📚 AS AQA Maths Paper 1 June 2022: Skills and Strategies | AS AQA 数学 Unit 1 2022年6月：核心技能与策略
The AQA AS Mathematics Paper 1 (7356/1), sat in June 2022, is a pure mathematics paper that tests a broad range of algebraic, geometric and calculus skills. This article breaks down the key topic areas, explains the methods you need, and highlights common pitfalls so you can approach similar questions with confidence.
AQA AS 数学 2022年6月试卷一（7356/1）是纯数学卷，考查代数、几何与微积分等多方面技能。本文将逐一拆解核心考点、说明解题方法并指出常见陷阱，帮助你在面对类似题目时更有信心。
1. Paper Overview | 试卷概览
Paper 1 is 90 minutes long and carries 80 marks. It is a calculator paper, and all questions are compulsory. Marks per question typically range from 2 to 11, with the longest questions involving multi-step reasoning.
试卷一考试时间为90分钟，满分80分。允许使用计算器，所有题目均为必答题。每题分值通常为2至11分，最长的题目需要多步骤推理。
The paper covers pure mathematics only: surds, quadratics, inequalities, coordinate geometry, trigonometry, polynomials, differentiation, integration, exponentials, logarithms, sequences and the binomial theorem.
本卷仅涵盖纯数学内容：根式、二次函数、不等式、坐标几何、三角学、多项式、微分、积分、指数、对数、数列与二项式定理。
Time management is critical. A useful rule is to aim to spend roughly one mark per minute and reserve the final 10 minutes for checking your work.
时间管理至关重要。一个实用的原则是每分钟完成约一分值的题目，并预留最后10分钟检查作答。
2. Algebraic Manipulation and Surds | 代数运算与根式
Algebraic simplification appears early in the paper, often as a low-mark starter question. You must be fluent with index laws and surd arithmetic.
代数化简通常出现在试卷开头的低分题目中。你必须熟练掌握指数法则与根式运算。
Recall the core index laws:
牢记核心指数法则：
aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ
For surds, the key simplifications are:
根式的核心化简规则如下：
√(ab) = √a × √b, √(a/b) = √a / √b
For example, to simplify (√8 + √18) / √2, first note that √8 = 2√2 and √18 = 3√2. The numerator becomes 5√2, so the expression equals 5.
例如，化简 (√8 + √18) / √2：先注意到 √8 = 2√2，√18 = 3√2，分子变为5√2，因此整个表达式等于5。
Rationalising the denominator is another frequent requirement:
有理化分母也是常见要求：
1 / (√3 + 1) = (√3 − 1) / ((√3 + 1)(√3 − 1)) = (√3 − 1) / 2
Multiply top and bottom by the conjugate, then simplify using the difference of two squares.
将分子分母同乘共轭式，再利用平方差公式化简。
3. Quadratics and Inequalities | 二次函数与不等式
Quadratics are the backbone of AS pure mathematics. Completing the square, the discriminant and solving quadratic inequalities all appear regularly.
二次函数是AS纯数学的基石。配方法、判别式以及解二次不等式都是常规考点。
Completing the square takes the form:
配方法的基本形式为：
x² + bx + c = (x + b/2)² − (b/2)² + c
If the coefficient of x² is not 1, factor it out first. For example:
若x²的系数不为1，先将其提出。例如：
2x² − 8x + 5 = 2(x² − 4x) + 5 = 2[(x − 2)² − 4] + 5 = 2(x − 2)² − 3
The discriminant b² − 4ac tells you the number of real roots. A tangent line to a curve corresponds to a discriminant of zero.
判别式 b² − 4ac 可判断实根个数。直线与曲线相切时判别式等于零。
For quadratic inequalities, sketch the graph. If a is positive, the curve is U-shaped. The solution to (x − p)(x − q) < 0 lies strictly between p and q, while (x − p)(x − q) > 0 lies outside the interval.
解二次不等式时先画草图。若a为正，曲线呈U形。不等式 (x − p)(x − q) < 0 的解位于p与q之间；而 (x − p)(x − q) > 0 的解位于区间之外。
A June 2022 style question might ask you to find the set of values of k for which a quadratic has no real roots. Set the discriminant less than zero and solve the resulting inequality.
2022年6月风格题目可能要求你求使二次函数无实根的k值范围。令判别式小于零并解所得不等式即可。
4. Coordinate Geometry of Circles | 圆的坐标几何
The circle is a central topic in AS coordinate geometry. The standard equation is:
圆是AS坐标几何的核心内容。其标准方程为：
(x − a)² + (y − b)² = r²
where (a, b) is the centre and r is the radius. You may be given the expanded form x² + y² + 2gx + 2fy + c = 0 and asked to complete the square to find the centre and radius.
其中(a, b)为圆心，r为半径。题目也可能给出展开形式 x² + y² + 2gx + 2fy + c = 0，要求你用配方法求圆心与半径。
To find the equation of a tangent to a circle at a given point, use the fact that the radius to the point of contact is perpendicular to the tangent. Calculate the gradient of the radius, then take the negative reciprocal.
求圆在某点处的切线方程时，利用”过切点的半径垂直于切线”这一性质。先计算半径的斜率，再取其负倒数。
A typical multi-part question asks you to verify that a point lies on a circle, find the tangent gradient, then write the tangent equation in the form ax + by + c = 0.
典型的多步骤题目会要求你先验证某点在圆上，再求切线斜率，最后将切线方程写成 ax + by + c = 0 的形式。
Remember that the perpendicular bisector of any chord passes through the centre of the circle. This property is useful when finding the centre from two points on the circumference.
请记住：任何弦的垂直平分线都经过圆心。当已知圆周上两点求圆心时，这一性质非常有用。
5. Trigonometry | 三角学
Trigonometry in AS AQA covers radians, exact values, graphs and solving trigonometric equations.
AS AQA 三角学涵盖弧度制、精确值、三角函数图像以及解三角方程。
You should know the exact values table for the special angles:
你需要熟记特殊角的三角函数精确值表：
| θ | π/6 (30°) | π/4 (45°) | π/3 (60°) |
| sin θ | ½ | √2/2 | √3/2 |
| cos θ | √3/2 | √2/2 | ½ |
| tan θ | √3/3 | 1 | √3 |
The negative angle rules are essential for solving equations:
负角关系式对解方程至关重要：
sin(π − θ) = sin θ, cos(2π − θ) = cos θ, tan(π + θ) = tan θ
To solve a trigonometric equation such as 2cos θ = 1 for 0 ≤ θ < 2π, first isolate cos θ, find the principal value, then use symmetry to find all solutions in the interval.
解三角方程如 2cos θ = 1（0 ≤ θ < 2π）时，先分离出cos θ，求出主值，再利用对称性找出区间内的所有解。
In the June 2022 paper, a typical trigonometric equation question rewarded method marks for showing the principal value and the quadrant analysis clearly.
在2022年6月试卷中，典型的三角方程题要求清晰写出主值和象限分析，才能获得方法分。
6. Polynomials and the Factor Theorem | 多项式与因式定理
The factor theorem states that if f(a) = 0, then (x − a) is a factor of f(x). The remainder theorem states that when f(x) is divided by (x − a), the remainder is f(a).
因式定理指出：若 f(a) = 0，则 (x − a) 是 f(x) 的一个因式。余数定理指出：f(x) 除以 (x − a) 所得的余数为 f(a)。
You may be asked to factorise a cubic polynomial completely. Try integer values ±1, ±2, ±3 with the factor theorem until you find a root, then divide out the linear factor.
题目可能要求你彻底分解三次多项式。先用因式定理尝试x = ±1、±2、±3等整数值找到根，再除以线性因式。
For example, to factorise f(x) = x³ − 4x² + x + 6:
例如，分解 f(x) = x³ − 4x² + x + 6：
f(2) = 8 − 16 + 2 + 6 = 0, so (x − 2) is a factor
Dividing by (x − 2) gives x² − 2x − 3, so the full factorisation is:
除以 (x − 2) 得到 x² − 2x − 3，因此完整分解为：
f(x) = (x − 2)(x − 3)(x + 1)
Once factorised, you can solve equations, locate roots of curves and find where a graph crosses the x-axis.
分解之后，你就可以解方程、确定曲线与x轴的交点位置。
7. Differentiation | 微分
Differentiation is the most heavily weighted calculus topic on the paper. The basic rule is:
微分是试卷中分值占比最高的微积分考点。基本法则为：
d/dx (xⁿ) = nxⁿ⁻¹
For a polynomial, differentiate term by term. For example:
对于多项式，逐项求导。例如：
f(x) = 3x² − 5x + 7 → f'(x) = 6x − 5
Differentiation from first principles is a guaranteed skill in AS AQA. The definition is:
用第一性原理求导是AS AQA必考技能。其定义为：
f'(x) = lim (h→0) [f(x + h) − f(x)] / h
For f(x) = x², expand (x + h)² = x² + 2xh + h², subtract x², divide by h, and take the limit as h approaches 0 to obtain 2x.
以 f(x) = x² 为例：展开 (x + h)² = x² + 2xh + h²，减去x²，除以h，再令h趋于0取极限，得到2x。
You also need to use differentiation to find the gradient of a tangent or normal, locate stationary points, and determine whether a point is a maximum or minimum.
你还需要利用微分求切线和法线的斜率、定位驻点，并判断该点是极大值还是极小值。
For stationary points, set f'(x) = 0 and solve. Use the second derivative test: if f”(a) < 0 then the point is a maximum; if f''(a) > 0 then it is a minimum.
求驻点时令 f'(x) = 0 并求解。用二阶导数判别法：若 f”(a) < 0，则该点为极大值；若 f''(a) > 0，则为极小值。
8. Integration | 积分
Integration is the reverse of differentiation. The indefinite integral is:
积分是微分的逆运算。不定积分为：
∫ xⁿ dx = xⁿ⁺¹ / (n + 1) + C, for n ≠ −1
Definite integrals evaluate the area between a curve and the x-axis:
定积分计算曲线与x轴之间的面积：
∫ₐᵇ f(x) dx = F(b) − F(a)
When the curve dips below the x-axis, the integral gives a negative value. If the question asks for the total enclosed area, split the interval at the x-intercepts and take absolute values.
当曲线位于x轴下方时，积分值为负。若题目要求总面积，应以x轴交点为界分段积分并取绝对值。
A June 2022 style question may give a quadratic curve and a straight line, asking for the area bounded between them. Subtract the line equation from the curve equation, find the intersection limits, then integrate.
2022年6月风格题目可能给出一条二次曲线和一条直线，求两者围成的面积。用曲线方程减去直线方程，求出交点作为积分上下限，再进行积分。
Always include the constant of integration C for indefinite integrals. Forgetting C loses an accuracy mark.
求不定积分时务必加上积分常数C。漏写C会丢失精确度分。
9. Exponentials and Logarithms | 指数与对数
The exponential function eˣ has the special property that its derivative is itself:
指数函数 eˣ 具有一个特殊性质：其导数等于自身：
d/dx (eˣ) = eˣ, ∫ eˣ dx = eˣ + C
Logarithms are the inverse of exponentials. The three key laws are:
对数是指数的逆运算。三个关键法则为：
logₐ(xy) = logₐx + logₐy, logₐ(x/y) = logₐx − logₐy, logₐ(xⁿ) = n logₐx
The change of base formula logₐb = log_c b / log_c a is also useful for solving equations with different bases.
换底公式 logₐb = log_c b / log_c a 在解不同底数方程时很有用。
To solve an equation like 2ˣ = 15, take the natural logarithm of both sides:
解方程如 2ˣ = 15 时，两边取自然对数：
x ln 2 = ln 15 → x = ln 15 / ln 2 ≈ 3.907
Exam questions often set logarithms in a practical context, such as exponential growth or decay. You may need to find the initial value, the rate constant, or the time taken to reach a certain quantity.
考试题常将对数置于实际情境中，如指数增长或衰减。你可能需要求初始值、速率常数，或达到某一数量所需的时间。
When modelling, remember that the natural logarithm of e is 1, and that log laws only apply when the arguments are positive.
建模时请注意：ln e = 1，且对数法则仅在其真数为正时适用。
10. Sequences and the Binomial Theorem | 数列与二项式定理
Arithmetic and geometric sequences appear as short-answer questions. For an arithmetic sequence:
等差与等比数列常以短答题形式出现。对于等差数列：
uₙ = a + (n − 1)d, Sₙ = n/2 [2a + (n − 1)d]
For a geometric sequence:
对于等比数列：
uₙ = arⁿ⁻¹, Sₙ = a(1 − rⁿ) / (1 − r)
The infinite sum of a geometric series exists only when |r| < 1 and equals a / (1 − r). Practice writing the first few terms and identifying common differences or ratios under exam pressure.
等比级数仅在 |r| < 1 时存在无穷和，其值为 a / (1 − r)。考试压力下，练习写出前几项并识别公差或公比非常重要。
The binomial theorem expands powers of a binomial expression:
二项式定理用于展开二项式的幂：
(a + b)ⁿ = aⁿ + ⁿC₁ aⁿ⁻¹b + ⁿC₂ aⁿ⁻²b² + … + bⁿ
You should know how to compute binomial coefficients using Pascal’s triangle or the nCr formula. A typical question asks for the term independent of x, or the coefficient of a particular power of x.
你应该会用杨辉三角或组合数公式计算二项式系数。典型题目要求求常数项或某一特定幂次的系数。
For example, the coefficient of x³ in (1 + 2x)⁵ is ⁵C₃ × 1² × (2x)³ = 5 × 4 / (3 × 2) × 8 = 80.
例如，(1 + 2x)⁵ 展开式中 x³ 的系数为 ⁵C₃ × 1² × (2x)³ = 5 × 4 / (3 × 2) × 8 = 80。
11. Common Mistakes and Exam Strategy | 常见错误与应试策略
Several errors recur across every sitting of this paper. Losing marks to sign errors is the most common: check every negative sign when expanding brackets or differentiating.
每次考试中都有几类反复出现的错误。因符号错误而失分最为常见：展开括号或求导时，请检查每一个负号。
Another frequent issue is confusing the equations for the tangent and the normal. The tangent gradient equals f'(a); the normal gradient is −1 / f'(a), provided f'(a) ≠ 0.
另一个常见问题是混淆切线与法线的方程。切线斜率等于 f'(a)；法线斜率为 −1 / f'(a)（前提是 f'(a) ≠ 0）。
Students also forget to give answers in the form requested. If the question says “give your answer to 3 significant figures” or “in the form a√b”, an unsimplified decimal answer will not gain full marks.
学生还会忘记按题目要求的形式作答。若题目要求”答案保留3位有效数字”或”写成 a√b 的形式”，未化简的小数答案无法获得满分。
For calculus questions, always show intermediate steps. AQA rewards method marks even if the final numerical answer is wrong.
对于微积分题目，务必写出中间步骤。即使最终数值答案错误，AQA仍会根据方法步骤给分。
Use your calculator wisely: solve equations numerically to check, but never substitute calculator output for written algebraic working that the mark scheme requires.
明智地使用计算器：可用数值方法检验方程的解，但绝不能用计算器输出替代评分标准所要求的书面代数步骤。
Finally, read every question twice. The June 2022 paper included several questions where a small word such as “positive” or “exact” changed the entire approach.
最后，每道题至少读两遍。2022年6月试卷中有若干题目，一个词如”正数”或”精确值”就完全改变了解题思路。
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