{
  "knowledge": [
    {
      "id": "K1-1-01",
      "name": "集合",
      "type": "概念",
      "definition": "把一些元素组成的总体叫做集合(set)，简称为集。",
      "prerequisite": []
    },
    {
      "id": "K1-1-02",
      "name": "元素",
      "type": "概念",
      "definition": "组成集合的研究对象统称为元素(element)。",
      "prerequisite": [
        "K1-1-01"
      ]
    },
    {
      "id": "K1-1-03",
      "name": "集合元素的特性",
      "type": "概念",
      "definition": "集合的元素具有确定性、互异性（不重复出现）。",
      "prerequisite": [
        "K1-1-01",
        "K1-1-02"
      ]
    },
    {
      "id": "K1-1-04",
      "name": "元素与集合的关系",
      "type": "概念",
      "definition": "如果a是集合A的元素，就说a属于集合A，记作a ∈ A；如果a不是集合A中的元素，就说a不属于集合A，记作a ∉ A。",
      "prerequisite": [
        "K1-1-01",
        "K1-1-02"
      ]
    },
    {
      "id": "K1-1-05",
      "name": "常用数集",
      "type": "概念",
      "definition": "非负整数集(自然数集)记作N；正整数集记作N*或N+；整数集记作Z；有理数集记作Q；实数集记作R。",
      "prerequisite": []
    },
    {
      "id": "K1-1-06",
      "name": "列举法",
      "type": "概念",
      "definition": "把集合的所有元素一一列举出来，并用花括号“{ }”括起来表示集合的方法。",
      "prerequisite": [
        "K1-1-01"
      ]
    },
    {
      "id": "K1-1-07",
      "name": "描述法",
      "type": "概念",
      "definition": "把集合中所有具有共同特征P(x)的元素x所组成的集合表示为{x ∈ A | P(x)}的方法。",
      "prerequisite": [
        "K1-1-01"
      ]
    },
    {
      "id": "K1-2-01",
      "name": "子集",
      "type": "概念",
      "definition": "对于两个集合A, B，如果集合A中任意一个元素都是集合B中的元素，就称集合A为集合B的子集，记作A ⊆ B (或 B ⊇ A)。",
      "prerequisite": [
        "K1-1-01"
      ]
    },
    {
      "id": "K1-2-02",
      "name": "集合相等",
      "type": "概念",
      "definition": "如果集合A的任何一个元素都是集合B的元素，同时集合B的任何一个元素都是集合A的元素，那么集合A与集合B相等，记作A=B。即 A ⊆ B 且 B ⊆ A。",
      "prerequisite": [
        "K1-2-01"
      ]
    },
    {
      "id": "K1-2-03",
      "name": "真子集",
      "type": "概念",
      "definition": "如果集合A ⊆ B，但存在元素x ∈ B，且x ∉ A，就称集合A是集合B的真子集，记作A ⊊ B。",
      "prerequisite": [
        "K1-2-01"
      ]
    },
    {
      "id": "K1-2-04",
      "name": "空集",
      "type": "概念",
      "definition": "不含任何元素的集合叫做空集，记为∅，并规定：空集是任何集合的子集。",
      "prerequisite": [
        "K1-2-01"
      ]
    },
    {
      "id": "K1-3-01",
      "name": "并集",
      "type": "概念",
      "definition": "由所有属于集合A或属于集合B的元素组成的集合，称为集合A与B的并集，记作A ∪ B。",
      "prerequisite": [
        "K1-1-01"
      ]
    },
    {
      "id": "K1-3-02",
      "name": "交集",
      "type": "概念",
      "definition": "由所有属于集合A且属于集合B的元素组成的集合，称为集合A与B的交集，记作A ∩ B。",
      "prerequisite": [
        "K1-1-01"
      ]
    },
    {
      "id": "K1-3-03",
      "name": "全集",
      "type": "概念",
      "definition": "如果一个集合含有所研究问题中涉及的所有元素，那么就称这个集合为全集，通常记作U。",
      "prerequisite": [
        "K1-1-01"
      ]
    },
    {
      "id": "K1-3-04",
      "name": "补集",
      "type": "概念",
      "definition": "对于一个集合A，由全集U中不属于集合A的所有元素组成的集合称为集合A相对于全集U的补集，简称为集合A的补集，记作C_U A。",
      "prerequisite": [
        "K1-3-03"
      ]
    },
    {
      "id": "K1-3-05",
      "name": "有限集元素个数公式",
      "type": "公式",
      "definition": "对任意两个有限集合 A, B, 有 card(A ∪ B) = card(A) + card(B) - card(A ∩ B)。",
      "prerequisite": [
        "K1-3-01",
        "K1-3-02"
      ]
    },
    {
      "id": "K1-4-01",
      "name": "命题",
      "type": "概念",
      "definition": "用语言、符号或式子表达的，可以判断真假的陈述句叫做命题。",
      "prerequisite": []
    },
    {
      "id": "K1-4-02",
      "name": "充分条件",
      "type": "概念",
      "definition": "如果“若p，则q”为真命题（p ⇒ q），则p是q的充分条件。",
      "prerequisite": [
        "K1-4-01"
      ]
    },
    {
      "id": "K1-4-03",
      "name": "必要条件",
      "type": "概念",
      "definition": "如果“若p，则q”为真命题（p ⇒ q），则q是p的必要条件。",
      "prerequisite": [
        "K1-4-01"
      ]
    },
    {
      "id": "K1-4-04",
      "name": "充要条件",
      "type": "概念",
      "definition": "如果“若p，则q”和它的逆命题“若q，则p”均为真命题（p ⇔ q），则p是q的充分必要条件，简称充要条件。",
      "prerequisite": [
        "K1-4-02",
        "K1-4-03"
      ]
    },
    {
      "id": "K1-5-01",
      "name": "全称量词",
      "type": "概念",
      "definition": "短语“所有的”、“任意一个”在逻辑中通常叫做全称量词，并用符号“∀”表示。",
      "prerequisite": [
        "K1-4-01"
      ]
    },
    {
      "id": "K1-5-02",
      "name": "全称量词命题",
      "type": "概念",
      "definition": "含有全称量词的命题，叫做全称量词命题。形式为 ∀x ∈ M, p(x)。",
      "prerequisite": [
        "K1-5-01"
      ]
    },
    {
      "id": "K1-5-03",
      "name": "存在量词",
      "type": "概念",
      "definition": "短语“存在一个”、“至少有一个”在逻辑中通常叫做存在量词，并用符号“∃”表示。",
      "prerequisite": [
        "K1-4-01"
      ]
    },
    {
      "id": "K1-5-04",
      "name": "存在量词命题",
      "type": "概念",
      "definition": "含有存在量词的命题，叫做存在量词命题。形式为 ∃x ∈ M, p(x)。",
      "prerequisite": [
        "K1-5-03"
      ]
    },
    {
      "id": "K1-5-05",
      "name": "全称量词命题的否定",
      "type": "规则",
      "definition": "全称量词命题 ∀x ∈ M, p(x) 的否定是存在量词命题 ∃x ∈ M, ¬p(x)。",
      "prerequisite": [
        "K1-5-02",
        "K1-5-04"
      ]
    },
    {
      "id": "K1-5-06",
      "name": "存在量词命题的否定",
      "type": "规则",
      "definition": "存在量词命题 ∃x ∈ M, p(x) 的否定是全称量词命题 ∀x ∈ M, ¬p(x)。",
      "prerequisite": [
        "K1-5-02",
        "K1-5-04"
      ]
    },
    {
      "id": "K2-1-01",
      "name": "不等式性质",
      "type": "定理",
      "definition": "不等式的基本性质，包括对称性、传递性、加法法则、乘法法则（分正数和负数两种情况）、同向不等式相加、同向同正不等式相乘、开方法则。",
      "prerequisite": []
    },
    {
      "id": "K2-2-01",
      "name": "基本不等式",
      "type": "定理",
      "definition": "对任意两个正数a, b，有 (a+b)/2 ≥ √ab，当且仅当 a=b 时等号成立。即两个正数的算术平均数不小于它们的几何平均数。",
      "prerequisite": [
        "K2-1-01"
      ]
    },
    {
      "id": "K2-3-01",
      "name": "一元二次不等式",
      "type": "概念",
      "definition": "含有一个未知数，并且未知数的最高次数是2的不等式，如 ax^2+bx+c>0 或 ax^2+bx+c<0 (a≠0)。",
      "prerequisite": []
    },
    {
      "id": "K3-1-1-01",
      "name": "函数",
      "type": "概念",
      "definition": "设A, B是非空的实数集，如果对于集合A中的任意一个数x，按照某种确定的对应关系f，在集合B中都有唯一确定的数y和它对应，那么就称f: A -> B为从集合A到集合B的一个函数，记作 y=f(x), x∈A。",
      "prerequisite": [
        "K1-1-01"
      ]
    },
    {
      "id": "K3-1-1-02",
      "name": "定义域",
      "type": "概念",
      "definition": "函数自变量x的取值范围A叫做函数的定义域。",
      "prerequisite": [
        "K3-1-1-01"
      ]
    },
    {
      "id": "K3-1-1-03",
      "name": "值域",
      "type": "概念",
      "definition": "函数值的集合 {f(x)|x∈A} 叫做函数的值域。",
      "prerequisite": [
        "K3-1-1-01"
      ]
    },
    {
      "id": "K3-1-2-01",
      "name": "函数的表示法",
      "type": "概念",
      "definition": "表示函数关系的方法，主要有解析法、列表法、图象法。",
      "prerequisite": [
        "K3-1-1-01"
      ]
    },
    {
      "id": "K3-1-2-02",
      "name": "分段函数",
      "type": "概念",
      "definition": "在定义域的不同部分，有不同的对应关系来表示的函数。",
      "prerequisite": [
        "K3-1-1-01"
      ]
    },
    {
      "id": "K3-2-1-01",
      "name": "函数的单调性",
      "type": "概念",
      "definition": "描述函数值随自变量增大而增大（单调递增）或减小（单调递减）的性质。",
      "prerequisite": [
        "K3-1-1-01"
      ]
    },
    {
      "id": "K3-2-1-02",
      "name": "函数的最大(小)值",
      "type": "概念",
      "definition": "函数在其定义域上所有函数值中的最大值或最小值。",
      "prerequisite": [
        "K3-1-1-01"
      ]
    },
    {
      "id": "K3-2-2-01",
      "name": "函数的奇偶性",
      "type": "概念",
      "definition": "函数的图象关于y轴对称（偶函数）或关于原点对称（奇函数）的性质。偶函数满足f(-x)=f(x)，奇函数满足f(-x)=-f(x)。",
      "prerequisite": [
        "K3-1-1-01"
      ]
    },
    {
      "id": "K3-3-01",
      "name": "幂函数",
      "type": "概念",
      "definition": "形如 y=x^α (α是常数) 的函数称为幂函数，其中x是自变量。",
      "prerequisite": [
        "K3-1-1-01"
      ]
    },
    {
      "id": "K4-1-1-01",
      "name": "n次方根",
      "type": "概念",
      "definition": "如果x^n = a，那么x叫做a的n次方根，其中n>1且n∈N*。",
      "prerequisite": []
    },
    {
      "id": "K4-1-1-02",
      "name": "分数指数幂",
      "type": "概念",
      "definition": "规定a^(m/n) = n√a^m (a>0, m, n∈N*, n>1) 和 a^(-m/n) = 1/a^(m/n) (a>0, m, n∈N*, n>1)。",
      "prerequisite": [
        "K4-1-1-01"
      ]
    },
    {
      "id": "K4-1-2-01",
      "name": "实数指数幂",
      "type": "概念",
      "definition": "将指数的取值范围从有理数推广到实数。无理数指数幂a^α (a>0) 是一个确定的实数。",
      "prerequisite": [
        "K4-1-1-02"
      ]
    },
    {
      "id": "K4-2-1-01",
      "name": "指数函数",
      "type": "概念",
      "definition": "函数 y=a^x (a>0, 且a≠1) 叫做指数函数，其中x是自变量，定义域是R。",
      "prerequisite": [
        "K4-1-2-01",
        "K3-1-1-01"
      ]
    },
    {
      "id": "K4-2-2-01",
      "name": "指数函数的图像与性质",
      "type": "定理",
      "definition": "当a>1时，指数函数是增函数；当0<a<1时，指数函数是减函数。图像都过点(0,1)，值域为(0, +∞)。",
      "prerequisite": [
        "K4-2-1-01"
      ]
    },
    {
      "id": "K4-3-1-01",
      "name": "对数",
      "type": "概念",
      "definition": "如果 a^x=N (a>0, 且a≠1)，那么数x叫做以a为底N的对数，记作x=log_a(N)。",
      "prerequisite": [
        "K4-1-2-01"
      ]
    },
    {
      "id": "K4-3-2-01",
      "name": "对数的运算法则",
      "type": "公式",
      "definition": "log_a(MN) = log_a(M) + log_a(N); log_a(M/N) = log_a(M) - log_a(N); log_a(M^n) = n*log_a(M)。",
      "prerequisite": [
        "K4-3-1-01"
      ]
    },
    {
      "id": "K4-3-2-02",
      "name": "换底公式",
      "type": "公式",
      "definition": "log_a(b) = log_c(b) / log_c(a) (a>0, a≠1; c>0, c≠1; b>0)。",
      "prerequisite": [
        "K4-3-1-01"
      ]
    },
    {
      "id": "K4-4-1-01",
      "name": "对数函数",
      "type": "概念",
      "definition": "函数 y = log_a(x) (a>0, 且a≠1) 叫做对数函数，其中x是自变量，定义域是(0, +∞)。",
      "prerequisite": [
        "K4-3-1-01",
        "K3-1-1-01"
      ]
    },
    {
      "id": "K4-4-2-01",
      "name": "对数函数的图像与性质",
      "type": "定理",
      "definition": "当a>1时，对数函数是增函数；当0<a<1时，对数函数是减函数。图像都过点(1,0)，值域为R。",
      "prerequisite": [
        "K4-4-1-01"
      ]
    },
    {
      "id": "K4-4-2-02",
      "name": "反函数",
      "type": "概念",
      "definition": "指数函数y=a^x与对数函数y=log_a(x)互为反函数，它们的图像关于直线y=x对称。",
      "prerequisite": [
        "K4-2-1-01",
        "K4-4-1-01"
      ]
    },
    {
      "id": "K4-5-1-01",
      "name": "函数零点",
      "type": "概念",
      "definition": "对于函数 y=f(x)，我们把使f(x)=0的实数x叫做函数y=f(x)的零点。",
      "prerequisite": [
        "K3-1-1-01"
      ]
    },
    {
      "id": "K4-5-1-02",
      "name": "函数零点存在定理",
      "type": "定理",
      "definition": "如果函数y=f(x)在区间[a,b]上的图象是一条连续不断的曲线，且有f(a)f(b)<0，那么，函数y=f(x)在区间(a,b)内至少有一个零点。",
      "prerequisite": [
        "K4-5-1-01"
      ]
    },
    {
      "id": "K5-1-1-01",
      "name": "任意角",
      "type": "概念",
      "definition": "角可以看成一条射线绕着它的端点旋转所形成的图形。按逆时针方向旋转形成的角叫正角，顺时针方向的叫负角，不旋转为零角。",
      "prerequisite": []
    },
    {
      "id": "K5-1-1-02",
      "name": "象限角",
      "type": "概念",
      "definition": "在直角坐标系中，使角的顶点与原点重合，始边与x轴非负半轴重合，角的终边在第几象限，就说这个角是第几象限角。",
      "prerequisite": [
        "K5-1-1-01"
      ]
    },
    {
      "id": "K5-1-1-03",
      "name": "终边相同的角",
      "type": "定理",
      "definition": "所有与角α终边相同的角，连同角α在内，可构成一个集合S = {β | β = α + k * 360°, k ∈ Z}。",
      "prerequisite": [
        "K5-1-1-01"
      ]
    },
    {
      "id": "K5-1-2-01",
      "name": "弧度制",
      "type": "概念",
      "definition": "长度等于半径长的圆弧所对的圆心角叫做1弧度的角。",
      "prerequisite": []
    },
    {
      "id": "K5-1-2-02",
      "name": "角度制与弧度制的换算",
      "type": "公式",
      "definition": "180° = π rad. 由此可得 1° = π/180 rad, 1 rad = (180/π)°.",
      "prerequisite": [
        "K5-1-2-01"
      ]
    },
    {
      "id": "K5-1-2-03",
      "name": "扇形弧长与面积公式（弧度制）",
      "type": "公式",
      "definition": "设扇形半径为R，圆心角为α rad，弧长为l，面积为S。则 l = αR, S = (1/2)αR^2 = (1/2)lR。",
      "prerequisite": [
        "K5-1-2-01"
      ]
    },
    {
      "id": "K5-2-1-01",
      "name": "三角函数定义",
      "type": "概念",
      "definition": "设α是一个任意角，它的终边与单位圆相交于点P(x,y)。则sinα=y, cosα=x, tanα=y/x (x≠0)。",
      "prerequisite": [
        "K5-1-1-01"
      ]
    },
    {
      "id": "K5-2-1-02",
      "name": "三角函数在各象限的符号",
      "type": "定理",
      "definition": "第一象限全为正；第二象限正弦为正；第三象限正切为正；第四象限余弦为正。",
      "prerequisite": [
        "K5-2-1-01"
      ]
    },
    {
      "id": "K5-2-2-01",
      "name": "同角三角函数的基本关系",
      "type": "公式",
      "definition": "sin²α + cos²α = 1; tanα = sinα / cosα。",
      "prerequisite": [
        "K5-2-1-01"
      ]
    },
    {
      "id": "K5-3-01",
      "name": "诱导公式",
      "type": "公式",
      "definition": "终边相同的角的同一三角函数的值相等。sin(α+2kπ)=sinα, cos(α+2kπ)=cosα, tan(α+2kπ)=tanα, (k∈Z)。以及其他关于π±α, -α, π/2±α, 3π/2±α的变换公式。",
      "prerequisite": [
        "K5-2-1-01"
      ]
    },
    {
      "id": "K5-4-1-01",
      "name": "正弦函数与余弦函数的图象",
      "type": "概念",
      "definition": "正弦函数的图象叫正弦曲线，余弦函数的图象叫余弦曲线，都是“波浪起伏”的连续光滑曲线。",
      "prerequisite": [
        "K5-2-1-01"
      ]
    },
    {
      "id": "K5-4-2-01",
      "name": "周期函数",
      "type": "概念",
      "definition": "设函数f(x)的定义域为D，如果存在一个非零常数T，使得对每一个x∈D都有x+T∈D，且f(x+T)=f(x)，那么函数f(x)就叫做周期函数，T叫做这个函数的周期。",
      "prarameters": [
        "K3-1-1-01"
      ]
    },
    {
      "id": "K5-4-2-02",
      "name": "正弦函数、余弦函数的性质",
      "type": "定理",
      "definition": "定义域为R，值域为[-1,1]，最小正周期为2π。正弦函数是奇函数，余弦函数是偶函数。在特定区间内具有单调性。",
      "prerequisite": [
        "K5-4-1-01",
        "K5-4-2-01",
        "K3-2-1-01",
        "K3-2-2-01"
      ]
    },
    {
      "id": "K5-4-3-01",
      "name": "正切函数的图象与性质",
      "type": "定理",
      "definition": "定义域为{x|x≠π/2+kπ, k∈Z}，值域为R，最小正周期为π，是奇函数。在每个开区间(-π/2+kπ, π/2+kπ)上单调递增。",
      "prerequisite": [
        "K5-2-1-01",
        "K5-4-2-01",
        "K3-2-1-01",
        "K3-2-2-01"
      ]
    },
    {
      "id": "K5-5-1-01",
      "name": "两角和与差的三角函数公式",
      "type": "公式",
      "definition": "cos(α±β) = cosαcosβ ∓ sinαsinβ; sin(α±β) = sinαcosβ ± cosαsinβ; tan(α±β) = (tanα ± tanβ) / (1 ∓ tanαtanβ).",
      "prerequisite": [
        "K5-2-1-01"
      ]
    },
    {
      "id": "K5-5-3-01",
      "name": "二倍角公式",
      "type": "公式",
      "definition": "sin(2α)=2sinαcosα; cos(2α)=cos²α-sin²α=2cos²α-1=1-2sin²α; tan(2α)=2tanα/(1-tan²α).",
      "prerequisite": [
        "K5-5-1-01"
      ]
    },
    {
      "id": "K5-6-01",
      "name": "函数y=Asin(ωx+φ)的图象",
      "type": "概念",
      "definition": "由正弦曲线y=sinx经过平移和伸缩变换得到。A是振幅，T=2π/ω是周期，f=1/T是频率，ωx+φ是相位，φ是初相。",
      "prerequisite": [
        "K5-4-1-01"
      ]
    }
  ],
  "methods": [
    {
      "id": "M2-3-01",
      "name": "一元二次不等式求解",
      "type": "解题方法",
      "steps": [
        "将不等式化为 ax^2+bx+c>0 (或<0) 且 a>0 的形式。",
        "计算判别式 Δ = b^2-4ac。",
        "根据Δ的符号判断对应方程 ax^2+bx+c=0 的根的情况。",
        "结合二次函数 y=ax^2+bx+c 的图象，确定不等式的解集。"
      ],
      "required_knowledge": [
        "K2-3-01"
      ]
    },
    {
      "id": "M3-2-01",
      "name": "函数单调性的证明方法（定义法）",
      "type": "证明方法",
      "steps": [
        "设x1, x2是区间I内的任意两个值，且x1 < x2。",
        "作差 f(x1) - f(x2)。",
        "对差式进行变形，通常是因式分解、配方等。",
        "判断差 f(x1) - f(x2) 的符号。",
        "根据定义得出结论：若f(x1) < f(x2)，则函数在I上单调递增；若f(x1) > f(x2)，则函数在I上单调递-减。"
      ],
      "required_knowledge": [
        "K3-2-1-01"
      ]
    },
    {
      "id": "M3-2-02",
      "name": "函数奇偶性的判断方法",
      "type": "解题方法",
      "steps": [
        "首先确定函数的定义域是否关于原点对称。",
        "计算f(-x)的表达式。",
        "比较f(-x)与f(x)的关系：如果f(-x)=f(x)，则是偶函数；如果f(-x)=-f(x)，则是奇函数；否则为非奇非偶函数。"
      ],
      "required_knowledge": [
        "K3-2-2-01"
      ]
    },
    {
      "id": "M4-5-01",
      "name": "二分法求方程近似解",
      "type": "解题方法",
      "steps": [
        "确定零点所在的初始区间[a,b]，验证f(a)f(b)<0。",
        "求区间(a,b)的中点c。",
        "计算f(c)。若f(c)=0，则c是零点；若f(a)f(c)<0，则令b=c；若f(c)f(b)<0，则令a=c。",
        "判断区间长度是否小于给定的精确度ε。若|a-b|<ε，则得到近似解；否则重复步骤2-4。"
      ],
      "required_knowledge": [
        "K4-5-1-01",
        "K4-5-1-02"
      ]
    },
    {
      "id": "M5-6-01",
      "name": "五点法画正弦/余弦函数简图",
      "type": "解题方法",
      "steps": [
        "令 z = ωx+φ。",
        "找出z取 0, π/2, π, 3π/2, 2π 时对应的五个关键点。",
        "计算这五个关键点对应的x值。",
        "计算这五个关键点对应的y值。",
        "在坐标系中描出这五个点，并用光滑的曲线连接起来。"
      ],
      "required_knowledge": [
        "K5-4-1-01",
        "K5-6-01"
      ]
    },
    {
      "id": "M5-6-02",
      "name": "由y=sinx图像得到y=Asin(ωx+φ)图像的变换步骤",
      "type": "解题方法",
      "steps": [
        "相位变换：将y=sinx的图像向左(φ>0)或向右(φ<0)平移|φ|个单位，得到y=sin(x+φ)的图像。",
        "周期变换：将y=sin(x+φ)的图像上所有点的横坐标缩短(ω>1)或伸长(0<ω<1)到原来的1/ω倍，得到y=sin(ωx+φ)的图像。",
        "振幅变换：将y=sin(ωx+φ)的图像上所有点的纵坐标伸长(A>1)或缩短(0<A<1)到原来的A倍，得到y=Asin(ωx+φ)的图像。"
      ],
      "required_knowledge": [
        "K5-6-01"
      ]
    }
  ],
  "problems": [
    {
      "id": "T1-1-E01",
      "type": "例题",
      "content": "用列举法表示下列集合：(1) 小于10的所有自然数组成的集合；(2) 方程x²=x的所有实数根组成的集合。",
      "knowledge": [
        "K1-1-05",
        "K1-1-06"
      ],
      "methods": [
        "M1-1-02"
      ]
    },
    {
      "id": "T1-1-E02",
      "type": "例题",
      "content": "试分别用描述法和列举法表示下列集合: (1) 方程 x²-2=0 的所有实数根组成的集合 A; (2) 由大于 10 且小于 20 的所有整数组成的集合 B.",
      "knowledge": [
        "K1-1-06",
        "K1-1-07"
      ],
      "methods": [
        "M1-1-02",
        "M1-1-03"
      ]
    },
    {
      "id": "T1-2-E01",
      "type": "例题",
      "content": "写出集合 {a, b} 的所有子集，并指出哪些是它的真子集。",
      "knowledge": [
        "K1-2-01",
        "K1-2-03",
        "K1-2-04"
      ],
      "methods": []
    },
    {
      "id": "T1-2-E02",
      "type": "例题",
      "content": "判断下列各题中集合 A 是否为集合 B 的子集，并说明理由：(1) A={1, 2, 3}, B={x|x 是 8 的约数}; (2) A={x|x 是长方形}, B={x|x 是两条对角线相等的平行四边形}。",
      "knowledge": [
        "K1-2-01"
      ],
      "methods": [
        "M1-2-01"
      ]
    },
    {
      "id": "T1-3-E01",
      "type": "例题",
      "content": "设 A={4, 5, 6, 8}, B={3, 5, 7, 8}, 求 A ∪ B.",
      "knowledge": [
        "K1-3-01"
      ],
      "methods": [
        "M1-3-01"
      ]
    },
    {
      "id": "T1-3-E02",
      "type": "例题",
      "content": "设集合 A={x | -1 < x < 2}, 集合 B={x | 1 < x < 3}, 求 A ∪ B.",
      "knowledge": [
        "K1-3-01"
      ],
      "methods": [
        "M1-3-01"
      ]
    },
    {
      "id": "T1-3-E03",
      "type": "例题",
      "content": "立德中学开运动会，设 A={x|x 是立德中学高一年级参加百米赛跑的同学}, B={x|x 是立德中学高一年级参加跳高比赛的同学}, 求 A ∩ B。",
      "knowledge": [
        "K1-3-02"
      ],
      "methods": [
        "M1-3-02"
      ]
    },
    {
      "id": "T1-3-E04",
      "type": "例题",
      "content": "设平面内直线l1上点的集合为L1，直线l2上点的集合为L2，试用集合的运算表示l1, l2的位置关系。",
      "knowledge": [
        "K1-3-02",
        "K1-2-04"
      ],
      "methods": [
        "M1-3-02"
      ]
    },
    {
      "id": "T1-3-E05",
      "type": "例题",
      "content": "设 U = {x | x 是小于 9 的正整数}, A = {1, 2, 3}, B = {3, 4, 5, 6}，求 C_U A, C_U B.",
      "knowledge": [
        "K1-3-04"
      ],
      "methods": [
        "M1-3-03"
      ]
    },
    {
      "id": "T1-3-E06",
      "type": "例题",
      "content": "设全集 U = {x | x 是三角形}, A = {x | x 是锐角三角形}, B = {x | x 是钝角三角形}，求 A ∩ B, C_U (A ∪ B).",
      "knowledge": [
        "K1-3-01",
        "K1-3-02",
        "K1-3-04"
      ],
      "methods": [
        "M1-3-01",
        "M1-3-02",
        "M1-3-03"
      ]
    },
    {
      "id": "T1-4-E01",
      "type": "例题",
      "content": "下列“若 p, 则 q”形式的命题中, 哪些命题中的 p 是 q 的充分条件? (1) 若四边形的两组对角分别相等, 则这个四边形是平行四边形; (2) 若两个三角形的三边成比例, 则这两个三角形相似; (3) 若四边形为菱形, 则这个四边形的对角线互相垂直; (4) 若 x²=1, 则 x=1; (5) 若 a=b, 则 ac=bc; (6) 若 x, y 为无理数, 则 xy 为无理数.",
      "knowledge": [
        "K1-4-02"
      ],
      "methods": [
        "M1-4-01"
      ]
    },
    {
      "id": "T1-4-E02",
      "type": "例题",
      "content": "下列“若p，则q”形式的命题中，哪些命题中的q是p的必要条件？(1) 若四边形为平行四边形，则这个四边形的两组对角分别相等；(2) 若两个三角形相似，则这两个三角形的三边成比例；(3) 若四边形的对角线互相垂直，则这个四边形是菱形；(4) 若x=1，则x²=1；(5) 若ac=bc，则a=b；(6) 若xy为无理数，则x,y为无理数。",
      "knowledge": [
        "K1-4-03"
      ],
      "methods": [
        "M1-4-01"
      ]
    },
    {
      "id": "T1-4-E03",
      "type": "例题",
      "content": "下列各题中，哪些 p 是 q 的充要条件? (1) p: 四边形是正方形，q: 四边形的对角线互相垂直且平分; (2) p: 两个三角形相似，q: 两个三角形三边成比例; (3) p: xy>0，q: x>0，y>0; (4) p: x=1 是一元二次方程 ax²+bx+c=0 的一个根，q: a+b+c=0 (a ≠ 0).",
      "knowledge": [
        "K1-4-04"
      ],
      "methods": [
        "M1-4-01"
      ]
    },
    {
      "id": "T1-4-E04",
      "type": "例题",
      "content": "已知：☉O 的半径为r，圆心O到直线l的距离为d。求证：d=r是直线l与☉O相切的充要条件。",
      "knowledge": [
        "K1-4-04"
      ],
      "methods": [
        "M1-4-01"
      ]
    },
    {
      "id": "T1-5-E01",
      "type": "例题",
      "content": "判断下列全称量词命题的真假: (1) 所有的素数都是奇数; (2) ∀ x ∈ R, |x|+1 ≥ 1; (3) 对任意一个无理数 x, x² 也是无理数.",
      "knowledge": [
        "K1-5-02"
      ],
      "methods": []
    },
    {
      "id": "T1-5-E02",
      "type": "例题",
      "content": "判断下列存在量词命题的真假: (1) 有一个实数 x，使 x²+2x+3=0; (2) 平面内存在两条相交直线垂直于同一条直线; (3) 有些平行四边形是菱形.",
      "knowledge": [
        "K1-5-04"
      ],
      "methods": []
    },
    {
      "id": "T1-5-E03",
      "type": "例题",
      "content": "写出下列全称量词命题的否定: (1) 所有能被3整除的整数都是奇数; (2) 每一个四边形的四个顶点在同一个圆上; (3) 对任意x ∈ Z, x²的个位数字不等于3.",
      "knowledge": [
        "K1-5-05"
      ],
      "methods": []
    },
    {
      "id": "T1-5-E04",
      "type": "例题",
      "content": "写出下列存在量词命题的否定: (1) ∃ x ∈ R, x+2 ≤ 0; (2) 有的三角形是等边三角形; (3) 有一个偶数是素数.",
      "knowledge": [
        "K1-5-06"
      ],
      "methods": []
    },
    {
      "id": "T1-5-E05",
      "type": "例题",
      "content": "写出下列命题的否定,并判断真假: (1) 任意两个等边三角形都相似; (2) ∃ x ∈ R, x²-x+1=0.",
      "knowledge": [
        "K1-5-05",
        "K1-5-06"
      ],
      "methods": []
    },
    {
      "id": "T2-1-E01",
      "type": "例题",
      "content": "比较(x+2)(x+3)和(x+1)(x+4)的大小.",
      "knowledge": [
        "K2-1-01"
      ],
      "methods": []
    },
    {
      "id": "T2-1-E02",
      "type": "例题",
      "content": "已知 a>b>0, c<0,求证 c/a > c/b.",
      "knowledge": [
        "K2-1-01"
      ],
      "methods": []
    },
    {
      "id": "T2-2-E01",
      "type": "例题",
      "content": "已知 x>0, 求 x + 1/x 的最小值。",
      "knowledge": [
        "K2-2-01"
      ],
      "methods": []
    },
    {
      "id": "T2-2-E02",
      "type": "例题",
      "content": "已知 x,y 都是正数，求证: (1) 如果积 xy 等于定值 P, 那么当 x=y 时，和 x+y 有最小值 2√P; (2) 如果和 x+y 等于定值 S, 那么当 x=y 时, 积 xy 有最大值 S²/4.",
      "knowledge": [
        "K2-2-01"
      ],
      "methods": []
    },
    {
      "id": "T2-2-E03",
      "type": "例题",
      "content": "(1) 用篱笆围一个面积为100 m²的矩形菜园, 当这个矩形的边长为多少时, 所用篱笆最短? 最短篱笆的长度是多少? (2) 用一段长为36 m的篱笆围成一个矩形菜园, 当这个矩形的边长为多少时, 菜园的面积最大? 最大面积是多少?",
      "knowledge": [
        "K2-2-01"
      ],
      "methods": []
    },
    {
      "id": "T2-2-E04",
      "type": "例题",
      "content": "某工厂要建造一个长方体形无盖贮水池,其容积为 4800 m³,深为 3 m.如果池底每平方米的造价为 150 元,池壁每平方米的造价为 120 元,那么怎样设计水池能使总造价最低?最低总造价是多少?",
      "knowledge": [
        "K2-2-01"
      ],
      "methods": []
    },
    {
      "id": "T2-3-E01",
      "type": "例题",
      "content": "求不等式 x²-5x+6>0 的解集.",
      "knowledge": [
        "K2-3-01"
      ],
      "methods": [
        "M2-3-01"
      ]
    },
    {
      "id": "T2-3-E02",
      "type": "例题",
      "content": "求不等式 9x²-6x+1>0 的解集.",
      "knowledge": [
        "K2-3-01"
      ],
      "methods": [
        "M2-3-01"
      ]
    },
    {
      "id": "T2-3-E03",
      "type": "例题",
      "content": "求不等式 -x²+2x-3>0 的解集.",
      "knowledge": [
        "K2-3-01"
      ],
      "methods": [
        "M2-3-01"
      ]
    },
    {
      "id": "T2-3-E04",
      "type": "例题",
      "content": "一家车辆制造厂引进了一条摩托车整车装配流水线, 这条流水线生产的摩托车数量 x (单位: 辆) 与创造的价值 y (单位: 元) 之间有如下的关系: y=-20x²+2200x. 若这家工厂希望在一个星期内利用这条流水线创收60000元以上, 则在一个星期内大约应该生产多少辆摩托车?",
      "knowledge": [
        "K2-3-01"
      ],
      "methods": [
        "M2-3-01"
      ]
    },
    {
      "id": "T2-3-E05",
      "type": "例题",
      "content": "某种汽车在水泥路面上的刹车距离 s (单位：m) 和汽车刹车前的车速 v (单位：km/h) 之间有如下关系：s=v/20+v²/180。在一次交通事故中，测得这种车的刹车距离大于 39.5 m，那么这辆汽车刹车前的车速至少为多少 (精确到 1 km/h)?",
      "knowledge": [
        "K2-3-01"
      ],
      "methods": [
        "M2-3-01"
      ]
    },
    {
      "id": "T3-1-E01",
      "type": "例题",
      "content": "试构建一个问题情境,使其中的变量关系可以用解析式 y=x(10-x) 来描述.",
      "knowledge": [
        "K3-1-1-01",
        "K3-1-1-02",
        "K3-1-1-03"
      ],
      "methods": []
    },
    {
      "id": "T3-1-E02",
      "type": "例题",
      "content": "已知函数 f(x)=√(x+3) + 1/(x+2)，(1) 求函数的定义域；(2) 求 f(-3), f(2/3) 的值；(3) 当 a ≥ 0 时,求 f(a), f(a-1) 的值。",
      "knowledge": [
        "K3-1-1-02"
      ],
      "methods": []
    },
    {
      "id": "T3-1-E03",
      "type": "例题",
      "content": "下列函数中哪个与函数 y=x 是同一个函数？(1) y=(√x)²; (2) u=∛(v³); (3) y=√x²; (4) m=n²/n.",
      "knowledge": [
        "K3-1-1-01",
        "K3-1-1-02"
      ],
      "methods": []
    },
    {
      "id": "T3-1-E04",
      "type": "例题",
      "content": "某种笔记本的单价是5元, 买 x (x ∈ {1, 2, 3, 4, 5}) 个笔记本需要 y 元. 试用函数的三种表示法表示函数 y=f(x).",
      "knowledge": [
        "K3-1-2-01"
      ],
      "methods": []
    },
    {
      "id": "T3-1-E05",
      "type": "例题",
      "content": "画出函数y=|x|的图象.",
      "knowledge": [
        "K3-1-2-01",
        "K3-1-2-02"
      ],
      "methods": []
    },
    {
      "id": "T3-1-E06",
      "type": "例题",
      "content": "给定函数 f(x)=x+1, g(x)=(x+1)², x ∈ R, 1. 在同一直角坐标系中画出函数 f(x), g(x) 的图象; 2. ∀ x ∈ R, 用 M(x) 表示 f(x), g(x) 中的最大者, 记为 M(x)=max{f(x), g(x)}. 请分别用图象法和解析法表示函数 M(x).",
      "knowledge": [
        "K3-1-2-01",
        "K3-1-2-02"
      ],
      "methods": []
    },
    {
      "id": "T3-1-E07",
      "type": "例题",
      "content": "表3.1-4 是某校高一(1)班三名同学在高一学年度六次数学测试的成绩及班级平均分表. 请你对这三位同学在高一学年的数学学习情况做一个分析.",
      "knowledge": [
        "K3-1-2-01"
      ],
      "methods": []
    },
    {
      "id": "T3-1-E08",
      "type": "例题",
      "content": "依法纳税是每个公民应尽的义务。根据2019年1月1日起的《中华人民共和国个人所得税法》规定，个税税额 = 应纳税所得额 × 税率 - 速算扣除数。应纳税所得额 = 综合所得收入额 - 60000 - 专项扣除 - 专项附加扣除 - 其他扣除。税率表如文所示。 (1) 设全年应纳税所得额为t，应缴纳个税税额为y，求y=f(t)，并画出图象； (2) 小王全年综合所得收入额为117600元，社保公积金占收入额20%，专项附加扣除9600元，其他扣除560元，求他全年应缴纳的个税。",
      "knowledge": [
        "K3-1-2-01",
        "K3-1-2-02"
      ],
      "methods": []
    },
    {
      "id": "T3-2-E01",
      "type": "例题",
      "content": "根据定义,研究函数 f(x)=kx+b(k ≠ 0) 的单调性.",
      "knowledge": [
        "K3-2-1-01"
      ],
      "methods": [
        "M3-2-01"
      ]
    },
    {
      "id": "T3-2-E02",
      "type": "例题",
      "content": "物理学中的玻意耳定律 p=k/V (k为正常数) 告诉我们,对于一定质量的气体,当其温度不变时,体积V减小,压强p将增大.试对此用函数的单调性证明.",
      "knowledge": [
        "K3-2-1-01"
      ],
      "methods": [
        "M3-2-01"
      ]
    },
    {
      "id": "T3-2-E03",
      "type": "例题",
      "content": "根据定义证明函数 y=x + 1/x 在区间 (1, +∞) 上单调递增.",
      "knowledge": [
        "K3-2-1-01"
      ],
      "methods": [
        "M3-2-01"
      ]
    },
    {
      "id": "T3-2-E04",
      "type": "例题",
      "content": "“菊花”烟花是最壮观的烟花之一, 制造时一般是期望在它达到最高点时爆裂. 如果烟花距地面的高度 h (单位: m) 与时间 t (单位: s) 之间的关系为 h(t)=-4.9t²+14.7t+18, 那么烟花冲出后什么时候是它爆裂的最佳时刻? 这时距地面的高度是多少 (精确到 1m)?",
      "knowledge": [
        "K3-2-1-02"
      ],
      "methods": []
    },
    {
      "id": "T3-2-E05",
      "type": "例题",
      "content": "已知函数 f(x) = 2/(x-1) (x ∈ [2, 6]), 求函数的最大值和最小值.",
      "knowledge": [
        "K3-2-1-01",
        "K3-2-1-02"
      ],
      "methods": [
        "M3-2-01"
      ]
    },
    {
      "id": "T3-2-E06",
      "type": "例题",
      "content": "判断下列函数的奇偶性：(1) f(x)=x⁴; (2) f(x)=x⁵; (3) f(x)=x+1/x; (4) f(x)=1/x².",
      "knowledge": [
        "K3-2-2-01"
      ],
      "methods": [
        "M3-2-02"
      ]
    },
    {
      "id": "T3-3-E01",
      "type": "例题",
      "content": "证明幂函数 f(x)=√x 是增函数。",
      "knowledge": [
        "K3-3-01",
        "K3-2-1-01"
      ],
      "methods": [
        "M3-2-01"
      ]
    },
    {
      "id": "T3-4-E01",
      "type": "例题",
      "content": "设小王的专项扣除比例、专项附加扣除金额、依法确定的其他扣除金额与3.1.2例8相同,全年综合所得收入额为x(单位:元),应缴纳综合所得个税税额为y(单位:元). (1) 求y关于x的函数解析式; (2) 如果小王全年的综合所得由117 600元增加到153 600元,那么他全年应缴纳多少综合所得个税?",
      "knowledge": [
        "K3-1-2-02"
      ],
      "methods": []
    },
    {
      "id": "T3-4-E02",
      "type": "例题",
      "content": "一辆汽车在某段路程中行驶的平均速率 v (单位: km/h) 与时间 t (单位: h) 的关系如图3.4-1 所示，(1) 求图3.4-1中阴影部分的面积，并说明所求面积的实际含义; (2) 假设这辆汽车的里程表在汽车行驶这段路程前的读数为2004 km，试建立行驶这段路程时汽车里程表读数 s (单位: km) 与时间 t 的函数解析式，并画出相应的图象.",
      "knowledge": [
        "K3-1-2-01",
        "K3-1-2-02"
      ],
      "methods": []
    },
    {
      "id": "T4-1-E01",
      "type": "例题",
      "content": "求下列各式的值: (1) ³√((-8)³); (2) √((-10)²); (3) ⁴√((3-π)⁴); (4) √((a-b)²).",
      "knowledge": [
        "K4-1-1-01"
      ],
      "methods": []
    },
    {
      "id": "T4-1-E02",
      "type": "例题",
      "content": "求值: (1) 8^(2/3); (2) (16/81)^(-3/4)。",
      "knowledge": [
        "K4-1-1-02"
      ],
      "methods": []
    },
    {
      "id": "T4-1-E03",
      "type": "例题",
      "content": "用分数指数幂的形式表示并计算下列各式(其中a>0): (1) a² · ³√a²; (2) √(a ³√a)。",
      "knowledge": [
        "K4-1-1-02"
      ],
      "methods": []
    },
    {
      "id": "T4-1-E04",
      "type": "例题",
      "content": "计算下列各式(式中字母均是正数): (1) (2a^(2/3)b^(1/2))(-6a^(1/2)b^(1/3)) ÷ (-3a^(1/6)b^(5/6)); (2) (m^(1/4)n^(-3/8))^8; (3) (³√a² - √a) ÷ ⁴√a².",
      "knowledge": [
        "K4-1-1-02"
      ],
      "methods": []
    },
    {
      "id": "T4-2-E01",
      "type": "例题",
      "content": "已知指数函数 f(x)=a^x (a>0, 且 a≠1), 且 f(3)=π, 求 f(0), f(1), f(-3) 的值.",
      "knowledge": [
        "K4-2-1-01"
      ],
      "methods": []
    },
    {
      "id": "T4-2-E02",
      "type": "例题",
      "content": "(1)在问题1中, 如果平均每位游客出游一次可给当地带来1000元(不含门票)的收入, A地景区的门票价格为150元, 比较这15年间 A, B两地旅游收入变化情况. (2)在问题2中, 某生物死亡10000年后, 它体内碳14的含量衰减为原来的百分之几?",
      "knowledge": [
        "K4-2-1-01"
      ],
      "methods": []
    },
    {
      "id": "T4-2-E03",
      "type": "例题",
      "content": "比较下列各题中两个值的大小：(1) 1.7^2.5, 1.7^3; (2) 0.8^-√2, 0.8^-√3; (3) 1.7^0.3, 0.9^3.1.",
      "knowledge": [
        "K4-2-2-01"
      ],
      "methods": []
    },
    {
      "id": "T4-2-E04",
      "type": "例题",
      "content": "如图4.2-7, 某城市人口呈指数增长。 (1) 根据图象，估计该城市人口每翻一番所需的时间 (倍增期)； (2) 该城市人口从80万人开始，经过20年会增长到多少万人？",
      "knowledge": [
        "K4-2-1-01"
      ],
      "methods": []
    },
    {
      "id": "T4-3-E01",
      "type": "例题",
      "content": "把下列指数式化为对数式，对数式化为指数式：(1) 5⁴=625; (2) 2⁻⁶=1/64; (3) (1/3)ᵐ=5.73; (4) log₁/₂16 = -4; (5) lg 0.01 = -2; (6) ln 10 = n.",
      "knowledge": [
        "K4-3-1-01"
      ],
      "methods": []
    },
    {
      "id": "T4-3-E02",
      "type": "例题",
      "content": "求下列各式中 x 的值: (1) log₆₄ x = -2/3; (2) logₓ 8 = 6; (3) lg 100 = x; (4) -ln e² = x.",
      "knowledge": [
        "K4-3-1-01"
      ],
      "methods": []
    },
    {
      "id": "T4-3-E03",
      "type": "例题",
      "content": "求下列各式的值: (1) lg ⁵√100; (2) log₂(4⁷ × 2⁵).",
      "knowledge": [
        "K4-3-2-01"
      ],
      "methods": []
    },
    {
      "id": "T4-3-E04",
      "type": "例题",
      "content": "用 lnx, lny, lnz 表示 ln(x²√y / ³√z).",
      "knowledge": [
        "K4-3-2-01"
      ],
      "methods": []
    },
    {
      "id": "T4-3-E05",
      "type": "例题",
      "content": "地震时释放出的能量E(单位:焦耳)与地震里氏震级M之间的关系为 lgE=4.8+1.5M. 2011年3月11日,日本东北部海域发生里氏9.0级地震,它所释放出来的能量是2008年5月12日我国汶川发生里氏8.0级地震的多少倍(精确到1)?",
      "knowledge": [
        "K4-3-1-01",
        "K4-3-2-01"
      ],
      "methods": []
    },
    {
      "id": "T4-4-E01",
      "type": "例题",
      "content": "求下列函数的定义域: (1) y=log₃x²; (2) y=logₐ(4-x) (a>0, 且 a≠1).",
      "knowledge": [
        "K4-4-1-01"
      ],
      "methods": []
    },
    {
      "id": "T4-4-E02",
      "type": "例题",
      "content": "假设某地初始物价为1，每年以5%的增长率递增，经过 t 年后的物价为 w。 (1) 该地的物价经过几年后会翻一番？ (2) 填写下表，并根据表中的数据，说明该地物价的变化规律。| 物价w | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | | 年数t | 0 | | | | | | | | | |",
      "knowledge": [
        "K4-4-1-01",
        "K4-2-1-01"
      ],
      "methods": []
    },
    {
      "id": "T4-4-E03",
      "type": "例题",
      "content": "比较下列各题中两个值的大小： (1) log₂3.4, log₂8.5; (2) log₀.₃1.8, log₀.₃2.7; (3) logₐ5.1, logₐ5.9 (a>0, 且 a≠1).",
      "knowledge": [
        "K4-4-2-01"
      ],
      "methods": []
    },
    {
      "id": "T4-4-E04",
      "type": "例题",
      "content": "溶液酸碱度的测量。pH的计算公式为pH=-lg[H⁺]，其中 [H⁺] 表示溶液中氢离子的浓度。 (1) 说明溶液酸碱度与溶液中氢离子的浓度之间的变化关系； (2) 已知纯净水中氢离子的浓度为 [H⁺]=10⁻⁷摩尔/升，计算纯净水的pH。",
      "knowledge": [
        "K4-4-2-01",
        "K4-3-2-01"
      ],
      "methods": []
    },
    {
      "id": "T4-5-E01",
      "type": "例题",
      "content": "求方程 ln(x) + 2x - 6 = 0 的实数解的个数。",
      "knowledge": [
        "K4-5-1-01",
        "K4-5-1-02",
        "K3-2-1-01"
      ],
      "methods": []
    },
    {
      "id": "T4-5-E02",
      "type": "例题",
      "content": "借助信息技术,用二分法求方程 2^x+3x=7 的近似解(精确度为0.1).",
      "knowledge": [
        "K4-5-1-01"
      ],
      "methods": [
        "M4-5-01"
      ]
    },
    {
      "id": "T4-5-E03",
      "type": "例题",
      "content": "使用马尔萨斯人口增长模型 y=y₀e^(rt) 解决以下问题：1. 根据中国1950年(55196万)和1959年(67207万)的人口数据，建立该期间的人口增长模型。2. 利用模型计算1951-1958年的人口，并与实际数据比较。3. 预测中国人口达到13亿的时间。",
      "knowledge": [
        "K4-2-1-01"
      ],
      "methods": []
    },
    {
      "id": "T4-5-E04",
      "type": "例题",
      "content": "2010年，考古学家对良渚古城水利系统中一条水坝的建筑材料(草裹泥)上提取的草茎遗存进行碳14年代学检测,检测出碳14的残留量约为初始量的55.2%,能否以此推断此水坝大概是什么年代建成的？(碳14半衰期为5730年)",
      "knowledge": [
        "K4-2-1-01",
        "K4-3-1-01"
      ],
      "methods": []
    },
    {
      "id": "T4-5-E05",
      "type": "例题",
      "content": "假设你有一笔资金用于投资, 现有三种投资方案供你选择, 这三种方案的回报如下: 方案一: 每天回报40元; 方案二: 第一天回报10元, 以后每天比前一天多回报10元; 方案三: 第一天回报0.4元, 以后每天的回报比前一天翻一番. 请问, 你会选择哪种投资方案?",
      "knowledge": [
        "K4-2-1-01"
      ],
      "methods": []
    },
    {
      "id": "T4-5-E06",
      "type": "例题",
      "content": "某公司为了实现1000万元利润的目标，准备制定一个激励销售人员的奖励方案：在销售利润达到10万元时，按销售利润进行奖励，且奖金y(万元)随销售利润x(万元)的增加而增加，但奖金总数不超过5万元，同时奖金不超过利润的25%。现有三个奖励模型：y=0.25x， y=log₇x+1， y=1.002^x，其中哪个模型能符合公司的要求？",
      "knowledge": [
        "K4-2-2-01",
        "K4-4-2-01",
        "K3-2-1-02"
      ],
      "methods": []
    },
    {
      "id": "T5-1-E01",
      "type": "例题",
      "content": "在0°到360°的范围内找出与-950°12'终边相同的角。",
      "knowledge": [
        "K5-1-1-03"
      ],
      "methods": []
    },
    {
      "id": "T5-1-E02",
      "type": "例题",
      "content": "写出终边在y轴上的角的集合S。",
      "knowledge": [
        "K5-1-1-02",
        "K5-1-1-03"
      ],
      "methods": []
    },
    {
      "id": "T5-1-E03",
      "type": "例题",
      "content": "写出终边在直线 y=x 上的角的集合 S. S 中满足不等式 -360° ≤ β < 720° 的元素 β 有哪些?",
      "knowledge": [
        "K5-1-1-02",
        "K5-1-1-03"
      ],
      "methods": []
    },
    {
      "id": "T5-1-E04",
      "type": "例题",
      "content": "按照下列要求，把 67°30' 化成弧度：(1) 精确值；(2) 精确到 0.001 的近似值。",
      "knowledge": [
        "K5-1-2-02"
      ],
      "methods": []
    },
    {
      "id": "T5-1-E05",
      "type": "例题",
      "content": "将 3.14 rad 换算成角度 (用度数表示, 精确到 0.001)。",
      "knowledge": [
        "K5-1-2-02"
      ],
      "methods": []
    },
    {
      "id": "T5-1-E06",
      "type": "例题",
      "content": "利用弧度制证明下列关于扇形的公式: (1) l=αR; (2) S=½αR²; (3) S=½lR.",
      "knowledge": [
        "K5-1-2-01",
        "K5-1-2-03"
      ],
      "methods": []
    },
    {
      "id": "T5-2-E01",
      "type": "例题",
      "content": "求 5π/3 的正弦、余弦和正切值。",
      "knowledge": [
        "K5-2-1-01"
      ],
      "methods": []
    },
    {
      "id": "T5-2-E02",
      "type": "例题",
      "content": "设α是一个任意角，它的终边上任意一点P(不与原点O重合)的坐标为(x,y)，点P与原点的距离为r。求证：sinα=y/r，cosα=x/r，tanα=y/x。",
      "knowledge": [
        "K5-2-1-01"
      ],
      "methods": []
    },
    {
      "id": "T5-2-E03",
      "type": "例题",
      "content": "求证: 角θ为第三象限角的充要条件是 {sinθ<0, tanθ>0}.",
      "knowledge": [
        "K1-4-04",
        "K5-2-1-02"
      ],
      "methods": [
        "M1-4-01"
      ]
    },
    {
      "id": "T5-2-E04",
      "type": "例题",
      "content": "确定下列三角函数值的符号，然后用计算工具验证: (1) cos 250°; (2) sin(-π/4); (3) tan(-672°); (4) tan 3π.",
      "knowledge": [
        "K5-2-1-02",
        "K5-3-01"
      ],
      "methods": []
    },
    {
      "id": "T5-2-E05",
      "type": "例题",
      "content": "求下列三角函数值: (1) sin 1480°10' (精确到 0.001); (2) cos(9π/4); (3) tan(-11π/6).",
      "knowledge": [
        "K5-3-01"
      ],
      "methods": []
    },
    {
      "id": "T5-2-E06",
      "type": "例题",
      "content": "已知 sin α = -3/5, 求 cos α, tan α 的值.",
      "knowledge": [
        "K5-2-2-01",
        "K5-2-1-02"
      ],
      "methods": []
    },
    {
      "id": "T5-2-E07",
      "type": "例题",
      "content": "求证: cos(x)/(1 - sin(x)) = (1 + sin(x))/cos(x).",
      "knowledge": [
        "K5-2-2-01"
      ],
      "methods": []
    },
    {
      "id": "T5-3-E01",
      "type": "例题",
      "content": "利用公式求下列三角函数值: (1) cos 225°; (2) sin(8π/3); (3) sin(-16π/3); (4) tan(-2040°).",
      "knowledge": [
        "K5-3-01"
      ],
      "methods": []
    },
    {
      "id": "T5-3-E02",
      "type": "例题",
      "content": "化简 (cos(180°+α)sin(α+360°))/(tan(-α-180°)cos(-180°+α)).",
      "knowledge": [
        "K5-3-01"
      ],
      "methods": []
    },
    {
      "id": "T5-3-E03",
      "type": "例题",
      "content": "证明: (1) sin(3π/2 - α) = -cosα; (2) cos(3π/2 + α) = sinα.",
      "knowledge": [
        "K5-3-01"
      ],
      "methods": []
    },
    {
      "id": "T5-3-E04",
      "type": "例题",
      "content": "化简 sin(2π-α)cos(π+α)cos(π/2+α)cos(11π/2-α) / [cos(π-α)sin(3π-α)sin(-π-α)sin(9π/2+α)].",
      "knowledge": [
        "K5-3-01"
      ],
      "methods": []
    },
    {
      "id": "T5-3-E05",
      "type": "例题",
      "content": "已知 sin(53°-α)=1/5, 且 -270°<α<-90°, 求 sin(37°+α) 的值.",
      "knowledge": [
        "K5-3-01",
        "K5-2-2-01"
      ],
      "methods": []
    },
    {
      "id": "T5-4-E01",
      "type": "例题",
      "content": "画出下列函数的简图: (1) y=1+sin x, x∈[0, 2π]; (2) y=-cos x, x∈[0, 2π].",
      "knowledge": [
        "K5-4-1-01"
      ],
      "methods": [
        "M5-6-01"
      ]
    },
    {
      "id": "T5-4-E02",
      "type": "例题",
      "content": "求下列函数的周期: (1) y=3sin x, x∈R; (2) y=cos 2x, x∈R; (3) y=2sin(x/2 - π/6), x∈R.",
      "knowledge": [
        "K5-4-2-01"
      ],
      "methods": []
    },
    {
      "id": "T5-4-E03",
      "type": "例题",
      "content": "下列函数有最大值、最小值吗? 如果有, 请写出取最大值、最小值时自变量 x 的集合, 并求出最大值、最小值. (1) y=cos x+1, x∈R; (2) y=-3sin 2x, x∈R.",
      "knowledge": [
        "K5-4-2-02"
      ],
      "methods": []
    },
    {
      "id": "T5-4-E04",
      "type": "例题",
      "content": "不通过求值,比较下列各组数的大小: (1) sin(-π/18) 与 sin(-π/10); (2) cos(-23π/5) 与 cos(-17π/4).",
      "knowledge": [
        "K5-4-2-02",
        "K5-3-01"
      ],
      "methods": []
    },
    {
      "id": "T5-4-E05",
      "type": "例题",
      "content": "求函数 y=sin(x/2+π/3), x∈[-2π, 2π] 的单调递增区间.",
      "knowledge": [
        "K5-4-2-02"
      ],
      "methods": []
    },
    {
      "id": "T5-4-E06",
      "type": "例题",
      "content": "求函数 y=tan(π/2*x+π/3) 的定义域、周期及单调区间。",
      "knowledge": [
        "K5-4-3-01"
      ],
      "methods": []
    },
    {
      "id": "T5-5-E01",
      "type": "例题",
      "content": "利用公式 C(α-β) 证明：(1) cos(π/2 - α) = sinα; (2) cos(π-α) = -cosα.",
      "knowledge": [
        "K5-5-1-01"
      ],
      "methods": []
    },
    {
      "id": "T5-5-E02",
      "type": "例题",
      "content": "已知 sinα=4/5, α∈(π/2, π), cosβ=-5/13, β是第三象限角, 求 cos(α-β) 的值。",
      "knowledge": [
        "K5-5-1-01",
        "K5-2-2-01"
      ],
      "methods": []
    },
    {
      "id": "T5-5-E03",
      "type": "例题",
      "content": "已知 sinα = -3/5，α 是第四象限角，求 sin(π/4 - α), cos(π/4 + α), tan(α - π/4) 的值.",
      "knowledge": [
        "K5-5-1-01",
        "K5-2-2-01"
      ],
      "methods": []
    },
    {
      "id": "T5-5-E04",
      "type": "例题",
      "content": "利用和(差)角公式计算下列各式的值: (1) sin72°cos42°-cos72°sin42°; (2) cos20°cos70°-sin20°sin70°; (3) (1+tan15°)/(1-tan15°).",
      "knowledge": [
        "K5-5-1-01"
      ],
      "methods": []
    },
    {
      "id": "T5-5-E05",
      "type": "例题",
      "content": "已知 sin2α=5/13, π/4 < α < π/2, 求 sin4α, cos4α, tan4α 的值.",
      "knowledge": [
        "K5-5-3-01",
        "K5-2-2-01"
      ],
      "methods": []
    },
    {
      "id": "T5-5-E06",
      "type": "例题",
      "content": "在△ABC中，cosA = 4/5, tanB = 2，求 tan(2A+2B) 的值.",
      "knowledge": [
        "K5-5-1-01",
        "K5-5-3-01",
        "K5-2-2-01"
      ],
      "methods": []
    },
    {
      "id": "T5-5-E07",
      "type": "例题",
      "content": "试以 cosα 表示 sin²(α/2), cos²(α/2), tan²(α/2).",
      "knowledge": [
        "K5-5-3-01"
      ],
      "methods": []
    },
    {
      "id": "T5-5-E08",
      "type": "例题",
      "content": "求证: (1) sinαcosβ = 1/2[sin(α+β)+sin(α-β)]; (2) sinθ+sinφ = 2sin((θ+φ)/2)cos((θ-φ)/2).",
      "knowledge": [
        "K5-5-1-01"
      ],
      "methods": []
    },
    {
      "id": "T5-5-E09",
      "type": "例题",
      "content": "化简：sin(x + π/3) + sin(x - π/3).",
      "knowledge": [
        "K5-5-1-01"
      ],
      "methods": []
    },
    {
      "id": "T5-5-E10",
      "type": "例题",
      "content": "如图，在 Rt△ABC 中，∠C=90°, AC=2, ∠BAC=α, (0<α<π/3), 求矩形 ABCD 的面积 S 的最大值.",
      "knowledge": [
        "K5-5-3-01"
      ],
      "methods": []
    },
    {
      "id": "T5-6-E01",
      "type": "例题",
      "content": "画出函数 y=2sin(3x-π/6) 的简图。",
      "knowledge": [
        "K5-6-01"
      ],
      "methods": [
        "M5-6-01",
        "M5-6-02"
      ]
    },
    {
      "id": "T5-6-E02",
      "type": "例题",
      "content": "已知函数 y=sin(x-π/4) 的部分图象如图，则 A,B,C,D,E 各点的坐标分别为 A(π/4, 0), B(3π/4, 1), C(5π/4, 0), D(7π/4, -1), E(9π/4, 0)。",
      "knowledge": [
        "K5-6-01",
        "K5-4-2-02"
      ],
      "methods": [
        "M5-6-01"
      ]
    },
    {
      "id": "T5-6-E03",
      "type": "例题",
      "content": "在长为1m的细绳一端系上一个小球,以另一端为圆心在竖直平面内做圆周运动,小球的起始位置在最低点,经过ts后,小球转过的角度为θ=t+π/2(rad). (1)经过多少时间,小球到达最高点? (2)经过多少时间,小球的高度第一次达到0.5m?",
      "knowledge": [
        "K5-6-01"
      ],
      "methods": []
    },
    {
      "id": "T5-6-E04",
      "type": "例题",
      "content": "摩天轮的半径为55m，最高点距地面120m。摩天轮运行一周约需30min。在摩天轮上，甲、乙两人从不同位置开始计时，经过ts后，甲、乙两人距离地面的高度分别为h_甲=65+55sin(πt/15 - π/2), h_乙=65+55sin(πt/15 + π/6)。求t为何值时，两人距离地面的高度差最大？",
      "knowledge": [
        "K5-6-01",
        "K5-5-1-01"
      ],
      "methods": []
    },
    {
      "id": "T5-7-P1",
      "type": "问题",
      "content": "某个弹簧振子在完成一次全振动的过程中，时间t(单位:s)与位移y(单位:mm)之间的对应数据如表5.7-1所示，试根据这些数据确定这个振子的位移关于时间的函数解析式。",
      "knowledge": [
        "K5-6-01"
      ],
      "methods": []
    },
    {
      "id": "T5-7-P2",
      "type": "问题",
      "content": "图5.7-2(1)是某次实验测得的交变电流i(单位:A)随时间t(单位:s)变化的图象。将测得的图象放大，得到图5.7-2(2)。(1)求电流i随时间t变化的函数解析式;(2)当t=0, 1/600, 1/150, 7/600, 1/60时，求电流i。",
      "knowledge": [
        "K5-6-01"
      ],
      "methods": []
    },
    {
      "id": "T5-7-E01",
      "type": "例题",
      "content": "如图5.7-3, 某地一天从6~14时的温度变化曲线近似满足函数 y=Asin(ωx+φ)+b. 1. 求这一天6~14时的最大温差; 2. 写出这段曲线的函数解析式.",
      "knowledge": [
        "K5-6-01"
      ],
      "methods": []
    },
    {
      "id": "T5-7-E02",
      "type": "例题",
      "content": "海水受日月的引力，在一定的时候发生涨落的现象叫潮。表5.7-2是某港口某天的时刻与水深关系的预报。 (1) 选用一个函数来近似描述这一天该港口的水深与时间的关系，给出整点时水深的近似数值。 (2) 一条货船的吃水深度为4m，安全间隙1.5m，该船这一天何时能进入港口？在港口能待多久？ (3) 某船吃水深度为4m，安全间隙1.5m，该船2:00开始卸货，吃水深度以0.3m/h减少，为了安全，需在水深与船所需安全水深相等时刻前0.4h停止卸货并离港，问何时停止卸货离港？",
      "knowledge": [
        "K5-6-01"
      ],
      "methods": []
    }
  ]
}