STPM Maths T Sem 1 Trial 2014 P1 Port Dickson
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Transcript of STPM Maths T Sem 1 Trial 2014 P1 Port Dickson
13
14
-1 S
MK
TIN
GG
I PO
RT
DIC
KS
ON
Sectio
n A
[45
mark
s]
An
swer a
ll qu
estion
s in th
is section
.
1.
Fu
nctio
ns f an
d g
, each w
ith d
om
ain R
, are defin
ed b
y f : x
→ x
+ 1
, and
g →
│x│
.
a) W
rite do
wn
the ex
pressio
n fo
r f -1(x), an
d state, g
ivin
g a reaso
n, w
heth
er g h
as an in
verse.
[ 2 m
arks ]
b)
Sk
etch th
e grap
h y
= g
of
[ 2 m
arks ]
c) F
ind
the so
lutio
n set o
f the eq
uatio
n g
of = fog
[ 2
ma
rks ]
2.
Th
e nth
term o
f an arith
metic p
rog
ression
is Tn .
a) S
ho
w th
at �� � �� ��2 �� �� �� is the n
th term
of a g
eom
etric pro
gressio
n.
[ 3
ma
rks]
b)
If Tn =
�� �17��14, evalu
ate ∑��
����.
[ 5
ma
rks ]
3.
a) Giv
en th
at zi =
5 +
i and
z2 =
– 2
+ 3
i
i) S
ho
w th
at │z
1 │2 =
2│z
2 │2
[ 1
ma
rk ]
ii) F
ind
arg (z
1 z2 ).
[ 3 m
arks ]
b)
Determ
ine th
e squ
are roo
ts of 1
6 –
30
i, in th
e form
of a
+ b
i.
[ 5
ma
rks ]
4.
a) Sh
ow
that th
e equ
ation
� 1 2 �3 2 6 �111 �2 7 �
� �� ! =
� "#$ !
has so
lutio
ns o
nly
if r + 2
q –
5p
= 0
. Describ
e the ty
pe o
f system
of eq
uatio
ns an
d so
lutio
ns.
[ 5 m
arks ]
b) H
ence, fin
d its so
lutio
ns if p
= r =
1 an
d q
= 2
.
[ 3
ma
rks ]
5.
a) Sk
etch th
e grap
h o
f the ellip
se with
equ
ation
of x
2 + 4
y2 =
1 [ 2
ma
rks ]
b) P
oin
t P lies o
n th
e ellipse an
d N
is the fo
ot o
f the p
erpen
dicu
lar from
P to
the lin
e x =
2.
Fin
d th
e equ
ation
of th
e locu
s of th
e mid
po
int o
f PN
wh
en P
mo
ves o
n th
e ellipse.
Describ
e the ty
pe o
f curv
e ob
tained
for th
e locu
s ob
tained
.
[8
ma
rks ]
6.
a) Fin
d th
e exp
ansio
n o
f ��%& ' (��)& *
in ascen
din
g p
ow
ers of x
un
til the term
s x3. [ 3
ma
rks ]
b)
If p=
– �+ , an
d q
lies in th
e interv
al [0, 9
], find
the larg
est po
ssible co
efficient o
f x3 .
[ 3
ma
rks ]
Sectio
n B
[15
mark
s]
An
swer an
y o
ne q
uestio
n in
this sectio
n.
7.
Giv
en th
at f(x) =
x3 +
px
2 + 7
x +
q, w
here p
, q are co
nstan
ts. Wh
en x
= –
1, f ’(x
) = 0
.
Wh
en f(x
) is div
ided
by (x
+ 1
), the rem
aind
er is –1
6 . F
ind
the v
alues o
f p an
d q
.
[ 4 m
arks ]
a) S
ho
w th
at f(x) =
0 h
as on
ly o
ne real ro
ot. F
ind
the set o
f x su
ch th
at f(x) >
0 . [ 7
ma
rks ]
b)
Ex
press &%,-�& in
partial fractio
ns.
[ 4
ma
rks ]
8.
Th
e po
sition
vecto
rs a, b
, and
c of th
ree po
ints A
, B an
d C
respectiv
ely are g
iven
by
a =
i + j +
k , b
= i +
2j +
3k
, c = i –
3j +
2k
.
a) Fin
d a u
nit v
ector p
arallel to a
+ b
+ c.
[3
mark
s]
b) C
alculate th
e acute an
gle b
etween
a an
d a
+ b
+ c. [3
mark
s]
c) Fin
d th
e vecto
r of th
e form
i + λ
j + µ
k p
erpen
dicu
lar to b
oth
a an
d b
. [2 m
arks]
d) D
etermin
e the p
ositio
n v
ector o
f the p
oin
t D w
hich
is such
that A
BC
D is a
parallelo
gram
hav
ing B
D as a d
iago
nal.
[3
mark
s]
e) Calcu
late the area o
f the p
arallelog
ram A
BC
D. [4
mark
s]