📚 B7F PP 1-40: Core Skills and Exam Strategy | B7F 试卷1-40题核心技能与备考策略
The first 40 questions of the B7F past paper cover a wide range of pure mathematics topics. This article groups them into the ten most important skill areas so that you can revise efficiently and understand exactly what the examiner is testing.
B7F 往年试卷的第 1-40 题覆盖了广泛的纯数学主题。本文将题目归纳为十个最重要的技能领域，帮助你高效复习，并准确理解考官的考查意图。
1. Algebraic Manipulation | 代数运算
Many early questions in the paper require you to expand products such as (x - 3)(2x + 5) and to collect like terms without losing negative signs. A common error is writing -3 multiplied by 5 as +15 instead of -15.
试卷前面的题目常常要求展开乘积，例如 (x - 3)(2x + 5)，并合并同类项且不能丢掉负号。常见错误是把 -3 乘以 5 写成 +15 而不是 -15。
, by grouping, and by recognising the difference of two squares: a² - b² = (a - b)(a + b).
你还应熟练运用公因式法、分组法以及平方差公式进行因式分解：a² - b² = (a - b)(a + b)。
When simplifying rational expressions, factorise the numerator and denominator first, then cancel only factors that are common to both. Do not cancel individual terms across a plus or minus sign.
在化简有理式时，应先对分子和分母进行因式分解，然后只约去二者共有的因式。不要跨过加号或减号约去单独的项。
tested frequently. The most useful forms are xᵃ × xᵇ = xᵃ⁺ᵇ, (xᵃ)ᵇ = xᵃᵇ, and x⁻ⁿ = 1/xⁿ.
指数法则也经常考查。最常用的形式为 xᵃ × xᵇ = xᵃ⁺ᵇ，(xᵃ)ᵇ = xᵃᵇ，以及 x⁻ⁿ = 1/xⁿ。
2. Quadratics and the Discriminant | 二次函数与判别式
riminant tells you how many real roots exist. The formula is central to many B7F questions.
对于二次方程 ax² + bx + c = 0，判别式可以告诉你它有多少个实数根。该公式是许多 B7F 题目的核心。
Δ = b² - 4ac
nct real roots, if Δ = 0 there is one repeated root, and if Δ < 0 there are no real roots.
若 Δ > 0，则有两个不同的实根；若 Δ = 0，则有一个重根；若 Δ < 0，则没有实根。
In the B7F questions, you may be asked to find unknown constants or to show that a curve lies completely above the x-axis by proving Δ < 0 and a > 0.
在 B7F 的题目中，可能会要求你求出未知常数，或通过证明 Δ < 0 且 a > 0 来说明曲线完全位于 x 轴上方。
Completing the square is another routine skill. For example, x² + 6x + 5 can be rewritten as (x + 3)² - 4, which immediately shows the vertex of the parabola.
配方法也是一项常规技能。例如，x² + 6x + 5 可以改写为 (x + 3)² - 4，这样可以直接看出抛物线的顶点。
3. Functions and Transformations | 函数与图像变换
f(x) = 2x - 3, f(x + 1) means replacing every x with x + 1, giving 2(x + 1) - 3 = 2x - 1.
理解函数记号至关重要。若 f(x) = 2x - 3，则 f(x + 1) 表示把每个 x 都替换为 x + 1，得到 2(x + 1) - 3 = 2x - 1。
x - a) shifts the graph a units to the right, while y = f(x) + b shifts it b units upward.
图像变换常常在同一道题中组合出现：y = f(x - a) 将图像向右平移 a 个单位，而 y = f(x) + b 将图像向上平移 b 个单位。
rizontally. For instance, y = f(2x) squeezes the graph toward the y-axis by a factor of 2.
函数内部的乘数会在水平方向压缩图像。例如，y = f(2x) 将图像向 y 轴方向压缩为原来的一半。
For composite functions, remember that fg(x) means apply g first, then f, so the order of operations matters. To find an inverse function, swap x and y and then solve for y.
对于复合函数，请记住 fg(x) 表示先作用 g，再作用 f，因此运算顺序很重要。求反函数时，交换 x 和 y，然后解出 y。
4. Coordinate Geometry | 坐标几何
Many questions between Q10 and Q20 test the relationship between gradient, distance, and midpoints. The distance formula is:
第 10 至 20 题中有许多题目考查斜率、距离和中点之间的关系。距离公式为：
d = √[(x₂ - x₁)² + (y₂ - y₁)²]
hrough two points is m = (y₂ - y₁) / (x₂ - x₁), and perpendicular lines satisfy m₁m₂ = -1.
过两点的直线斜率为 m = (y₂ - y₁) / (x₂ - x₁)，并且两条垂直直线的斜率满足 m₁m₂ = -1。
first calculate the gradient from the given curve or circle, then use y - y₁ = m(x - x₁).
求切线或法线方程时，先根据给定曲线或圆求出斜率，再使用点斜式 y - y₁ = m(x - x₁)。
b)² = r². A tangent to the circle is perpendicular to the radius at the point of contact.
圆心为 (a, b)、半径为 r 的圆方程为 (x - a)² + (y - b)² = r²。圆在切点处的切线与该点半径互相垂直。
5. Sequences and Series | 数列与级数
m is given by uₙ = a + (n - 1)d, where a is the first term and d is the common difference.
等差数列在纯数学试卷前半部分经常出现。第 n 项公式为 uₙ = a + (n - 1)d，其中 a 是首项，d 是公差。
The sum of the first n terms of an arithmetic series can be written in two useful forms:
等差数列前 n 项和可以写成两种实用的形式：
Published by TutorHao | Mathematics Revision Series | aleveler.com更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图