AEAS数学考试真题资料讲解

AEAS数学模拟题.pdf

AEAS考试真题1. What will it cost to carpet a room with indoor/outdoor carpet if the room is 10 feet wide and 12 feet long? The carpet costs 12.51 per square yard.A. $166.80B. $175.90C. $184.30D. $189.90E. $192.202. If the perimeter of a rectangular house is 44 yards, and the length is 36 feet, what is the width of the house?A. 10 yardsB. 18 yardsC. 28 feetD. 32 feetE. 36 yards3. What is the volume of the following cylinder?A. 210.91B. 226.20C. 75.36D. 904.32E. 28.264. What is the volume of a cube whose width is 5 inches?A. 15 cubic inchesB. 25 cubic inchesC. 64 cubic inchesD. 100 cubic inchesE. 125 cubic inches5. Sally has three pieces of material. The first piece is 1 yd. 2 ft. 6 in. long, the second piece is 2 yd. 1 ft. 5 in long, and the third piece is 4 yd. 2ft.8in long. How much material does Sally have?A. 7 yd. 1 ft. 8 in.B. 8 yd. 4 ft. 4 in.C. 8 yd. 11 in.D. 9 yd. 7 in.E. 10 yd.6. A can´s diameter is 3 inches, and its height is 8 inches. What is the volume of the can?A. 50.30B. 56.55C. 75.68D. 113.04E. 226.087.If the area of a square flowerbed is 16 square feet, then how many feet is the perimeter of the flo werbed?A. 4B. 12C. 16D. 20E. 248.Of the following units which would be more likely used to measure the amount of water in a bat htub?A. kilogramsB. litersC. millilitersD. centigramsE. volts9. If a match box is 0.17 feet long, what is its length in inches the most closely comparable to thefollowing?A. 5 1/16 inch highlighterB. 3 1/8 inch jewelry boxC. 2 3/4 inch lipstickD. 2 3/16 inch staple removerE. 4 1/2 inch calculator10. What is the cost in dollars to steam clean a room W yards wide and L yards long it the steam cleaners charge 10 cents per square foot?A. 0.9WLB. 0.3WLC. 0.1WLD. 9WLE. 3WLKEYS:1. (A) 2. (A) 3. (B) 4. (E) 5. (D) 6. (B) 7. (C) 8. (B) 9. (D) 10. (A)1. How many cubed pieces of fudge that are 3 inches on an edge can be packed into a Christmas tin that is 9 inches deep by 12 inches wide by 8 inches high with the lid still being able to be closed?A. 18B. 24C. 32D. 36E. 432. Sarah is twice as old as her youngest brother. If the difference between their ages is 15 years. How old is her youngest brother?A. 10B. 15C. 20D. 25E. 303. Which of the following fractions is equal to 5/6?A. 20/30B. 15/24C. 25/30D. 40/54E. 2/74. What will it cost to tile a kitchen floor that is 12 feet wide by 20 feet long if the tile cost $8.91 per square yard?A. $224.51B. $237.60C. $246.55D. $271.38E. $282.325. In a writing competition, the first place winner receives 1/2 of the prize money. The second runner up receives 1/2 of what the winner won. What was the total amount of prize money distributed if the winner receives $6,000?A. $6,000B. $8,500C. $12,000D. $15,000E. $18,5006. You are lying 120 ft away from a tree that is 50 feet tall. You look up at the top of the tree. Approximately how far is your hear from the top of the tree in a straight line?A. 50 feetB. 75 feetC. 120 feetD. 130 feetE. 150 feet7. A cyclist bikes x distance at 10 miles per hour and returns over the same path at 8 miles per hour.What is the cyclist’s average rate for the round trip in miles per hour?A. 8.1B. 8.3C. 8.6D. 8.9E. 9.08. If edging cost $2.32 per 12-inch stone, and you want a double layer of edging around your flower bed that is 6 yards by 1 yard. How much will edging you flower bed cost?A. $32.48B. $64.96C. $97.44D. $129.92E. $194.889. If 3x=6x-15 then x + 8=A. 5B. 10C. 11D. 12E. 1310. The number of milliliters in 1 liter isA. 10,000B. 1,000C. 0.1D. 0.01E. 0.001Answer Key 1. (B) 2. (B) 3. (C) 4. (B) 5. (C) 6. (D) 7. (D) 8. (E) 9. (E) 10. (B)。

AEAS数学考试真题

A E A S数学考试真题 Prepared on 24 November 2020AEAS考试真题1. What will it cost to carpet a room with indoor/outdoor carpet if the room is 10 feet wide and 12 feet long The carpet costs per square yard.A. $B. $C. $D. $E. $2. If the perimeter of a rectangular house is 44 yards, and the length is 36 feet, what is the width of the houseA. 10 yardsB. 18 yardsC. 28 feetD. 32 feetE. 36 yards3. What is the volume of the following cylinderA.B.C.D.E.4. What is the volume of a cube whose width is 5 inchesA. 15 cubic inchesB. 25 cubic inchesC. 64 cubic inchesD. 100 cubic inchesE. 125 cubic inches5. Sally has three pieces of material. The first piece is 1 yd. 2 ft. 6 in. long, the second piece is 2 yd. 1 ft. 5 in long, and the third piece is 4 yd. long. How much material does Sally haveA. 7 yd. 1 ft. 8 in.B. 8 yd. 4 ft. 4 in.C. 8 yd. 11 in.D. 9 yd. 7 in.E. 10 yd.6. A can′s diameter is 3 inches, and its height is 8 inches. What is the volume of the canA.B.C.D.E.7. If the area of a square flowerbed is 16 square feet, then how many feet is the perimeter of the flowerbedA. 4B. 12C. 16D. 20E. 248. Of the following units which would be more likely used to measure the amount of water in a bathtubA. kilogramsB. litersC. millilitersD. centigramsE. volts9. If a match box is feet long, what is its length in inches the most closely comparable to the followingA. 5 1/16 inch highlighterB. 3 1/8 inch jewelry boxC. 2 3/4 inch lipstickD. 2 3/16 inch staple removerE. 4 1/2 inch calculator10. What is the cost in dollars to steam clean a room W yards wide and L yards long it the steam cleaners charge 10 cents per square footA.B.C.D. 9WLE. 3WLKEYS:1. (A) 2. (A) 3. (B) 4. (E) 5. (D) 6. (B) 7. (C) 8. (B) 9. (D) 10. (A)1. How many cubed pieces of fudge that are 3 inches on an edge can be packed into a Christmas tin that is 9 inches deep by 12 inches wide by 8 inches high with the lid still being able to be closedA. 18B. 24C. 32D. 36E. 432. Sarah is twice as old as her youngest brother. If the difference between their ages is 15 years. How old is her youngest brotherA. 10B. 15C. 20D. 25E. 303. Which of the following fractions is equal to 5/6A. 20/30B. 15/24C. 25/30D. 40/54E. 2/74. What will it cost to tile a kitchen floor that is 12 feet wide by 20feet long if the tile cost $ per square yardA. $B. $D. $E. $5. In a writing competition, the first place winner receives 1/2 of the prize money. The second runner up receives 1/2 of what the winner won. What was the total amount of prize money distributed if the winner receives $6,000A. $6,000B. $8,500C. $12,000D. $15,000E. $18,5006. You are lying 120 ft away from a tree that is 50 feet tall. You look up at the top of the tree. Approximately how far is your hear from the top of the tree in a straight lineA. 50 feetB. 75 feetC. 120 feetD. 130 feetE. 150 feet7. A cyclist bikes x distance at 10 miles per hour and returns over the same path at 8 miles pe r hour. What is the cyclist’s average rate for the round trip in miles per hourA.B.C.D.8. If edging cost $ per 12-inch stone, and you want a double layer of edging around your flower bed that is 6 yards by 1 yard. How much will edging you flower bed costA. $B. $C. $D. $E. $9. If 3x=6x-15 then x + 8=A. 5B. 10C. 11D. 12E. 1310. The number of milliliters in 1 liter isA. 10,000B. 1,000C.D.E.Answer Key1. (B)2. (B)3. (C)4. (B)5. (C)6. (D)7. (D)8. (E)9. (E) 10. (B)。

2023年数学教师资格证考试真题及答案解析

2023年数学教师资格证考试真题及答案
解析
本文档提供了2023年数学教师资格证考试的真题及答案解析，旨在帮助备考的教师更好地准备考试。

一、选择题
1. 以下哪个不是二次函数的图象？（A）
A) 圆
B) 抛物线
C) 双曲线
D) 直线
解析：二次函数的图象是抛物线，而圆、双曲线和直线不是二
次函数的图象。

因此，选项A是正确答案。

2. 记数实验中的样本空间定义为什么？（B）
A) 所有可能观察到的事件
B) 所有可能结果的集合
C) 所有可能的信号集合
D) 所有可能的数据集合
解析：样本空间是指一个随机试验中所有可能结果的集合，因此选项B是正确答案。

二、解答题
1. 用分离变量法求微分方程 dy/dx = xy
解析：根据分离变量法，我们将dy和dx分离到等式的两边，得到dy/y = xdx。

然后对两边同时求不定积分，得到ln|y| = (1/2)x^2 + C，其中C为常数。

最后，去掉绝对值，得到y = Ce^(x^2/2)，C 为积分常数。

2. 某班级有40名学生，其中20名男生和20名女生。

从中随机选择两名学生，请问至少选择一名男生的概率是多少？
解析：首先计算至少选择一名男生的情况，即至少选择一个男生和一个女生的情况。

选择一个男生的概率是20/40，选择一个女
生的概率是20/39（因为选择男生之后，剩下的人数变成了39）。

因此，至少选择一名男生的概率是 (20/40) * (20/39)。

以上是2023年数学教师资格证考试的真题及答案解析。

希望对备考的教师有所帮助！祝您顺利通过考试！。

AEAS数学考试真题.docx

AEAS考试真题1. What will it cost to carpet a room with indoor/outdoor carpet if the room is 10 feet wide and 12 feet long? The carpet costs 12.51 per square yard.A. $166.80B. $175.90C. $184.30D. $189.90E. $192.202. If the perimeter of a rectangular house is 44 yards, and the length is 36 feet, what is the width of the house?A. 10 yardsB. 18 yardsC. 28 feetD. 32 feetE. 36 yards3. What is the volume of the following cylinder?A. 210.91B. 226.20C. 75.36D. 904.32E. 28.264. What is the volume of a cube whose width is 5 inches?A. 15 cubic inchesB. 25 cubic inchesC. 64 cubic inchesD. 100 cubic inchesE. 125 cubic inches5. Sally has three pieces of material. The first piece is 1 yd. 2 ft.6 in. long, the second piece is 2 yd. 1 ft. 5 in long, and the third piece is 4 yd. 2ft.8in long. How much material does Sally have?A. 7 yd. 1 ft. 8 in.B. 8 yd. 4 ft. 4 in.C. 8 yd. 11 in.D. 9 yd. 7 in.E. 10 yd.6. A can´s diameter is 3 inches, and its height is 8 inches. What is the volume of the can?A. 50.30B. 56.55C. 75.68D. 113.04E. 226.087. If the area of a square flowerbed is 16 square feet, then how many feet is the perimeter of the flowerbed?A. 4B. 12C. 16D. 20E. 248. Of the following units which would be more likely used to measure the amount of water in a bathtub?A. kilogramsB. litersC. millilitersD. centigramsE. volts9. If a match box is 0.17 feet long, what is its length in inches the most closely comparable to the following?A. 5 1/16 inch highlighterB. 3 1/8 inch jewelry boxC. 2 3/4 inch lipstickD. 2 3/16 inch staple removerE. 4 1/2 inch calculator10. What is the cost in dollars to steam clean a room W yards wide and L yards long it the steam cleaners charge 10 cents per square foot?A. 0.9WLB. 0.3WLC. 0.1WLD. 9WLE. 3WLKEYS:1. (A) 2. (A) 3. (B) 4. (E) 5. (D) 6. (B) 7. (C) 8. (B) 9. (D) 10. (A)1. How many cubed pieces of fudge that are 3 inches on an edge can be packed into a Christmas tin that is 9 inches deep by 12 inches wide by 8 inches high with the lid still being able to be closed?A. 18B. 24C. 32D. 36E. 432. Sarah is twice as old as her youngest brother. If the difference between their ages is 15 years. How old is her youngest brother?A. 10B. 15C. 20D. 25E. 303. Which of the following fractions is equal to 5/6?A. 20/30B. 15/24C. 25/30D. 40/54E. 2/74. What will it cost to tile a kitchen floor that is 12 feet wide by 20 feet long if the tile cost $8.91 per square yard?A. $224.51B. $237.60C. $246.55D. $271.38E. $282.325. In a writing competition, the first place winner receives 1/2 of the prize money. The second runner up receives 1/2 of what the winner won. What was the total amount of prize money distributed if the winner receives $6,000?A. $6,000B. $8,500C. $12,000D. $15,000E. $18,5006. You are lying 120 ft away from a tree that is 50 feet tall. You look up at the top of the tree. Approximately how far is your hear from the top of the tree in a straight line?A. 50 feetB. 75 feetC. 120 feetD. 130 feetE. 150 feet7. A cyclist bikes x distance at 10 miles per hour and returns over the same path at 8 miles per hour. What is the cyclist’s average rate for the round trip in miles per hour?A. 8.1B. 8.3C. 8.6D. 8.9E. 9.08. If edging cost $2.32 per 12-inch stone, and you want a double layer of edging around your flower bed that is 6 yards by 1 yard. How much will edging you flower bed cost?A. $32.48B. $64.96C. $97.44D. $129.92E. $194.889. If 3x=6x-15 then x + 8=A. 5B. 10C. 11D. 12E. 1310. The number of milliliters in 1 liter isA. 10,000B. 1,000C. 0.1D. 0.01E. 0.001Answer Key1. (B)2. (B)3. (C)4. (B)5. (C)6. (D)7. (D)8. (E)9. (E) 10. (B)。

AP Calculus AB考试2024历年题目全解

AP Calculus AB考试2024历年题目全解2024年的AP Calculus AB考试历年题目全解[注意：本文以解题步骤为主要内容，为了更好地呈现题目解析，会包含大量的公式和符号。

请确保在阅读时有一定的数学基础。

]1. 第一题题目描述：给定一个函数 f(x)，求其在区间 [a，b] 上的定积分。

解析：根据题目所给函数 f(x) 的表达式，我们可以使用不定积分的方法解决这个问题。

首先，先求 f(x) 的原函数 F(x)，然后计算 F(b) -F(a) 即可得到定积分的结果。

具体步骤如下：1. 求函数 f(x) 的原函数 F(x)。

2. 计算 F(b) - F(a) 的值，即为所求定积分的结果。

2. 第二题题目描述：计算函数 f(x) 的导数。

解析：根据题目所给函数 f(x) 的表达式，我们可以使用求导的方法来计算其导数。

根据导数的定义，导数 f'(x) 可以通过计算 f(x) 的极限来得到。

具体步骤如下：1. 对函数 f(x) 求导。

2. 化简导数的表达式，得到最终结果。

3. 第三题题目描述：给定一个函数 f(x)，求其在某点 x=c 处的极限。

解析：根据题目所给函数 f(x) 的表达式，我们可以使用极限的定义来计算其在某点 x=c 处的极限。

根据极限的定义，我们需要分别计算 x 无限接近 c 时 f(x) 的左极限和右极限。

具体步骤如下：1. 计算 f(x) 在 c 的左极限。

2. 计算 f(x) 在 c 的右极限。

3. 判断左极限和右极限是否存在且相等，若相等则极限存在，否则不存在。

4. 第四题题目描述：给定一个函数 f(x)，求其在某点 x=c 处的导数。

解析：根据题目所给函数 f(x) 的表达式，我们可以使用导数的定义来计算其在某点 x=c 处的导数。

导数 f'(x) 可以通过计算 f(x) 的极限来得到。

具体步骤如下：1. 对函数 f(x) 求导。

2. 将 x 的值替换为 c，计算导数的值，即为所求导数的结果。

考研高数a真题及答案解析

考研高数a真题及答案解析A真题及答案解析考研数学是研究生招生考试中不可或缺的一项科目，而高等数学A部分更是其中最为关键的知识点。

为了帮助考生更好地备考，我们将为大家提供一道典型的高数A真题及答案解析。

首先，让我们来看一下这道真题：【题目】设函数f(x)在区间[a,b]上可导，且f(a)=f(b)=0，证明存在一点ξ∈(a,b)，使得f′(ξ)=f(ξ)。

接下来，我们将逐步解析这道题：1. 首先，我们需对问题进行分析，理解题目的意思。

题目要求我们证明在函数f(x)在区间[a,b]上可导的前提下，存在一个点ξ使得f′(ξ)=f(ξ)。

2. 接下来，我们需要考虑如何使用中值定理来证明这个结论。

根据中值定理，如果一个函数在某个区间上可导，并且在区间的两个端点上取到相同的函数值，那么在这个区间内必定存在一个点，使得其导数等于函数值。

3. 现在，我们可以开始着手证明了。

首先，我们设h(x) = f(x) - f′(x)，这样我们的目标就是要证明在某个点ξ上，h(x)等于零。

4. 由于f(x)在区间[a,b]上可导，那么h(x)也在这个区间上可导。

根据导数的定义，我们可得h′(x) = f′(x) - f′′(x)。

5. 接下来，我们来观察h′(x)的值。

由于f′(x)可导，则h′(x)也可导。

我们在这里使用了函数的可导性的性质，即如果一个函数在某个点可导，则它在这个点的左右导数存在且相等。

6. 根据题设，我们知道f(a) = 0，因此h(a) = f(a) - f′(a) = 0 - f′(a) = - f′(a)。

同样地，我们可得h(b) = - f′(b)。

7. 现在，我们来分析h(a)和h(b)的符号。

根据题设可知f(a) = f(b) = 0，即f(a)和f(b)都等于零。

因此，我们可以得出结论：h(a)和h(b)的符号相反。

8. 根据导数的连续性，我们可以知道在区间[a,b]上，h(x)的符号一定发生了改变，即h(x)在这个区间内至少存在一个根。

aeas考试试题模拟

aeas考试试题模拟AEAS考试试题模拟一、听力理解（Listening Comprehension）Section A:1. 听录音，选择正确的图片。

- 录音一：描述了一种动物的特征。

- 录音二：描述了一个日常活动的场景。

2. 听对话，回答以下问题。

- 问题一：对话中提到了什么时间？- 问题二：对话中的人物计划做什么？Section B:3. 听短文，完成信息填空。

- 短文：介绍一个历史事件。

- 填空：请根据短文内容填写事件的名称、发生时间、主要人物。

二、阅读理解（Reading Comprehension）Passage 1:- 阅读一篇文章，关于环保的重要性。

- 问题一：文章中提到了哪些环保措施？- 问题二：作者认为环保对人类社会有何影响？Passage 2:- 阅读一篇科技文章，介绍一项新的科技发明。

- 问题一：这项发明的主要功能是什么？- 问题二：作者对这项发明的前景持什么态度？三、词汇与语法（Vocabulary and Grammar）Section A:4. 选择正确的单词填空。

- 例句：The _______ (inventor/discoverer) of the telephone was Alexander Graham Bell.Section B:5. 改写句子，使其意思相同或相近。

- 例句：Despite the heavy rain, they still went out for a walk.- 改写：Heavy rain didn't prevent them from going out for a walk.四、写作（Writing）Task 1:- 写一篇短文，描述你最喜欢的季节，并解释原因。

Task 2:- 根据给出的图表数据，写一篇报告，分析某地区过去五年的人口变化趋势。

五、数学问题（Mathematical Problems）Section A:6. 解决实际问题。

AP Calculus AB历年真题精选2024

AP Calculus AB历年真题精选2024 2024年的AP Calculus AB考试即将到来，对于准备参加考试的学生来说，熟悉历年真题是非常重要的。

在本文中，我将为大家精选几道2024年AP Calculus AB考试的历年真题，帮助大家更好地备考。

一、选择题1. 求函数f(x) = x^3 - 3x^2 + 2x的导数。

解析：根据导数的定义，对函数f(x)求导数，我们可以将每一项的指数乘以系数，并减少指数。

因此，f'(x) = 3x^2 - 6x + 2。

2. 设函数g(x) = (x - 2)^2 + 3，求g(x)的最小值。

解析：由于(x - 2)^2的值永远大于等于0，所以g(x)的最小值就是当(x - 2)^2的值为0时，即x = 2。

代入函数g(x)，可以得到最小值为3。

二、计算题1. 求函数y = x^2 + 4x + 3的定积分，区间为[-1, 2]。

解析：我们可以先对函数y = x^2 + 4x + 3求不定积分，得到原函数F(x) = (1/3)x^3 + 2x^2 + 3x + C。

然后，我们将上下限带入原函数，得到积分值为F(2) - F(-1) = 11。

2. 求曲线y = x^3在x = 2处的切线方程。

解析：首先，我们需要求得曲线y = x^3的导数，即y' = 3x^2。

然后，代入x = 2，得到斜率k = 12。

由于切线方程的一般形式为y = kx +b，我们可以代入已知的点(x, y) = (2, 8)，解得b = -4。

因此，切线方程为y = 12x - 4。

三、证明题证明：对于任意实数x和y，有(x + y)^2 ≥ 4xy。

解析：我们可以展开(x + y)^2，得到x^2 + 2xy + y^2。

根据平方项的非负性，我们可以得出x^2 ≥ 0和y^2 ≥ 0。

因此，x^2 + y^2 ≥ 0，进而有x^2 + y^2 + 2xy ≥ 2xy。

aeas考试试题

aeas考试试题A E A S 试题一、阅读理解（共两篇文章，每篇文章后有五个问题，每题2分，共计20分）文章一：随着全球化的不断推进，英语作为国际交流的通用语言，其重要性日益凸显。

AEAS（Australian Education Assessment Service）考试，作为评估非英语国家学生英语水平的一项重要考试，已经成为许多澳大利亚教育机构录取国际学生的必备条件。

AEAS考试不仅测试学生的英语语言能力，还包括对数学、科学等学科的基础知识进行评估。

通过这项考试，教育机构能够更好地了解申请者的学术背景和语言能力，从而为他们提供更加合适的学习环境和支持。

AEAS考试分为两个部分：笔试和口试。

笔试部分包括听力、阅读、写作和词汇与语法四个部分，而口试则是面对面进行，主要评估学生的口语交际能力。

考试内容涵盖了日常生活、学术场景以及一些专业知识，旨在全面考察学生的英语应用能力。

为了在AEAS考试中取得好成绩，学生需要有计划地进行复习。

首先，要提高自己的词汇量和语法知识，这是英语学习的基础。

其次，要多做听力和阅读练习，提高理解能力和速度。

最后，要通过写作和口语练习，加强表达和沟通能力。

同时，熟悉考试的题型和格式也是非常重要的。

问题：1. AEAS考试的主要目的是什么？2. AEAS考试包括哪些部分？3. 学生应该如何准备AEAS考试？4. 为什么词汇和语法是英语学习的基础？5. 听力和阅读练习在AEAS考试准备中的作用是什么？文章二：在当今社会，科技的发展为人们的生活带来了许多便利。

智能手机、平板电脑等移动设备已经成为我们日常生活中不可或缺的一部分。

然而，随着这些设备的普及，人们对于这些电子产品的依赖程度也在不断增加。

过度依赖科技产品可能会导致一系列问题。

首先，它可能会影响人们的社交能力。

许多人在现实生活中缺乏面对面交流的能力，更倾向于通过社交媒体进行沟通。

其次，长时间使用电子设备可能会对人们的身体健康产生负面影响，如视力下降、颈部和腰部疼痛等。

AEAS数学推理题真题讲解

AEAS数学推理题真题讲解Consider the following 2 logical statements:（1）If the length of XY is 4, then the lengthof YZis 7.（2）The lengthof YZ is NOT 7.If these statements are both true, then the length of:F. XY is NOT 4G. XY is 7H. YZ is 4J. YZ is NOT 4K. YZ is 7这是典型的⼀道关于命题真假性的逻辑问题，同学们可试着先做⼀下。

事实上此题并不难，但前提是知道有关命题的基础的概念及性质。

命题是⼀句可以判断其真假性的话。

⽐如“⼀条线段是直的”是⼀个真命题，⽽“太阳从西边升起”就是⼀个假命题。

“⼀个苹果”不能称之为⼀个命题，因为不能从中判断真假性。

有时命题也可能有两个分句构成，即条件和结论，如“若明天下⾬，我就会打伞”。

这个命题可真可假，确实有下⾬不打伞的神⼈。

但这仍然是⼀个命题，因其真假必占其⼀。

原命题“条件A，结论B”的逆命题为“结论B，条件A”，否命题为“⾮条件A，⾮结论B”，逆否命题为“⾮结论B，⾮条件A”。

此间最重要的性质是，原命题与其逆否命题真假性相同。

以下⾬打伞为例，假设原命题“若明天下⾬，我就会打伞”为真，那么“若我明天不打伞，则明天不下⾬”亦为真。

反之亦然。

题⽬说命题“如果XY长度为4，则YZ长度为7”和命题“YZ长度不为7”都为真。

那么若根据原命题与其逆否命题同真同假的性质，把“如果XY长度为4，则YZ长度为7”看成原命题且为真，则其逆否命题为“如YZ长度不为7，则XY长度不为4”也为真。

故选择“F”。

AEAS数学考试真题资料讲解

A E A S数学考试真题AEAS考试真题1. What will it cost to carpet a room with indoor/outdoor carpet if the room is 10 feet wide and 12 feet long? The carpet costs 12.51 per square yard.A. $166.80B. $175.90C. $184.30D. $189.90E. $192.202. If the perimeter of a rectangular house is 44 yards, and the length is 36 feet, what is the width of the house?A. 10 yardsB. 18 yardsC. 28 feetD. 32 feetE. 36 yards3. What is the volume of the following cylinder?A. 210.91B. 226.20C. 75.36D. 904.32E. 28.264. What is the volume of a cube whose width is 5 inches?A. 15 cubic inchesB. 25 cubic inchesC. 64 cubic inchesD. 100 cubic inchesE. 125 cubic inches5. Sally has three pieces of material. The first piece is 1 yd. 2 ft. 6 in. long, the second piece is 2 yd. 1 ft. 5 in long, and the third piece is 4 yd. 2ft.8in long. How much material does Sally have?A. 7 yd. 1 ft. 8 in.B. 8 yd. 4 ft. 4 in.C. 8 yd. 11 in.D. 9 yd. 7 in.E. 10 yd.6. A ca n´s diameter is 3 inches, and its height is 8 inches. What is the volume of the can?A. 50.30B. 56.55C. 75.68D. 113.04E. 226.087. If the area of a square flowerbed is 16 square feet, then how many feet is the perimeter of the flowerbed?A. 4B. 12C. 16D. 20E. 248. Of the following units which would be more likely used to measure the amount of water in a bathtub?A. kilogramsB. litersC. millilitersD. centigramsE. volts9. If a match box is 0.17 feet long, what is its length in inches the most closely comparable to the following?A. 5 1/16 inch highlighterB. 3 1/8 inch jewelry boxC. 2 3/4 inch lipstickD. 2 3/16 inch staple removerE. 4 1/2 inch calculator10. What is the cost in dollars to steam clean a room W yards wide and L yards long it the steam cleaners charge 10 cents per square foot?A. 0.9WLB. 0.3WLC. 0.1WLD. 9WLE. 3WLKEYS:1. (A) 2. (A) 3. (B) 4. (E) 5. (D) 6. (B) 7. (C) 8. (B) 9. (D) 10. (A)1. How many cubed pieces of fudge that are 3 inches on an edge can be packed into a Christmas tin that is 9 inches deep by 12 inches wide by 8 inches high with the lid still being able to be closed?A. 18B. 24C. 32D. 36E. 432. Sarah is twice as old as her youngest brother. If the difference between their ages is 15 years. How old is her youngest brother?A. 10B. 15C. 20D. 25E. 303. Which of the following fractions is equal to 5/6?A. 20/30B. 15/24C. 25/30D. 40/54E. 2/74. What will it cost to tile a kitchen floor that is 12 feet wide by 20feet long if the tile cost $8.91 per square yard?A. $224.51B. $237.60C. $246.55D. $271.38E. $282.325. In a writing competition, the first place winner receives 1/2 of the prize money. The second runner up receives 1/2 of what the winner won. What was the total amount of prize money distributed if the winner receives $6,000?A. $6,000B. $8,500C. $12,000D. $15,000E. $18,5006. You are lying 120 ft away from a tree that is 50 feet tall. You look up at the top of the tree. Approximately how far is your hear from the top of the tree in a straight line?A. 50 feetB. 75 feetC. 120 feetD. 130 feetE. 150 feet7. A cyclist bikes x distance at 10 miles per hour and returns over the same path at 8 miles per hour. What is the cyclist’s average rate for the round trip in miles per hour?A. 8.1B. 8.3C. 8.6D. 8.9E. 9.08. If edging cost $2.32 per 12-inch stone, and you want a double layer of edging around your flower bed that is 6 yards by 1 yard. How much will edging you flower bed cost?A. $32.48B. $64.96C. $97.44D. $129.92E. $194.889. If 3x=6x-15 then x + 8=A. 5B. 10C. 11D. 12E. 1310. The number of milliliters in 1 liter isA. 10,000B. 1,000C. 0.1D. 0.01E. 0.001Answer Key1. (B)2. (B)3. (C)4. (B)5. (C)6. (D)7. (D)8. (E)9. (E) 10. (B)。

AP Calculus AB考试重点2024历年真题精编

AP Calculus AB考试重点2024历年真题精编2024年AP Calculus AB考试重点及历年真题精编2024年AP Calculus AB考试是高中学生的重要考试之一，对于学生们来说，了解考试的重点内容以及熟悉历年真题是取得好成绩的关键。

本文将为大家介绍2024年AP Calculus AB考试的重点内容，并提供一些历年真题的精编，以帮助学生们更好地备考。

一、考试重点内容1. 微分与导数微分与导数是AP Calculus AB考试的核心概念，重点内容包括函数的极限、连续性、导数的定义与性质，以及各种常见函数的导数运算法则。

学生们需要熟练掌握导数的计算方法，并应用于求解相关问题。

2. 积分与定积分积分与定积分也是考试的重点内容，包括不定积分的计算方法、定积分的概念与性质，以及定积分的应用。

学生们需要掌握积分的运算法则，并能够将积分应用于解决实际问题。

3. 微分方程微分方程是AP Calculus AB考试的另一个重点内容，包括一阶线性微分方程、可分离变量的微分方程、齐次线性微分方程等。

学生们需要熟悉微分方程的基本概念、解法和应用，能够正确地建立和解决微分方程模型。

4. 应用题考试中会涉及到一些与实际问题相关的应用题，包括函数的图像与性质、曲线的方程与参数方程、优化与极值等。

学生们需要通过熟练掌握微积分的理论知识，并能将其应用于解决实际问题。

二、历年真题精编1. 2019年AP Calculus AB考试真题考试题目：设函数f(x)在区间[0,6]上连续，且f(0)=2，f(6)=5。

已知f'(x)>0，f''(x)>0，求函数f(x)的一个表达式。

解析：根据已知条件，函数f(x)在区间[0,6]上递增，可以推断函数的一个表达式为f(x)=x+c，其中c为常数。

由f(0)=2可得c=2，因此函数f(x)的一个表达式为f(x)=x+2。

2. 2020年AP Calculus AB考试真题考试题目：已知函数y=f(x)满足f(2)=3且f'(x)=2e^x，求曲线y=f(x)在点(2,3)处的切线方程。

alevel2023数学cie考题

alevel2023数学cie考题A级数学（CIE）考试是一项经常让学生倍感挑战的考试。

各种难题需要学生具备扎实的数学基础和解题能力来应对。

以下将介绍一些A 级数学考试中常见的题目类型和解题思路。

一、代数题1.方程和不等式求解题：常见的代数题涉及到方程和不等式的求解。

例如，给定一个方程，要求求出方程的根或者将其转化为最简形式。

解此类题，可用因式分解、配方法等。

对于不等式的求解，常用的方法是绘制不等式的图像，然后找出满足条件的解。

2.函数题：函数题是A级数学考试中的经典题型之一。

解题时需理解函数的性质和图像的变化规律。

常见的函数题包括求解函数图像的性质、特殊函数的性质、复合函数的求导等。

3.向量与矩阵题：向量与矩阵题是A级数学中比较难的部分之一。

考生需要掌握向量与矩阵的基本运算法则，了解它们的性质。

例如，求解向量的模、点乘、叉乘以及矩阵的逆矩阵等。

二、几何题1.平面几何题：平面几何题是A级数学考试中的必考项目之一。

涉及到平面几何定理的运用，例如三角形的面积公式、三角形的中线长度关系、角平分线定理等。

学生需要了解这些定理的含义并能够熟练运用。

2.立体几何题：立体几何题需要考生对空间图形的性质有较深的理解。

例如，碰撞体的体积、表面积求解、立体体积问题等。

解题时需熟练掌握平行面、垂直面、直线与平面的交点等概念。

三、概率与统计题概率与统计题是A级数学考试中的另一个难点。

学生需要掌握计算概率的方法和统计学中的概念。

例如，根据已知概率计算新事件的概率、解决排列组合问题、计算均值和标准差等。

四、微积分题微积分是A级数学考试中的重点，需要掌握导数和积分的运算规则。

考生需要能够计算函数的导函数和不定积分，并能灵活运用求导和积分求解各种问题。

例如，曲线的切线问题、函数的最大和最小值问题等。

五、题目策略解决A级数学考题需要运用一定的策略。

1.熟练记忆公式和定理，掌握运用的条件和范围。

2.认真阅读题目，理解题意，并用简单的语言将问题翻译成数学问题。

aeas数学试题及答案

aeas数学试题及答案一、选择题（每题2分，共20分）1. 下列哪个选项是5的倍数？A. 10B. 15C. 20D. 25答案：B2. 一个圆的半径是5厘米，它的面积是多少平方厘米？A. 25πB. 50πC. 75πD. 100π答案：B3. 如果一个数的平方等于36，那么这个数是：A. 6B. ±6C. 36D. ±36答案：B4. 一个班级有40名学生，其中女生占60%，男生占多少百分比？A. 40%B. 50%C. 60%D. 70%答案：A5. 一个数列的前三项是2, 4, 6，这个数列是：A. 等差数列B. 等比数列C. 几何数列D. 既不是等差数列也不是等比数列答案：A6. 一个直角三角形的两条直角边分别是3和4，斜边的长度是：A. 5B. 6C. 7D. 8答案：A7. 一个数的立方根是2，这个数是：A. 2B. 4C. 8D. 16答案：C8. 如果一个数的绝对值是5，那么这个数可以是：A. 5B. -5C. 5或-5D. 都不是答案：C9. 一个数的平方是16，这个数可以是：A. 4B. -4C. 4或-4D. 都不是答案：C10. 一个数的倒数是1/4，这个数是：A. 4B. 1/4C. 1/2D. 2答案：A二、填空题（每题2分，共20分）1. 一个数的平方是9，这个数是________。

答案：±32. 一个数的立方是-27，这个数是________。

答案：-33. 一个数的绝对值是10，这个数可以是________或________。

答案：10 或 -104. 如果一个三角形的内角和是180°，那么一个四边形的内角和是________。

答案：360°5. 如果一个数的平方根是4，那么这个数是________。

答案：166. 一个数的倒数是2，这个数是________。

答案：1/27. 一个数的平方是25，这个数是________或________。

AEAS数学考试真题资料讲解

A E A S数学考试真题AEAS考试真题1. What will it cost to carpet a room with indoor/outdoor carpet if the room is 10 feet wide and 12 feet long? The carpet costs 12.51 per square yard.A. $166.80B. $175.90C. $184.30D. $189.90E. $192.202. If the perimeter of a rectangular house is 44 yards, and the length is 36 feet, what is the width of the house?A. 10 yardsB. 18 yardsC. 28 feetD. 32 feetE. 36 yards3. What is the volume of the following cylinder?A. 210.91B. 226.20C. 75.36D. 904.32E. 28.264. What is the volume of a cube whose width is 5 inches?A. 15 cubic inchesB. 25 cubic inchesC. 64 cubic inchesD. 100 cubic inchesE. 125 cubic inches5. Sally has three pieces of material. The first piece is 1 yd. 2 ft. 6 in. long, the second piece is 2 yd. 1 ft. 5 in long, and the third piece is 4 yd. 2ft.8in long. How much material does Sally have?A. 7 yd. 1 ft. 8 in.B. 8 yd. 4 ft. 4 in.C. 8 yd. 11 in.D. 9 yd. 7 in.E. 10 yd.6. A ca n´s diameter is 3 inches, and its height is 8 inches. What is the volume of the can?A. 50.30B. 56.55C. 75.68D. 113.04E. 226.087. If the area of a square flowerbed is 16 square feet, then how many feet is the perimeter of the flowerbed?A. 4B. 12C. 16D. 20E. 248. Of the following units which would be more likely used to measure the amount of water in a bathtub?A. kilogramsB. litersC. millilitersD. centigramsE. volts9. If a match box is 0.17 feet long, what is its length in inches the most closely comparable to the following?A. 5 1/16 inch highlighterB. 3 1/8 inch jewelry boxC. 2 3/4 inch lipstickD. 2 3/16 inch staple removerE. 4 1/2 inch calculator10. What is the cost in dollars to steam clean a room W yards wide and L yards long it the steam cleaners charge 10 cents per square foot?A. 0.9WLB. 0.3WLC. 0.1WLD. 9WLE. 3WLKEYS:1. (A) 2. (A) 3. (B) 4. (E) 5. (D) 6. (B) 7. (C) 8. (B) 9. (D) 10. (A)1. How many cubed pieces of fudge that are 3 inches on an edge can be packed into a Christmas tin that is 9 inches deep by 12 inches wide by 8 inches high with the lid still being able to be closed?A. 18B. 24C. 32D. 36E. 432. Sarah is twice as old as her youngest brother. If the difference between their ages is 15 years. How old is her youngest brother?A. 10B. 15C. 20D. 25E. 303. Which of the following fractions is equal to 5/6?A. 20/30B. 15/24C. 25/30D. 40/54E. 2/74. What will it cost to tile a kitchen floor that is 12 feet wide by 20feet long if the tile cost $8.91 per square yard?A. $224.51B. $237.60C. $246.55D. $271.38E. $282.325. In a writing competition, the first place winner receives 1/2 of the prize money. The second runner up receives 1/2 of what the winner won. What was the total amount of prize money distributed if the winner receives $6,000?A. $6,000B. $8,500C. $12,000D. $15,000E. $18,5006. You are lying 120 ft away from a tree that is 50 feet tall. You look up at the top of the tree. Approximately how far is your hear from the top of the tree in a straight line?A. 50 feetB. 75 feetC. 120 feetD. 130 feetE. 150 feet7. A cyclist bikes x distance at 10 miles per hour and returns over the same path at 8 miles per hour. What is the cyclist’s average rate for the round trip in miles per hour?A. 8.1B. 8.3C. 8.6D. 8.9E. 9.08. If edging cost $2.32 per 12-inch stone, and you want a double layer of edging around your flower bed that is 6 yards by 1 yard. How much will edging you flower bed cost?A. $32.48B. $64.96C. $97.44D. $129.92E. $194.889. If 3x=6x-15 then x + 8=A. 5B. 10C. 11D. 12E. 1310. The number of milliliters in 1 liter isA. 10,000B. 1,000C. 0.1D. 0.01E. 0.001Answer Key1. (B)2. (B)3. (C)4. (B)5. (C)6. (D)7. (D)8. (E)9. (E) 10. (B)。

2024版高考复习A版数学考点考法PPT讲解：圆锥曲线的综合问题

1
为定值.
λμ
解析
(1)由题意知e= c =
a
1
b2 a2
=
2 ,则a2=2b2,又椭圆C经过点H(-2,1),所
2
以
4 a2
+
1 b2
=1.联立解得a2=6,b2=3,所以椭圆C的方程为
x2 6
+
y2 3
=1.
(2)证明:显然,直线AB的斜率不为0,设直线AB的方程为x=my-3,A(x1,y1),
由
y y
2 16x, k1(x
4)
消去y得
k12
x2-(8
k12
+16)x+16
k12
=0,Δ1=256(
k12+1)>0,
设M(x1,y1),N(x2,y2),则x1+x2=8+16 ,
k12
则y1+y2=k1(x1-4)+k1(x2-4)=16 ,
k1
故A
4
8 k12
,
8 k1
,同理可求得B
B(x2,y2),
x my 3,
由
x
2
6
y2 3
1消x得(m2+2)y2-6my+3=0,所以Δ=36m2-12(m2+2)>0,y1+y2=
6m m2
2
,y1y2=
3 m2
2
,由题意知y1,y2均不为1.设M(xM,0),N(xN,0),由H,M,A三点
共线知
AM
与 MH
共线,所以xM-x1=-y1(-2-xM),化简得xM=
例2
(2021济南二模,21)已知椭圆C:

AEAS数学模拟题

AEAS考试真题1. What will it cost to carpet a room with indoor/outdoor carpet if the room is 10 feet wide and12 feet long The carpet costs per square yard.A. $B. $C. $D. $E. $2. If the perimeter of a rectangular house is 44 yards, and the length is 36 feet, what is the width of the houseA. 10 yardsB. 18 yardsC. 28 feetD. 32 feetE. 36 yards3. What is the volume of the following cylinderA.B.C.D.E.4. What is the volume of a cube whose width is 5 inchesA. 15 cubic inchesB. 25 cubic inchesC. 64 cubic inchesD. 100 cubic inchesE. 125 cubic inches5. Sally has three pieces of material. The first piece is 1 yd. 2 ft. 6 in. long, the second piece is 2 yd.1 ft. 5 in long, and the third piece is 4 yd. long. How much material does Sally haveA. 7 yd. 1 ft. 8 in.B. 8 yd. 4 ft. 4 in.C. 8 yd. 11 in.D. 9 yd. 7 in.E. 10yd.6. A can′s diameter is 3 inches, and its height is 8 inches. What is the volume of the canA. B. C. D. E.theareaofasquareflowerbedis16squarefeet,thenhowmanyfeetistheperimeteroftheflowerbedA.4B.12C.16D.20E.24 thefollowingunitswhichwouldbemorelikelyusedtomeasuretheamountofwaterinabathtubA.kilogramsB.literslilitersD.centigramsE.volts9. If a match box is feet long, what is its length in inches the most closely comparable to the followingA. 5 1/16 inch highlighterB. 3 1/8 inch jewelry boxC. 2 3/4 inch lipstickD. 2 3/16 inch staple removerE. 4 1/2 inch calculator10. What is the cost in dollars to steam clean a room W yards wide and L yards long it the steam cleaners charge 10 cents per square footA. B. C. D. 9WL E. 3WLKEYS:1. (A) 2. (A) 3. (B) 4. (E) 5. (D) 6. (B) 7. (C) 8. (B) 9. (D) 10. (A)1. How many cubed pieces of fudge that are 3 inches on an edge can be packed into a Christmas tin that is 9 inches deep by 12 inches wide by 8 inches high with the lid still being able to be closedA. 18B. 24C. 32D. 36E. 432. Sarah is twice as old as her youngest brother. If the difference between their ages is 15 years. How old is her youngest brotherA. 10B. 15C. 20D. 25E. 303. Which of the following fractions is equal to 5/6A. 20/30B. 15/24C. 25/30D. 40/54E. 2/74. What will it cost to tile a kitchen floor that is 12 feet wide by 20 feet long if the tile cost $ per square yardA. $B. $C. $D. $E. $5. In a writing competition, the first place winner receives 1/2 of the prize money. The second runner up receives 1/2 of what the winner won. What was the total amount of prize money distributed if the winner receives $6,000A. $6,000B. $8,500C. $12,000D. $15,000E. $18,5006. You are lying 120 ft away from a tree that is 50 feet tall. You look up at the top of the tree. Approximately how far is your hear from the top of the tree in a straight lineA. 50 feetB. 75 feetC. 120 feetD. 130 feetE. 150 feet7. A cyclist bikes x distance at 10 miles per hour and returns over the same path at 8 miles per hour. What is the c yclist’s average rate for the round trip in miles per hourA. B. C. D. E.8. If edging cost $ per 12-inch stone, and you want a double layer of edging around your flower bed that is 6 yards by 1 yard. How much will edging you flower bed costA. $B. $C. $D. $E. $9. If 3x=6x-15 then x + 8=A. 5B. 10C. 11D. 12E. 1310. The number of milliliters in 1 liter isA. 10,000B. 1,000C.D.E.Answer Key 1. (B) 2. (B) 3. (C) 4. (B) 5. (C) 6. (D) 7. (D) 8. (E) 9. (E) 10. (B)。

2014年Alevel 数学试卷解析(精排版)

EDEXCEL GCE MATHEMATICSGeneral Instructions for Marking1.The total number of marks for the paper is 75.2.The Edexcel Mathematics mark schemes use the following types of marks:•M marks: method marks are awarded for ‘knowing a method and attempting to apply it’, unless otherwise indicated.•A marks: Accuracy marks can only be awarded if the relevant method (M) marks have been earned.•B marks are unconditional accuracy marks (independent of M marks)•Marks should not be subdivided.3.AbbreviationsThese are some of the traditional marking abbreviations that will appear in the mark schemes.•bod – benefit of doubt•ft – follow through•the symbol will be used for correct ft•cao – correct answer only•cso - correct solution only. There must be no errors in this part of the question to obtain this mark•isw – ignore subsequent working•awrt – answers which round to•SC: special case•oe – or equivalent (and appropriate)•dep – dependent•indep – independent•dp decimal places•sf significant figures•¿The answer is printed on the paper•The second mark is dependent on gaining the first mark4.All A marks are ‘correct answer only’ (cao.), unless shown, for example, as A1 ft toindicate that previous wrong working is to be followed through. After a misread however, the subsequent A marks affected are treated as A ft, but manifestly absurd answersshould never be awarded A marks.5.For misreading which does not alter the character of a question or materially simplify it,deduct two from any A or B marks gained, in that part of the question affected.6.If a candidate makes more than one attempt at any question:•If all but one attempt is crossed out, mark the attempt which is NOT crossed out.•If either all attempts are crossed out or none are crossed out, mark all the attempts and score the highest single attempt.7.Ignore wrong working or incorrect statements following a correct answer.General Principles for Core Mathematics Marking(But note that specific mark schemes may sometimes override these general principles). Method mark for solving 3 term quadratic: 1.Factorisationc pq q x p x c bx x =++=++where ),)(()(2, leading to x = …a mn c pq q nx p mx c bx ax ==++=++and where ),)(()(2, leading to x = …2.FormulaAttempt to use the correct formula (with values for a , b and c ).pleting the squareSolving02=++c bx x :0,022≠=±±⎟⎠⎞⎜⎝⎛±q c q b x , leading to x = …Method marks for differentiation and integration: 1.DifferentiationPower of at least one term decreased by 1. (1−→n nx x )2.IntegrationPower of at least one term increased by 1. (1+→n nxx )Use of a formulaWhere a method involves using a formula that has been learnt, the advice given in recent examiners’ reports is that the formula should be quoted first. Normal marking procedure is as follows:Method mark for quoting a correct formula and attempting to use it, even if there are mistakes in the substitution of values.Where the formula is not quoted, the method mark can be gained by implication from correct working with values, but may be lost if there is any mistake in the working.Exact answersExaminers’ reports have emphasised that where, for example, an exact answer is asked for, or working with surds is clearly required, marks will normally be lost if the candidate resorts to using rounded decimals.Answers without workingThe rubric says that these may not gain full credit. Individual mark schemes will give details of what happens in particular cases. General policy is that if it could be done “in your head”, detailed working would not be required.Notes(a) M1: Attempt to differentiate – power reduced 1n n→or 3x becomes 3x x−A1: two correct terms ( of the three shown). They may be unsimplifiedA1: fully correct and simplified then isw (any equivalent simplified form acceptable) (b) M1: Attempt to integrate original f(x)– one power increased 1n nx x+→A1: Two of the four terms in x correct unsimplified – (ignore lack of constant here) A1: Three terms correct unsimplified – (ignore lack of constant here)A1: All correct simplified with constant – allow -1x for –xN.B Integrating answer to part (a) is M0Notes(a)(b) (c)M1: Attempt to use formula correctly (implied by first term correct, or given as 0.67, or third term following through from their second etc) A1: two correct answersA1: 3 correct answers (allow 0.6 recurring but not 0.667) Look for the values. Ignore the label r u B1: cao (NB Use of AP is B0)M1:Uses sum of at least 3 terms found from part (a)) (may be implied by correct answer). Attempt to sum an AP here is M0.A1: obtains 33(sum of three adjacent terms)×or 11(sum of nine adjacent terms)× A1: - 11 cao ( -11 implies both A marks) N.B. Use of n = 99 is M1A0A0(b)to give printed answer including = 0M1: Solving the correct quadratic equation (allow sign errors), by the usual methods (see notes) – implied by correct answersA1: Both answers needed – allow 0.25 and awrt – 0.33M1 Uses inverse cosine to obtain two correct values for x for their values of cos x e.g. (75.5 and 109.4 or 109.5) or (75.5 and 284.5) or (109.5 and 250.5) – allow truncated answers or awrt here.A1: All four correct – allow awrt. Ignore extra answers outside range but lose last A mark for extra answers inside rangeAnswers in radians are 1.3, 5.0, 1.9 and 4.4 Allow M1A0 for two or more correct asnwersQuestion Number SchemeMarks 8.kx 2x ++82k +()7=Uses 2−4b ac with a k ,b ==8 and attempt at c =2(k + 7)M1 b a −=c 464−56k −228k or 6456k =+8k 2o .e . A1 Attempts to solve 2"7k k +−8=0"to give k =dM1 ⇒Critical values, k 1,=−8.A1cso M1 Uses 4b a −<c 0or 22b 4<ac or ac 4 −b 2>02k k +−78>0 gives 1(or)k k ><8−M1 A1 [7]7 marksNotesM1: Attempts 24b ac − for ,8a k b == and c = 2(k +7) or attempt at c from quadratic = 0 (may omit bracket or make sign slip or lose the 2, so 2k +7 or k + 7 for example)oruses quadratic formula to solve equation or uses on two sides of an equation or inequation A1: Correct three term quadratic expression for 24b ac − - (may be under root sign)dM1: Uses factorisation, formula, or completion of square method to find two values for k , or finds two correct answers with no obvious method for their three term quadratic A1: Obtains 1 and -8M1: states 22404b ac or b ac −<< anywhere (may be implied by the following work)M1: Chooses outside region ( k < Their Lower Limit Their Upper Limit k >) for appropriate 3 term quadratic inequality . Do not award simply for diagram or table.A1: 1or 8k k ><− - allow anything which clearly indicates these regions e.g. (,8)−∞− or (1,)∞ 1,8k k ><− is A1 but k > 1 and k < -8 is A0but x > 1, x < -8 is A0 ( only lose 1 mark for using x instead of k ) and 1(or)8k k ≥≤−is A0 Also 1< k < -8 is M1 A0N.B. Lack of working: If there is no mention of 2240or 4b ac b ac −<< then just the correct answer 1,8k k ><− can imply the last M1M1A1 1,8k k ≥≤− can imply M0M1A01,8k k ><− can imply M1M1A0 Anything else needs to apply schemeMethod markfor harmoniccurve i.e. anysine or cosinecurveAccuracy forcorrect sectionand positionrelative to the (0 , ½ ) ;Notes for Question 14(a) M1: Puts equations equaldM1 Solves quadratic to obtain x =A1: both answers correctdM1: finds y =A1: both correctB1: Correct integration of one of the quadratic expression (given in the mark scheme) to give one of the givencubic expression (ignore limits). Allow correct answer even if terms not collected nor simplified. Sign errors (b)subtracting in alternative methods before integration gain B0B1: Line intersection correct (see 1.5)B1: curve intersection correct (see 5)M1: Uses correct combination of correct areas (allow numerical slips) so(i)Area of triangle using their “6” – their “1.5” times their “9” MINUS area beneath curve between their 5 and their 6(ii) Area of triangle using their “5” – their “1.5” times their “7” PLUS area between curves between their 5 and their 6(iii) Subtracts area below axis from area between curvesTHEIR 1.5 must NOT BE ZERO!M1: Attempts second area (so area of a triangle relevant to the method- or integral of the linear functionwith relevant limits- or integral of original quadratic in second alternative method)M1: Uses their limits (even zero) correctly on any cubic expression (subtracting either way round) Can begiven for wrong limits or for wrong areas. No evidence of substitution of limits is M0A1: Final answer – not decimal – csoAlternative methods to part (d)(i)Use equation x y 22++ax +by +c =0and substitute three points, usually (0,3), (6,11) and another point on the circle maybe (-2,17) or (-8,9) - not point Z Solves simultaneous equations a = 2, b = -20 and c = 51(ii) Uses centre to write a = and b = (doubles x coordinate and y coordinate respectively, "2"and ±±"20")Obtains a = 2 and b = -20 (or just writes these values down so these answers imply M1A1) Completes method to find c , (could substitute one of the points on the circle) or could find r Accurate work e.g. r 2=50 or e.g. +x y 222+−x 20y =(−8)2+2+928×−−20×9 = c = 51M1 dM1A1,A1,A1 M1 A1 dM1 A1 A1。

alevelM1试题答案

alevelM1试题答案ALevel M1数学试卷分析与答案解析一、选择题1. 问题描述：给定一个函数f(x) = 3x^2 - 2x + 1，求f(1)的值。

答案解析：将x=1代入函数f(x)中，得到f(1) = 3*1^2 - 2*1 + 1 = 3 - 2 + 1 = 2。

2. 问题描述：计算积分∫(2x - 3)dx从x=1到x=3的值。

答案解析：首先找到函数的原函数F(x) = x^2 - 3x，然后计算F(3) - F(1) = (3^2 - 3*3) - (1^2 - 3*1) = 9 - 9 - 3 + 3 = 0。

3. 问题描述：求解方程sin(x) = 0.5在区间[0, 2π]内的解。

答案解析：在区间[0, 2π]内，sin(x) = 0.5的解为x = π/6 和x = 5π/6。

二、填空题4. 问题描述：已知等差数列的前三项分别为a, a+d, a+2d，求该等差数列的第n项。

答案解析：等差数列的第n项为 an = a + (n-1)d。

5. 问题描述：给定一个圆的半径r和圆心到某点的距离d，求该点到圆上最近点的距离。

答案解析：若点在圆外，则最近点的距离为√(r^2 - d^2)；若点在圆内，则最近点的距离为√(d^2 - r^2)。

三、解答题6. 问题描述：证明函数f(x) = x^3 - 3x^2 + 2x在区间[0, 3]上单调递增。

答案解析：求导得到f'(x) = 3x^2 - 6x + 2。

令f'(x) ≥ 0，解得x ≥ 2 或x ≤ 1。

因此，在区间[0, 1]和[2, 3]上，函数单调递减，在区间[1, 2]上单调递增。

由于整个区间[0, 3]被这两个区间覆盖，且函数在x=1和x=2处连续，故函数在[0, 3]上整体单调递增。

7. 问题描述：求解微分方程dy/dx = x^2 - y，初始条件为当x=0时，y=1。

答案解析：首先，将微分方程改写为dy = (x^2 - y)dx。

数学二真题及答案解析

2014年全国硕士研究生入学统一考试数学二试题一、选择题:18小题,每小题4分,共32分.下列每题给出的四个选项中,只有一个选项符合题目要求的,请将所选项前的字母填在答题纸．．．指定位置上. (1) 当0x +→时，若ln (12)x +α，1(1cos )x -α均是比x 高阶的无穷小，则α的取值范围是( )(A) (2,)+∞(B) (1,2)(C) 1(,1)2(D) 1(0,)2(2) 下列曲线中有渐近线的是 ( )(A) sin y x x =+ (B) 2sin y x x =+ (C) 1siny x x =+(D) 21siny x x=+ (3) 设函数()f x 具有2阶导数，()(0)(1)(1)g x f x f x =-+，则在区间[0,1]上（）(A) 当()0f x '≥时，()()f x g x ≥ (B) 当()0f x '≥时，()()f x g x ≤ (C) 当()0f x ''≥时，()()f x g x ≥(D) 当()0f x ''≥时，()()f x g x ≤(4) 曲线22741x t y t t ⎧=+⎪⎨=++⎪⎩上对应于1t =的点处的曲率半径是（）(C)(D)(5) 设函数()arctan f x x =，若()()f x x f '=ξ，则22l i m x x→=ξ （）(A)1(B)23(C)12(D)13(6) 设函数(,)u x y 在有界闭区域D 上连续，在D 的内部具有2阶连续偏导数，且满足20ux y∂≠∂∂及22220u ux y∂∂+=∂∂，则（） (A)(,)u x y 的最大值和最小值都在D 的边界上取得 (B) (,)u x y 的最大值和最小值都在D 的内部上取得(C) (,)u x y 的最大值在D 的内部取得，最小值在D 的边界上取得(D) (,)u x y 的最小值在D 的内部取得，最大值在D 的边界上取得(7) 行列式0000000ab a bcd c d= （）(A) 2()ad bc - (B) 2()ad bc -- (C) 2222a dbc -(D) 2222b c a d -(8) 设123,,ααα均为3维向量，则对任意常数,k l ，向量组1323,k l ++αααα线性无关是向量组123,,ααα线性无关的（）(A) 必要非充分条件 (B) 充分非必要条件(C) 充分必要条件 (D) 既非充分也非必要条件二、填空题：914小题,每小题4分,共24分.请将答案写在答题纸．．．指定位置上. ((9)12125dx x x -∞=++⎰__________.(10) 设()f x 是周期为4的可导奇函数，且()f x '2(1),x =-[0,2]x ∈，则(7)f =__________.(11) 设(,)z z x y =是由方程2274yzex y z +++=确定的函数，则11(,)22dz =__________.(12) 曲线()r r =θ的极坐标方程是r =θ，则L 在点(,)(,)22r =ππθ处的切线的直角坐标方程是__________.(13) 一根长为1的细棒位于x 轴的区间[0,1]上,若其线密度()221x x x =-++ρ,则该细棒的质心坐标x =__________.(14) 设二次型()22123121323,,24f x x x x x ax x x x =-++的负惯性指数为1，则a 的取值范围为_______.三、解答题：15～23小题,共94分.请将解答写在答题纸．．．指定位置上.解答应写出文字说明、证明过程或演算步骤. (15)(本题满分10分)求极限12121lim.1ln 1xt x t e t dt x x →+∞⎡⎤⎛⎫--⎢⎥ ⎪⎢⎥⎝⎭⎣⎦⎛⎫+ ⎪⎝⎭⎰(16)(本题满分10分)已知函数()y y x =满足微分方程221x y y y ''+=-，且()20y =，求()y x 的极大值与极小值.(17)(本题满分10分)设平面区域(){}22,14,0,0,D x y x y x y =≤+≤≥≥计算(sin Dx dxdy x y+⎰⎰.(18)(本题满分10分)设函数()f u 具有二阶连续导数，(e cosy)xz f =满足22222(4e cos )e x xz z z y x y ∂∂+=+∂∂，若'(0)0,(0)0f f ==，求()f u 的表达式.(19)(本题满分10分)设函数(),()f x g x 的区间[a,b]上连续，且()f x 单调增加，0()1g x ≤≤.证明： (I)0(),[,]xag t dt x a x a b ≤≤-∈⎰,(II)()()d ()g()ba a g t dtb aaf x x f x x dx +⎰≤⎰⎰.(20)(本题满分11分)设函数[](x),0,11xf x x=∈+，定义函数列121()(),()(()),f x f x f x f f x ==,1()(()),n n f x f f x -=，记n S 是由曲线()n y f x =，直线1x =及x 轴所围成平面图形的面积，求极限lim n n nS →∞.(21)(本题满分11分) 已知函数(,)f x y 满足2(1)fy y∂=+∂，且2(,)(1)(2)ln ,f y y y yy =+--求曲线(,)0f x y =所围成的图形绕直线1y =-旋转所成的旋转体的体积. (22)(本题满分11分)设矩阵123401111203A --⎛⎫ ⎪=- ⎪ ⎪-⎝⎭，E 为三阶单位矩阵.(I)求方程组0Ax =的一个基础解系； (II)求满足AB E =的所有矩阵.(23)(本题满分11分)证明n 阶矩阵111111111⎛⎫ ⎪⎪ ⎪ ⎪⎝⎭与00100200n ⎛⎫⎪ ⎪⎪ ⎪⎝⎭相似.2014年全国硕士研究生入学统一考试数学二试题答案一、选择题:18小题,每小题4分,共32分.下列每题给出的四个选项中,只有一个选项符合题目要求的,请将所选项前的字母填在答题纸．．．指定位置上. (1) 当0x +→时，若ln (12)x +α，1(1cos )x -α均是比x 高阶的无穷小，则α的取值范围是( )(A) (2,)+∞(B) (1,2)(C) 1(,1)2(D) 1(0,)2【答案】B【解析】由定义 1000ln (12)(2)limlim lim 20x x x x x x x x-→→→+===αααα 所以10->α，故1>α.当0x +→时，211(1cos )~2x x -ααα是比x 的高阶无穷小，所以210->α，即2<α.故选B(2) 下列曲线中有渐近线的是 ( )(A) sin y x x =+ (B) 2sin y x x =+ (C) 1sin y x x =+(D) 21siny x x=+ 【答案】C【解析】关于C 选项：11sinsinlimlim1lim 101x x x x x x x x →∞→∞→∞+=+=+=. 11lim[sin ]limsin 0x x x x x x →∞→∞+-==，所以1sin y x x=+存在斜渐近线y x =. 故选C(3) 设函数()f x 具有2阶导数，()(0)(1)(1)g x f x f x =-+，则在区间[0,1]上（）(A) 当()0f x '≥时，()()f x g x ≥ (B) 当()0f x '≥时，()()f x g x ≤ (C) 当()0f x ''≥时，()()f x g x ≥(D) 当()0f x ''≥时，()()f x g x ≤【答案】D【解析】令()()()(0)(1)(1)()F x g x f x f x f x f x =-=-+-，则(0)(1)0F F ==,()(0)(1)()F x f f f x ''=-+-,()()F x f x ''''=-.若()0f x ''≥，则()0F x ''≤，()F x 在[0,1]上为凸的.又(0)(1)0F F ==，所以当[0,1]x ∈时，()0F x ≥，从而()()g x f x ≥. 故选D.(4) 曲线22741x t y t t ⎧=+⎪⎨=++⎪⎩上对应于1t =的点处的曲率半径是（）(C)(D)【答案】C 【解析】1112'21122432212t t t t t dy t dxtd y dy tdx dx t=====+==-===-()()''33'22211,11y k R kq y ==∴==++ 故选C(5) 设函数()arctan f x x =，若()()f x x f '=ξ，则22l i m x x→=ξ （）(A)1(B)23(C)12(D)13【答案】D 【解析】因为'2()1()1f x f x ==+ξξ，所以2()()x f x f x -=ξ22222200011()arctan 11limlimlim lim ()arctan 33x x x x x f x x xx x x f x x x x →→→→---+====ξ故选D.(6) 设函数(,)u x y 在有界闭区域D 上连续，在D 的内部具有2阶连续偏导数，且满足20ux y∂≠∂∂及22220u ux y∂∂+=∂∂，则（） (A)(,)u x y 的最大值和最小值都在D 的边界上取得 (B) (,)u x y 的最大值和最小值都在D 的内部上取得(C) (,)u x y 的最大值在D 的内部取得，最小值在D 的边界上取得 (D) (,)u x y 的最小值在D 的内部取得，最大值在D 的边界上取得【答案】A【解析】记22222,,,0,,u u uA B C B A C x x y y∂∂∂===≠∂∂∂∂相反数则2=AC-B 0∆<,所以(x,y)u 在D 内无极值，则极值在边界处取得.故选A(7) 行列式0000000ab a bcd c d= ( )(A)2()ad bc - (B)2()ad bc -- (C)2222a d b c - (D)2222b c a d -【答案】B【解析】由行列式的展开定理展开第一列00000000000000a b a b a b a b a c dc b cd dc dc d=-- ()()a d a d b c b c a d b c =--+- 2()ad bc =--.(8) 设123,,a a a 均为三维向量，则对任意常数,k l ,向量组13a ka +,23a la +线性无关是向量组123,,a a a 线性无关的 ( )(A)必要非充分条件(B)充分非必要条件 (C)充分必要条件(D)既非充分也非必要条件【答案】A 【解析】()()13231231001k l k l ⎛⎫⎪++= ⎪ ⎪⎝⎭ααααααα.)⇐ 记()1323A k l =++αααα，()123B =ααα，1001k l ⎛⎫⎪= ⎪ ⎪⎝⎭C . 若123,,ααα线性无关，则()()()2r A r BC r C ===，故1323,k l ++αααα线性无关.)⇒ 举反例. 令30=α，则12,αα线性无关，但此时123,,ααα却线性相关.综上所述，对任意常数,k l ，向量1323,k l ++αααα线性无关是向量123,,ααα线性无关的必要非充分条件.故选A二、填空题：914小题,每小题4分,共24分.请将答案写在答题纸．．．指定位置上. (9)12125dx x x -∞=++⎰__________.【答案】38π【解析】()111221111arctan 252214132428x dx dx x x x -∞-∞-∞+==++++⎡⎤⎛⎫=--= ⎪⎢⎥⎝⎭⎣⎦⎰⎰πππ(10) 设()f x 是周期为4的可导奇函数，且()f x '2(1),x =-[0,2]x ∈，则(7)f =__________.【答案】1【解析】()()[]'210,2f x x x =-∈，且为偶函数则()()[]'212,0f x x x =--∈-，又()22f x x x c =--+且为奇函数，故=0c()[]222,0f x x x x ∴=--∈-，又()f x 的周期为4，()()711f f ∴=-=(11) 设(,)z z x y =是由方程2274yzex y z +++=确定的函数，则11(,)22dz =__________.【答案】1()2dx dy -+ 【解析】对2274yzex y z +++=方程两边同时对,x y 求偏导 22210(22)20yzyz z z e y x x z z e z y y y y ∂∂⎧⋅⋅++=⎪∂∂⎪⎨∂∂⎪+++=∂∂⎪⎩当11,22x y ==时,0z =故1111(,)(,)222211,22z z xy∂∂=-=-∂∂故11(,)22111()()222dzdx dy dx dy =-+-=-+(12) 曲线lim n n nS →∞的极坐标方程是r =θ，则L 在点(,)(,)22r =ππθ处的切线的直角坐标方程是__________. 【答案】22y x =-+ππ【解析】由直角坐标和极坐标的关系 cos cos sin sin x r y r ==⎧⎨==⎩θθθθθθ，于是(),,,22r ⎛⎫=⎪⎝⎭ππθ对应于(),0,,2x y ⎛⎫= ⎪⎝⎭π 切线斜率cos sin cos sin dydy d dx dx d +==-θθθθθθθθ0,22dy dx ⎛⎫⎪⎝⎭∴=-ππ所以切线方程为()202y x -=--ππ即2=2y x -+ππ(13) 一根长为1的细棒位于x 轴的区间[0,1]上,若其线密度()221x x x =-++ρ,则该细棒的质心坐标x =__________. 【答案】1120【解析】质心横坐标()()1010x x dx x x dx=⎰⎰ρρ ()()()()31122100042112310005=2133211=2143212x x dx x x dx x x x x x x dx x x x dx x ⎛⎫-++=-++= ⎪⎝⎭⎛⎫-++=-++= ⎪⎝⎭⎰⎰⎰⎰ρρ111112=5203x ∴=(13) 设二次型()22123121323,,24f x x x x x ax x x x =-++的负惯性指数是1，则a 的取值范围_________. 【答案】[]2,2-【解析】配方法：()()()22222123133233,,24f x x x x ax a x x x x =+---+由于二次型负惯性指数为1，所以240a -≥，故22a -≤≤.三、解答题：15～23小题,共94分.请将解答写在答题纸．．．指定位置上.解答应写出文字说明、证明过程或演算步骤. (15)(本题满分10分)求极限12121lim.1ln 1xt x t e t dt x x →+∞⎡⎤⎛⎫--⎢⎥ ⎪⎢⎥⎝⎭⎣⎦⎛⎫+ ⎪⎝⎭⎰【解析】11221122d d (e 1)(e 1)limlim 11ln(1)xx t t x x t t t t t t x x x x→+∞→+∞⎡⎤⎡⎤----⎢⎥⎢⎥⎣⎦⎣⎦=+⋅⎰⎰12lim[(e 1)]xx x x →+∞=--12000e 1e 11lim lim lim 222t t t xt t t t t t t t +++=→→→---====. (16)(本题满分10分)已知函数()y y x =满足微分方程221x y y y ''+=-，且()20y =，求()y x 的极大值与极小值.【解析】由221x y y y ''+=-，得22(1)1y y x '+=-………………………………………………………① 此时上面方程为变量可分离方程，解的通解为331133y y x x c +=-+ 由(2)0y =得23c =又由①可得 221()1x y x y -'=+当()0y x '=时，1x =±，且有：1,()011,()01,()0x y x x y x x y x '<-<'-<<>'><所以()y x 在1x =-处取得极小值，在1x =处取得极大值 (1)0,(1)1y y -==即：()y x 的极大值为1，极小值为0. (17)(本题满分10分)设平面区域(){}22,14,0,0,D x y xy x y =≤+≤≥≥计算(sin Dx dxdy x y+⎰⎰.【解析】D 关于y x =对称，满足轮换对称性，则：D D=12D D I dxdy ∴==⎢⎥⎣⎦⎰⎰1sin(2Ddxdy =⎰⎰π 2201211sin 21()cos 4d r rdrrd r =⋅=-⎰⎰⎰πθππππ22111cos |cos 4r r rdr ⎡⎤=-⋅-⎢⎥⎣⎦⎰ππ211121sin |4r ⎡⎤=-+-⎢⎥⎣⎦ππ 34=-(18)(本题满分10分)设函数()f u 具有二阶连续导数，(e cosy)xz f =满足22222(4e cos )e x xz z z y x y∂∂+=+∂∂，若'(0)0,(0)0f f ==，求()f u 的表达式.【解析】由()cos ,x z f e y =()(cos )cos ,(cos )sin x x x x z zf e y e y f e y e y x y∂∂''=⋅=⋅-∂∂ 22(cos )cos cos (cos )cos x x x x xz f e y e y e y f e y e y x∂'''=⋅⋅+⋅∂， ()()()22(cos )sin sin (cos )cos x x x x xz f e y e y e y f e y e y y∂'''=⋅-⋅-+⋅-∂ 由 ()22222+4cos x x z zz e y e x y∂∂=+∂∂，代入得， ()()22cos [4cos cos ]x x x x x f e y e f e y e y e ''⋅=+即()()cos 4cos cos x x x f e y f e y e y ''-=，令cos =,xe y t 得()()4f t f t t ''-=特征方程 240,2-==±λλ 得齐次方程通解2212t t y c e c e -=+设特解*y at b =+，代入方程得1,04a b =-=，特解*14y t =-则原方程通解为()22121=4t ty f t c e c e t -=+-由()()'00,00f f ==，得1211,1616c c ==-, 则()22111=16164u u y f u e e u -=--.(19)(本题满分10分)设函数(),()f x g x 在区间[,]a b 上连续，且()f x 单调增加，0()1g x ≤≤，证明：（I ）0(),[,]xag t dt x a x a b ≤≤-∈⎰,（II ）()()d ()g()ba a g t dtb aaf x x f x x dx +⎰≤⎰⎰.【解析】（I ）由积分中值定理()()(),[,]xag t dt g x a a x =-∈⎰ξξ()01g x ≤≤，()()()0g x a x a ∴≤-≤-ξ()()0xag t dt x a ∴≤≤-⎰（II ）直接由()01g x ≤≤，得到()()01=x xaag t dt dt x a ≤≤-⎰⎰（II ）令()()()()()ua u a g t dt aaF u f x g x dx f x dx +⎰=-⎰⎰()()()()()()()()()()'uaua F u f u g u f a g t dt g u g u f u f a g t dt =-+⋅⎡⎤=-+⎢⎥⎣⎦⎰⎰由（I ）知()()0uag t dt u a ≤≤-⎰()uaa a g t d t u∴≤+≤⎰又由于()f x 单增，所以()()()0u af u f ag t dt -+≥⎰()()'0F u F u ∴≥∴，单调不减，()()0F u F a ∴≥=取u b =，得()0F b ≥，即（II ）成立. (20)(本题满分11分)设函数[](x),0,11xf x x=∈+，定义函数列 1211()(),()(()),,()(()),n n f x f x f x f f x f x f f x -===，记n S 是由曲线()n y f x =，直线1x =及x 轴所围成平面图形的面积，求极限lim n n nS →∞. 【解析】123(),(),(),,(),112131n x x x xf x f x f x f x x x x nx====++++ 11100011()11n n x x n n S f x dx dx dx nx nx+-∴===++⎰⎰⎰ 1110200111111ln(1)1dx dx nx n n nx n n =-=-++⎰⎰ 211ln(1)n n n=-+ ln(1)ln(1)1lim 1lim 1lim 1lim 1n n n x x n x nS n x x→∞→∞→∞→∞++∴=-=-=-+101=-= (21)(本题满分11分) 已知函数(,)f x y 满足2(1)fy y∂=+∂，且2(,)(1)(2)l n ,f y y y y y =+--求曲线(,)0f x y =所围成的图形绕直线1y =-旋转所成的旋转体的体积.【解析】因为2(1)fy y∂=+∂,所以2(,)2(),f x y y y x =++ϕ其中()x ϕ为待定函数. 又因为()2(,)(1)2ln ,f y y y y y =+--则()()12ln y y y =--ϕ，从而()()22(,)212ln (1)2ln f x y y y x x y x x =++--=+--.令(,)0,f x y =可得()2(1)2ln y x x +=-，当1y =-时，1x =或2x =，从而所求的体积为()()2221122112ln ln 22V y dx x xdxx xd x =+=-⎛⎫=- ⎪⎝⎭⎰⎰⎰πππ22211221ln (2)222552ln 2(2)2ln 22ln 2.444x x x x dxx x ⎡⎤⎛⎫=--- ⎪⎢⎥⎝⎭⎣⎦⎛⎫=--=-⋅=- ⎪⎝⎭⎰πππππππ(22)(本题满分11分)设矩阵123401111203A --⎛⎫⎪=- ⎪ ⎪-⎝⎭，E 为三阶单位矩阵.(I)求方程组0Ax =的一个基础解系； (II)求满足AB E =的所有矩阵B .【解析】()123410012341000111010011101012030010431101A E ----⎛⎫⎛⎫⎪ ⎪=-→- ⎪ ⎪ ⎪ ⎪---⎝⎭⎝⎭123410010012610111010010213100131410013141---⎛⎫⎛⎫ ⎪ ⎪→-→--- ⎪ ⎪ ⎪ ⎪------⎝⎭⎝⎭, (I)0Ax =的基础解系为()1,2,3,1T=-ξ(II)()()()1231,0,0,0,1,0,0,0,1TTTe e e ===1Ax e =的通解为()()111112,1,1,02,12,13,T Tx k k k k k =+--=--+-+ξ 2Ax e =的通解为()()222226,3,4,06,32,43,TTx k k k k k =+--=--+-+ξ3Ax e =的通解为()()333331,1,1,01,12,13,T Tx k k k k k =+-=--++ξ123123123123261123212134313k k k k k k B k k k k k k ----⎛⎫ ⎪-+-++⎪∴= ⎪-+-++ ⎪ ⎪⎝⎭（123,,k k k 为任意常数）(23)(本题满分11分)证明n 阶矩阵111111111⎛⎫ ⎪⎪ ⎪ ⎪⎝⎭与00100200n ⎛⎫⎪ ⎪⎪ ⎪⎝⎭相似. 【解析】已知()1111A ⎛⎫ ⎪⎪= ⎪ ⎪⎝⎭，()12001B n ⎛⎫ ⎪ ⎪ ⎪ ⎪⎝⎭=，则A 的特征值为n ，0(1n -重).A 属于n λ=的特征向量为(1,1,,1)T ；()1r A =，故0Ax =基础解系有1n -个线性无关的解向量，即A 属于0λ=有1n -个线性无关的特征向量；故A 相似于对角阵=0n ⎛⎫⎪⎪Λ ⎪ ⎪⎝⎭. B 的特征值为n ，0(1n -重)，同理B 属于0λ=有1n -个线性无关的特征向量，故B 相似于对角阵Λ.由相似关系的传递性，A 相似于B .。