SEK
KEB MEN SULTAN
ISMAIL,
JOHOR
BAHRU
PreU
95412
MATIDMATICS T
TERM
2
TRI,AL
EXAMINATION
2OI4
MARKING
SCHEME
I
l.
The
functions/is
defined
as
:
_
1x+lxl
, x*0
flx):
I
z
12,
r=0
(a)
Without
using
graphs,
determine whether the
function/is
continuous
atx:
0.
(b) Skefth
the
graphs
of
the
firnctions/.
[5 marks][2
marks]Marks
(a)
(0'
.fli
='2,
(r,
r(0
r=0
r)0
I
, /(r)
=,\3
o:
o
lim
f(r)
=
lim
r:0
f+0*'' '
r+0*
lim
f(x)
=
lim.
f(x):0
x+O' ' x+Ol'
frg/(r):
o
fQ)=2
1
II
}gftr)
+f(o);;fis
not continuous
at
x
:
0
I
(b)
1+1
)
A
cylinder
is inscribed inside
a
sphere
of
radius
r.
Given
that the
volume
of
the
cylinderis
a
maximum,
show that
the
ratio (volume
of
sphere)
:
(volume
of
cylinder)
is
r/5 :
1.
[9
marks]
,':r'T
V=
ttxzh
v:nh(rz
+)
Y=
n(r'n
T)
1
To:o(
T)
IVolume
ma:rimum,
ff=
O
n(,'lfJ
=
o
h:
*4r
v3
'
h> 0,
h:
h,
I
1
o=h,,r=n(h4V#)
V
=
4=TEr3
3V3
I
d.zv
3
,
:hhz
2
o:h',#::G')
:{3r(<o)
.'. The
mocimurn volume of cylinder
is
ftzr3
I
I
(volume
of
sphere)
4z
Tr
3
1
\tr
(volume
of
cylinder)
4e
ftT
V3
a
:
VS
I
II
3.
Evaluate
th
following
@I:YO,
O)
f
6x3e'
dx
[3
marksJ
[5
marks]Marks
(a)
IiiUrolar:I%',
I
(ln
e)2
(lnl)2
=
22
I
1
2
1
z
3(b)
fi
ortr'
a*
:
li
3x27zxe* 1d,x
Letu=3x2,9*=6x
n:t2xex'dx=er'
=13x2e,'13
I:6xe, dx
l+l
:
l3xz
ex2

ger lloI
=
13(4)e4

3enl
[0

3eo]
I
=9e+
+3
1
4.
(a)
Find
the general
solution
ofthe
differential
equation
dv
fr:kdx+fi
where
t
is
a
constant,
giving
your
answer
in
the
formy
=
f(r).
[5
marks]
(b)
The
gradient
at
any
point
P(x,
.y)
of
a curve
is
proportional
to
the
surn
of
thecoordinates
of
P.
The curve
passes
through the pornt
(1,
2)
and
its
gradient at(1,
2)
is
4.
Find
the equation
of
the
curye.
[4
marksJ
(a)
*nr=
*
I
Integrating
factor
=
sl
n*c
=
etu
I
ek
9kvek*lsrkx
dx
d
Ero u)

k* o'
lekx
=
f
ftysx*6a
I
lekx=xetufetudx
1
lekx =
xew
 t, *,
=
X
 Y
gat'x
I
o)H <,
+
D,#=
Kx
+
y\
(1,
*2),#=
*0,
Kl
21=4
k=4
I
2: l


I
gs+(t)
4
3
u
4 n
II
=
x
i2t+1xt1
I
5.
By
using the
standard
Maclaurin
series,
find
the
first
fow
tenns in
the
series
expansion
tt(g).
State
the
set
of
the values
ofx
for which
the expansion
is
valid.
[4
marksl
(ff):]n1r+r4h(r
r)
1
1
=
)
3xry+g$+...
xry+g$+ '
I
=X*
lr,
*lr,
Tr*
*
,
+;*t
**.
:,
lr'
**r'
#rn*...
I
rtlr
lr
,
T
<
r
<
il
I
6.
Given
ttrat
the equation ln
x + x2 =
8
(a) Show that the equation
has
only
one
real
root.
[2 marks](b)
Verifi,
by
calculationthatthis
root
lies
betweenx=2
andr
=
3.
[2
marks](c) Use
the
NewtonRaphson method
to
find
the
of
the equation ln
x
*
xz

8o
giving
your
answer
correct to
two
decimal
places.
[4
marks]
(a)
lnx:
8

x2
.y

ktx
;
the equatisn
has
only
one
real
root.
I
I(b)
lnx+rz8:0
Letf
(x)
=
lnr
+
xz

8
f
(2)=
ln
2
+
22
8:
3.307
(<
o)
f(3)
:ln
3
+
32

8
=
2.099
P
o)
/is
continuous
on
[2,
3l,fQ)
and/(3)
have
opposite
signs,
;
The
root of this
equation lies
between
x
=
2
and
x =
3.
1
I
(c)
f
(x)
=
lnx
+
x
8
.f (x):L.*u
lnrr,
*
xnz

8
Xn+L
=
Xn
{*
z,^
ro
:3
xt
=
t

tt Jili
t
:2 6686
x
2 67
(2 d
p )
xz
=
2.6686

ln
2.6686
+
Z.668O2
8
*r*
+2(2s6g6)
:2.6506
x
2.65
(2
d.p.)
h
=
2.6505
x
2.65
(2
d.
p.)
:.
x:2.65
(2
d.p.)
I
I
1
I
+