官方SAT2数学真题解析 Subject Tests Answer Explanations Math
SAT 2 数学2 college board官网全部原题加答案讲解 SAT Subject Test Math Level 2
SAT Subject Test Practice - Results Summary Mathematics Level 21Your answer Omitted!What is the distance in space between the points with coordinates and ?(A)(B)(C)(D)(E)ExplanationDifficulty: EasyThe correct answer is D.The distance between the points with coordinates and is given by the distance formula: .Therefore, the distance between the points with coordinates and is:,which simplifies to .2Your answer Omitted!If , what value does approach as gets infinitely larger?(A)(B)(C)(D)(E)ExplanationDifficulty: EasyThe correct answer is E.One way to determine the value that approaches as gets infinitely larger is to rewrite the definition of the function to use only negative powers of and then reason about the behavior of negative powers of as gets infinitely larger. Since the question is only concerned with what happens to as gets infinitely larger, one can assume that is positive. For , theexpression is equivalent to the expression . As gets infinitely larger, the expression approaches the value , so as gets infinitely larger, the expression approaches the value . Thus, as gets infinitely larger, approaches .Alternatively, one can use a graphing calculator to estimate the height of the horizontal asymptote for the function . Graph the function on an interval with “large”, say, from to .By examining the graph, the all seem very close to . Graph the function again, from, say, to .The vary even less from . In fact, to the scale of the coordinate plane shown, the graph of the function is nearly indistinguishable from the asymptotic line . This suggests that as gets infinitely larger, approaches , that is, .Note: The algebraic method is preferable, as it provides a proof that guarantees that the value approaches is . Although the graphical method worked in this case, it does not provide a complete justification; for example, the graphical method does not ensure that the graph resembles a horizontal line for “very large”such as .3Your answer Omitted!If is a factor of , then(A)(B)(C)(D)(E)ExplanationDifficulty: EasyThe correct answer is A.By the Factor Theorem, is a factor of only when is a root ofthat is, , which simplifies to . Therefore, .Alternatively, one can perform the division of by and then find a value for so that the remainder of the division is .Since the remainder is , the value of must satisfy . Therefore, .4Your answer Omitted!Alison deposits into a new savings account that earns percent interest compounded annually. If Alison makes no additional deposits or withdrawals, how many years will it take for the amount in the account to double?(A)(B)(C)(D)(E)ExplanationDifficulty: MediumAfter year, the amount in the account is equal to . After years, the amount isequal to , and so on. After years, the amount is equal to . You needto find the value of for which . There are several ways to solve this equation. You can use logarithms to solve the equation as follows.Since , it will take more than years for the amount in the account to double. Thus, you need to round up to .Another way to find is to use your graphing calculator to graph and . From the answer choices, you know you need to set the viewing window with values from to about and values extending just beyond . The of the point of intersection is approximately . Thus you need to round up to .5Your answer Omitted!In the figure above, when is subtracted from , what is the length of the resultant vector?(A)(B)(C)(D)(E)ExplanationDifficulty: MediumThe resultant of can be determined by . The length of the resultant is:6Your answer Omitted!In the -plane, what is the area of a triangle whose vertices are , , and ?(A)(B)(C)(D)(E)ExplanationDifficulty: MediumIt is helpful to draw a sketch of the triangle:The length of the base of the triangle is and the height of the triangle is . Therefore, the area of the triangle is . The correct answer is B.7Your answer Omitted!A right circular cylinder has radius and height . If and are two points on its surface, what is the maximum possible straight-line distance between and ?(A)(B)(C)(D)(E)ExplanationDifficulty: MediumThe maximum possible distance occurs when and are on the circumference of opposite bases: You can use the Pythagorean Theorem:The correct answer is (B).8Your answer Omitted!Note: Figure not drawn to scale.In the figure above, and the measure of is . What is the value of ?(A)(B)(C)(D)(E)ExplanationDifficulty: MediumThere are several ways to solve this problem. One way is to use the law of sines. Since ,the measure of is and the measure of is . Thus, and . (Make sure your calculator is in degree mode.)You can also use the law of cosines:Since is isosceles, you can draw the altitude to the triangle.9Your answer Omitted!The function is defined by for .What is the difference between the maximum and minimum values of ?(A)(B)(C)(D)(E)ExplanationDifficulty: MediumIt is necessary to use your graphing calculator for this question. First graph the function. It is helpful to resize the viewing window so the -values go fromto . On this interval the maximum value of is and the minimum value of is. The difference between these two values is , which rounds to .10Your answer Omitted!Suppose the graph of is translated units left and unit up. If the resulting graph represents , what is the value of ?(A)(B)(C)(D)(E)ExplanationDifficulty: MediumIt may be helpful to draw a graph of and .The equation for is . Therefore,. The correct answer is B.11Your answer Omitted!A sequence is recursively defined by , for . If and , what is the value of ?(A)(B)(C)(D)(E)ExplanationDifficulty: MediumThe values for and are given. is equal to . is equal to. is equal to . is equal to.If your graphing calculator has a sequence mode, you can define the sequence recursively and findthe value of . Let , since the first term is . Define . Let , since we have to define the first two terms and . Then examining a graph or table, you can find .12Your answer Omitted!The diameter and height of a right circular cylinder are equal. If the volume of the cylinder is , what is the height of the cylinder?(A)(B)(C)(D)(E)ExplanationDifficulty: MediumThe correct answer is A.To determine the height of the cylinder, first express the diameter of the cylinder in terms of theheight, and then express the height in terms of the volume of the cylinder.The volume of a right circular cylinder is given by , where is the radius of the circular base of the cylinder and is the height of the cylinder. Since the diameter and height are equal, . Thus . Substitute the expression for in the volume formula to eliminate :. Solving for gives . Since the volume of the cylinder is , theheight of the cylinder is .13Your answer Omitted!If ,then(A)(B)(C)(D)(E)ExplanationDifficulty: MediumThe correct answer is E.One way to determine the value of is to apply the sine of difference of two angles identity: . Since and , the identity gives . Therefore, .Another way to determine the value of is to apply the supplementary angle trigonometric identity for the sine: . Therefore, .14Your answer Omitted!A line has parametric equations and , where is the parameter. The slope of the line is(A)(B)(C)(D)(E)ExplanationDifficulty: MediumThe correct answer is B.One way to determine the slope of the line is to compute two points on the line and then use the slope formula. For example, letting gives the point on the line, and letting gives the point on the line. Therefore, the slope of the line is equal to .Alternatively, one can express in terms of . Since and , it follows that . Therefore, the slope of the line is .15Your answer Omitted!What is the range of the function defined by ?(A) All real numbers(B) All real numbers except(C) All real numbers except(D) All real numbers except(E) All real numbers between andExplanationDifficulty: MediumThe correct answer is D.The range of the function defined by is the set of such thatfor some .One way to determine the range of the function defined by is to solve the equation for and then determine which correspond to at least one . To solve for , first subtract from both sides to get and then take the reciprocal of both sides to get . The equation shows that for anyother than , there is an such that , and that there is no such for . Therefore, the range of the function defined by is all real numbers except .Alternatively, one can reason about the possible values of the term . The expression can take on any value except , so the expression can take on any value except . Therefore, the range of the function defined by is all real numbers except .16Your answer Omitted!The table above shows the number of digital cameras that were sold during a three-day sale. The prices of models , , and were , , and , respectively. Which of the following matrix representations gives the total income, in dollars, received from the sale of the cameras for each of the three days?(A)(B)(C)(D)(E)ExplanationDifficulty: MediumThe correct answer is C.A correct matrix representation must have exactly three entries, each of which represents the total income, in dollars, for one of the three days. The total income for Day is given by the arithmetic expression , which is the single entry of the matrix product; in the same way, the total income for Day is given by, the single entry of ; and the total income for Day is given by , the single entry of. Therefore, the matrix representationgives the total income, in dollars, received from the sale of the cameras for each of the three days. Although it is not necessary to compute the matrix product in order to answer the question correctly, equals .17Your answer Omitted!The right circular cone above is sliced horizontally forming two pieces, each of which has the sameheight. What is the ratio of the volume of the smaller piece to the volume of the larger piece?(A)(B)(C)(D)(E)ExplanationDifficulty: HardIt is helpful to label the figure.The top piece is a cone whose height is one-half the height of the original cone . Using the properties of similar right triangles, you should realize the radii of these two cones must be in the same ratio. So if the top cone has radius , the original cone has radius .The volume of the top piece is equal to . The volume of the bottom piece is equal to the volume of the original cone minus the volume of the top piece.The ratio of the volume of the smaller piece to the volume of the larger piece is .18Your answer Omitted!In the figure above, is a regular pentagon with side of length . What is the -coordinate of ?(A)(B)(C)(D)(E)ExplanationDifficulty: HardThe sum of the measures of the interior angles of a regular pentagon is equal to . Each interior angle has a measure of . Using supplementary angles, has a measure of . You can use right triangle trigonometry to find the -coordinate of point .Since , is about . Since the length of each side of the pentagon is , the -coordinate of point is . Putting the information together tells us that the -coordinate of point is . The correct answer is (B).19Your answer Omitted!For a class test, the mean score was , the median score was , and the standard deviation of the scores was . The teacher decided to add points to each score due to a grading error. Which of the following statements must be true for the new scores?I. The new mean score is .II. The new median score is .III. The new standard deviation of the scores is .(A) None(B) only(C) only(D) and only(E) , , andExplanationDifficulty: HardFor this type of question you need to evaluate each statement separately. Statement is true. If you add to each number in a data set, the mean will also increase by . Statement is also true. The relative position of each score will remain the same. Thus, the new median score will be equal to more than the old median score. Statement is false. Since each new score is more than the old score, the spread of the scores and the position of the scores relative to the mean remain the same. Thus, the standard deviation of the new scores is the same as the standard deviation of the old scores.20Your answer Omitted!A game has two spinners. For the first spinner, the probability of landing on blue is . Independently, for the second spinner, the probability of landing on blue is What is the probability that the first spinner lands on blue and the second spinner does not land on blue?(A)(B)(C)(D)(E)ExplanationDifficulty: HardSince the two events are independent, the probability that the first spinner lands on blue and the second spinner does not land on blue is the product of the two probabilities. The first probability is given. Since the probability that the second spinner lands on blue is the probability that thesecond spinner does not land on blue is Therefore, . The correct answer is (E).21Your answer Omitted!In January the world’s population was billion. Assuming a growth rate of percent per year, the world’s population, in billions, for years after can be modeled by theequation . According to the model, the population growth from January to January was(A)(B)(C)(D)(E)ExplanationDifficulty: HardThe correct answer is C.According to the model, the world’s population in January was and in January was . Therefore, according to the model, the population growth from January to January , in billions, was , or equivalently,.22Your answer Omitted!What is the measure of one of the larger angles of a parallelogram in the that has vertices with coordinates , , and ?(A)(B)(C)(D)(E)ExplanationDifficulty: HardThe correct answer is C.First, note that the angle of the parallelogram with vertex is one of the two larger angles of the parallelogram: Looking at the graph of the parallelogram in the makes this apparent. Alternatively, the sides of the angle of the parallelogram with vertex are a horizontal line segment with endpoints and and a line segment of positive slope with endpoints and that intersects the horizontal line segment at its left endpoint , so the angle must measure more than Since the sum of the measures of the four angles of aparallelogram equals , the angle with vertex must be one of the larger angles.One way to determine the measure of the angle of the parallelogram with vertex is to apply the Law of Cosines to the triangle with vertices , , and . The length of the two sides of the angle with vertex are and; the length of the side opposite the angle is . Let represent the angle with vertex and apply the Law of Cosines: , so. Therefore, the measure of one of the larger angles of the parallelogram is .Another way to determine the measure of the angle of the parallelogram with vertex is to consider the triangle , , and . The measure of the angle of this triangle with vertex is less than the measure of the angle of the parallelogram with vertex . The angle of the triangle has opposite side of length and adjacent side of length , so the measure of this angle is . Therefore, the measure of the angle of the parallelogram withvertex is .Yet another way to determine the measure of the angle of the parallelogram with vertex is to use trigonometric relationships to find the measure of one of the smaller angles, and then use the fact that each pair of a larger and smaller angle is a pair of supplementary angles. Consider the angle of the parallelogram with vertex ; this angle coincides with the angle at vertex of the right triangle with vertices at , , and , with opposite side of lengthand adjacent side of length , so the measure of this angle is . This angle, together with the angle of the parallelogram with vertex , form a pair of interior angles on the same side of a transversal that intersects parallel lines, so the sum of the measures of the pair of angles equals . Therefore, the measure of the angle of the parallelogram with vertex is.23Your answer Omitted!For some real number , the first three terms of an arithmetic sequence are, and . What is the numerical value of the fourth term?(A)(B)(C)(D)(E)ExplanationDifficulty: HardThe correct answer is E.To determine the numerical value of the fourth term, first determine the value of and then apply the common difference.Since , and are the first three terms of an arithmetic sequence, it must be true that, that is, Solving for gives . Substituting for in the expressions of the three first terms of the sequence, one sees that they are , , and , respectively. The common difference between consecutive terms for this arithmetic sequence is , and therefore, the fourth term is .24Your answer Omitted!In a group of people, percent have brown eyes. Two people are to be selected at random from the group. What is the probability that neither person selected will have brown eyes?(A)(B)(C)(D)(E)ExplanationDifficulty: HardThe correct answer is A.One way to determine the probability that neither person selected will have brown eyes is to count both the number of ways to choose two people at random from the people who do not have brown eyes and the number of ways to choose two people at random from all people, and then compute the ratio of those two numbers.Since percent of the people have brown eyes, there are people with brown eyes, and people who do not have brown eyes. The number of ways of choosing two people, neither of whom has brown eyes, is : there are ways to choose a first person and ways to choose a second person, but there are ways in which that same pair of people could be chosen. Similarly, the number of ways of choosing two people at random from the people is . Therefore, the probability that neither of the two people selected has brown eyes is.Another way to determine the probability that neither person selected will have brown eyes is to multiply the probability of choosing one of the people who does not have brown eyes at random from the people times the probability of choosing one of the people who does not have brown eyes at random from the remaining people after one of the people who does not have brown eyes has been chosen.Since percent of the people have brown eyes, the probability of choosing one of the people who does not have brown eyes at random from the people is . If one of the people who does not have brown eyes has been chosen, there remain people who do not have brown eyes out of a total of people; the probability of choosing one of the people who does not have brown eyes at random from the people is . Therefore, if two people are to be selected from the group at random, the probability that neither person selected will have brown eyes is .25Your answer Omitted!If , what is ?(A)(B)(C)(D)(E)ExplanationDifficulty: HardThe correct answer is E.One way to determine the value of is to solve the equation for . Since , start with the equation , and cube both sides to get. Isolate to get , and apply the cube root to both sides of the equation to get .Another way to determine the value of is to find a formula for and then evaluate at Let and solve for : cubing both sides gives , so , and. Therefore, , and .26Your answer Omitted!Which of the following equations best models the data in the table above?(A)(B)(C)(D)(E)ExplanationDifficulty: HardThe correct answer is D.One way to determine which of the equations best models the data in the table is to use a calculator that has a statistics mode to compute an exponential regression for the data.The specific steps to be followed depend on the model of calculator, but can be summarized as follows: Enter the statistics mode, edit the list of ordered pairs to include only the four points givenin the table and perform an exponential regression. The coefficients are, approximately, for the constant and for the base, which indicates that the exponential equation is the result of performing the exponential regression. If the calculator reports a correlation, it should be a number that is very close , to which indicates that the data very closely matches the exponential equation. Therefore, of the given models, best fits the data.Alternatively, without using a calculator that has a statistics mode, one can reason about the data given in the table.The data indicates that as increases, increases; thus, options A and B cannot be candidates for such a relationship. Evaluating options C, D and E at shows that option D is the one that gives a value of that is closest to In the same way, evaluating options C, D and E at each of the other given data points shows that option D is a better model for that one data point than either option C or option E. Therefore, is the best of the given models for the data.27Your answer Omitted!The linear regression model above is based on an analysis of nutritional data from 14 varieties of cereal bars to relate the percent of calories from fat to the percent of calories from carbohydrates . Based on this model, which of the following statements must be true?I. There is a positive correlation between and .II. When percent of calories are from fat, the predicted percent of calories from carbohydrates is approximately .III. The slope indicates that as increases by , decreases by .(A) II only(B) I and II only(C) I and III only(D) II and III only(E) I, II, and IIIExplanationDifficulty: HardThe correct answer is D.Statement I is false: Since , high values of are associated with low values of which indicates that there is a negative correlation between and .Statement II is true: When percent of calories are from fat, and the predicted percent of calories from carbohydrates is .Statement III is true: Since the slope of the regression line is , as increases by , increases by ; that is, decreases by .28Your answer Omitted!The number of hours of daylight, , in Hartsville can be modeled by , where is the number of days after March . The day with the greatest number of hours of daylight has how many more daylight hours than May ? (March and May have days each. April and June have days each.)(A) hr(B) hr(C) hr(D) hr(E) hrExplanationDifficulty: HardThe correct answer is A.To determine how many more daylight hours the day with the greatest number of hours of daylight has than May , find the maximum number of daylight hours possible for any day and then subtract from that the number of daylight hours for May .To find the greatest number of daylight hours possible for any day, notice that the expressionis maximized when , which corresponds to , so. However, for this problem, must be a whole number, as it represents a count of days after March . From the shape of the graph of the sine function, either or corresponds to the day with the greatest number of hours of daylight, and since, the expression is maximized when days after March . (It is not required to find the day on which the greatest number of hours of daylight occurs, but it is days after March ,that is, June .)Since May is days after March , the number of hours of daylight for May is .Therefore, the day with the greatest number of hours of daylight hasmore daylight hours than May .。
数学二真题详细答案解析
数学二真题详细答案解析数学是一门抽象而又具有广泛应用的学科，它在现代社会中扮演着重要的角色。
无论是在科学研究，金融交易，还是在日常生活中，我们都会遇到各种与数学有关的问题。
因此，深入理解数学的原理和拓展能力对我们来说至关重要。
下面，我将对数学二真题进行详细解答，以帮助大家更好地掌握数学知识。
题目一：计算题某公司去年的年利润为100万元，今年的年利润比去年增长了20%。
今年的年利润是多少万元？解析：首先，我们需要知道“增长20%”意味着增加了原来的20%，即原数加上原数的1/5。
所以，今年的年利润为100万元 + 100万元/5 = 120万元。
题目二：代数题若函数f(x) = x^2 + 3x - 4, 求f(2)的值。
解析：将x=2代入函数f(x)的表达式中，可以得到f(2) = 2^2 + 3*2 - 4 = 4 + 6 - 4 = 6。
题目三：几何题已知正方形ABCD的边长为a，点E是AD边上一点，连接CE，并延长交BC于点F。
若BE=3，CF=5，求正方形ABCD的边长a。
解析：利用类似三角形的性质，我们可以发现三角形BEC与三角形CFD相似，因此可以得到BE/BC = CE/CF。
根据已知条件，可以得到3/（3+a）= （a+3）/5。
经过化简计算，可以得到a^2 - 11a + 12 = 0，这是一个一元二次方程。
求解方程，可以得到a = 1 或 a = 12。
但由于正方形的边长必须为正数，因此a = 12。
通过以上几道题目，我们可以看到数学在解决实际问题中所起到的重要作用。
通过运用数学的方法和原理，我们可以更好地理解和应对各种问题。
这样的能力不仅可以帮助我们解决日常生活中的困难，也对我们在学术和职业生涯中有所帮助。
在学习数学的过程中，我们还要注意培养抽象思维和逻辑推理能力。
数学不仅仅是一门死记硬背的学科，更是培养思维能力的工具。
在解答问题时，我们需要运用各种数学原理和方法，进行思维的灵活转换。
2023年全国统一高考数学试卷以及答案解析(全国2卷)
2023年全国统一高考数学试卷以及答案
解析(全国2卷)
简介
本文档为2023年全国统一高考数学试卷及答案解析提供了全
国2卷的详细内容。
试卷由相关教育机构编写，并经过严格审核确
保质量。
以下是试卷和答案解析的概要。
试卷内容
试卷分为多个部分，涵盖了数学的各个领域和知识点。
主要的
考查内容包括但不限于：代数、几何、概率与统计、函数与解析几
何等。
试卷设置了不同难度的题目，旨在全面考查学生的数学能力
和应试能力。
答案解析
答案解析部分为每个试题提供了详细的解题方法和步骤。
通过
阅读答案解析，学生能够理解每道题目的解题思路和方法。
答案解
析还包括常见错误的解释和注意事项，帮助学生避免犯同样的错误。
注意事项
1. 本文档提供的试卷及答案解析仅供研究和参考，不可作为学生高考成绩的依据。
2. 学生在参考本文档时应保持独立思考，不应完全依赖答案解析提供的答案。
3. 文档中提供的内容经过审核，但仍有可能存在错误或遗漏，敬请谅解。
结束语
希望本文档能为广大学生提供有价值的研究参考。
祝愿各位同学在2023年全国统一高考中取得优异的成绩！
---
该文档由Writing Documents助手编写。
如有疑问，请及时反馈。
SAT II Subject Test Math (Level 2)历年真题及其详解(代数和函数)【圣才出品】
代数和函数真题精选及详解◆代数和函数真题精选及详解（一）1. If 3a=8b and 15b=4c, then a/c= _____.(A) 4/5(B) 8/15(C) 32/45(D) 15/18(E) 11/9【答案】C【解析】Divide the equations: (3a)/(4c)=(8b)/(15b).Multiply both sides by 4/3：a/c=(8/15)×(4/3)=32/45.2. If x2=16, y2=25, and(x-4)(y+5)≠0, then x3+y3=?(A) 125(B) 61(C) 52(D) 20(E) 9【答案】B【解析】If x2=16, then x=4 or -4, and if y2=25, then y=5 or -5. But if (x-4)(y+5)≠0,then x≠4 and y≠-5. Therefore, x3 +y3 =(-4) 3 +53 =61.3. If f(x)=3x-5, and if the domain of x consists of all real numbers defined by the inequality -3<x<5, then the range of f(x) contains all of the following members except：(A) 0(B) 5(C) 10(D) -5(E)【答案】C【解析】f(-3)=3(-3)-5=-14. f(5)=3(5)-5=10. So -14<f(x)<10, or R={x|-14<x<10}.4. If g is a linear function, and if g(3)=4 and g(5)=2, then g(-3)= _____.(A) -36(B) -15(C) 81(D) 10(E) 12【答案】D【解析】The linear function g includes the two functional pairs(3, 4)and(5, 2), so its slope is m=(2-4)/(5-3)=1, then the equation of the line is y-2=-1(x-5), org(x)=y=-x+7. Hence, g(-3)=-(-3)+7=10.5. Let the function g be defined by g = {(-2, 4), (-1, 1), (0, 6), (1, -1), (2, 8)}. Which statement is true?(A) g(1) = g(-1)(B) g(2)+g(-2) = 0(C) g(-2) = (1/2)g(2)(D) g(0) = g(1)+g(-1)(E) g(g(-1)) = g(g(1))【答案】C【解析】For each ordered pair included in the set of ordered pairs that defines function g, the first member is x and the second member is g(x). For example, g(-2) = 4, since (-2,4) belongs to the function g. Check the statement in each choice until you find the one that is true:~ Choice (A): g(1) = -1 and g(-1) = 1. Hence, g(1) ≠ g(-1).~ Choice (B): g(2) = 8 and g(-2) = 4. Hence, g(2)+g(-2) ≠ 0.~ Choice (C):g(-2) = 4 and 1/2g(2) = 1/2(8)= 4. Hence, g(-2) = 1/2g(2). This is the correct choice.6. If the function k is defined by k(h) = (h + 1)2, then k(x-2) = _____.(A) x2-x(B) x2-2x(C) x 2-2x + 1(D) x 2 + 2x- 1(E) x 2-1【答案】C 【解析】If k(h) = (h+1)2, then k(x-2) = ((x-2)+1)2= (x-1)2= (x-1)(x-1) =x 2 -2x+1.7. If 22155x x -=, then _____. I. x=0II. x=1III. x = -1(A) I only(B) II only(C) I and II(D) I and III(E) II and III【答案】B【解析】Since 2211555x x --==, x 2-2x = -1. Then, x 2-2x +1 = 0, so (x - 1)(x - 1)= 0, which means x = 1 is the only root. Hence, only Roman numerical choice II is correct.8. Rosa has 35 coins that are worth a total of $2. 50. If the coins are all nickels and dimes, how many more nickels than dimes does she have?(A) 25(B) 20(C) 15(D) 10 (E) 5 【答案】E 【解析】Let x the number of nickels, then 35-x is the number of dimes. 5x+10(35-x)=250. x=20.9.If the broken curve above is the graph of the function ()f x x =, then the solid curve could be the graph of the function _____.(A) ()4g x x =+(B) ()4g x x =-(C) ()4g x x =-(D) ()4g x x =+ (E) ()4g x x =【答案】D 【解析】The solid graph represents the vertical shift of the broken graph 4 units vertically up so that the y-coordinate of each point on the solid graph is 4 more than the y-coordinate of its corresponding point on the broken graph. Hence, if the broken graph is the graph of ()f x x =, then the solid graph must be the graph of ()4g x x =+.10. The weekly profit for a magazine company is given by the function P(x ) = 3/2x -300, where x represents the number of magazine sold. What is the weekly profit if 400 magazines are sold?(A) $100。
SAT2数学level2经典考题讲解
解析：这道题目最本质的考点是：syntheticdivision(综合除法)。
但是对于考生而言，首先要理解的关键数学术语是zero，零点。
什么是零点呢?零点是使得方程的因变量(dependent variable, 即f(x))等于0 (即：图像与x轴相交)的x的值。
以下给出这个方程的图像，从中我们要找出蓝色的曲线和X轴(横轴)相交的三个点的x坐标值。
就本题而言，既然题目说-1是一个零点，那么就将x=-1带入方程，会发现f(x)确实等于0.那么，另外的使方程等于0的x的值应该怎么求得呢?方法1：画图求解，(考场上实际操作可能性为0)。
如上图所示，三个让方程与横轴相交的点的x值分别为-3.5，-1，3。
所以，-1给出来的情况下，另外两个点应该选择C。
但是，考场上根本不可能有时间和工具供我们画图，因此这个方法不是实际操作可行的方法，不予考虑。
方法2：代入法既然是选择题，就可以用带入法求解。
带入法求解的时候，要遵循以下两个原则：(1)带入常见的数值来检验是否正确;(2)在常见数值中带入易于计算的数值来看是否正确。
按照这两个原则，在5个选项中首先带入1来运算，因为好计算，还因为出现了3次。
当x=1时候，f(x)=-36，显然不是0，因此有1的ADE都不是答案。
下一步，在剩下的BC两个选项中，要检验的是-3，因为两项都有3，说明3一定是答案，而-3和-7/2中，-3更方便于计算，于是决定用-3来检验。
当x=-3时候，f(x)=12，显然不是0，因此答案不是B而是C.由此可见，代入法是一个smart strategy，是带有一点小聪明的策略。
在代入法的过程中耗时的长短体现了考生的判断和逻辑分析。
但是，即使可以凭借这种方法选择答案，这也不。
2023年全国卷Ⅱ数学试题及其参考答案
2023年全国卷Ⅱ数学试题及其参考答案第一题试题请计算下列数列的前n项和：\[a_n = 3n + 2\]参考答案数列\(\{a_n\}\)的前n项和可表示为：\[S_n = \sum_{k=1}^{n} a_k = \frac{n}{2}(a_1 + a_n)\]代入数列\(\{a_n\}\)的表达式，得：\[S_n = \frac{n}{2}(2a_1 + (n-1)d)\]其中，\(a_1\)为数列的首项，\(d\)为数列的公差。
根据题目中给定的数列，首项\(a_1 = 5\)，公差\(d = 3\)。
将这些值代入上述公式，可得数列\(\{a_n\}\)的前n项和的表达式为：\[S_n = \frac{n}{2}(5 + 3(n-1))\]第二题试题已知正整数\(n\)满足条件：\(\log_{n}{2} + \log_{2}{n} = 2\)，求\(n\)的值。
参考答案根据题目中给定的条件，我们可以转化为指数方程：\[\left(\frac{1}{n}\right)^2 + 2^{\log_{2}{n}} = 2\]化简可得：\[\frac{1}{n^2} + n = 2\]将该方程转化为二次方程：\[n^3 - 2n^2 + n - 2 = 0\]通过试算，我们发现\(n = 2\)是该方程的一个解。
通过带入韦达定理可知该方程有且仅有一个正整数解。
所以，\(n = 2\)。
第三题试题已知函数\(f(x) = x^3 + x^2\)，求函数\(g(x) = f^{-1}(x)\)的表达式。
参考答案对于函数\(f(x) = x^3 + x^2\)，我们需要求其反函数\(f^{-1}(x)\)的表达式。
首先，我们令\(y = f(x)\)，得：\(y = x^3 + x^2\)将该方程转化为关于\(x\)的二次方程：\(x^3 + x^2 - y = 0\)通过试算，我们发现\(x = -1\)是该方程的一个根。
普通高等学校招生全国统一考试(课标Ⅱ卷)理数答案解析(正式版)(解析版).docx
一、选择题：本大题共12小题，每小题5分，在每小题给出的四个选项中，只有一项是符合题目要求的。
1．已知集合21,01,2A =--｛，，｝，{}(1)(20B x x x =-+<,则A B =I （ ）A ．{}1,0A =-B ．{}0,1C ．{}1,0,1-D ．{}0,1,2 【答案】A考点：集合的运算．2．若a 为实数且(2)(2)4ai a i i +-=-，则a =（ ） A ．1- B ．0 C ．1 D ．2 【答案】B 【解析】试题分析：由已知得24(4)4a a i i +-=-，所以240,44a a =-=-，解得0a =，故选B ． 考点：复数的运算．3．根据下面给出的2004年至2013年我国二氧化硫排放量（单位：万吨）柱形图。
以下结论不正确的是( )A ．逐年比较，2008年减少二氧化硫排放量的效果最显著B ．2007年我国治理二氧化硫排放显现C ．2006年以来我国二氧化硫年排放量呈减少趋势D ．2006年以来我国二氧化硫年排放量与年份正相关 【答案】D 【解析】试题分析：由柱形图得，从2006年以来，我国二氧化硫排放量呈下降趋势，故年排放量与年份负相关，故选D ．2004年 2005年 2006年 2007年 2008年 2009年 2010年 2011年 2012年 2013年考点：正、负相关．4．等比数列｛a n ｝满足a 1=3，135a a a ++ =21，则357a a a ++= ( )A ．21B ．42C ．63D ．84 【答案】B考点：等比数列通项公式和性质．5．设函数211log (2),1,()2,1,x x x f x x -+-<⎧=⎨≥⎩,2(2)(log 12)f f -+=( )A ．3B ．6C ．9D ．12 【答案】C 【解析】试题分析：由已知得2(2)1log 43f -=+=，又2log 121>，所以22log 121log 62(log 12)226f -===，故2(2)(log 12)9f f -+=，故选C ．考点：分段函数．6．一个正方体被一个平面截去一部分后，剩余部分的三视图如右图，则截去部分体积与剩余部分体积的比值为( ) A ．81 B ．71 C ．61 D ．51【答案】D 【解析】试题分析：由三视图得，在正方体1111ABCD A B C D -中，截去四面体111A A B D -，如图所示，，设正方体棱长为a ，则11133111326A A B D V a a -=⨯=，故剩余几何体体积为3331566a a a -=，所以截去部分体积与剩余部分体积的比值为51，故选D ． 考点：三视图．A1A7．过三点(1,3)A ，(4,2)B ，(1,7)C -的圆交y 轴于M ，N 两点，则||MN =( ) A ．26 B ．8 C ．46 D ．10 【答案】C【解析】由已知得321143AB k -==--，27341CB k +==--，所以1AB CB k k =-，所以AB CB ⊥，即ABC ∆为直角三角形，其外接圆圆心为(1,2)-，半径为5，所以外接圆方程为22(1)(2)25x y -++=，令0x =，得2y =±，所以MN =C ． 考点：圆的方程．8．右边程序框图的算法思路源于我国古代数学名著《九章算术》中的“更相减损术”．执行该程序框图，若输入,a b 分别为14,18，则输出的a =( )【答案】B 【解析】 试题分析：程序在执行过程中，a ，b 的值依次为14a =，18b =；4b =；10a =；6a =；2a =；2b =，此时2a b ==程序结束，输出a 的值为2，故选B ． 考点：程序框图．9．已知A,B 是球O 的球面上两点，∠AOB=90,C 为该球面上的动点，若三棱锥O-ABC 体积的最大值为36，则球O 的表面积为( )A ．36π B.64π C.144π D.256π【答案】C 【解析】试题分析：如图所示，当点C 位于垂直于面AOB 的直径端点时，三棱锥O ABC -的体积最大，设球O 的半径为R ，此时2311136326O ABC C AOB V V R R R --==⨯⨯==，故6R =，则球O 的表面积为24144S R ππ==，故选C ．考点：外接球表面积和椎体的体积．10．如图，长方形ABCD 的边2AB =，1BC =，O 是AB 的中点，点P 沿着边BC ，CD 与DA 运动，记BOP x ∠=．将动P 到A 、B 两点距离之和表示为x 的函数()f x ，则()y f x =的图像大致为（ ）【答案】B 【解析】DPCBOAx考点：函数的图象和性质．11．已知A ，B 为双曲线E 的左，右顶点，点M 在E 上，∆ABM 为等腰三角形，且顶角为120°，则E 的离心率为（ ）A 5．2 C 32 【答案】D 【解析】试题分析：设双曲线方程为22221(0,0)x y a b a b-=>>，如图所示，AB BM =，0120ABM ∠=，过点M作MN x ⊥轴，垂足为N ，在Rt BMN ∆中，BN a =，3MN a =，故点M 的坐标为(23)M a a ，代入双曲线方程得2222a b a c ==-，即222c a =，所以2e =D ．考点：双曲线的标准方程和简单几何性质．12．设函数'()f x 是奇函数()()f x x R ∈的导函数，(1)0f -=，当0x >时，'()()0xf x f x -<，则使得()0f x >成立的x 的取值范围是（ ）A ．(,1)(0,1)-∞-UB ．(1,0)(1,)-+∞UC ．(,1)(1,0)-∞--UD ．(0,1)(1,)+∞U【答案】A 【解析】试题分析：记函数()()f x g x x=，则''2()()()xf x f x g x x -=，因为当0x >时，'()()0xf x f x -<，故当0x >时，'()0g x <，所以()g x 在(0,)+∞单调递减；又因为函数()()f x x R ∈是奇函数，故函数()g x 是偶函数，所以()g x 在(,0)-∞单调递减，且(1)(1)0g g -==．当01x <<时，()0g x >，则()0f x >；当1x <-时，()0g x <，则()0f x >，综上所述，使得()0f x >成立的x 的取值范围是(,1)(0,1)-∞-U ，故选A ．考点：导数的应用、函数的图象与性质．第II 卷（非选择题，共90分）本卷包括必考题和选考题两部分。
数学二真题及答案解析
数学二真题及答案解析在高考数学中，数学二是一项非常重要的科目。
它包括了数学的许多重要概念和方法，如函数、微积分、概率与统计等。
因此，对于考生来说，掌握数学二的知识和技巧是非常关键的。
本文将对数学二的真题及其答案进行解析和讨论，希望能够帮助考生更好地准备和应对数学二的考试。
一、函数与图像在数学二的考试中，函数与图像是一个重要的考点。
考生需要掌握函数的定义和性质，了解不同类型函数的特点和变化规律。
同时，他们还需要能够通过函数的图像进行解题，如确定函数的极值、最值等。
下面，我们来看一道典型的函数与图像题目。
题目：已知函数f(x) = x^3 - 3x^2 + 3x - 1，求函数f(x)的最小值。
解析：要求函数的最小值，我们可以使用求导方法来解决这个问题。
首先，对函数f(x)求导得到f'(x) = 3x^2 - 6x + 3。
然后，令f'(x) = 0，即可求出极值点。
解方程3x^2 - 6x + 3 = 0得到x = 1，将x = 1代入原函数f(x)中，得到f(1) = -1。
因此，函数f(x)的最小值为-1。
通过这道题目，我们可以看出数学二考试中，函数与图像的题目既需要运用基本的概念和性质，又需要运用一定的计算和推理能力。
二、微积分微积分是数学二考试中的重要部分。
它涉及到函数的导数、积分、微分方程等内容。
掌握微积分的原理和应用是考生取得高分的关键。
下面，我们来看一个与微积分相关的题目。
题目：已知函数f(x) = sin(x)，求函数f(x)在区间[0,π]上的定积分值。
解析：要求函数在一个区间上的定积分值，我们可以先求出函数积分的原函数，然后根据定积分的性质进行计算。
对于函数f(x) = sin(x)，它的原函数是F(x) = -cos(x)。
因此，函数f(x)在区间[0,π]上的定积分值为F(π) - F(0) = -cos(π) - (-cos(0)) = -(-1) - (-1) = 2。
- 1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
- 2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
- 3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。
SAT Subject Tests™
Answer Explanations to Practice Questions
from Getting Ready for the SAT Subject Tests
Visit the SAT® student website to get more practice and study tips for the Subject Tests in Mathematics Level 1 and Level 2.
pairs (10, 255.90) and (20, 261.80). The line has slope
261.80 − 255.90 20 − 10
=
0.59 and y-intercept 250, which is
the cost, in dollars, for the band to produce 0 CDs after
Getting Ready for the SAT Subject Tests: Answer Explanations to Practice Questions
1
SAT® Subject Tests in Mathematics Level 1 and Level 2
4. Difficulty: 3
Alternatively, one can invert the operations that were done to n. Apply the inverse of each operation, in the reverse order: First cube −0.5 to get −0.125, and then decrease this value by 8 to find that n = −0.125 − 8 = −8.125.
purchasing the equipment. The equation of this line is
y = 0.59n + 250, where n is the number of CDs used.
Alternatively, note that the total cost to produce 10 CDs
5. Difficulty: 3
Choice (C) is correct.
One way to determine all values of x for which 4 − x2 ≥ x − 2 is first to rewrite the inequality in an equivalent form that compares a factored expression to 0 and then reason about the arithmetic sign of the product of the factors. The inequality 4 − x2 ≥ x − 2 is equivalent to 4 − x2 − x + 2 ≥ 0, which in turn is equivalent to −x2 − x + 6 ≥ 0. Since −x2 − x + 6 factors as (−1)(x2 + x − 6) = (−1)(x + 3)(x − 2), the original inequality is equivalent to (−1)(x + 3)(x − 2) ≥ 0 or (x + 3)(x − 2) ≤ 0. To solve (x + 3)(x − 2) ≤ 0, notice that this inequality is satisfied by a value of x precisely when either x = −3, x = 2 or the product of the factors (x + 3) and (x − 2) is negative; this last condition is true for −3 < x < 2, as shown in the table below:
y°
x° z°
E
A
D
One way to determine x in terms of y and z is to find the measure of each of the angles of ∆EDC in terms of x, y and z and then apply the triangle sum theorem. Since ∠CED is supplementary to ∠BEC and the measure of ∠BEC is given to be x°, ,the measure of ∠CED is (180 − x)°. Since
( ) produce n CDs is
250 + 5.90
n 10
= 250 + 0.59n dollars.
2. Difficulty: 2
Choiceoint of intersection of lines AC and BD.
B
C
Alternatively, one can apply the interior angle sum theorem to pentagon ABECD. Since line AD is a transversal to the parallel lines AB and CD, it follows that ∠BAD and ∠ADC are supplementary; that is, the sum of the measures of these two angles is 180°. The measure of ∠BEC, interior to polygon ABECD, is (360 − x)°. The measure of ∠EBA is given to be y°, and the measure of ∠DCE is given to be z°. Therefore, the sum of the measures of the interior angles of pentagon ABECD is 180 + (360 − x) + y + z. The interior angle sum theorem applied to pentagon ABECD gives the equation 180 + (360 − x) + y + z = 540, which can be solved for x to arrive at x = y + z.
One way to determine the correct expression is to find
the slope-intercept form of the equation of the line that
passes through the two points given by the ordered
line AC is a transversal to the parallel lines AB and CD, the alternate interior angles ∠BDC and ∠DBA are of equal measure. Thus, the measure of ∠BDC is y, which is also the measure of ∠EDC. The measure of ∠DCE is given to be z°. Therefore, the sum of the angle measures of ∆EDC, in degrees, is (180 − x) + y + z. The triangle sum theorem applied to ∆EDC gives the equation (180 − x) + y + z = 180, which can be solved for x to arrive at x = y + z.
Choice (A) is correct.
To determine the value of b, apply the fact that two complex numbers are equal if and only if the real and pure imaginary parts are equal. Since (a + b) + 5i = 9 + ai, this gives the two equations a + b = 9 and 5i = ai;, that is, a + b = 9 and 5 = a. Therefore, 5 + b = 9, and b = 4.
The cost of the equipment is $250, and each package of 10 blank CDs costs $5.90. The total cost for the band to produce 10 CDs, that is, 1 package, is 250 + 5.90 = 255.90 dollars, and the total cost to produce 20 CDs, that is, 2 packages, is 250 + 5.90(2) = 250 + 11.80 = 261.80 dollars.
3. Difficulty: 2
Choice (C) is correct.