Matematik Tambahan SPM

SET 1 PAPER 2
3472/2
Rumus-rumus berikut boleh membantu anda menjawab soalan. Simbol-simbol yang diberi
adalah yang biasa digunakan.
The following formulae may be helpful in answering the questions. The symbols given are the
ones commonly used.
ALGEBRA
1. x =
a
acbb
2
42
−±−
2. am
× an
= am + n
3. am
÷ an
= am − n
4. (am
)n
= am n
5. loga mn = loga m + loga n
6. loga
n
m
= loga m − loga n
7. loga mn
= n loga m
8. balog =
a
b
c
c
log
log
9. nT = a + (n − 1)d
10. nS = { }dna
n
)1(2
2
−+
11. nT = 1−n
ar
12. nS =
1
)1(
−
−
r
ra n
=
r
ra n
−
−
1
)1(
, 1≠r
13. ∞S =
r
a
−1
, | r | < 1
KALKULUS (CALCULUS)
1. y = uv
dx
dy
= u
dx
dv
+ v
dx
du
2. y =
v
u
,
dx
dy
=
2
v
dx
dv
u
dx
du
v −
3.
dx
dy
=
du
dy
×
dx
du
4. Luas di bawah lengkung
(Area under a curve)
= ∫
b
a
dxy atau (or)
= ∫
b
a
dyx
5. Isipadu janaan (Volume of revolution)
= ∫
b
a
dxy2
π atau (or)
= ∫
b
a
dyx2
π
3472/2 © 2005 Hak Cipta Jabatan Pelajaran Terengganu [Lihat sebelah
SULIT
1

SET 1 PAPER 2
3472/2
STATISTIK (STATISTICS)
1. x =
N
x∑
2. x =
∑
∑
f
fx
3. σ =
N
xx∑ − 2
)( =
2
2
x
N
x
−
∑
4. σ =
∑
∑ −
f
xxf 2
)(
=
2
2
x
f
fx
−
∑
∑
5. m = CL
mf
FN








+
−2
1
6. I =
0
1
Q
Q
× 100
7. I =
∑
∑
i
ii
W
IW
8. r
n
P =
!)(
!
rn
n
−
9. r
n
C =
!!)(
!
rrn
n
−
10. P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
11. )( rXp = = rnr
r
n
qpC −
, p + q = 1
12. Min (Mean) = np
13. σ = npq
14. Z =
σ
µ−X
GEOMETRI (GEOMETRY)
1. Jarak (Distance)
= 2
21
2
21 )()( yyxx −+−
2. Titik tengah (Midpoint)
(x, y) = 




 ++
2
,
2
2121 yyxx
3. Titik yang membahagi suatu tembereng
garis
(A point dividing a segment of a line)
(x, y) = 1 2 1 2
,
nx mx ny my
m n m n
+ + 
 ÷+ + 
4. Luas segi tiga (Area of triangle) =
)()( 312312133221
2
1
yxyxyxyxyxyx ++−++
5. r = 22
yx +
6. rˆ = 22
yx
yx
+
+ ji
SULIT
2

SET 1 PAPER 2
3472/2
TRIGONOMETRI (TRIGONOMETRY)
1. Panjang lengkok, s = jθ
Arc length, s = rθ
2. Luas sektor, L = 2
1 2
j θ
Area of sector =
2
1 2
r θ
3. AA 22
kossin + = 1
AA 22
cossin + = 1
4. A2
sek = A2
tan1 +
A2
sec = A2
tan1 +
5. A2
kosek = A2
kot1 +
A2
cosec = A2
cot1 +
6 sin 2A = 2 sinA kosA
sin 2A = 2 sinA cosA
7. kos 2A = kos2
A − sin2
A
= 2 kos2
A − 1
= 1 − 2 sin2
A
cos 2A = cos2
A − sin2
A
= 2 cos2
A − 1
= 1 − 2 sin2
A
8. )(sin BA ± = sinA kosB ±kosA sinB
)(sin BA ± = sinA cosB ±cosA sinB
9. )(kos BA ± = kosA kosB  sinA sinB
)(cos BA ± = cosA cosB  sinA sinB
10. )(tan BA ± =
BA
BA
tantan1
tantan

±
11. tan 2A =
A
A
2
tan1
tan2
−
12.
A
a
sin
=
B
b
sin
=
C
c
sin
13. a2
= b2
+ c2
− 2bc kosA
a2
= b2
+ c2
− 2bc cosA
14. Luas segi tiga (Area of triangle)
= 2
1
ab sin C
SULIT
3

SET 1 PAPER 2
3472/2
Section A / Bahagian A
[40 marks / 40 markah]
Answer all questions from this section
Jawab semua soalan dalam bahagian ini.
1 Solve the equation 2p + k = p2
+ k2
− 5 = 5 [6 marks]
Selesaikan persamaan 2p + k = p2
+ k2
− 5 = 5.
2 A function is defined by :
3 2
m x
g x
x
+
→
+
, for all values of x except x = h, and m is a constant.
Satu fungsi ditakrifkan oleh :
3 2
m x
g x
x
+
→
+
, bagi semua nilai x kecuali x = h, dan m ialah
pemalar.
(a) Determine the value of h.
Tentukan nilai h.
(b) Given that 2 maps onto itself under the function g, find
Diberi nilai 2 dipetakan kepada dirinya sendiri di bawah fungsi g, cari
(i) the value of m,
nilai m,
(ii) the value of 1
( 1)g−
−
nilai bagi 1
( 1)g−
−
(iii) the value of p if g(2p) = 5
nilai p jika g(2p) = 5
[7 marks]
SULIT
4

SET 1 PAPER 2
3472/2
3 The straight line x + 4y = p is normal to the curve y = (2x – 1)2
+ 1 at the point Q.
Garis lurus x + 4y = p adalah normal kepada lengkung y = (2x – 1)2
+ 1 pada titik Q.
Find
Cari
(a) the coordinates of Q, [4 marks]
koordinat-koordinat Q,
(b) the value of p, [1 mark]
nilai p,
(c) the equation of the tangent at the point Q. [2 marks]
persamaan tangen pada titik Q.
Diagram 1
4 In Diagram 1, the equation of the straight line PQR is 2x – y + 4 = 0. If PQ : QR = 2 : 3,
find
Dalam Rajah 1, persamaan garis lurus PQR ialah 2x – y + 4 = 0. Jika PQ : QR = 2 : 3,
cari
(a) the coordinates of R, [3 marks]
koordinat-koordinat R,
(b) the equation of perpendicular bisector PR, [4 marks]
persamaan pembahagi dua sama serenjang PR.
SULIT
5
P
Q
R
•
•
•
x
y
0

SET 1 PAPER 2
3472/2
5 Solve each of the following equations :
Selesaikan setiap persamaan berikut :
(a) 4
1
3
9
27 −
−
= x
x
[3
marks]
(b) log5 3x – log5 (x – 2) = 1 [3 marks]
6 (a) The sum of the first n terms of an arithmetic progression is given by Sn =
)723(
2
n
n
− .
Hasil tambah n sebutan pertama suatu janjang aritmetik diberi oleh Sn =
)723(
2
n
n
− .
Find
Cari
(i) the first term and common difference,
sebutan pertama dan beza sepunya,
(ii) the sum of the all the terms from the 5th
term to the 10th
term.
hasil tambah sebutan ke 5 hingga sebutan ke 10.
[5 marks]
(b) Given that the 5th
and the 8th
terms of a geometric progression are 2 and 6
3
4
respectively, find the common ratio of this progression. [2 marks]
Diberi sebutan kelima dan sebutan kelapan suatu janjang geometri masing-masing
ialah 2 dan 6
3
4
, carikan nisbah sepunya janjang tersebut.
SULIT
6

SET 1 PAPER 2
3472/2
Section B / Bahagian B
Answer four questions from this section
Jawab empat soalan daripada bahagian ini.
Diagram 2 / Rajah 2
7 Diagram 2 shows two sectors of a circle, sector OCBA with centre O and sector PCOA
with centre P. ∆ OAP and ∆ OPC are two congruent equilateral triangle with sides 8 cm. Find
Rajah 2 menunjukkan dua sektor bulatan iaitu sektor OCBA yang berpusat di O dan sektor PCOA
yang berpusat di P. ∆ OAP dan ∆ OPC adalah dua segi tiga sama sisi yang kongruen dan
mempunyai sisi 8 cm. Cari
(a) the reflex angle AOC in radian, [1 mark]
sudut refleks AOC dalam radian,
(b) the perimeter of the shaded region, [3 marks]
perimeter kawasan berlorek,
(c) the area of the major sector OCBA, [2 marks]
luas sektor major OCBA,
(d) the area of the shaded region. [4 marks]
SULIT
7
P
A C
B
O

SET 1 PAPER 2
3472/2
luas kawasan berlorek.
[Use/ Gunakan π = 3.142]
x 1 3 5 7 8
y 10 42 90 154 192
Table 1 / Jadual 1
8 Table 1 shows the values of two variables, x and y which are related by the equation
y = ax (x + b) , where a and b are constants.
Jadual 1 menunjukkan nilai bagi dua pembolehubah x dan y yang dihubungkan oleh
persamaan y = ax (x + b) , dengan keadaan a dan b ialah pemalar.
(a) Reduce the equation y = ax (x + b) to the linear form. [1 mark]
Tukarkan persamaan y = ax (x + b) kepada persamaan bentuk linear.
(b) Plot the graph of
x
y
against x . [4 marks]
Lukiskan graf
x
y
melawan x .
(c) Use the graph from (a) to find
Gunakan graf anda dari (a) untuk mencari
(i) the value of a and of b
nilai a dan nilai b,
(ii) the value of y when x = 6 [5 marks]
nilai y apabila x = 6.
9 (a) A set of game score x1 , x2 , x3 , x4 and x5 has the mean 6 and standard deviation 1·2.
Satu set skor bagi suatu permainan x1 , x2 , x3 , x4 dan x5 mempunyai min 6 dan sisihan
piawai 1⋅2.
(i) Find the sum of the squares of the score, Σx2
[2 marks]
Cari hasil tambah kuasa dua skor itu, Σx2
(ii) If each score is multiplied by 3 and 2 is added to it, find the mean and variance of
the new score. [3 marks]
Jika setiap skor itu didarabkan dengan 3 dan ditambah dengan 2, cari min dan
varians bagi set skor yang baru.
SULIT
8

SET 1 PAPER 2
3472/2
Marks / Markah 1 – 20 21 – 40 41 – 60 61 – 80 81 – 100
Number of students /
Bilangan pelajar
4 10 13 8 5
Table 2 / Jadual 2
(b) Table 2 shows the marks obtained by a group of students in a particular test.
Jadual 2 menunjukkan markah yang diperoleh sekumpulan pelajar dalam satu ujian.
(i) Without drawing an ogive, find the median marks. [3 marks]
Tanpa melukis ogif, cari median bagi markah-markah itu.
(ii) Calculate the mean marks. [2 marks]
Hitungkan markah min.
10 Given the quadratic function f(x) = − 4x2
+ 8x − 1.
Diberi fungsi kuadratik f(x) = − 4x2
+ 8x − 1.
(a) Express the quadratic function f(x) in the form p(x + q)2
+ r , where p, q and r
are constants. Determine whether the function f(x) has a minimum or maximum
value and state its value. [3 marks]
Ungkapkan fungsi kuadratik f(x) dalam bentuk p(x + q)2
+ r dengan keadaan
p, q dan r ialah pemalar. Tentukan sama ada fungsi f(x) mempunyai nilai
maksimum atau nilai minimum dan nyatakan nilainya.
(b) Sketch the graph of the function f(x) for the domain − 1 ≤ x ≤ 3.
State the corresponding range for this domain. [3 marks]
Lakarkan graf fungsi f(x) untuk domain − 1 ≤ x ≤ 3.
Nyatakan julat yang sepadan dengan domain ini.
SULIT
9

SET 1 PAPER 2
3472/2
(c) Find the range of values of k such that f(x) = kx has no real roots. [4 marks]
Cari julat nilai k supaya f(x) = kx tidak mempunyai punca nyata.
11 (a) Given that ABCD is a parallelogram where BC
→
= 2i + 3j and CD
→
= − 2i – 2j, find
Diberi ABCD ialah sebuah segi empat selari dengan BC
→
= 2i + 3j dan CD
→
= − 2i – 2j,
cari
(i) AC
→
,
(ii) the unit vector in the direction of AB
→
. [4 marks]
vektor unit pada arah AB
→
.
Diagram 3 / Rajah 3
(b) Diagram 3 shows a triangle ABC where CP
→
= λ CB
→
. Given that AB
→
= a and AC
→
= b .
Rajah 3 menunjukkan sebuah segi tiga ABC dengan CP
→
= λ CB
→
. Diberi AB
→
= a
dan AC
→
= b .
(i) Show that AP
→
= λ a + (1 −λ )b
Tunjukkan bahawa AP
→
= λ a + (1 −λ )b
(ii) If BP
→
= − 3a + 3b, find the value of λ.
SULIT
10
A
P
C
B

SET 1 PAPER 2
3472/2
Jika BP
→
= − 3a + 3b, cari nilai λ.
[6 mark]
Section C / Bahagian C
Answer two questions from this section
Jawab dua soalan daripada bahagian ini.
12 A particle moves along a straight line from a fixed point O. Its displacement, S m is given
by S = 9t2 – t3 , where t is the time in seconds after leaving the point Q.
(Assume that motion to the right is positive)
Suatu zarah bergerak di sepanjang suatu garis lurus bermula dari satu titik tetap Q.
Sesarannya, S m, diberi oleh S = 9t2 – t3 , dengan keadaan t ialah masa, dalam saat,
selepas meninggalkan titik Q.
(Anggapkan gerakan ke arah kanan sebagai positif)
Find
Cari
(a) the distance travelled the 2nd
second, [2 marks]
jarak yang dilalui dalam saat kedua,
(b) the time at which the velocity of the particle is uniform, [3 marks]
masa ketika zarah mencapai halaju seragam,
(c) the value of t when the particle passes the point Q again, [2 marks]
nilai t apabila zarah itu melalui titik Q semula,
(d) the range of values of t during which the particle is moving towards the point Q after
instantaneous rest. [3 marks]
julat nilai t apabila zarah menuju ke titik Q selepas rehat seketika.
SULIT
11

SET 1 PAPER 2
3472/2
Jersey brand /
Jenama jersi
Price / Harga Price index / Indeks harga
2004(2003 =100)2003 2004
Adibas 750 x 110
Pumas y 740 125
Nikas 880 990 z
Table 3 / Jadual 3
13 Table 3 shows the price of a set of football jersey which has been worn by a particular team at Lion
Championship tournament in the year 2003 and 2004.
Jadual 3 menunjukkan harga bagi satu set jersi bola sepak yang dipakai oleh satu pasukan di
kejohanan Piala Lion untuk tahun 2003 dan 2004.
(a) Calculate the values of x, y and z. [4 marks]
Hitung nilai x, y dan z.
(b) Given that the index number of Adibas jersey is 120 in the year 2003 based on the year 2002,
find the price in the year 2002. [2 marks]
Diberi indeks harga untuk jersi Adibas ialah 120 pada tahun 2003 berasaskan tahun 2002, cari
harganya pada tahun 2002.
(c) If the price of Nikas jersey increases at a constant rate every year, determine the price in the
year 2005 to the nearest RM. [4 marks]
Jika harga jersi jenama Nikas bertambah dengan kadar tetap setiap tahun, tentukan harganya
pada tahun 2005 kepada RM terdekat.
SULIT
12

SET 1 PAPER 2
3472/2
14 Given that x and y are positive integers with the following conditions :
Diberi x dan y ialah dua integer positif dengan keadaan berikut :
I : The value of x is more than the value of y by 5 or more
Nilai x melebihi nilai y sebanyak 5 atau lebih.
II : The minimum value of 2x + 3y is 60
Nilai minimum bagi 2x + 3y ialah 60.
III : The maximum value of 4x + 3y is 5 times the minimum value of 2x + 3y
Nilai maksimum bagi 4x + 3y ialah 5 kali nilai minimum bagi 2x + 3y.
(a) Write down three inequalities which satisfy the above conditions. [3 marks]
Tulis tiga ketaksamaan bagi setiap keadaan di atas.
(b) Construct and shade the region R that satisfies all of the above conditions. [3 marks]
Bina dan lorekkan rantau R yang memuaskan syarat-syarat di atas.
(c) By using your graph from (b), answer the following questions :
Dengan menggunakan graf anda dari (b), jawab soalan-soalan berikut :
(i) Find the minimum value of the sum of the integers. [1 mark]
Cari nilai minimum bagi hasil tambah integer tersebut.
(ii) Given that x and y represents the number of item M and item N respectively
which was sold by the company. Find the total maximum sale obtained if the price
of one unit item M and N are RM15 and RM30 respectively. [3 marks]
SULIT
13

SET 1 PAPER 2
3472/2
Diberi bahawa x dan y masing-masing mewakili bilangan barang M dan barang N
yang dijual oleh sebuah syarikat. Cari jumlah jualan maksimum yang diperoleh
jika harga seunit barang M dan barang N masing-masing ialah RM15 dan RM30.
Diagram 4 / Rajah 4
15 (a) Diagram 4 shows a triangle PQR. Calculate ∠ QPR. [3 marks]
Rajah 4 menunjukkan sebuah segi tiga PQR. Hitungkan ∠ QPR.
SULIT
14
20°
50°
15 cm
13 cm
A
B
C
D
10 cm
P
Q
R
125°
10⋅5 cm
5 cm

SET 1 PAPER 2
3472/2
Diagram 5 / Rajah 5
(b) In Diagram 5, ABC and BCD are two triangles which attached at BC. Calculate
Dalam Rajah 5, ABC dan BCD ialah dua buah segi tiga yang bertemu pada garis BC.
Hitungkan
(i) the length of BC,
panjang BC,
(ii) BCD∠ ,
(iii) the area of whole diagram,
luas seluruh rajah.
[7 marks]
END OF QUESTIONS PAPER
KERTAS SOALAN TAMAT
SULIT
15

Matematik Tambahan SPM

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