澳門四高校聯合入學考試 ( 語言科及數學科 ) Joint Admission Examination for Macao Four Higher Education Institutions (Languages and Mathematics) 考試大綱 Syllabus Mathematics 數學正卷 1
考試時間 : 兩小時 1. 基本概念 : 實數系統 ; 集合和子集的概念 ; 集合的運算 : 併集 交集和補集 偉恩 (Venn) 圖 數學歸納法 2. 百分數 : 百分數的意義及其在日常生活中的應用 ; 盈利和虧蝕 折扣 單利息和複利息 增長及折舊 3. 變分 : 比 比例 ; 正變 反變 聯變及部分變 4. 多項式及有理分式 : 多項式的運算, 長除法及綜合除法 ; 因式分解 : 因式定理及餘式定理, 最高公因式 (H.C.F.) 及最低公倍式 (L.C.M.); 平方差公式, 立方和 ( 立方差 ) 公式, 部分分式 5. 二次方程及二次函數 : 一元二次方程的解與判別式的關係, 二次公式 ; 根與係數的關係 ; 二次函數的極值 配方法的應用 6. 指數及根式 : 指數定律 ; 根式的簡化與運算 7. 代數不等式 : 代數不等式和絕對不等式的運算及其解集 ; 解一元一次或二元一次不等式組, 包括用幾何方法求解 ; 在線性規劃問題的應用 8. 對數函數與指數函數 : 對數的性質, 換底公式, 自然指數函數 ; 在增長及衰變過程的應用 ( 包括連續複利息 ); 解指數方程及對數方程 9. 非線性方程 : 解分式方程及無理方程 10. 排列與組合 : 基本概念, 二項式定理 2
11. 數列 : 等差數列 等比數列及前 n 項和 ; 等比數列無限項之和 12. 直線圖形及圓 : (A) 直線圖形 : 三角形及凸多邊形內角和 ; 直線及角的性質和定理 ; 相似三角形 全等三角形 ; 畢氏定理 ; 正方形 矩形 菱形及平行四邊形的性質 ; 中點定理及截線定理 (B) 圓 : 圓 弦及弧的性質 ; 圓心角 圓周角 圓內接四邊形 外接圓 ; 弧長及扇形面積 13. 三角 : 角度制及弧度制的關係 ; 三角函數與三角恆等式, 複角公式及半角公式 ; 式子 a cos b sin 與輔助角公式 ; 三角形面積 ; 正弦定律, 餘弦定律 ; 反三角函數的定義 ; 含一個未知數的三角方程求解 14. 解析幾何 : (A) 直角座標系, 兩點的距離, 線段的定比分點 ; 直線的斜率及截距, 直線方程的不同表達式 ; 兩線平行與垂直 解不多於三個未知數的線性方程組 (B) 圓的標準方程 一般方程 圖形和性質 ; 橢圓 雙曲線 拋物線的定義和標準方程 圖形和性質 直線與圓錐曲線的相交 15. 函數圖形 : 一次 二次及三次函數, 有理函數 對數及指數函數, 正弦 餘弦及正切函數的描繪 ; 對稱 平移 伸展 收縮及反射等技巧的運用 16. 概率和統計 : 隨機試驗, 結果與事件 ; 概率加法規則和乘法規則 ; 集中趨勢的度量 : 算術平均數, 眾數及中位數 ; 離散度的度量 : 極差, 方差及標準差 3
Examination Duration: 2 hours 1. Fundamental Concepts: real number system; concept of sets and subsets; set operations: union, intersection and complement. Venn diagrams. Mathematical induction. 2. Percentage: its meaning and applications to daily life problems. Profit and loss, discount, simple and compound interest, growth and depreciation. 3. Variations: ratio, proportion; direct, inverse, joint and partial variations. 4. Polynomial and Rational Fraction: manipulation of polynomials, long division and synthetic division, factorization of polynomials: the factor theorem and the remainder theorem; highest common factors (H.C.F.) and least common multiple (L.C.M.); formula for the difference of two squares, formulae for the sum of two cubes and the difference of two cubes; partial fractions. 5. Quadratic Equations and Quadratic Functions: the relation between the solution of a quadratic equation in one variable and its discriminant, the quadratic formula; relations between roots and coefficients; the extreme value of a quadratic function applying the method of completing the square. 6. Indices and Surds: laws of indices; simplification and operations of surds. 7. Algebraic Inequalities: manipulation of algebraic inequalities and absolute inequalities, and their solution sets; solving system of linear inequalities in one or two variables, including graphical solutions; applications to linear programming problems. 8. Logarithmic and Exponential Functions: properties of logarithms, change of bases of logarithms; natural exponential functions; applications in growth and decay processes (including continuous compounding of interest); solving equations of indices and equations of logarithms. 9. Nonlinear equations: solving fractional equations and irrational equations. 10. Permutation and Combination: basic concepts, binomial theorem. 4
11. Sequences: arithmetic sequence, geometric sequence, and sum of the first n terms; sum of geometric sequence with an infinite number of terms. 12. Rectilinear Figures and Circles: (A) Rectilinear Figures: the sum of interior angles of triangles and convex polygons; properties and theorems of lines and angles; similar triangles, congruent triangles; Pythagoras theorem; properties of squares, rectangles, rhombuses, and parallelograms; mid-point theorem and intercept theorem. (B) Circles: properties of circles, arcs and chords; angles of chord, angles of circumference, cyclic quadrilaterals, circumcircles; arc lengths and area of sectors. 13. Trigonometry: relation between degree measure and radian measure; trigonometric functions and trigonometric identities, compound angle formulae and half-angle formulae; the expression a cos a cos b sin and the auxiliary angle formula; area of a triangle; the Sine Law, the Cosine Law; the definitions of inverse trigonometric functions; solving trigonometric equations in one unknown. 14. Coordinate Geometry: (A) Rectangular Cartesian coordinate system, distance between two points; point of division of a line segment in a given ratio; the slope and intercept of a straight line, different forms of equations of a straight line; parallel and perpendicular lines. Solving system of linear equations with at most three variables. (B) The standard form of a circle, its general form, its graph and its properties; the definitions and standard forms of ellipse, hyperbola, and parabola, their graphs and their properties. Intersection of lines and conic. 15. Graphs of functions: sketching of linear, quadratic, cubic, rational, logarithmic, exponential, sine, cosine, and tangent functions; application of the techniques of symmetry, translation, stretching, shrinking, and reflection. 16. Probability and Statistics: random experiment, outcomes and events; addition rule and multiplication rule of probabilities; measures of central tendency: mean, mode, and median; measures of dispersion: range, variance and standard deviation. 5