題記：「知識百寶箱」系列是「寶仁工作室」為了實踐2021-2022年度工作願景，而特別設立。目的旨在加快「寶仁工作室」的轉型，全面成為以「知識型專欄」為基礎之「知識主導型」的網誌。期望以協助提升大眾的學術素養為信條，並配合STEM的發展。除了普及科學知識外，也負起潛移默化為大家的人生有所改變的重責大任。將以學術專題探討、學習筆記為內容主體，回饋社會，服務讀者。
内容介紹：本篇屬重啟篇章，會跟讀者分析會考必修數學課程，本人將透過多份歷史資料，配合本人對高等數學課程的了解，客觀分析會考必修數學課程的歷年沿革。以協助讀者對會考必修數學課程的歷年沿革有基本了解，從而幫助有讀者可以選用合適的Past Paper去備試，爭取好成績。以及協助教育工作者在進行數學教育工作時，有更多參考資源可供參考。
各位大家好，本篇為《2024年第13篇：香港中學會考數學科目分析（必修數學篇）》的附錄，以中英雙語形式，整理並列出當年的會考數學課程。建議讀者連同正篇一同閲讀，從而令大家更清晰了解歷年會考數學課程之發展。
在本人撰寫正篇及附錄期間，曾研讀大量參考讀物，包括：
Ø 梁操雅和羅天佑合著的《香港考評文化的承與變：從強調篩選到反映能力》
Ø 歷年會考之Regulations and
Syllabues
Ø Syllabues for Secondary Schools - Mathematics Syllabus A (From I-V)
(1982)
Ø Syllabues for Secondary Schools - Mathematics Syllabus B (From I-V)
(1982)
Ø 中學課程綱要－數學科課程綱要 (中一至中五適用) (1985)
Ø 中學課程綱要－數學科 (中一至中五) (1999)
這些Regulations and Syllabues，以及上述列出之參考讀物，可以在香港中央圖書館、港大參考圖館和中大圖書館中找到。這都幫助本人對會考數學課程有如此透徹的了解，特此鳴謝。
附錄1：1937年會考數學課程
1.
Arithmetic算術 (Group 3 Part 2)
Four Rules – Simple and Compound – Prime Numbers – HCF and LCM –
Fractions – Decimals – Metric System – Unitary Method – Easy problems on Work
and Time, and Time and Distance – Averages – Percentages – Simple and Compound
Interest – Areas and Volumes – Square Root – Use of logarithm tables – Graphs.
（中文參考譯本）四則運算；簡單和複合單位；質數；最大公因數及最小公倍數；分數；小數；公制單位；歸一法；工作和時間、時間和距離的簡單問題；平均數；百分數；單利息和複利息；面積和體積；平方根；對數表的使用；圖像。
(Covered by Hall &
Stevens’ Arithmetic – Excluding Chap. X, Chap. XVIII (p.p. 366-370), Chap. XIX,
Chap. XX.)
2.
Algebra代數 (Group 3 Part 2)
Four Rules – Brackets – Symbolic Expression and Use of Formula –
Simple Equations in one or two unknowns and quadratic equations in one unknown,
and problems involving the solution of such equations – Remainder Theorem –
Easy factors – Easy Identities – HCF and LCM. Fraction – Square Root – Graphs.
（中文參考譯本）四則運算；括號；符號表達式和公式的使用；一元或二元簡易方程及一元二次方程，以及涉及此類方程解的問題；餘式定理；簡單因式；簡單恆等式；最大公因數及最小公倍數；分數；平方根；圖像。
(Covered by Hall &
Knights Algebra, Part I).
3.
Geometry幾何 (Group 3 Part 2)
Definitions – Axioms – Postulates – The subject matter of Euclid
Book I – Easy deductions, problems on loci, and geometrical constructions.
（中文參考譯本）定義；公理；公設；《歐幾里得第一卷》的主題；簡單演繹法；軌跡問題和幾何結構。
(Covered by Hall &
Stevens Geometry, Parts I & II, and to page 171).
4.
Trigonometry三角 (Group 3 Part 2)
The Measurement of Angular magnitude – The circular functions, their
relations and properties, and graphical representation of the circular
functions of a single angle – Easy Identities and Equations – Right-angled
triangles – The sine and cosine rules for any triangle – Area of a triangle –
Easy problems on heights and distance – Use of logarithmic and trigonometrical
tables.
（中文參考譯本）角大小的測量；三角函數、它們的關係和性質，以及單一角度的三角函數的圖像表示；簡單恆等式和方程；直角三角形；任意三角形的正弦法則和餘弦法則；三角形的面積；關於高度和距離的簡單問題；使用對數和三角表。
(Covered by Hall &
Knights Trigonometry, Chaps. I-XV, excluding Chaps. XI, XII, Chap. XIII, Arts 147-150).
5.
In order to pass in Part 2,
candidates must satisfy the examiners in Section (i) and in one of Sections
(ii), (iii), or (iv).
（中文參考譯本）為了通過第 2 部分，考生必須滿足第 (i) 節以及第 (ii)、(iii) 或 (iv) 節之一的考官要求。
附錄2：1952年至1974年會考數學課程
Version 1: 1952, 1953, 1954, 1955, 1956,
1957, 1958, 1959, 1960, 1961, 1962, 1963, 1964
Mathematics (including Elementary
Mathematics) (1962-1963
Change to Lower Mathematics)
Paper I: Arithmetic
and Algebra (2 hours)
Paper
II: Geometry and Trigonometry (2 hours)
Since
1962, Lower Mathematics only one paper of 2.5 hours
Arithmetic
1.
Elementary principles and
processes; first four rules; factors, HCF and LCM.
2.
British and Metric systems of
weights and measures. Pounds, shillings and pence; dollars and cents; piculs,
catties and taels.
3.
Vulgar and decimal fractions,
excluding recurring decimals. Square roots.
4.
Averages; ratio and proportion;
percentages, profit and loss.
5.
Simple interest, direct and
inverse. Calculation of compound interest. Elementary stocks and shares,
excluding brokerage.
6.
Elementary mensuration,
including simple formulae for determining volumes and areas connected with the
rectangular block, circular cylinder and sphere.
7.
Use of logarithm table. (1957 need to include in
Elementary Mathematics)
8.
Significant figures. (The use of contracted methods of multiplication and division will
not be compulsory in any question.)
Algebra
1.
Elementary operations;
symbolical expression; interpretation and evaluation of formulae. Change of
subject of a formula.
2.
Factors of expressions of
simple types, including a^3±b^3. HCF and LCM by factors. Easy fractions. Elementary applications
of the Remainder Theorem.
3.
Solution of simple equations;
of simultaneous linear equations with two unknowns; of quadratic equations with
one unknown (by factors only in Elementary Mathematics); and of simultaneous
equations, one linear and one quadratic, with two unknowns. Simple problems
leading to such equations. Easy literal equations.
4.
Graphs and their simple
applications. (until 1961)
Graphs and their simple
applications (linear functions only for Lower Mathematics). (1962)
5.
The use of negative and
fractional indices.
Note:
In 1962, for Lower Mathematics, quadratic equations will not be required, and
for graphs, linear functions only will be required.
Geometry
1.
The paper will contain
questions on Practical and Theoretical Geometry.
2.
The questions on Practical
Geometry will be set on the constructions contained in Schedule A, together
with easy extensions of them. Constructions need not be described, nor proofs
of constructions given, unless specifically required by the question. All
figures must be drawn accurately and distinctly, with construction lines
clearly visible.
3.
Candidates will be required to
bring to the examination a ruler, graduated in inches and tenths and in
centimetres and millimetres, a set square, protractor, compasses and a fairly
hard pencil. Questions may be set in which the use of certain instruments is
forbidden.
4.
The questions on Theoretical
Geometry will consist of theorems contained in Schedule B, together with simple
applications of these to the examiners to form part of a systematic treatment
of the subject; the order in which the theorems are stated in Schedule B is not
imposed as a sequence of their treatment. In Elementary/Lower Mathematics
proofs of theorems will not be required, but simple problems depending on them
will be set. The use of algebraic symbols and of hypothetical constructions
will be permitted.
Trigonometry
1.
The trigonometrical ratios. Solution
of right-angled triangles, with simple applications. Easy problems (in two of three
dimensions) soluble by analysis into right-angled triangles.
2.
The use of logarithmic
and trigonometrical tables.
(1957 need to include in Elementary Mathematics)
3.
Radian measure; length
of arc, area of sectors.
4.
Simple relations between the
ratios. Identities.
5.
Angles of any magnitude.
Easy equations (solutions from 0 to 360 only). Graphs of simple trigonometrical
functions.
6.
The Sine and Cosine
Rules for a triangle. The formulae 1/2 bc sin A, sqrt(s(s-a)(s-b)(s-c)) (the
proof of the latter formula will not be required).
7.
The solution of
triangles.
數學（包括初級數學）
試卷一：算術和代數（2 小時）
試卷二：幾何與三角學（2 小時）
自1962年以來，初級數學只有一張2.5小時之試卷
算術
1.
基本原理和過程；四則運算；
因子，最大公因數和最小公倍數。
2.
英制和公制的度量衡系統。
英鎊、先令和便士；
美元和美分；
匹、斤、兩。
3.
常用分數（即現在的分數）及十進分數（即現在的小數），不包括循環小數。
平方根。
4.
平均數；
比和比例；
百分數、盈利和虧損。
5.
單利息、正比例和反比例。
複利息的計算。 基本股票和股份，不包括經紀業務。
6.
基本測量，包括確定長方體、圓柱體和球體的體積和面積的簡單公式。
7.
對數表的使用。（1957年需納入初級數學考試範圍）
8.
有效數字。（在任何問題中都不強制使用乘法和除法的約定方法。）
代數
1.
基本運算；
符號表達；
公式的闡釋和計算。
主項變換。
2.
簡單代數式的因子，包括a^3±b^3。最大公因式及最小公倍式。
簡單分數。
餘式定理的基本應用。
3.
簡單方程式的解；
具有兩個未知量的聯立線性方程；
具有一個未知量的二次方程式（初級數學僅需使用因式法）；
以及聯立方程，一為線性方程和一為二次方程，有兩個未知量。需要使用方程之簡單問題。
簡單文字方程。
4.
圖像及其簡單應用。
（直至1961）
圖像及其簡單應用（初級數學只需認識線性函數）。
（1962）
5.
負數和分數指數的使用。
註：在1962年，初級數學不需要學習二次方程，圖像只需要了解線性函數。
幾何
1.
試卷將包含實用幾何和理論幾何的問題。
2.
實用幾何的問題將根據附表A中包含的結構以及它們的簡單擴展進行設定。 除非問題有特別要求，否則不需要描述結構，也不需要給予結構證明。
所有圖形必須繪製準確、清晰，結構線清晰可見。
3.
考生參加考試時必須攜帶一把以英吋和十分之一、公分和毫米為單位的尺、三角尺、量角器、圓規和一支硬身鉛筆。
可能會提出禁止使用某些工具的問題。
4.
理論幾何問題將包含附表 B 的定理，以及對考官的簡單應用，構成該科目系統處理的一部分； 附表 B 中陳述定理的順序並未強調為處理它們的順序。 在初級數學中，不需要定理證明，但會設定依賴定理的簡單問題。允許使用代數符號和假設結構。
三角學
1.
三角比。
直角三角形的解法，簡單應用。
透過分析直角三角形可以解決簡單的問題（在二維或三維空間）。
2.
對數表和三角表的使用。（1957年需納入初級數學考試範圍）
3.
弧度法；
弧長、扇形面積。
4.
三角比之間的簡單關係。恒等式。
5.
任何大小的角度。
簡單方程（僅限0度到 360度的解）。 簡單三角函數圖像。
6.
三角形的正弦法則和餘弦法則。
公式 1/2 bc sin A, sqrt(s(s-a)(s-b)(s-c)) （後一個公式不需要證明）。
7.
三角形的解法。
Geometry Schedules for Mathematics and
Elementary Mathematics
Schedule A (Practical)
1.
Bisection of angles and of
straight lines.
2.
Construction of perpendiculars
to straight lines.
3.
Construction of an angle equal
to a given angle.
4.
Construction of angles of 60,
45 and 30.
5.
Construction of parallels to a
given straight line.
6.
Simple cases of the
construction from sufficient data of triangles and quadrilaterals.
7.
Division of straight lines into
a given number of equal parts or into parts in any given proportions.
8.
Construction of a triangle
equal in area to a given polygon.
9.
Construction of tangents to a
circle and of common tangents to two circles.
10.
Construction of circumscribed,
inscribed and escribed circles of a triangle.
11.
Simple cases of the
construction of circles from sufficient data.
Schedule B (Theoretical)
(Proofs of theorems shown in brackets will
not be required for Mathematics; no proofs of theorems will be required for
Elementary Mathematics.)
Angle at a Point
1.
(If a straight line stands on
another straight line, the sum of the two angles so formed is equal to two
right angles; and the converse.)
2.
(If two straight lines intersect,
the vertically opposite angles are equal).
Parallel Straight Lines
1.
(When a straight line cuts two
other straight lines, if
(i) a pair of alternate angles are equal,
or (ii) a pair of corresponding angles are equal,
or (iii) a pair of interior angles on the same side of the cutting
line are together equal to two right angles,
then the two straight lines are parallel, and the converse.)
2.
(Straight lines which are
parallel to the same straight line are parallel to one another).
Triangles & Rectilinear Figures
1.
The sum of the angles of a
triangle is equal to two right angles.
2.
In a polygon of n sides,
the sum of the interior angles in equal to 2n-4 right angles.
3.
If the sides of a convex
polygon are produced in order, the sum of the angles so formed is equal to four
right angles.
4.
(If two triangles have
two sides of the one equal to two sides of the other, each to each, and also
the angles contained by those sides equal, the triangles are congruent.)
5.
(If two triangles have
two angles of the one equal to two angles of the other, each to each, and also
one side of the one equal to the corresponding side of the other, the triangles
are congruent.)
6.
If two sides of a triangle are
equal, the angles opposite to these sides are equal; and the converse.
7.
(If two triangles have the
three sides of the one equal to the three sides of the other, each to each, the
triangles are congruent.)
8.
(If two right-angled triangles
have their hypotenuses equal, and one side of the one equal to one side of the
other, the triangle are congruent.)
9.
(If two sides of a
triangle are unequal, the greater side has the greater angle opposite to it,
and the converse.)
10.
(Of all the straight
lines that can be drawn to a given straight line from a given point outside it,
the perpendicular is the shortest.)
11.
The opposite sides & angles
of a parallelogram are equal, each diagonal bisects the parallelogram, and the
diagonals bisect each other.
12.
If a pair of opposite sides of
a quadrilateral are equal and parallel, it is a parallelogram.
13.
The straight line drawn through
the middle point of one side of a triangle parallel to another side bisects the
third side.
14.
The straight line joining the
middle points of two sides of a triangle is parallel to the third side, and
equal to one-half of it.
15.
If there are three or
more parallel straight lines, and the intercepts made by them on any straight
line that cuts them are equal, then the corresponding intercepts on any other
straight line that cuts them are also equal.
Areas
1.
Parallelograms one the same
base and between the same parallels are equal in area.
2.
The area of a triangle is half
the area of the rectangle on the same base and with the same altitude.
3.
(Triangles on the same or equal
bases and of the same altitude are equal in area.)
4.
(Equal triangles on the same or
equal bases are of the same altitude.)
5.
In a right-angled triangle, the
square described on the hypotenuse is equal to the sum of the squares described
on the sides containing the right angle (and the converse).
6.
The square on a side of
a triangle is greater or less than the sum of the squares on the other two
sides, according as the angle contained by those sides is obtuse or acute. The
difference is twice the rectangle contained by one of the two sides and the
projection on it of the other.
7.
In any triangle, the sum
of the squares on any two sides is equal to twice the square on half the third
side together with twice the square on the median which bisects the third side.
Loci
1.
The locus of a point
which is equidistant from two fixed points is the perpendicular bisector of the
straight line joining the two fixed points.
2.
The locus of a point which
is equidistant from two intersecting straight lines consists of the pair of
straight lines which bisect angles between the two given lines.
3.
(The perpendicular
bisectors of the sides of a triangle concurrent.)
4.
(The bisectors of the angles
of a triangle are concurrent.)
The Circle
1.
A straight line drawn from the
centre of a circle to bisect a chord which is not a diameter, is at right
angles to the chord; conversely, the perpendicular to a chord from the centre
bisects the chord.
2.
(There is one circle, and one
only, which passes through three given points not in a straight line.)
3.
Equal chords of a circle are
equidistant from the centre; and the converse.
4.
(The tangent at any point of a
circle and the radius through the point are perpendicular to each other.)
5.
If two circles touch, the point
of contact lies on the straight line through the centres.
6.
The angle which an arc
of a circle subtends at the centre is double that which is subtends at any
points on the remaining part of the circumference.
7.
Angles in the same
segment of a circle are equal; (and, if the line joining two points subtends
equal angles at two other points on the same side of it, the four points lie on
a circle.)
8.
The angle in a semicircle is a
right angle.
9.
(The circle described on the
hypotenuse of a right-angled triangle as diameter passes through the opposite
vertex.)
10.
The opposite angles of
any quadrilateral inscribed in a circle are supplementary; (and the converse.)
11.
(In equal circles (or,
in the same circle) (i) if two arcs subtend equal angles at the centre, they
are equal; (ii) conversely, if two arcs are equal, they subtend equal angles at
the centre).
12.
(In equal circles (or,
in the same circle) (i) if two chords are equal, they cut off equal arcs; (ii)
conversely, if two arcs are equal, the chords of the arcs are equal.)
13.
If a straight line touch
a circle, and from the point of contact a chord be drawn, the angles which this
chord makes with the tangent are equal to the angles in the alternate segments.
（中文參考譯本）
附表A（實用）
1.
角和直線的平分。
2.
垂直於直線的繪製。
3.
繪製一個等於給定角的角。
4.
60°、45°、30°角的繪製。
5.
繪製與給定直線的平行線。
6.
利用足夠的三角形和四邊形資料去繪製簡單案例。
7.
將直線分成給定數量的等份或依任意給定比例的份數。
8.
繪製面積與給定多邊形相等的三角形。
9.
建構圓的切線和兩個圓的公切線。
10.
三角形的外接圓、內切圓和外切圓的繪製。
11.
根據足夠的數據建立圓形的簡單案例。
附表 B（理論）
（數學不需要括號中所示的定理證明；初級數學不需要定理證明。）
直線上的角
1.
（如果一條直線與另一條直線重疊，則所形成的兩個角和等於兩個直角；反之亦然。）[直線上的鄰角]
2.
（如果兩條直線相交，則垂直的對角相等）。[對頂角]
平行的直線
1.
（當一條直線與另外兩條直線相交時，如果
(i) 一對交錯角相等，[內錯角相等]
或 (ii) 一對對應的角度相等，[同位角相等]
(iii) 截線同一側的一對內角合起來等於兩個直角，[同旁內角互補]
則兩條直線平行，反之亦然。）
2.
（與同一條直線平行的直線彼此平行）。
三角形和直線圖形
1.
三角形的內角和等於兩個直角。[三角形內角和]
2.
在 n 條邊的多邊形中，內角和等於 2n-4 個直角。[多邊形內角和]
3.
若凸多邊形的邊依序生成，則所形成的角和等於四個直角。[多邊形外角和]
4.
（如果兩個三角形的兩邊與另一個三角形的兩邊相等，而這些邊所含的角度也相等，則這兩個三角形全等。）[SAS]
5.
（如果兩個三角形的兩個角等於另一個三角形的兩個角，且一個三角形的一邊等於另一個三角形的對應邊，則這兩個三角形全等。）[ASA、AAS]
6.
如果三角形的兩邊相等，則這兩邊的對角也相等；
反之亦然。[全等三角形的對應邊、全等三角形的對應角]
7.
（如果兩個三角形的一個的三邊等於另一個的三邊，則這兩個三角形全等。）[SSS]
8.
（如果兩個直角三角形的斜邊相等，並且一個直角三角形的一側等於另一個直角三角形的一側，則該三角形全等。）[RHS]
9.
（如果三角形的兩邊不相等，則較大的邊與其相對的角度較大，反之亦然。）[大邊對大角、大角對大邊]
10.
（從給定直線外部的給定點可以繪製到給定直線的所有直線中，垂線是最短的。）
11.
平行四邊形的對邊和角相等，每條對角線平分平行四邊形，對角線互相平分。[平行四邊形對邊、平行四邊形對角、平行四邊形對角線]
12.
如果四邊形的一對對邊相等且平行，則它是平行四邊形。[對邊平行且相等]
13.
穿過三角形一邊的中點與另一邊平行的直線平分第三邊。[中點定理]
14.
連接三角形兩邊中點的直線平行於第三邊，並且等於它的二分之一。[中點定理]
15.
如果有3條或3條以上平行直線，且它們在與它們相交的任何一條直線上的截距相等，則在與它們相交的任何其他直線上相應的截距也相等。[截線定理]
面積
1.
同底且同平行線之間的平行四邊形面積相等。
2.
三角形的面積是同底同高的長方形面積的一半。
3.
（等底、等高的三角形面積相等。）
4.
（相同或等底的等三角形的高相同。）
5.
在直角三角形中，斜邊上的平方等於直角邊的平方和（反之亦然）。[畢氏定理、畢氏定理逆定理]
6.
三角形的一邊的平方大於或小於另外兩邊的平方和，取決於這些邊所含的角度是鈍角還是銳角。其差值是兩邊之一和另一邊在其上的投影所包含的矩形的兩倍。
7.
在任何三角形中，任兩邊的平方和等於第三邊一半的平方的兩倍加上平分第三邊的中線的平方的兩倍。[中線定理]
軌跡
1.
與兩個固定點等距的點的軌跡是連接兩個固定點的直線的垂直平分線。
2.
與兩條相交直線等距的點的軌跡由平分兩條給定直線之間的角度的一對直線組成。
3.
（三角形各邊的垂直平分線相交。）
4.
（三角形的角平分線是重合的。）
圓
1.
從圓心平分非直徑弦的直線與弦成直角；
相反，從中心到弦的垂線平分該弦。[圓心至弦的垂線平分弦、圓心至弦中點的連線垂直弦]
2.
（有一個圓，而且只有一個圓，它穿過三個給定的點，但不在一條直線上。）
3.
圓的等弦距圓心等距；
反之亦然。[等弦對等弦心距、等弦心距對等弦]
4.
（圓任一點的切線和經過該點的半徑互相垂直。）[切線⊥半徑]
5.
如果兩個圓相接觸，則接觸點位於通過圓心的直線上。
6.
弧在中心所對的角度是圓周其餘部分任意點所對的角度的兩倍。[圓心角兩倍於圓周角]
7.
圓內同一弓形的角相等；
（並且，如果連接兩點的線與同一側的另外兩個點所成的角度相等，則這四個點位於一個圓上。）[同弓形內的圓周角、同弓形內的圓周角的逆定理]
8.
半圓內的角是直角。[半圓上的圓周角]
9.
（在直角三角形的斜邊上描繪的圓，直徑穿過對角頂點。）
10.
圓內切四邊形的對角互補；
（反之亦然。）[圓內接四邊形內角、對角互補]
11.
（在等圓中（或在同一個圓中）（i）如果兩個圓弧在中心所對的角相等，則它們相等；（ii）反之，如果兩個圓弧相等，則它們在中心所對的角相等） 。[等角對等弧、等弧對等角]
12.
（在等圓中（或者，在同一個圓中）（i）如果兩個弦相等，則它們截取的弧相等；（ii）反之，如果兩個弧相等，則這兩個弧的弦相等。）[等弦對等弧、等弧對等弦]
13.
若一條直線與一個圓相交，並從接觸點引一條弦，則該弦與切線所成的角度等於相交線段中的角度。[交錯弓形的圓周角]
Version 2: 1965, 1966
Mathematics [cancel Lower Mathematics, content
not change]
Paper I: Arithmetic and Algebra (2 hours)
Paper II: Geometry and Trigonometry (2
hours)
Arithmetic
1.
Elementary principles and
processes; first four rules; factors, HCF and LCM.
2.
British and Metric systems of
weights and measures. Pounds, shillings and pence; dollars and cents; piculs,
catties and taels.
3.
Vulgar and decimal fractions,
excluding recurring decimals. Square roots.
4.
Averages; ratio and proportion;
percentages, profit and loss.
5.
Simple interest, direct and
inverse. Calculation of compound interest. Elementary stocks and shares,
excluding brokerage.
6.
Elementary mensuration,
including simple formulae for determining volumes and areas connected with the
rectangular block, circular cylinder and sphere.
7.
Use of logarithm table.
8.
Significant figures. (The use
of contracted methods of multiplication and division will not be compulsory in
any question.)
Algebra
1.
Elementary operations;
symbolical expression; interpretation and evaluation of formulae. Change of
subject of a formula.
2.
Factors of expressions of
simple types, including a^3±b^3. HCF and LCM by factors. Easy fractions. Elementary applications
of the Remainder Theorem.
3.
Solution of simple equations;
of simultaneous linear equations with two unknowns; of quadratic equations with
one unknown (by factors only in Elementary Mathematics); and of simultaneous
equations, one linear and one quadratic, with two unknowns. Simple problems
leading to such equations. Easy literal equations.
4.
Graphs and their simple
applications.
5.
The use of negative and
fractional indices.
Geometry
1.
The paper will contain
questions on Practical and Theoretical Geometry.
2.
The questions on Practical
Geometry will be set on the constructions contained in Schedule A, together
with easy extensions of them. Constructions need not be described, nor proofs
of constructions given, unless specifically required by the question. All figures
must be drawn accurately and distinctly, with construction lines clearly
visible.
3.
Candidates will be required to
bring to the examination a ruler, graduated in inches and tenths and in
centimetres and millimetres, a set square, protractor, compasses and a fairly
hard pencil. Questions may be set in which the use of certain instruments is
forbidden.
4.
The questions on Theoretical
Geometry will consist of theorems contained in Schedule B, together with simple
applications of these to the examiners to form part of a systematic treatment
of the subject; the order in which the theorems are stated in Schedule B is not
imposed as a sequence of their treatment.
Trigonometry
1.
The trigonometrical ratios.
Solution of right-angled triangles, with simple applications. Easy problems (in
two of three dimensions) soluble by analysis into right-angled triangles.
2.
The use of logarithmic and
trigonometrical tables.
3.
Radian measure; length of arc,
area of sectors.
4.
Simple relations between the
ratios. Identities.
5.
Angles of any magnitude. Easy
equations (solutions from 0 to 360 only). Graphs of simple trigonometrical
functions.
6.
The Sine and Cosine Rules for a
triangle. The formulae 1/2 bc sin A, sqrt(s(s-a)(s-b)(s-c)) (the proof of the
latter formula will not be required).
7.
The solution of triangles.
（中文參考譯本）
數學 [取消初級數學，內容不變]
試卷一：算術和代數（2 小時）
試卷二：幾何與三角學（2 小時）
算術
1.
基本原理和過程；四則運算；
因子，最大公因數和最小公倍數。
2.
英制和公制的度量衡系統。
英鎊、先令和便士； 美元和美分； 匹、斤、兩。
3.
常用分數（即現在的分數）及十進分數（即現在的小數），不包括循環小數。
平方根。
4.
平均數；
比和比例； 百分數、盈利和虧損。
5.
單利息、正比例和反比例。
複利息的計算。 基本股票和股份，不包括經紀業務。
6.
基本測量，包括確定長方體、圓柱體和球體的體積和面積的簡單公式。
7.
對數表的使用。
8.
有效數字。（在任何問題中都不強制使用乘法和除法的約定方法。）
代數
1.
基本運算；
符號表達； 公式的闡釋和計算。
主項變換。
2.
簡單代數式的因子，包括a^3±b^3。最大公因式及最小公倍式。
簡單分數。 餘式定理的基本應用。
3.
簡單方程式的解；
具有兩個未知量的聯立線性方程； 具有一個未知量的二次方程式（初級數學僅需使用因式法）； 以及聯立方程，一為線性方程和一為二次方程，有兩個未知量。需要使用方程之簡單問題。
簡單文字方程。
4.
圖像及其簡單應用。
5.
負數和分數指數的使用。
幾何
1.
試卷將包含實用幾何和理論幾何的問題。
2.
實用幾何的問題將根據附表A中包含的結構以及它們的簡單擴展進行設定。
除非問題有特別要求，否則不需要描述結構，也不需要給予結構證明。 所有圖形必須繪製準確、清晰，結構線清晰可見。
3.
考生參加考試時必須攜帶一把以英吋和十分之一、公分和毫米為單位的尺、三角尺、量角器、圓規和一支硬身鉛筆。
可能會提出禁止使用某些工具的問題。
4.
理論幾何問題將包含附表 B 的定理，以及對考官的簡單應用，構成該科目系統處理的一部分；
附表 B 中陳述定理的順序並未強調為處理它們的順序。
三角學
1.
三角比。
直角三角形的解法，簡單應用。 透過分析直角三角形可以解決簡單的問題（在二維或三維空間）。
2.
對數表和三角表的使用。
3.
弧度法；
弧長、扇形面積。
4.
三角比之間的簡單關係。恒等式。
5.
任何大小的角度。
簡單方程（僅限0度到 360度的解）。 簡單三角函數圖像。
6.
三角形的正弦法則和餘弦法則。
公式 1/2 bc sin A,
sqrt(s(s-a)(s-b)(s-c)) （後一個公式不需要證明）。
7.
三角形的解法。
Geometry Schedules for Mathematics and
Elementary Mathematics
1.
Bisection of angles and of
straight lines.
2.
Construction of perpendiculars
to straight lines.
3.
Construction of an angle equal
to a given angle.
4.
Construction of angles of 60,
45 and 30.
5.
Construction of parallels to a
given straight line.
6.
Simple cases of the
construction from sufficient data of triangles and quadrilaterals.
7.
Division of straight lines into
a given number of equal parts or into parts in any given proportions.
8.
Construction of a triangle
equal in area to a given polygon.
9.
Construction of tangents to a
circle and of common tangents to two circles. Construction of circumscribed,
inscribed and escribed circles of a triangle.
10.
Simple cases of the
construction of circles from sufficient data.
Schedule
B (Theoretical)
(Proofs
of theorems shown in brackets will not be required for Mathematics; no proofs
of theorems will be required for Elementary Mathematics.)
Angle
at a Point
1.
(If a straight line stands on
another straight line, the sum of the two angles so formed is equal to two
right angles; and the converse.)
2.
(If two straight lines
intersect, the vertically opposite angles are equal).
Parallel
Straight Lines
1.
(When a straight line cuts two
other straight lines, if
(i) a pair of alternate
angles are equal,
or (ii) a pair of
corresponding angles are equal,
or (iii) a pair of interior
angles on the same side of the cutting line are together equal to two right
angles,
then the two straight lines
are parallel, and the converse.)
2.
(Straight lines which are
parallel to the same straight line are parallel to one another).
Triangles
& Rectilinear Figures
1.
The sum of the angles of a
triangle is equal to two right angles.
2.
In a polygon of n sides, the
sum of the interior angles in equal to 2n-4 right angles.
3.
If the sides of a convex
polygon are produced in order, the sum of the angles so formed is equal to four
right angles.
4.
(If two triangles have two
sides of the one equal to two sides of the other, each to each, and also the
angles contained by those sides equal, the triangles are congruent.)
5.
(If two triangles have two
angles of the one equal to two angles of the other, each to each, and also one
side of the one equal to the corresponding side of the other, the triangles are
congruent.)
6.
If two sides of a triangle are
equal, the angles opposite to these sides are equal; and the converse.
7.
(If two triangles have the
three sides of the one equal to the three sides of the other, each to each, the
triangles are congruent.)
8.
(If two right-angled triangles
have their hypotenuses equal, and one side of the one equal to one side of the
other, the triangle are congruent.)
9.
(If two sides of a triangle are
unequal, the greater side has the greater angle opposite to it, and the
converse.)
10.
(Of all the straight lines that
can be drawn to a given straight line from a given point outside it, the
perpendicular is the shortest.)
11.
The opposite sides & angles
of a parallelogram are equal, each diagonal bisects the parallelogram, and the
diagonals bisect each other.
12.
If a pair of opposite sides of
a quadrilateral are equal and parallel, it is a parallelogram.
13.
The straight line drawn through
the middle point of one side of a triangle parallel to another side bisects the
third side.
14.
The straight line joining the
middle points of two sides of a triangle is parallel to the third side, and
equal to one-half of it.
15.
If there are three or more
parallel straight lines, and the intercepts made by them on any straight line
that cuts them are equal, then the corresponding intercepts on any other
straight line that cuts them are also equal.
Areas
1.
Parallelograms one the same
base and between the same parallels are equal in area.
2.
The area of a triangle is half
the area of the rectangle on the same base and with the same altitude.
3.
(Triangles on the same or equal
bases and of the same altitude are equal in area.)
4.
(Equal triangles on the same or
equal bases are of the same altitude.)
5.
In a right-angled triangle, the
square described on the hypotenuse is equal to the sum of the squares described
on the sides containing the right angle (and the converse).
6.
The square on a side of a
triangle is greater or less than the sum of the squares on the other two sides,
according as the angle contained by those sides is obtuse or acute. The
difference is twice the rectangle contained by one of the two sides and the projection
on it of the other.
7.
In any triangle, the sum of the
squares on any two sides is equal to twice the square on half the third side
together with twice the square on the median which bisects the third side.
Loci
1.
The locus of a point which is
equidistant from two fixed points is the perpendicular bisector of the straight
line joining the two fixed points.
2.
The locus of a point which is
equidistant from two intersecting straight lines consists of the pair of
straight lines which bisect angles between the two given lines.
3.
(The perpendicular bisectors of
the sides of a triangle concurrent.)
4.
(The bisectors of the angles of
a triangle are concurrent.)
The
Circle
1.
A straight line drawn from the
centre of a circle to bisect a chord which is not a diameter, is at right
angles to the chord; conversely, the perpendicular to a chord from the centre
bisects the chord.
2.
(There is one circle, and one
only, which passes through three given points not in a straight line.)
3.
Equal chords of a circle are
equidistant from the centre; and the converse.
4.
(The tangent at any point of a
circle and the radius through the point are perpendicular to each other.)
5.
If two circles touch, the point
of contact lies on the straight line through the centres.
6.
The angle which an arc of a
circle subtends at the centre is double that which is subtends at any points on
the remaining part of the circumference.
7.
Angles in the same segment of a
circle are equal; (and, if the line joining two points subtends equal angles at
two other points on the same side of it, the four points lie on a circle.)
8.
The angle in a semicircle is a
right angle.
9.
(The circle described on the
hypotenuse of a right-angled triangle as diameter passes through the opposite
vertex.)
10.
The opposite angles of any
quadrilateral inscribed in a circle are supplementary; (and the converse.)
11.
(In equal circles (or, in the
same circle) (i) if two arcs subtend equal angles at the centre, they are
equal; (ii) conversely, if two arcs are equal, they subtend equal angles at the
centre).
12.
(In equal circles (or, in the
same circle) (i) if two chords are equal, they cut off equal arcs; (ii)
conversely, if two arcs are equal, the chords of the arcs are equal.)
13.
If a straight line touch a
circle, and from the point of contact a chord be drawn, the angles which this
chord makes with the tangent are equal to the angles in the alternate segments.
（中文參考譯本）
附表A（實用）
1.
角和直線的平分。
2.
垂直於直線的繪製。
3.
繪製一個等於給定角的角。
4.
60°、45°、30°角的繪製。
5.
繪製與給定直線的平行線。
6.
利用足夠的三角形和四邊形資料去繪製簡單案例。
7.
將直線分成給定數量的等份或依任意給定比例的份數。
8.
繪製面積與給定多邊形相等的三角形。
9.
建構圓的切線和兩個圓的公切線。三角形的外接圓、內切圓和外切圓的繪製。
10.
根據足夠的數據建立圓形的簡單案例。
附表 B（理論）
（數學不需要括號中所示的定理證明；初級數學不需要定理證明。）
直線上的角
1.
（如果一條直線與另一條直線重疊，則所形成的兩個角和等於兩個直角；反之亦然。）[直線上的鄰角]
2.
（如果兩條直線相交，則垂直的對角相等）。[對頂角]
平行的直線
1.
（當一條直線與另外兩條直線相交時，如果
(i) 一對交錯角相等，[內錯角相等]
或 (ii) 一對對應的角度相等，[同位角相等]
(iii) 截線同一側的一對內角合起來等於兩個直角，[同旁內角互補]
則兩條直線平行，反之亦然。）
2.
（與同一條直線平行的直線彼此平行）。
三角形和直線圖形
1.
三角形的內角和等於兩個直角。[三角形內角和]
2.
在 n 條邊的多邊形中，內角和等於 2n-4 個直角。[多邊形內角和]
3.
若凸多邊形的邊依序生成，則所形成的角和等於四個直角。[多邊形外角和]
4.
（如果兩個三角形的兩邊與另一個三角形的兩邊相等，而這些邊所含的角度也相等，則這兩個三角形全等。）[SAS]
5.
（如果兩個三角形的兩個角等於另一個三角形的兩個角，且一個三角形的一邊等於另一個三角形的對應邊，則這兩個三角形全等。）[ASA、AAS]
6.
如果三角形的兩邊相等，則這兩邊的對角也相等；
反之亦然。[全等三角形的對應邊、全等三角形的對應角]
7.
（如果兩個三角形的一個的三邊等於另一個的三邊，則這兩個三角形全等。）[SSS]
8.
（如果兩個直角三角形的斜邊相等，並且一個直角三角形的一側等於另一個直角三角形的一側，則該三角形全等。）[RHS]
9.
（如果三角形的兩邊不相等，則較大的邊與其相對的角度較大，反之亦然。）[大邊對大角、大角對大邊]
10.
（從給定直線外部的給定點可以繪製到給定直線的所有直線中，垂線是最短的。）
11.
平行四邊形的對邊和角相等，每條對角線平分平行四邊形，對角線互相平分。[平行四邊形對邊、平行四邊形對角、平行四邊形對角線]
12.
如果四邊形的一對對邊相等且平行，則它是平行四邊形。[對邊平行且相等]
13.
穿過三角形一邊的中點與另一邊平行的直線平分第三邊。[中點定理]
14.
連接三角形兩邊中點的直線平行於第三邊，並且等於它的二分之一。[中點定理]
15.
如果有3條或3條以上平行直線，且它們在與它們相交的任何一條直線上的截距相等，則在與它們相交的任何其他直線上相應的截距也相等。[截線定理]
面積
1.
同底且同平行線之間的平行四邊形面積相等。
2.
三角形的面積是同底同高的長方形面積的一半。
3.
（等底、等高的三角形面積相等。）
4.
（相同或等底的等三角形的高相同。）
5.
在直角三角形中，斜邊上的平方等於直角邊的平方和（反之亦然）。[畢氏定理、畢氏定理逆定理]
6.
三角形的一邊的平方大於或小於另外兩邊的平方和，取決於這些邊所含的角度是鈍角還是銳角。其差值是兩邊之一和另一邊在其上的投影所包含的矩形的兩倍。
7.
在任何三角形中，任兩邊的平方和等於第三邊一半的平方的兩倍加上平分第三邊的中線的平方的兩倍。[中線定理]
軌跡
1.
與兩個固定點等距的點的軌跡是連接兩個固定點的直線的垂直平分線。
2.
與兩條相交直線等距的點的軌跡由平分兩條給定直線之間的角度的一對直線組成。
3.
（三角形各邊的垂直平分線相交。）
4.
（三角形的角平分線是重合的。）
圓
1.
從圓心平分非直徑弦的直線與弦成直角；
相反，從中心到弦的垂線平分該弦。[圓心至弦的垂線平分弦、圓心至弦中點的連線垂直弦]
2.
（有一個圓，而且只有一個圓，它穿過三個給定的點，但不在一條直線上。）
3.
圓的等弦距圓心等距；
反之亦然。[等弦對等弦心距、等弦心距對等弦]
4.
（圓任一點的切線和經過該點的半徑互相垂直。）[切線⊥半徑]
5.
如果兩個圓相接觸，則接觸點位於通過圓心的直線上。
6.
弧在中心所對的角度是圓周其餘部分任意點所對的角度的兩倍。[圓心角兩倍於圓周角]
7.
圓內同一弓形的角相等；
（並且，如果連接兩點的線與同一側的另外兩個點所成的角度相等，則這四個點位於一個圓上。）[同弓形內的圓周角、同弓形內的圓周角的逆定理]
8.
半圓內的角是直角。[半圓上的圓周角]
9.
（在直角三角形的斜邊上描繪的圓，直徑穿過對角頂點。）
10.
圓內切四邊形的對角互補；
（反之亦然。）[圓內接四邊形內角、對角互補]
11.
（在等圓中（或在同一個圓中）（i）如果兩個圓弧在中心所對的角相等，則它們相等；（ii）反之，如果兩個圓弧相等，則它們在中心所對的角相等） 。[等角對等弧、等弧對等角]
12.
（在等圓中（或者，在同一個圓中）（i）如果兩個弦相等，則它們截取的弧相等；（ii）反之，如果兩個弧相等，則這兩個弧的弦相等。）[等弦對等弧、等弧對等弦]
13.
若一條直線與一個圓相交，並從接觸點引一條弦，則該弦與切線所成的角度等於相交線段中的角度。[交錯弓形的圓周角]
Version 3: 1967 (named Mathematics
Syllabus A), 1968, 1969, 1970, 1971
Paper
1: Arithmetic and Trigonometry (2 hours)
Paper
2: Algebra (1.5 hours)
Paper
3: Geometry (2 hours)
Since
1970,
Paper
1: Short questions in Arithmetic, Trigonometry and Algebra (1 hour)
Paper
2: Further questions in Arithmetic, Trigonometry and Algebra (2 hours)
Paper
3: Geometry (2 hours)
Arithmetic
Knowledge
of Primary School Mathematics is assumed. (Add in 1968)
Candidates
should be familiar with
Ø the British, Metric and Chinese systems of weights and measure and
with the following monetary systems: pounds, shillings and pence; dollars and
cents.
Ø Addition, subtraction, multiplication and division applied to
numerical calculations.
Ø Fractions and decimals.
Ø Proportion and proportional parts.
Ø Extraction of square roots, by factors and by tables, the long rule
not being required.
Ø Averages, percentages, simple and compound interest; profit and
loss.
Ø Elementary mensuration of the triangle, the circle, the rectangular
block, the cylinder, the sphere, and the right circular cone.
Ø Problems on speeds.
Ø Graphs from numerical and statistical data.
Ø The use of logarithm tables.
Candidate
may be required to give results to a specified degree of approximation, but the
use of contracted methods of multiplication and division is not essential.
Trigonometry
1.
The six trigonometrical ratios,
(including their graphs). Solution of right-angled triangles, with simple
applications. Easy problems (in two and three dimensions) soluble by analysis
into right-angled triangles.
2.
The use of logarithmic and
trigonometrical tables.
3.
Radian measure (involving
simple numerical examples only); length of arc, area of sector.
4.
Simple relations between ratios
(excluding multiple and compound angles).
5.
Angle from 0 to 360. Easy
equations (solutions from 0 to 360 only).
6.
Use of the sine and cosine
formulae for a triangle and of the formula 1/2 bc sinA.
Algebra
Elementary
algebraic operations; formulae expressing arithmetical generalisation; change
of subjects of a formula; factors or expressions of simple types, including
trinomials and a^3±b^3; easy fractions.
Ø The use of fractional and negative indices.
Ø Calculation by logarithms to base 10 with the use of four-figure
tables.
Ø Solution of linear equations involving not more than two unknowns
and quadratic equations, one linear and one quadratic, involving two unknowns;
and simple problems leading to such equations.
Ø The use of the remainder theorem; ratio and proportion; variation.
Ø Graphs and their simple applications.
Ø Arithmetical and finite geometrical progressions.
Geometry
The
paper in Geometry will contain questions on Practical and on Theoretical
Geometry.
Practical
Geometry
The
questions on Practical Geometry will be set on the construction contained in
Schedule A, together with easy extensions of them. In cases where the validity
of a construction is not obvious, the reasoning by which it is justified may be
required. Every candidate must provide himself with a ruler graduated in inches
and tenths of an inch, and in centimetres and millimetres, a set square, a
protractor, compasses, and a fairly hard pencil.
Schedule
A. For Practical Geometry
1.
Bisection of angles and of
straight lines.
2.
Construction of perpendiculars
to straight lines.
3.
Construction of an angle equal
to a given angle.
4.
Construction of angles of 60,
45 and 30.
5.
Construction of parallels to a
given straight line.
6.
Simple cases of the
construction from sufficient data of triangles and quadrilaterals.
7.
Division of straight lines into
a given number of equal parts or into parts in any given proportions.
8.
Construction of a triangle
equal in area to a given polygon.
9.
Construction of tangents to a
circle and of common tangents to two circles.
10.
Construction of circumscribed,
inscribed and escribed circles of a triangle.
11.
Simple cases of the
construction of circles from sufficient data.
12.
Construction of a square equal
in area to a given polygon.
13.
Construction of a fourth
proportional to three given straight lines and a mean proportional to two given
straight lines.
14.
Construction of regular figures
of 3, 4, 6 or 8 sides in or about a given circle.
Theoretical
Geometry
The
questions on Theoretical Geometry will consist of problems (i.e. Riders) on the
theorems contained in Schedule B. Formal proofs of these theorems will not be
asked, but the problems will require a thorough knowledge of the ground to be
covered.
Schedule
B. For Theoretical Geometry
Angles
at a Point
1.
If a straight line stands on
another straight line, the sum of the two angles so formed is equal to two
right angles; and the converse.
2.
If two straight lines
intersect, the vertically opposite angles are equal.
Parallel
Straight Lines
1.
When a straight line cuts two
other straight lines, if
(i) a pair of alternate
angles are equal,
or (ii) a pair of
corresponding angles are equal,
or (iii) a pair of interior
angles on the same side of the cutting line are together equal to two right
angles,
then the two straight lines
are parallel, and the converse.
2.
Straight lines which are
parallel to the same straight line are parallel to one another.
Triangles
and Rectilinear Figures
1.
The sum of the angles of a
triangle is equal to two right angles.
2.
In a polygon of n sides, the
sum of the interior angles in equal to 2n-4 right angles.
3.
If the sides of a convex
polygon are produced in order, the sum of the angles so formed is equal to four
right angles.
4.
If two triangles have two sides
of the one equal to two sides of the other, each to each, and also the angles
contained by those sides equal, the triangles are congruent.
5.
If two triangles have two
angles of the one equal to two angles of the other, each to each, and also one
side of the one equal to the corresponding side of the other, the triangles are
congruent.
6.
If two sides of a triangle are
equal, the angles opposite to these sides are equal; and the converse.
7.
If two triangles have the three
sides of the one equal to the three sides of the other, each to each, the
triangles are congruent.
8.
If two right-angled triangles
have their hypotenuses equal, and one side of the one equal to one side of the
other, the triangle are congruent.
9.
If two sides of a triangle are
unequal, the greater side has the greater angle opposite to it, and the
converse.
10.
Of all the straight lines that
can be drawn to a given straight line from a given point outside it, the
perpendicular is the shortest.
11.
The opposite sides & angles
of a parallelogram are equal, each diagonal bisects the parallelogram, and the
diagonals bisect each other.
12.
If a pair of opposite sides of
a quadrilateral are equal and parallel, it is a parallelogram.
13.
The straight line drawn through
the middle point of one side of a triangle parallel to another side bisects the
third side.
14.
The straight line joining the
middle points of two sides of a triangle is parallel to the third side, and
equal to one-half of it.
15.
If there are three or more
parallel straight lines, and the intercepts made by them on any straight line
that cuts them are equal, then the corresponding intercepts on any other
straight line that cuts them are also equal.
Areas
1.
Parallelograms one the same
base and between the same parallels are equal in area.
2.
Triangles on the same or equal
bases and of the same altitude are equal in area.
3.
Equal triangles on the same or
equal bases are of the same altitude.
4.
In a right-angled triangle, the
square described on the hypotenuse is equal to the sum of the squares described
on the sides containing the right angles; and the converse.
>>>
5.
The square on a side of a
triangle is greater or less than the sum of the squares on the other two sides,
according as the angle contained by those sides is obtuse or acute. The
difference is twice the rectangle contained by one of the two sides and the projection
on it of the other.
6.
In any triangle, the sum of the
squares on any two sides is equal to twice the square on half the third side
together with twice the square on the median which bisects the third side.
Loci
1.
The locus of the point which is
equidistant from two fixed points is the perpendicular bisector of the straight
line joining the two fixed points.
2.
The locus of a point which is
equidistant from two intersecting straight lines consists of the pair of
straight lines which bisect angles between the two given lines.
The
Circle
1.
A straight line drawn from the
centre of a circle to bisect a chord which is not a diameter, is at right
angles to the chord; conversely, the perpendicular to a chord from the centre
bisects the chord.
2.
There is one circle, and one
only, which passes through three given points not in a straight line.
3.
Equal chords of a circle are
equidistant from the centre; and the converse.
4.
The tangent at any point of a
circle and the radius through the point are perpendicular to each other.
5.
The tangent to a circle from an
external point are equal.
6.
If two circles touch, the point
of contact lies on the straight line through the centres.
7.
The angle which an arc of a
circle subtends at the centre is double that which it subtends at any points on
the remaining part of the circumference.
8.
Angles in the same segment of a
circle are equal; and, if the line joining two points subtends equal angles at
two other points on the same side of it, the four points lie on a circle.
9.
The angle in a semicircle is a
right angle; and the converse.
10.
The opposite angles of any
quadrilateral inscribed in a circle are supplementary; and the converse.
>>>
11.
In equal circles (or, in the
same circle) (i) if two arcs subtend equal angles at the centre, they are
equal; (ii) conversely, if two arcs are equal, they subtend equal angles at the
centre.
12.
In equal circles (or, in the
same circle) (i) if two chords are equal, they cut off equal arcs; (ii)
conversely, if two arcs are equal, the chords of the arcs are equal.
13.
If a straight line touch a
circle, and from the point of contact a chord be drawn, the angles which this
chord makes with the tangent are equal to the angles in the alternate segments.
14.
If two chords of a circle
intersect either inside or outside the circle the rectangle contained by the
parts of the one is equal to the rectangle contained by the parts of the other.
Proportion:
Similar Triangles
1.
If a straight line is drawn
parallel to one side of a triangle, the other two sides are divided
proportionally; and the converse.
2.
If two triangles are
equiangular their corresponding sides are proportional; and the converse.
3.
If two triangles have one angle
of the one equal to one angle of the other and the sides about these equal
angles proportional, the triangles are similar.
4.
If a perpendicular is drawn
from the right angle of a right-angled triangle to the hypotenuse, the
triangles on each side of the perpendicular are similar to the whole triangle
and to each other.
5.
The internal bisector of an
angle of a triangle divides the opposite side internally in the ratio of the
sides containing the angle, and likewise the external bisector externally.
6.
The ratio of the areas of
similar triangles is equal to the ratio of the squares on corresponding sides.
（中文參考譯本）
將有三份試卷。
試卷一：算術與三角學（2小時）
試卷二：代數（1.5 小時）
試卷三：幾何學（2 小時）
自 1970 年以來，
試卷1：算術、三角學和代數短問題（1 小時）
試卷2：算術、三角學和代數的進一步問題（2 小時）
試卷3：幾何學（2 小時）
算術
假定具備小學數學知識。（1968年新增）
考生應熟悉
Ø 英制、公制和中國的度量衡體系以及以下貨幣體系：英鎊、先令和便士； 美元和美分。
Ø 加法、減法、乘法和除法應用於數值計算。
Ø 分數和小數。
Ø 比例和比例部分。
Ø 透過因式和四位數表提取平方根，不需要長式。
Ø 平均數、百分數、單利息和複利息；盈利與虧損。
Ø 三角形、圓形、長方體、圓柱體、球體、直圓錐的初步測量。
Ø 速度問題。
Ø 來自數值和統計資料的圖表。
Ø 對數表的使用。
考生可能需要給出指定程度的近似結果，但乘法和除法的簡化方法的使用並不是必需的。
三角學
1.
六個三角比（包括它們的圖像）。
直角三角形的解法，簡單應用。 透過分析直角三角形可以解決簡單的問題（二維和三維）。
2.
對數表和三角表的使用。
3.
弧度法（僅涉及簡單數值範例）；弧長、扇形面積。
4.
簡單的三角比關係（不包括倍角和複角）。
5.
角度從0度到360度。簡單的方程式（僅限0度到360度的解）。
6.
使用三角形的正弦公式和餘弦公式以及公式 1/2 bc sinA。
代數
基本代數運算；表達概括算術的公式；
主項變換； 簡單類型的因式或代數式，包括三項式和a^3±b^3； 簡單分數。
Ø 使用分數和負指數。
Ø 使用四位數表進行以10為底的對數計算。
Ø 解涉及不超過兩個未知數的線性方程式和涉及兩個未知數的一元一二次方程式； 以及導致此類方程式的簡單問題。
Ø 餘式定理的運用； 比和比例；變分。
Ø 圖像及其簡單應用。
Ø 等差和有限等比級數。
幾何學
幾何試卷將包含實用幾何和理論幾何的問題。
實用幾何
實用幾何的問題將圍繞附表 A 中包含的結構以及它們的簡單擴展進行設定。
如果解釋的有效性不明顯，則可能需要證明其合理性的推理。 每個候選人都必須為自己準備一把以英寸和十分之一英寸為單位、以厘米和毫米為單位的尺子、一個三角尺、一個量角器、圓規和一支硬身鉛筆。
附表 A. 實用幾何
1.
角和直線的平分。
2.
垂直於直線的繪製。
3.
繪製一個等於給定角的角。
4.
60°、45°、30°角的繪製。
5.
繪製與給定直線的平行線。
6.
利用足夠的三角形和四邊形資料來繪製簡單案例。
7.
將直線分成給定數量的等份或依任意給定比例的份數。
8.
繪製面積與給定多邊形相等的三角形。
9.
繪製圓的切線和兩個圓的公切線。
10.
三角形的外接圓、內切圓和外切圓的繪製。
11.
根據足夠的數據建立圓的簡單案例。
12.
繪製一個面積等於給定多邊形的正方形。
13.
繪製與三條給定直線成比例的第四條線和與兩條給定直線成比例的平均值。
14.
在給定圓內或圍繞給定圓構造 3、4、6 或 8 條邊的規則圖形。
理論幾何
理論幾何問題將包括附表 B 中包含的定理問題（即附加題）。不會提問這些定理的正式證明，但這些問題需要對所涉及的基礎有透徹的了解。
附表 B. 理論幾何
直線上的角
1.
如果一條直線與另一條直線重疊，則所形成的兩個角和等於兩個直角；反之亦然。[直線上的鄰角]
2.
如果兩條直線相交，則垂直的對角相等。[對頂角]
平行的直線
1.
當一條直線與另外兩條直線相交時，如果
(i) 一對交錯角相等，[內錯角相等]
或 (ii) 一對對應的角度相等，[同位角相等]
(iii) 截線同一側的一對內角合起來等於兩個直角，[同旁內角互補]
則兩條直線平行，反之亦然。
2.
與同一條直線平行的直線彼此平行。
三角形和直線圖形
1.
三角形的內角和等於兩個直角。[三角形內角和]
2.
在 n 條邊的多邊形中，內角和等於 2n-4 個直角。[多邊形內角和]
3.
若凸多邊形的邊依序生成，則所形成的角和等於四個直角。[多邊形外角和]
4.
如果兩個三角形的兩邊與另一個三角形的兩邊相等，而這些邊所含的角度也相等，則這兩個三角形全等。[SAS]
5.
如果兩個三角形的兩個角等於另一個三角形的兩個角，且一個三角形的一邊等於另一個三角形的對應邊，則這兩個三角形全等。[ASA、AAS]
6.
如果三角形的兩邊相等，則這兩邊的對角也相等；反之亦然。[全等三角形的對應邊、全等三角形的對應角]
7.
如果兩個三角形的一個的三邊等於另一個的三邊，則這兩個三角形全等。[SSS]
8.
如果兩個直角三角形的斜邊相等，並且一個直角三角形的一側等於另一個直角三角形的一側，則該三角形全等。[RHS]
9.
如果三角形的兩邊不相等，則較大的邊與其相對的角度較大，反之亦然。[大邊對大角、大角對大邊]
10.
從給定直線外部的給定點可以繪製到給定直線的所有直線中，垂線是最短的。
11.
平行四邊形的對邊和角相等，每條對角線平分平行四邊形，對角線互相平分。[平行四邊形對邊、平行四邊形對角、平行四邊形對角線]
12.
如果四邊形的一對對邊相等且平行，則它是平行四邊形。[對邊平行且相等]
13.
穿過三角形一邊的中點與另一邊平行的直線平分第三邊。[中點定理]
14.
連接三角形兩邊中點的直線平行於第三邊，並且等於它的二分之一。[中點定理]
15.
如果有3條或3條以上平行直線，且它們在與它們相交的任何一條直線上的截距相等，則在與它們相交的任何其他直線上相應的截距也相等。[截線定理]
面積
1.
同底且同平行線之間的平行四邊形面積相等。
2.
等底、等高的三角形面積相等。
3.
相同或等底的等三角形的高相同。
4.
在直角三角形中，斜邊上的平方等於直角邊的平方和；反之亦然。[畢氏定理、畢氏定理逆定理]
>>>
5.
三角形的一邊的平方大於或小於另外兩邊的平方和，取決於這些邊所含的角度是鈍角還是銳角。其差值是兩邊之一和另一邊在其上的投影所包含的矩形的兩倍。
6.
在任何三角形中，任兩邊的平方和等於第三邊一半的平方的兩倍加上平分第三邊的中線的平方的兩倍。[中線定理]
軌跡
1.
與兩個固定點等距的點的軌跡是連接兩個固定點的直線的垂直平分線。
2.
與兩條相交直線等距的點的軌跡由平分兩條給定直線之間的角度的一對直線組成。
圓
1.
從圓心平分非直徑弦的直線與弦成直角；
相反，從中心到弦的垂線平分該弦。[圓心至弦的垂線平分弦、圓心至弦中點的連線垂直弦]
2.
有一個圓，而且只有一個圓，它穿過三個給定的點，但不在一條直線上。
3.
圓的等弦距圓心等距；
反之亦然。[等弦對等弦心距、等弦心距對等弦]
4.
圓任一點的切線和經過該點的半徑互相垂直。[切線⊥半徑]
5.
從外點到圓的切線相等。[切線性質]
6.
如果兩個圓相接觸，則接觸點位於通過圓心的直線上。
7.
弧在中心所對的角度是圓周其餘部分任意點所對的角度的兩倍。[圓心角兩倍於圓周角]
8.
圓內同一弓形的角相等；並且，如果連接兩點的線與同一側的另外兩個點所成的角度相等，則這四個點位於一個圓上。[同弓形內的圓周角、同弓形內的圓周角的逆定理]
9.
半圓內的角是直角。[半圓上的圓周角]
10.
圓內切四邊形的對角互補；反之亦然。[圓內接四邊形內角、對角互補]
>>>
11.
在等圓中（或在同一個圓中）（i）如果兩個圓弧在中心所對的角相等，則它們相等；（ii）反之，如果兩個圓弧相等，則它們在中心所對的角相等。[等角對等弧、等弧對等角]
12.
在等圓中（或者，在同一個圓中）（i）如果兩個弦相等，則它們截取的弧相等；（ii）反之，如果兩個弧相等，則這兩個弧的弦相等。[等弦對等弧、等弧對等弦]
13.
若一條直線與一個圓相交，並從接觸點引一條弦，則該弦與切線所成的角度等於相交線段中的角度。[交錯弓形的圓周角]
14.
如果圓的兩條弦在圓的內部或外部相交，則其中一個的部分所包含的矩形等於另一個的部分所包含的矩形。
比例：相似三角形
1.
若平行於三角形的一條邊畫一條直線，則按比例分割另外兩條邊；
反之亦然。
2.
如果兩個三角形是等角，則它們的對應邊成比例；
反之亦然。[AAA、三邊成比例]
3.
如果兩個三角形的一個角等於另一個角，並且這些等角的邊成比例，則這兩個三角形相似。[兩邊成比例及夾角相等]
4.
如果從直角三角形的直角到斜邊引一條垂線，則垂線兩側的三角形與整個三角形相似，並且彼此相似。
5.
三角形角的內平分線以包含該角的邊的比率在內部平分對邊，同樣，外平分線在外部。
6.
相似三角形的面積之比等於對應邊上的正方形之比。
Version 4: 1972, 1973, 1974 (change
named Mathematics Alternative Syllabus B)
Paper 1: Multiple Choice questions in
Arithmetic, Algebra, Trigonometry and Geometry (1.5 hours)
Paper 2: Further questions in Arithmetic
and Algebra (1.5 hours)
Paper 3: Further questions in Trigonometry
and Geometry (1.5 hours)
Since 1974,
Paper 1: Arithmetic & Algebra (1.5
hours)
Paper 2: Short questions in Arithmetic,
Algebra, Trigonometry and Geometry (1.5 hours)
Paper 3: Trigonometry & Geometry (1.5
hours)
1.
Knowledge of Primary School
Mathematics is assumed.
2.
Use of British, Metric and
Chinese systems of weights and measures. Decimalized monetary systems. (until
1973)
Use of SI units of weights
and measures. (1974)
3.
Primes and factorizations of
natural numbers. Principles of simple divisibility tests for 2, 3, 4, 5, 8, 9,
11 LCM and HCF including the general principles of finding the HCF. (Euclidean
algorithm). (until 1973)
Primes and factorizations
of natural numbers. Principles of simple divisibility tests for 2, 3, 4, 5, 8,
9, 11 LCM and HCF including the general principles of finding the HCF including
the use of the Euclidean algorithm. (1974)
4.
Problems in simple and compound
interest, profit and loss, percentage, averages, speed and relative speeds
including the use of fractions and decimals. (1972)
Problems in simple and
compound interest, profit and loss, percentage, averages, speed and relative
speeds. (1973, 1974)
5.
Elementary mensuration of the rectangle,
triangle, parallelogram, trapezium, circle, rectangular block, prism, pyramid,
right circular cone and sphere.
6.
Elementary algebraic
operations; formulae expressing arithmetical generalization; change of subject
of a formula, factorization, HCF and LCM of polynomials of simple types in one
or two variables including trinomials and 
; easy algebraic fractions. (1972)
Ø Elementary algebraic operations including easy fractions;
factorization of simple polynomials in one or two variables including
trinomials and sum & difference of two cubes; HCF and LCM of such
polynomials. (1973, 1974)
Ø Formulae expressing arithmetical generalization; change of subject
of a formula. (1973, 1974)
7.
Solution of linear equations in
one unknown and of simultaneous linear equations in two unknowns and quadratic
equations involving only one unknown including knowledge of the relations
between the sum and product of the roots and the coefficients of a quadratic
equation; the solution of simultaneous equations, one linear and one quadratic,
involving two unknowns; simple problems leading to such equations.
8.
Linear and quadratic
inequalities and their solutions.
9.
Graphical from numerical and
statistical data. Graphs of simple algebraic functions of not more than the
third degree.
10.
Ratio, proportion, proportional
parts and variation.
11.
Laws of non-negative integral
indices with extension to fractional and negative indices; and logarithms
including use of the root sign. (
to
represent the positive square root of “a” where “a” is any positive number).
Calculation by logarithms to base 10 with the use of four-figure tables.
Extraction of square roots, by factors and by tables. (The long rule is not
required).
12.
Arithmetical and finite
geometrical progressions.
Trigonometry
(1972)
1.
The six trigonometrical ratios
of angles from 0 to 360 and simple relation between them (excluding multiple
and compound angles). Solution of right-angled triangles, with simple
applications. Easy problems in two and three dimensions soluble by analysis into
right-angled triangles.
2.
The use of trigonometrical and
logarithmic trigonometrical tables.
3.
Easy equations (solutions from
0 to 360 only). Graphs of linear functions of sine and cosine in the range from
0 to 360.
4.
Radian measure involving simple
numerical examples only; length of arc and area of sector.
5.
Sine and cosine formulae and
their application to the solution of a triangle, including the greater angle
opposite the greater side and vice versa; area of a triangle as 1/2bc sinA.
(1973,
1974)
1.
Measure of angles in degrees in
the range 
and in
radians in the range 
. Length of arc and area of sector of a circle.
2.
The six trigonometrical ratios
and simple relation between them (excluding multiple and compound angles).
Solution of right-angled triangles, with simple applications. Easy problems in
two and three dimensions soluble by analysis into right-angled triangles.
3.
Easy equations (solution in the
range ). Graph of linear functions of sine and cosine
in the range .
4.
Sine and cosine formulae and
their application to the solution of a triangle, including the greater angle
opposite the greater side and vice versa; area of a triangle as 1/2bc sinA.
(Candidates may be required to give results to a specified degree of
approximation, but the use of contracted methods of multiplications and
division is not essential.)
Geometry
(until 1973)
Every
candidate must provide himself with a ruler graduated in inches and tenths of
an inch, and in centimetres and millimetres, a protractor, compasses, and a
fairly hard pencil. The paper in Geometry will contain question on Practical
and on Theoretical Geometry.
Practical
Geometry
The
questions on Practical Geometry will be set on the constructions by
straight-edge and compasses contained in Schedule A, together with easy
extensions of them. In cases where the validity of a construction is not
obvious, the reasoning by which it is justified may be required. Construction
lines should be shown.
Schedule
A. For Practical Geometry
1.
Bisection of angles and of line
segments.
2.
Construction of perpendiculars
to straight lines.
3.
Construction of an angle equal
to a given angle.
4.
Construction of angles of 60,
45 and 30.
5.
Construction of parallels to a
given straight line.
6.
Simple cases of the
construction from sufficient data of triangles and quadrilaterals.
7.
Division of straight lines into
a given number of equal parts.
8.
Construction of tangents to a
circle.
9.
Construction of circumscribed,
inscribed and escribed circles of a triangle.
10.
Construction of a mean
proportional to two given line segments.
Theoretical
Geometry
The
questions on Theoretical Geometry will consist of problems (i.e. riders) on the
theorems contained in Schedule B. Formal proofs of these theorems will not be
asked, but the problems will require a thorough knowledge of the ground to be
covered.
Schedule
B. For Theoretical Geometry
Angles
at a Point
1.
If a straight line stands on
another straight line, the sum of the two angles so formed is equal to two
right angles; and the converse.
2.
If two straight lines
intersect, the vertically opposite angles are equal.
Parallel
Straight Lines
1.
When a straight line cuts two
other straight lines, if
(i) a pair of alternate
angles are equal,
or (ii) a pair of
corresponding angles are equal,
or (iii) a pair of interior
angles on the same side of the cutting line are together equal to two right
angles,
then the two straight lines
are parallel, and the converse.
2.
Straight lines which are
parallel to the same straight line are parallel to one another.
Triangles
and Rectilinear Figures
1.
The sum of the angles of a
triangle is equal to two right angles.
2.
In a polygon of n sides, the
sum of the interior angles in equal to (2n-4) right angles.
3.
If the sides of a convex
polygon are produced in order, the sum of the angles so formed is equal to four
right angles.
4.
If two triangles have two sides
of the one equal to two sides of the other, each to each, and also the angles
contained by those sides equal, the triangles are congruent.
5.
If two triangles have two
angles of the one equal to two angles of the other, each to each, and also one
side of the one equal to the corresponding side of the other, the triangles are
congruent.
6.
If two sides of a triangle are
equal, the angles opposite to these sides are equal; and the converse.
7.
If two triangles have the three
sides of the one equal to the three sides of the other, each to each, the
triangles are congruent.
8.
If two right-angled triangles
have their hypotenuses equal, and one side of the one equal to one side of the
other, the triangle are congruent.
9.
Of all the straight lines that
can be drawn to a given straight line from a given point outside it, the
perpendicular is the shortest.
10.
The opposite sides & angles
of a parallelogram are equal, each diagonal bisects the parallelogram, and the
diagonals bisect each other.
11.
If a pair of opposite sides of
a quadrilateral are equal and parallel, it is a parallelogram.
12.
The straight line drawn through
the middle point of one side of a triangle parallel to another side bisects the
third side.
13.
The straight line joining the
middle points of two sides of a triangle is parallel to the third side, and
equal to one-half of it.
14.
If there are three or more
parallel straight lines, and the intercepts made by them on any straight line
that cuts them are equal, then the corresponding intercepts on any other
straight line that cuts them are also equal.
Areas
1.
Parallelograms one the same
base and between the same parallels are equal in area.
2.
Triangles on the same or equal
bases and of the same altitude are equal in area.
3.
Equal triangles on the same or
equal bases are of the same altitude.
4.
In a right-angled triangle, the
square described on the hypotenuse is equal to the sum of the squares described
on the sides containing the right angles; and the converse.
Loci
1.
The locus of the point which is
equidistant from two fixed points is the perpendicular bisector of the straight
line joining the two fixed points.
2.
The locus of a point which is
equidistant from two intersecting straight lines consists of the pair of
straight lines which bisect angles between the two given lines.
The
Circle
1.
A straight line drawn from the
centre of a circle to bisect a chord which is not a diameter, is at right
angles to the chord; conversely, the perpendicular to a chord from the centre
bisects the chord.
2.
There is one circle, and one
only, which passes through three given points not in a straight line.
3.
Equal chords of a circle are
equidistant from the centre; and the converse.
4.
The tangent at any point of a
circle and the radius through the point are perpendicular to each other.
5.
The tangent to a circle from an
external point are equal.
6.
If two circles touch, the point
of contact lies on the straight line through the centres.
7.
The angle which an arc of a
circle subtends at the centre is double that which it subtends at any points on
the remaining part of the circumference.
8.
Angles in the same segment of a
circle are equal; and, if the line joining two points subtends equal angles at
two other points on the same side of it, the four points lie on a circle.
9.
The angle in a semicircle is a
right angle; and the converse.
10.
The opposite angles of any
quadrilateral inscribed in a circle are supplementary; and the converse.
11.
In equal circles (or, in the
same circle) (i) if two arcs subtend equal angles at the centre, they are
equal; (ii) conversely, if two arcs are equal, they subtend equal angles at the
centre.
12.
In equal circles (or, in the
same circle) (i) if two chords are equal, they cut off equal arcs; (ii)
conversely, if two arcs are equal, the chords of the arcs are equal.
13.
If a straight line touch a
circle, and from the point of contact a chord be drawn, the angles which this
chord makes with the tangent are equal to the angles in the alternate segments.
Proportion:
Similar Triangles
1.
If a straight line is drawn
parallel to one side of a triangle, the other two sides are divided
proportionally; and the converse.
2.
If two triangles are
equiangular their corresponding sides are proportional; and the converse.
Plane
Geometry (1974)
Knowledge
and application of the following*:
Ø Angles at a point;
Ø parallel lines;
Ø sum of angles of a triangle and other convex polygons;
Ø congruent triangles, isosceles triangles;
Ø the distance of a point from a straight line;
Ø parallelograms;
Ø mid-point and intercept theorems;
Ø areas of rectangles, parallelograms and triangles;
Ø Pythagoras’ theorem and its converse;
Ø perpendicular bisector and angle bisector loci;
Ø circles – arcs, chords (excluding the rectangular properties),
angles in a segment, cyclic quadrilaterals, tangents, angle in the alternate
segment;
Ø proportional division of transversals by parallel lines;
Ø similar triangles.
Formal proofs of theorem as they appear in Geometry text-books will
not be asked.
Ability
to do the following, showing clearly all construction lines:
1.
Bisection of angles and line
segments.
2.
Construction of perpendiculars
to straight lines.
3.
Construction of an angle equal
to a given angle.
4.
Simple cases of the
construction from sufficient data of triangles and quadrilaterals.
5.
Division of a line segment into
a number of equal parts.
6.
Construction of a tangent to a
circle.
7.
Construction of circumscribed
and inscribed circles of a triangle.
8.
Easy extensions of the above.
*Although
the format of the syllabus has been changed, no essential change in the content
of the previous syllabus is intended.
（中文參考譯本）
試卷一：算術、代數、三角學和幾何多項選擇題（1.5 小時）
試卷二：算術和代數的進一步問題（1.5 小時）
試卷三：三角學和幾何進一步問題（1.5 小時）
自1974年以來，
試卷一：算術與代數（1.5 小時）
試卷二：算術、代數、三角學和幾何短問題（1.5 小時）
試卷三：三角學與幾何（1.5 小時）
算術和代數
1.
假定具備小學數學知識。
2.
使用英制、公制和中國的度量衡系統。
十進制貨幣體系。 （至1973 年）
使用國際單位制重量和測量單位。（1974）
3.
自然數的質數和因式分解。 2、3、4、5、8、9、11 LCM 和 HCF 的簡單整除性測試原理，包括找出最大公因數的一般原理。
（歐幾里得算法）。 （直至1973）
自然數的質數和因式分解。 2、3、4、5、8、9、11 LCM 和 HCF 的簡單整除性測試原理，包括找出最大公因數的一般原理，包括使用歐幾里得算法。（1974）
4.
單利息和複利息、盈利與虧損、百分數、平均數、速度和相對速度（包括分數和小數的使用）方面的問題。（1972）
單利息和複利息、盈利與虧損、百分數、平均數、速度和相對速度的問題。 （1973、1974）
5.
長方形、三角形、平行四邊形、梯形、圓形、長方體、稜柱、稜錐、直立圓錐、球體的初步測量。
6.
基本代數運算；
表達算術概括的公式； 一或兩個變數（包括三項式和 a^3±b^3）的簡單類型多項式；主項變換、因式分解、多項式的HCF 和 LCM； 簡單的代數分式。 （1972）
Ø 基本代數運算，包括簡單的分數； 將簡單多項式分解為一個或兩個變量，包括三項式和兩個立方的和與差； 此類多項式的 HCF 和 LCM。 （1973、1974）
Ø 表達算術概括的公式；主項變換。 （1973、1974）
7.
解一個未知數中的線性方程、兩個未知數中的聯立線性方程以及僅涉及一個未知數的二次方程，包括了解二次方程的根與係數的和與積之間的關係；
聯立方程式的解，一個線性方程和一個二次方程，涉及兩個未知數； 導致此類方程式的簡單問題。
8.
線性和二次不等式及其解。
9.
來自數字和統計數據的圖形。
三次以下的簡單代數函數的圖像。
10.
比、比例、比例部分和變分。
11.
非負整數指數定律並擴展到分數指數和負指數；和對數，包括使用根符號。
（√a 表示「a」的正平方根，其中「a」是任意正數）。 使用四位數表，以10 為底的對數進行計算。 透過因式和數表提取平方根。
（不需要長式）。
12.
等差和有限等比級數。
三角學
(1972)
1.
0度到360度角的六種三角比及其之間的簡單關係（不包括倍角和複角）。
直角三角形的解法，簡單應用。 透過分析直角三角形可以解決二維和三維的簡單問題。
2.
三角函數和對數三角函數表的使用。
3.
簡單的方程式（僅限0度到360度的解）。0度到360度範圍內的正弦和餘弦線性函數圖像。
4.
僅涉及簡單的數值範例之弧度法；
弧長和扇形面積。
5.
正弦和餘弦公式及其在三角形解中的應用，包括與較大邊相對的較大角度，反之亦然；
三角形面積為 1/2bc sinA。
（1973、1974）
1.
以度為單位測量 0°≤θ≤360° 範圍內的角度，以弧度為單位測量 0≤θ≤2π 範圍內的角度。 圓的弧長和扇形面積。
2.
六種三角比及其之間的簡單關係（不包括倍角和複角）。
直角三角形的解法，簡單應用。 透過分析直角三角形可以解決二維和三維的簡單問題。
3.
簡單方程式（0°≤θ≤360°範圍內的解）。 0°≤θ≤360° 範圍內的正弦和餘弦線性函數圖像。
4.
正弦和餘弦公式及其在三角形解中的應用，包括與較大邊相對的較大角度，反之亦然；
三角形面積為 1/2bc sinA。
（可能會要求考生給出指定程度的近似結果，但使用乘法和除法的簡化方法並不是必需的。）
幾何學（至1973）
每個候選人都必須為自己準備一把以英寸和十分之一英寸為單位、以厘米和毫米為單位的尺子、一個量角器、圓規和一支硬身鉛筆。
幾何試卷將包含實用幾何和理論幾何的問題。
實用幾何
實用幾何的問題將針對附表A中所包含的直尺和圓規的構造以及它們的簡單擴展進行設定。
如果解釋的有效性不明顯，則可能需要證明其合理性的推理。 應顯示施工線。
附表 A. 實用幾何
1.
角和直線的平分。
2.
垂直於直線的繪製。
3.
繪製一個等於給定角的角。
4.
60°、45°、30°角的繪製。
5.
繪製與給定直線的平行線。
6.
利用足夠的三角形和四邊形資料來繪製簡單案例。
7.
將直線分成給定數量的相等部分。
8.
繪製圓的切線。
9.
三角形的外接圓、內切圓和外切圓的繪製。
10.
繪製兩條給定直線成比例的平均值。
理論幾何
理論幾何問題將包括附表 B 中包含的定理問題（即附加題）。不會提問這些定理的正式證明，但這些問題需要對所涉及的基礎有透徹的了解。
附表 B. 理論幾何
直線上的角
1.
如果一條直線與另一條直線重疊，則所形成的兩個角和等於兩個直角；反之亦然。[直線上的鄰角]
2.
如果兩條直線相交，則垂直的對角相等。[對頂角]
平行的直線
1.
當一條直線與另外兩條直線相交時，如果
(i) 一對交錯角相等，[內錯角相等]
或 (ii) 一對對應的角度相等，[同位角相等]
(iii) 截線同一側的一對內角合起來等於兩個直角，[同旁內角互補]
則兩條直線平行，反之亦然。
2.
與同一條直線平行的直線彼此平行。
三角形和直線圖形
1.
三角形的內角和等於兩個直角。[三角形內角和]
2.
在 n 條邊的多邊形中，內角和等於
(2n-4) 個直角。[多邊形內角和]
3.
若凸多邊形的邊依序生成，則所形成的角和等於四個直角。[多邊形外角和]
4.
如果兩個三角形的兩邊與另一個三角形的兩邊相等，而這些邊所含的角度也相等，則這兩個三角形全等。[SAS]
5.
如果兩個三角形的兩個角等於另一個三角形的兩個角，且一個三角形的一邊等於另一個三角形的對應邊，則這兩個三角形全等。[ASA、AAS]
6.
如果三角形的兩邊相等，則這兩邊的對角也相等；反之亦然。[全等三角形的對應邊、全等三角形的對應角]
7.
如果兩個三角形的一個的三邊等於另一個的三邊，則這兩個三角形全等。[SSS]
8.
如果兩個直角三角形的斜邊相等，並且一個直角三角形的一側等於另一個直角三角形的一側，則該三角形全等。[RHS]
9.
從給定直線外部的給定點可以繪製到給定直線的所有直線中，垂線是最短的。
10.
平行四邊形的對邊和角相等，每條對角線平分平行四邊形，對角線互相平分。[平行四邊形對邊、平行四邊形對角、平行四邊形對角線]
11.
如果四邊形的一對對邊相等且平行，則它是平行四邊形。[對邊平行且相等]
12.
穿過三角形一邊的中點與另一邊平行的直線平分第三邊。[中點定理]
13.
連接三角形兩邊中點的直線平行於第三邊，並且等於它的二分之一。[中點定理]
14.
如果有3條或3條以上平行直線，且它們在與它們相交的任何一條直線上的截距相等，則在與它們相交的任何其他直線上相應的截距也相等。[截線定理]
面積
1.
同底且同平行線之間的平行四邊形面積相等。
2.
等底、等高的三角形面積相等。
3.
相同或等底的等三角形的高相同。
4.
在直角三角形中，斜邊上的平方等於直角邊的平方和；反之亦然。[畢氏定理、畢氏定理逆定理]
軌跡
1.
與兩個固定點等距的點的軌跡是連接兩個固定點的直線的垂直平分線。
2.
與兩條相交直線等距的點的軌跡由平分兩條給定直線之間的角度的一對直線組成。
圓
1.
從圓心平分非直徑弦的直線與弦成直角；
相反，從中心到弦的垂線平分該弦。[圓心至弦的垂線平分弦、圓心至弦中點的連線垂直弦]
2.
有一個圓，而且只有一個圓，它穿過三個給定的點，但不在一條直線上。
3.
圓的等弦距圓心等距；
反之亦然。[等弦對等弦心距、等弦心距對等弦]
4.
圓任一點的切線和經過該點的半徑互相垂直。[切線⊥半徑]
5.
從外點到圓的切線相等。[切線性質]
6.
如果兩個圓相接觸，則接觸點位於通過圓心的直線上。
7.
弧在中心所對的角度是圓周其餘部分任意點所對的角度的兩倍。[圓心角兩倍於圓周角]
8.
圓內同一弓形的角相等；並且，如果連接兩點的線與同一側的另外兩個點所成的角度相等，則這四個點位於一個圓上。[同弓形內的圓周角、同弓形內的圓周角的逆定理]
9.
半圓內的角是直角。[半圓上的圓周角]
10.
圓內切四邊形的對角互補；反之亦然。[圓內接四邊形內角、對角互補]
11.
在等圓中（或在同一個圓中）（i）如果兩個圓弧在中心所對的角相等，則它們相等；（ii）反之，如果兩個圓弧相等，則它們在中心所對的角相等。[等角對等弧、等弧對等角]
12.
在等圓中（或者，在同一個圓中）（i）如果兩個弦相等，則它們截取的弧相等；（ii）反之，如果兩個弧相等，則這兩個弧的弦相等。[等弦對等弧、等弧對等弦]
13.
若一條直線與一個圓相交，並從接觸點引一條弦，則該弦與切線所成的角度等於相交線段中的角度。[交錯弓形的圓周角]
比例：相似三角形
1.
若平行於三角形的一條邊畫一條直線，則按比例分割另外兩條邊；
反之亦然。
2.
如果兩個三角形是等角，則它們的對應邊成比例；
反之亦然。[AAA、三邊成比例]
平面幾何 (1974)
以下知識與應用*：
Ø 直線上的角；
Ø 平行線；
Ø 三角形和其他凸多邊形的內角和；
Ø 全等三角形、等腰三角形；
Ø 一點到直線的距離；
Ø 平行四邊形；
Ø 中點定理和截線定理；
Ø 長方形、平行四邊形和三角形的面積；
Ø 畢氏定理及其逆命題；
Ø 垂直平分線和角平分線軌跡；
Ø 圓－弧、弦（不含矩形特性）、線段內的角、圓內接四邊形、切線、交錯弓形內的角；
Ø 平行線按比例劃分橫斷面；
Ø 相似三角形。
不會詢問幾何教科書中出現的定理的正式證明。
能夠執行以下操作，清楚顯示所有構造線：
1.
角和線段的平分。
2.
垂直於直線的繪製。
3.
繪製一個等於給定角的角。
4.
利用充足的三角形和四邊形資料繪製簡單案例。
5.
將線段分成許多相等的部分。
6.
繪製圓的切線。
7.
三角形的外接圓和內切圓的繪製。
8.
簡易擴充上述內容。
*雖然課程大綱的格式發生了變化，但原課程大綱的內容並未發生本質變化。
附錄3：1975年至1982年會考數學課程
1975 (change named Mathematics Alternative
Syllabus), 1976, 1977, 1978, 1979, 1980, 1981 (change named Mathematics
Syllabus 2), 1982
Paper 1: Elementary type question, harder
questions (2.5 hours)
Paper 2: Multiple Choice Questions (1.5
hours)
Arithmetic and Algebra
1.
Primes and factorizations of
natural numbers. Principles of simple divisibility tests for 2, 3, 4, 5, 8, 9,
11 LCM and HCF including the general principles of finding the HCF including
the use of the Euclidean algorithm. (1978 cancel this part)
2.
Problems in simple and compound
interest, profit and loss, percentage, averages, speed and relative speeds.
(1975-1978)
Problems in simple and compound interest, growth and depreciation,
profit and loss, percentage, averages, speeds. (1979-1982)
3.
Elementary mensuration of the
rectangle, triangle, parallelogram, trapezium, circle, rectangular block,
prism, pyramid, right circular cone and sphere. (until 1978)
Elementary mensuration of the rectangle, triangle, parallelogram,
trapezium, circle, rectangular block, prism, pyramid, right circular cone and
sphere. Similar plane figures and solids; relation of areas and volumes to
their corresponding dimensions. (1979, 1980, 1981, 1982)
4.
Polynomials in one variable and
their fundamental operations. Factorization of 
, 
, 
, ; 
where
h, k, m and n are integers. LCM and HCF. Simple algebraic fractions.
5.
Formulae/ Formulas (1981):
their manipulation and numerical applications.
6.
Solution of linear equations in
one unknown and of simultaneous linear equations in two unknowns and quadratic
equations involving only one unknown including knowledge of the relations
between the sum and product of the roots and the coefficients of a quadratic
equation; the solution of simultaneous equations, one linear and one quadratic,
involving two unknowns. Simple problems leading to such equations. Distinction
between equations and identities.
7.
Ratio, proportion and
variation.
8.
Graph of numerical data, travel
graphs, graphs of linear and quadratic functions.
9.
Linear and quadratic inequalities
in one variable and their solutions. (until 1978)
Linear inequalities in one or two variables and their applications.
Quadratics inequalities in one variable. Graphical representation of linear
inequalities with applications to simple practical problems such as Linear
Programming. (1979, 1980, 1981, 1982)
10.
Laws of rational indices.
Calculation using common logarithms. Equations with unknown indices.
11.
Surds, rationalization/rationalization
(since 1978), equations with the unknown under a radical sign.
12.
Functional notation. Remainder
theorem.
13.
Arithmetic and geometric
progressions. (until 1978)
Arithmetic and geometric progressions. Insertion of arithmetic and
geometric means, sum to n terms, summation of geometric progression to
infinity. (1979, 1980, 1981, 1982)
Trigonometry
1.
Measure of angles in degrees
and in radians. Length of arc and area of sector of a circle. The function
sine, cosine, tangent and their graphs in the interval 0 to 
. The relations 
, 
.
2.
Solution of right-angled
triangles, with simple applications. Easy problems in three dimensions soluble
by analysis into right-angled triangles. Sine and cosine formulae and their
applications. Area of a triangle as 
.
3.
Easy trigonometric equations
(solution in the interval 0 to ).
Plane Geometry
Knowledge and application of the following:
Ø Angles at a point;
Ø parallel lines;
Ø sum of angles of a triangle and other convex polygons;
Ø congruent triangles, isosceles triangles;
Ø the distance of a point from a straight line;
Ø parallelograms;
Ø mid-point and intercept theorems;
Ø areas of rectangles, parallelograms and triangles;
Ø Pythagoras’ theorem and its converse;
Ø perpendicular bisector and angle bisector loci;
Ø circles;
Ø arcs, chords (excluding the rectangular properties), angles in a
segment, cyclic quadrilaterals, tangents, angle in the alternate segment;
Ø proportional division of transversals by parallel lines;
Ø similar triangles.
Formal proofs of theorem as they appear
in Geometry text-books will not be asked.
(1975-1977, 1978 cancel the following part)
Ability to do the following, showing
clearly all construction lines:
1.
Bisection of angles and line
segments.
2.
Construction of perpendiculars
to straight lines.
3.
Construction of an angle equal
to a given angle.
4.
Simple cases of the
construction from sufficient data of triangles and quadrilaterals.
5.
Division of a line segment into
a number of equal parts.
6.
Construction of a tangent to a
circle.
7.
Construction of circumscribed
and inscribed circles of a triangle.
8.
Easy extensions of the above.
（中文版本）
算術及代數
1.
質數及自然數之因子分解，以2，3，4，5，8，9，11，整除之簡單檢驗。最大公約和最小公倍，包括輾轉刪除法之運用。（1978取消此部份）
2.
單利息及複利息問題，盈利及賠本，百分率，平均值，速度及相對速度。（1975-1978）
單利息及複利息問題，增長及折舊，盈利及賠本，百分率，平均值，速率。（1979-1982）
3.
面積及體積之計算，包括矩形，三角形，平行四邊形，梯形，多邊形，圓形，長方體，柱體，圓柱體，錐體，圓錐體和球體。（直到1978）
面積及體積之計算，包括矩形、三角形、平行四邊形、梯形、多邊形、圓形、長方體、角柱、圓柱、角錐、直立圓錐及球體。相似形及相似體，其面積及體積。（1979-1982）
4.
單元多項式及基本運算。下列各式之因子分解，，，，，，其中h，k，m，n為整數，最大公約及最少公倍。簡單分式。
5.
公式：計算上之應用。
6.
一元線性方程，二元綫方程組，一元二次方程，根與系數之關係。二元聯立方程組（一為一次，一為二次）。各類方程之應用題。方程與恒等式之區別。
7.
單變元綫性與二次不等式及其解法。（直至1978）
單變元及雙變元之線性不等式及其應用。單變元之二次不等式。線性不等式之圖像及其在簡易實際問題中之應用，如線性規劃。（1979-1982）
8.
數據之圖解，行程圖解，綫性及二次函數之圖解。
9.
比，比例及變數法。
10.
有理指數定律，用常用對數之計算，指數方程。
11.
根式及其有理化，無理方程式。
12.
函數之符號，餘式定理。
13.
等差級數及等比級數。（直至1978）
等差級數及等比級數。等差中項及等比中項之插入，n項和，等比級數無限項之和。（1979-1982）
三角
1.
角之量度：以度或徑為單位，弧長及扇形面積，正弦，餘弦，正切，在0至區間之函數及圖像，三角函數之基本關係：，及。
2.
直角三角形解法及其應用，三維空間之簡易應用題，只限於能用直角三角形求解者。正弦定律，餘弦定律及其應用。以求三角形面積。
3.
簡易三角方程（答案在0至區間）。
平面幾何
下列項目的知識及應用：
同頂角，平行線，三角形及多邊形之內角和，全等三角形，等腰三角形，點以直綫之距離，平行四邊形，中點定理及截綫定理。矩形，平行四邊形及三角形之面積，畢氏定理及其逆定理，垂直等分線軌跡與分角線軌跡。圓：弧及弦（相交弦之矩形性質除外），對同弧圓周角，圓内接四邊形，切綫，弧切角，平行線組分各截綫為定此綫段，相似三角形。
本試不予測驗幾何定理之形式化證明，一如一般幾何課本所刊載者。
（1975-1977，1978取消此部份）
考生須具下列作圖知識，並於作圖時顯示所有輔助綫。
1.
等分已知角及已知線段。
2.
作已知直綫之垂綫。
3.
作一角等於已知角。
4.
三角形或四邊形之簡易作圖。
5.
分一直綫為若干等長部份。
6.
作圖之切線。
7.
三角形之外接圖及內接圓之作法。
8.
以上項目之簡易推廣。
附錄4：1980年至1987年會考數學課程
1983 (remain syllabus 3 to syllabus A
and combined syllabus 1 and 2 to syllabus B), 1984-1987
Mathematics (Syllabus 3) 1980-1982
>>> [Syllabus A] 1983-1987
Paper 1 - 2.5 hours for 1980, 2 hours for
1981
1.
Problems in simple and compound
interest, growth and depreciation, profit and loss, percentage.
2.
Elementary mensuration of and
formulas for rectangle, triangle, parallelogram, trapezium, polygon, circle,
rectangular block, prism, cylinder, pyramid, right circular cone and sphere.
Similar figures and solids; relation of areas and volume to their corresponding
dimensions. (1980-1981 only)
Elementary
mensuration of and formulas for rectangle, triangle, parallelogram, trapezium,
polygon, circle, rectangular block, prism, cylinder, pyramid, right circular
cone and sphere. Similar figures and solids, their areas and volumes. (since
1982)
3.
Formulas: their manipulation
and numerical applications.
4.
Polynomials in one variable and
their fundamental operations. Expansion 
for
any positive integer n less than 6. Factorization of 
, , , , 
where h, k, m, n are integers. Notation for
functions, e.g. 
. Remainder
theorem. LCM and HCF. Simple algebraic fractions.
5.
Graphs of linear and quadratic
functions. Slope (gradient) of a curve at a point and its application in
sketching graphs of polynomials in one variable od degree not higher than 3.
(1980-1981 only)
Graphs of 
and 
. Slope (gradient) of a curve at
a points and its application in sketching graphs of polynomials in one variable
of degree not higher than three. (since 1982)
6.
Solution of linear equations in
one unknown, of simultaneous linear equations in two unknowns, and of quadratic
equations in one unknown. Relations between the roots and the coefficients of
quadratic equations in one unknown. Solution of simultaneous equations, one
linear and one quadratic, in two unknowns. Simple problems leading to such
equations. Distinction between equations and identities. Approximate solution
of simple equations in one variable by graphical method (including the method of magnification [added 1984-1987]).
7.
Ratio and proportion.
8.
Linear inequalities in one or
two variables and their graphical representation; applications to simple
practical problems such as linear programming. Quadratic inequalities in one
variable.
9.
Laws of rational indices and
simple properties of logarithm.
10.
Quadratic surds,
rationalisation, easy equations with the unknown under a radical sign.
11.
Arithmetic and geometric
progressions. Insertion of arithmetic and geometric means, sum to n terms; sum
to infinity of geometric progression.
12.
Simple ideas of probability
with applications of the addition law and multiplication law to easy problems.
13.
Graphical representation of
statistical data by bar charts, pie charts, histograms, frequency polygons and
curves, cumulative frequency polygons and curves. Mean, mode and median.
Measures of dispersion: range, mean deviation, standard deviation; calculation
and use of standard deviation. (old
description 1980-1984)
Collection of
organization of numerical data, and their graphical representation by bar
charts, pie charts, histograms, frequency polygons and curves, cumulative frequency
polygons and curves. Mean, mode and median. Measures of dispersion: range, mean
deviation, standard deviation; calculation and use of standard deviation. (new
description 1985-1987)
14.
Measure of angles in degrees
and in radians. Length of arc and area of sector of a circle. The functions
sine, cosine, tangent and their graphs in the interval 0 to radians.
15.
Solution of right-angled
triangles, with simple applications. Easy problems in three dimensions soluble
by analysis into right-angled triangles. Sine and cosine formulas and their
applications. Area of triangles as .
16.
Easy trigonometric equations
(solutions in the interval 0 to radians).
17.
Rectangular coordinates in the
2-dimensional space. Distance between two points. Internal division of a line
segment in a given ratio. Equations of a straight line; slope (gradient) of a
straight line; perpendicularity; intersection. Coordinate treatment of loci for
straight line and circle, and parabola in standard position.
Intersection of
straight line with a circle. (added 1985-1987)
18.
Easy application of the
following in plane geometry. Lengthy proofs will not be required but brief
reasons may be required.
Angles at a point, sum of angles of a triangle and of other convex
polygons; parallel lines; similar triangles; congruent triangles; isosceles
triangles; parallelograms; mid-point and intercept theorems; Pythagoras’
Theorem and its converse; perpendicular bisector and angle bisector; circles:
arcs, chords, angles in a segment, cyclic quadrilaterals, tangents, angle in
the alternate segment.
（中文版本）
1.
單利息及複利息問題，增長及折舊。盈利及賠本，百分率。
2.
面積及體積之公式及計算，包括矩形、三角形、平行四邊形、梯形、多邊形、圓形、長方體、角柱、圓柱、角錐、直立圓錐及球體。相似形及相似體，面積、體積與對應線段之比。（只限1980-1981）
面積及體積之公式及計算，包括矩形、三角形、平行四邊形、梯形、多邊形、圓形、長方體、角柱、圓柱、角錐、直立圓錐及球體。相似形及相似體，其面積及體積。（自1982）
3.
公式：計算上之應用。
4.
單變元多項式及其中基本運算。之展式，其中n為小於6之正整數。下列各項之因子分解：，，，，，其中h，k，m，n為整數。函數之符號，如。餘式定理。HCF及LCM。簡單分式。
5.
線性及二次函數之圖形。曲線於某點之斜率及其應用於描繪次數不超過三之單變元多項式之圖形。（只限1980-1981）
之圖形。曲線於某點之斜率及其應用於描繪次數不超過三之單變元多項式之圖形。（自1982）
6.
一元線性方程，二元聯立線性方程組及一元二次方程之解法。一元二次方程之根與系數關係。二元聯立方程組（一為一次、一為二次）之解法。以上各類方程之簡易應用題。方程與恒等式之區別。應用圖解法以求簡易一元方程之近似解（包括放大法[1984-1987加入]）。
7.
比及比例。
8.
單變元及雙變元之線快不等式及其圖像，及其在簡易實際問題之應用，如線性規劃。單變元之二次不等式。
9.
有理數指數定律及對數之簡易性質。
10.
二次根式及其有理化，簡易無理方程。
11.
等差級數。等比級數。等差中項及等比中項之插入，n項和，等比級數無限項之和。
12.
簡易概率。加法定律，乘法定律及其簡易應用。
13.
數據之圖示：條線圖，圓瓣圖，直方圖，頻率多邊形及曲線，累積頻率直方圖及曲線。平均值、中位數及眾數。分佈之量度：分類域、平均偏差及標準差；標準差之計算及應用。（1983-1984）
數據之搜集，組織及其表示法，條線圖，圓瓣圖，直方圖，頻率多邊形及曲線，累積頻率直方圖及曲線。平均值、中位數及眾數。分佈之量度：分類域，平均偏差及標準差；標準差之計算及應用。（1985-1987）
14.
角之量度：以度及弧度或單位。弧長及扇形面積。正弦、餘弦、正切在0至弧度區間之函數及圖形。三角函數之基本關係：及.。
15.
直角三角形解法求其簡易應用。三維空間之簡易應用題，只限能用直角三角形求解者。正弦定律，餘弦定律及其應用。以求三角形面積。
16.
簡易三角方程（在0至弧度區間之解）。
17.
平面直角座標系。兩點距離，以定比將線段內分，直線斜率，垂直之條件，兩直線交點。以坐標方法處理下列軌跡問題：直線、圓及在標準位置上之抛物線。
直線與圓之交點。（1985-1987加入）
18.
下列各項在平面幾何上之簡易應用（不要求冗長之證明，但考生須舉出簡略之理由）：
同頂角，三角形及多邊凸形之内角和，平行線，相似三角形，全等三角形，等腰三角形，平行四邊形。中點定理及截線定理。畢氏定理及其逆定理。垂直等分線與分角線。圓：弧及弦，對同弧圓周角，圓内接四邊形，切線，弦切角。
[Syllabus B] 1983-1987
1.
Problems in simple and compound
interest, growth and depreciation, profit and loss, percentage, averages,
speeds.
2.
Elementary mensuration of and
formulas for rectangle, triangle, parallelogram, trapezium, polygon, circle,
rectangular block; prism, cylinder, pyramid, right circular cone and sphere.
Similar figures and solids, relation of areas and volumes to their
corresponding dimensions.
3.
Formulas: their manipulation
and numerical applications.
4.
Polynomials in one variable and
their fundamental operations. Factorization of , , , , where h, k, m, n are integers. Notation for
functions, e.g. . Remainder
theorem. LCM and HCF. Simple algebraic fractions.
5.
Graphs of linear and quadratic
functions, travel graphs. (1983-1984)
Graphs of 
and , travel
graphs. (since 1985)
6.
Solution of linear equations in
one unknown, of simultaneous linear equations in two unknowns, and of quadratic
equations in one unknown. Relations between the roots and the coefficients of
quadratic equations in one unknown. Solution of simultaneous equations, one
linear and one quadratic, in two unknowns. Simple problems leading to such
equations. Distinction between equations and identities.
7.
Ratio, proportion and variation.
8.
Linear inequalities in one or
two variables and their graphical representation; applications to simple
practical problems such as linear programming. Quadratic inequalities in one
variable.
9.
Laws of rational indices.
Calculation using common logarithm, Equations with unknown indices.
10.
Quadratic surds,
rationalisation, easy equations with the unknown under a radical sign.
11.
Arithmetic and geometric
progressions. Insertion of arithmetic and geometric means, sum to n terms,
summation of geometric progression to infinity.
12.
Simple ideas of probability
with applications of the addition law and multiplication law to easy problems.
13.
Collection and organization of
numerical data, and their graphical representation by bar charts, pie charts,
histograms, frequency polygons and curves, cumulative frequency polygons and
curves. Calculation of the mean. Determination of the median. Calculation and
use of standard deviation as a measure of dispersion.
14.
Measure of angles in degrees
and in radians. Length of arc and area of sector of a circle. The functions
sine, cosine, tangent and their graphs in the interval 0 to radians.
The relations , .
15.
Solution of right-angled
triangles, with simple applications. Easy problems in three dimensions soluble
by analysis into right-angled triangles. Sine and cosine formulas and their
applications. Area of triangles as .
16.
Easy trigonometric equations
(solutions in the interval 0 to radians).
17.
Rectangular coordinates in the
2-dimensional space. Ordered pair. Distance between two points. Points dividing
line segments in a given ratio. Equations of a straight line; slope (gradient)
of a straight line; perpendicularity; intersection. Equations of circles;
coordinates of centre; length of radius. Intersection of straight lines and
circles.
18.
Easy application of the
following:
Angles at a point, sum of angles of a triangle and of other convex
polygons; parallel lines; proportional division of transversals by parallel
lines; similar triangles; congruent triangles; isosceles triangles; the
distance of a point from a straight line (1985 cancelled); parallelograms;
mid-point and intercept theorems; Pythagoras’ Theorem and its converse;
perpendicular bisector and angle bisector; circles: arcs, chords, angles in a
segment, cyclic quadrilaterals, tangents, angle in the alternate segment.
（中文版本）
1.
單利息及複利息問題，增長及折舊。盈利及賠本，百分率，平均值，速率。
2.
面積及體積之公式及計算，包括矩形、三角形、平行四邊形、梯形、多邊形、圓形、長方體、角柱、圓柱、角錐、直立角錐及球體。相似形及相似體，面積、體積與對應線段之比。
3.
公式：計算上之應用。
4.
單變元多項式及基本運算。下列各項之因子分解：，，，，，其中h，k，m，n為整數。函數之符號，如。餘式定理。LCM及HCF。簡單分式。
5.
線性及二次函數之圖形，行程圖形。（1983-1984）
6.
一元線性方程、二元聯立線性方程組及一元二次方程之解法。一元二次方程之根與系數關係。二元聯立方程組（一為一次、一為二次）之解法。各類方程之簡易應用題。方程與恒等式之區別。
7.
比、比例及變數法。
8.
單變元及雙變元之線性不等式及其圖形，及其在簡易實際之應用，如綫性規劃。單變元之二次不等式。
9.
有理指數定律。應用常用對數之計算。指數方程。
10.
二次根式及其有理化，簡易無理方程。
11.
等差級數。等比級數。等差中項及等比中項之插入，n項和，等比級數無限項之和。
12.
簡易概率。加法定律，乘法定律及其簡易應用。
13.
數據之搜集，組織及其表示法，條綫圖，圓瓣圖，直方圖，頻率多邊形及曲線，累積頻率直方圖及曲綫。平均值之計算。中位數之確定；標準差作為分佈之量度，其計算及應用。
14.
角之量度：以度及弧度為單位。弧長及扇形面積。正弦、餘弦、正切，在0至2π弧度區間之函數及圖形。三角函數之基本關係：及.。
15.
直角三角形解法及其簡易應用。三位空間之簡易應用題，只限於能用直角三角形求解者。正弦定律，餘弦定律及其應用。以求三角形面積。
16.
簡易三個方程（在0至2π弧度區間之解）。
17.
平面之直個座標系。序偶。兩點距離，分綫段為定比之點，直綫方程，直綫斜率，垂直之條件，兩直線交點。圓之方程，圓心坐標，半徑長，直綫與圓之交點。
18.
下列各項之簡易應用：
同頂角，三角形多邊凸形之外角和，平行綫，平行綫系分各截綫為定比綫，相似三角形，全等三角形，等腰三角形，點與直綫之距離（1985取消）。平行四邊形，中點定理及截綫定理。畢氏定理及其逆定理。垂直等分綫與分角線。圓：弧及弦，對同弧圓周角，圓内接四邊形，切線，弦切角。
附錄5：1988年至2011年會考數學課程
1988
1.
Simple and compound interest,
growth and depreciation, profit and loss, percentages.
2.
Elementary mensuration of and
formulas for rectangle, triangle, parallelogram, trapezium, polygon, circle,
rectangular block, prism, cylinder, pyramid, right circular cone and sphere.
Similar plane figures and solids, their areas and volumes.
3.
Formulas: their manipulation
and numerical applications.
4.
Polynomials in one variable and
their fundamental operations. Factorization of , , , , where h, k, m, n are integers. Notation for
functions, e.g. . Remainder
theorem. LCM and HCF. Simple algebraic fractions.
5.
Graphs of and .
6.
Solution of linear equations in
one unknown, of simultaneous linear equations in two unknowns, and of quadratic
equations in one unknown. Relations between the roots and the coefficients of
quadratic equations in one unknown. Solution of simultaneous equations, one
linear and one quadratic, in two unknown. Simple problems leading to such
equations. Distinction between equations and identities. Approximate solution
of simple equations in one unknown by the graphical method and improvement of
accuracy of the method of bisection.
7.
Ratio, proportion and
variation.
8.
Linear inequalities in one or
two variables and their graphical representation; applications to simple
practical problems such as linear programming. Quadratic inequalities in one
variable.
9.
Law of rational indices and
simple properties of logarithms. Rationalization of expression of the form 
10.
Arithmetic and geometric
progressions. Insertion of arithmetic and geometric means, sum to n terms; sum
to infinity of geometric progressions.
11.
Simple ideas of probability
with applications of the addition law and multiplication law to easy problems.
12.
Collection and organization of
numerical data, and their graphical representation by bar charts, pie charts,
histograms, frequency polygons and curves, cumulative frequency polygons and
curves. Measure of central tendency: mean, mode and median. Measure of
dispersion; range, inter-quartile range, mean deviation, standard deviation.
13.
Measure of angles in degrees
and in radians. Length of arcs and area of sectors of a circle. The function
sine, cosine tangent and their graph in the interval 0 to radians. The relations , .
14.
Solution of right-angled
triangles, with simple applications. Easy problems in three dimensions soluble
by analysis into right-angled triangles. Sine and cosine formulas and their
applications. Area of triangles as .
15.
Easy trigonometric equations
(solutions in the interval 0 to radians, i.e. 0 to 360).
16.
Rectangular coordinates in
2-dimensional space. Distance between two points. Internal division of a line
segment in a given ratio. Equations of a straight line; slope (gradient) of a
straight line; perpendicularity; intersection. Equation of a straight line with
a circle.
17.
Easy applications of the
following (lengthy proofs will not be required but brief reasons may be
required):
Angles at a point, sum of angles of a triangle and of other convex
polygons; parallel lines; similar triangles; congruent triangles; isosceles
triangles; parallelograms; mid-point and intercept theorems; Pythagoras;
Theorem and its converse; perpendicular bisector and angle bisector; circles:
arcs, chords, angles in a segment, cyclic quadrilaterals, tangents, angle in
the alternate segment.
（中文版本）
1.
單利息及複利息，增長及折舊。盈利及賠本，百分率。
2.
面積及體積之公式及計算，包括矩形、三角形、平行四邊形、梯形、多邊形、圓形、長方體、角柱、圓柱、角錐、直立圓錐及球體。相似形及相似體，其面積及體積。
3.
公式：計算上之應用。
4.
單變元多項式及其基本運算。下列各項之因子分解：，，，，，其中h，k，m，n為整數。函數之符號，如。餘式定理。LCM及HCF。簡單分式。
5.
及之圖形。
6.
一元線性方程、二元聯立線性方程組及一元二次方程之解法。一元二次方程之根與系數關係。二元聯立方程組（一為一次、一為二次）之解法。以上各類方程之簡易應用題。方程與恒等式之區別。應用圖解法以求簡易一元方程之近似解及以分半方法提高近似以解之準確度。
7.
比、比例及變數法。
8.
單變元及雙變元線性不等式及其圖形，及其在簡易實際問題中之應用，如線性規劃。單變元之二次不等式。
9.
有理指數定律及對數之簡易性質。根式形如之有理化。
10.
等差級數。等比級數。等差中項及等比中項之插入，n項和，等比級數無限項之和。
11.
簡易概率。加法定律，乘法定律及其簡易應用。
12.
數據之搜集，組織及其表示法，條線圖，圓瓣圖，直方圖，頻率多邊形及曲線，累積頻率直方圖及曲線。集中趨勢之量度：平均值、眾數及中位數。分佈之量度：分佈域，四分位數間距，平均偏差及標準差。
13.
角之量度：以度及弧度為單位。弧長及扇形面積。正弦、餘弦、正切在0至弧度區間之函數及圖形。三角函數之基本關係：及。
14.
直角三角形解法及其簡易應用。三維空間之簡易應用題，只限於能用直角三角形求解者。正弦定律，餘弦定律及其應用。以求三角形面積。
15.
簡易三角方程[在0至弧度（即0至360）區間之解]。
16.
平面的直角座標系。兩點距離，以定比將線段內分，直線方程，直線斜率，垂直之條件，兩直線交點。圓之方程，圓心坐標，半徑長。直線與圓之交點。
17.
下列各項之簡易應用（不要求冗長之證明，但考生需能對舉出簡略之理由）：同頂角，三角形及凸多邊形之内角和，平行線，相似三角形，全等三角形，等腰三角形，平行四邊形，中點定理及截線定理。畢氏定理及其逆定理。垂直等分線與分角線。圓：弧及弦，對同弧圓周角，圓内接四邊形，切線，弧切角。
1989 (give detail information), 1990-1997,
1998 (set non-foundation part), 1999-2005
|
(English Version) 1.
Percentages. 2.
Manipulation of formulas.* Factorization of simple expressions. LCM and HCF. Simple algebraic fractions.* Polynomials in one variable. Notation for functions. Remainder theorem.* Graph of 3.
Linear equations in one
unknown, Quadratic equations in one unknown.* Simultaneous equations in two unknowns.* Simple problems leading to the above equations. Distinction between equations and identities. 4.
Approximate solution of
simple equations in one unknown by the graphical method and improvement of
accuracy by the method of bisection.* 5.
Linear inequalities in one or
two variables and their graphical representation.* Quadratic inequalities in one variable and their
graphical representation. 6.
Laws of rational indices. Simple properties of logarithms.* Simple manipulations of surds including rationalizing
the denominator of expressions of the form 7.
Ratio, proportion and
variation.* 8.
Arithmetic and geometric
sequences.* 9.
Simple problems in
probability. 10.
Organization and
representation of numerical data.* Measures of central tendency: mean, mode and median.* Measures of dispersion: range, inter-quartile, mean
deviation, standard deviation.* Applications of standard deviation. 11.
Measure of angles in degrees
and radians.* The functions sine, cosine, tangent in the interval 0
to The relations 12.
Easy trigonometric
equations.* 13.
Sine and cosine formulas.
Problems in two dimensions. Easy problems in three dimensions.* 14.
Mensuration of common plane
figures and solids. Length of arcs and area of sectors of a circle.* Similar plane and solids.* 15.
Plane rectangular
coordinates. Distance between two points. Internal division of a
line segment.* Slope (gradient) of a straight line. Equation of a
straight line in different forms.* Condition for two lines to be parallel or to be
perpendicular. Intersection of lines. Equation of a circle.* Intersection of a circle and a straight line.* 16.
Knowledge and applications of
the following topics in elementary plane geometry. Angles. Isosceles and equilateral triangles. Similar and
congruent triangles. Pythagoras’ theorem and its converse. Parallelograms, rectangles, squares and rhombuses. Mid-point and intercept theorems.* Circles.* Cyclic quadrilaterals.* *with non-foundation part |
（中文版本） 1.
百分率。 2.
公式之運算。* 簡易因式分解。 最低公倍式及最高公因式。 簡單代數分式。 單變元多項式。* 函數之記號。 餘式定理。*
3.
一元一次方程。 一元二次方程。* 二元聯立方程組。* 以上各類方程之簡易應用題。 方程與恒等式之區別。 4.
用圖解法求簡易一元方程之近似解及以分半方法提高近似解之準確度。* 5.
單變元及雙變元之線性不等式及其圖示。* 單變元之二次不等式及其圖示。 6.
有理指數定律。 對數之簡易性質。*
7.
比、比例及變數法。* 8.
等差級數及等比級數。* 9.
簡易概率問題。 10.
數據之組織及其表示法：* 集中趨势之量度：平均值、眾數及中位數。* 分佈之量度：分佈域、四分位數間距、平均偏差及標準差。* 11.
角之量度：以度及弧度為單位。* 在0至
12.
簡易三角方程。* 13.
正弦定律及餘弦定律。平面三角問題。三維空間之簡易問題。* 14.
普通平面及立體圖形之面積及體積的計算。 弧長及扇形面積。* 相似平面及立體圖形。* 平面直角坐標。 兩點間距離。線段之內分。* 直線之斜率。直線方程之各種形式。* 兩直線平行或垂直之條件。 直線之相交。 圓之方程。* 直線與圓之相交。* 15.
對下列各項簡易平面幾何綱目之認識及應用： 等腰及等邊三角形。相似及全等三角形。畢氏定理及其逆定理。 平行四邊形、矩形、正方形及菱形。 中點定理及截線定理。* 圓。* 圓内接四邊形。* *含非基礎部分 |
2006-2011
|
(English Version) 1.
Percentages. Rate and ratio. Variations. Estimation. 2.
Polynomials. Remainder theorem.* 3.
Laws of indices.* 4.
Sequences.* 5.
Equations in one unknowns.* Simultaneous equations in two unknowns.* 6.
Formulas. Functions and graphs.* 7.
Inequalities.* 8.
Exponential and logarithmic
functions.* 9.
Mensuration of common plane
figures and solids. Length of an arc and area of a sector of a circle. Similar plane figures and solids. 10.
Deductive reasoning of
geometry. Angle and straight lines.* Triangles.* Pythagoras’ theorem. Quadrilaterals and polygons.* Circles.* Transformation and symmetry in 2-D figures. 3-D figures. 11.
Introduction to coordinates.* Coordinate geometry of straight lines. Coordinate geometry of circles.* 12.
Measure of angles. Trigonometric ratios.* Simple trigonometric equations.* Applications of trigonometric ratios.* 13.
Probability.* 14.
Organization and
representation of numerical data. Measures of central tendency. Measure of dispersion.* Uses and abuses of statistics.* *with non-foundation part |
（中文版本） 1.
百分法。 率及比。 變數法。 估算。 2.
多項式。 餘式定理。* 3.
指數定律。* 4.
數列。* 5.
一元方程。* 二元聯立方程。* 6.
公式。 函數及圖像。* 7.
不等式。* 8.
指數函數及對數函數。* 9.
常見平面及立體圖形的求積。 弧長及扇形面積。 相似平面圖形及立體圖形。 10.
演繹推理幾何。 角和直線。* 三角形。* 畢氏定理。 四邊形及多邊形。* 圓形。* 平面圖形之變換及對稱性。 立體圖形。 11.
坐標簡介。* 直線之坐標幾何。 圓之坐標幾何。* 12.
角之量度。 三角比。* 簡單三角方程。* 三角比之應用。* 13.
概率。* 14.
數據之組織及其表示法。 集中趨勢之量度。 離差之量度。* 統計的應用及誤用。* *含非基礎部分 |
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