GAC 수학

A price can fall by 20% and then rise by 20% without returning to its starting value. Mathematics I helps you explain such results with a method, units and a clear interpretation. The six units connect arithmetic, algebra, graphs, geometry, trigonometry and exponential models.

Your centre's current assessment brief is the authority for tasks, weights, permitted tools and deadlines. An in-class test 课堂测验 is completed under stated classroom conditions. Projects 项目 can involve collecting data, calculating and reporting; an examination 考试 and coursework 平时作业 may have different instructions. Do not infer an official assessment pattern from these practice sheets.

A terminology logbook 术语记录本 can record the English term, a meaning in your own words and a small example. Follow your centre's instructions if it is submitted or assessed.

Read the command word 指令词: solve asks for values satisfying a condition, simplify for an equivalent expression, and justify for a reason. Show a valid method so a reader can check how the result follows. The mark scheme for each task determines its marks.

The six accompanying practice sheets are original GAC004-aligned material, not official papers. Their solutions award marks for stated mathematical steps rather than for the length of a written report.

The vocabulary of number is assumed by every later unit.

An integer 整数 is a whole number, including negative whole numbers and zero. A rational number 有理数 can be written as a fraction of two integers with a nonzero denominator; an irrational number 无理数 cannot, and $\pi$ and $\sqrt{2}$ are the standard examples. A prime number 质数 is a positive integer greater than 1 with exactly two positive factors, itself and 1.

The order of operations 运算顺序 fixes what a written expression means: brackets, then indices, then multiplication and division, then addition and subtraction, working left to right within a level.

• A factor 因数 divides a number exactly; a multiple 倍数 is what you get by multiplying it. 6 is a factor of 24, and 24 is a multiple of 6.
• The highest common factor (HCF) 最大公因数 and lowest common multiple (LCM) 最小公倍数 come from the prime factors: $24 = 2^3 \times 3$ and $36 = 2^2 \times 3^2$, so the HCF is $2^2 \times 3 = 12$ and the LCM is $2^3 \times 3^2 = 72$.
• A percentage 百分比 is a fraction with denominator 100. An increase of 15% multiplies by $1.15$; a decrease of 15% multiplies by $0.85$. Reversing a percentage change means dividing, never subtracting the same percentage back.
• Significant figures 有效数字 are the digits retained to express a value at a stated precision. They do not alone establish the accuracy of a measurement. $0.004\,072$ to three significant figures is $0.004\,07$.
• Standard form 科学记数法 writes a number as $a \times 10^n$ with $1 \le |a| < 10$ for a nonzero number, with integer n. It is how a calculator shows a very large or very small answer.

Worked example. a decrease and increase use different bases

Known: an invented price of 80 units falls by 20%, then rises by 20%. Why use successive multipliers? Each change is calculated from the current price.

The price is not back at 80. The decrease was 16, while the later increase was 12.8, because the bases differ.

Continue with practice sheet 1.1. Solve before opening the solutions, and check both the method and the final units.

Algebra is arithmetic with letters standing for numbers, and it has its own vocabulary.

In $5x^2 - 3x + 7$, the whole thing is an expression 表达式; $5x^2$, $-3x$ and $7$ are its terms 项; 5 is the coefficient 系数 of $x^2$; and 7 is a constant 常数.

An equation 方程 says two expressions are equal and is solved for a value. An identity 恒等式 is true for every value in its domain. An inequality 不等式 compares two expressions with $<$, $\le$, $>$ or $\ge$.

To expand 展开 is to remove brackets; to factorise 因式分解 is to put them back.

• Solving a linear equation 一元一次方程 means doing the same operation to both sides until the letter stands alone.
• A quadratic equation 一元二次方程 has the form $ax^2+bx+c=0$ with $a\ne0$. Factorise when suitable, or use $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
• Simultaneous equations 联立方程 are two equations in two unknowns, solved by substitution or by elimination. For two linear equations, a unique solution is their intersection. Parallel distinct lines have no solution; identical lines have infinitely many.
• ⚠ An inequality reverses 反向 when you multiply or divide both sides by a negative number: from $-2x > 6$ it follows that $x < -3$.

Worked example. reverse the inequality sign when dividing by a negative

Known: solve $7-3x>1$. Subtract 7 from both sides, then divide by negative 3. The last operation reverses the inequality.

The open circle excludes 2. Check $x=1$: $7-3(1)=4>1$; the boundary $x=2$ gives equality and is excluded.

Continue with practice sheet 1.2. Solve before opening the solutions, and check both the method and the final units.

A graph turns an equation into a picture, and coordinate geometry measures that picture.

A nonvertical straight line can be written $y = mx + c$, where $m$ is the gradient 斜率 and $c$ the y-intercept y 轴截距. The gradient is the rise divided by the run.

• Distinct nonvertical lines are parallel 平行 when their gradients are equal. Two nonvertical lines are perpendicular 垂直 when their gradients multiply to $-1$. Vertical lines have undefined gradient; a vertical and a horizontal line are perpendicular.
• The midpoint 中点 of a segment is the average of the endpoints, and the distance 距离 between two points comes from Pythagoras: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
• A quadratic graphs as a parabola 抛物线. Its vertex 顶点 is the turning point, and the roots 根 are the x-values where it meets the $x$-axis, the solutions of $y=0$. A repeated root touches the axis without crossing; some quadratics have no real roots.
• To solve two equations graphically 用图像求解, draw both and read the coordinates of the intersection. The answer is only as accurate as the drawing, which is why an algebraic check matters.

Worked example. a rising line through two given points

Known: $A=(1,2)$ and $B=(4,8)$. Use gradient because the line is nonvertical.

The line equation is $y=mx+c$. Substitute A to find the intercept.

Thus $y=2x$. The diagram's horizontal change is 3 and vertical change is 6.

Continue with practice sheet 1.3. Solve before opening the solutions, and check both the method and the final units.

Plane geometry is about flat shapes, solid geometry about three-dimensional ones, and Euclidean geometry is the reasoning that connects them.

Angles on a straight line add to $180°$, angles around a point to $360°$, and the interior angles of a plane Euclidean triangle to $180°$. In a simple polygon 多边形 of $n$ sides the interior angles add to $(n - 2) \times 180°$.

• Congruent 全等 shapes are identical in size and shape; similar 相似 shapes have the same shape with all lengths in one ratio.
• Pythagoras' theorem 勾股定理 holds in a right-angled triangle: $a^2 + b^2 = c^2$, where $c$ is the hypotenuse 斜边.
• Area 面积 is measured in square units and volume 体积 in cubic units. A cylinder has volume $\pi r^2 h$; a sphere has volume $\tfrac{4}{3}\pi r^3$ and surface area $4\pi r^2$.
• ⚠ Scaling is not linear. If every length of a solid is doubled, its area is multiplied by $2^2 = 4$ and its volume by $2^3 = 8$.

Worked example. a triangular prism has a constant cross-section

Known: perpendicular triangle sides are 3 and 4 cm; prism length is 10 cm. Find the cross-sectional area, then multiply by the prism length.

The sloping triangle side is not its perpendicular height. Volume is not the area of the triangular end.

Continue with practice sheet 1.4. Solve before opening the solutions, and check both the method and the final units.

Trigonometry connects the angles of a triangle to its sides.

In a right-angled triangle, with $\theta$ one of the acute angles, the three ratios are $\sin \theta = \dfrac{\text{opposite}}{\text{hypotenuse}}$, $\cos \theta = \dfrac{\text{adjacent}}{\text{hypotenuse}}$ and $\tan \theta = \dfrac{\text{opposite}}{\text{adjacent}}$.

For a nondegenerate plane Euclidean triangle, the sine rule 正弦定理 and the cosine rule 余弦定理 apply:

• An angle can be measured in degrees 度 or in radians 弧度, where $\pi$ radians is $180°$. Check which mode your calculator is in before every question.
• The unit circle 单位圆 extends the ratios beyond $90°$ and explains why $\sin$ and $\cos$ repeat every $360°$: they are periodic 周期的.
• A bearing 方位角 is measured clockwise from north and always written with three figures, such as $075°$.
• Angles of elevation and depression 仰角与俯角 are measured from the horizontal, upwards and downwards respectively. Add the observer or instrument height if the answer needs height above the ground. With side-side-angle data, the sine rule may allow two triangles: check both supplementary angles against the remaining angle sum.

Worked example. height from a horizontal distance and angle

Known: horizontal distance $d=12$ m and elevation $\theta=30^\circ$. Height h is opposite and d is adjacent, so use tangent in degree mode.

This is the height above the observer's horizontal sight level, not automatically the height above the ground.

Continue with practice sheet 1.5. Solve before opening the solutions, and check both the method and the final units.

These two functions describe growth and decay, and they undo each other.

An exponential function 指数函数 has the form $y=a^x$ with $a>0$ and $a\ne1$: the variable is in the exponent 指数. Each unit step multiplies its value by a. It grows when $a>1$ and decays when $0. Its y-intercept is 1 and its values stay positive; the x-axis is a horizontal asymptote.

A logarithm 对数 answers the reverse question. $\log_a y = x$ means exactly $a^x = y$, so $\log_{10}1000=3$. For real logarithms the argument must be positive, and the base must be positive and different from 1. The graph $y=\log_a x$ passes through $(1,0)$ and has vertical asymptote $x=0$.

• For positive arguments and the same valid base, the laws turn multiplication into addition, which is what makes a logarithm useful for solving an equation with the unknown in the exponent.
• Natural logarithms 自然对数 use the base $e \approx 2.718$ and are written $\ln$.
• Compound interest 复利 is exponential: $A = P(1 + r)^n$ after $n$ periods.
• Exponential decay 指数衰减 has a base between 0 and 1, and can describe a retained proportion each period under stated assumptions. It does not guarantee that a real process follows that model indefinitely.

Worked example. repeated reductions multiply rather than subtract a fixed amount

Known: an invented quantity starts at 100 units and decreases by 20% per whole period. The retained proportion is 0.8, so use a multiplicative model.

The first decrease is 20 units and the second is 16. A fixed subtraction of 20 each period would be a different model.

Continue with practice sheet 1.6. Solve before opening the solutions, and check both the method and the final units.

An average speed does not tell you every instantaneous speed, and a signed displacement does not always equal total distance. Calculus makes those distinctions precise.

GAC016 develops differentiation 微分, integration 积分 and their applications. Your centre's current brief determines assessment tasks and grading requirements; these original practice sheets do not establish a university credit decision or an official examination pattern.

State the independent variable, domain and units. The notation $dy/dx$ is a derivative with respect to x; a dot over y normally denotes a derivative with respect to time. Those notations describe the same operation only when their independent variables agree.

For applications, describe what a rate or accumulated value means in the stated model. A negative volume-change rate of 3 cubic centimetres per second means volume is falling at that rate, rather than that the volume itself is negative.

A derivative 导数 is a rate of change: how fast $y$ changes as $x$ changes. Geometrically it is the gradient of the tangent 切线斜率 at a point.

That limit is the definition. Differentiation from first principles 从定义求导 uses the difference quotient for nonzero h and then takes its limit. Unequal one-sided limits, as for $|x|$ at zero, prevent a derivative even when the function is continuous.

• The power rule 幂法则, on a domain where the power is differentiable: if $y = x^n$ then $\dfrac{dy}{dx} = nx^{\,n-1}$.
• The product rule 乘积法则: $(uv)' = u'v + uv'$.
• The quotient rule 商法则, where $v\ne0$: $\left(\dfrac{u}{v}\right)' = \dfrac{u'v - uv'}{v^2}$.
• The chain rule 链式法则: if $y = f(g(x))$ then $\dfrac{dy}{dx} = f'(g(x)) \cdot g'(x)$.
• Stationary points 驻点 occur where $f'(x) = 0$. The second derivative 二阶导数 then classifies a stationary point when it is nonzero: negative gives a local maximum 极大值 and positive a local minimum 极小值. A zero second derivative is inconclusive; use derivative signs or other evidence. A stationary inflection can be neither an extremum.

Worked example. a tangent slope differs from a secant slope

Known: $f(x)=x^2$. The secant between x equal to 1 and 2 has slope 3, but the derivative at x equal to 1 is 2.

The average slope over a finite interval need not equal the instantaneous slope at either endpoint.

Use original practice sheet 3.1 to test the method, domain and interpretation against its solutions.

Indefinite integration finds antiderivatives. Definite integration measures signed accumulation, which can differ from geometric area. For a continuous integrand, the fundamental theorem of calculus 微积分基本定理 connects a definite integral to the difference of antiderivative values at its limits.

• An indefinite integral 不定积分 has no limits and needs the constant of integration 积分常数: $\displaystyle\int x^n\,dx = \frac{x^{\,n+1}}{n+1} + c$ for $n \neq -1$.
• A definite integral 定积分 has limits and gives a number: $\displaystyle\int_a^b f(x)\,dx = F(b) - F(a)$.
• Integration by substitution 换元积分法 reverses the chain rule.
• ⚠ Forgetting $+c$ on an indefinite integral is the single most frequent lost mark in the module, and it is lost on questions you have otherwise answered correctly.

Worked example. negative and positive areas can cancel

Known: $f(x)=x$ from negative 1 to 1. Its definite integral is zero because the negative and positive contributions cancel.

Geometric area instead adds the magnitudes of the two triangular regions.

Use original practice sheet 3.2 to test the method, domain and interpretation against its solutions.

• Optimisation 最优化 finds the largest or smallest value of a quantity: write the quantity as a function of one variable on a stated feasible domain, differentiate, find candidates and justify the required optimum. Check included boundaries and integer constraints where relevant.
• Rates of change 变化率 chain together: $\dfrac{dV}{dt} = \dfrac{dV}{dr} \times \dfrac{dr}{dt}$.
• Area between two curves 两曲线间面积 is $\displaystyle\int_a^b (f(x) - g(x))\,dx$ where $f$ is the upper curve on that whole interval. Split at crossings if their order changes.
• Marginal cost and revenue 边际成本与边际收益 are derivatives of stated continuous cost and revenue models. They are instantaneous rates, not automatically exact discrete-unit changes.

Worked example. a fixed perimeter leaves one area variable

Known: a rectangle has perimeter 20, so length plus width is 10. Let width be x; length is $10-x$ and $0.

The stationary candidate is $x=5$. Since $A''(x)=-2<0$ and the area approaches zero at both domain ends, this candidate is the global maximum. It is a 5 by 5 square with area 25.

Use original practice sheet 3.3 to test the method, domain and interpretation against its solutions.

A repeated set member is counted once, a binary carry may exceed a fixed width, and the fewest-edge route may not have the smallest weight. Discrete mathematics makes those rules explicit.

GAC024 covers sets, counting systems, binary logic, algorithms and networks. Your centre's current brief determines assessment tasks, tools, weights and deadlines. These original practice sheets do not establish official marking rules or a university credit decision.

State the universe, representation width, allowed inputs or graph assumptions before solving. Show enough working for another reader to reproduce the result and distinguish a mathematical model from its real implementation.

• A set 集合 is a collection of distinct objects. Order and repetition do not matter.
• Union 并集 $A \cup B$ is everything in either; intersection 交集 $A \cap B$ is what is in both; the set complement 补集 is everything in the stated universe but outside the set.
• A subset 子集 has all its elements inside another set.
• A relation 关系 pairs elements of two sets. A function 函数 is a relation where each input in its stated domain has exactly one output. Different inputs may share an output; an inverse relation is a function only when outputs uniquely identify their inputs.
• A Venn diagram 韦恩图 turns a set problem into a picture, and can show the disjoint regions and their counts. Check that those regions add to the supplied universe total.

The inclusion-exclusion principle 容斥原理 subtracts the twice-counted overlap once: $|A\cup B|=|A|+|B|-|A\cap B|$.

Worked example. subtract an overlap only once

Known: 30 learners, 18 study French, 15 German, and 7 both. The overlap is included in both subject totals.

French-only is $18-7=11$ and German-only $15-7=8$. The four disjoint regions sum to 30.

Practice sheet 4.1 includes progressively harder problems and independently checked solutions.

• A positional number base 进制 b uses digits from zero to b minus one and place weights $b^i$. Decimal 十进制 uses ten, binary 二进制 two, hexadecimal 十六进制 sixteen.
• Every digit's value is its place value 位值: in binary the places are 1, 2, 4, 8, 16 and so on.
• Hexadecimal is shorthand for binary: one hex digit is exactly four bits, so conversion can group a stated-width binary pattern into four-bit blocks. Leading zeros preserve width while leaving the unsigned value unchanged.

For n unsigned bits, values run from zero to $2^n-1$. Distinguish an unrestricted sum from a stored fixed-width result; a wraparound rule, if explicitly given, keeps the low n bits.

Worked example. place weights determine the decimal value

Known numeral $1101_2$. Use weights from right to left: 1, 2, 4 and 8.

The same value is D in hexadecimal. Leading zeros would not change this nonnegative value but can record an intended width.

Practice sheet 4.2 includes progressively harder problems and independently checked solutions.

• Binary arithmetic 二进制运算 adds like decimal, carrying at 2 instead of at 10.
• A bit 位 is one binary digit; a byte 字节 is eight.
• Boolean algebra 布尔代数 works on true and false with AND, OR and NOT.
• A truth table 真值表 lists every Boolean input combination and output. Matching every row proves equivalence for the same finite Boolean inputs; it does not prove physical circuit timing or real-system security.
• Logic gates 逻辑门 implement stated operations, and a logic circuit 逻辑电路 connects them. Trace the abstract logic according to its connections and input conventions.

Use inclusive OR and explicit brackets. De Morgan gives $\neg(A\land B)=(\neg A)\lor(\neg B)$. Bitwise NOT inverts only the stated width, not an unspecified infinite representation.

Worked example. an OR output is inverted by NOT

Known: $Y=\neg(A\lor B)$. Inclusive OR is false only when both inputs are false; NOT reverses that result. In row order $(A,B)=(0,0),(0,1),(1,0),(1,1)$, the output column is 1, 0, 0, 0. De Morgan gives equivalent expression $(\neg A)\land(\neg B)$.

Practice sheet 4.3 includes progressively harder problems and independently checked solutions.

• An algorithm 算法 describes unambiguous steps for a task. A procedure solving the stated finite task must terminate and give the required result for its allowed inputs.
• A flowchart 流程图 draws it: a decision is a diamond, a process a rectangle.
• Pseudocode 伪代码 represents its steps without requiring a particular implementation language. State assignment, loop bounds and index conventions before tracing.
• Tracing 追踪 an algorithm — a table with one column per variable and one row per step — records its actual updates. A trace checks the chosen input; a claim for all allowed inputs also needs a correctness argument.
• Efficiency 效率 matters: a linear search can stop early but may inspect all n items. Binary search repeatedly discards half of an ordered search range; its logarithmic comparison count requires the sorted-data and bound conventions.

Worked example. repeat a remainder step until the second number is zero

Known: start with positive integers a equal to 10 and b equal to 6. While b is nonzero, compute r as a MOD b, then set a to b and b to r. Pairs after complete iterations are (6,4), (4,2), (2,0), giving output 2. Temporary r preserves the remainder before a and b change. Each nonzero remainder is smaller than the previous positive b, supporting termination.

Practice sheet 4.4 includes progressively harder problems and independently checked solutions.

• A graph 图 is a set of vertices 顶点 joined by edges 边. It models anything with connections: roads, friendships, dependencies.
• For a simple undirected graph with no loops or repeated edges, the degree 度 counts incident edges. Every edge contributes two to the total degree sum.
• A tree 树 is a connected graph with no cycles, and a finite tree with n vertices has n minus 1 edges. Some hierarchical models use trees, but actual systems can also contain cross-links or cycles.
• A shortest path 最短路径 problem asks for the cheapest route between two vertices, by total weight under the stated constraints, rather than by the number of edges alone. A minimum spanning tree instead connects every vertex without cycles and minimises total included edge weight.

Worked example. compare total route weight, not the number of edges

Known edge weights are AB = 2, BC = 3, AC = 8 and CD = 1. The path A-C-D has weight 9, while A-B-C-D has weight 6. Therefore the three-edge path is shorter by weight despite having more edges. The minimum spanning tree for this small network uses AB, BC and CD with total 6; the agreement of totals here does not make the tasks identical.

Practice sheet 4.5 includes progressively harder problems and independently checked solutions.