PART I · PAPER 1 SECTION A第一部分 · 第一卷 A 节No calculator · short response · 25 marks不可使用计算器 · 简答题 · 25 分
Section A · Short ResponseA 节 · 简答题
Quote each formula on its own line before substituting numerical values. Leave answers exact (in terms of $\pi$ or as surds) unless the question asks for a decimal. No calculator permitted.代值之前先单独一行写出公式。除题目要求小数外，答案保留精确形式（带 $\pi$ 或带根号）。不可使用计算器。
Q1EASYPaper 1AC1.2 Volume of a Cone[4 marks]
A right circular cone has base radius $r = 3$ and vertical height $h = 4$. Find its volume, leaving the answer in terms of $\pi$.一正圆锥底半径 $r = 3$、铅直高 $h = 4$。求其体积（以 $\pi$ 表示）。
(a)Quote the volume formula for a right circular cone.写出正圆锥体积公式。[1]
(b)Substitute $r = 3$ and $h = 4$ and simplify.代入 $r = 3$、$h = 4$ 并化简。[2]
(c)State the answer in the exact form $V = k\pi$ for an integer $k$.以 $V = k\pi$（$k$ 为整数）的形式给出答案。[1]
Q2MEDIUMPaper 1AC1.5 Arc Length and Sector Area[6 marks]
A sector of a circle has radius $r = 10$ and central angle $\theta = \dfrac{\pi}{3}$ radians.一扇形半径 $r = 10$，圆心角 $\theta = \dfrac{\pi}{3}$ 弧度。
(a)Quote the arc-length and sector-area formulas (radian forms).写出弧长与扇形面积公式（弧度形式）。[2]
(b)Compute the arc length $s$ in exact form.求弧长 $s$ 的精确值。[2]
(c)Compute the sector area $A$ in exact form.求扇形面积 $A$ 的精确值。[2]
Q3MEDIUMPaper 1AC1.3 Surface Area of a Closed Cylinder[7 marks]
A closed right circular cylinder has radius $r = 5$ and height $h = 12$.一封闭正圆柱半径 $r = 5$、高 $h = 12$。
(a)State the area of one circular end.写出一个圆形端面的面积。[1]
(b)Explain why the curved (lateral) surface unrolls into a rectangle of width $h$ and length $2\pi r$, and hence find its area.说明侧面（曲面）展开为宽 $h$、长 $2\pi r$ 的矩形，并求其面积。[3]
(c)Add the three pieces (two ends plus curved surface) and give the total in the form $S = k\pi$.合并三部分（两端面加侧面），以 $S = k\pi$ 的形式给出总和。[3]
Q4HARDPaper 1AC1.1 + C1.4 3D Distance and Angle to a Plane[8 marks]
Consider the points $A = (1, 2, -1)$ and $B = (4, 6, 3)$ in three-dimensional space.考虑空间中两点 $A = (1, 2, -1)$ 与 $B = (4, 6, 3)$。
(a)Find the distance $AB$ in exact form.求距离 $AB$ 的精确值。[3]
(b)Find the length of the projection of $\overrightarrow{AB}$ onto the $xy$-plane (that is, the horizontal distance between $A$ and $B$).求 $\overrightarrow{AB}$ 在 $xy$ 平面上投影的长度（即 $A$ 与 $B$ 的水平距离）。[2]
(c)Hence show that the angle $\theta$ that the line $AB$ makes with the $xy$-plane satisfies $\tan\theta = \dfrac{4}{5}$.由此证明直线 $AB$ 与 $xy$ 平面所成的角 $\theta$ 满足 $\tan\theta = \dfrac{4}{5}$。[3]
PART II · PAPER 1 SECTION B第二部分 · 第一卷 B 节No calculator · extended response · 12 marks不可使用计算器 · 长答题 · 12 分
Section B · Extended ResponseB 节 · 长答题
Decompose composite solids into named pieces (hemisphere, cylinder, cone, disc) before applying any formula. State which faces are present and which are absent. Leave answers in exact form.先把组合立体分解为已命名的部件（半球、圆柱、圆锥、圆盘），再代入公式。指明哪些面存在、哪些面缺失。答案保留精确形式。
Q5HARDPaper 1BC1.2 + C1.3 Composite Solid (Hemisphere on Cylinder)[12 marks]
A solid trophy consists of a solid hemisphere of radius $r$ joined to the top of a solid right circular cylinder of the same radius $r$ and height $r$. The bottom face of the cylinder is a closed disc; the hemisphere sits flush on the top face of the cylinder (no gap, and the shared circular face is internal).一立体奖杯由一实心半球与一同半径的实心正圆柱组合而成，半球安置于圆柱顶面、与之齐平（无空隙，公共圆面为内部，不计入表面）；圆柱底面为封闭圆盘。半球与圆柱半径均为 $r$，圆柱高为 $r$。
(a)Show that the total volume of the trophy is $V = \dfrac{5}{3}\pi r^{3}$.证明奖杯的总体积为 $V = \dfrac{5}{3}\pi r^{3}$。[3]
(b)List the external faces (curved hemisphere, curved cylinder lateral, bottom disc) and show that the total external surface area is $S = 5\pi r^{2}$.列出外表面（半球曲面、圆柱侧面、底面圆盘），并证明总外表面积为 $S = 5\pi r^{2}$。[4]
(c)Evaluate $V$ and $S$ when $r = 3$, giving both answers in the form $k\pi$ for an integer $k$.取 $r = 3$，分别求 $V$ 与 $S$，并以 $k\pi$（$k$ 为整数）的形式给出。[2]
(d)Show that $V/S = r/3$ for every $r > 0$, and verify that this implies $V = S$ at $r = 3$ (consistent with part (c)). State the geometric meaning of the ratio $V/S$.证明对每个 $r > 0$ 有 $V/S = r/3$，并验证由此得 $r = 3$ 时 $V = S$（与 (c) 一致）。说明比值 $V/S$ 的几何意义。[3]
PART III · PAPER 2第三部分 · 第二卷Calculator · mixed response · 23 marks可使用计算器 · 混合题型 · 23 分
Paper 2 · Calculator Permitted第二卷 · 允许使用计算器
A graphing calculator is required. Round final numerical answers to three significant figures unless an exact form is requested. Show every formula before substituting; the method marks live on those lines.需要图形计算器（GDC）。除题目要求精确形式外，最终数值答案取三位有效数字。代值前每个公式都要写出来 — 方法分就在这些行上。
A right circular cone has slant height $l = 13 \text{ cm}$ and base radius $r = 5 \text{ cm}$.一正圆锥斜高 $l = 13 \text{ cm}$，底半径 $r = 5 \text{ cm}$。
(a)Find the vertical height $h$ of the cone.求圆锥的铅直高 $h$。[2]
(b)Find the total surface area of the closed cone (base plus curved surface), giving the answer in exact form.求该封闭圆锥的总表面积（底面加侧面），结果取精确形式。[3]
(c)Find the volume of the cone in exact form, then give the decimal value to three significant figures.求圆锥体积的精确值，再给出三位有效数字的小数值。[2]
(d)A solid sphere has the same volume as the cone. Find the radius of the sphere, giving the answer to three significant figures.某实心球的体积与该圆锥相等。求该球的半径，结果取三位有效数字。[3]
An open-top cylindrical can has radius $r$, height $h$, and fixed interior volume $V_{0} = 1000 \text{ cm}^{3}$ (so $V_{0}$ is a constant). The can has a closed circular base and a curved lateral surface, but no top.一无盖圆柱形罐半径为 $r$、高为 $h$、内体积固定为 $V_{0} = 1000 \text{ cm}^{3}$（$V_{0}$ 为常数）。罐有封闭的圆形底面与曲面侧壁，但无顶。
(a)Use the volume constraint to express $h$ in terms of $r$ and $V_{0}$.由体积约束，将 $h$ 用 $r$ 与 $V_{0}$ 表示。[1]
(b)Show that the total external surface area is $S(r) = \pi r^{2} + \dfrac{2 V_{0}}{r}$.证明外表面积为 $S(r) = \pi r^{2} + \dfrac{2 V_{0}}{r}$。[2]
(c)Differentiate $S(r)$ with respect to $r$, set the derivative to zero, and show that the critical radius is $r^{*} = \left(\dfrac{V_{0}}{\pi}\right)^{1/3}$.对 $r$ 求导 $S(r)$，令导数为 $0$，证明临界半径为 $r^{*} = \left(\dfrac{V_{0}}{\pi}\right)^{1/3}$。[4]
(d)Show that, at this critical radius, the optimal height satisfies $h^{*} = r^{*}$.证明在该临界半径处，最优高度满足 $h^{*} = r^{*}$。[2]
(e)Use the second-derivative test to confirm that $r^{*}$ gives a minimum (not a maximum) of $S$.用二阶导数检验确认 $r^{*}$ 给出 $S$ 的极小值（非极大值）。[2]
(f)For $V_{0} = 1000 \text{ cm}^{3}$, compute $r^{*}$, $h^{*}$, and the minimum surface area $S(r^{*})$, each to three significant figures.取 $V_{0} = 1000 \text{ cm}^{3}$，分别求 $r^{*}$、$h^{*}$、$S(r^{*})$，各取三位有效数字。[2]