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ppr maths nbk
SEKTOR SEKOLAH BERASRAMA PENUH BAHAGIAN SEKOLAH
KEMENTERIAN PELAJARAN MALAYSIA
PEPERIKSAAN DIAGNOSTIK TINGKATAN EMPAT 2007
Pemeriksa
Bahagian Soalan Markah Penuh
Markah Diperoleh
1 4
MATEMATIK
Kertas 2
Dua jam tiga puluh minit
2 4
3 5
4 4
5 3
6 4
7 6
8 6
9 4
10 7
A
11 5
12 12
13 12
14 12
15 12
B
16 12
JANGAN BUKA KERTAS SOALAN INI
SEHINGGA DIBERITAHU
1. Kertas soalan ini mengandungi dua bahagian : Bahagian A dan Bahagian B. Jawab semua soalan daripada Bahagian A dan empat soalan dalam Bahagian B.
2. Jawapan hendaklah ditulis dengan jelas dalam
ruang yang disediakan dalam kertas soalan. Tunjukkan langkah-langkah penting. Ini boleh membantu anda untuk mendapatkan markah.
3. Rajah yang mengiringi soalan tidak dilukis
mengikut skala kecuali dinyatakan. 4. Satu senarai rumus disediakan di halaman 2 & 3. 5. Anda dibenarkan menggunakan kalkulator
saintifik yang tidak boleh diprogram. Jumlah
Kertas soalan ini mengandungi 24 halaman bercetak.
SULIT 1449/2 Matematik Kertas 2 Oktober 2007
212 jam
1449/2
1449/2 © 2007 Hak Cipta Sektor SBP [Lihat sebelah SULIT
NAMA :
TINGKATAN :
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MATHEMATICAL FORMULAE The following formulae may be helpful in answering the questions. The symbols given are the ones commonly 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⎟⎟⎠
⎞⎜⎜⎝
⎛−
−acbd
5 P ( A ) = )()(
SnAn
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 Theorem c2 = a2 + b2
13 m = 12
12
xxyy
−−
14 -intercept-intercept
ymx
=
distance travelled time taken
sum of data number of data
sum of (class mark × frequency) sum of frequencies
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SHAPES AND SPACE
1 Area of trapezium = 21 × sum of parallel sides × height
2 Circumference of circle = πd = 2πr 3 Area of circle = πr2 4 Curved surface area of cylinder = 2πrh 5 Surface area of sphere = 4πr2 6 Volume of right prism = cross sectional area × length 7 Volume of cylinder = πr2h
8 Volume of cone = 31 πr2h
9 Volume of sphere = 34πr3
10 Volume of right pyramid = 31 × base area× height
11 Sum of interior angles of a polygon = ( n – 2) × 180˚
12 arc length angle subtended at centrecircumference of circle 360
=o
13 area of sector angle subtended at centrearea of circle 360
=o
14 Scale factor , k = PAPA'
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 equation 3
343
−−
=x
xx
[4 marks]
Answer :
2 Calculate the value of p and of q that satisfy the following simultaneous linear equations:
394632−=+
=−qpqp
[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 sequence above.
[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 π = 722 , calculate the volume, in cm3, of the solid
[4 marks]
Answers:
J K
L M
P Q
R S T
N 8 cm
7 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,
ξ = P ∪ Q ∪ R.
On the diagrams in the answer space, shade
(a) P′ ∩ R′
(b) Q ∩ R ∪ P
[3 marks] Answer :
(a) (b)
P
Q R
P
Q R
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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
C D
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.
[6 marks]
Answer:
(a)
(b)
(c)
x O
K R
S
L
DIAGRAM 3
y
P
Q(−3, −3) •
•
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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 5>x , 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 is 73 .
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 π = 722 , 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 big
walk 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
Frequency Upper
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 = 221 cm and ∠COP = 60°.
Using π = 722 , 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
B C
Q R
P O
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, ξ = 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
7 2x
5
18 – 2x 10 + 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
=−
ww .
[4 marks] (b) Calculate the values of x and y that satisfy the following simultaneous linear equations: 2x + y = 3 4x - 3y = 11
[4 marks] (c) (i) Form a true compound statement by combining the two statements given below. (a) 52 = 10
(b) 41 = 0.25
(ii) Form a general conclusion by induction based on the numerical sequence below.
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 true statement. ‘………………. prime numbers are odd numbers.’
[4 marks]
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Answer:
16 (a)
(b)
(c) (i)
(ii)
(iii)