Add Maths Final Form 4 2011 P1
Add Maths Final Form 4 2011 P1
Add Maths Final Form 4 2011 P1
r
r
r a
r
r a
n n
13 1 ,
1
<
r
r
a
S
CALCULUS (KALKULUS)
1
dx
du
v
dx
dv
u
dx
dy
uv y + = = , 4 Area under a curve
(Luas di bawah lengkung)
=
}
b
a
dx y or (atau)
2
2
v
dx
dv
u
dx
du
v
dx
dy
v
u
y
= = , =
}
b
a
dy x
3
dx
du
du
dy
dx
dy
= 5 Volume generated
(Isipadu janaan)
=
}
b
a
dx y
2
t or (atau)
=
}
b
a
dy x
2
t
SULIT 3472/1
3472/1 2011 Hak Cipta PPDKT SULIT
4
STATISTICS (STATISTIK)
1
N
x
x
=
2
=
f
f x
x
3 = =
N
x x
2
) (
2
2
x
N
x
4 = =
N
x x f
2
) (
2
2
x
f
f x
5 C
m
f
F N
L m
(
(
(
(
+ =
2
1
6 I = 100
0
1
Q
Q
7 I
=
i
i i
W
I W
8
)! (
!
r n
n
r
P
n
=
9
! )! (
!
r r n
n
r
C
n
=
10 ) ( ) ( ) ( ) ( B A P B P A P B A P + =
11 1 , ) ( = +
= = q p
r n
q
r
p
r
C
n
r X P
12 Min = np
13 npq o =
14
o
=
X
Z
GEOMETRY (GEOMETRI)
1 Distance (Jarak)
= ( ) ( )
2 2
1 2 1 2
x x y y +
2 Midpoint (Titik tengah)
( x , y ) = |
.
|
\
| + +
2 2
2 1 2 1
y y x x
,
3 A point dividing a segment of a line
(Titik yang membahagi suatu tembereng
garis)
(x , y ) =
|
|
.
|
\
|
+
+
+
+
n m
my ny
n m
mx nx
2 1 2 1
,
4 Area of triangle (Luas segitiga)
( ) ( )
1 2 2 3 3 1 2 1 3 2 1 3
1
2
x y x y x y x y x y x y = + + + +
5
2 2
x y = + r
6
2 2
xi yj
r
x y
+
=
+
SULIT 3472/1
3472/1 2011 Hak Cipta PPDKT [ Lihat sebelah
SULIT
5
TRIGONOMETRY (TRIGONOMETRI)
1 Arc length , s = r
Panjang lengkok , s = j
2 Area of sector ,
2
1
2
A r = u
Luas sektor ,
2
1
2
L j = u
3 sin
2
A + cos
2
A = 1
sin
2
A + kos
2
A = 1
4 sec
2
A = 1 + tan
2
A
sek
2
A = 1 + tan
2
A
5 cosec
2
A = 1 + cot
2
A
kosek
2
A = 1 + kot
2
A
6 sin 2A = 2 sin A cos A
sin 2A = 2 sin A kos A
7 cos 2A = cos
2
A - sin
2
A
= 2 cos
2
A 1
= 1 2 sin
2
A
kos 2A = kos
2
A - sin
2
A
= 2 kos
2
A 1
= 1 2 sin
2
A
8 sin (A B) = sin A cos B cos A sin B
sin (A B) = sin A kos B kos A sin B
9 cos (A B) = cos A cos B sin A sin B
kos (A B) = kos A kos B sin A sin B
10 tan (A B) =
tan tan
1 tan tan
A B
A B
11 tan 2 A =
2 tan
2
1 - tan
A
A
12
C
c
B
b
A
a
sin sin sin
= =
13 a
2
= b
2
+ c
2
2 bc cos A
a
2
= b
2
+ c
2
2 bc kos A
14 Area of triangle (Luas segitiga)
=
2
1
ab sin C
SULIT 3472/1
3472/1 [Lihat sebelah
SULIT
Answer all questions.
1 Diagram 1 shows the relation between set X and set Y in the graph form.
Rajah 1 menunjukkan hubungan antara set X dan set Y dalam bentuk graf.
State
Nyatakan
(a) the images of 3.
imej-imej bagi 3.
(b) the type of the relation. [2 marks]
jenis hubungan itu. [2 markah]
Answer/ Jawapan :
(a)
(b)
2 Given the functions 2 3 : x x f and 4 2 :
2
+ x x x g , find
Diberi fungsi 2 3 : x x f dan 4 2 :
2
+ x x x g
,
cari
(a) ) 3 (
1
f
(b) ) (x gf [4 marks]
[4 markah]
Answer/ Jawapan :
(a)
(b)
For
examiners
use only
Set Y
Set X
Diagram 1
Rajah 1
0
5 4 3
2
1
5
4
3
2 1
2
1
4
2
SULIT 7 3472/1
3472/1 2011 Hak Cipta PPDKT [Lihat sebelah
SULIT
3 Given the function 1 2 ) ( + = x x f and kx x g = 3 ) ( , find
Diberi fungsi 1 2 ) ( + = x x f
dan kx x g = 3 ) ( , cari
(a) ) 2 ( f
(b) the value of k such that 7 ) 2 ( = gf [3 marks]
nilai k jika 7 ) 2 ( = gf [3 markah]
Answer / Jawapan :
(a)
(b)
4 Given that and are the roots of the quadratic equation 2x
2
+ 5x + 10 = 0. Form a
quadratic equation with roots + 1 and + 1 .
Diberi bahawa dan adalah punca-punca bagi persamaan kuadratik 2x
2
+ 5x + 10 = 0
Bentukkan persamaan kuadratik baru dengan punca-punca + 1 dan + 1 .
[4 marks]
[4 markah]
Answer/ Jawapan :
5 Given that 3 is one of the roots of the quadratic equation 0 3 2
2
= + px x , find the
value of p.
Diberi bahawa 3 ialah satu daripada punca persamaan kuadratik 0 3 2
2
= + px x ,
cari nilai p.
[2 marks]
[2 markah]
Answer / Jawapan :
For
examiners
use only
2
5
3
3
4
4
SULIT 8 3472/1
3472/1 2011 Hak Cipta PPDKT SULIT
6 Given that the quadratic equation 4 2
2
= + + h hx x has two equal roots. Find the
possible value of h.
Diberi bahawa persamaan kuadratik 4 2
2
= + + h hx x mempunyai dua punca sama.
Cari nilai h yang mungkin.
[2 marks]
[2 markah]
Answer/ Jawapan :
7 Diagram 7 shows the graph of a quadratic function ().
Rajah 7 menunjukkan graf fungsi kuadratik ().
Diagram 7
Rajah 7
State
Nyatakan
(a) the roots of the equation if ()
punca-punca bagi persamaan jika ()
(b) the equation of the axis of symmetry of the curve. [3 marks]
persamaan paksi simetri lengkung itu. [3 markah]
Answer / Jawapan:
(a)
(b)
For
examiners
use only
y
1 4
x
y = f (x)
- -
4 -
3
7
2
6
SULIT 9 3472/1
3472/1 2011 Hak Cipta PPDKT [Lihat sebelah
SULIT
8 Diagram 8 shows the graph of a quadratic function () ( )
2
3, where p is a
constant.
Rajah 8 menunjukkan graf fungsi kuadratik () ( )
2
3 dengan keadaan p
ialah pemalar.
Diagram 8
Rajah 8
The curve () has a minimum point (-1, q), where q is a constant.
Lengkung () mempunyai titik minimum (-1, q) dengan keadaan q ialah
pemalar.
State
Nyatakan
(a) the value of p,
nilai p,
(b) the value of q,
nilai q,
(c) the equation of the axis of symmetry.
persamaan paksi simetri.
[3 marks]
[3 markah]
Answer / Jawapan:
(a)
(b)
(c)
For
examiners
use only
y
x
y = f (x)
(-1, q)
O
-
3
8
SULIT 10 3472/1
3472/1 2011 Hak Cipta PPDKT SULIT
9 Find the range of values of x for which 2x
2
[3 marks]
Cari julat nilai x bagi 2x
2
[3 markah]
Answer / Jawapan:
10 Given 625 25 5
1 3
=
n n
, find the value of n. [ 3 marks ]
Diberi 625 25 5
1 3
=
n n
, cari nilai n. [ 3 markah ]
Answer / Jawapan:
11 Solve the equation . 1 2 2
3 2
=
x x
[ 3 marks ]
Selesaikan persamaan . 1 2 2
3 2
=
x x
[ 3 markah ]
Answer / Jawapan:
For
examiners
use only
3
10
3
11
3
9
SULIT 11 3472/1
3472/1 2011 Hak Cipta PPDKT [Lihat sebelah
SULIT
12 Given that y x
y
x
3 3
4
3
log log 2 2 log + + =
|
|
.
|
\
|
, express y in terms of x. [4 marks]
Diberi y x
y
x
3 3
4
3
log log 2 2 log + + =
|
|
.
|
\
|
, ungkapkan y dalam sebutan x. [4 markah]
Answer/ Jawapan :
13 Solve
( ) 3 4 log 4 log
2 2
= x x
[3 marks]
Selesaikan
( ) 3 4 log 4 log
2 2
= x x
[3 markah]
Answer/ Jawapan :
14 Given k m =
3
log , express
3
27
9
log m
m
in terms of k . [4 marks]
Diberi k m =
3
log , ungkapkan
3
27
9
log m
m
dalam sebutan k . [4 markah]
Answer/ Jawapan :
4
12
For
examiners
use only
3
13
4
14
SULIT 12 3472/1
3472/1 2011 Hak Cipta PPDKT SULIT
15 Given that 2 3 log log
2 4
+ = x . Find the value of x. [ 4 marks ]
Diberi 2 3 log log
2 4
+ = x . Cari nilai bagi x. [4 markah]
Answer/ Jawapan :
16 A straight line passes through A( 4, 4) and B(2, 8).
Suatu garis lurus melalui A( 4, 4) and B(2, 8).
(a) Find the midpoint of the straight line AB.
Cari titik tengah garis lurus AB.
(b) Point D divides the line segment AB in the ratio 1 : 2.
Find the coordinates of D.
Titik D membahagikan tembereng garis AB dalam nisbah 1 : 2.
Cari koordinat D.
[4 marks]
[4 markah]
Answer/ Jawapan:
(a)
(b)
For
examiners
use only
4
16
4
15
SULIT 13 3472/1
3472/1 2011 Hak Cipta PPDKT [Lihat sebelah
SULIT
17 The vertices of a triangle are A(m, 3), B(2, 4) and C(6, 1).
Given that the area of the triangle is 10 units
2
, find the values of m.
Bucu-bucu sebuah segitiga ialah A(m, 3), B(2, 4) and C(6, 1).
Diberi luas segi tiga itu ialah 10 unit
2
, cari nilai-nilai m.
[ 3 marks]
[ 3 markah ]
Answer/ Jawapan :
18 Point P moves such that its distance is always 7 units from point R(5, 2).
Find the equation of the locus of P. [ 3 marks ]
Titik P bergerak dengan keadaan jaraknya sentiasa 7 unit dari titik R (5, 2).
Cari persamaan lokus bagi P. [ 3 markah ]
Answer/ Jawapan :
19 A set of data consists of 4 , 3 , 3 , 6 , 9 , 10 and 12 .
Determine the interquartile range of the data. [ 3 marks ]
Suatu set data terdiri daripada 4 , 3 , 3 , 6 , 9 , 10 dan 12 .
Tentukan julat antara kuartil bagi data itu. [ 3 markah ]
Answer/ Jawapan :
3
17
For
examiners
use only
3
18
3
19
SULIT 14 3472/1
3472/1 2011 Hak Cipta PPDKT SULIT
20 A set of seven numbers has a mean of 6.
Satu set data yang terdiri daripada tujuh nombor mempunyai min 6
(a) Find the sum of the numbers.
Carikan jumlah hasil tambah nombor-nombor tersebut.
(b) When a number m is added to this set, the new mean is 7.5.
Apabila satu nombor m ditambah kepada set ini, min baru ialah 7.5
Find the value of m. [ 3 marks ]
Cari nilai m. [ 3 markah ]
Answer/ Jawapan :
(a)
(b)
21 Calculate the standard deviation of the set of data 15 , 17 , 21 , 24 and 31.
Kira sisihan piawai bagi data 15 , 17 , 21 , 24 dan 31.
[4 marks]
[4 markah ]
Answer/ Jawapan :
For
examiners
use only
3
20
4
21
SULIT 15 3472/1
3472/1 2011 Hak Cipta PPDKT [Lihat sebelah
SULIT
22 Diagram 22 shows a circle with centre O.
Rajah 22 menunjukkan sebuah bulatan berpusat O.
Diagram 22
Rajah 22
The length of the minor arc AB is 15 cm and the angle of the major sector AOB is 300
o
.
Panjang lengkok minor AB ialah 15 cm dan sudut sektor major AOB ialah 300
o
.
Using = 3.142, find
Dengan menggunakan = 3.142, carikan
(a) the value of u , in radian
(Give your answer correct to four significant figures)
nilai u , dalam radian
(Berikan jawapan anda betul kepada empat angka bererti.)
(b) the length, in cm, of the radius of the circle. [ 3 marks ]
panjang, dalam cm, jejari bulatan itu. [ 3 markah ]
Answer/ Jawapan :
(a)
(b)
For
examiners
use only
O
u
A
B
3
22
SULIT 16 3472/1
3472/1 2011 Hak Cipta PPDKT SULIT
23 Diagram 23 shows a semicircle ABC with centre O and a sector DBE of a circle with
centre E.
Rajah 23 menunjukkan sebuah semi bulatan ABC berpusat O dan sektor DBE bagi
sebuah bulatan berpusat E
Diagram 23
Rajah 23
Given that ZBED = 0.5 radian , OC = 10 cm and BE = 12 cm , find the area of the
shaded region. [ Use = 3.142 ] [3 marks]
Diberi ZBED = 0.5 radian , OC = 10 cm dan BE = 12 cm , cari luas kawasan yang
berlorek. [Guna = 3.142] [3 markah]
Answer/ Jawapan :
24 Given that ( )
( )
2
5 3
1
=
x
x h , evaluate ( ) 2
"
h [ 3 marks ]
Diberi ( )
( )
2
5 3
1
=
x
x h , nilaikan ( ) 2
"
h [3 markah]
Answer/ Jawapan :
4
16
3
23
3
24
A D O E C
0.5 rad
B
SULIT 17 3472/1
3472/1 2011 Hak Cipta PPDKT [Lihat sebelah
SULIT
25 The volume of water, V cm
3
, in a container is given by h h V 5
3
1
3
+ = . Given that the
volume increases at a constant rate of 12 cm
3
s
-1
. Find the rate of change of h, in
cm
3
s
-1
, when h = 3 cm.
Isipadu air, V cm
3
, dalam satu bekas diberi oleh h h V 5
3
1
3
+ = . Diberi isipadu air
bertambah dengan kadar malar 12 cm
3
s
-1
. Carikan kadar perubahan h, dalam cm
3
s
-1
,
pada ketika h = 3 cm.
[ 4 marks ]
[ 4 markah ]
Answer/ Jawapan :
END OF QUESTION PAPER.
For
examiners
use only
4
25