La Salle College Examination
F.2
Number
Final Examination 2005-2006
Mathematics
Section A
Multiple Choice questions
Question Book
1. Write your Examination Number in the
spaces provided on this cover.
2. The total mark of the section is 30. Each
correct answer carries 1.5 marks.
3. Answer ALL questions in this section.
4. Mark all your answers on the Multiple
Choice Answer Sheet provided.
5. Use HB pencils in marking your answers.
6. NO MARK will be given if you mark more
than one answer in each question
7. The diagrams in this question book are not
necessarily drawn to scale.
1
Section A: Multiple Choice questions (30 marks)
1. The lower limit and the upper limit of the measurement 127.5 cm, correct to the nearest 0.1 cm, are
respectively
A. 127 cm and 128 cm.
B. 127.4 cm and 127.6 cm.
C. 127.45 cm and 127.55 cm.
D. 127.49 cm and 127.51 cm.
2. The length and the breadth of a rectangle are 6.4 cm and 8.6 cm respectively when each
measurement is rounded off correct to 1 decimal place. Which of the following cannot be the
actual area of the rectangle?
A. 54.5 cm2
B. 55.3 cm2
C. 55.7 cm2
D. 56.2 cm2
1 1
3. Simplify 2 .
ab b 2
a ab
1
A.
ab
a b
B.
ab
1
C.
ab(a b)
a b
D.
ab(a b)
I 4
4. If 2 , then r =
e R r
I
A. R2 .
4e
4e
B. . R2
I
4e
C. .
IR 2
IR 2
D. .
4e
2
a: b: c =
1 :: 5
1 1
5. 1: : =
1x10
2 5
5x10
= :
x10 :
A. 1 : 2 : 5
B. 5 : 2 : 1
C. 2 : 5 : 10
O
D. 10 : 5 : 2
1
2 y 1
x
.
1
y2
x
A. x = 2, y = 1
B. x = 2, y = 1
C. x = 1, y = 1
D. x = 1, y = 1
&
7. Referring to the cumulative frequency distribution table, what is the frequency of the third
class interval?
A. 17
Mark less than 15 20 25 30 35
B. 22.5 Cumulative frequency 9 20 37 51 60
C. 27.5
D. 37
8. Referring to the diagram, which of the followings must be true?
d
A. b + d + e = 180 c
B. a + b + d + e = 180
b
C. a + b = d + e
e
a
D. a + b + c + d + e = 360
3
A. 18
B. 16
C. 14
D. 12
I. AB // CD >
II. EG // FH
III. AB // CD and EG // FH
>
A. I only
B. II only
C. III only
D. None of the above
Given conditions: (i) The sum of all interior angles of a triangle is 180
(ii) Equilateral triangle is a kind of triangle.
O
A. The sum of all interior angles of an equilateral triangle is 180.
B. All interior angles of an equilateral triangle are 60
↑
C. All triangles are equilateral triangles.
D. The interior angles of an equilateral triangle are equal.
12. If 2, 3 and x are the lengths of a right-angled triangle, what is/are the possible value(s) of x?
* I. 4
II. 5
~ III. 13
A. I only
B. II only
C. III only
8
D. II and III only
4
13. Given that QR2 – PR2 = PQ2, which of the followings about PQR must be true?
*
I. R = 90
z II. QR is the longest side.
III. QR = PQ + PR
A. I only
②B. II only
C. III only
D. I and II only
14. Referring to the diagram, a + b + c + d + e + f = 3688
A. 270
a f
8
B. 360
Sa + b +e
C. 540
b C e
D. 720 a+ 2
c d
sin
15. Referring to the diagram, = A
sin
a
A. d
b b
b
B.
a
B D
ab
C.
d2 a
d2
D. C
ab
16. Referring to the diagram, DB = C
A. d sin β cos α
d
B. d cos β sin α
C. d sin β tan α
A B
D
D. d cos β tan α
5
&17. Given that 12 sin – 5 cos = 0, find the value of sin .
A. 0
5
B.
12
5
C.
13
12
D.
13
-
18. Which of the followings is/are identity/identities?
I. 2 sin2 – 1 = sin2 – cos2
1
II. sin + tan (90 – ) =
sin
III. cos2 (90 – ) = 1 – cos2
A. I and III only
B. I and II only
C. II and III only
D. All of the above
19. Two metallic cubes of side 8 cm are melted and recast into a cylinder with base radius 4 cm. Find
the height of the cylinder formed, correct to the nearest 0.1 cm.
A. 10.2 cm
B. 13.3 cm
C. 20.4 cm
D. 40.7 cm
Ev zr
20. If A and C stand for the area and the circumference of a circle respectively, then
A
A. C = C22 A =
v2
B. C = 2A = ~
C. C = 4A
:
2AnE
OD. C =2 A
(E) = .
End of Section A
6
La Salle College Examination
Number
F.2 Final Examination 2005-2006
Questions Marks
Mathematics
21
Section B
Conventional questions 22
Question - Answer Book
23
1. Write your Examination Number in the
24
spaces provided on this cover.
25
2. The total mark of this section is 70.
26
3. Answer ALL questions in this section. 27
4. Supplementary answer sheets will be supplied 28
on request. Write your Examination number
29
on each supplementary answer sheet, if any.
30
5. Show ALL necessary working steps in this
Total
section.
6. Unless otherwise specified, numerical
answers should either be exact or correct to 3
significant figures.
7. The diagrams in this question book are not
necessarily drawn to scale.
1/9
Section B: Conventional questions (70 marks)
21.
(a) Factorize the polynomial 8ax + 6bx – 4ay – 3by + 12a + 9b. (2 marks)
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
1 1 2 2
(b) Expand by using identities. (3 marks)
x y x y
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
(c) If x2 + Ax + 2 ≡ (x – 2) 2 + B, where A and B are constants, find A and B. (3 marks)
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
2/9
22. The total cost of 8 school badges and 3 school ties is $73, and the total cost of 2 school badges and
1 school tie is $21. What is the total cost of one school badge and one school tie? (6 marks)
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
3/9
23. A pet shop has three kinds of animals, namely, dogs, cats and tortoises. The ratio of the number
of dogs to that of cats is 4 : 3; the ratio of the number of the dogs to that of tortoises is 3 : 5.
(a) Find the ratio of the number of dogs : the number of cats : the number of tortoises. (2 marks)
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
(b) If there are altogether 123 animals in the pet shop, find the number of cats. (2 marks)
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
-
The heights of 30 objects
Cumulative
frequency
Height
(cm)
(a) How many objects have a height less than 16 cm? (2 marks)
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
(b) How many objects have a height between 14 cm and 18 cm? (2 marks)
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
(c) If there are 4 objects with a height more than x cm, find the value of x. (2 marks)
………………………………………………………………………………………………………………
4/9
25. In the figure, AB // CD, CE bisects AEF and BF bisects EFD.
A
E
B
C
(a) Prove that CE // FB. (6 marks)
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
(b) If ECF = 38˚, find EBF. (4 marks)
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
5/9
26. Each interior angle of a regular polygon is greater than the exterior angle by 135˚. Find the
number of sides of this regular polygon. (5 marks)
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
27. The distance between city A and city B is 800 km and that between city B and city C is 1000 km.
The locations of the three cities form a right-angled triangle and ABC = 90˚. In the past, to
travel from city A to city C, one must pass through city B. Now, an airline company introduces a
direct flight between cities A and C. Find the distance shortened with the introduction of the
direct flight, correct to the nearest km. (6 marks)
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
6/9
28. A ladder of length 5 m is put against a vertical wall with the lower end on a horizontal ground.
The angle between the ladder and the ground is 75˚. The ground is slippery and the ladder slides
down.
(a) If the ladder slides down to half of its original height, draw a diagram to show the original and new
positions of the ladder. (Label your diagram appropriately) (2 marks)
(b) Find the angle between the ladder at the new position and the ground, correct to the nearest degree.
(4 marks)
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
(c) Find the distance moved by the lower end of the ladder along the ground, correct to 3 significant
figures. (4 marks)
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
7/9
&
29.
4
(a) Given that tan α. = , find the value of cos2 α – sin2 α by using Pythagoras’ Theorem. (3 marks)
5
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
1
(b) Given that sin β = , find the value of 3 cos2 β + sin β – 1 by using trigonometric identities.
4
(3 marks)
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
1
(c) Prove that sin θ sin(90˚ – θ) ≡ . (3 marks)
tan tan90
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
8/9
30. A cylindrical container is 12 cm high. The outer diameter of the container is 6 cm and the inner
diameter is 5 cm, and the container is of uniform thickness.
(a) If all the surfaces of the container are painted by paint at a cost of $4 per cm2, find the total cost for
painting the whole container, correct to the nearest dollar. (3 marks)
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
(b) If the container is made of metal weighing 5.8 g per cm3, find the weight of the container correct to
the nearest g. (3 marks)
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
End of Section B
9/9
