I. Using alcebraic laws ofsct. prove that I(Au B) C]u[(Au B) C'] B = AB. [6 marb]
2.
.Ii
Find 1 3x dx
ZTx' +1
[4 marb]
3. On the same diagram, sketch the graphs of y = x: I and Y  2x  J. [3 morks)
A region R is bounded by the two graphs and the yaxis. Find the area of region R.(7 marh]
A solid is fonned by rotating the region R completely about yaxis. Find the volume ofthc
sol id fonned. [5 marb]
4. The functionfis defined by fIx) = { ~ . x ~
I.
Using alcebraiclaws
ofsct.
prove
that
I
(A
u
B)
C
]u[
(A
u
B)
C
']
B
=
AB.
[6
marb]
2.
.Ii
Find
1
3x
dx
ZTx+1
[4
marb]
3.
On
thesame diagram,sketchthegraphs
of
y
=
x:
I
and
Y

2x

J.
[3
morks)
A
region
R
isbounded
by
thetwo graphs
and
theyaxis.
Find
the area
of
region
R.(7
marh]
Asol
id
is
fonnedbyrotating the region R
com
pl
etely aboutyaxis. Findthe volume
ofthc
sol
id
fonned.
[5
marb]
4.
The
functionfis
definedby
fIx)
=
{
~
.
x
~
3.
O
x
=3
(i)Find
! ~ T 
fIx)
.
! ~ T 
f<x)
nndhencedctennine
if
the
functionfis
continuous
atx
3.
[3
marb]
(ii)Sketch the
graph
o
ff
[2
marb]
S.
Prove
that.
for
aU
yaJues
of
k.
the
line
kx
+
y

8k
is atangent
to
the
hyperbola
xy
=
16
.
Hence,find
the
coordinates
of
the point
of
contact.
(S
marks]
6.Sketchthe graphs
of
y=
....!.
andY
=I
2x
llonlhesamediagram
.Hence,
so
lv
cthe
Ixl
inequality
IZx
I
I>
~
.
[7
marb]
Ixl
7.
[Y2
Fin
dthevaluesofx.yandzif
4..1'
;4
x'
4<
1
x issymmetric.
2:
1
[5
marb)
8.Giventhatf(x)
...
f
+
a:/
+
8x
+
b,
where
a
and
b
areconstant.
f
'(x)

0when
x
z:
2.When
1(
..1')
is
divided
by
x
+
2,the remainder is2.
9.Dctenninethe\'alues
of
a
and
b.
(8)
Show
that theequati
onf(x
)
0 has
on
ly
one
real root and
find
theset
of
values
of
x
suchthatf(x}
>
0(b) Express
x
+
4
in
partial
fructions.
fIx)
(I I
Given
thematricesA
P 4
73Find
the
values
of
p.q
and
r.
I)
(I
and2
II
[5
marb]
papercollection
954
/1
10
.
Hence.so
lv
e
the
simultaneousequations
x+yz
::2
2y
+
Z
a.
5.r
9
7x

3y
:
14
Giventhat
y
z>
,
showthat
(I+X2
)d
1y
+
3 x ~
=
O
dx'dx
2009
[4
marks)
) I.Express
_
1
_
aspartialfractio
n.
(r
+
I
)(r
+
2)
(4
marks)
,.J
I
Hence
or otherwise.
find
the
value
of
)

.:=,
(r
+l
)(r
+2)
,..2..
I
Find
alsothe
value
of
lim
L

.......
.
...
41
(r
+
l
)(r
+
2)
(6
marks)
[4
marks)
12
.By usingsketchi
ng
graph.
show
that
theupproximatevalue
of
the root
can
be
derived by the
following NewtonRaphson
it
erativefonnula
X
.
.1
=
X
.

f(x
,,
)
.
[4
marks]
f(x
.)
Sh
ow
that
the equation
Ian
x
2.r
hasa root
in
the
interval(;.
~).
[3
marks]
Hence.
find
the
root
of
me equation for
xe
( ~
.
%).
correcttothree decimalplaces.
[7
mor.tsJ
papercollection
1.
Ex
press
3sinOcosO
inthefonn
Rsin{Oa)
,with
R>O
andO
o
<a
<
90
°.
Henccsol
veth
eequation
3sinOcosB
=I.
for
0°
<
0<360°.
18
J
2.
Byusingthesubstitution
y
=
vx.
show
that
thegeneral solution forthe differentialequations
~
=
x
+
2}'
is
AX
l
=
x
_
y)
l
,where A
is
a arbitrary constant
I
5
I
dx
3x
3.
In
a chemica1process
to
obtain substanceBfrom substance
A.
everymolecule
of
A
is
directlychange to molecule
of
B.Therate
of
increment
of
molecule Batany instantarc directly
proportional
to
the number
of
molecule Aat thatinstant. Initially the number
of
molecule
of
A
is
/,1
andthe number
of
molecule Bat
t
minutesafterthe processbeganis
x.
By
assuming
x
as
the
continuousvariable.Sbow
that
the differentialequation representing
th
t'
chemical process
is
~
=
*(M

x)
,
withkasaeonstant
dl
If
x
Owhent
=
O.showthat
x=M(1
e
b)
Sketch thegraph ofx versus
t.
18
I
4.
Ship A moveswitbvclocity (
30~+40j)
kmh'
and shipB moveswith velocity
(15~+12j
)
kmh'.
Initially.the
po
sitionvector
of
shipBrelative
to
ship Ais
(5
~
1

12;')
km.
Calculate themagnitudeanddirection
of
ship A relative
to
shipBandthe nearest distance
bctwccp
ship A andshipB
1
5
1
5.
(a)If
!
and
!!...
aretheposition vectors
of
twopoints A and B
re
spectively.andPis a poi
nt
no
+mb
which divides
AB
in the ratio m:
n.
Show thatP has
th
eposition vector
r
=

m+n
PJ
(b)
Findavectorof theequation
oflhe
straight linepassing
through
the
point A with position
vCClOr
a
=
2
i
+.i
and is parallel
to
thevector
~
=5
~
2
j
I'I
papercollection
6.
lbe
figure
shows aquadrilateralwhichinscribed two isosceles triangles
ABC
and
ADE
withbase
AB
and AErespectively. Each trianglcsh
asthe
base
angles
of75
o.
Be
and
DE
areparallcland
equal
in lengths.
~ , ~    
                  ~ ;
Sh
ow
that
(a)
Le
BE
=
LBED
=
90'
(b)
ACD
is
an
equilateral triangle[4)
r
31
7.
The random
variableX is normally distributed
with
mean ã
jJ
ã
and
variance
400
.
It
iskn
own
that
I'(X
>
1159)
,;
0.123and
P(X
>
769)
~
0.881.Determinethe range
of
the value
of
I'.
15)8.
In
a certain
city
.records forburgla
ri
es
s
how
th
at
theprobability
of
an arrest
in
one
week is 0.42.l
bc
probability of
an
arrest and
co
nviction
in
that week is0.35.(a)Findtheprobability thataperson arrested for aparticularweekforburglary
will
be
co
nvicted.
(b)
What is
the
probability
thatthere
wi
ll
be
3
wee
ks
of
su
ccessf
ularrested and conviction inonc
particular month. .[5}
9. The
continuous
randomvariable Xhasprobability dcnsityfunctiongivenby
{
k
(x+I),
O';x
S2
lex)
=
2k
2S
x
S
b.
()
otherwise
Where
k
and
b
arc constants.(a) Giventhat
P(X
>
2.5)
=
.!..
find
the
value
of
theconstants
k
and
b.
6
(b)
Sketch
the
graph
ofthc
probabilitydensityfunction
[(x).
(e)Find
1'(1
.5
S
X
S2.5)
16)
1
31
12
1
papercollection