| {"question": "把研究的对象统称为元素，把一些元素组成的总体叫做集合。请问以下哪个不属于集合三要素？", "options": ["无序性", "有序性", "确定性", "互异性"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "如果构成两个集合的元素是一样的，那么就可以称这两个集合？", "options": ["一致", "相等", "无关", "相似"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "$N^{*} $或者$N_{+}$表示什么类型的集合？", "options": ["负整数集合", "无理数集合", "正整数集合", "复数集合"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "整数集合用什么字母表示？", "options": ["A", "S", "Z", "D"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "有理数集合用什么字母表示？", "options": ["R", "W", "Q", "E"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "R表示什么集合？", "options": ["无理数集合", "整数集合", "实数集合", "复数集合"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "以下哪个不是集合的表示方法？", "options": ["区间法", "图示法", "类比法", "描述法"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "对于两个集合A、B，如果集合A中任意一个元素都是B中的元素，则称集合A是集合B的?", "options": ["交集", "空集", "子集", "真子集"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "集合A是集合B的子集，如何表示？", "options": ["$A \\in B$", "$A \\le B$", "$A \\subseteq B$", "$A \\approx B$"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "如果集合$A \\sqsubseteq B$，但存在元素$x \\in B$，且$x \\notin A$，则称集合A是集合B的?", "options": ["真子集", "子集", "交集", "并集"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "集合A是集合B的真子集，如何表示?", "options": ["$A \\subseteq B$", "$A \\sqsupseteq B$", "$A \\subsetneeq B$", "$A = B$"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "不含任何元素的集合叫做？", "options": ["空集", "非空真子集", "交集", "并集"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "空集的表示方法是？", "options": ["$ \\ni $", "$ \\subset $", "$ \\iff $", "$ \\varnothing $"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "任何集合都有的子集是？", "options": ["非空集合", "复数集合", "实数集合", "空集合"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "如果集合A中含有n个元素，则集合A中有多少个子集？", "options": ["${n}^{2}$", "${n}^{3}$", "$2^{n-1}$", "$2^{n}$"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "如果集合A中含有n个元素，则集合A中有多少个真子集？", "options": ["${n}^{3}-1$", "$2^{n-1}$", "${n}^{2}$", "$2^{n}-1$"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "设A、B是非空的数集，如果按照某种确定的对应关系f，使对于集合A中的任意一个数x，在集合B中都有惟一确定的数f(x)和它对应，那么就称$f: A \\to B$为集合A到集合B的一个，又是如何表示的？", "options": ["向量，$y \\approx f(x), x \\in A$", "反射，$y \\times f(x), x \\in A$", "函数，$y = f(x), x \\in A$", "映射，$y = f(x), x \\notin A$"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "以下哪个不是函数的构成要素？", "options": ["熵", "定义域", "对应关系", "值域"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "如果两个函数的定义域相同，并且对应关系完全一致，则称这两个函数?", "options": ["相似", "相同", "一致", "相等"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "一般地，设A、B是两个非空集合，如果按照某种对应法则，对于A中的每一个元素，在B中都有唯一的元素与之对应，那么这样的单值对应叫做从集合A到集合B的，如何表示？", "options": ["合集，$f: A \\to B$", "映射，$f: A \\to B$", "函数，$f: A + B$", "反映，$f: A\\nsubseteq B$"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "什么是基本事件？", "options": ["随机数字次数试验中可能出现的每一个基本结果", "一次试验中可能出现的每一个基本结果", "多次次试验中可能出现的每一个基本结果", "一次试验中可能出现的两个结果组合成的基本结果"], "answer": "B", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "以下哪个不是古典概型的特点?", "options": ["每个基本事件都是等可能发生", "都是", "所有的基本事件只有有限个", "所有的基本事件有无限个"], "answer": "D", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "如果一次试验的等可能基本事件共有n个，事件A包含了其中的m个基本事件，则事件A发生的概率P(A) = ?", "options": ["$m{\\div} n$", "$m \\setminus n$", "$m + n$", "$m \\pm n$"], "answer": "A", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "什么是互斥事件?", "options": ["不可能同时发生的两个事件称为互斥事件", "一定同时发生的两个事件称为互斥事件", "同时发生的两个事件称为互斥事件", "可能同时发生的两个事件称为互斥事件"], "answer": "A", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "如果事件${A}_{1},{A}_{2},{A}_{3}…{A}_{n}$任意两个都是互斥事件，则称事件${A}_{1},{A}_{2},{A}_{3}…{A}_{n}$ ？", "options": ["彼此类似", "彼此互斥", "彼此相同", "彼此不一致"], "answer": "B", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "如果事件A, B互斥，那么事件A+B发生的概率，等于事件A, B发生的概率的__, 即：", "options": ["乘积, $P(A+B) = P(A)\\times P(B)$", "商, $P(A+B) = P(A) {\\div} P(B)$", "差, $P(A+B) = P(A) - P(B)$", "和, $P(A+B) = P(A)+P(B)$"], "answer": "D", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "如果事件${A}_{1},{A}_{2},{A}_{3}…{A}_{n}互斥$, 则 $P({A}_{1},{A}_{2},{A}_{3}…{A}_{n}) 与P({A}_{1}) + P({A}_{2})+P({A}_{3})+…+P({A}_{n})$有什么关系？", "options": ["小于", "大于", "等于", "约定于"], "answer": "C", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "什么是对立事件?", "options": ["两个互斥事件中必全部不发生。", "两个互斥事件必全部发生。", "两个互斥事件中至少有一个要发生。", "两个互斥事件中必有一个要发生。"], "answer": "D", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "事件A的对立事件记作$\\bar{A} $, 则$P(A)+P(\\bar{A} )= __, P(\\bar{A} )=__$", "options": ["1, $2 - P(A)$", "2, $1 + P(A)$", "2, $1 - P(A)$", "1, $1 - P(A)$"], "answer": "D", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "对立事件和互斥事件的关系是什么？", "options": ["对立事件一定是互斥事件，互斥事件未必是对立事件", "对立事件不一定是互斥事件，互斥事件一定是对立事件", "对立事件不一定是互斥事件，互斥事件未必是对立事件", "对立事件一定是互斥事件，互斥事件一定是对立事件"], "answer": "A", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "${\\sin }^{2}\\alpha + {\\cos }^{2}\\alpha $= ?", "options": ["0.5", "1", "3", "2"], "answer": "B", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "$\\tan \\alpha = ?$", "options": ["$ \\sin \\alpha + \\cos \\alpha $", "$ \\sin \\alpha \\times \\cos \\alpha $", "$ \\sin \\alpha {\\div} \\cos \\alpha $", "$ \\sin \\alpha \\pm \\cos \\alpha $"], "answer": "C", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "一般地，如果对于函数$f(x)$的定义域内任意一个x，都有$f(-x)=f(x)$，那么函数$f(x)$被称为?", "options": ["奇函数", "不具有奇偶性", "同时具有奇偶两种性质", "偶函数"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "偶函数图象关于什么对称？", "options": ["y=-x", "y轴", "y=x", "x轴"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "一般地，如果对于函数$f(x)$的定义域内任意一个x，都有$f(-x)=-f(x)$，那么函数$f(x)$被称为?", "options": ["奇函数", "单调减函数", "单调增函数", "偶函数"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "奇函数图象关于什么对称？", "options": ["x轴", "原点", "y轴", "y=x"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "$y=0$的奇偶性是什么？", "options": ["既是奇函数，也是偶函数", "只能是偶函数", "只能是奇函数", "既不是奇函数，也不是偶函数"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "$y=0$的定义域有多少个？", "options": ["1个", "2个", "无数个", "0个"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "$f(M)=f(N)$，如果$M+N=d$，d是常数，则$f(x)$的对称轴是", "options": ["$x=\\frac{d}{4} $", "$x=\\frac{d}{2} $", "$x=\\frac{2d}{3} $", "$x=\\frac{d}{2} $"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "$f(x)=g(x)$关于原点对称，那么$f(x)$和$g(x)$本身也关于原点对称？", "options": ["是的", "不知道", "不是", "不一定"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "$f(kx+b)$是偶函数，则$f(x)$的对称轴是", "options": ["以上都不是", "$x=\\frac{2b}{k}$", "$x=\\frac{b}{k}$", "$x=\\frac{b}{2k}$"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "$f(kx+b)$是奇函数，则$f(x)$的对称中心是", "options": ["$(\\frac{b}{k},0)$", "$(0,\\frac{b}{k})$", "$(\\frac{b}{2k},0)$", "$(\\frac{2b}{k},0)$"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "对于函数$f(x)$，如果存在一个非零常数T，使得当x取定义域内的每一个值时，都有$f(x+T)=f(x)$，那么这个函数叫做？", "options": ["周期函数", "常函数", "单调函数", "奇函数"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "对于函数$f(x)$，存在一个非零常数T，使得当x取定义域内的每一个值时，都有$f(x+T)=f(x)$，那么这个函数的周期是？", "options": ["1", "0", "2T", "T"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "函数$f(x)=f(x+T)$的周期是", "options": ["2T", "T", "3T", "都不是"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "函数$f(x)=-f(x+T)$的周期是", "options": ["T", "3T", "都不是", "2T"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "函数$f(x)=-\\frac{1}{f(x+T)} $的周期是", "options": ["都不是", "T", "3T", "2T"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "函数$ff(x)=\\frac{1}{f(x+T)} $的周期是", "options": ["T", "3T", "都不是", "2T"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "如果函数关于$x=a$、$x=b$或者$(a,0)$、$(b,0)$对称，则它的周期是", "options": ["都不是", "$2\\left | b-a \\right |$", "$\\left | b-a \\right |$", "$4\\left | b-a \\right |$"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "如果函数关于$x=a$、$(b,0)$或者$x=b$、$(a,0)$对称，则它的周期是", "options": ["$2\\left | b-a \\right |$", "都不是", "$\\left | b-a \\right |$", "$4\\left | b-a \\right |$"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "如果函数$f(x)$是偶函数，关于$x=a$对称，则它的周期是", "options": ["2a", "3a", "都不是", "a"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "如果函数$f(x)$是奇函数，关于$x=a$对称，则它的周期是", "options": ["a", "2a", "都不是", "4a"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "定义域连续的常数函数是否是周期函数？", "options": ["否", "是", "我不知道", "不一定"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "定义域连续的常数函数是否具有最小正周期？", "options": ["我不知道", "是", "不一定", "否"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "如何将函数y的图像变换到$y+b$？", "options": ["向上平移b", "向下平移b", "向左平移b", "向右平移b"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "如何将函数y的图像变换到$-y$？", "options": ["都不是", "平移", "上下互换", "左右互换"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "如何将函数y的图像变换到$\\left | y \\right |$", "options": ["将在x轴上方的图像翻折到下方", "将在x轴下方的图像翻折到上方", "将在y轴右侧的图像翻折到左方", "将在y轴左侧的图像翻折到右方"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "一般地，如果$x^{n} =a$，其中$n>1$，$n\\in N_{+} $，那么x叫做a的", "options": ["n次幂", "都不是", "n次方根", "n倍"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "当n为奇数时，$\\sqrt[n]{x^{n}}$ 等于", "options": ["a^2", "都不是", "-a", "a"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "当n为偶数时，$\\sqrt[n]{x^{n}}$等于", "options": ["a", "$\\left | a \\right | $", "都不是", "-a"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "$n>0$，$a^{-n}$可改写为", "options": ["$\\frac{1}{a^{-n}}$", "$\\frac{1}{a^{n}}$", "$-\\frac{1}{a^{n}}$", "$-\\frac{1}{a^{-n}}$"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "如果$a>0$、$r,s \\in Q$，则$a^{r}a^{s}$等于", "options": ["都不是", "$a^{r-s}$", "$a^{rs}$", "$a^{r+s}$"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "如果$a>0$、$r,s \\in Q$，则${\\left ( a^{r} \\right ) }^{s} $等于", "options": ["$a^{r-s} $", "都不是", "$a^{r+s} $", "$a^{rs} $"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "一般地,由一个平面多边形沿某一方向平移形成的空间几何体叫做？", "options": ["棱柱", "圆柱", "棱锥", "棱台"], "answer": "A", "topic": "High--Intermediate Geometry--Solid Geometry"} |
| {"question": "用一个平行于棱锥底面的平面去截棱锥，底面与截面之间的多面体叫做？", "options": ["棱锥", "棱台", "圆台", "棱柱"], "answer": "B", "topic": "High--Intermediate Geometry--Solid Geometry"} |
| {"question": "圆柱侧面积的计算公式是什么？", "options": ["$S=4\\pi \\cdot r \\cdot l$", "$S=2\\pi \\cdot r \\cdot l$", "都不是", "$S=\\pi \\cdot r \\cdot l$"], "answer": "B", "topic": "High--Intermediate Geometry--Solid Geometry"} |
| {"question": "圆锥侧面积的计算公式是什么？", "options": ["都不是", "$S=\\pi \\cdot r \\cdot l$", "$S=4\\pi \\cdot r \\cdot l$", "$S=2\\pi \\cdot r \\cdot l$"], "answer": "B", "topic": "High--Intermediate Geometry--Solid Geometry"} |
| {"question": "菱形的面积如何计算？", "options": ["对角线乘积的一半", "边长乘积", "对角线乘积", "边长乘积的一半"], "answer": "A", "topic": "High--Intermediate Geometry--Solid Geometry"} |
| {"question": "正方体外接球、内切球的球心位于哪里？", "options": ["外接球球心位于棱边中心，内切球球心位于体对角线中点", "外接球球心位于体对角线中点，内切球球心位于棱边中心", "均位于棱边的中心", "均位于体对角线的中点"], "answer": "D", "topic": "High--Intermediate Geometry--Solid Geometry"} |
| {"question": "正四面体各面都是什么图形？", "options": ["等边三角形", "普通三角形", "等腰三角形", "四边形"], "answer": "A", "topic": "High--Intermediate Geometry--Solid Geometry"} |
| {"question": "正三棱锥的底面都是什么图形？", "options": ["普通三角形", "四边形", "等边三角形", "等腰三角形"], "answer": "C", "topic": "High--Intermediate Geometry--Solid Geometry"} |
| {"question": "若正四面体棱长为a，则其外接球半径为？", "options": ["$ \\frac{\\sqrt{3}}{4} a $", "$ \\frac{\\sqrt{6}}{4} a $", "$ \\frac{\\sqrt{6}}{12} a $", "$ \\frac{\\sqrt{3}}{6} a $"], "answer": "B", "topic": "High--Intermediate Geometry--Solid Geometry"} |
| {"question": "若正四面体棱长为a，则其内切球半径为?", "options": ["$ \\frac{\\sqrt{6}}{12} a $", "$ \\frac{\\sqrt{3}}{6} a $", "$ \\frac{\\sqrt{3}}{4} a $", "$ \\frac{\\sqrt{6}}{4} a $"], "answer": "A", "topic": "High--Intermediate Geometry--Solid Geometry"} |
| {"question": "三角形的重心是三条什么的交点？", "options": ["等分线", "中垂线", "中线", "高线"], "answer": "C", "topic": "High--Intermediate Geometry--Solid Geometry"} |
| {"question": "$\\sin \\left ( \\alpha + 2k \\pi \\right ) $=？", "options": ["$-\\sin \\alpha $", "$-\\sin 2\\alpha $", "$\\sin 2\\alpha $", "$\\sin \\alpha $"], "answer": "D", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "$\\cos \\left ( \\alpha + 2k \\pi \\right ) $=？", "options": ["$\\cos \\alpha $", "$-\\cos \\alpha $", "$-\\cos 2\\alpha $", "$\\cos 2\\alpha $"], "answer": "A", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "$\\sin \\left (\\pi + \\alpha \\right) $=?", "options": ["$-\\cos \\alpha $", "$\\sin \\alpha $", "$\\cos \\alpha $", "$- \\sin \\alpha $"], "answer": "D", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "$\\cos \\left (\\pi + \\alpha \\right) $=？", "options": ["$\\sin \\alpha $", "$\\cos \\alpha $", "$-\\sin \\alpha $", "$-\\cos \\alpha $"], "answer": "D", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "$\\sin \\left (- \\alpha \\right) $=？", "options": ["$\\sin 2\\alpha $", "$\\sin \\alpha $", "$-\\sin 2\\alpha $", "$-\\sin \\alpha $"], "answer": "D", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "$\\tan \\left ( -\\alpha \\right ) $=？", "options": ["$-\\tan \\left ( 2\\alpha \\right ) $", "$\\tan \\left ( \\alpha \\right ) $", "$-\\tan \\left ( \\alpha \\right ) $", "$-\\cot \\left ( \\alpha \\right ) $"], "answer": "C", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "$\\sin \\left (\\frac{\\pi}{2}+ \\alpha \\right) $=？", "options": ["$-\\cos \\alpha $", "$-\\sin \\alpha $", "$\\sin \\alpha $", "$\\cos \\alpha $"], "answer": "D", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "$\\cos \\left (\\frac{\\pi}{2} + \\alpha \\right) $=?", "options": ["$\\cos \\alpha $", "$-\\sin \\alpha $", "$\\sin \\alpha $", "$-\\cos \\alpha $"], "answer": "B", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "$y = \\sin x$ 的值域是？", "options": ["$R$", "不存在值域", "$\\left [ -1,1 \\right ] $", "$[0,1]$"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "$y=\\cos x$的奇偶性是？", "options": ["偶函数", "奇函数", "既是奇函数，也是偶函数", "既不是奇函数，也不是偶函数"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "如果一个数列从第2项起，每一项与它前一项的差等于同一个常数，那么这个数列叫做？", "options": ["等比数列", "单减数列", "单增数列", "等差数列"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Sequences"} |
| {"question": "下面哪个选项不是等差数列前n项和？", "options": ["$ \\frac{\\left ( a_1+a_n \\right ) n}{2} $", "$\\frac{a_n-a_1}{d} +1$", "$ na_1+ \\frac{nd\\left ( n-1 \\right ) }{2} $", "$An^2+Bn$，且$A=\\frac{d}{2} ,$B=a_1-\\frac{d}{2} $"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Sequences"} |
| {"question": "在等差数列中，下标为等差数列的项仍然组成？", "options": ["单减数列", "单增数列", "等比数列", "等差数列"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Sequences"} |
| {"question": "等差数列${a_{n}}$的公差是d，当d大于0时，${a_{n}}$是什么数列？", "options": ["常", "递减", "递增", "振荡"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Sequences"} |
| {"question": "在等差数列{a_{n}}中，$a_{m}=n$，$a_{n}=m$，且$m \\ne n$，则$a_{m+n}$=?", "options": ["0", "m", "n", "m+n"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Sequences"} |
| {"question": "若$S_{m}=S_{n}$，且$m \\ne n$，则$S_{m+n}$等于?", "options": ["n", "m", "0", "$m+n$"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Sequences"} |
| {"question": "若$S_{m}=n$，$S_{n}=m$，且$m \\ne n$，则$S_{m+n}$等于？", "options": ["m", "$-m-n$", "n", "$m+n$"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Sequences"} |
| {"question": "如果一个不含0的数列从第2项起，每一项与它的前一项的比等于同一个常数，那么这个数列叫做？", "options": ["常数列", "等比数列", "等差数列", "单增数列"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Sequences"} |
| {"question": "在等比数列中，下标成等差数列的项组成的数列是什么数列？", "options": ["常数列", "等差数列", "单增数列", "等比数列"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Sequences"} |
| {"question": "在公比为q的等比数列中，若每一项都乘以不等于零的常数&\\lambda $，那么现在该数列的公比为？", "options": ["&\\lambda $", "q", "$q^2$", "$q\\cdot \\lambda $"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Sequences"} |
| {"question": "已知公比为q的正项等比数列$\\left \\{ a_n \\right \\} $，则数列$\\left \\{ \\lg_{}{a_n} \\right \\} $为什么数列?", "options": ["等比数列", "既不是等差也不是等比数列", "既是等差也是等比数列", "等差数列"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Sequences"} |
| {"question": "已知公比为q的正项等比数列$\\left \\{ a_n \\right \\} $，则数列$\\left \\{ \\lg_{}{a_n} \\right \\} $为公差是什么的等差数列?", "options": ["$\\lg_{}{q} $", "$q$", "$q_{}^{2} $", "$\\ln_{}{q} $"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Sequences"} |
| {"question": "简单随机抽样是从个体数为$N$的总体中怎么取出n个个体作为样本，且每个个体都有相同的机会被抽到？", "options": ["随机有放回地", "逐个不放回地", "逐个放回地", "随机无放回地"], "answer": "B", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "简单随机抽样的方法包括抽签法和什么方法？", "options": ["排列法", "分层抽样法", "随机数表法", "随机抽签法"], "answer": "C", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "要选择几个数作为随机数表法的开始？", "options": ["三个及以上", "一个", "随机数量", "两个"], "answer": "B", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "在$N$个个体的总体中抽取出$n$个个体组成样本，每个个体被抽到的机会（概率）均是什么？", "options": ["$\\frac{N}{n}$", "$\\frac{1}{n}$", "$\\frac{n}{N}$", "$\\frac{1}{N}$"], "answer": "C", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "总体分布的密度曲线与横轴围成的面积为多少？", "options": ["由分布决定", "100", "10", "1"], "answer": "D", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "在频率分布表中计算平均数时，应该取每组的哪个值作为计算基础？", "options": ["最大值", "组中值", "中位数", "平均值"], "answer": "B", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "在总体特征数的估计中，如果取值为$x_1, x_2, \\ldots, x_n$的频率分别为 $p_1, p_2, \\ldots, p_n$，那么计算平均数的公式是什么？", "options": ["$\\bar{X} = \\frac{\\sum_{i=1}^{n}p_i}{n}$", "$\\bar{X} = \\frac{\\sum_{i=1}^{n}x_i}{n}$", "$\\bar{X} = \\frac{\\sum_{i=1}^{n}x_i \\cdot p_i}{n}n$", "$\\bar{X} = \\frac{\\sum_{i=1}^{n}p_i \\cdot x_i}{\\sum_{i=1}^{n}p_i}$"], "answer": "C", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "平均数的变化公式$E(aX+b)$中，正确的表达是什么？", "options": ["$E(aX)+E(b)$", "$aE(X)+b$", "$a \\cdot E(X+b)$", "$E(aX+b)$"], "answer": "B", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "标准差如何表示？", "options": ["$s^2 = \\sqrt{\\frac{\\sum_{i=1}^{n} (x_i - \\bar{x})^2}{n-1}}$", "$s = \\frac{\\sum_{i=1}^{n} (x_i - \\bar{x})}{n-1}$", "$s = \\sqrt{\\frac{\\sum_{i=1}^{n} (x_i - \\bar{x})^2}{n}}$", "$s^2 = \\frac{\\sum_{i=1}^{n} (x_i - \\bar{x})^2}{n}$"], "answer": "C", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "方差的变化$V(aX+b)$ 等于以下哪个表达式？", "options": ["$aV(X)$", "$V(aX+b)$", "$a^2V(X)$", "$aX+b$"], "answer": "C", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "什么是众数？", "options": ["数据集中的最小值", "频率最高的区间内中位数", "数据集中的最大值", "频率最低的区间内中位数"], "answer": "B", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "什么是中位数？", "options": ["一列数里面从小到大最中间的数", "一列数里面从大到小最中间的数", "数据集中的平均值", "数据集中的众数"], "answer": "A", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "如果数据数量为偶数，如何取中位数？", "options": ["取中间两个数的最小值", "取中间两个数的最大值", "取中间两个数的平均值", "取最中间的数"], "answer": "C", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "在求直方图中的中位数时，若设中位数为Q，满足方程$(f_1 + f_2 + \\ldots + f_n) + (Q - a) \\cdot x_i = 0.5$。其中$a$是中位数的左端点值，$x_i$是中位数频率和组距比值。这个方程应该如何理解$f_1+ \\ldots + f_n$这一部分的含义？", "options": ["$f_1+ \\ldots + f_n$表示中位数两侧的频率之和", "$f_1+ \\ldots + f_n$表示中位数右侧的频率之和", "$f_1+ \\ldots + f_n$表示整个数据集的频率之和", "$f_1+ \\ldots + f_n$表示中位数左侧的频率之和"], "answer": "D", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "变量之间的两类关系主要包括哪两种？", "options": ["正相关关系与负相关关系", "线性关系与非线性关系", "直线关系与曲线关系", "函数关系与相关关系"], "answer": "D", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "相关系数r绝对值越接近1，相关性越强。且$r \\in (0, 1]$。关于相关系数r的性质，以下哪个说法是正确的？", "options": ["相关系数r可以取任意实数值", "相关系数r绝对值越接近0，相关性越强", "相关系数r绝对值越接近1，相关性越强", "相关系数r绝对值越接近1，相关性越弱"], "answer": "C", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "线性回归直线是否一定经过定点$(\\bar{x}, \\bar{y})$？", "options": ["只在特殊情况下", "不一定", "一定", "取决于数据的分布"], "answer": "C", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "相关系数的主要作用是什么？", "options": ["判断两个变量是否存在线性关系", "计算两个变量的平均值", "衡量两个变量的差异", "描述两个变量的分布形状"], "answer": "A", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "两变量间有线性关系是否意味着两变量间有一次函数关系？", "options": ["是", "只在特殊情况下", "否", "取决于数据的分布"], "answer": "C", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "一般地，由所有属于集合A或集合B的元素组成的集合，称为集合A与B的什么？记作什么？", "options": ["并集；$A\\cup B$", "并集；$A\\cap B$", "交集；$A\\cup B$", "交集：$A\\cap B$"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "一般地，由所有属于集合A或集合B的元素组成的集合，称为集合A与B的什么？记作什么？", "options": ["并集；$A\\cap B$", "并集；$A\\cup B$", "交集；$A\\cup B$", "交集：$A\\cap B$"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "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A相对于全集U的补集，如何用公式表示？", "options": ["${C}_{U}A=\\left \\{ x\\mid x\\in U\\right \\}$", "${C}_{U}A=\\left \\{ x\\mid x\\notin U$, 且$x\\in A\\right \\}$", "${C}_{U}A=\\left \\{ x\\mid x\\in U$, 且$x\\notin A\\right \\}$", "${C}_{U}A=\\left \\{ x\\mid x\\notin U\\right \\}$"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "若$f\\left ( x \\right ) \\le f\\left ( x_{0} \\right ) $，则$f\\left ( x_{0} \\right ) $为$f\\left ( x \\right ) $什么值？", "options": ["最小值", "最大值", "初始值", "中值"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "是否可以根据$f\\left ( x \\right ) \\le M$，从而推导M为$f\\left ( x \\right ) $的最大值？", "options": ["可以，若$f\\left ( x \\right ) \\le M$，则M为$f\\left ( x \\right ) $的最大值。", "不可以，除非M在$f\\left ( x \\right ) $上。", "不可以，若$f\\left ( x \\right ) \\le M$，则M为$f\\left ( x \\right ) $的初始值。", "不可以，若$f\\left ( x \\right ) \\le M$，则M为$f\\left ( x \\right ) $的最小值。"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "指数函数过哪一定点？", "options": ["指数函数过定点$\\left ( 1,0 \\right )$", "指数函数过定点$\\left ( 0,1 \\right )$", "指数函数过定点$\\left ( 1,1 \\right )$", "指数函数过定点$\\left ( 0,0 \\right )$"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "设指数函数$y= a^{x} \\left ( a> 0, a\\ne 1 \\right ) $，当$a> 1$时，函数单调性如何？", "options": ["单调递增", "单调递减", "先增后减", "先减后增"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "设指数函数$y= a^{x} \\left ( a> 0, a\\ne 1 \\right ) $，当$0< a< 1$时，函数单调性如何？", "options": ["单调递减", "单调递增", "先增后减", "先减后增"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "假设$a^x=N$,那么如何用a和N表示x？", "options": ["$x=a^N$", "$x=\\log_N a$", "$x=\\log_a N$", "$x=N^a$"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "$a^{\\log_a N}=$", "options": ["$a$", "$a^N$", "$N$", "$N^a$"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "$\\log_a 1=$ ( ), $\\log_a a=$ ( ).", "options": ["1,1", "1,0", "0,1", "0,0"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "当$a>0, a\\neq 1，M>0, N>0$时，$\\log_a (MN)=\\log_a M +( )$, $\\log_a(\\frac{M}{N})=\\log_a M-( )$, $\\log_a M^n=$( )$\\log_a M$.", "options": ["$\\log_a N$, $\\log_N a$, n", "$\\log_N a$, $\\log_a N$, n", "$\\log_N a$, $\\log_N a$, n", "$log_a N$, $\\log_a N$, n"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "当$a>0,a\\neq 1,c>0,c\\neq 1,b>0$时， $\\log_a b =\\log_c b /( )$", "options": ["$\\log_b a$", "$\\log_a c$", "$\\log_c a$", "$\\log_a b$"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "当$a>0, a\\neq 1, b>0, b\\neq 1$时，$log_a b$的倒数是什么？", "options": ["$a^b$", "$\\log_a b$", "$b^a$", "$\\log_b a$"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "通常情况下常用对数是以( )为底，如lg2等，自然对数是以( )为底，如ln2等。", "options": ["10,e", "e,2", "2,e", "e,10"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "对数函数$y=\\log_a x$图像具有以下性质： 过定点(1,1)，$a>1$时的单调性是（ ），$0<a<1$时的单调性是（ ）。", "options": ["单调递增，单调递增", "单调递减，单调递增", "单调递减，单调递减", "单调递增，单调递减"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "对数函数$y=\\log_a x$的a越大，图像越落下还是越升高？", "options": ["落下", "不清楚", "升高", "不变"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "函数$y=e^x$和$y=\\ln x$的图像关于直线<<<Answer>>>对称。", "options": ["$y=0$", "$y=-x$", "$y=x$", "$x=0$"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "一条直线经过$(x_1,y_1),(x_2,y_2)$两个点，这条直线的q倾斜角为$\\alpha$, 那么斜率k的计算公式是？", "options": ["$k = \\tan \\alpha = \\frac{x_2-x_1}{y_2-y_1}$", "$k = \\cos \\alpha = \\frac{y_2-y_1}{x_2-x_1}$", "$k = \\sin \\alpha = \\frac{y_2-y_1}{x_2-x_1}$", "$k = \\tan \\alpha = \\frac{y_2-y_1}{x_2-x_1}$"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "$\\alpha,\\beta$均为第一象限角，若$\\alpha > \\beta$, $\\tan \\alpha $和$\\tan \\beta$的大小关系是？", "options": ["$\\tan \\alpha > \\tan \\beta$", "$\\tan \\alpha = \\tan \\beta$", "$\\tan \\alpha < \\tan \\beta$", "无法确定"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "直线方程点斜式的表达式是？", "options": ["$y-y_0 = k(x_0-x)$", "$y-y_0 = k(x-x_0)$", "$y-y_0 = x-kx_0$", "$y-y_0 = kx-x_0$"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} |
| {"question": "直线方程斜截式的表达式是？", "options": ["$\\frac{y-y_1}{x-x_1}=\\frac{y_2-y_1}{x_2-x_1}$", "$\\frac{x}{a}+\\frac{y}{b}=1$", "$y = kx+b$", "$y-y_0 = k(x-x_0)$"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} |
| {"question": "直线方程两点式的表达式是？", "options": ["$y-y_0 = k(x-x_0)$", "$y = kx+b$", "$Ax+By+C=0$", "$\\frac{y-y_1}{x-x_1}=\\frac{y_2-y_1}{x_2-x_1}$"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} |
| {"question": "直线方程截距式的表达式是？", "options": ["$y-y_0 = k(x-x_0)$", "$y = kx+b$", "$\\frac{y-y_1}{x-x_1}=\\frac{y_2-y_1}{x_2-x_1}$", "$\\frac{x}{a}+\\frac{y}{b}=1$"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} |
| {"question": "直线方程一般式的表达式是?", "options": ["$\\frac{x}{a}+\\frac{y}{b}=1$", "$y = kx+b$", "$Ax+By+C=0$", "$\\frac{y-y_1}{x-x_1}=\\frac{y_2-y_1}{x_2-x_1}$"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} |
| {"question": "对于直线$l_1:y=k_1 x+b_1, l_2:y=k_2 x+b_2$, 若有$l_1//l_2$，那么$k_1$和$k_2$的大小关系是？$b_1$和$b_2$的大小关系是？", "options": ["$k_1=k_2, b_1 = b_2$", "$k_1=k_2, b_1\\neq b_2$", "$k_1\\neq k_2, b_1\\neq b_2$", "$k_1\\neq k_2, b_1 = b_2$"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} |
| {"question": "对于直线$l_1:y=k_1 x+b_1, l_2:y=k_2 x+b_2$, 若有：$l_1 \\perp l_2 ，那么$k_1,k_2$的数值关系是？", "options": ["$k_1 = k_2 $", "$k_1 k_2 = 0$", "$k_1 k_2 = 1$", "$k_1 k_2 = -1$"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} |
| {"question": "对于直线$l_1: A_1 x+B_1 y+C_1=0, l_2: A_2x+B_2y+C_2=0$, 讨论直线是否平行最好的办法是运用结论：$l_1 // l_2 \\Leftrightarrow$", "options": ["$A_1B_2\\neq A_2B_1, B_1C_2 \\neq B_2C_1$", "$A_1B_2=A_2B_1, B_1C_2 = B_2C_1$", "$A_1B_2=A_2B_1, B_1C_2 \\neq B_2C_1$", "$A_1B_2 \\neq A_2B_1, B_1C_2 = B_2C_1$"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} |
| {"question": "对于直线$l_1: A_1 x+B_1 y+C_1=0, l_2: A_2x+B_2y+C_2=0$, 讨论直线是否垂直最好的办法是运用结论：", "options": ["$A_1A_2 - B_1B_2=0$", "$A_1A_2 - B_1B_2=1$", "$A_1A_2+B_1B_2=1$", "$A_1A_2+B_1B_2=0$"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} |
| {"question": "给出坐标系中两点$(x_1,y_1),(x_2,y_2)$, 两点间的距离公式是什么？", "options": ["$|P_1P_2|=\\sqrt{(x_2+x_1)^2-(y_2+y_1)^2}$", "$|P_1P_2|=\\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$", "$|P_1P_2|=\\sqrt{(x_2-x_1)^2-(y_2-y_1)^2}$", "$|P_1P_2|=\\sqrt{(x_2+x_1)^2+(y_2+y_1)^2}$"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} |
| {"question": "给出点坐标$(x_0,y_0)$和直线$Ax+By+C=0$, 点到直线距离公式是什么？", "options": ["$d=\\frac{|Ax_0+By_0-C|}{\\sqrt{A^2+B^2}}$", "$d=\\frac{|Ax_0+By_0+C|}{\\sqrt{A^2+B^2}}$", "$d=\\frac{|Ax_0+By_0+C|}{\\sqrt{A^2+B^2}}$", "$d=\\frac{|Ax_0+By_0+C|}{(A^2+B^2)}$"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} |
| {"question": "两平行线$Ax+By+C_1=0, Ax+By+C_2=0$, 两平行线间的距离公式是什么？", "options": ["$d=\\frac{C_1-C_2}{\\sqrt{A^2-B^2}}$", "$d=\\frac{C_1-C_2}{(A^2+B^2)}$", "$d=\\frac{C_1-C_2}{\\sqrt{A^2+B^2}}$", "$d=\\frac{C_1+C_2}{\\sqrt{A^2+B^2}}$"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} |
| {"question": "长度等于半径长的弧所对的圆心角叫做<<<Answer>>>的角？", "options": ["1周期", "1度", "1°", "1弧度(rad)"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} |
| {"question": "1rad=<<<Answer>>>度", "options": ["180", "$\\frac{pi}{\\180}$", "$\\pi$", "$\\frac{180}{\\pi}$"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} |
| {"question": "角$\\alpha$的弧度数公式是什么？", "options": ["$|\\alpha|=\\frac{\\pi}{r}$", "$|\\alpha|=\\frac{r}{l}$", "$|\\alpha|=\\frac{l}{r}$", "$|\\alpha|=\\frac{l}{\\pi}$"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} |
| {"question": "弧长公式是什么？（角度制）", "options": ["$l=\\frac{n\\pi R}{180}$", "$l=\\frac{180}{n\\pi R}$", "$l=\\frac{n R}{180}$", "$l=\\frac{n R}{\\pi}$"], "answer": "A", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} |
| {"question": "弧长公式是什么？（弧度制）", "options": ["$l=|\\alpha|R$", "$l=\\frac{|\\alpha|}{R}$", "$l=|\\alpha|\\pi$", "$l=|\\pi|R$"], "answer": "A", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} |
| {"question": "设$\\alpha$是一个任意角$,它的终边与单位圆交于点$P(x,y)$，那么：\n$\\sin\\alpha=$<<<Answer1>>>, $\\cos\\alpha=$<<<Answer2>>>, $\\tan\\alpha=$<<<Answer3>>>", "options": ["$y, x, \\frac{y}{x}$", "$x, y, \\frac{y}{x}$", "$y, \\frac{y}{x}, x$", "$\\frac{y}{x}, x, y$"], "answer": "A", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} |
| {"question": "设点$A(x,y)$为角$\\alpha$终边上任意一点，设$r=\\sqrt{x^2+y^2}$,那么\n$\\sin\\alpha=$<<<Answer1>>>, $\\cos\\alpha=$<<<Answer2>>>, $\\tan\\alpha=$<<<Answer3>>>", "options": ["$\\frac{x}{r}, \\frac{y}{x}, \\frac{y}{r}$", "$\\frac{x}{r}, \\frac{y}{r}, \\frac{y}{x}$", "$\\frac{y}{r}, \\frac{x}{r}, \\frac{y}{x}$", "$\\frac{y}{x}, \\frac{x}{r}, \\frac{y}{r}$"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} |
| {"question": "圆的标准方程为：$(x-a)^2+(y-b)^2=r^2$,其中圆心为?", "options": ["$(a^2$,b^2$)$", "$(b,a)$", "$(a,b)$", "$(b^2$,a^2$)$"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} |
| {"question": "圆的标准方程为：$(x-a)^2+(y-b)^2 =r^2$,其中半径为？", "options": ["-2r", "2r", "-r", "r"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} |
| {"question": "圆的一般方程为:$x^2+y^2+Dx+Ey+F=0$，其中圆心为？", "options": ["$(2D,2E)$", "$(E,D)$", "$(D,E)$", "$\\left(-\\frac{D}{2},-\\frac{E}{2}\\right)$"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} |
| {"question": "圆的一般方程为：$x^2+y^2+Dx+Ey+F=0$，其中半径为？", "options": ["$D+E-F$", "$D+E+F$", "$\\frac{1}{2} \\sqrt{D^{2}+E^{2}-4 F}$", "$D^2+E^2+F^2$"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} |
| {"question": "给出点 A$(x_1, y_1)$ 和点 B$(x_2, y_2)$ ，以AB为直径的圆的方程可以写成？", "options": ["$(x - y_1)(x - y_2) + (y - x_1)(y - x_2) = 0$", "$(x - x_1)(x - x_2) + (y - y_1)(y - y_2) = 0$", "$(x - y_1)(x - y_2) + (y - y_1)(y - y_2) = 0$", "$(x - x_1)(x - x_2) + (y - x_1)(y - x_2) = 0$"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} |
| {"question": "直线$ Ax + By + C = 0 $与圆 $(x-a)^2+(y-b)^2=r^2$相离时，有？", "options": ["$d>r$,此时$\\Delta<0$", "$d=r$,此时$\\Delta=0$", "$d<r$,此时$\\Delta>0$", "d和r无关"], "answer": "A", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} |
| {"question": "直线$ Ax + By + C = 0 $与圆 $(x-a)^2+(y-b)^2=r^2$相切时，有？", "options": ["$d>r$,此时$\\Delta<0$", "$d=r$,此时$\\Delta=0$", "$d<r$,此时$\\Delta>0$", "d和r无关"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} |
| {"question": "直线$ Ax + By + C = 0 $与圆 $(x-a)^2+(y-b)^2=r^2$相交时，有？", "options": ["$d=r$,此时$\\Delta=0$", "d和r无关", "$d<r$,此时$\\Delta>0$", "$d>r$,此时$\\Delta<0$"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} |
| {"question": "如果已知圆的r和d，r是圆的半径，d是圆心到弦的中点的距离，如何计算这两个点所形成的弦长？", "options": ["$l=2\\sqrt{r - d}$", "$l=sqrt{r^2 - d^2}$", "$l=2\\sqrt{r^2 - d^2}$", "$l=3\\sqrt{r^2 - d^2}$"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} |
| {"question": "如果已知$(x_1,y_1)$ 和 $(x_2, y_2)$ 是弦的两个端点的坐标，k是这条直线的斜率，如何计算这两个点所形成的弦长？", "options": ["$l=\\sqrt{1+k^2} \\sqrt{(x_1+x_2)^2-4x_1x_2}$", "$l=\\sqrt{1+k^2} \\sqrt{(x_1+x_2)^2-2x_1x_2}$", "$l=\\sqrt{1+k^2} \\sqrt{(x_1+x_2)-4x_1x_2}$", "$l=\\sqrt{1+k} \\sqrt{(x_1+x_2)^2-4x_1x_2}$"], "answer": "A", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} |
| {"question": "若$d=|O_1O_2|$,O_1与O_2内切时，d与两者半径有何关系？", "options": ["$R-r<d<R+r$", "$d<R-r$", "$d=R+r$", "$d=R-r$"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} |
| {"question": "过一个定点P$(a,b)$作已知圆的弦，求弦的中点的轨迹方程。解题思路：垂径定理，找出直角三角形，什么便是直径。", "options": ["三角形周长", "较短的一条边", "直角边", "斜边"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} |
| {"question": "一个动点到两定点距离成比例，则该动点的轨迹为?", "options": ["椭圆", "圆", "矩形", "双曲线"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} |
| {"question": "命题是什么？", "options": ["陈述性语句", "以上都不是", "可以判断真假的语句", "推断性语句"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "以下哪个不是逻辑连接词？", "options": ["而", "非", "或", "且"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "简单命题是什么？", "options": ["至少含一个逻辑联结词的命题", "含一个逻辑联结词的命题", "不含逻辑联结词的命题", "含两个逻辑联结词的命题"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "复合命题是什么？", "options": ["以上都不是", "由简单命题与逻辑联结词构成的命题", "至少有两个逻辑联结词的命题", "多个命题"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "如果两个命题互为逆否命题，则这两个命题：", "options": ["没有真假性关系", "一真一假", "同真同假", "均为错"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "如果两个命题为互逆命题或者互否命题，则这两个命题：", "options": ["均为错", "同真同假", "没有真假性关系", "一真一假"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "一般地，如果已知$p \\Rightarrow q$，则", "options": ["p是q的充分必要条件", "以上都不是", "p是q的充分条件，q是p的必要条件", "p是q的必要条件，q是p的充分条件"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "若$p \\Leftrightarrow q$，则", "options": ["以上都不是", "p是q的必要条件，q是p的充分条件", "p是q的充分必要条件", "p是q的充分条件，q是p的必要条件"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "一般地，如果已知$p \\Rightarrow q$，则", "options": ["p是q的必要条件，q是p的充分条件", "以上都不是", "p是q的充分必要条件", "p是q的充分条件，q是p的必要条件"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "若$p \\Rightarrow q$，但$q \\nRightarrow p$，则", "options": ["p是q充分而不必要条件", "p是q必要而不充分条件", "p是q充要条件", "以上都不对"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "若$p \\nRightarrow q$，但$q \\Rightarrow p$，则", "options": ["以上都不对", "p是q充分而不必要条件", "p是q必要而不充分条件", "p是q充要条件"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "若$p \\Rightarrow q$且$q \\Rightarrow p$，则p是q的什么条件？", "options": ["充要条件", "必要非充分条件", "充分非必要条件", "既非充分也非必要条件"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "若$p \\nRightarrow q$且$q \\nRightarrow p$，则p是q的什么条件？", "options": ["既非充分也非必要条件", "必要非充分条件", "充要条件", "充分非必要条件"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "以下哪个选项不是复合命题的形式？", "options": ["$p \\land q$", "$p \\lor q$", "$B\\not\\subset A$", "$\\neg p$"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "复合命题\"p或q\"在什么情况下为假？", "options": ["p为真q为假", "p为真q为真", "p为假q为真", "p为假q为假"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "复合命题\"p且q\"在什么情况下为真？", "options": ["p为真q为假", "p为假q为真", "p为真q为真", "p为假q为假"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "全称命题是什么？", "options": ["用符号\"$ \\phi $\"表示的命题", "用符号\"$ \\forall $\"表示的命题", "用符号\"$ \\exists $\"表示的命题", "用符号\"$ \\alpha $\"表示的命题"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "以下哪个是全称量词？", "options": ["$ \\forall $", "$ \\alpha $", "$ \\phi $", "$ \\exists $"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "特称命题是什么？", "options": ["用符号\"$ \\phi $\"表示的命题", "用符号\"$ \\alpha $\"表示的命题", "用符号\"$ \\exists $\"表示的命题", "用符号\"$ \\forall $\"表示的命题"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "以下哪个是存在量词？", "options": ["$ \\forall $", "$ \\alpha $", "$ \\exists $", "$ \\phi $"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "全称命题p：$\\forall x \\in M , p(x)$，它的否定$\\neg p$是什么？", "options": ["$\\exists {x}_{0}\\in M, p({x}_{0})$", "$\\forall x \\in M , p(x)$", "$\\forall x \\in M ,\\neg p(x)$", "$\\exists {x}_{0}\\in M,\\neg p({x}_{0})$"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "特称命题p：$\\exists {x}_{0}\\in M, p({x}_{0})$，它的否定$\\neg p$是什么？", "options": ["$\\forall x \\in M ,\\neg p(x)$", "$\\exists {x}_{0}\\in M, p({x}_{0})$", "$\\forall x \\in M , p(x)$", "$\\exists {x}_{0}\\in M,\\neg p({x}_{0})$"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "全称命题的否定是什么？", "options": ["特称命题", "以上都不对", "存在命题", "全称命题"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "特称命题的否定是什么？", "options": ["全称命题", "特称命题", "以上都不对", "存在命题"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "一条直线上至少几个点在一个平面内，那么这条直线所有点都在这个平面内？", "options": ["3个", "4个", "2个", "1个"], "answer": "C", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "如果一条直线上两点在一个平面内，那么这条直线几个点在这个平面内？", "options": ["5个", "所有的点", "3个", "4个"], "answer": "B", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "如果两个平面有一个公共点，那么它们还会有几个其他公共点？", "options": ["0个", "1个", "无数个", "2个"], "answer": "C", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "如果两个平面有一个公共点，那么它们还有其他公共点，这些公共点的集合是", "options": ["与经过公共点直线平行的一个面", "与经过公共点直线相交的一个面", "经过这个公共点的一条直线", "零散的点"], "answer": "C", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "经过不在同一直线上的三点，有几个平面？", "options": ["无数个", "3个", "2个", "1个"], "answer": "D", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "经过一条直线和这条直线外的一点，有几个平面？", "options": ["2个", "3个", "无数个", "1个"], "answer": "D", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "经过一条直线和哪里的一点，有且只有一个平面？", "options": ["无法找到该点", "任意", "直线上", "这条直线外"], "answer": "D", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "经过两条相交直线，有几个平面？", "options": ["1个", "3个", "2个", "无数个"], "answer": "A", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "经过两条平行直线，有几个平面？", "options": ["1个", "3个", "无数个", "2个"], "answer": "A", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "平行于同一条直线的两条直线有什么关系？", "options": ["相交", "平行", "异面", "重合"], "answer": "B", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "两条都平行于什么会平行？", "options": ["均不对", "同一个球面", "同一条直线", "同一个平面"], "answer": "C", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "如果一个角的两边和另一个角的两边分别平行且方向相同，那么这两个角有什么关系？", "options": ["重合", "互余", "相等", "互补"], "answer": "C", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "两个角的两条边有什么关系则两个角相等？", "options": ["分别相交", "分别平行且有一条边方向相反", "都不是", "分别平行且方向相同"], "answer": "D", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "过平面内一点与平面外一点的直线，和这个平面内不经过该点的直线是什么关系？", "options": ["异面", "重合", "平行", "相交"], "answer": "A", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "过平面内一点与平面外一点的直线，和以下哪种直线是异面直线？", "options": ["这个平面内不经过该点的直线", "这个平面内所有直线", "和这个平面平行的直线", "这个平面外的所有直线"], "answer": "A", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "以下哪个不是线面位置关系？", "options": ["直线在平面内", "直线和平面相交", "直线和平面平行", "直线和平面不相交也不平行"], "answer": "D", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "以下哪个不是面面位置关系？", "options": ["平行", "均是", "既不相交也不平行", "相交"], "answer": "C", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "如果平面外一条直线和这个平面内一条直线平行，那么这条直线和这个平面的关系是什么？", "options": ["异面", "相交", "直线在平面内", "平行"], "answer": "D", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "平面外一条直线和哪种直线平行则这条直线和这个平面平行？", "options": ["都不是", "和这个平面相交的任意一条直线", "这个平面内一条直线", "这个平面外的一条直线"], "answer": "C", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "如果一条直线和一个平面平行，经过这条直线的平面和这个平面相交，那么这条直线和交线有什么位置关系？", "options": ["平行", "相交", "异面", "重合"], "answer": "A", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "如果一个平面内的两条相交直线都平行于另一个平面，那么这两个平面有什么位置关系？", "options": ["相交", "平行", "都不是", "异面"], "answer": "B", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "如果两个平行平面同时和第三个平面相交，那么所得的两条交线有什么位置关系？", "options": ["相交", "平行", "异面", "都不是"], "answer": "B", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "两个有什么位置关系的平面同时和第三个平面相交，那么所得的两条交线平行？", "options": ["都不是", "平行", "相交", "异面"], "answer": "B", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "如果一条直线垂直于一个平面内任意一条直线，那么就说这条直线和这个平面的关系是？", "options": ["异面", "垂直", "平行", "都不是"], "answer": "B", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "什么情况下，可以说这条直线和这个平面垂直", "options": ["一条直线垂直于一个平面内两条直线", "这条直线垂直于一个平面内任意一条直线", "一条直线垂直于一个平面内一条直线", "都不是"], "answer": "B", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "过一点有几条直线与已知平面垂直？", "options": ["1条", "无数条", "2条", "3条"], "answer": "A", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "过一点有几个平面与已知直线垂直？", "options": ["1个", "无数个", "3个", "2个"], "answer": "A", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "如果一条直线和一个平面内的两条相交垂直，那么这条直线和平面的位置关系是什么？", "options": ["垂直", "异面", "都不是", "平行"], "answer": "A", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "一条直线和一个平面内怎样的两条直线垂直，则这条直线和平面垂直？", "options": ["都不是", "异面", "相交", "平行"], "answer": "C", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "两条直线都垂直于什么，那么这两条直线平行？", "options": ["同一个球面", "同一个平面", "都不是", "同一条直线"], "answer": "B", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "两条直线与同一个平面有怎样的位置关系则两条直线平行？", "options": ["相交", "平行", "在同一个平面上", "垂直"], "answer": "D", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "两个平面相交，如果它们所成的二面角是直二面角，那么这两个平面的位置关系是什么？", "options": ["都不是", "互相平行", "互相垂直", "互相异面"], "answer": "C", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "两个平面相交，它们所成的二面角在什么情况下，就说这两个平面互相垂直", "options": ["锐二面角", "钝二面角", "都不是", "直二面角"], "answer": "D", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "如果一个平面经过另一个平面的一条怎样的直线，那么这两个平面垂直", "options": ["平行线", "都不是", "垂线", "任意直线"], "answer": "C", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "公式$y = M \\sin(\\omega x + \\phi) + N$中，$N$为？", "options": ["振幅", "初始相位角", "角频率", "轮心到地面的距离"], "answer": "D", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "正弦的两角和公式$\\sin(A + B) = \\sin A \\cdot \\cos B + $______?", "options": ["$2 \\sin A \\cdot \\cos A$", "$\\cos A \\cdot \\sin B$", "$\\sin A \\cdot \\sin B$", "$\\tan A \\cdot \\tan B$"], "answer": "B", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "$\\cos(A + B) = \\cos A \\cdot \\cos B - \\sin A \\cdot \\sin B$是下列哪一函数公式？", "options": ["余弦的两角差公式", "正弦的两角和公式", "正切两角和公式", "余弦的两角和公式"], "answer": "D", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "正切两角和公式:$\\tan(A + B) = \\frac{\\tan A + \\tan B}$_____?", "options": ["${\\tan A + \\tan B}$", "${\\tan A - \\tan B}$", "${1 - \\tan A \\cdot \\tan B}$", "${1 + \\tan A \\cdot \\tan B}$"], "answer": "C", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "正弦的两角差公式:$\\sin(A - B) = \\sin A \\cdot \\cos B - $______?", "options": ["$\\tan A \\cdot \\tan B$", "$\\cos A \\cdot \\sin B$", "$\\sin A \\cdot \\cos B$", "$ \\cos A \\cdot \\cos B $"], "answer": "B", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "$\\cos A \\cdot \\cos B + \\sin A \\cdot \\sin B=$______?", "options": ["$\\cos(2A)$", "$\\cos(A + B) $", "$\\cos(A - B)$", "$tan(A+B) $"], "answer": "C", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "$\\tan(A - B)=\\frac{\\tan A - \\tan B}{1 + \\tan A \\cdot \\tan B}$是什么公式？", "options": ["正切的二角和公式", "余弦的二角差公式", "正切的两角差公式", "正切的二倍角公式"], "answer": "C", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "正弦的二倍角公式：$\\sin(2A) = $_________?", "options": ["$2 \\cos^2 A - 1$", "$1 - 2 \\sin^2 A$", "$2 \\sin A \\cdot \\cos A$", "$\\cos^2 A - \\sin^2 A$"], "answer": "C", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "$\\cos^2 A - \\sin^2 A = 2 \\cos^2 A - 1 = 1 - 2 \\sin^2 A=$_______?", "options": ["$\\sin(2A)$", "$\\cos^2 A$", "$\\cos(2A)$", "$\\sin^2 A$"], "answer": "C", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "$\\frac{2 \\tan A}{1 - \\tan^2 A}=$______?", "options": ["$\\tan^2 A$", "$\\cos^2 A$", "$\\tan(2A)$", "$\\cos(2A)$"], "answer": "C", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "$ \\frac{ \\sin 2A}{1 + \\cos 2A} = \\frac{ 1 - \\cos 2A}{\\sin 2A}=$______?", "options": ["$\\tan(A) $", "$\\tan(2A)$", "$\\tan^2 A - 1$", "$\\tan^2 A$"], "answer": "A", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "在三角形ABC中，三个角正切值的和与积的关系是？", "options": ["相等", "和大于积", "没有关系", "和小于积"], "answer": "A", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "升幂降角$\\sin^2 x$可以转换为？", "options": ["$\\cos A \\cdot \\cos B + \\sin A \\cdot \\sin B$", "$\\frac{1 - \\cos(2x)}{2}$", "\\frac{1 - \\cos(2x)}{2}", "$ \\cos^2 x - \\sin^2 x$"], "answer": "B", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "升角降幂$\\cos(2x)$可以转换为？", "options": ["$\\frac{1 - \\cos(2x)}{2}$", "\\frac{1 - \\cos(2x)}{2}", "$\\cos A \\cdot \\cos B + \\sin A \\cdot \\sin B$", "$ \\cos^2 x - \\sin^2 x$"], "answer": "D", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "$\\sqrt{A^2 + B^2} \\cdot \\sin\\left(x + \\phi\\right) = A \\sin x + $_______?\n其中$\\phi = \\arctan\\left(\\frac{B}{A}\\right)$", "options": ["$\\sin A \\cdot \\sin B$", "$\\sin\\left(x + \\phi\\right)$", "$ \\cos^2 x - \\sin^2 x$", "$B \\cos x$"], "answer": "D", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "$y = A\\sin^2\\alpha+B\\sin\\alpha+C$换元转化成二次函数，第一步就是将$\\sin^2\\alpha$转换为?", "options": ["$ \\cos^2 x - \\sin^2 x$", "$\\sin A \\cdot \\sin B$", "$\\frac{1 - \\cos 2\\alpha}{2}$", "$ \\cos^2 x - \\sin^2 x$"], "answer": "C", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "运用公式$y =\\frac{\\sin\\theta-a}{\\cos\\theta-b}$求导$\\theta_0$的斜率时，可以将上述公式转化为:$\\text{斜率} = \\frac{1 - b\\cos(\\theta_0) + a\\sin(\\theta_0)}{____}$?", "options": ["${(\\cos(\\theta_0) - b)^2}$", "$b\\cos(\\theta_0)$", "${\\cos\\theta-b}$", "$(n-2) \\cdot \\pi$"], "answer": "A", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "n边形的内角和等于?", "options": ["$(n-2) \\cdot \\pi$", "$^2n \\cdot \\pi$", "$(2n +1) \\cdot \\pi$", "$2n \\cdot \\pi$"], "answer": "A", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "$\\sin15=\\cos75=$______?", "options": ["$\\frac{\\sqrt6+\\sqrt2}{4}$", "$\\frac{\\sqrt6-\\sqrt2}{4}$", "$2$", "$\\sin A \\cdot \\sin B$"], "answer": "B", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "$\\sin75=\\cos15=$_______?", "options": ["$\\sin A \\cdot \\sin B$", "$\\frac{\\sqrt6+\\sqrt2}{4}$", "$2$", "$\\frac{\\sqrt6-\\sqrt2}{4}$"], "answer": "B", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "下列排列数公式错误的是？", "options": ["$A_{n}^{m}=n(n-1)(n-2)\\cdots(n-m)$", "$A_{n}^{m}=n(n-1)(n-2)\\cdots(n-m+1)$", "$A_{n}^{m}={\\frac{n!}{\\left(n-m\\right)!}}$", "$A_{n}^{n}=n!$，规定$0!=1$"], "answer": "A", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "下列组合数公式错误的是？", "options": ["$C_{n}^{m}={\\frac{n(n-1)(n-2)\\cdots(n-m)}{m!}}$", "$C_{n}^{m}={\\frac{n(n-1)(n-2)\\cdots(n-m+1)}{m!}}$", "$C_{n}^{m}=\\frac{n!}{m!\\left(n-{m}\\right)!}$", "$C_{n}^{m}=C_{n}^{n-m}$，规定$C_{n}^{0}=1$"], "answer": "A", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "排列与组合的联系为$A_{n}^{m}=C_{n}^{m}\\cdot A_{m}^{m}$，即排列就是?", "options": ["全排列", "全组合", "先组合再全排列", "先排列再全组合"], "answer": "C", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "下面哪个排列性质是正确的？", "options": ["$A_{n+1}^{m}=A_{n+1}^{m}+m A_{n}^{m-1}$", "$A_{n+1}^{m}=A_{n}^{m}+m A_{n}^{m-1}$", "$A_{n+1}^{m}=A_{n}^{m}+m A_{n-1}^{m-1}$", "$A_{n}^{m}=A_{n}^{m}+m A_{n}^{m-1}$"], "answer": "B", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "下面哪个组合性质是正确的？", "options": ["$C_{n+1}^{m}=C_{n}^{m}+C_{n}^{m-1}$", "$C_{n}^{m}=C_{n}^{m}+C_{n}^{m-1}$", "$C_{n+1}^{m}=C_{n}^{m}+C_{n-1}^{m-1}$", "$C_{n+1}^{m}=C_{n+1}^{m}+C_{n}^{m-1}$"], "answer": "A", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "杨辉三角除每行最左侧与最右侧的数字以外，每个数字等于?", "options": ["它的左上方与右上方两个数字之差", "它的左上方与右上方两个数字之商", "它的左上方与右上方两个数字之积", "它的左上方与右上方两个数字之和"], "answer": "D", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "二项式展开公式为？", "options": ["$(a+b)^{n}=C_{n}^{0}a^{n}+C_{n}^{1}a^{n-1}b+C_{n}^{2}a^{n-2}b^{2}+\\cdots+C_{n}^{r}a^{n-r}b^{r}+\\cdots+C_{n}^{n}b^{n}(n\\in N_{+})$", "$(a+b)^{n}=C_{n}^{0}a^{n}+C_{n}^{1}a^{n-1}b+C_{n}^{2}a^{n-2}b^{2}+\\cdots+C_{n}^{r}a^{n-r}b^{r}+\\cdots+C_{n}^{n}b^{n-1}(n\\in N_{+})$", "$(a+b)^{n}=C_{n}^{0}a^{n}+C_{n}^{1}a^{n-1}b+C_{n}^{2}a^{n-2}b^{2}+\\cdots+C_{n}^{r}a^{n-r}b^{r}+\\cdots+C_{n}^{n}b^{n+1}(n\\in N_{+})$", "$(a+b)^{n}=C_{n}^{0}a^{n}+C_{n}^{1}a^{n-1}b+C_{n}^{2}a^{n-2}b^{2}+\\cdots+C_{n}^{r}a^{n-r}b^{r}+\\cdots+C_{n}^{n}b^{n+2}(n\\in N_{+})$"], "answer": "A", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "二项式展开式的通项公式为？", "options": ["$T_{r}=C_{n}^{r}a^{n-r}b^{r}\\left(0\\leq r\\leq n,r\\in N,n\\in N_{+}\\right)$", "$T_{r+1}=C_{n}^{r-1}a^{n-r}b^{r}\\left(0\\leq r\\leq n,r\\in N,n\\in N_{+}\\right)$", "$T_{r+1}=C_{n}^{r+1}a^{n-r}b^{r}\\left(0\\leq r\\leq n,r\\in N,n\\in N_{+}\\right)$", "$T_{r+1}=C_{n}^{r}a^{n-r}b^{r}\\left(0\\leq r\\leq n,r\\in N,n\\in N_{+}\\right)$"], "answer": "D", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "二项式展开式的通项公式主要用途为？", "options": ["求项的差", "求项的合", "求指定的项", "求项的个数"], "answer": "C", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "当项的系数和二项式系数相等时，二项式的两个项的系数为？", "options": ["-1", "2", "1", "0"], "answer": "C", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "$(1+x)^{n}$的展开式为?", "options": ["$\\bigl(1+x\\bigr)^{n}=C_{n}^{0}x^{n}+C_{n}^{1}x^{n-1}+C_{n}^{2}x^{n-2}+\\dots+C_{n}^{n}x^{3}$", "$\\bigl(1+x\\bigr)^{n}=C_{n}^{0}x^{n}+C_{n}^{1}x^{n-1}+C_{n}^{2}x^{n-2}+\\dots+C_{n}^{n}x^{1}$", "$\\bigl(1+x\\bigr)^{n}=C_{n}^{0}x^{n}+C_{n}^{1}x^{n-1}+C_{n}^{2}x^{n-2}+\\dots+C_{n}^{n}x^{2}$", "$\\bigl(1+x\\bigr)^{n}=C_{n}^{0}x^{n}+C_{n}^{1}x^{n-1}+C_{n}^{2}x^{n-2}+\\dots+C_{n}^{n}x^{0}$"], "answer": "D", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "$(1+x)^{n}$的展开式为$\\bigl(1+x\\bigr)^{n}=C_{n}^{0}x^{n}+C_{n}^{1}x^{n-1}+C_{n}^{2}x^{n-2}+\\dots+C_{n}^{n}x^{0}$，若令$x=1$，则有？", "options": ["$\\left(1+1\\right)^{n}=2^{n}=C_{n}^{0}+C_{n}^{1}+C_{n}^{2}+\\dots+C_{n}^{n}$", "$\\left(1+1\\right)^{n}=2^{n}=C_{n}^{0}+C_{n}^{1}+C_{n}^{2}+\\dots+C_{n}^{n-2}$", "$\\left(1+1\\right)^{n}=2^{n}=C_{n}^{0}+C_{n}^{1}+C_{n}^{2}+\\dots+C_{n}^{n-3}$", "$\\left(1+1\\right)^{n}=2^{n}=C_{n}^{0}+C_{n}^{1}+C_{n}^{2}+\\dots+C_{n}^{n-1}$"], "answer": "A", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "二项式系数的对称性为", "options": ["与首末两端“等距离”的两个二项式系数不相等，即$C_{n}^{m}=C_{m}^{n}$", "与首末两端“等距离”的两个二项式系数相等，即$C_{n}^{m}=C_{n}^{n-m}$", "与首末两端“等距离”的两个二项式系数相等，即$C_{n}^{m}=C_{m}^{n}$", "与首末两端“等距离”的两个二项式系数不相等，即$C_{n}^{m}=C_{n}^{n-m}$"], "answer": "B", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "设第$r$项的系数$A_{\\mathrm{r}}$最大，要确定$r$，选用的不等式组为？", "options": ["$\\begin{cases}A_{\\mathrm{r}}\\geq A_{\\mathrm{r-1}}\\\\A_{\\mathrm{r}}\\geq A_{\\mathrm{r-2}}\\end{cases}$", "$\\begin{cases}A_{\\mathrm{r}}\\geq A_{\\mathrm{r-2}}\\\\A_{\\mathrm{r}}\\geq A_{\\mathrm{r+2}}\\end{cases}$", "$\\begin{cases}A_{\\mathrm{r}}\\geq A_{\\mathrm{r-1}}\\\\A_{\\mathrm{r}}\\geq A_{\\mathrm{r+1}}\\end{cases}$", "$\\begin{cases}A_{\\mathrm{r}}\\geq A_{\\mathrm{r-2}}\\\\A_{\\mathrm{r}}\\geq A_{\\mathrm{r+1}}\\end{cases}$"], "answer": "C", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "二项式性质$A_{n+1}^{m}=$?", "options": ["$A_{n}^{m}-mA_{n}^{m-1}$", "$A_{n}^{m}+mA_{n}^{m+1}$", "$A_{n}^{m+1}+mA_{n}^{m-1}$", "$A_{n}^{m}+mA_{n}^{m-1}$"], "answer": "D", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "当$A,B$是互斥事件时，那么事件$A+B$发生（即$A,B$中有一个发生）的概率，等于事件$A,B$分别发生的概率的和，即？", "options": ["$P(A-B)=P(A)+P(B)$", "$P(A+B)=P(A)+P(B)$", "$P(A+B)=P(A)-P(B)$", "$P(A-B)=P(A)-P(B)$"], "answer": "B", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "对立事件的概率和等于1,记作？", "options": ["$P({\\overline{{A}}})=P(A)-1$", "$P({\\overline{{A}}})=1+P(A)$", "$P({\\overline{{A}}})=1-P(A)$", "$P({\\overline{{A}}})=P(A)$"], "answer": "C", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "事件A（或B）是否发生对事件B（或A）发生的概率没有影响，（即其中一个事件是否发生对另一个事件发生的概率没有影响）.这样的两个事件叫做?", "options": ["相互独立事件", "相互事件", "不独立事件", "独立事件"], "answer": "A", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "当$A,B$是相互独立事件时，那么事件$A,B$同时发生的概率等于事件分别发生的概率为？", "options": ["$P(A\\cdot B)=P(A)-P(B)$", "$P(A\\cdot B)=P(A)+P(B)$", "$P(A\\cdot B)=P(A)\\div P(B))$", "$P(A\\cdot B)=P(A)\\cdot P(B)$"], "answer": "D", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "独立重复试验的概率公式为？", "options": ["$P_{n}(k)=C_{n+1}^{k}\\,p^{k}\\,(1-p)^{n-k}\\quad\\left(k=0,1,2,\\cdots n\\right)$", "$P_{n}(k)=C_{n}^{k-1}\\,p^{k}\\,(1-p)^{n-k}\\quad\\left(k=0,1,2,\\cdots n\\right)$", "$P_{n}(k)=C_{n}^{k}\\,p^{k}\\,(1-p)^{n-k}\\quad\\left(k=0,1,2,\\cdots n\\right)$", "$P_{n}(k)=C_{n-1}^{k}\\,p^{k}\\,(1-p)^{n-k}\\quad\\left(k=0,1,2,\\cdots n\\right)$"], "answer": "C", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "A发生的条件下B发生的概率公式为？", "options": ["$P(B|A)={\\frac{P(A B)}{P(A)}},P(A)>0$", "$P(B|A)={\\frac{P(B)}{P(A)}},P(A)>0$", "$P(B|A)={\\frac{P(A B)}{P(A)}},P(A)<0$", "$P(B|A)={\\frac{P(A B)}{P(B)}},P(A)>0$"], "answer": "A", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "如果随机试验的结果可用一个变量来表示，那么这样的变量叫做?", "options": ["特殊变量", "特殊常量", "常量", "随机变量"], "answer": "D", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "对于随机变量可能取的值，可以按一定次序一一列出，这样的随机变量叫做?", "options": ["连续型常量", "离散型常量", "离散型随机变量", "连续型随机变量"], "answer": "C", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "对于随机变量可能取的值，可以取某一区间内的一切值，这样的变量就叫做？", "options": ["离散型随机变量", "连续型常量", "连续型随机变量", "离散型常量"], "answer": "C", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "概率分布的性质为？", "options": ["${\\cal P}_{i}\\geq0,i=1,2,...n;\\sum_{i=1}^{n}{\\cal P}_{i}=3$", "${\\cal P}_{i}\\geq0,i=1,2,...n;\\sum_{i=1}^{n}{\\cal P}_{i}=1$", "${\\cal P}_{i}\\geq0,i=1,2,...n;\\sum_{i=1}^{n}{\\cal P}_{i}=4$", "${\\cal P}_{i}\\geq1,i=1,2,...n;\\sum_{i=1}^{n}{\\cal P}_{i}=2$"], "answer": "B", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "X服从两点分布，则成功概率为?", "options": ["$p=P(X=1)$", "$p=P(X=2)$", "$p=P(X=3)$", "$p=P(X=0)$"], "answer": "A", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "随机变量X服从二项分布,记作？", "options": ["$X\\sim B(n,1)$", "$X\\sim B(1,p)$", "$X\\sim B(n,p)$", "$X\\sim B(p,n)$"], "answer": "C", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "判断一个随机变量是否服从二项分布，关键有哪三点？", "options": ["不对立性,重复性,等概率性", "对立性,重复性,不等概率性", "对立性,不重复性,等概率性", "对立性,重复性,等概率性"], "answer": "D", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "在含有M件次品的N件产品中，任取n件,其中恰有X件次品数,则事件{X=k}发生的概率为$P(X=k)={\\frac{C_{M}^{k}C_{N-M}^{n-k}}{C_{N}^{n}}}(k=0,1,2,\\cdots,m)$，于是得到随机变量X的概率分布$m=\\operatorname*{min}\\left\\{M,n\\right\\},n\\leq N,M\\leq N,n,M,N\\in N^{\\ast } $,我们称这样的随机变量X的分布列为超几何分布列,且称随机变量X服从？", "options": ["两点分布", "超几何分布", "几何分布", "二项分布"], "answer": "B", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "下列离散型随机变量的均值性质错误的为？", "options": ["若$X\\sim H\\{n,M,N\\}$，则$E(X)\\!=\\!{\\frac{{M}}{N}}\\!\\cdot\\!n$", "$E(a X+b)=a E(X)+b$", "若X服从两点分布，则$E(X)=p$", "$E(a X+b)=a E(X)-b$"], "answer": "D", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "离散型随机变量的方差性质为?", "options": ["DX越小，X的稳定性越高，波动越小，取值越集中；DX越大，X的稳定性越差，波动越大，取值越分散", "DX越大，X的稳定性越高，波动越小，取值越集中；DX越小，X的稳定性越差，波动越大，取值越分散", "DX越小，X的稳定性越差，波动越小，取值越集中；DX越大，X的稳定性越高，波动越大，取值越分散", "DX越小，X的稳定性越高，波动越大，取值越集中；DX越大，X的稳定性越差，波动越小，取值越分散"], "answer": "A", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "下列关于正态分布性质正确的是？", "options": ["$X< \\mu $时曲线下降,$X> \\mu$时曲线上升", "$S_{总} =2$", "$\\sigma $越大，曲线越尖陡，$\\sigma $越小，曲线越扁平", "关于$X= \\mu $对称"], "answer": "D", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "独立性检验公式为?", "options": ["$K^{2} =\\frac{n(ad-bc)^{3}}{(a+b)(a+c)(b+d)(c+d)} $（其中$n=a+b+c+d$为样本容量）", "$K^{2} =\\frac{n(bc-ad)^{2}}{(a+b)(a+c)(b+d)(c+d)} $（其中$n=a+b+c+d$为样本容量）", "$K^{2} =\\frac{n(ad-bc)^{2}}{(a+b)(a+c)(b+d)(c+d)} $（其中$n=a+b+c+d$为样本容量）", "$K^{2} =\\frac{n(ad+bc)^{2}}{(a+b)(a+c)(b+d)(c+d)} $（其中$n=a+b+c+d$为样本容量）"], "answer": "C", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "既有<<<Answer 1>>>又有<<<Answer 2>>>的量叫做向量", "options": ["大小,速度", "力，大小", "方向,速度", "大小,方向"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} |
| {"question": "带有方向的线段叫做有向线段，有向线段包含的三个要素是？", "options": ["[”起点“、”方向“、”长度“]", "[”起点“、”终点“、”长度“]", "[”起点“、”方向“、”终点“]", "[”终点“、”方向“、”长度“]"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} |
| {"question": "向量$\\vec{AB}$的大小，也就是向量$\\vec{AB}$的长度（或称模），记作<<<Answer 1>>>；长度为零的向量叫做<<<Answer 2>>>；长度等于1个单位的向量叫做<<<Answer 3>>>.", "options": ["$\\lvert\\vec{AB} \\rvert$,零向量,单位向量", "$\\lvert \\vec{AB} \\rvert$,单位向量,零向量", "$\\vec{AB} $,零向量,单位向量", "$\\lvert \\vec{AB} \\rvert$,平行向量,单位向量"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} |
| {"question": "方向相同或相反的非零向量叫做<<<Answer>>>.", "options": ["相反向量", "单位向量", "零向量", "平行向量"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} |
| {"question": "<<<Answer>>>与任意向量平行", "options": ["平行向量", "零向量", "相反向量", "单位向量"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} |
| {"question": "$\\vec{a}$,$\\vec{b}$夹角为\\theta，则在上投影的长度为<<<Answer>>>", "options": ["$ \\lvert \\vec{a} \\rvert \\ccdot \\sin \\theta$", "$ \\lvert \\vec{a} \\rvert \\ccdot \\cot \\theta$", "$ \\lvert \\vec{a} \\rvert \\ccdot \\tan \\theta$", "$ \\lvert \\vec{a} \\rvert \\ccdot \\cos \\theta$"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} |
| {"question": "长度相等且方向相同的向量叫做<<<Answer>>>", "options": ["零向量", "单位向量", "相反向量", "相等向量"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} |
| {"question": "$ \\lvert \\vec{a}+\\vec{b} \\rvert $<<<Answer>>>$ \\lvert \\vec{a} \\rvert+\\lvert \\vec{b} \\rvert $", "options": ["$ > $", "$ \\lep $", "$ \\geq $", "$ < $"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} |
| {"question": "与$ \\vec{a} $长度相等方向相反的向量叫做\\vec{a}的<<<Answer>>>", "options": ["相反向量", "零向量", "单位向量", "相等向量"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} |
| {"question": "实数$\\Lambda$与向量$a$的积是一个向量，这种运一个实数和一个向量的运算叫做向量的数乘.记作：$\\Lambda \\cdot \\vec{a}$. 当$\\Lambda$<<<Answer 1>>>$0 $ 时，$\\Lambda \\cdot \\vec{a}$ 的方向与$ \\vec{a} $的方向相同；当\\Lambda$<<<Answer 2>>>$0$ 时，$\\Lambda \\cdot \\vec{a}$ 的方向与$ \\vec{a} $的方向相反.", "options": ["$>$,$<$", "$<$,$>$", "$<$,$<$", "$>$,$>$"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} |
| {"question": "向量$ \\vec{a} (\\vec{a} \\neq \\vec{0} $与 $\\vec{b} $共线，有几个实数$\\Lambda$，使$ \\vec{b}=\\Lambda \\cdot \\vec{a}$", "options": ["一个", "四个", "三个", "二个"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} |
| {"question": "如果$ \\vec{e1} $,$ \\vec{e2} $是同一平面内的两个不共线向量，那么对于这一平面内任一向量$ \\vec{a} $,有几对实数$\\Lambda1$,$\\Lambda2$,使$ \\vec{a}=\\Lambda1 \\cdot \\vec{e1}+\\Lambda2 \\cdot \\vec{e2}$", "options": ["二对", "三对", "四对", "一对"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} |
| {"question": "$ \\vec{OC}=\\Lambda \\cdot \\vec{OA}+\\mu \\cdot \\vec{OB}$,如果$A$、$B$、$C$三点共线，那么$\\Lambda+\\mu=$<<<Answer>>>?", "options": ["$1$", "$0$", "$5$", "$2$"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} |
| {"question": "$\\vec{a}=x \\cdot \\vec{i}+y \\cdot \\vec{j}=$<<<Answer>>>", "options": ["$(i,j)$", "$(x,j)$", "$(i,y)$", "$(x,y)$"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} |
| {"question": "设$\\vec{a}=（x_1,y_1)$,$\\vec{b}=（x_2,y_2)$,则：\n$\\vec{a}+\\vec{b}=$<<<Answer>>>.", "options": ["$(y_1+x_2,x_1+y_2)$", "$(x_1+x_2,y_1+y_2)$", "$(x_1+y_2,y_1+x_2)$", "$(x_1-x_2,y_1-y_2)$"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} |
| {"question": "设$\\vec{a}=（x_1,y_1)$,$\\vec{b}=（x_2,y_2)$,则：\n$\\vec{a}-\\vec{b}=$<<<Answer>>>", "options": ["$(x_1-x_2,y_1-y_2)$", "$(x_1+x_2,y_1+y_2)$", "$(x_1+x_2,y_1-y_2)$", "$(x_1-x_2,y_1+y_2)$"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} |
| {"question": "设$\\vec{a}=（x_1,y_1)$,$\\vec{b}=（x_2,y_2)$,则：\n$\\Lambda \\cdot \\vec{a}=$<<<Answer>>>,", "options": ["$(\\Lambda \\cdot x_2,\\Lambda \\cdot y_2)$", "$(\\Lambda \\cdot x_2,\\Lambda \\cdot y_1)$", "$(\\Lambda \\cdot x_1,\\Lambda \\cdot y_2)$", "$(\\Lambda \\cdot x_1,\\Lambda \\cdot y_1)$"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} |
| {"question": "设$\\vec{a}=（x_1,y_1)$,$\\vec{b}=（x_2,y_2)$,则：\n$\\vec{a} \\parallel \\vec{b} \\Leftrightarrow x_1 \\cdot y_2=$<<<Answer>>>.", "options": ["$ x_2 \\cdot y_2 $", "$ y_2 \\cdot x_1 $", "$ x_1 \\cdot y_1 $", "$ x_2 \\cdot y_1 $"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} |
| {"question": "设$A(x_1,y_1)$,$B(x_2,y_2),则:\n$\\vec{AB}=$<<<Answer>>>.", "options": ["$(x_2-x_1,y_1-y_2)$", "$(x_2-x_1,y_2-y_1)$", "$(x_1-x_2,y_2-y_1)$", "$(x_1-x_2,y_1-y_2)$"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} |
| {"question": "设$A(x_1,y_1)$,$B(x_2,y_2)$,$C(x_3,y_3)$,则线段$AB$中点坐标为<<<Answer>>>.", "options": ["$(\\frac{x_1+x_2}{2},\\frac{y_1+y_2}{2})$", "$(\\frac{x_1+y_2}{2},\\frac{y_1+x_2}{2})$", "$(\\frac{y_1+x_2}{2},\\frac{x_1+y_2}{2})$", "$(\\frac{x_1+x_3}{2},\\frac{y_3+y_2}{2})$"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} |
| {"question": "设$A(x_1,y_1)$,$B(x_2,y_2)$,$C(x_3,y_3)$,则$\\triangleABC$的重心坐标为<<<Answer>>>.", "options": ["$(\\frac{x_1+x_2+x_3}{2},\\frac{y_1+y_2+y_3}{2})$", "$(\\frac{x_1+x_2+y_3}{3},\\frac{y_1+y_2+x_3}{3})$", "$(\\frac{x_1+x_2+x_3}{3},\\frac{y_1+y_2+y_3}{3})$", "$(\\frac{x_1+y_2+x_3}{3},\\frac{y_1+x_2+y_3}{3})$"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} |
| {"question": "$\\vec{a}\\cdot \\vec{b}=\\lvert \\vec{a} \\rvert \\cdot \\lvert \\vec{b} \\rvert $<<<Answer>>>", "options": ["$\\cot \\theta$", "$\\sin \\theta$", "$\\tan \\theta$", "$\\cos \\theta$"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} |
| {"question": "$\\vec{a}$在$\\vec{b}$方向上的投影是:$\\lvert \\vec{a} \\rvert$<<<Answer>>>(可以为负)", "options": ["$\\cos \\theta$", "$\\tan \\theta$", "$\\cot \\theta$", "$\\sin \\theta$"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} |
| {"question": "$\\vec{a}^2$<<<Answer>>>$\\lvert \\vec{a} \\rvert^2$", "options": ["$\\neq$", "$>$", "$=$", "$<$"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} |
| {"question": "$\\lvert \\vec{a} \\rvert=$\\sqrt{<<<Answer>>>}$", "options": ["$\\vec{a}^3$", "$\\vec{a}^2$", "$\\vec{a}$", "$\\vec{a}^4$"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} |
| {"question": "$\\vec{a} \\bot \\vec{b} \\Leftrightarrow vec{a} \\cdot vec{b}=$", "options": ["$0$", "$5$", "$10$", "$1$"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} |
| {"question": "向量角的范围是？", "options": ["$\\left ( 0,\\frac{\\pi}{2} \\right ] $", "$\\left [ 0,\\pi \\right ] $", "$\\left [ 0,\\pi \\right ] $", "$\\left [ 0,\\frac{\\pi }{2} \\right ] $"], "answer": "B", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "线线角（异面直线所成的角）的范围是？", "options": ["$\\left ( 0,\\frac{\\pi}{2} \\right ] $", "$\\left [ 0,\\frac{\\pi }{2} \\right ] $", "$\\left [ 0,\\pi \\right ] $", "$\\left [ 0,\\pi \\right ] $"], "answer": "A", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "线面角的范围是？", "options": ["$\\left [ 0,\\frac{\\pi }{2} \\right ] $", "$left ( 0,\\frac{\\pi}{2} \\right ] $", "$\\left [ 0,\\pi \\right ] $", "$\\left [ 0,\\pi \\right ] $"], "answer": "A", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "二面角的范围是？", "options": ["$\\left [ 0,\\pi \\right ] $", "$\\left [ 0,\\frac{\\pi }{2} \\right ] $", "$\\left [ 0,\\pi \\right ] $", "$\\left ( 0,\\frac{\\pi}{2} \\right ] $"], "answer": "A", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "设向量a和向量b不共线。根据共面定理，下列哪个条件是判断向量p与向量a、b共面的正确表述？", "options": ["存在实数x和y，使得$p=xa+yb$", "向量p可以表示为向量a和向量b的和", "向量a和向量b必须是单位向量", "向量p必须垂直于向量a和向量b"], "answer": "A", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "若 A、B是直线l上的任意两点，则$\\overrightarrow{AB} $以及与$\\overrightarrow{AB} $平行的任意非零向量为直线l的什么向量？", "options": ["法向量", "垂直向量", "单位向量", "方向向量"], "answer": "D", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "如果$\\overrightarrow{n} \\perp \\alpha $，那么向量$\\overrightarrow{n} $叫做平面$\\alpha $的什么向量？", "options": ["垂直向量", "方向向量", "单位向量", "法向量"], "answer": "D", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "设直线 $l_{1} $ 和直线 $l_{2} $ 的方向向量分别是 $\\overrightarrow{a} $ 和 $\\overrightarrow{b} $。要证明两条直线 $l_{1} $ 和 $l_{2} $ 平行，应当证明下列哪项条件？", "options": ["向量 $\\overrightarrow{a}$ 和 $\\overrightarrow{b}$ 不能表示为彼此的线性组合", "存在实数 k 使得 $\\overrightarrow{a} = k \\overrightarrow{b} $", "向量 $\\overrightarrow{a}$ 和 $\\overrightarrow{b}$ 垂直", "向量 $\\overrightarrow{a}$ 和 $\\overrightarrow{b}$ 的长度相等"], "answer": "B", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "若两直线平行或重合，当且仅当满足什么条件？", "options": ["两直线的方向向量垂直", "两直线的方向向量不能表示为彼此的线性组合", "两直线的方向向量的长度相等", "两直线的方向向量共线"], "answer": "D", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "设直线 l 的方向向量是 $\\overrightarrow{a} $，平面 $\\alpha $ 的法向量是 $\\overrightarrow{u} $。要证明直线 l 平行于平面 $\\alpha $，应当证明下列哪项条件？", "options": ["$\\overrightarrow{a} $ 和 $\\overrightarrow{u} $ 是共线的", "$\\overrightarrow{a} $ 和 $\\overrightarrow{u} $ 的长度相等", "$\\overrightarrow{a} \\cdot \\overrightarrow{u} = 0 $", "$\\overrightarrow{a} \\cdot \\overrightarrow{u} = 1 $"], "answer": "C", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "若直线与平面平行，当且仅当满足什么条件？", "options": ["直线的方向向量与该平面的法向量的长度相等", "直线的方向向量与该平面的法向量是共线的", "直线的方向向量与该平面的法向量的方向相同或相反", "直线的方向向量与该平面的法向量垂直且直线在平面外"], "answer": "D", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "若平面 $\\alpha $ 的法向量为 $\\overrightarrow{u} $，平面 $\\beta $ 的法向量为 $\\overrightarrow{v} $，要证明平面 $\\alpha $ 与平面 $\\beta $ 平行，应当证明下列哪项条件？", "options": ["$\\overrightarrow{u} $ 和 $\\overrightarrow{v} $ 垂直", "$\\overrightarrow{u} $ 和 $\\overrightarrow{v} $ 不共线", "存在实数 $\\lambda $ 使得 $\\overrightarrow{u} = \\lambda \\overrightarrow{v} $", "$\\overrightarrow{u} $ 和 $\\overrightarrow{v} $ 的长度相等"], "answer": "C", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "若两平面平行或重合，当且仅当满足什么条件？", "options": ["两平面的法向量共线", "两平面的法向量不共线", "两平面的法向量垂直", "两平面的法向量的长度相等"], "answer": "A", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "设直线 $l_{1} $ 和直线 $l_{2} $ 的方向向量分别是 $\\overrightarrow{a} $ 和 $\\overrightarrow{b} $。要证明两条直线 $l_{1} $ 和 $l_{2} $ 垂直，应当证明下列哪项条件？", "options": ["$\\overrightarrow{a} $ 和 $\\overrightarrow{b} $ 的长度相等", "$\\overrightarrow{a} \\cdot \\overrightarrow{b} = 0 $", "$\\overrightarrow{a} $ 和 $\\overrightarrow{b} $ 是共线的", "$\\overrightarrow{a} \\cdot \\overrightarrow{b} = 1 $"], "answer": "B", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "若两直线垂直，当且仅当满足什么条件？", "options": ["两直线的方向向量垂直", "两直线的方向向量是共线的", "两直线的方向向量的方向相同或相反", "两直线的方向向量的长度相等"], "answer": "A", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "设直线 l 的方向向量是 $\\overrightarrow{a} $，平面 $\\alpha $ 的法向量是 $\\overrightarrow{u} $。要证明直线 l 垂直于平面 $\\alpha $，应当证明下列哪项条件？", "options": ["$\\overrightarrow{a} $ 和 $\\overrightarrow{u} $ 垂直", "$\\overrightarrow{a} $ 和 $\\overrightarrow{u} $ 平行，即存在实数 $\\lambda $ 使得 $\\overrightarrow{a} = \\lambda \\overrightarrow{u} $", "$\\overrightarrow{a} $ 和 $\\overrightarrow{u} $ 的长度相等", "$\\overrightarrow{a} $ 和 $\\overrightarrow{u} $ 不共线"], "answer": "B", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "若直线与平面垂直，当且仅当满足什么条件？", "options": ["直线的方向向量与平面的法向量的长度相等", "直线的方向向量与平面的法向量垂直", "直线的方向向量与平面的法向量共线，或者直线的方向向量与平面内两条不共线直线的方向向量都垂直", "直线的方向向量与平面的法向量不共线"], "answer": "C", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "若平面 $\\alpha $ 的法向量为 $\\overrightarrow{u} $，平面 $\\beta $ 的法向量为 $\\overrightarrow{v} $，要证明平面 $\\alpha $ 与平面 $\\beta $ 垂直，应当证明下列哪项条件？", "options": ["$\\overrightarrow{u} $ 和 $\\overrightarrow{v} $ 是共线的", "$\\overrightarrow{u} $ 和 $\\overrightarrow{v} $ 的长度相等", "$\\overrightarrow{u} \\cdot \\overrightarrow{v} = 0 $", "$\\overrightarrow{u} \\cdot \\overrightarrow{v} = 1 $"], "answer": "C", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "若两平面垂直，当且仅当满足什么条件？", "options": ["两平面的法向量垂直", "两平面的法向量的方向相同或相反", "两平面的法向量是共线的", "两平面的法向量的长度相等"], "answer": "A", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "已知 a 和 b 为两异面直线，A 和 C 分别是直线 a 上的任意两点，B 和 D 分别是直线 b 上的任意两点。a 和 b 所成的角为 $\\theta $。根据给定的信息，$\\cos \\theta $ 等于下列哪个表达式？", "options": ["$\\frac{\\overrightarrow{AC} + \\overrightarrow{BD}}{\\left | \\overrightarrow{AC} \\right | \\cdot \\left | \\overrightarrow{BD} \\right | } $", "$\\frac{\\overrightarrow{AC} \\cdot \\overrightarrow{BD}}{\\left | \\overrightarrow{AC} \\right | \\cdot \\left | \\overrightarrow{BD} \\right | } $", "$\\frac{\\left | \\overrightarrow{AC} + \\overrightarrow{BD} \\right | }{\\left | \\overrightarrow{AC} \\right | + \\left | \\overrightarrow{BD} \\right | } $", "$\\frac{\\left | \\overrightarrow{AC} \\cdot \\overrightarrow{BD} \\right | }{\\left | \\overrightarrow{AC} \\right | \\cdot \\left | \\overrightarrow{BD} \\right | } $"], "answer": "D", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "下列哪项描述正确地表达了斜线与平面所成角的概念？", "options": ["斜线在平面上的射影与平面的法线之间的角", "斜线与与其平行的平面上的任意直线之间的角", "斜线与平面的法线之间的角", "斜线与其在平面上的射影之间的锐角"], "answer": "D", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "关于二面角的描述，下列哪项是正确的？", "options": ["二面角是由平面上任意两点连线形成的角", "二面角是由两条相交直线构成的图形", "二面角是由两个平行直线之间的空间构成的图形", "二面角是由一条直线出发的两个半平面所组成的图形"], "answer": "D", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "设二面角 $\\alpha - l - \\beta$ 的两个半平面的法向量分别为 $\\overrightarrow{m}$ 和 $\\overrightarrow{n}$，这两个向量的夹角为 $\\varphi $，二面角 $\\alpha - l - \\beta $ 的平面角为 $\\theta $。根据二面角的平面角和法向量夹角的关系，下列哪项描述是正确的？", "options": ["如果 $\\theta $ 是钝角，则 $\\cos \\theta = \\cos \\varphi = \\frac{\\overrightarrow{m} \\cdot \\overrightarrow{n} }{\\left | \\overrightarrow{m} \\right | \\left | \\overrightarrow{n} \\right | } $", "如果 $\\theta $ 是锐角，则 $\\cos \\theta = \\left | \\cos \\varphi \\right | = \\frac{\\left | \\overrightarrow{m} \\cdot \\overrightarrow{n} \\right | }{\\left | \\overrightarrow{m} \\right | \\left | \\overrightarrow{n} \\right | } $", "如果 $\\theta $ 是锐角，则 $\\cos \\theta = \\cos \\varphi = \\frac{\\overrightarrow{m} \\cdot \\overrightarrow{n} }{\\left | \\overrightarrow{m} \\right | \\left | \\overrightarrow{n} \\right | } $", "如果 $\\theta $ 是钝角，则 $\\cos \\theta = \\left | \\cos \\varphi \\right | = \\frac{\\left | \\overrightarrow{m} \\cdot \\overrightarrow{n} \\right | }{\\left | \\overrightarrow{m} \\right | \\left | \\overrightarrow{n} \\right | } $"], "answer": "B", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "在平面几何中，如果一条直线与平面内的任意两条相交直线所成的角 $\\theta $ 相同，那么这条直线的射影在平面上的性质是什么？", "options": ["这条直线的射影是这两条相交直线所成角的角平分线", "这条直线的射影是这两条相交直线所成角的外角平分线", "这条直线的射影与这两条相交直线垂直", "这条直线的射影与这两条相交直线平行"], "answer": "A", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "空间几何体的外接球的球心到该几何体各个顶点的距离相等。关于这个外接球和几何体面上的外接圆的关系，下列哪项描述是正确的？", "options": ["外接球在几何体的每个面上形成的是内切圆", "外接球在各面上的投影就是这个面的外接圆", "外接球的直径等于几何体各面外接圆的直径", "外接球的半径等于几何体最长边的长度"], "answer": "B", "topic": "High--Intermediate Geometry--Lines and Planes in Space"} |
| {"question": "关于圆锥曲线的描述，下列哪项是正确的？", "options": ["圆锥曲线是圆锥被平面截得的边缘轮廓", "圆锥曲线是圆锥底面的轮廓", "圆锥曲线仅包括圆和椭圆", "圆锥曲线是圆锥上的任意一条直线"], "answer": "A", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "当圆锥被一个平面截面所截，并且截面与圆锥的轴 $PO $ 所成角 $\\theta $ 小于 $PO $ 与侧面所成角 $\\delta $ 时，圆锥被平面截得的边缘轮廓是什么曲线？", "options": ["抛物线", "椭圆", "圆", "双曲线"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "当圆锥被一个平面截面所截，并且截面与圆锥的轴 $PO $ 所成角 $\\theta $ 等于 $PO $ 与侧面所成角 $\\delta $ 时，圆锥被平面截得的边缘轮廓是什么曲线？", "options": ["双曲线", "抛物线", "椭圆", "圆"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "当圆锥被一个平面截面所截，并且截面与圆锥的轴 $PO $ 所成角 $\\theta $ 大于 $PO $ 与侧面所成角 $\\delta $ 但小于 $\\frac{\\pi}{2} $ 时，圆锥被平面截得的边缘轮廓是什么曲线？", "options": ["圆", "双曲线", "椭圆", "抛物线"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "当圆锥被一个平面截面所截，并且截面与圆锥的轴 $PO $ 所成角 $\\theta $ 等于 $\\frac{\\pi}{2} $ 时，圆锥被平面截得的边缘轮廓是什么曲线？", "options": ["双曲线", "抛物线", "椭圆", "圆"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "如果与一定点$F$和一条定直线$l$的距离相等的点形成了一个轨迹，那么这个轨迹叫什么？", "options": ["直线", "圆", "抛物线", "椭圆"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "抛物线是一个什么样的轨迹？", "options": ["抛物线平面内到定点$F1$、$F2$的距离之和等于常数(大于$|F1F2|$）的动点$P$的轨迹", "均不是", "抛物线是与一定点$F$和一条定直线$l$的距离相等的点形成的轨迹", "抛物线是一个点在平面或空间沿着一定方向和其相反方向运动的轨迹"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "当$p>0$时，抛物线哪个标准方程是正确的？", "options": ["$y=x$", "$x^2=2py$", "$x^2=-2py$", "$y^2=2px$"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "当$p<0$时，抛物线哪个标准方程是正确的？", "options": ["$y^2=2px$", "$y^2=-2px$", "$x^2=2py$", "$y=x$"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "标准抛物线的离心率$e$是多少？", "options": ["3", "2", "1", "4"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "标准抛物线的顶点坐标是多少？", "options": ["(2,2)", "(0,0)", "(3,3)", "(1,1)"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "标准抛物线的对称轴可能是哪一个轴？", "options": ["$x$轴", "$z=2$轴", "$z=1$轴", "$z$轴"], "answer": "A", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "标准抛物线方程中，焦半径$|MF|$的值如果是两个值的加和，一个值是$p/2$，另一个值是什么？", "options": ["x_0", "x_0+2", "x_0+3", "x_0+1"], "answer": "A", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "标准抛物线方程中，通径的定义是什么？", "options": ["过抛物线的焦点且成45度角于对称轴的弦称为通径", "过抛物线的焦点且平行于对称轴的弦称为通径", "过抛物线的焦点且垂直于对称轴的弦称为通径", "过抛物线的焦点且成60度角于对称轴的弦称为通径"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "标准抛物线方程中，焦点弦长公式是什么？", "options": ["$|AB|=x_1+x_2+p+3$", "$|AB|=x_1+x_2+p+2$", "$|AB|=x_1+x_2+p$", "$|AB|=x_1+x_2+p+1$"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "标准抛物线方程中，参数$p$的几何意义是什么？", "options": ["参数$p$表示焦点到准线的距离，$p$的值与开口大小之间的关系不确定", "参数$p$表示焦点到准线的距离，$p$的值与开口大小无关", "参数$p$表示焦点到准线的距离，$p$越小，开口越阔", "参数$p$表示焦点到准线的距离，$p$越大，开口越阔"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "标准抛物线方程中，焦半径范围是什么？", "options": ["$[p/2，+∞)$", "$[p/2，1)$", "$[p/2，2)$", "$[p/2，0)$"], "answer": "A", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "标准抛物线方程中，已知$Δ=0$，对于$x^2=±2py$和$y=±x^2/2p$的条件下，$y’$和$k$的关系等式是什么？", "options": ["$y?=k+2$", "$y?=k+3$", "$y?=k$", "$y?=k+1$"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "标准抛物线方程中，已知$Δ=0$，对于$y^2=±2px$和$x=±y^2/2p$的条件下，$y’$和$k$的关系等式是什么？", "options": ["$y?=4/k$", "$y?=3/k$", "$y?=1/k$", "$y?=2/k$"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "标准抛物线方程中，$AB⊥y$时，$AB$取什么极值？", "options": ["极大值", "最大值", "极小值", "最小值"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "对于分式$Q=\\frac{\\sqrt{(x-a)^2+(y-b)^2} }{\\frac{|Am+By+C|}{\\sqrt{A^2+B^2} } } $，它的几何意义的实质是什么？", "options": ["该式子实质上就是点$(m,n)$到点$(a,b)$的距离和", "该式子实质上就是点$(m,n)$到到直线$l：Ax+By+C=0$的距离", "该式子实质上就是点$(m,n)$到点$(a,b)$的距离和点$(m,n)$到直线$l：Ax+By+C=0$的距离之比", "该式子实质上就是直线$l：Ax+By+C=0$到点$(m,n)$到点$(a,b)$的距离的距离之比"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "对于分式$Q=\\frac{\\sqrt{(x-a)^2+(y-b)^2} }{\\frac{|Am+By+C|}{\\sqrt{A^2+B^2} } } $，当$0<Q<1$时，$(x,y)$的轨迹为？", "options": ["抛物线", "双曲线", "直线", "椭圆"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "对于分式$Q=\\frac{\\sqrt{(x-a)^2+(y-b)^2} }{\\frac{|Am+By+C|}{\\sqrt{A^2+B^2} } } $，当$Q=1$时，$(x,y)$的轨迹为？", "options": ["双曲线", "直线", "抛物线", "椭圆"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "对于分式$Q=\\frac{\\sqrt{(x-a)^2+(y-b)^2} }{\\frac{|Am+By+C|}{\\sqrt{A^2+B^2} } } $，当$Q>1$时，$(x,y)$的轨迹为？", "options": ["直线", "抛物线", "椭圆", "双曲线"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "对点$(M,N)$,有$AM^2+BN^2=1$,结合三角函数定义，则$M$的值是什么？", "options": ["$\\frac{cos?\\theta}{\\sqrt{A} } $", "cos?\\theta", "$\\frac{sin\\theta}{\\sqrt{B} } $", "sin?\\theta"], "answer": "A", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "对点$(M,N)$,有$AM^2+BN^2=1$,结合三角函数定义，则$N$的值是什么？", "options": ["cos?\\theta", "$\\frac{cos?\\theta}{\\sqrt{A} } $", "sin?\\theta", "$\\frac{sin\\theta}{\\sqrt{B} } $"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "在圆的参数方程中，$x$的值是什么？", "options": ["sin?\\theta", "cos?\\theta", "$b+rsin\\theta$", "$a+rcos?\\theta$"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "在圆的参数方程中，$y$的值是什么？", "options": ["cos?\\theta", "$a+rcos?\\theta$", "sin?\\theta", "$b+rsin\\theta$"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "在圆的参数方程中，$a$的几何意义是什么？", "options": ["圆心纵坐标", "圆心横坐标", "曲率", "半径"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "在圆的参数方程中，$b$的几何意义是什么？", "options": ["半径", "圆心横坐标", "圆心纵坐标", "曲率"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "在圆的参数方程中，$r$的几何意义是什么？", "options": ["曲率", "半径", "圆心纵坐标", "圆心横坐标"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "在焦点在$x$轴的椭圆的参数方程中，$x$的表达式是什么？", "options": ["$a^3+cos?\\theta$", "$a^2+cos?\\theta$", "$b+sin\\theta$", "$a+cos?\\theta$"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "在焦点在$x$轴的椭圆的参数方程中，$y$的表达式是什么？", "options": ["$a^2+cos?\\theta$", "$a^3+cos?\\theta$", "$b+sin\\theta$", "$a+cos?\\theta$"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "在焦点在$x$轴的椭圆的参数方程中，$a$的几何意义是什么？", "options": ["实轴", "面积", "虚轴", "投影"], "answer": "A", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "在焦点在$x$轴的椭圆的参数方程中，$b$的几何意义是什么？", "options": ["虚轴", "投影", "面积", "实轴"], "answer": "A", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "复数的虚数单位一般用什么表示？", "options": ["$i$", "$c$", "$a$", "$b$"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Complex Numbers"} |
| {"question": "复数的代数形式一般是什么样的？", "options": ["$z=a^2+bi(a,b\\in R)$", "$z=a+bi(a,b\\in R)$", "$z=a^3+bi(a,b\\in R)$", "$z=a^4+bi(a,b\\in R)$"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Complex Numbers"} |
| {"question": "在复数的代数表达式$z=a+bi(a,b\\in R)$中，若$b=0$,则复数为什么？", "options": ["无限基数", "纯虚数", "实数", "非纯虚数"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Complex Numbers"} |
| {"question": "在复数的代数表达式$z=a+bi(a,b\\in R)$中，若$a=0$,$b\\ne0$,则复数为什么？", "options": ["纯虚数", "无限基数", "非纯虚数", "实数"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Complex Numbers"} |
| {"question": "在复数的代数表达式$z=a+bi(a,b\\in R)$中，若$$a\\ne0$,$b\\ne0$,则复数为什么？", "options": ["非纯虚数", "无限基数", "实数", "纯虚数"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Complex Numbers"} |
| {"question": "如果$a+bi=c+di$,则$a,b,c,d$之间有什么等式关系？", "options": ["$a^2=b=c=d$", "$a^3=b=c=d$", "$a=b$且$c=d$", "$a=b=c=d$"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Complex Numbers"} |
| {"question": "如果$a+bi=0$,则$a,b$之间有什么等式关系？", "options": ["$a=b=2$", "$a=b=1$", "$a=b=0$", "$a=b=3$"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Complex Numbers"} |
| {"question": "如果$|z|=|a+bi|$,则$z,a,b$之间有什么等式关系？", "options": ["|z|=\\sqrt{a^5+b^2}", "|z|=\\sqrt{a^2+b^2}", "|z|=\\sqrt{a^3+b^2}", "|z|=\\sqrt{a^4+b^2}"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Complex Numbers"} |
| {"question": "如果$z=a+bi$，则$\\bar{z}$的值该怎么表示 ？", "options": ["$\\bar{z}=a^4-bi$", "$\\bar{z}=a-bi$", "$\\bar{z}=a^2-bi$", "$\\bar{z}=a^3-bi$"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Complex Numbers"} |
| {"question": "在复数的加减法中，$(a+bi)\\pm (c+di)$的值等于什么？", "options": ["$(a\\pm c)+(b\\pm d)i$", "$(a^4\\pm c)+(b\\pm d)i$", "$(a^3\\pm c)+(b\\pm d)i$", "$(a^2\\pm c)+(b\\pm d)i$"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Complex Numbers"} |
| {"question": "在复数的乘法中，$(a+bi) (c+di)$的值等于什么？", "options": ["$(ac-bd)i+(bd+ad)$", "$(ac-bd)+(bd-ad)i$", "$(ac-bd)+(bd+ad)i$", "$(ac+bd)+(bd+ad)i$"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Complex Numbers"} |
| {"question": "在复数的除法中，$\\frac{a+bi}{c+di} $的值等于什么？", "options": ["$\\frac{ac+bd}{c^3+d^2} +\\frac{bc-ad}{c^2+d^2} i$", "$\\frac{ac+bd}{c^4+d^2} +\\frac{bc-ad}{c^2+d^2} i$", "$\\frac{ac+bd}{c^5+d^2} +\\frac{bc-ad}{c^2+d^2} i$", "$\\frac{ac+bd}{c^2+d^2} +\\frac{bc-ad}{c^2+d^2} i$"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Complex Numbers"} |
| {"question": "复平面是指什么？", "options": ["一个柱面坐标系", "一个双连通区域", "一个单连通区域", "用来表示复数的直角坐标系"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Complex Numbers"} |
| {"question": "复平面的$x$轴叫做什么？", "options": ["对称轴", "复平面的实轴", "中心轴", "复平面的虚轴"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Complex Numbers"} |
| {"question": "复平面的$y$轴叫做什么？", "options": ["复平面的实轴", "中心轴", "复平面的虚轴", "对称轴"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Complex Numbers"} |
| {"question": "复数$z=a+bi$和复平面的哪个点一一对应?", "options": ["$Z(a^4,b)$", "$Z(a,b)$", "$Z(a^3,b)$", "$Z(a^2,b)$"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Complex Numbers"} |
| {"question": "复数$z=a+bi$和哪个平面向量一一对应?", "options": ["$\\overrightarrow{ZO}$", "$\\overrightarrow{OZ}$", "(b,a)", "(a,b)"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Complex Numbers"} |
| {"question": "复数$|z_1-z_2|$的几何意义是什么?", "options": ["复平面中两个复数之间的角度", "表示两个复数在实轴上的投影", "两个复数在虚轴上的投影", "复平面$Z_1$、$Z_2$之间的距离"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Complex Numbers"} |
| {"question": "对于方程$x^2+mx+n=0 (\\Delta <0)$，它的两个根是虚根，则这两个虚根$x_1$和$x_2$的关系是什么？", "options": ["相等", "互为相反数", "互为质数", "共轭"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Complex Numbers"} |
| {"question": "$\\vec{a}$=(5,6),$\\vec{b}$=(0,8),则$\\vec{a}*vec{b}$=", "options": ["48", "39", "32", "29"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} |
| {"question": "设$A$(X1,Y1),$B$(X2,Y2)，则$\\lvert\\vec{AB}\\rvert$=", "options": ["\\sqrtX2^2+Y1^2", "\\sqrtX1^2+Y2^2", "\\sqrt(X2+X1)^2+(Y2+Y1)^2", "\\sqrt(X2-X1)^2+(Y2-Y1)^2"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} |
| {"question": "$cos(theta)$ =", "options": ["$frac{{mathbf{a}}}{{ |mathbf{b}|}}$", "$frac{{mathbf{a} cdot mathbf{b}}}{{|mathbf{a}| |mathbf{b}|}}$", "$frac{{mathbf{a} }}{{|mathbf{a}| }$", "$frac{{ mathbf{b}}}{{ |mathbf{b}|}}$"], "answer": "B", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "S表示内切圆的面积,r表示内切圆的半径,s表示三角形的半周长,三角形的内切圆的面积公式可以表示为", "options": ["S = r \\mp s", "S = r \\otimes s", "S = r \\times s", "S = r \\div s"], "answer": "C", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "设矩形的四个顶点分别为$A$,$B$,$C$,$D$,矩形外的一点为$P$,则有以下等式成立", "options": ["$AP^2$ + $CP^2$ =$DP^2$ - $BP^2$", "$AP^2$ - $CP^2$ =$DP^2$ - $BP^2$", "$AP^2$ + $CP^2$ =$DP^2$ + $BP^2$", "$AP^2$ - $CP^2$ =$DP^2$ + $BP^2$"], "answer": "C", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "三角形ABC中，三条重心线共点于重心$G$，并且重心到三个顶点的距离满足以下关系", "options": ["$\\overrightarrow{GA} + \\overrightarrow{GB} - \\overrightarrow{GC}= \\vec{0}\\$", "$\\overrightarrow{GA} - \\overrightarrow{GB} + \\overrightarrow{GC}= \\vec{0}\\$", "$\\overrightarrow{GA} + \\overrightarrow{GB} \\times \\overrightarrow{GC}= \\vec{0}\\$", "$\\overrightarrow{GA} + \\overrightarrow{GB} + \\overrightarrow{GC}= \\vec{0}\\$"], "answer": "D", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "设三角形的周长为$P$，内切圆的半径为$r$，三条角平分线的交点为$O$，$a$、$b$、$c$分别为三边的长度。 则三角形的面积为：", "options": ["$S = Pr$", "$S = \\frac{P}{2}-\\frac{a+b+c}{2}$", "$S = \\frac{1}{4}Pr$", "$S = \\frac{1}{2}Pr$"], "answer": "D", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "假设向量 $BA$的坐标为(x1,y1)，向量 $BC$的坐标为(x2,y2)，三角形ABC的面积公式如下：", "options": ["$S$ = $frac{1}{2}\\lvert(x1y2 + x2y1) \\rvert$", "$S$ = $frac{1}{2}\\lvert(x1y2 - x2y1) \\rvert$", "$S$ = $frac{1}{2}\\lvert(x1y1 - x2y2) \\rvert$", "$S$ = $frac{1}{4}\\lvert(x1y2 - x2y1) \\rvert$"], "answer": "B", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "$a$，$b$，$c$分别表示三角形的三边的长度，$A$，$B$，$C$ 分别表示三角形的三个角的度数,则", "options": ["$frac{a}{sin A}$ =$ frac{b}{sin B} $=$ frac{c}{sin C}$", "$frac{a}{sin A}$ =$ frac{c}{sin B} $=$ frac{b}{sin C}$", "$frac{b}{sin A}$ =$ frac{c}{sin B} $=$ frac{a}{sin C}$", "$frac{c}{sin A}$ =$ frac{b}{sin B} $=$ frac{a}{sin C}$"], "answer": "A", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "$c$表示三角形的第三边的长度，$a$和$b$ 表示三角形的两边的长度，$C$表示这两边所夹的角的度数。", "options": ["$c^2 = a^2 + b^2 - 2ab cos C$", "$b^2 = a^2 + b^2 - 2ac cos C$", "$c^2 = c^2 + b^2 - 2ab cos C$", "$a^2 = c^2 + b^2 - 2ab cos C$"], "answer": "A", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "$S$为三角形$ABC$的面积，$a$和$b$分别是三角形的两条边的长度，$C$是它们之间的夹角，用正弦定理来表示三角形的面积，则", "options": ["$S$= \\frac{1}{2}$ab$\\sin$C$", "$S$= \\frac{1}{2}$ab$\\sin$A$", "$S$= \\frac{1}{2}$ab$\\sin$B$", "$S$= \\frac{1}{2}$ac$\\sin$C$"], "answer": "A", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "根据三角形的内角和，三角形$ABC$中，$2C$=", "options": ["$2\\pi-2(A-B)$", "$2\\pi-2(A-C)$", "$2\\pi-2(A+B)$", "$2\\pi-2(A+C)$"], "answer": "C", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "在三角形$ABC$中，若$a > b$，那么,", "options": ["$\\sin A > \\sin B$", "$\\sin A > \\sin C$", "$\\sin A < \\sin C$", "$\\sin A < \\sin B$"], "answer": "A", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "对数函数的底数的取值范围是？", "options": ["底数小于0", "底数大于0", "底数小于1", "底数大于1"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "对于对数函数$f(x)$和$f(y)$，下列哪一项正确？", "options": ["$f(x)+f(y)=f(xy)$", "$f(x)-f(y)=f(xy)$", "$f(x)+f(y)=f(y)$", "$f(x)+f(y)=f(x)$"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "下列哪一个函数是幂函数？", "options": ["$y = x^{a}$", "$y = 3x^{a}$", "$y = 2x^{a}$", "$y = 4x^{a}$"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "我们把形如$y = x^{a}$的函数称为什么函数？", "options": ["幂函数", "三角函数", "反比例函数", "常数函数"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "幂函数的次数大于零时，在自变量大于零的区间上，幂函数的单调性如何，当次数小于零时，单调性又如何？", "options": ["单调递增；单调递增", "单调递增；单调递减", "单调递减；单调递增", "单调递减；单调递减"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "幂函数恒过定点？", "options": ["$(1,0)$", "$(1,1)$", "$(0,1)$", "$(0,0)$"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "三角函数$y = A\\sin(wx + \\phi)+B$，其中A大于0，w大于0，该三角函数的振幅怎么表示，周期T怎么表示，初相怎么表示，相位怎么表示，频率怎么表示？", "options": ["$\\frac{\\pi}{|w|}$；$\\phi$；$wx + \\phi$；$\\frac{w}{\\pi}$", "$\\frac{2\\pi}{|w|}$；$\\phi$；$wx + \\phi$；$\\frac{w}{2\\pi}$", "$\\frac{\\pi}{|w|}$；$\\phi$；$wx + \\phi$；$\\frac{w}{2\\pi}$", "$\\frac{2\\pi}{|w|}$；$\\phi$；$wx + \\phi$；$\\frac{w}{\\pi}$"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "由三角函数$y = \\sin(x)$变换到三角函数$y = \\sin(x + \\phi)$，反映在图像时是原函数图像左右平移了多少个单位长度？", "options": ["$2|\\varphi|$", "$|\\varphi|$", "$\\varphi$", "1"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "由三角函数$y = \\sin(x + \\phi)$变换到三角函数$y = \\Asin(x + \\phi)$，反映在图像时是原函数图像纵坐标变为原来多少倍？", "options": ["不变", "A", "3A", "2A"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "由三角函数$y = \\sin(x + \\phi)$变换到三角函数$y = \\Asin(wx + \\phi)$，反映在图像时是原函数图像横坐标变为原来的多少倍？", "options": ["$\\frac{1}{|w|}$", "$\\frac{2}{w}$", "$\\frac{2}{|w|}$", "$\\frac{1}{w}$"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "由三角函数$y = A\\sin(wx + \\phi)$变换到三角函数$y = A\\sin(wx + \\phi) +B$，反应在图像时是原函数图像上下平移了多少个单位长度？", "options": ["B", "$|A|$", "$|B|$", "A"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "由三角函数$y = A\\sin(wx)$变换到三角函数$y = A\\sin(wx + \\phi)$，反应在图像时是原函数图像左右平移了多少个单位长度？", "options": ["$\\frac{2|\\varphi|}{|w|}$", "$\\frac{|\\varphi|}{|w|}$", "$\\frac{|\\varphi|}{w}$", "$\\frac{\\varphi}{|w|}$"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "对三角函数$y = A\\sin(wx + \\phi)$和三角函数$y = A\\cos(wx + \\phi)$来说，对称中心和对称轴分别与什么相联系？", "options": ["零点；最值点", "零点；零点", "原点；最值点", "最值点；最值点"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "假设三角函数$y = A\\sin(wx + \\phi)$的对称中心是$(m,0)$，对称轴是x=n，则m，n可分别表示为？", "options": ["$\\frac{2k\\pi + \\frac{\\pi}{2} - \\varphi}{\\w}$；$\\frac{2k\\pi - \\varphi}{w}$，k为整数", "$\\frac{k\\pi + \\frac{\\pi}{2} - \\varphi}{\\w}$；$\\frac{2k\\pi - \\varphi}{w}$，k为整数", "$\\frac{k\\pi + \\frac{\\pi}{2} - \\varphi}{\\w}$；$\\frac{k\\pi - \\varphi}{w}$，k为整数", "$\\frac{2k\\pi + \\frac{\\pi}{2} - \\varphi}{\\w}$；$\\frac{k\\pi - \\varphi}{w}$，k为整数"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "对三角函数$y = A\\sin(wx + \\phi) +B$来说，如果已知函数最大值和最小值$ymax$，$ymin$，则A，B分别用最值来表示是？", "options": ["$(\\max(y) - \\min(y)) / 2$；$(\\max(y) + \\min(y)) / 2$", "$(\\max(y) + \\min(y)) / 2$；$(\\max(y) + \\min(y)) / 2$", "$(\\max(y) - \\min(y)) / 2$；$(\\max(y) - \\min(y)) / 2$", "$(\\max(y) + \\min(y)) / 2$；$(\\max(y) - \\min(y)) / 2$"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "对三角函数$y = A\\sin(wx + \\phi) +B$来说，在对称中心，对称轴的函数值分别满足什么条件", "options": ["函数值是最值；函数值是最值", "函数值是0；函数值是最值", "函数值是最值；函数值是0", "函数值是0；函数值是0"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "对于焦点在x轴上的椭圆，标准方程为？", "options": ["$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1(a>b>0)$", "$\\dfrac{y^2}{a^2}+\\dfrac{x^2}{b^2}=1(a>b>0)$", "$\\dfrac{y^2}{a^2}-\\dfrac{x^2}{b^2}=1(a>b>0)$", "$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1(a>b>0)$"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "对于焦点在y轴上的椭圆，标准方程为？", "options": ["$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1(a>b>0)$", "$\\dfrac{y^2}{a^2}-\\dfrac{x^2}{b^2}=1(a>b>0)$", "$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1(a>b>0)$", "$\\dfrac{y^2}{a^2}+\\dfrac{x^2}{b^2}=1(a>b>0)$"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "对于焦点在x轴上的椭圆，x、y的取值范围是？", "options": ["$-b\\leq x\\leq b\\text{ 且}-a\\leq y\\leq a$", "$x\\leq-a\\text{ 或 }x\\geq a\\mathrm{~,~}y\\in R$", "$-a\\leq x\\leq a\\text{ 且}-b\\leq y\\leq b$", "$y\\leq-a\\text{ 或 }y\\geq a\\mathrm{~,~}x\\in R$"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "对于焦点在y轴上的椭圆，x、y的取值范围是？", "options": ["$y\\leq-a\\text{ 或 }y\\geq a\\mathrm{~,~}x\\in R$", "$x\\leq-a\\text{ 或 }x\\geq a\\mathrm{~,~}y\\in R$", "$-b\\leq x\\leq b\\text{ 且}-a\\leq y\\leq a$", "$-a\\leq x\\leq a\\text{ 且}-b\\leq y\\leq b$"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "对于焦点在x轴上的椭圆，顶点是？", "options": ["${A}_1\\left(0,-a\\right)\\text{、A}_2\\left(0,a\\right)$", "$\\begin{aligned}\\mathrm{A}_1&\\left(-a,a\\right)\\text{、A}_2\\left(a,a\\right)\\\\\\\\\\mathrm{B}_1&\\left(0,-b\\right)\\text{、B}_2\\left(0,b\\right)\\end{aligned}$", "$\\begin{aligned}\\mathrm{A}_1&\\left(-a,0\\right)\\text{、A}_2&\\left(a,0\\right)\\\\\\\\\\mathrm{B}_1&\\left(0,-b\\right)\\text{、B}_2&\\left(0,b\\right)\\end{aligned}$", "${A}_1\\left(-a,0\\right)\\text{、A}_2\\left(a,0\\right)$"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "椭圆的对称性是？", "options": ["关于y=$x^2$对称", "关于x轴、y轴对称，关于原点中心对称", "关于y=1对称", "关于y=x对称"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "对于焦点在x轴上的椭圆，焦点是？", "options": ["$F_1(-c,c)$、$F_2(c,0)$", "$F_1(-c,0)$、$F_2(c,0)$", "$F_1(-c,0)$、$F_2(c,c)$", "$F_1(0,-c)$、$F_2(0,c)$"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "对于焦点在y轴上的椭圆，焦点是？", "options": ["$F_1(-c,c)$、$F_2(c,0)$", "$F_1(-c,0)$、$F_2(c,0)$", "$F_1(-c,0)$、$F_2(c,c)$", "$F_1(0,-c)$、$F_2(0,c)$"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "对于标准方程为$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1(a>b>0)$的椭圆，焦距$\\left|F_1F_2\\right|=？(c^2=a^2-b^2)$", "options": ["c", "2c", "2b", "2a"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "对于标准方程为$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1(a>b>0)$的椭圆，离心率e为？(0<e<1)", "options": ["$e=\\frac ca=\\sqrt{\\frac{c^2}{a^2}}=\\sqrt{\\frac{a^2+b^2}{a^2}}=\\sqrt{1+\\frac{b^2}{a^2}}$", "$e=\\frac ca=\\sqrt{\\frac{c^2}{a^2}}=\\sqrt{\\frac{a^2-b^2}{a^2}}=\\sqrt{1-\\frac{c^2}{a^2}}$", "$e=\\frac ca=\\sqrt{\\frac{c^2}{a^2}}=\\sqrt{\\frac{a^2+b^2}{a^2}}=\\sqrt{1+\\frac{c^2}{a^2}}$", "$e=\\frac ca=\\sqrt{\\frac{c^2}{a^2}}=\\sqrt{\\frac{a^2-b^2}{a^2}}=\\sqrt{1-\\frac{b^2}{a^2}}$"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "对于焦点在x轴上的椭圆，准线方程为？", "options": ["$y=\\pm\\frac{a^2}c$", "$y=\\pm\\frac{b^2}c$", "$x=\\pm\\frac{a^2}c$", "$x=\\pm\\frac{b^2}c$"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "对于焦点在y轴上的椭圆，准线方程为？", "options": ["$x=\\pm\\frac{a^2}c$", "$y=\\pm\\frac{a^2}c$", "$x=\\pm\\frac{b^2}c$", "$y=\\pm\\frac{b^2}c$"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "对于标准方程为$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1(a>b>0)$的椭圆，焦点三角形面积为？($\\theta=\\angle F_1MF_2$)", "options": ["$S_{\\Delta MF_1F_2}=a^2\\tan\\frac\\theta2$", "$S_{\\Delta MF_1F_2}=b^2\\tan\\frac\\theta2$", "$S_{\\Delta MF_1F_2}=b^2\\tan\\theta$", "$S_{\\Delta MF_1F_2}=b^2\\cdot\\frac1{\\tan\\frac\\theta2}$"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "对于标准方程为$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1(a>b>0)$的椭圆，过焦点且垂直于长轴的弦叫通径，通径$\\begin{vmatrix}HH^{\\prime}\\end{vmatrix}=$？", "options": ["$\\frac{b^2}a$", "$\\frac{a^2}b$", "$\\frac a b$", "$\\frac b a$"], "answer": "A", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "对于标准方程为$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1(a>b>0)$的椭圆，椭圆上两点$A(x_{1,}y_1),B(x_{2,}y_2)$，弦长公式为？", "options": ["$\\left|AB\\right|=\\sqrt{1+k}\\left|x_1-x_2\\right|=\\sqrt{1+k}\\sqrt{\\left(x_1+x_2\\right)^2-4x_1x_2}$", "$\\left|AB\\right|=\\sqrt{1+k^2}\\left|x_1-x_2\\right|=\\sqrt{1+k^2}\\sqrt{\\left(x_1+x_2\\right)^2-2x_1x_2}$", "$\\left|AB\\right|=\\sqrt{1+k^2}\\left|x_1-x_2\\right|=\\sqrt{1+k^2}\\sqrt{\\left(x_1+x_2\\right)^2-4x_1x_2}$", "$\\left|AB\\right|=\\sqrt{1-k^2}\\left|x_1-x_2\\right|=\\sqrt{1-k^2}\\sqrt{\\left(x_1+x_2\\right)^2-4x_1x_2}$"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "任意直线和椭圆相交于 A、B 两点,P 为AB 中点,则AB 和 OP 斜率之积为？", "options": ["$-\\frac{a^2}{b^2}$", "$-\\frac{b^2}{a^2}$", "$\\frac{a^2}{b^2}$", "$\\frac{b^2}{a^2}$"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "求椭圆的方程方法？", "options": ["类比法", "定义法、消参法", "几何构造法", "总结归纳法"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "对于焦点在x轴上的双曲线，标准方程为？", "options": ["$\\dfrac{y^2}{a^2}-\\dfrac{x^2}{b^2}=1(a>b>0)$", "$\\dfrac{y^2}{a^2}+\\dfrac{x^2}{b^2}=1(a>b>0)$", "$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1(a>b>0)$", "$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1(a>b>0)$"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "对于焦点在x轴上的双曲线，x、y的取值范围是？", "options": ["$-b\\leq x\\leq b\\text{ 且}-a\\leq y\\leq a$", "$y\\leq-a\\text{ 或 }y\\geq a\\mathrm{~,~}x\\in R$", "$-a\\leq x\\leq a\\text{ 且}-b\\leq y\\leq b$", "$x\\leq-a\\text{ 或 }x\\geq a\\mathrm{~,~}y\\in R$"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "对于焦点在y轴上的双曲线，x、y的取值范围是？", "options": ["$-a\\leq x\\leq a\\text{ 且}-b\\leq y\\leq b$", "$y\\leq-a\\text{ 或 }y\\geq a\\mathrm{~,~}x\\in R$", "$x\\leq-a\\text{ 或 }x\\geq a\\mathrm{~,~}y\\in R$", "$-b\\leq x\\leq b\\text{ 且}-a\\leq y\\leq a$"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "对于焦点在x轴上的双曲线，顶点是？", "options": ["${A}_1\\left(-a,0\\right)\\text、{A}_2\\left(a,0\\right)$", "${A}_1\\left(0,-a\\right)\\text、{A}_2\\left(0,a\\right)$", "${A}_1\\left(-a,a\\right)\\text、{A}_2\\left(a,0\\right)$", "${A}_1\\left(-a,0\\right)\\text、{A}_2\\left(a,a\\right)$"], "answer": "A", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "对于焦点在y轴上的双曲线，顶点是？", "options": ["${A}_1\\left(0,-a\\right)\\text、{A}_2\\left(0,a\\right)$", "${A}_1\\left(-a,0\\right)\\text、{A}_2\\left(a,a\\right)$", "${A}_1\\left(-a,a\\right)\\text、{A}_2\\left(a,0\\right)$", "${A}_1\\left(-a,0\\right)\\text、{A}_2\\left(a,0\\right)$"], "answer": "A", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "对于标准方程为$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1(a>b>0)$的双曲线，实轴的长为?", "options": ["b", "2a", "2b", "a"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "双曲线的对称性是？", "options": ["关于y=1对称", "关于y=$x^2$对称", "关于x轴、y轴对称，关于原点中心对称", "关于y=x对称"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "对于焦点在x轴上的双曲线，焦点是？", "options": ["$F_1(-c,0)、F_2(c,0)$", "$F_1(-c,c)、F_2(c,0)$", "$F_1(-c,0)、F_2(c,c)$", "$F_1(0,-c)、F_2(0,c)$"], "answer": "A", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "对于焦点在y轴上的双曲线，焦点是？", "options": ["$F_1(-c,c)、F_2(c,0)$", "$F_1(-c,0)、F_2(c,0)$", "$F_1(-c,0)、F_2(c,c)$", "$F_1(0,-c)、F_2(0,c)$"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "对于标准方程为$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$(a>b>0)的双曲线，焦距$\\left|F_1F_2\\right|=？(c^2=a^2+b^2)$", "options": ["2b", "c", "2c", "2a"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "对于标准方程为$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$(a>b>0)的双曲线，离心率e为？(e>1)", "options": ["$e=\\frac ca=\\sqrt{\\frac{c^2}{a^2}}=\\sqrt{\\frac{a^2-b^2}{a^2}}=\\sqrt{1-\\frac{c^2}{a^2}}$", "$e=\\frac ca=\\sqrt{\\frac{c^2}{a^2}}=\\sqrt{\\frac{a^2+b^2}{a^2}}=\\sqrt{1+\\frac{c^2}{a^2}}$", "$e=\\frac ca=\\sqrt{\\frac{c^2}{a^2}}=\\sqrt{\\frac{a^2-b^2}{a^2}}=\\sqrt{1-\\frac{b^2}{a^2}}$", "$e=\\frac ca=\\sqrt{\\frac{c^2}{a^2}}=\\sqrt{\\frac{a^2+b^2}{a^2}}=\\sqrt{1+\\frac{b^2}{a^2}}$"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "对于焦点在x轴上的双曲线，准线方程为？", "options": ["$y=\\pm\\frac{b^2}c$", "$y=\\pm\\frac{a^2}c$", "$x=\\pm\\frac{a^2}c$", "$x=\\pm\\frac{b^2}c$"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "对于焦点在y轴上的双曲线，准线方程为？", "options": ["$y=\\pm\\frac{b^2}c$", "$y=\\pm\\frac{a^2}c$", "$x=\\pm\\frac{a^2}c$", "$x=\\pm\\frac{b^2}c$"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "对于焦点在x轴上的双曲线，渐进线方程为？", "options": ["$y=\\pm\\frac a b x$", "$y=\\pm\\frac{b^2}c x$", "$y=\\pm\\frac b a x$", "$y=\\pm\\frac{a^2}c x$"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "对于焦点在y轴上的双曲线，渐进线方程为？", "options": ["$y=\\pm\\frac{a^2}c x$", "$y=\\pm\\frac a b x$", "$y=\\pm\\frac b a x$", "$y=\\pm\\frac{b^2}c x$"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "对于焦点在x轴上的双曲线，焦半径为？", "options": ["$\\begin{gathered}M\\text{ 在右支}\\begin{cases}\\text{左焦:}\\left|MF_1\\right|=ex_0+a\\\\\\text{右焦:}\\left|MF_2\\right|=ex_0-a\\end{cases}\\\\M\\text{ 在左支}\\begin{cases}\\text{左焦}:\\left|MF_1\\right|=-ex_0-a\\\\\\text{右焦:}\\left|MF_2\\right|=-ex_0+a\\end{cases}\\end{gathered}$", "$\\begin{gathered}M\\text{ 在上支}\\begin{cases}\\text{左焦:}\\left|MF_1\\right|=ey_0+a\\\\\\text{右焦:}\\left|MF_2\\right|=ey_0-a\\end{cases} \\\\M\\text{ 在下支}\\begin{cases}\\text{左焦:}\\left|MF_1\\right|=-ey_0-a\\\\\\text{右焦:}\\left|MF_2\\right|=-ey_0+a\\end{cases}\\end{gathered}$", "$\\begin{gathered}M\\text{ 在上支}\\begin{cases}\\text{左焦:}\\left|MF_1\\right|=ey_0-a\\\\\\text{右焦:}\\left|MF_2\\right|=ey_0+a\\end{cases} \\\\M\\text{ 在下支}\\begin{cases}\\text{左焦:}\\left|MF_1\\right|=-ey_0+a\\\\\\text{右焦:}\\left|MF_2\\right|=-ey_0-a\\end{cases}\\end{gathered}$", "$\\begin{gathered}M\\text{ 在右支}\\begin{cases}\\text{左焦:}\\left|MF_1\\right|=ex_0-a\\\\\\text{右焦:}\\left|MF_2\\right|=ex_0+a\\end{cases}\\\\M\\text{ 在左支}\\begin{cases}\\text{左焦}:\\left|MF_1\\right|=-ex_0+a\\\\\\text{右焦:}\\left|MF_2\\right|=-ex_0-a\\end{cases}\\end{gathered}$"], "answer": "A", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "对于焦点在y轴上的双曲线，焦半径为？", "options": ["$\\begin{gathered}M\\text{ 在右支}\\begin{cases}\\text{左焦:}\\left|MF_1\\right|=ex_0+a\\\\\\text{右焦:}\\left|MF_2\\right|=ex_0-a\\end{cases}\\\\M\\text{ 在左支}\\begin{cases}\\text{左焦}:\\left|MF_1\\right|=-ex_0-a\\\\\\text{右焦:}\\left|MF_2\\right|=-ex_0+a\\end{cases}\\end{gathered}$", "$\\begin{gathered}M\\text{ 在上支}\\begin{cases}\\text{左焦:}\\left|MF_1\\right|=ey_0-a\\\\\\text{右焦:}\\left|MF_2\\right|=ey_0+a\\end{cases} \\\\M\\text{ 在下支}\\begin{cases}\\text{左焦:}\\left|MF_1\\right|=-ey_0+a\\\\\\text{右焦:}\\left|MF_2\\right|=-ey_0-a\\end{cases}\\end{gathered}$", "$\\begin{gathered}M\\text{ 在上支}\\begin{cases}\\text{左焦:}\\left|MF_1\\right|=ey_0+a\\\\\\text{右焦:}\\left|MF_2\\right|=ey_0-a\\end{cases} \\\\M\\text{ 在下支}\\begin{cases}\\text{左焦:}\\left|MF_1\\right|=-ey_0-a\\\\\\text{右焦:}\\left|MF_2\\right|=-ey_0+a\\end{cases}\\end{gathered}$", "$\\begin{gathered}M\\text{ 在右支}\\begin{cases}\\text{左焦:}\\left|MF_1\\right|=ex_0-a\\\\\\text{右焦:}\\left|MF_2\\right|=ex_0+a\\end{cases}\\\\M\\text{ 在左支}\\begin{cases}\\text{左焦}:\\left|MF_1\\right|=-ex_0+a\\\\\\text{右焦:}\\left|MF_2\\right|=-ex_0-a\\end{cases}\\end{gathered}$"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "对于标准方程为$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$(a>b>0)的双曲线，焦点三角形面积为？", "options": ["$S_{\\Delta MF_1F_2}=a^2\\tan\\frac\\theta2$", "$S_{\\Delta MF_1F_2}=b^2\\tan\\frac\\theta2$", "$S_{\\Delta MF_1F_2}=b^2\\tan\\theta$", "$S_{\\Delta MF_1F_2}=b^2\\cdot\\frac1{\\tan\\frac\\theta2}$"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "对于标准方程为$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$(a>b>0)的双曲线，过焦点且垂直于长轴的弦叫通径，通径$\\begin{vmatrix}HH^{\\prime}\\end{vmatrix}=$？", "options": ["$\\frac{a^2}b$", "$\\frac{b^2}a$", "$\\frac a b$", "$\\frac b a$"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "对于双曲线的$PF_{1},PF_{2}$差值问题要讨论P点所在范围!\n焦点在x轴上：$PF_{1}-PF_{2}$ > 0,则P点在？", "options": ["上支", "下支", "右支", "左支"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "对于标准方程为$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1(a>b>0)$的双曲线，焦半径范围？", "options": ["$[b-c,+\\infty)$", "$[c-b,+\\infty)$", "$[a-c,+\\infty)$", "$[c-a,+\\infty)$"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "对于标准方程为$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1(a>b>0)$的双曲线，焦点到渐近线距离为？", "options": ["b-a", "a", "a-b", "b"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "过双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$上一点P作平行线，它们和双曲线的两个交点以及P、O构成的四边形面积为？", "options": ["定值ab", "定值$\\frac{ab}{2} $", "定值$\\frac{b}{2} $", "定值$\\frac{a}{2} $"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "不等式$a>b>0$可以根据不等式的哪个性质改写为$a^{n} >b^{n}$ ,($n\\in N$,且n>1)?", "options": ["同向可加性", "倒数法则", "平方法则", "对称性"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Applications of Inequalities"} |
| {"question": "若$a^{n} >b^{n}$ ,($n\\in N$,且n>1)，则a与b的关系是什么？", "options": ["$a=b<0$", "$a>b>0$", "$a=b>0$", "$a<b<0$"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Applications of Inequalities"} |
| {"question": "已知$a>b>0$，则$\\frac{1}{a} $与$\\frac{1}{b}$的大小关系如何？", "options": ["=", ">", "不确定", "<"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Applications of Inequalities"} |
| {"question": "不等式$a^{2} +b^{2 }\\ge 2ab $、$\\frac{a+b}{2} \\ge \\sqrt{ab} $、$\\frac{2}{a^{-1}+b^{-1} } \\le \\sqrt{ab} \\le \\frac{a+b}{2} \\le \\sqrt{\\frac{a^{2}+b^{2} }{2} }$($a,b\\in R^{+}$ )的取等条件是：", "options": ["$a>b$", "$b=0$", "$a=0$", "$a=b$"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Applications of Inequalities"} |
| {"question": "根据绝对值三角不等式，$\\left | a \right | -\\left | b \right | \\le \\left | a\\pm b \right | \\le \\left | a \right | +\\left | b \right |$以下哪个说法是正确的？", "options": ["$a>b$时等式成立", "$a=b$时等式成立", "$a=0$时等式成立", "无论ab关系都成立"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Applications of Inequalities"} |
| {"question": "平均不等式$\\frac{2}{a^{-1}+b^{-1} } \\le \\sqrt{ab} \\le \\frac{a+b}{2} \\le \\sqrt{\\frac{a^{2}+b^{2} }{2} }$($a,b\\in R^{+}$ ) （当且仅当a=b时等式成立）的变形公式除了$ab\\le (\\frac{a+b}{2} )^{2} \\le \\frac{a^{2} +b^{2} }{2} $，还有可能是哪个？", "options": ["$a^{4} +b^{2} \\ge \\frac{(a+b)^{2} }{2}$", "$a^{5} +b^{2} \\ge \\frac{(a+b)^{2} }{2}$", "$a^{3} +b^{2} \\ge \\frac{(a+b)^{2} }{2}$", "$a^{2} +b^{2} \\ge \\frac{(a+b)^{2} }{2}$"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Applications of Inequalities"} |
| {"question": "下列关于不等式$ax^{2} +bx+c>0$或（$<0$）($a \\not = 0$,$\\Delta =b^{2} -4ac>0$)说法错误的是？", "options": ["五解集：根据图像写出不等式的解集", "一化：化二次项前的系数为负", "二判：判断对应方程的根", "四画：画出对应函数的图像"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Applications of Inequalities"} |
| {"question": "下列对于穿根法叙述错误的是？", "options": ["偶穿奇不穿", "系数化正", "右上切入", "奇穿偶回"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Applications of Inequalities"} |
| {"question": " 分式不等式可以等价转化为整式不等式进行求解，因此可以将$\\frac{f(x)}{g(x)} > 0$转化为整式不等式什么？", "options": ["$f(x) \\cdot g(x) > 0$", "$f(x) + g(x) > 0$", "$f(x) - g(x) > 0$", "$f(x) / g(x) > 0$"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Applications of Inequalities"} |
| {"question": "下列对含绝对值不等式解法叙述错误的是：", "options": ["$\\left | x\\right | \\le a=-a\\le x\\le a(a\\ge 0)$", "$|f(x)| \\le |g(x)|=f^2(x) \\le g^2(x)$", "含有两个（或两个以上）绝对值的不等式的解法是找零点、划区间、分段讨论去绝对值、每段中取交集，最后取各段的并集。", "$\\left |f(x ) \\right | \\ge g(x)=-g(x)\\le f(x)\\le g(x) (g(x)\\ge0 )$"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Applications of Inequalities"} |
| {"question": "下列对含绝对值不等式解法叙述正确的是：", "options": ["$\\left | x\\right | \\ge a$可以转换成 $x\\le a 或 x\\ge -a (a\\ge 0)$", "$\\left| f(x) \\right| \\geq g(x)$可以转换成$f(x)\\le g(x) 或 f(x)\\ge -g(x) (g(x)\\ge0 )$", "$\\left |f(x ) \\right | \\le g(x)$可以转化成$f(x) \\leq g(x)$", "$ |x| \\leq a $可以转换成 $-a\\le x\\le a(a\\ge 0)$"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Applications of Inequalities"} |
| {"question": "对于含有两个（或两个以上）绝对值的不等式，其解法通常包括哪些步骤？", "options": ["找零点、划区间、分段讨论去绝对值、每段中取交集，最后取各段的并集。", "直接计算绝对值，然后比较大小。", "将不等式转化为等式求解。", "使用图形方法直接在坐标系中表示。"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Applications of Inequalities"} |
| {"question": "在处理含有参数的不等式问题时，通常首先考虑的是：", "options": ["直接求解不等式，不考虑参数的影响。", "将不等式转化为最值问题，通过参变分离的方法求解。", "通过代数方法直接找到不等式的解集。", "使用图形方法来直观表示不等式的解。"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Applications of Inequalities"} |
| {"question": "下列不属于不等式相关问题的解决思路的是：", "options": ["1的代换", "轮换", "改头换面", "奇穿偶回"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Applications of Inequalities"} |
| {"question": "在解决含有轮换对称性的不等式时，以下哪种情况允许你通过将两个变量相等代入已知式子，然后求出最值？", "options": ["当要求的式子与已知式子完全相同时。", "当已知式子中只有一个变量时。", "当两个变量在已知式子中相等时。", "当两个变量在要求的式子中交换位置后，式子保持不变。"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Applications of Inequalities"} |
| {"question": "下列关于不等式恒成立的条件说法错误的是？", "options": ["函数$f(x)<a$恒成立，那么$f(x)_{max}$与a的关系是$f(x)_{max}<a$", "当$a \\ne 0$时，不等式$ax^2+bx+c>0$的解集是全体实数（或恒成立）的条件是 $a>0$, $\\Delta <0$", "不等式$ax^{2} +bx+c<0$的解集是全体实数（或恒成立）的条件是当$a=0$时，$b=0,c<0$。当$a\\not = 0$时，$a<0，\\Delta<0$", "当$a=0$时，不等式$ax^2+bx+c>0$的解集是全体实数（或恒成立）的条件是 $b=0$, $c <0$"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Applications of Inequalities"} |
| {"question": "对于一元二次不等式 $ ax^2 + bx + c < 0 $，使其解集包含全体实数（即恒成立）的条件是什么？", "options": ["当 $a \\neq 0$且$ a > 0$，且判别式$ \\Delta > 0 $。", "当 $a \\neq 0 $ 时$ a > 0 $，且判别式 $\\Delta < 0$。", "当 $a =0 $ 时 $ b = 0 $，$ c < 0 $。", "当 $a = 0$ 且 $ b > 0$，$ c < 0$。"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Applications of Inequalities"} |
| {"question": "如果函数 $ f(x) \\leq a$恒成立，那么$f(x)_{\\text{max}}$与a的关系是？", "options": ["$ f(x)_{\\text{max}} = a $", "$ f(x)_{\\text{max}} > a $", "$ f(x)_{\\text{max}} \\leq a $", "$ f(x)_{\\text{max}} \\geq a $"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Applications of Inequalities"} |
|
|
