SEKTOR SEKBA
KEMENTER
PEPERTING
MATEMATIK
Kertas 2
Dua jam tiga puluh minit
JANGAN BUKA KERTAS SOALASEHINGGA DIBERITAHU
1. Kertas soalan ini mengandungi dua baBahagian A dan Bahagian B. Jawabsoalan daripada Bahagian A dan empadalam Bahagian B.
2. Jawapan hendaklah ditulis dengan jelaruang yang disediakan dalam kertasTunjukkan langkah-langkah penting. Imembantu anda untuk mendapatkan mar
3. Rajah yang mengiringi soalan tidakmengikut skala kecuali dinyatakan.
4. Satu senarai rumus disediakan di halam
2 & 3.
5. Anda dibenarkan menggunakan kasaintifik yang tidak boleh diprogram.
Kertas soalan in
SULIT1449/2MatematikKertas 2Oktober2007
2
12 jam
1449/2
1449/2 2007 Hak Cipta Sektor SBP
NAMA :
TINGKATAN :
OLAH BERASRAMHAGIAN SEKOLAHIAN PELAJARAN M
IKSAAN DIAGNOSTKATAN EMPAT 200
Peme
Bahagian
A
B
N INI
hagian :semua
t soalan
s dalamsoalan.
ni bolehkah.
dilukis
an
lkulator
i mengandungi 24 halama
A PENUH
ALAYSIA
IK7
riksa
SoalanMarkahPenuh
MarkahDiperoleh
1 4
2 4
3 5
4 4
5 3
6 4
7 6
8 6
9 4
10 7
11 5
12 12
13 12
14 12
15 12
16 12
Jumlah
n bercetak.
[Lihat sebelahSULIT
SULIT 1449/2
1449/2 SULIT2
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MATHEMATICAL FORMULAE
The following formulae may be helpful in answering the questions. The symbols given are the onescommonly used.
RELATIONS
1 am x an = a m+ n
2 am an = a m – n
3 ( am )n = a mn
4 A-1 =bcad
1
ac
bd
5 P ( A ) =)(
)(
Sn
An
6 P ( A ) = 1 P(A)
7 Distance = 2 21 2 1 2( ) ( )x x y y
8 Midpoint, ( x, y ) =
2,
22121 yyxx
9 Average speed =
10 Mean =
11 Mean =
12 Pythagoras Theoremc2 = a2 + b2
13 m =12
12
xx
yy
14-intercept
-intercept
ym
x
distance travelledtime taken
sum of datanumber of data
sum of (class mark × frequency)sum of frequencies
SULIT 1449/2
1449/2 [Lihat sebelahSULIT
3
UntukKegunaanPemeriksaSHAPES AND SPACE
1 Area of trapezium =2
1× sum of parallel sides × height
2 Circumference of circle = d = 2r
3 Area of circle = r2
4 Curved surface area of cylinder = 2rh
5 Surface area of sphere = 4r2
6 Volume of right prism = cross sectional area × length
7 Volume of cylinder = r2h
8 Volume of cone =3
1r2h
9 Volume of sphere =3
4r3
10 Volume of right pyramid =3
1× base area× height
11 Sum of interior angles of a polygon = ( n – 2) × 180˚
12arc length angle subtended at centre
circumference of circle 360
13area of sector angle subtended at centre
area of circle 360
14 Scale factor , k =PA
PA'
15 Area of image = k 2 × area of object
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Section A[52 marks]
Answer all questions in this section.
1 Solve the quadratic equation3
3
4
3
x
xx
[4 marks]
Answer :
2 Calculate the value of p and of q that satisfy the following simultaneous linear equations:
394
632
qp
qp
[4 marks]
Answer :
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3 (a) State whether the following statement is true or false.
63and326 2
(b) Write down two implications based on the sentence below.
4m 20 if and only if m 5
(c) Given the number sequence 5, 18, 39, 68, … which follows the pattern
5 = 1 + 4(12)
18 = 2 + 4(22)
39 = 3 + 4(32)
68 = 4 + 4(42)
……………..……………..
Form a general conclusion by using the induction method for the numerical sequenceabove.
[5 marks]
Answer :
(a)
(b) Implication 1 : _________________________________________________
Implication 2 : _________________________________________________
(c)
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4 Diagram 1 shows a combined solid consists of a cuboid JKLMPQRS and half cylinder
JMNTSP.
Using =7
22, calculate the volume, in cm3, of the solid
[4 marks]
Answers:
J K
LM
P Q
RS
T
N
8 cm7 cm
16 cm
DIAGRAM 1
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5 The Venn diagram in the answer space shows sets P, Q and R such that the universal set,
= P Q R.
On the diagrams in the answer space, shade
(a) P R
(b) Q R P
[3 marks]
Answer :
(a)
(b)
P
QR
P
QR
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6 Diagram 2 shows a cuboid with horizontal rectangular base, ABCD.
Calculate the angle between the plane ABE and the base ABCD.
[4 marks]Answer :
A B
CD
E
F G
H
DIAGRAM 2
10 cm
8 cm
6 cm
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7 In Diagram 3, O is the origin. KL, PQ and RS are straight lines. PQ is parallel to RS.The equation of the straight line KL is 62 xy .
Find the
(a) coordinate of P,
(b) gradient of the straight line PQ,
(c) equation of the straight line RS.
Answer:
(a)
(b)
(c)
x
K R
S
DIA
y
P
Q(3, 3)
O9
GRAM 3
[Lihat sebelahSULIT
[6 marks]
L
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8 (a) Identify the antecedent and consequent in the following implication.
‘ If a triangle has two equal sides, then it is an isosceles triangle.’
(b) State the converse of the following implication.
If 5x , then x2 > 25
(c) Complete the premise in the following argument.
Premise 1 : If x is an angle in a semicircle, then x = 90°
Premise 2 : _________________________________________________
Conclusion : x = 90°
(d) Complete the following argument
Premise 1 : _________________________________________________
Premise 2 : M N ≠ M
Conclusion : M N
[6 marks]
Answer:
(a) Antecedent : …………………………………………………………………
Consequent : ………………………………………………………………….
(b) …………………………………………………………………………………
(c) Premise 2 : ……………………………………………………………………
(d) Premise 1 : …………………………………………………………………….
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9 Ten cards are placed in an empty box.
(a) If a card is selected at random from the box, state the probability that the card marked
C is selected.
(b) A number of cards marked C are added into the box. If a card is selected at random
from the box, the probability of selecting a card marked C is7
3.
Find the number of cards that has been added into the box.
[4 marks]Answer :
(a)
(b)
E X C E L L E N C E
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10 In Diagram 4, ABOD is a square and PQR is a semicircle with centre O.
OR = 7 cm. Using =7
22, calculate
(a) the perimeter, in cm, of the whole diagram,
(b) the area, in cm2, of the shaded region.
[7 marks]
Answer :
(a)
(b)
A B
D O
P
Q
R
DIAGRAM 4
14 cm
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11 Diagram 5 shows a right pyramid with rectangular base PQRS .
The apex V is 5 cm vertically above the point T.
Calculate the angle between the line VP and the base PQRS.[5 marks]
Answer:
R
8 cm
V
P Q
S
6 cmT
DIAGRAM 5
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Section B
[48 marks]
Answer any four questions in this section.
12 Table 1 in the answer space shows the distribution of the ages of 200 participants in a bigwalk event.
(a) Using the data in Table 1, complete the table provided in the answer space.[4 marks]
(b) By using a scale of 2 cm to 5 years on the x-axis and 2 cm to 20 participants on
the y-axis , draw an ogive for the data.[5 marks]
(c) Based on your ogive in (b),
(i) find the inter quartile,
(ii) explain briefly the meaning of the third quartile.
[3 marks]
Answer:
(a)Age (years) Frequency Cumulative
FrequencyUpper
Boundary
15 19 10
20 24 20
25 29 50
30 34 60
35 39 36
40 44 18
45 49 6
(b) Refer graph on page 15.
(c) (i)
(ii)
TABLE 1
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Graph for Question 12
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13 (a) Diagram 6 shows sectors OTQR, OABC and OCP with the same centre O.TAOP and OCR are straight lines.
OR = 2OC , OC =2
21cm and COP = 60.
Using =7
22, calculate
(i) the perimeter, in cm, of the whole diagram,[3 marks]
(ii) the area, in cm2, of the shaded region.[4 marks]
(b) Diagram 7 shows some number cards.
A card is picked at random. State the probability of choosing a card containing
(i) digit 5,
(ii) a multiple of 5,
(iii) a number such that when divided by 5, the remainder is 1.
[5 marks]
T A
BC
QR
PO
DIAGRAM 6
45 46 47 48 49 50 51 52
53 54 55 56 57 58 59 60
DIAGRAM 7
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Answer:
13 (a) (i)
(ii)
(b) (i)
(ii)
(iii)
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14 The data in Diagram 8 show the marks of 40 students for the Mathematics monthly test.
46 53 44 60 42 38 31 55
35 37 54 32 46 56 40 60
52 40 34 45 52 35 50 36
47 38 40 48 45 42 53 44
50 44 58 51 36 48 56 32
(a) Using the data in Diagram 8 and a class interval of 5 marks, complete Table 2
in the answer space.
[4 marks]
(b) Based on Table 2 in (a),
(i) state the modal class,
(ii) calculate the mean mark of the Mathematics monthly test and give your answer
correct to 2 decimal places.
[4 marks]
(c) For this part of the question, use the graph paper provided on page 19.
By using a scale of 2 cm to 5 marks on the horizontal axis and 2 cm to 1 student on the
vertical axis, draw a histogram for the data.
[4 marks]
DIAGRAM 8
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Answer:
14 (a)
Class interval Midpoint Frequency
31 – 35
36 - 40
(b) (i)
(ii)
(c) Refer to the graph on page 20.
TABLE 2
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Graph for Question 14
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15 (a) Diagram 9 is a Venn diagram showing the number of students in a class who play at least
one of the three games. Given the universal set, = F R H.
Given that set F = { students who play football }
set R = { students who play rugby } and
set H = { students who play hockey}.
(i) Find the number of students who play hockey.
(ii) If the number of students who play football is 30, find
(a) the number of students who play rugby,
(b) the number of students who play one game only.
[6 marks]
Answer:
(a) (i)
(ii) (a)
(b)
R
F
H
x
72x
5
18 – 2x10 + x 10
DIAGRAM 9
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15 (b) In Diagram 10 , TQRS, NR and OQ are straight lines drawn on a Cartesan plane. O is the
origin and OQ is parallel to NR. The gradient of NR = 2.
Find
(i) the value of p,
(ii) the coordinate of R,
(iii) the equation of the straight line ST.
[6 marks]
Answer:
(b) (i)
(ii)
(iii)
y
x
R
N(-6, 0)
Q(p, 3)
T
0
DIAGRAM 10
S
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16 (a) Solve the equation 15
32 2
w
w.
[4 marks](b) Calculate the values of x and y that satisfy the following simultaneous linear
equations:2x + y = 34x - 3y = 11
[4 marks]
(c) (i) Form a true compound statement by combining the two statementsgiven below.
(a) 52 = 10
(b)4
1= 0.25
(ii) Form a general conclusion by induction based on the numerical sequencebelow.
2, 9, 16, 23 …
2 = 2 + 7 (0)9 = 2 + 7 (1)16 = 2 + 7 (2)23 = 2 + 7 (3)……………….
(iii) Complete the following sentence using a suitable quantifier to make it a truestatement.
‘………………. prime numbers are odd numbers.’
[4 marks]
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Answer:
16 (a)
(b)
(c) (i)
(ii)
(iii)
SULIT 1449/21449/2MatematikKertas2PeraturanPemarkahanOktober2007
SEKT
KEME
Peraturan
1449/2
OR SEKOLAH BERASRAMA PENUHBAHAGIAN SEKOLAH
NTERIAN PELAJARAN MALAYSIA
PEPERIKSAAN DIAGNOSTIKTINGKATAN 4 2007
MATEMATIK
Kertas 2
PERATURAN PEMARKAHAN
pemarkahan ini mengandungi 10 halaman.
[Lihat sebelahSULIT
Section A[ 52 marks ]
No Marking Scheme Marks
1
3,3
4
0)3)(43(
012`53 2
xx
xx
xx K1
K1
N1N1 4
2
2
3
1
15101515
18961264
p
q
porq
qporqp K1
K1
N1
N1 4
3(a) False
(b) Implication 1 : If 4m 20, then m 5Implication 2 : If m 5, then 4m 20
(c) n + 4n2 , n = 1,2, 3, ….
P1
P1P1
P1, P1 5
4 8 x 7 x 16
162
7
2
7
7
22
2
1
8 x 7 x 16 + 162
7
2
7
7
22
2
1
1204
K1
K1
K1
N1 4
5 (a) PR (b) QRP
6
2
Identify EAD
Tan EAD =8
6
EAD = 36.87 or 3652
P1 P2
3
P1
K2
N1 4
3
No Marking Scheme Marks
7 (a) P(3,0)
(b) mPQ =30
33
= 2
(c) mRS = 2
20
6
x
yor equivalent
y = 2x + 6
P1
K2
N1
K1
K1
N1 6
8 (a) Antecedent : A triangle has two equal sidesConsequent: It is an isoceles triangle
(b) If2x > 25, then x > 5
(c) x is an angle in a semicircle
(d) If M N, then MN = M
P1P1
P1
P1
P1 7
9 (a)5
1
(b)7
3=
x
x
10
2
x = 4
P1
K2
N1 4
10(a) 7
7
222
2
1
14(4) + 77
222
2
1
78
(b) 777
22
2
1
14147
22
4
1
777
22
2
1 + ( 14 X 14 1414
7
22
4
1 )
119
K1
K1
N1
K1
K1
K1
N1
7
4
No Marking Scheme Marks
11 VPR or VPT
22 86 ( seen )
Tan VPR =5
5
VPR = 45°
P1
K1
K2
N15
Section B[48 marks]
No Marking Scheme Marks
12 (a)
Age (years) Frequency Cumulative frequency Upper boundary
10 – 14 0( I )
0( II )14.5
15 - 19 10 10 19.520 – 24 20 30 24.525 – 29 50 80 29.530 – 34 60 140 34.535 - 39 36 176 39.540 - 44 18 194 44.545 - 49 6 200 49.5
All values in Column ( I ) correct excluding Row I correct.All values in Column ( II ) correct excluding Row I correct.
(b) x-axis is drawn with the right direction and in uniform scale from14.5 ≤ x ≤ 49.5
y-axis is drawn with the right direction and in uniform scale from0 ≤ y ≤ 200
All eight points* plotted correctly.Note : Seven or six points* plotted correctly. 1 P1
(14.5, 0) plotted or passed throughAll the right eight points plotted correctly and ogive is drawn smoothly passingthrough all the points.
(c) (i) 35.5 – 27.58 – 9
(ii) 150 participants aged 35.5 0.5
P1P2
K1
K1
K2
K1N1
K1N1
N1 12
13(a)
360
120x 2 x
7
22x 21
360
60x 2 x
7
22x
2
21
360
120x 2 x
7
22x 21 + 21 + 21 +
360
60x 2 x
7
22x
2
21
97
K1
K1
K1
N1
5
No Marking Scheme Marks
(b)360
120x
7
22x 21 x 21
360
120x
7
22x
2
21x
2
21
360
120x
7
22x 21 x 21
360
120x
7
22x
2
21x
2
21+
360
60x
7
22x
2
21x
2
21
4044
1
(b) (i)16
11
(ii)4
1
(iii)16
3
K1
K1
K1
N1
P1
P1
P212
14 (a)
Class interval Midpoint Frequency
31 – 35 33 6
36 - 40 38 5
41 – 45 43 10
46 – 50 48 6
51 – 55 53 8
56 - 60 58 5
Column 1Column 2Column 3
(b) (i) 41 – 45
(ii)5861056
)5(58)8(53)6(48)10(43)5(38)6(33
45.5
(c) Refer to the graphAxes drawn in the correct direction , uniform scale for 5.605.30 x and
100 y .
Horizontal axis labeled using midpoint / boundary / class interval.6 bars drawn correctly
P1P1P2
P1
K2
N1
K1
K1N2 12
15 (a) (i) 2 x + 10 + 5 + 18 - 2 x33
(ii) 7 + 10 + 12 + 1645
(iii) 6 + 6 +1628
K1N1
K1N1
K1N1
6
No Marking Scheme Marks
(b) (i)0
03
p= 2
p =2
3
(ii) 2)6(0
0
y
R (0,12)
(iii) m ST = 6
126 xy
K1
N1
K1
N1
K1N1
12
16a 2
2w - 5 w - 3 = 0
(2 w +1)( w -3) =0
w = -2
1, 3
b 4 x + 2 y = 6 or equivalent
5 y = - 5
y = -1
x = 2
c i orii 2+7n, n = 0,1,2,3,…
iii some
K1K1
N1N1
K1K1N1N1
P1P2P1 12
7
Graph for number 12
20
Cu
mu
lati
ve
Fre
qu
ency
40
60
80
100
120
140
160
180
200
14.5 19.5 24.5 29.5 34.5 39.5 44.5
Ages49.5
8
Graph for number 14
0 33 38 43 48 53 Marks
1
2
3
4
5
6
7
8
9Nu
mb
erof
stu
den
ts
10
58