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CSCA Mathematics: Advanced Syllabus, Mock Exams & Prep GuideCSCA 数学考试：权威复习大纲、历年真题与备考指南

Excel in CSCA Math: Calculus, Algebra, Vectors & Analytic Geometry. Practice realistic past-paper questions with KaTeX on CSCA Space.高效通关 CSCA 数学考试。深入解析微积分、线性代数、解析几何及向量考点。提供包含 LaTeX 公式渲染的历年真题模拟与手写演练。

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CSCA Mathematics Syllabus & Key Topics/CSCA 数学考试大纲与核心考点

01
Limits & Continuity: Evaluating numerical and algebraic limits, L'Hopital's rule, asymptote behavior.
极限与连续性：数值与代数极限求解、洛必达法则、渐近线行为分析。
02
Differential Calculus: Derivatives of trigonometric, exponential, and logarithmic functions, chain rule, optimization.
微分学：三角函数、指数与对数函数的导数、链式法则、极值最优化问题。
03
Integral Calculus: Definite and indefinite integrals, integration techniques (substitution, parts), area calculation.
积分学：不定积分与定积分、积分方法（换元积分、分部积分）、面积与体积计算。
04
Linear Algebra & Vectors: Matrices, systems of linear equations, dot and cross products of vectors, spatial geometry.
线性代数与向量：矩阵运算、线性方程组求解、向量点积与叉积、空间解析几何。

How to Prepare for CSCA Math (Study Guide)/CSCA 数学如何高效备考与复习

Master LaTeX mathematical derivation steps to follow complex algebraic calculations.
掌握标准 LaTeX 公式推导步骤，紧跟复杂的代数计算过程。
Solve sample questions using the active stylus drawing pad to check your intermediate algebraic steps.
利用手写草稿纸解答样题，边写边查中间代数步骤，减少笔误。
Target speed-running calculations: practice solving mock questions under 90 seconds.
训练答题速度：练习在90秒内快速理清思路并求解出数学模拟题。
Review past paper topics regularly to familiarize yourself with the pattern of Chinese college entrance math.
定期总结历年真题题型，熟悉中国高等教育数学入学考试的命题规律。
ADAPTIVE SAMPLE
The domain of the function is:
函数 的定义域是：
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CSCA Past Papers & Simulation

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Frequently Asked Questions

All you need to know about the CSCA assessment & prep

CSCA Mathematics is based on the standard curriculum for undergraduate university admissions in China, covering Sets & Inequalities, Functions & Sequences, Conic Sections & Solid Geometry, Vectors & Complex Numbers, and Probability & Statistics.

CSCA 数学考试基于中国高校本科入学考试标准大纲，考查内容涵盖集合与不等式、函数与数列、圆锥曲线与立体几何、平面向量与复数，以及概率与统计。

No, calculators are strictly prohibited. Questions are designed for analytical factoring, geometric visualization, and concise algebraic evaluation.

严禁使用计算器。试卷所有题目均围绕代数因式分解、几何直观与解析化简设计，考查学生的基础数学核心素养。

The official CSCA Mathematics examination contains 48 single-choice questions with a total duration of 60 minutes (100 total points), requiring an average pacing of 75 seconds per question.

官方 CSCA 数学考试标准题量为 48 道单项选择题，考试总时长为 60 分钟（满分 100 分），建议答题配速为每题 75 秒。

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help_outlineSyllabus & Exam Governance

Frequently Asked Questions: CSCA Mathematics Prep — Past Papers & Syllabus

Authoritative answers on CSCA MATH examination scope, calculator policies, time budgeting, and grading rubrics.

What is the difficulty level of CSCA Math?
+

CSCA Mathematics is based on the standard curriculum for undergraduate university admissions in China, covering Sets & Inequalities, Functions & Sequences, Conic Sections & Solid Geometry, Vectors & Complex Numbers, and Probability & Statistics.

CSCA 数学考试难度如何？: CSCA 数学考试基于中国高校本科入学考试标准大纲，考查内容涵盖集合与不等式、函数与数列、圆锥曲线与立体几何、平面向量与复数，以及概率与统计。

Are calculators allowed in the CSCA Math exam?
+

No, calculators are strictly prohibited. Questions are designed for analytical factoring, geometric visualization, and concise algebraic evaluation.

CSCA 数学考试中允许使用计算器吗？: 严禁使用计算器。试卷所有题目均围绕代数因式分解、几何直观与解析化简设计，考查学生的基础数学核心素养。

How many questions are in the Math section?
+

The official CSCA Mathematics examination contains 48 single-choice questions with a total duration of 60 minutes (100 total points), requiring an average pacing of 75 seconds per question.

数学部分有多少道题？: 官方 CSCA 数学考试标准题量为 48 道单项选择题，考试总时长为 60 分钟（满分 100 分），建议答题配速为每题 75 秒。

CSCA Mathematics Prep — Past Papers & Syllabus Official Curriculum Masterclasses

Direct access to all accredited syllabus nodes, lesson breakdowns, and examiner traps.

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Ch.1.1.1Beginner

Sets, Venn Diagrams & Logical Quantifiers

Subset, intersection, union, complement in universal sets $\mathbb{R}$, existential $\exists$ & universal $\forall$ quantifiers, and necessary vs sufficient conditions.

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Ch.1.1.2Intermediate

Quadratic, Fractional Inequalities & AM-GM

Root-sign method for polynomials, fractional transformation $\frac{f(x)}{g(x)} > 0 \iff f(x)g(x) > 0$, and arithmetic-geometric mean inequality $a+b \ge 2\sqrt{ab}$.

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Ch.2.2.1Intermediate

Function Properties: Monotonicity, Parity & Periodicity

Domain symmetry, even/odd test $f(-x) = \pm f(x)$, strictly increasing/decreasing criteria, and periodic patterns $f(x+T) = f(x)$.

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Ch.2.2.2Intermediate

Exponential, Logarithmic & Trigonometric Functions

Laws of exponents, logarithm identities, change-of-base, and fundamental trigonometric identities.

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Ch.2.2.3Intermediate

Sequences: Arithmetic & Geometric Progressions

General terms, common difference/ratio, sum of first n terms formulas, and recurrence relations.

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Ch.2.2.4Advanced

Derivatives: Tangent Lines, Extrema & Monotonicity

Power rule, product/quotient rules, chain rule, tangent lines, critical points, and optimization.

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Ch.3.3.1Advanced

Conic Sections: Ellipse, Hyperbola & Parabola

Standard equation of lines, circles, focal definitions, eccentricity $e = c/a$, asymptotes $y = \pm \frac{b}{a}x$, and focal chord properties.

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Ch.3.3.2Beginner

Plane Vectors: Linear Operations & Dot Product

Vector coordinates, collinearity condition $x_1 y_2 - x_2 y_1 = 0$, dot product $\mathbf{a}\cdot\mathbf{b}$, angle formula, and orthogonality.

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Ch.3.3.3Beginner

Complex Numbers: Arithmetic & Modulus

Standard form $z = a + bi$, imaginary unit $i^2 = -1$, conjugate $\bar{z} = a - bi$, modulus $|z| = \sqrt{a^2+b^2}$, and complex plane representation.

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Ch.3.3.4Intermediate

Solid Geometry: Line-Plane Positional Relations & Volume

Parallelism and perpendicularity criteria for lines and planes, spatial angles, surface area and volume of prisms, pyramids, cylinders, cones, spheres.

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Ch.4.4.1Intermediate

Permutations, Combinations & Classical Probability

Addition and multiplication counting principles, permutations $A_n^m = \frac{n!}{(n-m)!}$, combinations $C_n^m = \frac{n!}{m!(n-m)!}$, and equally likely outcomes.

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Ch.4.4.2Beginner

Statistics: Mean, Variance & Normal Distribution

Sample mean $\bar{x}$, sample variance $s^2$, standard deviation, and normal distribution curve symmetry ($3\sigma$ rule).

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CSCA Mathematics Prep — Past Papers & Syllabus Formula Permapages & Proofs (30)

Canonical LaTeX definitions, SymPy-verified mathematical derivations, and misconception traps.

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Quadratic FormulaSets & Inequalities

一元二次方程求根公式

$\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]$→
Union of SetsSets & Inequalities

集合的并集

$\[ A \cup B = \{ x \mid x \in A \text{ or } x \in B \} \]$→
Intersection of SetsSets & Inequalities

集合的交集

$\[ A \cap B = \{ x \mid x \in A \text{ and } x \in B \} \]$→
Complement of a SetSets & Inequalities

集合的补集

$\[ \complement_{U} A = \{ x \in U \mid x \notin A \} \]$→
Rational Inequality EquivalenceSets & Inequalities

分式不等式的等价转化

$\[ \frac{ax + b}{cx + d} \le 0 \iff (ax + b)(cx + d) \le 0 \quad \text{and} \quad cx + d \ne 0 \]$→
AM-GM InequalitySets & Inequalities

基本均值不等式

$\[ \frac{a + b}{2} \ge \sqrt{ab} \quad (a, b \ge 0) \]$→
Change of Base FormulaFunctions

对数的换底公式

$\[ \log_a b = \frac{\log_c b}{\log_c a} \quad (a, b, c > 0, \, a, c \ne 1) \]$→
Logarithmic PropertiesFunctions

对数的基本运算性质

$\[ \log_a (xy) = \log_a x + \log_a y \quad \text{and} \quad \log_a \left(\frac{x}{y}\right) = \log_a x - \log_a y \]$→
Trigonometric Pythagorean IdentityFunctions

同角三角关系（平方关系）

$\[ \sin^2 \theta + \cos^2 \theta = 1 \]$→
Double-Angle Formula (Sine)Functions

正弦的二倍角公式

$\[ \sin 2\theta = 2 \sin \theta \cos \theta \]$→
Double-Angle Formula (Cosine)Functions

余弦的二倍角公式

$\[ \cos 2\theta = \cos^2 \theta - \sin^2 \theta \]$→
Function Parity (Odd & Even)Functions

函数的奇偶性

$\[ \text{Even: } f(-x) = f(x), \quad \text{Odd: } f(-x) = -f(x) \]$→
Arithmetic Sequence Nth TermSequences

等差数列通项公式

$\[ a_n = a_1 + (n - 1)d \]$→
Arithmetic Sequence SumSequences

等差数列前n项和公式

$\[ S_n = \frac{n(a_1 + a_n)}{2} \]$→
Geometric Sequence Nth TermSequences

等比数列通项公式

$\[ a_n = a_1 q^{n-1} \]$→
Geometric Sequence SumSequences

等比数列前n项和公式

$\[ S_n = \frac{a_1(1 - q^n)}{1 - q} \quad (q \ne 1) \]$→
Power Rule for DerivativesDerivatives

导数的幂函数求导法则

$\[ \frac{d}{dx} x^n = n x^{n-1} \]$→
Tangent Line EquationDerivatives

曲线切线方程公式

$\[ y - y_0 = f'(x_0)(x - x_0) \]$→
Distance FormulaPlane Analytic Geometry

两点间距离公式

$\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]$→
Equation of a CirclePlane Analytic Geometry

圆的标准方程

$\[ (x - a)^2 + (y - b)^2 = R^2 \]$→
Equation of an EllipsePlane Analytic Geometry

椭圆的标准方程

$\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \quad (a > b > 0) \]$→
Equation of a HyperbolaPlane Analytic Geometry

双曲线的标准方程

$\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \quad (a > 0, \, b > 0) \]$→
Equation of a ParabolaPlane Analytic Geometry

抛物线的标准方程

$\[ y^2 = 2px \quad (p > 0) \]$→
Equation of a Line (Slope-Intercept & General Form)Plane Analytic Geometry

直线的方程（斜截式与一般式）

$\[ y = kx + b \quad \text{and} \quad Ax + By + C = 0 \]$→
Vector Dot ProductVectors & Complex Numbers

平面向量的数量积（点积）

$\[ \mathbf{a} \cdot \mathbf{b} = a_x b_x + a_y b_y = |\mathbf{a}| |\mathbf{b}| \cos \theta \]$→
Modulus of a Complex NumberVectors & Complex Numbers

复数的模

$\[ |z| = \sqrt{a^2 + b^2} \quad (z = a + bi) \]$→
Volume and Surface Area of a SphereSolid Geometry

球的体积与表面积公式

$\[ V = \frac{4}{3}\pi R^3 \quad \text{and} \quad S = 4\pi R^2 \]$→
Classical Probability FormulaProbability & Statistics

古典概型概率计算公式

$\[ P(A) = \frac{N(A)}{N} \]$→
Variance and Standard DeviationProbability & Statistics

样本方差与标准差公式

$\[ s^2 = \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2 \]$→
Normal Distribution (3-Sigma Rule)Probability & Statistics

正态分布（3σ原则 / 68-95-99.7准则）

$\[ P(\mu - \sigma \le X \le \mu + \sigma) \approx 0.6826 \quad \text{and} \quad P(\mu - 2\sigma \le X \le \mu + 2\sigma) \approx 0.9544 \]$→