1. The document contains a mathematics exam paper for Form 4 students with 22 pages. It has two sections (Section A and B) and includes questions, formulas, and diagrams. 2. Section A contains 11 multiple choice questions worth 52 marks total. Section B contains 4 extended response questions worth 48 marks total. Students must answer all questions in Section A and 4 questions from Section B. 3. The exam paper tests students on topics like algebra, geometry, trigonometry, statistics, and probability. Formulas are provided that may help with answering questions.

1. ppr maths nbk
1449/2
Matematik
Kertas 2
Okt
2006
2 ½ jam
SEKTOR SEKOLAH BERASRAMA PENUH
BAHAGIAN SEKOLAH
KEMENTERIAN PELAJARAN MALAYSIA
PEPERIKSAAN AKHIR TAHUN
TINGKATAN 4 2006
Kertas soalan ini mengandungi 22 halaman bercetak
1449/2 @ 2006 Hak Cipta SBP
MATEMATIK
Kertas 2
Dua jam tiga puluh minit
JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU
Pemeriksa
Bahagian Soalan
Markah
Penuh
Markah
Diperoleh
1 4
2 4
3 4
4 5
5 5
6 4
7 7
8 5
9 3
10 5
A
11 6
12 12
13 12
14 12
15 12
B
16 12
1. Kertas soalan ini mengandungi dua
bahagian : Bahagian A dan Bahagian
B.
2. Jawab semua soalan Bahagian A dan
empat soalan daripada Bahagian B.
3. Jawapan hendaklah ditulis dengan jelas
dalam ruang yang disediakan dalam
kertas soalan.
4. Tunjukkan langkah-langkah penting
dalam kerja mengira anda. Ini boleh
membantu anda untuk mendapatkan
markah.
5. Rajah yang mengiringi soalan tidak
dilukis mengikut skala kecuali
dinyatakan.
6. Satu senarai rumus disediakan di
halaman 2 dan 3.
7. Anda dibenarkan menggunakan
kalkulator saintifik yang tidak boleh
diprogram.
Jumlah
NAMA : …………………………………………TINGKATAN : ………………

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The following formulae may be helpful in answering the questions. The symbols
given are the ones commonly used.
RELATIONS
1 am
× an
= am + n
2 am
÷ an
= am - n
3 (am
)n
= am n
4 A-1
= ⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
−
− ac
bd
bcad
1
5 P(A) =
( )
( )Sn
An
6 P(A’) = 1 – P(A)
7 Distance = ( )2
12
2
12 )( yyxx −+−
8 Midpoint
(x, y) = ⎟
⎠
⎞
⎜
⎝
⎛ ++
2
,
2
2121 yyxx
9 Average speed =
10 Mean =
11 Mean =
12 Pythagoras Theorem
c2
= a2
+ b2
13
12
12
xx
yy
m
−
−
=
14
erceptintx
erceptinty
m
−
−
=
distance traveled
time taken
sum of data
number of data
Sum of (midpoint of interval × frequency)
Sum of frequencies

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SHAPE AND SPACE
1 Area of trapezium =
2
1
× 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 = πr2
h
8 Volume of cone =
3
1
πr2
h
9 Volume of sphere =
3
4
πr3
10 Volume of right pyramid =
3
1
× base area × height
11 Sum of interior angles of a polygon = (n – 2) × 180°
12
13
14 Scale factor, k =
PA
PA'
15 Area of image = k2
× area of object.
arc length angle subtended at centre
circumference of circle 360°
area of sector angle subtended at centre
area of circle 360°
=
=

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Section A
[52 marks]
Answer all questions in this section.
1 Solve the quadratic equation
2
1
3
52 2
=
−
p
p
.
[4 marks]
Answer:
2 Calculate the value of p and of q that satisfy the following simultaneous linear
equations:
3p – 4q = –2
132
2
1
=+ qp
[4 marks]
Answer:
For
Examiner’s
Use

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3 Diagram 1 shows a right prism. Right angled triangle PQR is the uniform cross-
section of the prism.
DIAGRAM 1
(a) Name the angle between the plane STP and the plane STQR,
(b) Calculate the angle between the plane STP and the plane STQR.
[4 marks]
Answer:
(a)
(b)
For
Examiner’s
Use
[Lihat sebelah
P
Q R
ST
U
9 cm
18 cm
12 cm

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4 A straight line ST is parallel to line y = –2x + 5 and passes through point (4, –2).
Find
(a) the gradient of the straight line ST,
(b) the equation of the straight line ST and hence, state its y-intercept.
. [5 marks]
Answer:
(a)
(b)
5 A bag contains 50 pens. 15 of them are blue and the rest are red and green.
A pen is chosen at random from the bag.
(a) Find the probability that a blue pen is chosen.
(b) The probability of choosing a green pen is
5
1
. How many green pens
are there in the bag?
(c) Another 10 green pens are put into the bag. Find the probability that a
green pen is chosen from the bag.
[5 marks]
Answer:
(a)
(b)
(c)

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6 Diagram 2(a) shows a container in the form of a right pyramid fully filled with
water. Diagram 2(b) shows an empty cylindrical container. The height of the
pyramid is 15 cm.
DIAGRAM 2(a) DIAGRAM 2(b)
All the water from the right pyramid container is poured into the cylindrical
container.
By using π =
7
22
, calculate the height, in cm, of the water level in the cylinder.
[4 marks]
Answer:
For
Examiner’s
Use
[Lihat sebelah
14 cmA B
CD
E
12 cm
8 cm

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7 In Diagram 3, OPQS is a quadrant with the centre O and OSQR is a semicircle
with the centre S.
DIAGRAM 3
Given that OP = 14 cm. Using π =
7
22
, calculate
(a) the area, in cm2
, of the shaded region,
(b) the perimeter, in cm, of the whole diagram.
[6 marks]
Answer:
(a)
(b)
O P
Q
R S
60°
T
For
Examiner’s
Use

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8 (a) Determine whether the following is a statement or not. Give a reason to
your answer.
(b) Rewrite the following statement by inserting the word ‘not’ into the
original statement. State the truth value of your new statement.
(c) Construct a true statement using a suitable quantifier for the given object
and the property.
[5 marks]
Answer:
(a) ……………………………………………………………………………..
(b) ……………………………………………………………………………..
(c) …………………………………………………………………………….
For
Examiner’s
Use
[Lihat sebelah
64 = 42
3 is the factor of 15
Object : Triangles
Property : Have a right angle

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9 The Venn diagram in the answer space shows set P, set Q and set R with the
universal set ξ = P ∪ Q ∪ R.
On the diagrams provided in the answer space, shade
(a) the set Q ∩ (P ∪ R),
(b) the set Q ∪ R’ ∩ P.
. [3 marks]
Answer:
(a) (b)
10 In Diagram 4 , AE is a tangent to the circle centre O, at A. CDE is a straight line.
DIAGRAM 4
Find the value of
(a) x
(b) y
(c) z
[5 marks]
Answer:
(a)
(b)
(c)
For
Examiner’s
Use
P
R
Q
A
20°
B
C
D
70°
z°
x°
y° O E
P
R
Q

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11 (a) Diagram 5 shows a unit circle.
DIAGRAM 5
Find the value of
(i) sin v°
(ii) cos v° + sin 270°
[3 marks]
(b) Diagram 6 shows a rhombus PQRS.
DIAGRAM 6
It is given that TSQ is a straight line and tan x° =
12
5
− .
Find
(i) the length of RP
(ii) sin x°
[3 marks]
Answer:
(a) (i)
(ii)
(b) (i)
(ii)
For
Examiner’s
Use
[Lihat sebelah
–1 1
–1
1
y
x
v°
(– 0.42, – 0.91)
O
P
Q
R
S
T
x°
26 cm

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Section B
[48 marks]
Answer four questions in this section.
12 (a) In Diagram 7, a straight line RS is parallel to the straight line PQ. The
equation of PQ is 2y = x – 4.
DIAGRAM 7
Find
(i) the y-intercept of PQ,
(ii) the value of m,
(iii) the equation of RS.
[6 marks]
Answer:
(a) (i)
(ii)
(iii)
For
Examiner’s
Use
y
xO
R Q(8, m)
P
S(4, 5)

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(b) In Diagram 8, EFGH is a parallelogram and O is the origin. The gradient
of EF is 2.
DIAGRAM 8
Find
(i) the equation of line EF,
(ii) the coordinates of H,
(iii) the gradient of HF.
[6 marks]
Answer:
(b) (i)
(ii)
(iii)
[Lihat sebelah
For
Examiner’s
Use
H
xO G
E
F(4, 10)y
12

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13 (a) Diagram 9 shows a right prism with a horizontal rectangular base PQRS.
DIAGRAM 9
Given that E and F are midpoints of WV and SR respectively.
(i) Find the length of PF,
(ii) Calculate the angle between the line PE and the plane PQRS,
(iii) Name the angle between the plane PQVE and the plane PQUT.
[6 marks]
Answer:
(a) (i)
(ii)
(iii)
For
Examiner’s
Use
8 cm
5 cm
7 cm
12 cm
P
Q
R
U
W
T
E
F
S
V

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(b) In Diagram 10, MP and LQ are two vertical flagpoles standing on
horizontal ground KLM .
DIAGRAM 10
Given that the angle of depression of P from Q is 38°, MP = 5 m,
LQ = h m and KM = 10 m.
Calculate
(i) the angle of elevation of P from K,
(ii) the value of h.
[6 marks]
Answer:
(b) (i)
(ii)
For
Examiner’s
Use
400
5 m
P
70° M
Q
h m
K
L
13
[Lihat sebelah

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14 (a) (i) Fill the blanks with ‘and’ or ‘or’ in order to form a true statement.
(a) sin 30˚ =
2
1
cos 30˚ =
2
1
(b) All the multiples of 3 are multiples of 6 all multiples
of 6 are multiples of 3.
(ii) Complete the argument below.
Premise 1 : If a ∈ A, then a ∈ A ∪ B.
Premise 2 : __________________________________
Conclusion : 5 ∉ A
(iii) Make a general conclusion by induction for the sequence
13, 28, 49, 76,… which follows the following pattern:
13 = 3 (2)2
+ 1
28 = 3 (3)2
+ 1
49 = 3 (4)2
+ 1
76 = 3 (5)2
+ 1
………………
………………
[6 marks]
Answer:
(a) (i) (a) ……………………………………………………………….
(b) ……………………………………………………………….
(ii) ..........................................................................................................
(iii) ..........................................................................................................
For
Examiner’s
Use

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(b) Diagram 11 is an incomplete Venn diagram. A group of 45 students
involved in their favourite games.
DIAGRAM 11
Given that ξ = B ∪ S ∪ T and the number of students who involve in
badminton only is the same as the students who involve in tennis only.
Find the number of students who involve in
(i) all the three games,
(ii) neither badminton nor tennis,
(iii) two games only.
(iv) tennis only.
[6 marks]
Answer:
(b) (i)
(ii)
(iii)
(iv)
For
Examiner’s
Use
[Lihat sebelah
B = Badminton
2
5
3
6
11
T = Tennis
S = Squasy
14

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15 The daily distances, in km, travelled in 40 days by a salesman are shown in
Diagram 12.
DIAGRAM 12
(a) Using data in Diagram 12 and a class interval of 10 km, complete Table 1
in the answer space.
[4 marks]
Answer:
(a)
TABLE 1
(b) Determine the range of the given data. [1 mark]
(c) Based on your table in (a), calculate the estimated mean distance
travelled.
[3 marks]
Answer:
(b)
(c) (i)
(ii)
(d) For this part of the question, use the graph paper on page 19.
By using a scale of 2 cm to 10 km on the x-axis and a scale of 2 cm to
1 day on the y-axis, draw a frequency polygon for the data above.
[4 marks]
Answer:
(d) Refer graph on page 19.
42 62 51 37 70 14 12 93
53 45 54 27 70 76 91 72
68 58 57 65 21 18 22 87
64 63 57 59 70 19 41 55
36 56 50 57 73 75 13 39
Distance (km) Midpoint Frequency
11 – 20
21 – 30
31 – 40
For
Examiner’s
Use

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Graph for Question 15(d)

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16 The histogram in Diagram 13 shows the marks achieved in a test by a group of
students in a particular school.
DIAGRAM 13
(a) Find the total number of students who sat for the test. [1 mark]
(b) Find the modal class. [1 mark]
(c) Based on the histogram, complete the Table 3 below. [3 marks]
Answer:
(a)
(b)
(c)
Marks Upper boundary Cumulative
frequency
20 – 29 29.5 0
30 – 39 39.5 4
TABLE 3
(d) For this part of the question, use the graph paper on page 22.
By using the scale of 2 cm to 10 marks on the x-axis, and 2 cm to 10
students on the y-axis, draw an ogive for the data.
[4 marks]
(e) From the ogive, find
(i) the median,
(ii) the number of students who passed the test if the passing mark is
45%. [3 marks]
For
Examiner’s
Use
29.5 39.5 49.5 59.5 69.5 79.5 89.5 99.5
4
8
12
16
20
Marks
Frequency

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Answer:
(d) Refer graph on page 22.
(e) (i)
(ii)
For
Examiner’s
Use

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Graph for Question 16(d)
END OF QUESTION PAPER