
8.The polynomial
p(x)
=
x·
+
ax'
7x'
4ax
+b
bas.
factor
x+
3 and, when divided by
x3,
.bas
remainder 60.Findthevalues
of
a
and
b,
andf.ctorise
p(x)
completely.[9]
Using the
s u b s t i t u t i o ~
'
y
.!
,
solve
the equation
12y
_8
y
3
_7y
l
+
2y+
1
=
O.
x
[3]
A=(~
2
l
.
The matrix
Ais
given
by
3
2
(a)
Find
A'.
[3)
(b)
Given
that
A'
=
rnA
+
nI
,
where
m,
values
of
m
and
n.
n
are
integers
and
lis
a
3
x
3
identity matrix,
find the
[3]
(0)
Hence,or otherwise find tha
inverse
ofA
.
[4)
10.
(a)
Find
the
sum
of
he
following
series:
I
x
4+2x
7
+3
x
10+
............
..
...+
n(Jn+
1). [6)
(b)
In
a
geOmetric
progression, the first term
is12 and
the
fourth
tenn
is
~
.Find
the leastvalue
2
ofn
for
which the magniiude
of
the
difference between
S,
and
S.
is
less than
0.001.[4]
'
11
.·
Expand
(1
+
x)~
in
ascending power
of
x
until the
tenn
in
Xl
.
By
takingx
=
_1_ ,
findthe
40
I
I
I
approximation
for
32.S;
correct
to
Jour
decimal
places.
If
the
expansion
of
+ax
and
(1
+
xl
i
1
+bx
..
~
20
3
are
the same until
the
term
in
x',
find
thevalues
of
a
and
b.
Hence, show
th.t
32.8'
~
.
12]
10112.
Sketch
in
separate
diagnuns,
the
graphs
of
(a)
)I=3x'
+4~+2
.1
(b)
)I
=
.;'::=
3x'
+4.+2
stating,
in
each
case,
the coordinates
of
my
turning points,
and
for
(b),
show the shape
of
the
curve
for
Jarge,
positive
values
of
x
and
large
,negative values of
x.
By
sketching an additional graph onone
of
the
above
graphs, or otherwise,show that
the
.equation
3x'
+
4x'
+
2x
=
6
has
one,
and
only one,real root.
Show
also
that
thisroot
Ues
between 0
and
1.
END
OF
THE
QUESTION
PAPER
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[12)