2004-CE
math
Paper I
LIU PO SHAN MEMORIAL COLLEGE
(2003 - 2004)
MOCK EXAMINATION
SECONDARY
FIVE
MATHEMATICS PAPER I
Class:
(
)
Date : 17 – 3 – 2004
Name:
Time allowed: 2 hours
1. This paper must be answered in English.
2. Write your name, class and class number in the spaces provided on this cover.
3. This paper consists of THREE sections, A(1), A(2) and B. Section A(1) carries 34 marks. Sections
A(2) and B carry 33 marks.
4. Attempt ALL questions in Sections A(1) and A(2), and any THREE questions in Section B. Write
your answers in the spaces provided in this Question-Answer Book. Supplementary answer sheets
will be supplied on request. Write your name, class and class number on each sheet.
5. Unless otherwise specified, all working must be clearly shown.
6. Unless otherwise specified, numerical answers should either be exact or correct to 3 significant
figures.
7. The diagrams in this paper are not necessarily drawn to scale.
FORMULAS FOR REFERENCE
SPHERE
CYLINDER
Surface area
=
4π r 2
Volume
=
4
π r3
3
Area of curved surface =
2π rh
Volume
CONE
= π r 2h
Area of curved surface = π r l
Volume
=
1
π r 2h
3
PRISM
Volume
=
base area × height
PYRAMID
Volume
=
1
× base area × height
3
P.1
SECTION A(1) (34 marks)
Answer ALL questions in this section and write your answers in the spaces provided.
(a −2 b) 3
1.
Simplify
2.
Make x the subject of the formula a =
a2
and express your answer with positive indices.
(3 marks)
bx − 2
.
x
(3 marks)
3. Let f(x) = x3 + 3x2 – 4x – 12 .
a) Find f(–2) .
b) Factorize f(x) .
(3 marks)
P.2
4. Find the perimeter of the sector in Figure 1.
(3 marks)
Figure 1
5. Find the range of values of x which satisfy both 5 – 2x > – 4 + x and x + 4 > 0 and represent
the solution in Figure 2.
(4 marks)
Figure 2
6. In Figure 3, A, B, C, D are points on a circle. Find x.
(4 marks)
Figure 3
P.3
7. The radius of a sphere is 6 cm. A new sphere is formed by decreasing the radius by 20%.
a) Find the surface area of the new sphere.
b) Find the percentage decrease in the surface area of the sphere.
(4 marks)
8. An auditorium has 20 rows of seats. All seats are numbered in numerical order from the first row to
the last row, and from left to right, as shown in Figure 4. The first row has 30 seats. The second row
has 33 seats. Each succeeding row has 3 more seats than the previous one.
Figure 4
a) How many seats are there in the last row ?
(2 marks)
b) Determine in which row the seat numbered 500 is located.
(3 marks)
P.4
9. In Figure 5, find BC and the area of △ABC.
(5 marks)
Figure 5
P.5
SECTION A(2) (33 marks)
Answer ALL questions in this section and write your answers in the spaces provided.
10. P is the sum of two parts. One part varies directly as t and the other part varies inversely as t .
When t = 2, P = 22 and when t = 3, P = 23 .
a) Express P in terms of t.
(4 marks)
b) Find the value(s) of t when P = 34.
(2 marks)
P.6
11. In Figure 6, ABCD is a parallelogram. E and F are points on AB and CD respectively and
∠ADE=∠CBF。
Figure 6
a) Prove that ∠EDF = ∠FBE。
(2 marks)
b) Prove that EBFD is a parallelogram.
(4 marks)
P.7
12. In Figure 7(a), a piece of wood in the form of a right circular cone is cut into two portions by a plane
parallel to its base. The lower portion is a frustum with height 12 cm, and the radii of the two
parallel faces are 5 cm and 9 cm respectively. The pen-stand shown in Figure 7(b) is made from
the frustum by drilling a hole in the middle. The hole consists of a cylindrical upper part of radius
4 cm and a hemispherical lower part of the same radius. The depth of the hole is 10 cm.
Figure 7(a)
Figure 7(b)
a)
Find the capacity of the hole.
(2 marks)
b)
Find the volume of wood in the pen-stand.
(4 marks)
P.8
13. The marks scored by 10 students in a mathematics quiz are as follows:
10 20 30 45 50 60 65 65 65 70
a) Find
(i) the mode,
(ii) the median,
(iii) the mean,
(iv) the standard deviation
and (v) the inter-quartile range
of the above marks.
(5 marks)
b) The teacher found that the marks were too low. If 10 marks are added to each of the above
marks, what will be the new mean and standard deviation ?
(2 marks)
P.9
14. In Figure 8, L is the straight line passing through A(0, 4) and B(4,2).
a) Find the equation of L.
(2 marks)
b) L1 is the line passing through (5,5) and perpendicular to the line L. Find the equation of L1.
(3 marks)
c) If L intersects the x-axis at C, find the coordinates of C and the area of △OAC.
Figure 8
P.10
(3 marks)
SECTION B (33 marks)
Answer any THREE questions in this section and write your answers in the spaces provided.
question carries 11 marks.
Each
15. Amy lives in Tuen Mun. She travels to school either by West Rail or by bus. The probability of
being late for school is 0.08 if she travels by West Rail and 0.2 if she travels by bus.
a) In a certain week, Amy travels to school by West Rail on Monday and Tuesday. Find the
probability that
(i) she will be late on all these two days;
(ii) she will not be late on all these two days.
(4 marks)
b) In the same week, Amy travels to school by bus on Wednesday, Thursday and Friday. Find the
probability that
(i) she will be late on Thursday and Friday only in these three days;
(ii) she will not be late on only one of these three days.
(4 marks)
c) Amy travels to school by bus on Saturday. The bus fare is $11. She does not have an
Octopus card but has two10-dollar coins, three 5-dollar coins and four 1-dollar coins. If she
randomly takes out two coins, what is the probability that the total value of these coins is exactly
$11 ?
(3 marks)
P.11
P.12
16. a)
Figure 9
(i) On the graph paper provided, draw the following straight lines:
4x + y = 80
x + y = 50
x + 3y = 80 .
(ii) On the same graph paper, shade the region that satisfies all the following constraints:
⎧ x≥0
⎪ y≥0
⎪⎪
⎨ 4 x + y ≥ 80
⎪ x + y ≥ 50
⎪
⎪⎩ x + 3 y ≥ 80
.
(4 marks)
P.13
b) A company had two workshops A and B. Workshop A produces 80 electric fans, 10 refrigerators
and 10 air conditioners each day; Workshop B produces 20 electric fans, 10 refrigerators and 30
air conditioners each day. The expenditures to operate Workshop A and Workshop B are
respectively $45000 and $30000 each day. The company gets an order for 1600 electric fans,
500 refrigerators and 800 air conditioners. Using the result of (a)(ii) to find the number of days
each workshop should operate to meet the order if the total expenditure in operating the
workshops is to be kept to a minimum.
(Denote the number of days that Workshops A and B
should operate by x and y respectively.)
(7 marks)
P.14
17. Figure 10 shows two circles C1 and C2 touching each other externally. The center of C1 is (–5,0) and
the equation of C2 is (x + 11)2 + (y – 8)2 = 36.
Figure 10
a) Find the equation of C1.
(3 marks)
b) Find the equations of the two tangents to C1 from the origin.
(4 marks)
c)
One of the tangents in (b) cuts C2 at two distinct points P and Q. Find the coordinates of the
mid-point of PQ.
(4 marks)
P.15
P.16
18.
Figure 11(a)
Figure 11(b)
In Figure 11(a), ABCD is a thin square metal sheet of side 2.5 m. The metal sheet is folded along
BD and the edges AD and CD of the folded metal sheet are placed on a horizontal plane with B
2 metres vertically above the plane as shown in Figure 11(b). E is the foot of the perpendicular
from B to the plane.
a) Find the lengths of BD, ED and AE.
(3 marks)
b) Find ∠ADE.
(2 marks)
c) Find the angle between BD and the plane.
(2 marks)
d) Find the angle between the planes ABD and CBD.
(4 marks)
P.17
END OF PAPER
P.18