📚 Review Set 7A – Non‑Calculator | 复习集 7A – 非计算器
Review sets are a powerful way to consolidate mathematical skills without relying on technology. In IB Mathematics, non‑calculator papers demand fluency in algebra, exact trigonometric values, logarithmic laws, and calculus techniques performed entirely by hand. Review Set 7A has been crafted to target these competencies, offering a carefully balanced mixture of short‑response and structured problems that mirror the style of Paper 1. The exercises revisit key concepts from the first half of the course, pushing you to think critically and present clear, logical working. By engaging with this set, you strengthen your mental arithmetic, deepen your understanding of function behaviour, and learn to spot common pitfalls before the actual examination.
复习集是巩固数学技能而不依赖技术的有力方式。在 IB 数学中，非计算器试卷要求学生熟练掌握代数、精确三角值、对数运算法则以及完全通过手工进行的微积分技巧。复习集 7A 专为训练这些能力而设计，提供了精心平衡的简答题和结构化问题，模仿卷一试题的风格。这些练习回顾了上半年课程中的关键概念，促使你进行批判性思考并呈现清晰、逻辑严谨的解题过程。通过完成这个复习集，你将提高心算能力，加深对函数行为的理解，并学会在实际考试前识别常见的陷阱。
1. Purpose of a Non‑Calculator Review Set | 非计算器复习集的目的
Paper 1 tasks in the IB Analysis & Approaches and Applications & Interpretation courses explicitly forbid calculators. The rationale is to assess whether students can manipulate symbols, reason abstractly, and produce exact answers. Review Set 7A simulates this environment, removing the safety net of a graphing display or automatic simplification. Every question forces you to rely on factorisation, rationalising denominators, exact trigonometric ratios, and derivative rules memorised rather than computed. This builds the kind of mathematical confidence that cannot be obtained by typing expressions into a screen. It also highlights the elegance of exact answers—leaving a solution as ln 4 or √3/2 is often much more informative than a decimal approximation.
IB 分析与方法和应用与解释课程的卷一明确禁止使用计算器。其目的是评估学生是否能够操作符号、进行抽象推理并给出精确答案。复习集 7A 模拟了这一环境，移除了图形显示或自动化简的安全网。每一道题都迫使你依赖因式分解、分母有理化、精确三角比以及记忆而非计算得出的求导法则。这能培养那种仅凭在屏幕上输入表达式无法获得的数学自信。它同时也凸显了精确答案的优雅——把解写成 ln 4 或 √3/2 往往比给出小数近似值更具信息量。
2. Core Skills Tested | 测试的核心技能
The set weaves together multiple strands of the syllabus. You will encounter polynomial equations that require synthetic division, rational expressions that must be simplified by finding a common denominator, and inequalities whose sign diagrams need careful construction. Geometry problems ask for exact lengths or areas derived from trigonometry, while calculus questions expect you to find turning points and integrals without numerical crutches. There is also a dedicated section on combinatorics and basic probability, where you must count systematically and express outcomes as simplified fractions. The intentional mix mirrors the interconnected nature of mathematics and discourages compartmentalised learning.
这个复习集交织了课程大纲的多个分支。你将遇到需要综合除法来求解的多项式方程、必须通过通分进行化简的有理式，以及需要精心构建符号图的不等式。几何问题要求根据三角学推导出精确的长度或面积，而微积分题目则期望你在没有数值辅助的情况下找到驻点和积分。其中还专门设有组合数学和基础概率的部分，要求你系统地计数并将结果表示为最简分数。这种有意的混合反映了数学的内在联系，有助于避免孤立地学习各个知识点。
3. Algebraic Manipulation | 代数运算
Algebraic fluency is the backbone of non‑calculator success. In Review Set 7A you are asked to expand products such as (2x – 3)(x² + x – 1) and to factorise cubics by guessing a root and performing polynomial long division. Simplifying complex fractions like
(x/(x – 2)) – (3/(x + 1))
demands a solid command of equivalent fractions and attention to sign changes. Another common task is solving equations of the form
|2x + 5| = 3x – 1
which requires considering separate cases and checking for extraneous solutions. Without a calculator, every line of working must be precise; one sign error can propagate and invalidate a multi‑step proof. Practice with these manipulations rewards you with the speed and accuracy needed for the timed examination.
代数流畅性是非计算器考试成功的基石。在复习集 7A 中，你需要展开诸如 (2x – 3)(x² + x – 1) 的乘积，并通过猜测一个根并进行多项式长除法来对三次式进行因式分解。简化复杂分数（如
(x/(x – 2)) – (3/(x + 1))
）要求你熟练掌握等值分数并注意符号变化。另一类常见任务是解形如
|2x + 5| = 3x – 1
的方程，这需要分情况讨论并检查增根。没有计算器时，每一步推算都必须精确；一个符号错误就可能传播并使多步证明无效。通过练习这些运算，你将获得在限时考试中所需的速度和准确性。
4. Functions and Graphs | 函数与图像
Understanding the anatomy of functions is tested heavily. You might be given f(x) = 1/(x – 2) and asked to write the equations of its vertical and horizontal asymptotes, or to sketch g(x) = 2 sin (3x – π/2) + 1 by identifying amplitude, period, and phase shift. Composite functions such as f ∘ g(x) appear together with domains that must be restricted to avoid division by zero or negative square roots. Transformations including reflections, stretches, and translations are revised through questions that link the graph of y = f(x) to that of y = 2f(–x) + 3. The non‑calculator format demands that you label key points with exact coordinates, often expressed in multiples of π or with square roots, which reinforces a genuine grasp of function behaviour rather than button‑pressing pattern recognition.
对函数结构的理解是重点考查内容。你可能会遇到 f(x) = 1/(x – 2)，并要求写出其垂直和水平渐近线的方程，或者通过识别振幅、周期和相移来画出 g(x) = 2 sin (3x – π/2) + 1 的图像。复合函数（如 f ∘ g(x)）会与必须限制定义域以避免除以零或负平方根的情况一起出现。通过将 y = f(x) 的图像与 y = 2f(–x) + 3 的图像联系起来的问题，可以复习包括对称、伸缩和平移在内的变换。非计算器的形式要求你用精确坐标标注关键点，这些坐标通常用 π 的倍数或平方根表示，这有助于真正掌握函数的行为，而不是依靠按键识别模式。
5. Trigonometry without a Calculator | 无计算器的三角学
hallmark of IB Paper 1. Review Set 7A expects you to recall, without hesitation, that sin (π/3) = √3/2, cos (π/6) = √3/2, and tan (π/4) = 1. Questions extend to solving equations like
2 sin² θ – sin θ – 1 = 0, 0 ≤ θ ≤ 2π
by treating them as quadratic in sin θ. The sine and cosine rules are revisited in triangle problems, where side lengths and angles must be found exactly, sometimes requiring rationalisation of surds. Radian measure is used throughout, and you must be comfortable converting between degrees and radians mentally for common angles. The unit circle provides the framework for visualising the periodic nature of these functions and solving inequalities like cos x ≥ 1/2 for x in [0, 2π]. This section hones the quick‑fire recall that can make the difference between finishing the paper and leaving questions unattempted.
精确的三角函数值是 IB 卷一的标志。复习集 7A 要求你不假思索地回想起 sin (π/3) = √3/2，cos (π/6) = √3/2，以及 tan (π/4) = 1。题目扩展至将诸如
2 sin² θ – sin θ – 1 = 0, 0 ≤ θ ≤ 2π
的方程视为关于 sin θ 的二次方程进行求解。三角形问题中会回顾正弦定理和余弦定理，必须精确求出边长和角度，有时还需要将根式有理化。全程使用弧度制，你必须能够熟练地对常见角度进行度与弧度的心理转换。单位圆为理解这些函数的周期性以及在 [0, 2π] 内求解如 cos x ≥ 1/2 的不等式提供了框架。这一部分打磨了快速回忆的能力，这可能是完成试卷与遗留题目的差别所在。
6. Exponents and Logarithms | 指数与对数
Laws of indices and logarithms are manipulated repeatedly. Typical problems involve rewriting 8²ˣ⁻¹ = 4ˣ⁺³ with a common base and solving for x. Logarithmic equations such as
log₂ (x + 3) + log₂ (x – 1) = 3
require combining logs into a single expression and converting to exponential form. You also need to evaluate expressions like log₅ 25 – log₅ (1/5) without a calculator by recognising the underlying powers. The relationship between exponential growth and its inverse is tested through modelling contexts—for instance, finding the time taken for a population to double given a growth rate per unit time. Exact simplification using ln and e is expected; an answer of ln(5/2) is perfectly acceptable, but a decimal such as 0.916 is not. This builds the habit of leaving results in surd or logarithmic form, which is a consistent requirement across IB mathematics assessment.
指数和对数的运算法则会被反复使用。典型问题包括将 8²ˣ⁻¹ = 4ˣ⁺³ 写成同底数形式并求解 x。形如
log₂ (x + 3) + log₂ (x – 1) = 3
的对数方程需要将多个对数合并为单一表达式并转换为指数形式。你还需要在不使用计算器的情况下，通过识别底层的幂来计算 log₅ 25 – log₅ (1/5) 这样的式子。指数增长与其反函数之间的关系通过建模情境进行考查——例如，给定单位时间的增长率，求某一总体数量翻倍所需的时间。要求能够使用 ln 和 e 进行精确化简；写成 ln(5/2) 的答案完全可以接受，而 0.916 这样的小数则不行。这有助于养成将结果保留为根式或对数形式的习惯，这也是 IB 数学评估中始终如一的要求。
7. Calculus by Hand | 手工微积分
Differentiation and integration without a calculator test your memory of standard results and the chain, product, and quotient rules. In Review Set 7A you differentiate functions like
f(x) = (3x² – 2)⁴ and g(x) = e²ˣ sin x
and subsequently find the equation of a tangent at a specific point. Integration questions range from simple polynomial antiderivatives to definite integrals requiring substitution, such as
∫₀¹ 2x (x² + 1)³ dx
where the limits must be changed or reversed correctly. Area between curves and volume of revolution problems are presented with algebraic boundaries, so the final integral remains exact, often yielding a neat fraction multiple of π. Working by hand reinforces the fundamental theorem of calculus and the geometric meaning of the derivative, encouraging you to visualise the gradient of a curve rather than just computing numbers.
不使用计算器进行微分和积分，考验你对标准结果以及链式法则、积法则和商法则的记忆。在复习集 7A 中，你将求诸如
f(x) = (3x² – 2)⁴ 和 g(x) = e²ˣ sin x
等函数的导数，并进而求出某一点处的切线方程。积分题目从简单的多项式原函数到需要换元的定积分，例如
∫₀¹ 2x (x² + 1)³ dx
其中需要正确地变换或颠倒积分上下限。曲线围成的面积以及旋转体体积问题都会给出代数边界，因此最终的积分保持精确，结果常常是 π 的整数倍或一个整洁的分数。手工计算强化了对微积分基本定理和导数几何意义的理解，促使你直观地想象曲线的斜率，而不仅仅是计算数字。
8. Sequences and Series | 数列与级数
Arithmetic and geometric sequences appear in multiple formats. You are given the first few terms and asked to derive the nth term formula, or you must find the sum of the first 15 terms without summing individually. For geometric series, the condition for convergence |r| < 1 is examined, and the sum to infinity S∞ = a/(1 – r) is derived from first principles. Word problems translate real‑life scenarios—loan repayments, population decline, bouncing ball heights—into series, requiring careful identification of the common difference or ratio. Sigma notation, ∑ (3k – 2) st your ability to interpret and manipulate sums algebraically. Because no calculator is allowed, you must handle fractions and surds that emerge from the formulae, which deepens your familiarity with algebraic simplification.
等差数列和等比数列会以多种形式出现。你可能会拿到数列的前几项并被要求推导出第 n 项的公式，或者需要求出前 15 项的和而不逐项相加。对于等比数列，会考查收敛条件 |r| < 1，并且无穷和 S∞ = a/(1 – r) 是从基本原理推导出来的。应用题将现实情境——贷款偿还、人口衰减、弹跳球高度——转化为数列，需要仔细识别公差或公比。求和符号 ∑ (3k – 2)（k 从 1 到 n）被广泛使用，以检验你通过代数方式解读和处理和式的能力。由于不允许使用计算器，你必须处理公式中出现的分数和根式，从而加深你对代数化简的熟悉程度。
9. Vector Geometry | 向量几何
Non‑calculator vector problems emphasise magnitude, direction, and dot product calculations that produce exact results. You might be given two vectors a = i – 2j + 3k and b = –i + j + 2k and asked to find the angle between them using cos θ = (a·b)/(|a||b|). The arithmetic involves square roots of integers, and the cosine value often turns out to be a nice rational number that corresponds to a familiar angle. Equations of lines in 3D and the distance from a point to a line are tackled through parametric forms. Questions about parallel, intersecting, or skew lines require solving simultaneous vector equations by hand, without the benefit of a matrix solver. This process sharpens spatial intuition and comfort with the algebraic manipulation of components.
非计算器的向量问题强调模长、方向以及能产生精确结果的数量积计算。你可能会拿到两个向量 a = i – 2j + 3k 和 b = –i + j + 2k，并要求使用 cos θ = (a·b)/(|a||b|) 求它们之间的夹角。算术涉及整数的平方根，而余弦值往往是一个漂亮的、对应常见角度的有理数。三维空间中直线的方程以及点到直线的距离通过参数形式来解决。关于平行、相交或异面直线的问题需要手工求解联立向量方程，而没有矩阵求解器的帮助。这一过程能增强空间直觉，并让你更加习惯于对分量的代数操作。
10. Probability and Counting | 概率与计数
Combinatorics problems in Review Set 7A explore permutations, combinations, and the multiplication principle. You may be asked how many 4‑digit numbers can be formed from a set of digits without repetition, or how many different committees of 3 can be chosen from 10 students. Probability questions build on these counts, requiring you to calculate probabilities of compound events using the formula
P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
Conditional probability and tree diagrams are constructed entirely by hand, with final probabilities expressed as simplified fractions. A typical item: “Given that two marbles are drawn without replacement from a bag containing 5 red and 4 blue, find the probability that at least one is red.” Drawing the tree and multiplying along branches develops a systematic approach that is far more robust than an ad‑hoc guess, and it prepares you for the abstract reasoning needed in statistical inference.
复习集 7A 中的组合数学问题探讨了排列、组合和乘法原理。你可能被问到用一组不重复的数字可以组成多少个不同的四位数，或者从 10 名学生中选出 3 人组成委员会有多少种不同的选择。概率问题建立在这些计数的基础上，要求你使用公式
P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
计算复合事件的概率。条件概率和树状图完全通过手工构建，最终的概率表示为最简分数。一个典型的题目是：“从一个装有 5 个红球和 4 个蓝球的袋子中不放回地抽取两个球，求至少抽到一个红球的概率。” 绘制树状图并沿着分支相乘能够培养一种系统性的方法，这比临时猜测要可靠得多，并且能为统计推断所需的抽象推理做好准备。
11. Tips for Success | 成功技巧
To get the most out of Review Set 7A, attempt every problem under timed, closed‑book conditions without peeking at the answers. Write every line of working as if an examiner will read it—clarity matters. When stuck, ask yourself: “Can I rewrite this expression using an identity, factorisation, or a different representation?” Master the exact values of sin, cos, and tan for 0, π/6, π/4, π/3, π/2 and their multiples, as these recur endlessly. Memorise the quadratic formula, the discriminant conditions, and the shapes of common graphs. For calculus, practice switching between integral and derivative forms of ln x, eˣ, and trig functions until the rules become second nature. Finally, always check your solution by substituting back into the original equation or by using an alternative method where possible. The discipline of verifying without a calculator is a powerful learning tool.
为了从复习集 7A 中获得最大收益，请在定时、闭卷且不看答案的条件下尝试每一道题。像考官会阅读你答卷那样写下每一步求解过程——清晰性至关重要。当你卡住时，问问自己：“我能否用恒等式、因式分解或其他表示方法来重写这个表达式？” 熟练掌握 0、π/6、π/4、π/3、π/2 及其倍数的正弦、余弦和正切精确值，因为它们在试题中反复出现。牢记求根公式、判别式条件以及常见图形的大致形状。在微积分方面，练习在 ln x、eˣ 和三角函数的积分与导数形式之间切换，直到这些规则成为你的第二天性。最后，务必通过将解代入原方程，或者在可能的情况下使用另一种方法来检查你的答案。在不借助计算器的情况下进行验证，这种自律本身就是一种强大的学习工具。
12. Conclusion | 结语
Review Set 7A is more than a collection of questions; it is a structured rehearsal for the demands of the non‑calculator examination. Each exercise pushes you to connect ideas across algebra, geometry, trigonometry, and calculus, fostering the kind of agile thinking that distinguishes a confident IB mathematician. As you work through the set, pay attention not just to correct answers, but to the efficiency and elegance of your methods. Consistent practice in this format will reduce anxiety on the day of Paper 1, leaving you with the mental stamina to tackle even the most challenging problems. Trust the process, refine your technique, and remember that every line of exact working brings you closer to mastery.
复习集 7A 不仅仅是一组题目；它是对非计算器考试要求的一次结构化彩排。每个练习都推动你将代数、几何、三角和微积分中的思想联系起来，培养那种让一位自信的 IB 数学学生脱颖而出的敏捷思维。当你完成这个复习集时，不仅要关注正确答案，还要留意自己方法的效率和优雅性。以这种形式持续练习将减少在卷一考试当天的焦虑，让你有足够的脑力去攻克最具挑战性的问题。相信这个过程，打磨你的技巧，并记住，每一行精确的推导都会让你离精通更近一步。
Published by TutorHao | IB Mathematics Revision Series | aleveler.com
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