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1.y=2x2+7x – 3x y本体答案不确定。
我开始做的时候被题目放了烟雾弹，以为要经过什么特别处理，像配方啥的。
但配着配着就感觉不对劲，x与y只有函数关系，没有绝对的大小关系嘛。
汗2.The figure above shows the graph of a function f, defined by f(x)=|2x|+4for all numbers x. For which of the following functions g defined for allnumbers x does the graph of g intersect the graph of f ?A g(x)=x–2B g(x)=x+3C g(x)=2x – 2D g(x)=2x+3E g(x)=3x – 2这个题就是问两个函数图象相交的问题，根据观察，要与这个折线相交的直线的斜率满足的条件是要么是正斜率比2大，要么是负斜率比-2小，因此只能选E了。
3.Each employee of a certain company is in either Department X orDepartment Y, and there are more than twice as many employees inDepartment X as in Department Y. The average (arithmetic mean) salaryis $25,000 for the employees in Department X and is $35,000 for theemployees in Department Y. Which of the following amounts could bethe average salary for all of the employees in the company?Indicate all such amounts.A $26,000B $28,000C $29,000D $30,000E $31,000F $32,000G $34,000本题我们要注意到题目中的more than twice的字眼，我们当做twice算的时候，结果是（25000*2+35000）/3约等于28333，因为有more than，所以选的答案要么是比前面这个数大，要么小，又由于more的是X部门的，那么肯定会将平均工资给拉低了，因此，选比前面那个数小的答案，AB都可以了。
2020年GRE模拟试题及答案（卷五）Investors are grateful that the attorney general has stepped in to pursue inquiries into misfeasance on the financial markets, given that the regulators officially charged with policing the industry have been _____.A：diffidentB：meticulousC：straightforwardD：implacableE：tenacious答案：AA significant element of the Gothic genre, the literary grotesque is sometimes inaccurately discussed as if it were a _____Gothic or, conversely, as if it were something entirely different from Gothic.A：synonym forB：characteristic ofC：counterweight toD：ramification ofE：deviation from答案：AIn his study of Senegambian cleric Shaykh Mass Kah (1827-1936), Bala S. K. Saho notes that it is difficult to fully (i) _____many of the oral accounts on which the study relies. Saho’s work shows, however, thatdespite this absence of (ii)_____ , oral history can provide useful sources from which historians can reconstruct the past.Blank (I)Blank (II)A：comprehend D：partialityB：reproduce E：argumentationC：validate F：corroboration答案：CFThe city’s traffic-planning department has been working hard to (i) _____drivers. Closely spaced stop lights have been added on roads into town, causing delays. Pedestrian underpasses designed to allow traffic to flow freely across major intersections have been (ii)_____ .Blank (I)Blank (II)A：assist D：enhancedB：calm E：stabilizedC：discourage F：removed答案：CFThe (i) _____current-generation solar cells are (ii)_____ : although experimental cells have reached efficiencies greater than 40 percent, most commercially available cells in the early part of the twenty-first century still struggle to get past about 20 percent.Blank (I)Blank (II)A：attractions of D：clearB：limitations of E：unmatchedC：improvements in F：misunderstood答案：BDEven those observers who are the most _____about genetic privacy issues would have to concede that genetic discrimination is rare: there have only been two cases of any notoriety.A：sanguineB：zealousC：candidD：objectiveE：apathetic答案：BShe constantly _____herself for not living up to her own ideals—for not working hard enough or not having motives that were pure enough.A：exaltedB：coddledC：excoriatedD：mollifiedE：deluded答案：CAncient cart ruts found on Malta were created in soft limestone that begins to dissolve when exposed to rainfall. Their forms thus necessarilybecome (i)_____over time and their original features are (ii)_____ .Blank (I)Blank (II)A：solidifed D：refinedB：degraded E：replicatedC：conspicuous F：obscured答案：BFMaking loans and fighting poverty are normally two of the least glamorous pursuits around, but remarkably enough put the two together, and you have an economic innovation that has become not just (i) _____but downright (ii) _____.Blank (I)Blank (II)A：popular D：chicB：pointless E：unfathomableC：dangerous F：sensible答案：ADConvinced of the gravity of her poetry, Voigt must have found the reviews of her most recent collection (i)_____ reading: one amused reviewer thought that it was unrecognizable as poetry but decidedly (ii)_____ .Blank (I)Blank (II)A：tempting D：inspiringB：depressing E：irritatingC：thrilling F：diverting答案：BFCommon and easily accessible resources (prey for predators or hosts for parasites) should be, all other things being equal, used frequently, yet in some environments apparently accessible and suitable resources remain _____.A：vulnerableB：unobtainableC：sustainableD：depletedE：unexploited答案：EEven though the authors repeatedly_____ their own shrewdness, they show a remarkable credulousness toward far-fetched ideas such as carbon-eating trees and cloud-making machinery.A：soft-pedalB：extolC：deprecateD：broadenE：compromise答案：BThe (i)_____ of molecular oxygen on Earth-sized planets aroundother stars in the universe would not be (ii)_____sign of life: molecular oxygen can be a signature of photosynthesis(a biotic process) or merely of the rapid escape of water from the upper reaches of a planetary atmosphere (an abiotic process).Blank (I)Blank (II)A：dearth D：a controversialB：presumption E：an unambiguousC：detection F：a possible答案：CEIn reviewing cases decided by lower courts, Supreme Court justices search for precedents to justify their arguments. Reliance on precedent (i)_____ judicial restraint: the precedent (ii)_____ a judge’s ability to determine the outcome of a case in a way that he or she might choose if there were no precedent.Blank (I)Blank (II)A：promotes D：establishesB：compromises E：constraintsC：promulgates F：prioritizes答案：AEAs he has matured as a scholar, Felmar has come to see the merit of qualification. His conclusions, which early in his career he (i) _____, are now often (ii) _____.Blank (I)Blank (II)A：stated as absolute D：hedgedB：refused to reveal E：simplifiedC：backed up extensively F：reiterated答案：ADThe inconspicuous location and lack of striking vistas that characterize the villa reflect the_____ aspect of its creator’s personality.A：volatileB：grandioseC：gregariousD：self-effacingE：imperious答案：DThe chairman, faced with the need to forge a consensus on a number of proposals,acknowledged that it would be difficult to reconcile the push for a radical overhaul with the stance of those who want _____change.A：deliberateB：indiscriminateC：genuineD：immediateE：wholesale答案：ARecently released statistics on the prevalence of heart disease in the United States, while (i)_____ , nevertheless reflect a decline from heights reached in the 1960s, before health officials began publicly (ii)_____ people to guard against heart disease.Blank (I)Blank (II)A：definite D：entreatingB：sobering E：defyingC：implausible F：absolving答案：BDThe humor in this play derives from its (i)_____ . The new production, however, inexplicably goes in the opposite direction; it is so (ii)_____ that the audience does not even seem to realize that the play is supposed to be a comedy.Blank (I)Blank (II)A：verbal nimbleness D：accessibleB：political allusions E：ploddingC：deadpan dialogue F：implausible答案：AETo say the actors were (i)_____ their director is an understatement: a director who is visibly bored by his cast and their performances is hard to (ii)_____ .Blank (I)Blank (II)A：disappointed in D：lambastB：accepting of E：displeaseC：motivated by F：suffer答案：BFWhen pulsars were first discovered, some of the astrophysicists involved briefly (i)_____ these types of stars might be (ii)______extraterrestrial intelligence. Specifically, the pulsar’s periodic radiation pattern was initially interpreted as (iii)______ rather than—as it turned out to be—the natural product of the spinning of a neutron star.Blank (I)Blank (II)Blank (III)A：ignored whether D：suggestive of G：a deliberate hoaxB：thought that E：unrelated to H：an episodic phenomenonC：disavowed that F：evidence against I：a constructed code答案：BDIThere has been great enthusiasm in the United States for reducing fossil fuel dependence by increasing production of biofuels from crops such as corn and switchgrass, but this (i)_____ about biofuel’s potential should be (ii)_____ by a realistic appraisal of the costs and challenges of biofuel production.Blank (I)Blank (II)A：forbearance D：temperedB：exuberance E：delineatedC：obduracy F：exacerbated答案：BDUntil now,old snapshots and home movies faded and crumbled and were eventually (i)_____.Only a few precious mementos were preserved and passed along. But as photography moves into the digital realm, family albums and home videos seem capable of (ii)_____: our capacity to store them is, for all practical purposes, approaching the infinite. Is such a transformation a good thing? The natural world teaches us that (iii)_____ are vital to ecological health. Does a similar principle apply to communal memory?Blank (I)Blank (II)Blank (III)A：archived D：transmission G：death and decayB：discarded E：immortality H：predator and preyC：reproduced F：revolution I：reproduction and renewal答案：BEGThe threat of litigation makes the art authentication industry _____ realm: connoisseurs refuse to communicate in writing and confidential agreements bind authenticators to silence.A：an opportunisticB：a clandestineC：a moralizingD：a xenophobicE：an anarchistic答案：BRecent research runs counter to the long-cherished notion that a small drop in body temperature during and after surgery is either _____ or actually protects the patient by slowing metabolism and reducing the body’s demand for blood and oxygen.A：beneficialB：immaterialC：inconsequentialD：preventiveE：prophylacticF：redundant答案：BCThe automation of many of the functions performed at the factory, although initially inspiring_____ in many of the company’s employees, has had none of the deleterious effects forecast either within or beyond the organization.A：indifferenceB：optimismC：ambitionD：arroganceE：trepidation答案：EOne thing both authors have in common is a striking amount of _____: they claim to know how massive institutions, some of them richly endowed, all of them central to American society and culture, should be reshaped.A：hubrisB：proprietyC：biasD：prescienceE：indolence答案：AMaking the shift to the 90-nanometer manufacturing process has been (i) _____for semiconductor companies. This process effectively doubles the manufacturing capacity of the industry, but it (ii)_____ enormous technical challenges because some components of the new semiconductor chips are no more than five to seven molecules thick.Blank (I)Blank (II)A：an unanticipated boon D：circumventsB：a routine accomplishment E：entailsC：a significant struggle F：resolves答案：CEThere has been (i) _____elephant’s fabled mental capacities until recently, when these behavioral observations have begun to be (ii) _____by brain science. MRI scans of an elephant’s brain suggest that even relative to its overall size it has a large hippocampus, the component in the mammalian brain linked to memory and an important part of its limbic system, which is involved in processing emotions.Blank (I)Blank (II)A：surprising credence given to D：buttressedB：a widespread dismissal of E：anticipatedC：only anecdotal evidence for F：overwhelmed答案：CDThe description of Green’s scholarship as (i)_____ is grossly misleading: while her research on interstellar particles is not especially novel, the conclusions she draws from her data are (ii)_____ .Blank (I)Blank (II)A：esoteric D：remarkably pioneeringB：tendentious E：dubiously supportedC：derivative F：strangely comforting答案：CDThe brain has become, for many people, _____ the biologicalmachinations of the self, and the self-knowledge promised by neuroscience has ignited a hunger to understand how new findings weigh in on age-old questions.A：tantamount toB：synonymous withC：implicated inD：divorced fromE：detached fromF：subservient to答案：ABIf you follow your intuition, you will more often than not err by misclassifying a random event as (i)_____. We are far too willing to (ii)_____ the belief that much of what we see in life is random.Blank (I)Blank (II)A：uncharacteristic D：countenanceB：systematic E：rejectC：arbitrary F：champion答案：BEWithin the culture as a whole, the natural sciences have been so successful that the word “scientific”is often used in (i)_____ manner: it is often assumed that to call something “scientific”is to imply that its reliability has been (ii)_____ by methods whose results cannotreasonably be (iii)_____.Blank (I)Blank (II)Blank (III)A：an ironic D：maligned G：exaggeratedB：a literal E：challenged H：anticipatedC：an honorific F：established I：disputed答案：CFIThe researcher found that in assessing others, many people hold an unconscious view that competence and warmth are (i)_____: when they perceive a person to be highly capable, they infer that he or she must have a tendency to be (ii)_____.Blank (I)Blank (II)A：equally important D：ambitiousB：mutually reinforcing E：unfeelingC：inversely related F：disingenuous答案：CEThe era’s examples of _____ that are cited by the author can be balanced in part by certain examples of dissent during the same period.A：diversityB：authoritarianismC：forbearanceD：volatilityE：lucidity答案：BThe researcher noted that microbes, although _____, make up far more of the living protoplasm on Earth than all humans, animals and plants combined.A：invisibleB：omnipresentC：diminutiveD：ubiquitousE：minusculeF：ethereal答案：CEAerial viewings of the gigantic stone horse attributed to the Native American Quechuan people fail to _____ the considerable artistry required to create the piece: the horse appears crudely constructed unless carefully examined from the ground.A：revealB：justifyC：manifestD：mitigateE：diminishF：undercut答案：ACHarper Lee’s narration in To Kill a Mockingbird is _____, mixing an adult’s and a child’s perspective according to no logic other than the immediate exigencies of the plot.A：a paradigmB：a hodgepodgeC：a modelD：an innovationE：a patchworkF：an embarrassment答案：BEChanges made to ecosystem in order to achieve a goal, such as food production or flood control, often _____ significant unforeseen trade-offs between other important products and service the ecosystems provide.A：predictB：delayC：foretellD：obscureE：yieldF：engender答案：EFThe (i)_____ of molecular oxygen on Earth-sized planets aroundother stars in the universe would not be (ii)_____ sign of life: molecular oxygen can be a signature of photosynthesis (a biotic process) or merely of the rapid escape of water from the upper reaches of a planetary atmosphere (an abiotic process).Blank (I)Blank (II)A：dearth D：a controversialB：presumption E：an unambiguousC：detection F：a possible答案：CEThe painter’s problem, like that of an author whose early literary masterpiece exhausts the themes it embodies, is how to_____his first highly acclaimed efforts with works of comparable significance and presence.A：combineB：illuminateC：realizeD：amendE：follow答案：ESince many African farmers face a soil fertility problem, providing funding for fertilizer seems_____ ; closer examination of the data raises some troubling questions, however.A：imprudentB：expensiveC：unimpeachableD：modernE：worrisome答案：CIn years prior to the Civil War, Philadelphia’s African American press encouraged readers to be vaccinated against the smallpox.This journalistic campain was initially (i)_____,appealing to readers’sense of communal duty, and became even more(ii)_____once the war started, as smallpox outbreaks began to occur on Philadelphia’s city skirts.Blank (I)Blank (II)A：hortatory D：controversialB：ineffective E：urgentC：widespread F：inopportune答案：AEA hallmark of certain nineteenth-century mystery novels was the reform agenda of their authors, who ostensibly sought to expose economic injustice while depicting the seamy underside of urban life. In reality, however,these claims to a radical political agenda were often (i)_____, meant to give lurid thrillers the appearance of (ii)_____.Blank (I)Blank (II)A：authentic D：escapist appealB：complicated E：high-minded purposeC：disingenuous F：gripping suspense答案：CEIt remains a mystery how Theobroma cacao, which scholars believe originated in lowland Amazonia, was introduced to tropical Mesoarnerica. It must have been a (i)_____process: cacao trees do not easily sprout from seed,need years to mature, and grow only in humid lowland forest. Moreover, in a tropical climate, cacao pods spoil quickly,rendering their transport from lowland Amazonia to tropical Mesoamerica in a single journey by foot or canoe (ii)_____.Blank (I)Blank (II)A：clandestine D：unlikelyB：protracted E：redundantC：spontaneous F：expeditious答案：BDUnlike some mammals—cows and sheep, for instance—that are notably _____, lions have a wide range of facial expressions.A：tractableB：impassiveC：solitaryD：socialE：sluggish答案：BThe research on otters’environmental requirements is surprisingly (i)_____. One reason for this has to do with the estimation of how much they use different areas. Doing so may be (ii)_____ in some kinds of terrain, such as Shetland where the Eurasian otters are active in daytime and have clear individual markings. There it is possible to identify the individuals over stretches of coast of a few kilometers and to see what kinds of coast they use. However, the field conditions are (iii)_____.Blank (I)Blank (II)Blank (III)A：straightforward D：quite problematic G：routineB：controversial E：relatively simple H：deceptiveC：difficult F：largely unnecessary I：exceptional答案：CEIEarly practitioners of the natural sciences developed methods to remove distortions caused by either the research environment or the researcher. Such methods, especially with respect to the researcher, were considered to (i)_____ those (ii)_____ subjectivity whose unbridled expression was thought to (iii)_____ research.Blank (I)Blank (II)Blank (III)A：restrain D：incursions of G：corruptB：reveal E：restrictions on H：justificationC：disguise F：acknowledgements of I：expedite答案：ADGAlthough one can adduce myriad of examples of ecosystem disruption by nonindigenous species, nevertheless most introduced species that survive in fact appear to have quite _____ effects on the ecosystem they have invaded.A：minimalB：triflingC：markedD：conspicuousE：intriguingF：deleterious答案：ABAt Cerro Portezuelo, the task of separating grinding tools from the larger collection of excavated stone objects was (i)_____ the ancient practice of recycling grinding tool fragments for building materials, hammer stones, and other purposes that (ii)_____ their original use.Blank (I)Blank (II)A：complicated by D：complementedB：important to E：obscuredC：independent from F：underscored答案：AEThe researcher noted that microbes, although _____, make up far more of the living protoplasm on Earth than all humans, animals and plants combined.A：invisibleB：omnipresentC：diminutiveD：ubiquitousE：minusculeF：ethereal答案：CEAerial viewings of the gigantic stone horse attributed to the Native American Quechuan people fail to _____ the considerable artistry required to create the piece: the horse appears crudely constructed unless carefully examined from the ground.A：revealB：justifyC：manifestD：mitigateE：diminishF：undercut答案：ACHarper Lee’s narration in To Kill a Mockingbird is _____, mixing anadult’s and a child’s perspective according to no logic other than the immediate exigencies of the plot.A：a paradigmB：a hodgepodgeC：a modelD：an innovationE：a patchworkF：an embarrassment答案：BEChanges made to ecosystem in order to achieve a goal, such as food production or flood control, often _____ significant unforeseen trade-offs between other important products and service the ecosystems provide.A：predictB：delayC：foretellD：obscureE：yieldF：engender答案：EFThe (i)_____ of molecular oxygen on Earth-sized planets around other stars in the universe would not be (ii)_____ sign of life: molecularoxygen can be a signature of photosynthesis (a biotic process) or merely of the rapid escape of water from the upper reaches of a planetary atmosphere (an abiotic process).Blank (I)Blank (II)A：dearth D：a controversialB：presumption E：an unambiguousC：detection F：a possible答案：CEOne of the peculiarities of humans is that we irrationally gravitate to the predictable and avoid risk, whatever the reasons for this _____, it is hardly a sound basis for dealing with complex, long-term problems A：eccentricityB：predilectionC：vacillationD：proclivityE：waveringF：cowardice答案：BDIn protoscientific (for example, in ancient Greece), claims about the physical world were often accepted as true if they were reasonable; experimental verification, if thought necessary at all, was _____.A：utilitarianB：perfunctoryC：egregiousD：empiricalE：inductive答案：BAlthough scientific progress leads to constant revision of ideas, one observation that has remained _____ over the years is that there are a lot of insects in the world: some 950,000 species have been identified.A：robustB：significantC：strongD：perplexingE：confoundingF：obscure答案：ACIn the discussions on international patent law, many (i)_____ issues will probably be pushed far into the future. This cautious approach makes diplomatic sense, since attempts over the past 15 years to reach international agreements on patents have (ii)_____ just such sensitive issuesBlank (I)Blank (II)A：contentious D：approachedB：stimulating E：articulatedC：subjective F：foundered on答案：AFIn the northeastern United States, beaver populations had been critically reduced or even _____ in large areas at the end of nineteenth century; as a result, several states instituted prohibitions on beaver trapping.A：diminishedB：extirpatedC：eliminatedD：devaluedE：weakenedF：underrated答案：BCViewing people as ‘’social atoms’’that obey rather simple rule (which are not unlike the laws of physics), one may discover certain (i)_____. Take, for example, the way channels emerge when people move in crowds. In the midst of initially (ii)_____ movements, one person begins to follow another—in an effort to avoid collisions—and streams of movement emerge. As more people join in, there is greater pull on others to join the flow, and the particular channels become (iii)_____.Blank (I)Blank (II)Blank (III)A：apparent contradictions D：inflexible G：self-defeatingB：unproductive tendencies E：straightforward H：self-aggrandizing C：lawlike regularities F：chaotic I：self-perpetuating答案：CFIGiven children’s active fantasy lives, one might think of truthfulness as (i)_____ virtue in young children, but it turns out that lying is the more (ii)_____ skill. A child who is going to lie must recognize the truth, intellectually conceive of an alternate reality, and be able to convincingly sell that new reality to someone else. Therefore, lying (iii)_____ cognitive development and social skills in a way that honesty simply does not.Blank (I)Blank (II)Blank (III)A：an instinctive D：advanced G：underminesB：an acquired E：practical H：forgoesC：a conscious F：mundane I：demands答案：BDIAfter rising continuously over the summer, commodity prices fell, leaving analyst wondering whether the downward trend is a turning point or merely a _____ before demand picks up in the winter months.A：spikeB：upsurgeC：harbingerD：portentE：lullF：respite答案：EFThe juxtaposition of fertile alluvial soils originating in the Andes with the infertile inland soils of central Amazon Basin is one example of _____ of the soil conditions that can be found in the tropics.A：intractabilityB：heterogeneityC：incompatibilityD：disparatenessE：instabilityF：fragility答案：BDHis new role gives the normally clownish actor achance to impress audiences with his (i)_____. He is among the most uninhibited comic performers around, but here he buttons his lip and stares straight ahead.Perhaps without quite knowing it, the audience waits for a wink, a hint that some of the (ii)_____ spirit that animated his previous movies might be lurking inside the (iii)_____ manner he presents in this film.Blank (I)Blank (II)Blank (III)A：raillery D：madcap G：facetiousB：effusiveness E：lugubrious H：reserved C：forbearance F：edifying I：beguiling答案：CDH。
GRE考试数学历年真题全景解析2024GRE考试是许多申请研究生学位的学生必须参加的考试之一。
数学部分是其中一个重要的组成部分。
为了帮助考生更好地掌握GRE数学考试的内容和解题技巧，本文将对2024年的GRE数学历年真题进行全景解析。
一、整数与基本运算整数是数学的基础，GRE数学考试经常涉及整数的概念和运算。
在2024年的数学部分真题中，有一道题目如下：1. 若a和b都是整数，且a>b>0，则a^2 - b^2等于多少？解析：首先，我们可以利用差平方公式将a^2 - b^2进行分解。
根据差平方公式，我们有a^2 - b^2 = (a+b)(a-b)。
根据题目中的条件a>b>0，我们可以确定a+b>a-b>0。
因此，答案是a^2 - b^2 = (a+b)(a-b)。
二、代数与方程代数和方程是GRE数学考试中的另一个重要主题。
在2024年的数学部分真题中，有一道关于代数与方程的题目如下：2. 对于方程2x + 3y = 12，下列哪个点是其解？(A) (1, 5)(B) (3, 4)(C) (6, 0)(D) (-2, 6)(E) (0, 4)解析：我们可以将选项依次代入方程2x + 3y = 12中，看哪个选项满足等式。
对于选项(A)，代入x=1，y=5后，我们得到2(1) + 3(5) = 2+ 15 = 17，不满足等式。
同样地，对于选项(B)、(C)、(D)、(E)，都不能满足等式。
所以，这个方程没有整数解。
三、概率与统计概率与统计是GRE数学考试中的另一个考点。
在2024年的数学部分真题中，有一道关于概率与统计的题目如下：3. 一组学生的平均年龄是20岁，如果其中5名学生的年龄为22岁，另外3名学生的年龄为18岁，其他学生的年龄保持不变，则平均年龄变为多少岁？解析：首先，我们可以计算这组学生的总年龄。
根据题目，我们可以知道总年龄为20 * (5+3+n)，其中n代表其他学生的数量。
GRE（QUANTITATIVE）综合模拟试卷20(题后含答案及解析) 题型有：1. 计算题1．When the positive integer n is divided by 3, the remainder is 2 and when n is divided by 5, the remainder is 1. What is the least possible value of n?正确答案：112．Ellen has received the following scores on 3 exams: 82, 74, and 90. What score will Ellen need to receive on the next exam so that the average(arithmetic mean)score for the 4 exams will be 85?正确答案：94解析：Let x represent the score on Ellens next exam. This initial step of assigning a variable to the quantity that is sought is an important beginning to solving the problem. Then in terms of x, the average of the 4 exams iswhich is supposed to equal 85. Now simplify the expression and set it equal to 85:Solving the resulting linear equation for x, you get246 + x = 340 x = 94Therefore, Ellen will need to attain a score of 94 on the next exam.3．A mixture of 12 ounces of vinegar and oil is 40 percent vinegar, where all of the measurements are by weight. How many ounces of oil must be added to the mixture to produce a new mixture that is only 25 percent vinegar?正确答案：7.2解析：Let x represent the number of ounces of oil to be added. Then the total number of ounces of the new mixture will be 12 + x, and the total number of ounces of vinegar in the new mixture will be(0.40)(12). Since the new mixture must be 25 percent vinegar,Therefore,(0.40)(12)=(12+ x)(0.25)4.8 = 3 + 0.25x 1.8 = 0.25x7.2 = xThus, 7.2 ounces of oil must be added to produce a new mixture that is 25 percent vinegar.4．In a driving competition, Jeff and Dennis drove the same course at average speeds of 51 miles per hour and 54 miles per hour, respectively. If it took Jeff 40 minutes to drive the course, how long did it take Dennis?正确答案：37.8 minutes解析：Let x be the time, in minutes, that it took Dennis to drive the course. The distance d, in miles, is equal to the product of the rate r, in miles per hour, and the time t, in hours; that is,d=rtNote that since the rates are given in miles per hour, it is necessary to express the times in hours; for example, 40 minutes equalsof an hour.Thus, the distance traveled by Jeff is the product of his speed and his time,(51)()miles, and the distance traveled by Dennis is similarly represented by(54)()miles. Since the distances are equal, Thus, it took Dennis approximately 37.8 minutes to drive the course.5．Working alone at its constant rate, machine A takes 3 hours to produce a batch of identical computer parts. Working alone at its constant rate, machine B takes 2 hours to produce an identical batch of parts. How long will it take the two machines, working simultaneously at their respective constant rates, to produce an identical batch of parts?正确答案：6/5 hours, or 1 hour 12 minutes解析：Since machine A takes 3 hours to produce a batch, machine A can produce 1/3 of the batch in 1 hour. Similarly, machine B can produce - of the batch in 1 hour. If we let x represent the number of hours it takes both machines, working simultaneously, to produce the batch, then the two machines will produce - of the job in 1 hour. When the two machines work together, adding their individual production rates, 1/3 and 1/2, gives their combined production rate 1/x. Therefore,Thus, working together, the machines will take 6/5 hours, or 1 hour 12 minutes, to produce a batch of parts.6．At a fruit stand, apples can be purchased for $0.15 each and pears for $0.20 each. At these rates, a bag of apples and pears was purchased for $3.80. If the bag contained 21 pieces of fruit, how many of the pieces were pears?正确答案：13解析：If a represents the number of apples purchased and p represents the number of pears purchased, the information can be translated into the following system of equations.0.15a+ 0.20p = 3.80(total cost)a+p = 21(total number of fruit)From the second equation, a = 21-p. Substituting 21-p into the first equation fora gives0.15(21-p)+ 0.20p = 3.80(0.15)(21)- 0.15p + 0.20p = 3.803.15-0.15p + 0.20p =3.800.05p = 0.65p=13Thus, of the 21 pieces of fruit, 13 were pears.7．To produce a particular radio model, it costs a manufacturer $30 per radio, and it is assumed that if 500 radios are produced, all of them will be sold. What must be the selling price per radio to ensure that the profit(revenue from the sales minus the total production cost)on the 500 radios is greater than $8,200?正确答案：$46.40解析：If y represents the selling price per radio, then the profit is 500(y - 30). Therefore, we set500(y-30)> 8,200Solving the inequality, we get500y- 15,000 > 8,200 500y> 23,200y > 46.4Thus, the selling price must be greater than $46.40 to ensure that the profit is greater than $8,200.Some applications involve computing interest earned on an investment during a specified time period. The interest can be computedas simple interest or compound interest.Simple interest is based only on the initial deposit, which serves as the amount on which interest is computed, called the principal, for the entire time period. If the amount P is invested at a simple annual interest rate of r percent, then the value V of the investment at the end of t years is given by the formulaV = P(1+)where P and V are in dollars.In the case of compound interest, interest is added to the principal at regular time intervals, such as annually, quarterly, and monthly. Each time interest is added to the principal, the interest is said to be compounded. After each compounding, interest is earned on the new principal, which is the sum of the preceding principal and the interest just added. If the amount P is invested at an annual interest rate of r percent, compounded annually, then the value V of the investment at the end of t years is given by the formulaV =If the amount P is invested at an annual interest rate of r percent, compounded n times per year, then the value V of the investment at the end of t years is given by the formulaV =8．If $10,000 is invested at a simple annual interest rate of 6 percent, what is the value of the investment after half a year?正确答案：$10,300解析：According to the formula for simple interest, the value of the investment after 1/2 year is=$10,000(1.03)= $10,3009．If an amount P is to be invested at an annual interest rate of 3.5 percent, compounded annually, what should be the value of P so that the value of the investment is $1,000 at the end of 3 years?正确答案：$901.94解析：According to the formula for 3.5 percent annual interest, compounded annually, the value of the investment after 3 years isP(1 + 0.035)3and we set it to be equal to $1,000P(l + 0.035)3 = $1,000To find the value of P, we divide both sides of the equation by(1 + 0.035)3.P =≈$901.94 Thus, an amount of approximately $901.94 should be invested.10．A college student expects to earn at least $1,000 in interest on an initial investment of $20,000. If the money is invested for one year at interest compounded quarterly, what is the least annual interest rate that would achieve the goal?正确答案：4.91 percent解析：According to the formula for r percent annual interest, compounded quarterly, the value of the investment after 1 year isBy setting this value greater than or equal to $21,000 and solving for r, we getWe can use the fact that taking the positive fourth root of each side of an inequality preserves the direction of the inequality. This is also true for the positive square root or any other positive root.To compute the fourth root, we can use the fact thatfor x ≥0; that is, we can compute afourth root by taking a square root twice:So the least annual interest rate is approximately 4.91 percent.11．Find an algebraic expression to represent each of the following.(a)The square of y is subtracted from 5, and the result is multiplied by 37.(b)Three times x is squared, and the result is divided by 7.(c)The product of(x + 4)and y is added to 18.正确答案：(a)37(5-y2), or 185-37y2(b)(c)18 +(x + 4)(y), or 18 + xy + 4y12．Simplify each of the following algebraic expressions.(a)3x2-6 + x + 11-x2 +5x(b)3(5x-1)-x + 4(c), where x≠4(d)(2x + 5)(3x-l)正确答案：(a)2x2 + 6x + 5(b)14x+l(c)x + 4(d)6x2+13x-513．(a)What is the value of f(x)= 3x2 -7x + 23 when x = - 2 ?(b)What is the value of h(x)= x3-2x2 + x-2 when x = 2?(c)What is the value of k(x)=x-7 when x = 0?正确答案：(a)49(b)0(c)-714．If the function g is defined for all nonzero numbers y by g(y)=, find the value of each of the following. (a)g(2)(b)g(-2)(c)g(2)- g(-2)正确答案：(a)1(b)-1(c)215．Use the rules of exponents to simplify the following.(a)(n5)(n-3)(b)(s7)(t7)(e)(w5)-3(f)(50)(d3)正确答案：(a)n2(b)(sf)7(c)r8(d)(e)(f)d3(g)(h)9x2y316．Solve each of the following equations for x.(a)5x-7 = 28(b)12-5x = x + 30(c)5(x + 2)= l-3x(d)(x + 6)(2x-l)= 0(e)x2+5x-14 = 0(f)x2-x-l = 0正确答案：(a)7(b)-3(c)(d)-6, 1/2(e)-7, 2(f)17．Solve each of the following systems of equations for x and y.(a)x + y = 24x-y = 18(b)3x-y = -5 x + 2y = 3(c)15x-18-2y = -3x + 7 10x + 7y + 20 = 4x + 2 正确答案：(a)x=21y=3(b)x=-1 y=2(c)x=1/2 y=-318．Solve each of the following inequalities for x.(a)-3x>7 + x(b)25x + 16≥10-x(c)16 + x>8x-12正确答案：(c)x<419．For a given two-digit positive integer, the tens digit is 5 more than the units digit. The sum of the digits is 11. Find the integer.正确答案：8320．If the ratio of 2x to 5y is 3 to 4, what is the ratio of x to y?正确答案：15 to 821．Kathleens weekly salary was increased by 8 percent to $237.60. What was her weekly salary before the increase?正确答案：22022．A theater sells children’s tickets for half the adult ticket price. If 5 adult tickets and 8 children’s tickets cost a total of $27, what is the cost of an adult ticket?正确答案：323．Pat invested a total of $3,000. Part of the money was invested in a money market account that paid 10 percent simple annual interest, and the remainder of the money was invested in a fund that paid 8 percent simple annual interest. If the interest earned at the end of the first year from these investments was $256, how much did Pat invest at 10 percent and how much at 8 percent?正确答案：$800 at 10% and $2,200 at 8%24．Two cars started from the same point and traveled on a straight course in opposite directions for exactly 2 hours, at which time they were 208 miles apart. If one car traveled, on average, 8 miles per hour faster than the other car, what was the average speed of each car for the 2-hour trip?正确答案：48 mph and 56 mph25．A group can charter a particular aircraft at a fixed total cost. If 36 people charter the aircraft rather than 40 people, then the cost per person is greater by $12.(a)What is the fixed total cost to charter the aircraft?(b)What is the cost per person if 40 people charter the aircraft?正确答案：(a)$4,320(b)$108。
gre数学考试关于⼏何的重点试题gre数学考试关于⼏何的重点试题1. If the areas of three of the faces of a rectangular solid are 6,10 and 15, what is the volume of the solid?A.30B.90C.150D.300E.450答案：A2.△RST lies in the XY-plane and points R and T have (x, y) coordinates (0,0) and (6,0), respectively. The area of △RST is 12.The x-coordinate of RThe y-coordinate of S答案：D3.What is the maximum number of nonover-lapping regions into which 3 lines can divide the interior of a circle?A.4B.6C.7D.8E.9答案:C4.The rectangular rug shown in the figure above has a floral border 1 foo t wide on all sides. What is the area, in square feet, of that portion of the rug that excludes the border?A.28B.40C.45D.48E.53答案：A5. The diagram represents a rectangular garden. The shaded regions are planted in flowers, and the unshaded region is a walk 2 feet wide. All angles are right angles.The sum of the areas of the shaded regions2,800 square feet答案：D6.What are the coordinates of point B in the xy-plane above?A.(6,12)B.(6,28)C.(8,20)D.(12,20)E.(14,28)答案：B7.If 9 trees are originally planted in a circular pattern as shown above, what is the least number of trees that must be transplanted so that the 9 trees will be in 2 straight rows?A.4B.5C.6D.7E.8答案：B8.If the sum of the measures of two angles is 180o,each angle is a supplement of the other, whereas, if the sum oftheir measures is 90o, each is a complement of the other.The measure of an angle with a supplement that measures 130o The measure of an angle with a supplement that measures 40o 答案：C9.If a solid pyramid has 4 vertices and 4 faces, how many edges does the pyramid have?A.2B.3C.4D.6E.8答案：D10.A rectangular floor with an area of 12 square meters is drawn to scale with 2 centimeters representing 1 meter.The area of the scale drawing of the floor24 square centimeters答案：A11.In the rectangular coordinate system, line k passes through the point (0,0) and (4,8); line m passes through the points (0,1) and (4,9).The slope of line kThe slope of line m答案：C12.In the stop sign shown in the figure above, all sides have equal length and all angles have equal measure. If the figure could be rotated 225 degrees in a clockwise direction, point G would be in the position of point.A.IB.JC.KD.LE.M答案：B13.What is the ratio of the perimeter of a pentagon with each side of length 6 to the perimeter of an octagon with each side of length 6?A.5/6B.4/5C.3/4D.2/3E.5/8答案：E14.A circular flower bed is inscribed in a flat, square garden plot that measures x meters on each side.The fraction of the garden plot area that is not part of the flower bed if x=50 metersThe fraction of the garden plot area that is not part of the flower bed if x=45 meters答案：C15.The dimensions, in centimeters, of rectangular box R are 6 by 8 by 10. Which of the following cannot be the total surface area, in square centimeters, of two faces of R?A.96B.120C.128D.160E.180答案：E16.The figure above shows a large square formed by fitting three L- shaped tiles and one small square tile together. If a rectangular floor 10 feet by 12 feet is to be tiled in large squares of this design, how many L-shaped tiles will be needed?A.810B.405C.270D.135E.45答案：A17.A certain rectangle has perimeter 54. If the ratio of the length of the rectangle to the width is 5 to 4, what is the length of the rectangle?A.30B.27C.24D.18E.15答案：E18.In the rectangular coordinate system above, the coordinates of points A and B are shown. If the dotted lines representa second pair of coordinate axes with origin at A, and if the scale is the same on both pairs of axes, what are the coordinat es of point B with respect to the second pair of axes?A.(-33,88)B.(33,48)C.(33,-48)D.(97,88)19. If the perimeter of the rectangle above is 36, then l=A.9B.14C.16D.28E.32答案：B20.If the perimeter of a triangle is 18, then the length of one of the sides cannot beA.1B.3C.6D.8E.9答案：E21.In the figure above, if ABCD is a parallelogram with A at the origin, what are the coordinates of C?A.(6,5)B.(6,6)C.(7,4)D.(8,5)E.(8,6)答案：A22.What fraction of a cubic meter is the volume of a cubic block of stone with edge of length 10 centimeters?(1 meter=100centimeters)A.3/10D.1/100E.1/1000答案：E23.In the rectangular coordinate system above, for how many of the points that lie inside or on the boundary of the shaded region are both coordinates integers?A.sixB.sevenC.eightD.nineE.ten答案：E24. A rectangular tabletop consists of a piece of laminated wood bordered by a thin metal strip along its four edges. The surface area of the tabletop if x square feet, and the total length of the strip before it was attached was x feet. If the tabletop is3 feet wide, what is its approximate length, in feet?A.12B.10C.9D.8E.6答案：E25.In the coordinate plane, points A and B are endpoints of a diameter of a circle with center (-2,1). If the coordinates of A are (-5,5), what are the coordinates of B?A.(0,0)B.(3,-2)C.(3,-1)D.(2,-2)E.(1,-3)答案：E26.Three solid cubes of lead, each with edges 10 centimeters long, are melted together in a level, r ectangular shaped pan. The base of the pan has inside dimensions of 20 centimeters by 30 centimeters, and the pan is 15 centimeters deep. If the volume of the solid lead is approximately the same as the volume of the melted lead, approximately how many cen timeters deep is the melted lead in the pan?A.2.5B.3C.5D.7.5E.9答案：C27.In the figure above, if x, y, and z are integers such that xA.59 and 91B.59 and 135C.91 and 178D.120 and 135E.120 and 178答案：C28.In the rectangular coordinate system above, if P, not shown, is a point AB and if the x-coordinate of P is 1, what is the y-coordinate of P?A.4/3B.3/2C.7/3D.5/2E.8/3答案：A29.There are six marked points on the circle above. How many different lines can be drawn that contain two of the marked points?A.5B.6C.12D.15E.30答案：D30.In the figure above, ABCD is a square. What are the coordinates of point B?A.(-4,2)B.(-2,4)C.(-2,6)D.(4,-6)E.(6,-2) 答案：C。
gre考试题库及答案GRE考试题库及答案1. 填空题：在下列句子中填入合适的词汇。
- The professor's lecture was so _______ that it left the audience in a state of awe.- The _______ of the new policy was met with mixedreactions from the public.- Despite the _______ of the evidence, the jury remained unconvinced.答案：1. 1) captivating 2) implementation 3) overwhelming2. 阅读理解：阅读以下段落，并回答问题。
"In the realm of environmental science, the concept of sustainability has become increasingly important. It refersto the ability to meet the needs of the present without compromising the ability of future generations to meet their own needs. This involves a delicate balance between economic growth, social development, and environmental protection."问题：根据上文，可持续发展的定义是什么？答案：可持续发展是指在不损害后代满足自身需求的能力的前提下，满足当前需求的能力。
这涉及到经济增长、社会发展和环境保护之间的微妙平衡。
3. 数学问题：解决以下方程。
2024 GRE考试必备数学历年真题练习在GRE数学部分的备考过程中，历年真题的练习是非常重要的一环。
通过针对性的练习，考生可以熟悉考试题型，了解考点，提升解题速度和准确性。
本文将为大家提供2024年GRE考试的数学历年真题练习，帮助考生更好地备考。
1. 整数1.1 题目选择下列哪个数是正偶数？(A) -12(B) -5(C) 0(D) 9(E) 271.2 解析正偶数是指能够被2整除的正整数。
从选项中排除负数，在0和正数中，只有0能够被2整除，因此答案选(C)。
2. 几何2.1 题目下图中，正方形ABCD的边长为3。
点E是线段BC的中点，点F是线段BD上的一点，且AF的长度为3。
求射线AF与线段CE的交点P到点E的距离。
[图片描述：一个正方形ABCD，边长为3，线段BC的中点为E，线段BD上的一点为F，AF的长度为3]2.2 解析首先，可以得出正方形ABCD的对角线AC的长度为3的开平方乘以2，即AC=3乘以根号2。
由于AE与CF平行且等长，射线AF可以看作与线段BE平行且等长。
因此，三角形BEP是等腰直角三角形，所以BP = EP = EC的一半。
又因为BC=3，所以EC=3/2。
因此，点P到点E的距离为1.5个单位。
3. 概率与统计3.1 题目某次测试的成绩服从正态分布，平均成绩为80分，标准差为5分。
已知一个学生的成绩在85分以上的概率为0.841，求这个学生的成绩。
3.2 解析根据正态分布的性质，均值加上标准差得到的分数对应的概率是大约0.841。
因此，这个学生的成绩应该在平均成绩80分加上标准差5分的位置，即85分。
通过以上三个部分的例题，希望能够帮助到考生更好地了解2024年GRE考试数学部分的题型和解题思路。
在备考过程中，考生还需深入学习数学知识，掌握解题技巧，并进行大量真题练习，提升解题能力。
祝愿各位考生在考试中取得好成绩！。
2020年GRE模拟试题及答案（卷七）Retrofitted with stabilizing devices, some of which _____ its aesthetics, the bridge has been reopened, no longer prone to excessive swaying but not quitethe breathtaking structure it originally was.A：impairB：resistC：improveD：enhanceE：restoreF：compromise答案：AFAlthough cosmic objects have struck Earth since the planet’s very formation, humanity has only recently become aware of these events: two centuries ago the idea that objects orbiting the Sun could collide with Earth was widely_____ .A：ridiculedB：doubtedC：disseminatedD：promulgatedE：marginalizedF：disbelieved答案：BFThat people _____ the musical features of birdsongs suggests that despite the vast evolutionary gulf between birds and mammals, songbirds and humans share some common auditory perceptual abilities.A：mimicB：recognizeC：relishD：are confounded byE：can make outF：are puzzled by答案：BETorpey’s study has turned a seemingly _____ topic, the passport, into a fascinating one by making an original contribution to the sociology of the state.A：ironicB：banalC：provocativeD：wittyE：insipidF：stimulating答案：BEBritain is attractive to worldwide advertisers because it is _____market, so there is no need to tailor advertisements for different parts ofthe country.A：a globalB：an uncomplicatedC：a vastD：a homogeneousE：a uniformF：an immense答案：DEInstant celebrity is often (i) _____ asset because if there is no (ii) _____ to interest the public—no stage or screen triumphs,no interesting books, no heroic exploits—people quickly become bored.Blank (I)Blank (II)A：a fleeting D：competing attractionB：an incomparable E：continuity of exposureC：an untapped F：real achievement答案：AFAt their best, (i) _____book reviews are written in defense of value and in the tacit hope that the author, having had his or her (ii) _____ pointed out, might secretly agree that the book could be improved.Blank (I)Blank (II)A：abstruse D：strengthsB：adverse E：transgressionsC：hortatory F：assumptions答案：BEThe gaps in existing accounts of the playwright’s life are not (i) _____ , since much of the documentary evidence on which historians have relied is (ii) _____Blank (I)Blank (II)A：trifling D：credibleB：obvious E：extantC：implicit F：incomplete答案：AFThat today’s students of American culture tend to (i) _____ classical music is understandable. In our own time, America’s musical high culture has degenerated into a formulaic entertainment divorced from the contemporary moment. Thus, to miss out on what our orchestras are up to is not to (ii) _____ much. In the late Gilded Age, however, music was widely esteemed as the “queen of the arts.”Classical music was in its American heyday, (iii) _____ the culture at large.Blank (I)Blank (II)Blank (III)A：promote D：sacrifice G：antagonistic towardB：reinterpret E：appreciate H：generally rejected byC：ignore F：malign I：centrally embedded in答案：CDIThe serious study of popular culture by intellectuals is regularly credited with having rendered obsolete aonce-dominant view that popular culture is inherently inferior to high art. Yet this alteration of attitudes may be somewhat (i) _____ . Although it is now academically respectable to analyze popular culture, the fact that many intellectuals feel compelled to rationalize their own (ii) _____ action movies or mass-market fiction reveals,perhaps unwittingly, their continued (iii) _____ the old hierarchy of high and low culture.Blank (I)Blank (II)Blank (III)A：counterproductive D：penchant for G：aversion toB：underappreciated E：distaste for H：investment inC：overstated F：indifference to I：misunderstanding of答案：CDHStem borers are insect pests that often ruin North American corn crops. On some other continents, crop damage by stem borers is controlled by a certain species of wasp. Since these wasps eat nothing but stem borers, importing them into North America will keep crop damage from stem borers under control without endangering other North American insect species.Which of the following is an assumption on which the argument depends?(A) Corn is the principal food of stem borers that live on continents other than North America.(B) The wasps are capable of surviving in North America long enough to eat significant numbers of stem borers.(C) No wasp in North America is closely related to the species of wasp that eats stem borers.(D) On continent other than North America, the wasps control stem borers more effectively than does any other pest control measure.(E) Corn crops on continents other than North America are not damaged by any insect pests other than stem borers.答案：BFrom a newspaper editorial:Many people who are addicted to heroin will eventually attempt to overcome their addiction, principally for two reasons:the expense of maintaining a heroin addiction and the fear of arrest. If heroin were legalized and made available cheaply, as some people advocate, neither of these reasons would apply.The considerations above can best serve as part of an argument that(A) legalizing the sale of heroin would cause the price of this drug to go down(B) making it easier for heroin addicts to obtain treatment for their addiction would encourage many heroin addicts to attempt to overcometheir addiction(C) legalizing the sale of heroin would increase the number of crimes committed by heroin addicts to support their addiction(D) making heroin available legally and cheaply would make it less likely that heroin addicts will attempt to overcome their addiction(E) decreasing the severity of penalties for individuals who use heroin would not increase the number of new heroin addicts 答案：DSome recent historians have argued that life in the British colonies in America from approximately 1763 to 1789 was marked by internal conflicts among colonists. Inheritors of some of the viewpoints of early twentieth-century Progressive historians such as Beard and Becker, these recent historians have put forward arguments that deserve evaluation.The kind of conflict most emphasized by these historians is class conflict. Yet with the Revolutionary War dominating these years, how does one distinguish class conflict within that larger conflict? Certainly not by the side a person supported. Although many of these historians have accepted the earlier assumption that Loyalists represented an upper class, new evidence indicates that Loyalists, like rebels, were drawn from all socioeconomic classes. (It is nonetheless probably true that a larger percentage of the well-to-do joined the Loyalists than joined the rebels.) looking at the rebel side, we find little evidence for thecontention that lower-class rebels were in conflict with upper-class rebels. Indeed, the war effort against Britain tended to suppress class conflicts. Where it did not, the disputing rebels of one or another class usually became Loyalists. Loyalism thus operated as a safety valve to remove socioeconomic discontent that existed among the rebels. Disputes occurred, of course, among those who remained on the rebel side, but the extraordinary social mobility of eighteenth-century American society (with the obvious exception of slaves) usually prevented such disputes from hardening along class lines. Social structure was in fact so fluid-though recent statistics suggest a narrowing of economic opportunity as the latter half of the century progressed-that to talk about social classes at all requires the use of loose economic categories such as rich, poor, and middle class, or eighteenth-century designations like "the better sort." Despite these vague categories, one should not claim unequivocally that hostility between recognizable classes cannot be legitimately observed. Outside of New York, however, there were very few instances of openly expressed class antagonism.Having said this, however, one must add that there is much evidence to support the further claim of recent historians that sectional conflicts were common between 1763 and 1789. The "Paxton Boys" incident and the Regulator movement are representative examples of the widespread, and justified, discontent of western settlers againstcolonial or state governments dominated by eastern interests. Although undertones of class conflict existed beneath such hostility, the opposition was primarily geographical. Sectional conflict-which also existed between North and South-deserves further investigation.In summary, historians must be careful about the kind of conflict they emphasize in eighteenth-century America. Yet those who stress the achievement of a general consensus among the colonists cannot fully understand that consensus without understanding the conflicts that had to be overcome or repressed in order to reach it.The author suggests which of the following about the representativeness of colonial or state governments in America from 1763 to 1789?A.The governments inadequately represented the interests of people in western regions.B.The governments more often represented class interests than sectional interests.C.The governments were less representative than they had been before 1763D.The governments were dominated by the interests of people of an upper socioeconomic class.E.The governments of the northern colonies were less representative than were the governments of the southern colonies.According to the passage, which of the following is a true statement about sectional conflicts in America between 1763 and 1789?A.These conflicts were instigated by eastern interests against western settlers.B.These conflicts were the most serious kind of conflict in America.C.The conflicts eventually led to openly expressed class antagonism.D.These conflicts contained an element of class hostility.E.These conflicts were motivated by class conflicts.It can be inferred from the passage that the author would be most likely to agree with which of the following statements regarding socioeconomic class and support for the rebel and Loyalist causes during the American Revolutionary War?A.Identifying a person's socioeconomic class is the least accurate method of ascertaining which side that person supported.B.Identifying a person as a member of the rebel or of the Loyalist side does not necessarily reveal that person's particular socioeconomic class.C.Both the rebel and the Loyalist sides contained members of all socioeconomic classes, although there were fewer disputes among socioeconomic classes on the Loyalist side.D.Both the rebel and the Loyalist sides contained members of all socioeconomic classes, although the Loyalist side was made up primarilyof members of the upper classes.E.Both the rebel and the Loyalist sides contained members of all socioeconomic classes, although many upper-class rebels eventually joined the Loyalists.The passage suggests that the author would be likely to agree with which of the following statements about the social structure of eighteenth-century American society?I. It allowed greater economic opportunity than it did social mobility.II. It permitted greater economic opportunity prior to 1750 than after 1750.III. It did not contain rigidly defined socioeconomic divisions.IV. It prevented economic disputes from arising among members of the society.A.I and IV onlyB.II and III onlyC.III and IV onlyD.I, II, and III onlyE.I, II, III, and IVThe author most likely refers to “historians such as Beard and Becker”(lines 5-6) in order toA.isolate the two historians whose work is most representative ofthe viewpoints of Progressive historiansB.emphasize the need to find connections between recent historical writing and the work of earlier historiansC.make a case for the importance of the views of the Progressive historians concerning eighteenth-century American lifeD.suggest that Progressive historians were the first to discover the particular internal conflicts in eighteenth-century American life mentioned in the passageE.point out historians whose views of history anticipated some of the views of the recent historians mentioned in the passageAccording to the passage, Loyalism during the American Revolutionary War served the function ofA.eliminating the disputes that existed among those colonists who supported the rebel causeB.drawing upper, as opposed to lower, socioeconomic classes away from the rebel causeC.tolerating the kinds of socioeconomic discontent that were not allowed to exist on the rebel sideD.channeling conflict that existed within a socioeconomic class into the war effort against the rebel causeE.absorbing members of socioeconomic groups on the rebel side who felt themselves in contention with members of other socioeconomicgroupsThe author considers the contentions made by the recent historians discussed in the passage to beA.potentially verifiableB.partially justifiedC.logically contradictoryD.ingenious but flawedE.capricious and unsupported正确答案:A D B B E E BSince 1953, many experimental attem-pts to synthesize the chemical consti-tuents of life under "primitive Earth condi-tions" have been performed, but none of the these experiments has produced anything approaching complexity of the simplest organism. They have demonstrated, however, that a variety of the complex molecules currently making up living organisms could have been present in the early ocean and atmosphere, with only one limitation: such molecules are synthesized far less readily when oxygen-containing compounds dominate the atmosphere. Therefore some scientists postulate that the Earth's earliest atmosphere, unlike that of today, was dominated by hydrogen, methane, and ammonia.From these studies, scientists have concluded that the surface of the primitive Earth was covered with oceans containing the moleculesfundamental to life. Although, at present, scientists cannot explain how these relatively small molecules combined to produce larger, more complex molecules, some scientists have precipitously ventured hypotheses that attempt to explain the development, from lager molecules, of the earliest self-duplicating organisms.The primary purpose of the passage is toA.point out that theories about how life developed on Earth have changed little since 1953B.warn of increasing levels of hydrogen, methane, and ammonia in the Earth's atmosphereC.describe the development since 1953 of some scientists' understanding of how life began on EarthD.demonstrate that the synthesis of life in the laboratory is too difficult for modern technologyE.describe how primitive atmospheric conditions produced the complex molecules of living organismsIt can be inferred from the passage that “some scientists”assume which of the following concerning “larger, more complex molecules”(line 20)?A.The earliest atmosphere was formed primarily of these molecules.B.Chemical processes involving these molecules proceeded much more slowly under primitive Earth conditions.C.The presence of these molecules would necessarily precede the existence of simple organisms.D.Experimental techniques will never be sufficiently sophisticated to produce in the laboratory simple organisms from these chemical constituents.E.Explanations could easily be developed to explain how simple molecules combined to form these more complex ones.The author's reaction to the attempts that have been made to explain the development of the first self-duplication organisms can best be described as one ofA.enthusiasmB.expectationC.dismayD.skepticismE.antipathyAccording to the passage, which of the following can be inferred about the process by which the chemical constituents of life were synthesized under primitive Earth conditions?A.The synthesis is unlikely to occur under current atmospheric conditions.B.The synthesis is common in modern laboratories.C.The synthesis occurs more readily in the atmosphere than in theocean.D.The synthesis easily produces the most complex organic molecules.E.The synthesis is accelerated by the presence of oxygen-containing compounds.正确答案:C C D AIsadora Duncan's masterly writings on the dance reveal the depth of her determination to create a lyric form of the art which was free of characterization, storytelling, and the theatrical exhibition of skills. She wished to discard the traditional methods and established vocabularies of such dance forms as ballet and to explore the internal sources of human expressiveness. She shunned bodily ornamentation and strove to use only the natural movements of her body, undistorted by acrobatic exaggeration and stimulated only by internal compulsion. In her recitals Duncan danced to the music of Beethoven, Wagner, and Gluck, among others, but, contrary to popular belief, she made no attempt to visualize or to interpret the music; rather, she simply relied on it to provide the inspiration for expressing inner feelings through movement. She did not regard this use of music as ideal, however, believing that she would someday dispense with music entirely. That day never came.The author is primarily concerned with Duncan'sA.masterful lyricism as expressed in her writings on the danceB.concerted efforts to subdue the natural movements of the danceC.belated recognition that she could not actually fulfill all of her ideals for the danceD.basic standards for the dance form that she wished to create and performE.continuous responsiveness to a popular misconception about the nature of her new art formThe author implies that Duncan relied on music in her recitals in order toA.interpret musical works solely by means of natural body movementsB.foster the illusion that music serves as an inspiration for the danceC.inspire the expression of inner feeling when she dancedD.validate the public belief that music inspires the expression of feeling through movementE.counter the public belief that she made no attempt to visualize musicAccording to the passage, Duncan intended to develop an art form that would do all of the following EXCEPTA.avoid the use of standard ballet techniquesB.revitalize an earlier established vocabularyC.draw on internal sources of human expressivenessD.create intended effects without the use of acrobatic exaggerationE.derive inspiration solely from inner feelingsIt can be inferred from the passage that which of the following endeavors is LEAST compatible with Duncan's ideals for the dance?ing music to stimulate the inspiration to danceB.Attempting to free an art form of both characterization and storytellingC.Minimizing the theatrical exhibition of skillsD.Being inspired to express inner feeling through movementE.Creating a lyric art form by drawing on inner personal resources正确答案:D C B AThe recent, apparently successful prediction by mathematical models of an appearance of EI Nino- the warm ocean current that periodically develops along the Pacific coast of South America-has excited researchers. Jacob Bjerknes pointed out over 20 years ago how winds might create either abnormally warm or abnormally cold water in the eastern equatorial Pacific. Nonetheless, until the development of the models no one could explain why conditions should regularly shift from one to the other, as happens in the periodic oscillations between appearances of the warm EI Nino and the cold so-called anti-El Nino. The answer, at least if the current model that links the behavior of the ocean to that of the atmosphere is correct, is to be found in the ocean.It has long been known that during an El Nino, two conditions exist: (1) unusually warm water extends along the eastern Pacific, principally along the coasts of Ecuador and Peru, and (2) winds blow from the west into the warmer air rising over the warm water in the east. These winds tend to create a feedback mechanism by driving the warmer surface water into a "pile" that blocks the normal upwelling of deeper, cold water in the east and further warms the eastern water, thus strengthening the wind still more. The contribution of the model is to show that the winds of an El Nino, which raise sea level in the cast, simultaneously send a signal to the west lowering sea level. According to the model, that signal is generated as a negative Rossby wave, a wave of depressed, or negative, sea level, that moves westward parallel to the equator at 25 to 85 kilometers per day. Taking months to traverse the Pacific, Rossby waves march to the western boundary of the Pacific basin, which is modeled as a smooth wall but in reality consists of quite irregular island chains, such as, the Philippines and Indonesia.When the waves meet the western boundary, they are reflected, and the model predicts that Rossby waves will be broken into numerous coastal Kelvin waves carrying the same negative sea-level signal. These eventually shoot toward the equator, and then head eastward along the equator propelled by the rotation of the Earth at a speed of about 250 kilometers per day. When enough Kelvin waves of sufficient amplitudearrive from the western Pacific, their negative sea-level signal overcomes the feedback mechanism tending to raise the sea level, and they begin to drive the system into the opposite cold mode. This produces a gradual shift in winds, one that will eventually send positive sea-level Rossby waves westward, waves that will eventually return as cold cycle-ending positive Kelvin waves, beginning another warming cycle.Which of the following, if true, would most seriously undermine the validity of the model of El Nino that is presented in the passage?A.During some years El Nino extends significantly farther along the coasts of Ecuador and Peru than during other years.B.During periods of unusually cool temperatures along the eastern Pacific, an El Nino is much colder than normal.C.The normal upwelling of cold water in the eastern Pacific depends much more on the local characteristics of the ocean than on atmospheric conditions.D.The variations in the time it takes Rossby waves to cross the Pacific depend on the power of the winds that the waves encounter.E.The western boundary of the Pacific basin is so irregular that it impedes most coastal Kelvin waves from heading eastward.It can be inferred from the passage that which of the following would result fairly immediately from the cessation of the winds of an El Nino?I. Negative Rossby waves would cease to be generated in the eastern Pacific.II. The sea level in the eastern Pacific would fall.III. The surface water in the eastern Pacific would again be cooled by being mixed with deep water.A.I onlyB.II onlyC.I and II onlyD.I and III onlyE.I, II, and IIIAccording to the model presented in the passage, which of the following normally signals the disappearance of an El Nino?A.The arrival in the eastern Pacific of negative sea-level Kelvin waves.B.A shift in the direction of the winds produced by the start of an anti-El Nino elsewhere in the Pacific.C.The reflection of Kelvin waves after they reach the eastern boundary of the Pacific, along Ecuador and Peru.D.An increase in the speed at which negative Rossby waves cross the Pacific.E.The creation of a reservoir of colder, deep ocean water trapped under the pile of warmer, surface ocean water.The primary function of the passage as a whole is toA.introduce a new explanation of a physical phenomenonB.explain the difference between two related physical phenomenaC.illustrate the limitations of applying mathematics to complicated physical phenomenaD.indicate the direction that research into a particular physical phenomenon should takeE.clarify the differences between an old explanation of a physical phenomenon and a new model of itWhich of the following best describes the organization of the first paragraph?A.A theory is presented and criticized.B.A model is described and evaluated.C.A result is reported and its importance explained.D.A phenomenon is noted and its significance debated.E.A hypothesis is introduced and contrary evidence presented.According to the passage, which of the following features is characteristic of an El Nino?A.Cold coastal water near PeruB.Winds blowing from the westC.Random occurrenceD.Worldwide effectsE.Short durationThe passage best supports the conclusion that during an anti-El Nino the fastest-moving signal waves areA.negative Rossby waves moving east along the equatorB.positive Rossby waves moving west along the equatorC.negative Kelvin waves moving west along the equatorD.positive Kelvin waves moving west along the equatorE.positive Kelvin waves moving east along the equator正确答案:E E A A C B EPresent-day philosophers usually envision their discipline as an endeavor that has been, since antiquity, distinct from and superior to any particular intellectual discipline, such as theology or science. Such philosoph- ical concerns as the mind-body problem or, more generally, the nature of human knowledge they believe, are basic human questions whose tentative philosophical solutions have served as the necessary foundations on which all other intellectual speculation has rested.The basis for this view, however, lies in a serious misinterpretation of the past, a projection of modern concerns onto past events. The idea of an autonomous discipline called "philosophy," distinct from and sitting in judgment on such pursuits as theology and science turns out, on close examination, to be of quite recent origin. When, in the seventeenth century, Descartes and Hobbes rejected medieval philosophy, they didnot think of themselves, as modern philosophers do, as proposing a new and better philosophy, but rather as furthering "the warfare between science and theology." They were fighting, albeit discreetly, to open the intellectual world to the new science and to liberate intellectual life from ecclesiastical philosophy and envisioned their work as contributing to the growth, not of philosophy, but of research in mathematics and physics. This link between philosophical interests and scientific practice persisted until the nineteenth century, when decline in ecclesiastical power over scholarship and changes in the nature of science provoked the final separation of philosophy from both.The demarcation of philosophy from science was facilitated by the development in the early nineteenth century of a new notion, that philosophy's core interest should be epistemology, the general explanation of what it means to know something. Modern philosophers now trace that notion back at least to Descartes and Spinoza, but it was not explicitly articulated until the late eighteenth century, by Kant, and did not become built into the structure of academic institutions and the standard self-descriptions of philosophy professors until the late nineteenth century. Without the idea of epistemology, the survival of philosophy in an age of modern science is hard to imagine. Metaphysics, philosophy's traditional core-considered as the most general description of how the heavens and the earth are put together-had been renderedalmost completely meaningless by the spectacular progress of physics. Kant, however, by focusing philosophy on the problem of knowledge, managed to replace metaphysics with epistemology, and thus to transform the notion of philosophy as "queen of sciences" into the new notion of philosophy as a separate, foundational discipline. Philosophy became "primary" no longer in the sense of "highest" but in the sense of "underlying." After Kant, philosophers were able to reinterpret seventeenth-and eighteenth-century thinkers as attempting to discover "How is our knowledge possible?" and to project this question back even on the ancients.Which of the following best expresses the author's main point?A.Philosophy's overriding interest in basic human questions is a legacy primarily of the work of Kant.B.Philosophy was deeply involved in the seventeenth-century warfare between science and religion.C.The set of problems of primary importance to philosophers has remained relatively constant since antiquity.D.The status of philosophy as an independent intellectual pursuit is a relatively recent development.E.The role of philosophy in guiding intellectual speculation has gradually been usurped by science.According to the passage, present-day philosophers believe that the。
GRE数学精选试题练习参考GRE数学精选试题练习参考1、To find the median of data given by percentage给了不同年龄range, 和各个range的percentage, 问median 落在哪个range 里. 把percentage加到50%就是median的range了.担小心一点，range首先要保证是有序排列.Example for this:Given: 10～20 = 20%, 30～50 = 30%, 0～10 = 40%, 20～30 = 10%, 问median在哪个range里.分析：千万不要上来就加，要先排序，切记!重新排序为： 0～10 = 40%, 10～20 = 20%, 20～30 = 10%, 30～50 = 40%. 然后从小开始加， median落在 10～20这个range里.如果觉得比较玄乎，我的方法，GRE大部分的题都可以这么搞。
0～10岁40匹ETS猪，10～20岁20匹ETS猪，20～30岁匹ETS 猪，30～50岁匹ETS猪，这100匹ETS猪按着年龄排下来，你说第五十匹ETS猪的年龄落在那个范围。
2、比较,当n1时，n，1，2 和1，2,3的标准方差谁大standard error 和standard variation 都是用来衡量一组数据的离散程度的'统计数值,只不过由于standard error中涉及绝对值,在数学上是很难处里的所以,都用标准方差,实际上standard error更合理一些,它代表了数据和平均值的平均距离.很明显题目中如果n=0的话,0,1,2的离散程度应该和1,2,3的离散程度相同.如果 n0,则n,1,2,的离散程度大于后者,而0Key: n是整数，前〉=后，2，3，4，5。
2024 GRE考试专题数学历年真题集锦GRE考试是对申请美国研究生院的学生进行综合能力测试的重要方式之一。
其中，数学部分是考生必须重点准备和应对的内容之一。
为了帮助考生更好地备考，本文整理了2024年GRE考试数学部分的历年真题集锦，供考生参考和复习使用。
一、整数和1. 如果a和b都是正整数，且a + b = 8，那么a和b可能的取值有哪些？答案：(1, 7), (2, 6), (3, 5), (4, 4)2. 如果m和n都是正整数，且m - n = 14，那么m和n的最小公倍数是多少？答案：28二、几何1. 一根长为10英尺的绳子被剪成两段，使得其中一段的长度是另外一段的2倍。
求较短一段的长度。
答案：4英尺2. 一个半径为5的圆与一个半径为8的圆相切于一点，并且两个圆的圆心之间的距离为13。
求两个圆相切点之间的距离。
答案：12三、代数1. 如果a是一个正整数，并且a^2 - 5a + 6 = 0，那么a的值是多少？答案：2或32. 设a、b、c是正整数，且满足a + b = c。
若a能被5整除，c能被9整除，那么b能被几整除？答案：4四、概率与统计1. 在一个有10个数字的集合中，每一个数字都是从1到10中随机选取的，那么从这个集合中选取一个数字，并且这个数字是偶数的概率是多少？答案：1/22. 一张标准扑克牌中红桃的数量为13，黑桃的数量为13。
从扑克牌中随机抽取一张牌，那么这张牌为红桃或黑桃的概率是多少？答案：26/52 = 1/2通过以上历年真题的集锦，考生可以更好地理解GRE数学部分的题目类型和解题思路。
每个题目的解答都提供了详细的答案，考生可以通过演算来验证答案的正确性，并进行自我评估。
为了顺利完成GRE考试，考生需要做到以下几点：首先，掌握基础知识。
GRE数学部分主要考察考生对数学基本概念和方法的理解和运用能力。
因此，考生需要熟悉数学基本概念和公式，同时要能够快速准确地运用这些概念和公式解题。
2020年名校学术联盟高考数学模拟试卷(理科)(二)一、单项选择题（本大题共12小题，共60.0分）1. 已知集合M ={x ∈R|x 2+2x =0}，N ={2,0}，则M ∩N =( )A. {0}B. {2}C. ⌀D. {−2,0，2}2. 已知向量a ⃗ =(λ+2,λ)，b ⃗ =(λ,1)，若a ⃗ ⊥b ⃗ ，则实数λ的值为( )A. 0或3B. −3或0C. 3D. −33. 已知α终边上一点P(1,2)，则cos2α=( )A. −35B. −45C. 35D. 454. 命题p ：∃x ∈R ，2x +1>0的否定为( )A. ∀x ∈R ，2x +1<0B. ∃x ∈R ，2x +1≤0C. ∀x ∈R ，2x +1>0D. ∀x ∈R ，2x +1≤05. p 为椭圆x 29+y 24=1上的一点，F 1，F 2分别为左、右焦点，且∠F 1PF 2=60° 则|PF 1|⋅|PF 2|=( )A. 83B. 163C. 4√33 D. 8√336. 如图，已知AP ⃗⃗⃗⃗⃗ =43AB ⃗⃗⃗⃗⃗ ，用OA ⃗⃗⃗⃗⃗ ,OB ⃗⃗⃗⃗⃗⃗ 表示OP ⃗⃗⃗⃗⃗ ，则OP ⃗⃗⃗⃗⃗ 等于( )A. 13OA ⃗⃗⃗⃗⃗−43OB ⃗⃗⃗⃗⃗⃗ B. 13OA ⃗⃗⃗⃗⃗+43OB ⃗⃗⃗⃗⃗⃗ C. −13OA ⃗⃗⃗⃗⃗+43OB ⃗⃗⃗⃗⃗⃗ D. −13OA ⃗⃗⃗⃗⃗−43OB ⃗⃗⃗⃗⃗⃗ 7. 若函数f(x)=cos 2 x +asin x +b 在[0,π2]上的最大值为M ，最小值为m ，则M −m 的值( )．A. 与a 有关，且与b 有关B. 与a 无关，且与b 有关C. 与a 有关，且与b 无关D. 与a 无关，且与b 无关8. ABCD 是正方形，P 是平面ABCD 外一点，PD ⊥AD ，PD =AD =2，二面角P −AD −C 为60°，则P 到AB 的距离是( )A. 2√2B. √3C. 2D. √79. 函数f(x)=x −log 12x 的零点个数为( ) A. 0个 B. 1个C. 2个D. 无数多个10. 设0<a <1，则函数f(x)=log a |x−1x+1|( )A. 在(−∞,−1)上单调递减，在(−1,1)上单调递增B. 在(−∞,−1)上单调递增，在(−1,1)上单调递减C. 在(−∞,−1)上单调递增，在(−1,1)上单调递增D. 在(−∞,−1)上单调递减，在(−1,1)上单调递减11. 若函数f(x)=(2a −1)lnx −x 在(0,1)上为减函数，则实数a 的取值范围是( )A. a <1B. a >12C. a ≥1D. a ≤1212. 设点A ，B ，C 是半径为2的球O 的球面上的三个不同的点，且OA ⊥BC ，BC =3，∠BAC =120°，则三棱锥O −ABC 的体积为( )A. √34B. √32C. 3√34D. √3二、填空题（本大题共4小题，共20.0分）13. 记S n 为等比数列{a n }的前n 项和，若数列{S n −2a 1}也为等比数列，则S4S 3=_______．14. 椭圆x 225+y 29=1的离心率为______ ．15. 已知实数x 、y 满足{x ≥2x +y −6≤02y −x ≥0，则z =y+1x−6的最大值为____．16. 已知正项数列{a n }满足：n(n +1)a n 2+(n 2+n −1)a n −1=0，其前n 项和为S n ，则2019S 2018=______．三、解答题（本大题共6小题，共72.0分）17. 已知函数f(x)=13x 3+ax 2−bx(a,b ∈R).若y =f(x)的图象上的点(1,−113)处切线的斜率为−4，(1)求a ，b 的值． (2)y =f(x)的极大值．18.已知在△ABC中，A，B，C所对的边分别为a，b，c，若a2+b2−c2=8，△ABC的面积为2√3．(1)求角C的大小；(2)若c=2√3，求sinA+sinB的值．19.如图，△ABC为正三角形，且BC=CD=2，，将△ABC沿BC翻折．(1)当AD=2时，求证：平面ABD⊥平面BCD；(2)若点A的射影在△BCD内(包括边界)，且直线AB与平面ACD所成角为60°，求AD的长．20.已知圆C：x2+y2+2ax−4y+a2−9a+6=0．(Ⅰ)求a的取值范围；(Ⅱ)若圆C与直线x+y−a−2=0交于M，N两点，CM⊥CN，求a的值．21.已知数列{a n}的前n项和S n=n2，数列{b n}为等比数列，且b1=a1，b2=a2．(1)求数列{a n}，{b n}的通项公式；(2)数列{c n}满足c n=a n，求{c n}的前项和T n．b n22.求函数f(x)=−x4+2x2+3，x∈[−3,2]上的最值．【答案与解析】1.答案：A解析：解方程求出集合M，由交集的运算求出M∩N．本题考查交集及其运算，属于基础题．解：由题意知，M={x∈R|x2+2x=0}={−2,0}，又N={2,0}，则M∩N={0}，故选A．2.答案：B解析：解：∵a⃗⊥b⃗ ；∴a⃗⋅b⃗ =(λ+2)λ+λ=0；解得λ=−3，或0．故选：B．根据a⃗⊥b⃗ 得，a⃗⋅b⃗ =0，进行数量积的坐标运算即可求出实数λ的值．考查向量垂直的充要条件，向量数量积的坐标运算．3.答案：A解析：本题主要考查任意角的三角函数的定义和二倍角公式，属于基础题．利用任意角的三角函数的定义，求得cosα的值，再利用二倍角公式即可得到答案．解：因为P(1,2)，所以r=|OP|=√5，所以，则cos 2α=2cos2α−1=2×15−1=−35．故选A．4.答案：D解析：解：命题为特称量词命题，则命题的否定为：∀x ∈R ，2x +1≤0， 故选：D ．根据含有量词的命题的否定即可得到结论． 本题主要考查含有量词的命题的否定，比较基础．5.答案：B解析：解：设|PF 1|=m ，|PF 2|=n ， 由椭圆的定义可知m +n =2a =6， ∴m 2+n 2+2nm =36， ∴m 2+n 2=36−2nm 由余弦定理可知cos60°=m 2+n 2−202mn=12求得mn =163故选B ．先设出|PF 1|=m ，|PF 2|=n ，利用椭圆的定义求得n +m 的值，平方后求得mn 和m 2+n 2的关系，利用余弦定理中求得mn 的值．本题主要考查了椭圆的应用，椭圆的简单性质和椭圆的定义．考查了考生对所学知识的综合运用．6.答案：C解析：OP ⃗⃗⃗⃗⃗=OA ⃗⃗⃗⃗⃗ +AP ⃗⃗⃗⃗⃗ =OA ⃗⃗⃗⃗⃗ +43AB ⃗⃗⃗⃗⃗ =OA ⃗⃗⃗⃗⃗ +43(OB ⃗⃗⃗⃗⃗⃗ −OA ⃗⃗⃗⃗⃗ )=−13OA ⃗⃗⃗⃗⃗ +43OB ⃗⃗⃗⃗⃗⃗ ，选C 7.答案：C解析：本题考查是二次函数的图象和性质，三角函数的最值，熟练掌握二次函数的图象和性质，是解答的关键．设sinx =t ，则0≤t ≤1，则y =−t 2+at +b +1，结合二次函数的图象和性质，其最大值和最小值均在端点或对称轴处取值，比较这些值即可得到答案．解：函数f(x)=cos 2 x +asin x +b 在[0,π2]上的最大值为M ，最小值为m ， 设sinx =t ，则0≤t ≤1，∴y=1−t2+at+b=−t2+at+b+1=−(t−a2)2+a24+b+1，当t=a2时，y=a24+b+1;当t=0时，y=b+1；当t=1时，y=a+b；当a变化时，最大值M和最小值m均在这三个函数值中取值，而这三个函数值之间的差均与b无关，与a有关，∴M−m的值与a有关，但与b无关．故选C．8.答案：D解析：本小题主要考查空间线面关系、二面角的度量、点线面距离的技计算等知识，考查数形结合、化归与转化的数学思想方法，以及空间想象能力，要求同学们熟练掌握．要想求P到AB的距离要先证明AB⊥平面PEF，即PF⊥AB，根据题中已知条件求出PE的长度，再根据勾股定理便可求出PF的长度．解：过P作PE⊥CD，过E作EF//BC，连接PF，∵AD⊥CD，PD⊥AD，∴AD⊥平面PDC，又∵PE在平面PDC上，∴AD⊥PE，又∵PE⊥CD，∴PE⊥平面ABCD，∴PE⊥AB∵EF//BC，∴AB⊥EF，∴AB⊥平面PEF，∴PF⊥AB，∴PF即为P到AB的距离，∵∠PDC=60°，PD=2，∴PE=√3，∵EF=AD=2，由勾股定理可得PF=√3+4=√7．故选D．9.答案：B解析：解：函数f(x)=x−log 12x的零点个数，就是函数y=x与y=log 12x，两个函数的图象的交点个数，如图：可知函数的图象只有一个交点．函数f(x)=x−log 12x的零点个数为：1个．故选：B．画出两个函数的图象，判断交点个数，即可得到选项．本题考查函数的零点个数的判断，考查数形结合思想的应用，是基础题．10.答案：A解析：解：函数f(x)的定义域为{x|x≠±1}，当x<−1时，t=|x−1x+1|=x−1x+1=1−2x+1，单调递增，而0<a<1，所以y=log a t单调递减，所以f(x)在(−∞,−1)上单调递减；当−1<x<1时，t=|x−1x+1|=−x−1x+1=−1+2x+1，单调递减，而0<a<1，所以y=log a t单调递减，所以f(x)在(−1,1)上单调递增，故选A．求出函数f(x)的定义域，先判断x<−1及−1<x<1时t=|x−1x+1|的单调性，再根据y=log a t单调递减及复合函数单调性的判定方法可知f(x)的单调性．本题考查对数函数单调性及复合函数单调性的判定，熟记基本函数的单调性为解决该类题目提供了简捷方法．11.答案：D解析：解：∵f(x)=(2a−1)lnx−x，f′(x)=2a−1x −1=(2a−1)−xx，若f(x)在(0,1)上为减函数，则(2a−1)−x≤0在x∈(0,1)恒成立，即a≤(x+12)min=12，故选：D．求出函数的导数，得到(2a−1)−x≥0在x∈(0,1)恒成立，分离参数，求出a的范围即可．本题考查了函数的单调性问题，考查导数的应用，是一道中档题．12.答案：A解析：解：如图，取BC中点G，连接OG，AG，则OG⊥BC，又∵OA⊥BC，∴BC⊥平面OAG，则BC⊥AG，在△ABC中，由BC=3，∠BAC=120°，可得AG=√32，在△OBC中，由OB=OC=2，BC=3，可得OG=√72，在△OGA中，由OA=2，AG=√32，OG=√72，可得cos∠OGA=34+74−42×√32×√72=−√217．∴sin∠OGA=2√77．∴V O−ABC=13×12×√72×√32×2√77×3=√34，故选：A．由题意画出图形，取BC 中点G ，连接OG ，AG ，则OG ⊥BC ，又∵OA ⊥BC ，可得BC ⊥平面OAG ，则BC ⊥AG ，然后求出三角形OAG 的面积，利用等积法求得三棱锥O −ABC 的体积． 本题考查棱锥体积的求法，考查数形结合的解题思想方法，是中档题．13.答案：1514解析：解：根据题意，设等比数列{a n }的公比为q ，对于等比数列{S n −2a 1}，其前三项为：−a 1，a 2−a 1，a 3+a 2−a 1，则有(−a 1)(a 3+a 2−a 1)=(a 2−a 1)2，变形可得：−(q 2+q −1)=(q −1)2， 解可得：q =12或0(舍)，则q =12， 则S4S 3=a 1(1−q 4)1−q a 1(1−q 3)1−q=1−q 41−q 3=1514；故答案为：1514．根据题意，设等比数列{a n }的公比为q ，求出数列{S n −2a 1}得前三项，由等比数列的定义分析可得q 的值，进而由等比数列的前n 项和公式分析可得答案．本题考查等比数列前n 项和公式的应用，涉及等比数列的定义以及应用，属于基础题．14.答案：45解析：解：由椭圆x 225+y 29=1方程可知，a =5，b =3，c =4，∴离心率e =c a =45，故答案为：45． 由椭圆x 225+y 29=1方程可知，a ，b ，c 的值，由离心率e =ca 求出结果．本题考查椭圆的标准方程，以及椭圆的简单性质的应用，求出a 、c 的值是解题的关键．15.答案：−12解析：本题主要考查线性规划的应用，根据直线斜率的几何意义结合数形结合是解决本题的关键． 解：z =y+1x−6的几何意义是区域内的点到定点D(6,−1)的斜率，作出不等式组对应的平面区域，由图象知AD 的斜率最大，BD 的斜率最小， 由{x =22y −x =0解得A(2,1)， 此时z =1+12−6=−12． 故答案为−12．16.答案：2018解析：解：∵正项数列{a n }满足：n(n +1)a n 2+(n 2+n −1)a n −1=0，∴[n(n +1)a n −1)(a n +1)=0,∵a n >0，∴a n =1n(n+1)=1n −1n+1， ∴S 2018=1−12+12−13+⋯+12018−12019=1−12019=20182019．∴2019S 2018=2018． 故答案为：2018．正项数列{a n }满足：n(n +1)a n 2+(n 2+n −1)a n −1=0，因式分解为[n(n +1)a n −1)(a n +1)=0，由a n >0，可得a n =1n(n+1)=1n −1n+1，利用裂项求和方法即可得出．本题考查了数列递推关系、裂项求和方法，考查了推理能力与计算能力，属于中档题．17.答案：解：(Ⅰ)∵f(x)=13x 3+ax 2−bx(a ，b ∈R)，∴f′(x)=x 2+2ax −b ，∵y =f(x)图象上的点(1,−113)处的切线斜率为−4，∴f(1)=−113，且f′(1)=−4，∴{13+a−b=−1131+2a−b=−4，∴{a=−1b=3；(Ⅱ)由(Ⅰ)可知，a=−1，b=3，∴f(x)=13x3−x2−3x，∴f′(x)=x2−2x−3=(x+1)(x−3)，令f′(x)=0，解得x=−1，x=3，∴当x变化时，f′(x)，f(x)的变化情况如下表：x(−∞,−1)−1(−1,3)3(3,+∞) f′(x)+0−0+ f(x)↗极大值↘极小值↗∴当x=−1时，f(x)取极大值53，∴y=f(x)的极大值为53．解析：(Ⅰ)根据题意，切点(1,−113)在函数f(x)的图象上，可得f(1)=−113，再根据导数的几何意义，即可得f′(1)=−4，求解方程组，即可求得a，b的值；(Ⅱ)由(Ⅰ)可得f(x)的解析式，求出f′(x)=0的根，判断根左右的单调性，结合极大值的定义，即可求得y=f(x)的极大值．本题考查了利用导数研究曲线上某点切线方程，函数在某点处取得极值的条件．导数的几何意义即在某点处的导数即该点处切线的斜率，解题时要注意运用切点在曲线上和切点在切线上．解题时要注意运用极值点必定是导函数对应方程的根，而导函数对应方程的根不一定是极值点．求函数极值的步骤是：先求导函数，令导函数等于0，求出方程的根，确定函数在方程的根左右的单调性，根据极值的定义，确定极值点和极值．过程中要注意运用导数确定函数的单调性，一般导数的正负对应着函数的单调性．属于中档题．18.答案：解：(1)由△ABC的面积为2√3可得，由a2+b2−c2=8及余弦定理可得2abcosC=8，所以，故，又C∈(0,π)，；即C=π3(2)∵C=π，2abcosC=8，3∴ab=8，又a2+b2−c2=8，c=2√3，则(a+b)2−2ab−12=8，即(a+b)2=36，可得a+b=6．由正弦定理，得．解析：本题考查正弦定理，余弦定理及三角形面积公式，属于中档题．(1)根据三角形面积公式和余弦定理求解即可；(2)根据正弦定理求解即可．19.答案：解：(1)取BD中点E，BC中点O，连接AE，OE，因为AB=AD=2，则AE⊥BD，△ABC是正三角形，∴BC⊥AO，∵BC=CD=2，CD⊥BC，BD=√2，∴BD=2√2，AE=12CD=1，又OE=12△ABC为正三角形中，AO=√3，所以AE2+OE2=AO2所以AE⊥OE，又AE⊥BD，BD与OE相交，所以AE⊥平面BCD，AE⊂平面ABD，所以平面ABD⊥平面BCD．(2)以O 为原点，以BC 为x 轴，以BE 为y 轴，以平面BCD 的过O 的垂线为z 轴建立空间直角坐标系，如图所示： 设二面角D −BC −A 为θ，则A(0,√3cosθ,√3sinθ)，B(−1,0，0)， C(1,0，0)，D(1,2，0)．∴BA ⃗⃗⃗⃗⃗ =(1,√3cosθ,√3sinθ)，CD ⃗⃗⃗⃗⃗ =(0,2，0)， CA⃗⃗⃗⃗⃗ =(−1,√3cosθ,√3sinθ)， 设平面ACD 的法向量为n⃗ =(x,y ，z)，则{n ⃗ ⋅CD ⃗⃗⃗⃗⃗ =0n ⃗ ⋅CA⃗⃗⃗⃗⃗ =0． ∴{2y =0−x +√3cosθy +√3sinθz =0, 令z =1得n ⃗ =(√3sinθ,0，1)． 直线AB 与平面ACD 所成角为60° ∴cos <n ⃗ ⋅BA ⃗⃗⃗⃗⃗ >=√3sinθ2⋅√3sin 2θ+1=√32.解得sinθ=1．∴A(0,0，√3)，又D(1,2，0)．∴|AD|=√1+22+√32=2√2．解析：本题考查了面面垂直的证明、利用空间向量进行空间角及空间距离的计算，考查推理能力和计算能力，属于中档题．(1)取BD 中点E ，BC 中点O ，连接AE ，OE ，结合已知证明：AE ⊥平面BCD ，即可证明平面ABD ⊥平面BCD ；(2)以O 为原点建立空间坐标系，设二面角D −BC −A 为θ，用θ表示出A 的坐标，求出BA ⃗⃗⃗⃗⃗ 和平面ACD的法向量n ⃗ ，令|cos <n ⃗ ，BA ⃗⃗⃗⃗⃗ >|=√32得出sinθ，从而得出A 点坐标，代入两点间的距离公式求出|AD|． 20.答案：解：(Ⅰ)将圆C 配方，得(x +a )2+(y −2)2=9a −2，由9a −2>0，得a >29，故a 的取值范围为(29,+∞)；(Ⅱ)由圆C 与直线l ：x +y −a −2=0交于M ，N 两点，，CM ⊥CN ， 知ΔCMN 为等腰直角三角形，所以圆心C (−a,2)到直线l 的距离d =√22r ，即√2=√22·√9a −2，化简，得4a 2−9a +2=0，解得a =2,或a =14， 又2和14均大于29，故a =2,或a =14．解析：本题考查圆的方程，考查直线与圆的位置关系，考查点到直线距离公式的运用，属于中档题． (1)把圆C 的方程化为标准形式，根据半径大于零，求得m 的范围．(2)由题意可得，弦心距等于半径的√22倍，再利用点到直线的距离公式，求得a 的值．21.答案：解：(1)当n ≥2时，a n =S n −S n−1=n 2−(n −1)2=2n −1，当n =1时，a 1=S 1=1，也适合上式． ∴a n =2n −1．设等比数列{b n }的公比为q ，∵b 1=a 1=1，b 2=a 2=3． ∴q =31=3． ∴b n =3n−1．(2)数列{c n }满足c n =a n b n=2n−13n−1，∴{c n }的前项和T n =1+33+532+⋯…+2n−13n−1，∴13T n =13+332+⋯…+2n−33n−1+2n−13n，相减可得：23S n =1+2(13+132+⋯…+13n−1)−2n−13n=1+2×13(1−13n−1)1−13−2n−13n，化为：S n =3−n+13n−1．解析：(1)当n ≥2时，a n =S n −S n−1=n 2−(n −1)2=2n −1，当n =1时，a 1=S 1，可得a n .设等比数列{b n }的公比为q ，b 1=a 1=1，b 2=a 2=3.可得q =3.可得b n ．(2)数列{c n}满足c n=a nb n =2n−13n−1，利用错位相减法即可得出．本题考查了数列递推关系、等比数列的通项公式与求和公式、错位相减法，考查了推理能力与计算能力，属于中档题．22.答案：解：f′(x)=−4x3+4x，令f′(x)=−4x(x+1)(x−1)=0，得x=−1，x=0，x=1．当x变化时，f′(x)及f(x)的变化怀情况如图下表：所以当x=−3时，f(x)取最小值−60；当x=−1或x=1时，f(x)取最大值4．解析：本题主要考查利用导数研究闭区间上的最值．。
2020年高考数学模拟试卷 (19)一、选择题（本大题共12小题，共60.0分）1.若复数z的共轭复数在复平面所对应的点的坐标是(2,−1)，则z(1−i)=()A. 3+iB. 3−iC. −3+iD. −3−i2.集合A={x||x+1|≤3},B={y|y=√x,0≤x≤4}.则下列关系正确的是()A. A∪B=RB. A⊆∁R BC. B⊆∁R AD. ∁R A⊆∁R B3.在区间[0,π]上随机取一实数x，使得的概率为()A. 1πB. 2πC. 13D. 234.已知a是实数，则1a<1是a>1的()A. 既不充分又不必要条件B. 充要条件C. 充分不必要条件D. 必要不充分条件5.数列{a n}满足a n+1+a n=2n−3，则a8−a4=()A. 7B. 6C. 5D. 46.《九章算术》中描述的“羡除”是一个五面体，其中有三个面是梯形，另两个面是三角形．已知一个羡除的三视图如图粗线所示，其中小正方形网格的边长为１，则该羡除的体积为()A. 20B. 24C. 28D. 327.漳州某公园举办水仙花展，有甲、乙、丙、丁4名志愿者，随机安排2人到A展区，另2人到B展区维持秩序，则甲、乙两人同时被安排到A展区的概率为()A. 112B. 16C. 13D. 128.已知函数f(x)是定义在R上的奇函数，当x∈(−∞,0)时，f(x)=x3−2x2，则f(3)=()A. 9B. −9C. 45D. −459.在三棱锥A−BCD中，△BCD是边长为2的等边三角形，∠ABC=∠ABD=π3，AB=3，则三棱锥A−BCD的外接球的表面积为()A. 19π2B. 19π C. 7√5π6D. √7π10.若a，b均为正实数，则ab+ba2+b2+1的最大值为()A. 23B. √22C. √2D. 211.点P是抛物线y2=4x上一点，记P到抛物线准线的距离为d1，到直线x−2y+10=0的距离为d2，则d1+d2的最小值为()A. 12√55+1 B. 11√55C. 5D. 不存在12. 已知不等式x ⋅lnx −ax −1≥0在x ∈[1,4]恒成立，则实数a 的最大值是( )A. −1B. 0C. 1D. 2 二、填空题（本大题共4小题，共20.0分）13. 已知(1+ax)6的展开式中，含x 3项的系数等于160，则实数a = ______ ． 14. 已知△ABC 是等腰直角三角形，AC =BC =2，则AB ⃗⃗⃗⃗⃗ ⋅BC ⃗⃗⃗⃗⃗ = ______ ．15. 若递增数列{a n }满足a n +a n+1+a n+2=3n −6，且a 2=12a 1，则a 1的取值范围是________． 16. 已知S n 为数列{a n }的前n 项和，a 1=1，2S n =(n +1)a n ，若存在唯一的正整数n 使得不等式a n 2−ta n −2t 2≤0成立，则实数t 的取值范围为________． 三、解答题（本大题共7小题，共82.0分）17. 已知在△ABC 中，sin(A +B)=2sin(A −B)．(1)若B =π6，求A ；(2)若tanA =2，求tan B 的值．18. 如图，四棱锥P −ABCD 的底面是正方形，PD ⊥底面ABCD ，PD =DC =2，E 是PC 的中点．(1)求直线DC 与平面PBD 所成角的大小； (2)求二面角E −BD −P 的大小的余弦值．19. 为实现有效利用扶贫资金，增加贫困村民的收入，扶贫工作组结合某贫困村水质优良的特点，决定利用扶贫资金从外地购买甲、乙、丙三种鱼苗在鱼塘中进行养殖试验，试验后选择其中一种进行大面积养殖，已知鱼苗甲的自然成活率为0.8.鱼苗乙，丙的自然成活率均为0.9，且甲、乙、丙三种鱼苗是否成活相互独立．(1)试验时从甲、乙，丙三种鱼苗中各取一尾，记自然成活的尾数为X，求X的分布列和数学期望；(2)试验后发现乙种鱼苗较好，扶贫工作组决定购买n尾乙种鱼苗进行大面积养殖，为提高鱼苗的成活率，工作组采取增氧措施，该措施实施对能够自然成活的鱼苗不产生影响.使不能自然成活的鱼苗的成活率提高了50%.若每尾乙种鱼苗最终成活后可获利10元，不成活则亏损2元，且扶贫工作组的扶贫目标是获利不低于37.6万元，问需至少购买多少尾乙种鱼苗？20.已知椭圆C：x2a2+y2b2=1(a>b>0)的左，右焦点分别为F1，F2，离心率为12，P是C上的一个动点．当P为C的上顶点时，△F1PF2的面积为√3．(1)求C的方程；(2)设斜率存在的直线PF2与C的另一个交点为Q.若存在点T(t,0)，使得|TP|=|TQ|，求t的取值范围．21.已知函数f(x)=x+1x2+1e−x．(1)求f(x)的最大值；(2)证明：当f(x1)=f(x2)(x1≠x2)时，x1+x2>0．22. 在直角坐标系xOy 中，直线l 的参数方程为{x =−2+12ty =√32t(t 为参数)，以坐标原点为极点，x 轴的正半轴为极轴建立极坐标系，圆C 的极坐标方程为ρ=√10． (1)若l 与C 相交于A ，B 两点P(−2,0)，求|PA|⋅|PB|；(2)圆M 的圆心在极轴上，且圆M 经过极点，若l 被圆M 截得的弦长为1，求圆M 的半径．23. 已知a >0，b >0，且a 2+b 2=1．(1)证明：(1a +1b )(a 5+b 5)≥1；(2)若1a 2+4b 2≥|2x −1|−|x −1|恒成立，求x 的取值范围．-------- 答案与解析 --------1.答案：B解析：【分析】本题考查复数的四则运算，共轭复数以及复数的几何意义，属于基础题．由复数z 的共轭复数在复平面的点坐标求出z ，再根据复数的乘法运算求解即可． 【解答】解：复数z 的共轭复数在复平面所对应的点的坐标是(2,−1)， 则z =2−i ，z =2+i ，则z(1−i)=(2+i )(1−i )=3−i ． 故选B ． 2.答案：D解析：解：∵A ={x||x +1|≤3}={x|−4≤x ≤2}， B ={y|0≤y ≤2}={x|0≤x ≤2}，∴A ∪B ={x|−4≤x ≤2}，∁R A ={x|x >2或x <−4}， ∁R B ={x|x >2或x <0}， ∴∁R A ⊆∁R B . 故选D ．根据绝对值不等式的解法化简集合A ，利用函数的值域化简集合B ，求出两集合的并集或集合的补集，最后利用集合间的包含关系的概念能够得到结果．本题考查不等式的解法，函数的值域，集合的包含关系的判断和应用等，是基础题．解题时要认真审题，仔细解答． 3.答案：C解析：【分析】本题考查解三角函数与几何概型等知识，由题意，本题属于几何概型的运用，已知区间的长度为π，满足sinx ∈[0,12]的x ∈[0,π6]∪[5π6,π]，求出区间长度，由几何概型公式解答． 【解答】解：在区间[0,π]上，当x ∈[0,π6]∪[5π6,π]时，sinx ∈[0,12]， 由几何概型知，符合条件的概率为．故选C ． 4.答案：D解析：解：解不等式1a <1得：a <0或a >1， 故1a <1是a >1的必要不充分条件，故选：D．解出关于a的不等式，结合充分必要条件的定义，从而求出答案．本题考查了充分必要条件，考查解不等式问题，是一道基础题．5.答案：D解析：【分析】本题考查了学生的化简运算能力及整体思想的应用．由a n+1+a n=2n−3得到a n+2−a n=2，从而求出答案，属基础题．【解答】解：依题意得(a n+2+a n+1)−(a n+1+a n)=[2(n+1)−3]−(2n−3)，即a n+2−a n=2，所以a8−a4=(a8−a6)+(a6−a4)=2+2=4．故选D．6.答案：B解析：【分析】本题考查多面体体积的求法，训练了利用分割补形法及等积法求多面体的体积，是中档题．连接CE，BE，DB，由已知利用多面体体积V=V E−ABCD+V C−BEF求解．【解答】解：如图连接CE，BE，DB，则V E−ABCD=13×12×(6+2)×4×3=16，V C−BEF=13×12×4×3×4=8．∴这个羡除的体积V=V E−ABCD+V C−BEF=16+8=24．故选：B．7.答案：B解析：解：有甲、乙、丙、丁4名志愿者，随机安排2人到A展区，另2人到B展区维持秩序，基本事件总数n=C42C22=6，甲、乙两人同时被安排到A展区包含的基本事件个数m=C22C22=1，∴甲、乙两人同时被安排到A展区的概率p=mn =16．故选：B．基本事件总数n=C42C22=6，甲、乙两人同时被安排到A展区包含的基本事件个数m=C22C22=1，由此能求出甲、乙两人同时被安排到A展区的概率．本题考查概率的求法，考查古典概型等基础知识，考查运算求解能力，考查函数与方程思想，是基础题．8.答案：C解析：【分析】本题考查了函数的奇偶性，属于基础题．【解答】∵函数f(x)是定义在R上的奇函数，∴f(x)=−f(−x)，∵当时，f(x)=x3−2x2，∴f(3)=−f(−3)=−(−27−18)=45．故选C．9.答案：A解析：【分析】本题考查的知识要点：面面垂直的判定，勾股定理的应用，球心的确定，球的表面积公式的应用，属于难题．首先判定E⊥AE，进一步设出球心，再利用勾股定理确定球的半径，最后求出球的表面积．解析：解：如图所示：设点O为球心，半径为r．G为△BCD的中心，连接BG并延长交CD于点E，再连接AE，过O作OF⊥AE，由于三棱锥A−BCD中，△BCD是边长为2的等边三角形，∠ABC=∠ABD=π3，AB=3，在△ABC中，由余弦定理可得，代入数据可得AC=√7，同理可得AD=√7，因为G为△BCD的中心，所以E为CD中点，故在等腰三角ADC 中，AE ⊥CD ，在Rt △ACE 中，易得AE =√AC 2−CE 2=√6，而在△ABE 中，根据线段长度关系，可得AB 2=BE 2+AE 2， 所以BE ⊥AE ，则四边形OGEF 为矩形． 设：OG =x ，在△OGC 中，利用勾股定理：r 2=(2√33)2+x 2，在△OFA 中，利用勾股定理：r 2=(√33)2+(√6−x)2，解得：x =2√6 所以：r 2=198则：S =4π⋅198=192π．故选A ．10.答案：B解析：解：因为a ，b 均为正实数， 则ab+ba 2+b 2+1=a+1a 2+1b+b≤2√b×b=2√a 2+1=12√a 2+2a+1a 2+1=12√1+2a a 2+1≤12√1+2√a 2×1=√22，即则ab+b a 2+b 2+1的最大值为√22，故选：B ．由重要不等式可得：则ab+b a 2+b 2+1=a+1a 2+1b+b≤2√a 2+1b×b=2√a 2+1=12√a 2+2a+1a 2+1=12√1+2aa 2+1≤12√12√a 2×1=√22，得解． 本题考查了基本不等式及其应用，属中档题． 11.答案：B解析：解：点P 到准线的距离等于点P 到焦点F 的距离，过焦点F 作直线x −2y +10=0的垂线，此时d 1+d 2最小， ∵F(1,0)，则d 1+d 2=22=11√55． 故选：B ．点P 到准线的距离等于点P 到焦点F 的距离，过焦点F 作直线x −2y +10=0的垂线，此时d 1+d 2最小，根据抛物线方程求得F ，进而利用点到直线的距离公式求得d 1+d 2的最小值．本题主要考查了抛物线的简单性质，两点距离公式的应用，直线与抛物线的简单性质的应用，是基础题．12.答案：A解析：由题意知x ⋅lnx −1≥ax ，即lnx −1x ≥a ，函数y =lnx −1x 在[1,4]上递增，所以a ≤−1．13.答案：2解析：【分析】本题主要考查二项式定理的应用，求展开式中某项的系数，属于基础题．先求出二项式展开式的通项公式，再令x 的幂指数等于3，求得r 的值，即可求得展开式中的含x 3项的系数，再根据含x 3项的系数等于160，求得实数a 的值． 【解答】解：∵(1+ax)6的展开式为T r+1=C 6r⋅(ax)r ， 令r =3，可得含x 3项的系数等于a 3⋅C 63=160， 解得a =2， 故答案为：2． 14.答案：−4解析：【解答】解：∵△ABC 是等腰直角三角形，AC =BC =2，∴AB =2√2，<AB ⃗⃗⃗⃗⃗ ⋅BC ⃗⃗⃗⃗⃗ >=135∘，AB⃗⃗⃗⃗⃗ ⋅BC ⃗⃗⃗⃗⃗ =|AB ⃗⃗⃗⃗⃗ |×|BC ⃗⃗⃗⃗⃗ |cos135°=2√2×2×(−√22)=−4 故答案为：−4 【分析】由已知得AB =2√2，<AB⃗⃗⃗⃗⃗ ⋅BC ⃗⃗⃗⃗⃗ >=135∘，AB ⃗⃗⃗⃗⃗ ⋅BC ⃗⃗⃗⃗⃗ =|AB ⃗⃗⃗⃗⃗ |×|BC ⃗⃗⃗⃗⃗ |cos135°，代入计算即可得到所求值．本题考查了向量的数量积运算，属于基础题。
算术新gre改革，很多考生都觉得难，如果觉得复习GRE数学考试比较难不妨从这些题目练习开始，每篇都是按照分类给大家列举了一些典型题目，本篇主要是针对GRE数学考试算术方面的典型题目及答案。
比较大小A. If the quantity in column A is greater;B. If the quantity in column B is greater;C. If the two quantities are equal;D. if the relationship cannot be determined from the information given.题目：1.How many positive whole numbers less than 81 are not equal squares of whole numbers?A.9B.70C.71D.72E.73答案：D2. 23(784)24(783)答案：B3.A printer numbered consecutively the pages of a book, beginning with 1 on the first page. In numbering the pages, he printed a total of 189 digits.The number of pages in the book100答案：B4. m,p, and x are positive integers and mp=x.mx答案：D5.n=7*193The number of distinct positive factors of n10答案：B6.Seven is equal to how many thirds of seven?A.1/3B.1C.3D.7E.21答案：C7.How many positive integers less than 20 are equal to the sum of a positive multiple of 3 and a positive multiple of 4?A.2B.5C.7D.10答案：D8.5/2 percent of 112022*7答案：C9.The figure above consists of 25 squares. If the figure were folded along the dotted diagonal to forn a flat triangle, then 26 minus the number in the square that would coincide with the square containing 26 would beA.13B.14C.15D.16E.17答案：D10.What is the remainder when 63 is divided by 8?A.5B.3C.2E.0答案：E每道题目都是关于gre数学考试的典型题目，考生可以根据题目来复习知识点，这样更有针对性。
2020年高考数学全真模拟试卷(4)(2月份)一、填空题（本大题共14小题，共70.0分）1. 已知集合A ={x|1<x <3}，B ={x|2<x <4}，则A ∪B =________．2. 已知a ，b ∈R ，i 是虚数单位．若a +i =2−bi ，则(a +bi)2= ______ ．3. 从总体中抽取一个样本：3、7、4、6、5，则总体标准差的点估计值为______ ．4. 执行如图所示的框图，若输出的结果为12，则输入的实数x 的值是______．5. 平行四边形ABCD 中，AB =5，AD =3，|BA ⃗⃗⃗⃗⃗ +BC ⃗⃗⃗⃗⃗ |=4，则AB ⃗⃗⃗⃗⃗ ⋅BC ⃗⃗⃗⃗⃗ =______．6. 各项均为正数的等比数列{a n }中，若a 2a 3a 4=a 2+a 3+a 4，则a 3的最小值为 ．7. 用数字1，2，3，4，5组成没有重复数字的五位数，其中奇数的个数为_____．8. 已知sinβ=35(π2<β<π)，且sin(α+β)=cosα，则tan(α+β)=______．9. 函数f(x)=2x+1x+1的值域为________．10. 若a =(23)x ，b =x 2，c =log 23x ，则当x >1时，a ，b ，c 的大小关系是________(从小到大)． 11. 正方体ABCD −A 1B 1C 1D 1的棱长为4，点P 是棱BB 1上一点，若异面直线AC 1与PD 所成角的余弦值为√1133，则BP =______．12. 已知的定义域为R ，则实数a 的取值范围是______．13. 已知f(x)是定义在R 上的奇函数，且在[0,+∞)上单调递增，若f(lg x)<0，则x 的取值范围是________． 14. 过双曲线x 2a 2−y 2b 2=1(a >0,b >0)的一个焦点F 作渐近线的垂线l ，垂足为M ，l 交y 轴于点E ，若FM ⃗⃗⃗⃗⃗⃗ =2ME⃗⃗⃗⃗⃗⃗ ，则该双曲线的渐近线方程为______． 二、解答题（本大题共11小题，共124.0分）15. 在△ABC 中，已知cosC +cosAcosB −√3sinAcosB =0(Ⅰ)求角B 的大小；(Ⅱ)若a +c =1，求b 的取值范围．16. 如图，在四棱锥P −ABCD 中，已知底面ABCD 为矩形，且AB =√2，BC =1，E ，F 分别是AB ，PC 的中点，PA ⊥DE ．(1)求证：EF//平面PAD ；(2)求证：平面PAC ⊥平面PDE ．17.已知椭圆C：x2a2+y2b2=1(a>b>0)的一个顶点为(0,−1)，离心率e=√22．(1)求椭圆C的方程；(2)过M(0,m)(−1<m<0)的直线l交椭圆C于A、B两点，试问：在椭圆C上是否存在定点T，使得无论直线l如何转动，以AB为直径的圆恒过定点T？若存在，求出m的值及点T的坐标；若不存在，请说明理由．18.如图，PQ为某公园的一条道路，一半径为20米的圆形观赏鱼塘与PQ相切，记其圆心为O，切点为G.为参观方便，现新修建两条道路CA、CB，分别与圆O相切于D、E两点，同时与PQ 分别交于A、B两点，其中C、O、G三点共线且满足CA=CB，记道路CA、CB长之和为L．(1)①设∠ACO=θ，求出L关于θ的函数关系式L(θ)；②设AB=2x米，求出L关于x的函数关系式L(x)．(2)若新建道路每米造价一定，请选择(1)中的一个函数关系式，研究并确定如何设计使得新建道路造价最少．x3(x∈R)19.已知函数f(x)=4x+ax2−23(1)当a=1时，求函数的单调区间；(2)若函数在区间[1,+∞)上是减函数，求实数a的取值范围．20.数列{a n}中，a1=2，a n+1=a n+cn(c是常数，n=1，2，3，…)，且a1，a2，a3成公比不为1的等比数列．(1)求c的值；(2)求{a n}的通项公式．21.在平面直角坐标系xOy中，设点P(x,5)在矩阵M=[1234]对应的变换下得到点Q(y−2,y)，求M−1[y x]．22.[选修4—4：坐标系与参数方程]在直角坐标系xOy中，直线C1：x+y=4，曲线C2：{x=1+cosθ,y=sinθ(θ为参数)，以坐标原点为极点，x轴正半轴为极轴建立极坐标系．(1)写出直线C1与曲线C2的极坐标方程；(2)若射线l：θ=α(ρ>0)分别交C1与C2于A，B两点，求|OB||OA|的取值范围．23.已知f(x)=x2+a|x|−b．(1)当a =−1，b =2时，解不等式f(x)≥0；(2)若|f(1)|≤1，|f(3)|≤1，求|f(2)|的最大值．24. 已知f(x)=(1+ax)n =a 0+a 1x 1+a 2x 2+⋯+a n x n ，若对于任意n ∈N ∗，都有∑a i n i=0=(23)n ． (1)求实数a 的值；(2)若[f(x)]2=b 0+b 1x +b 2x 2+⋯+b 2n x 2n ，求13b 1+132b 2+133b 3+⋯+132n b 2n 的值．25. 2018年2月22日，在韩国平昌冬奥会短道速滑男子500米比赛中，中国选手武大靖以连续打破世界纪录的优异表现，为中国代表队夺得了本届冬奥会的首枚金牌，也创造了中国男子冰上竞速项目在冬奥会金牌零的突破.根据短道速滑男子500米的比赛规则，运动员自出发点出发进入滑行阶段后，每滑行一圈都要依次经过4个直道与弯道的交接口A k (k =1,2,3,4).已知某男子速滑运动员顺利通过每个交接口的概率均为34，摔倒的概率均为14.假定运动员只有在摔倒或到达终点时才停止滑行，现在用X表示该运动员滑行最后一圈时在这一圈内已经顺利通过的交接口数．(1)求该运动员停止滑行时恰好已顺利通过3个交接口的概率；(2)求X的分布列及数学期望E(X)．【答案与解析】1.答案：{x|1<x<4}解析：本题主要考查函数的并集及其运算，属于较易题．解：本题考查集合的并集运算．由并集定义可得A∪B={x|1<x<4}．2.答案：3−4i解析：解：由a，b∈R，且a+i=2−bi，得{a=2−b=1，即a=2，b=−1．∴a+bi=2−i．∴(a+bi)2=(2−i)2=3−4i．故答案为：3−4i．由已知等式结合复数相等的条件求得a，b的值，则复数a+bi可求，然后利用复数代数形式的乘法运算得答案．本题考查了复数代数形式的乘法运算，考查了复数相等的条件，是基础题．3.答案：√2解析：解：样本数据：3、7、4、6、5的平均数为：×(3+7+4+6+5)=5，x=15×[(3−5)2+(7−5)2+(4−5)2+(6−5)2+(5−5)2]=2，方差为s2=15所以标准差为s=√2．故答案为：√2．根据平均数与方差、标准差的计算公式，即可求出结论．本题考查了平均数，方差与标准差的计算问题，是基础题目．4.答案：√2解析：解：若执行y =x −1，由x −1=12，即x =32∉(−∞,1],∴不成立，若执行y =log 2x ，由log 2x =12，得x =√2∈(1,+∞)，成立故答案为：√2本题主要考查的是条件函数f(x)={x >1,x ≤1log 2x,x >1，根据函数表达式进行计算即可得到结论． 本题主要考查程序框图的识别和判断，根据条件得到函数f(x)的表达式是解决本题的关键，比较基础． 5.答案：−9解析：解：由题意得，AB =5，AD =3，BD =4∴cos∠ABC =−35∴AB ⃗⃗⃗⃗⃗ ⋅BC ⃗⃗⃗⃗⃗ =5×3×(−3)=−9 故答案为−9．运用数量积的定义和直角三角形的知识可解决．本题考查平面向量的数量积和平行四边形法则．6.答案：√3解析：本题考查了等比数列的概念，等比数列的性质和利用基本不等式求最值．利用等比数列的概念得q >0，再利用等比数列的性质得a 32=1q +q +1，最后利用基本不等式求最值得结论．解：因为数列{a n }是各项均为正数的等比数列，所以公比q >0．由a 2a 3a 4=a 2+a 3+a 4得a 33=a 3q +a 3+a 3q ，即a 32=1q +q +1，所以a32≥3，即a3≥√3．因此a3的最小值为√3．故答案为√3．7.答案：72解析：用1、2、3、4、5组成无重复数字的五位奇数，可以看作是填5个空，要求个位是奇数，其它位置无条件限制，因此先从3个奇数中任选1个填入，其它4个数在4个位置上全排列即可．解：组成无重复数字的五位奇数，则个位只能排1，3，5中的一个数，共有3种排法，然后还剩4个数，剩余的4个数可以在十位到万位4个位置上全排列，共有A44=24种排法．由分步乘法计数原理得，由1、2、3、4、5组成的无重复数字的五位数中奇数有3×24=72个．故答案为72．8.答案：−2解析：解：∵sinβ=35，π2<β<π，∴cosβ=−√1−sin2β=−45，又sin(α+β)=cosα=cos[(α+β)−β]=cos(α+β)cosβ+sin(α+β)sinβ=−45cos(α+β)+35sin(α+β)，∴25sin(α+β)=−45cos(α+β)，∴tan(α+β)=−2．故答案为：−2．依题意，可求得cosβ=−45，由于α=(α+β)−β，巧用两角差的余弦即可求得tan(α+β)的值． 本题考查两角和与差的余弦函数，“凑角”是关键，考查三角函数间的关系式的应用，属于中档题．9.答案：(−∞,2)∪(2,+∞)解析：本题考查函数的值域，属于基础题． 解：∵f(x)=2x+1x+1=2(x+1)−1x+1=2−1x+1，1x+1≠0，∴f(x)≠2，∴函数f(x)的值域为(−∞,2)∪(2,+∞)． 故答案为(−∞,2)∪(2,+∞)．10.答案： c <a <b解析：本题考查指对数函数，幂函数性质的应用，属于中档题． 利用特值及指对数，幂函数的性质求解大小． 解：方法一：由题意x >1，∴当x =2，a =(23)2=49，b =4，c =log 232<0， 则c <a <b ， 故答案为c <a <b ． 方法二：y =(23)x 、y =x 2、三个函数图象如下图，由图象可知，x >1时，c <a <b ． 故答案为c <a <b ．11.答案：1解析：解：设BP =t ，以D 为原点，DA 为x 轴，DC 为y 轴，DD 1为z 轴，建立空间直角坐标系，则A(4,0，0)，C 1(0,4，4)，P(4,4，t)，D(0,0，0)， AC 1⃗⃗⃗⃗⃗⃗⃗ =(−4,4，4)，DP ⃗⃗⃗⃗⃗ =(4,4，t)， ∵异面直线AC 1与PD 所成角的余弦值为√1133，∴|cos <AC 1⃗⃗⃗⃗⃗⃗⃗ ，DP ⃗⃗⃗⃗⃗ >|=|AC 1⃗⃗⃗⃗⃗⃗⃗⃗ ⋅DP ⃗⃗⃗⃗⃗⃗||AC 1⃗⃗⃗⃗⃗⃗⃗⃗ |⋅|DP ⃗⃗⃗⃗⃗⃗ |=4t √48⋅√32+t 2=√1133， 由t >0，解得t =1， ∴BP =1． 故答案为：1．设BP =t ，以D 为原点，DA 为x 轴，DC 为y 轴，DD 1为z 轴，建立空间直角坐标系，利用向量法能求出结果．本题考查异面直线所成角的余弦值的求法，考查空间中线线、线面、面面间的位置关系等基础知识，考查运算求解能力，是中档题．12.答案：[−1,0]解析：本题考查函数的定义域及其求法，考查转化化归思想，是中档题． 把f(x)=√3x 2+2ax−a −1的定义域为R 转化为3x2+2ax−1−1≥0对任意x ∈R 恒成立，即x 2+2ax −a ≥0对任意x ∈R 恒成立，再由判别式小于等于0求解． 解：∵f(x)=√3x 2+2ax−a −1的定义域为R ， ∴3x2+2ax−1−1≥0对任意x ∈R 恒成立， 即3x2+2ax−a≥1=30恒成立，即x 2+2ax −a ≥0对任意x ∈R 恒成立， ∴△=4a 2+4a ≤0，则−1≤a ≤0． 故答案为[−1,0]．13.答案：(0,1)解析：本题主要考察了利用函数性质解不等式的相关问题，熟练掌握相关知识点是解决此类问题的关键． 解：由奇函数y =f(x)在[0,+∞)上递增，且f(0)=0， 得函数y =f(x)在(−∞,0)上递增，， 由f(lgx)<0，则lgx <0，得0<x <1， 即x 的取值范围是(0,1) ， 故答案为(0,1)．14.答案：y =±√2x解析：本题考查双曲线渐近线方程，难度一般，属中档题．由双曲线的标准方程可得右焦点F ，渐近线方程，利用FM ⃗⃗⃗⃗⃗⃗ =2ME ⃗⃗⃗⃗⃗⃗ ，求出M 的坐标，代入渐近线y =ba x ，即可得出结论．熟练掌握双曲线的标准方程及其性质、确定M 的坐标是解题的关键． 解：如图所示．取右焦点F(c,0)，渐近线y =ba x.∵FM ⊥OM ，∴可得直线FM 的方程为y =−ab (x −c)， 令x =0，解得y =acb，∴E(0,acb ). ∵FM ⃗⃗⃗⃗⃗⃗ =2ME ⃗⃗⃗⃗⃗⃗ ， ∴M(c 3,2ac3b)，又M 在渐近线y =ba x 上，∴2ac 3b=b a ×c3，解得√2a =b ．∴该双曲线的渐近线方程为：y =±√2x ． 故答案为：y =±√2x ．15.答案：解：(Ⅰ)由已知得cosAcosB +cosC =√3sinAcosB ，即cosAcosB +cos[π−(A +B)]=√3sinAcosB ． cosAcosB −cos(A +B)=√3sinAcosB ．所以sinAsinB =√3sinAcosB ，两边除以sin A cos B ，得，tanB =√3， ∴B =π3，(Ⅱ)由余弦定理可得b 2=a 2+c 2−2ac ⋅cosB =a 2+c 2−ac =(a +c)2−3ac =1−3ac ． ∵a +c =1≥2√ac ， ∴ac ≤14．∴b 2=1−3ac ≥14，即b ≥12．再由b <a +c =1，可得 12≤b <1，故边b 的取值范围是[12,1)．解析：(Ⅰ)利用两角和的余弦公式，将cosAcosB +cosC =√3sinAcosB ，变形为sinAsinB =√3sinAcosB ，即可求B ．(Ⅱ)由余弦定理可得b 2=1−3ac ，利用基本不等式求出b ≥12，再由b <a +c =1，求出边b 的取值范围．本题考查三角函数公式，余弦定理、基本不等式的综合灵活应用，考查转化变形、计算能力，属于中档题．16.答案：证明：(1)取PD中点G，连AG，FG，因为F，G分别为PC，PD的中点，所以FG//CD，且FG=12CD.又因为E为AB中点，所以AE//CD，且AE=12CD．所以AE//FG，AE=FG.故四边形AEFG为平行四边形．所以EF//AG，又EF⊄平面PAD，AG⊂平面PAD，故EF//平面PAD.(2)设AC∩DE=H，由△AEH∽△CDH及E为AB中点得AHCH =AECD=12，又因为AB=√2，BC=1，所以AC=√3，AH=13AC=√33．所以AHAE =ABAC=√2√3，又∠BAC为公共角，所以△HAE∽△BAC．所以∠AHE=∠ABC=90°，即DE⊥AC.又DE⊥PA，PA∩AC=A，所以DE⊥平面PAC，又DE⊂平面PDE，所以平面PAC⊥平面PDE．解析：本题考查线面平行的判定，考查面面垂直的判定，属于中档题． (1)由题意得到EF//AG ，利用线面垂直的判定即可证明； (2)由题意可得DE ⊥平面PAC ，再利用面面垂直的判定证明即可．17.答案：解：(1)由题意，b =1，c a =√22，∴a =√2，b =1，c =1， ∴椭圆C 的方程为x 22+y 2=1；(2)①当直线l 的斜率不存在时，以AB 为直径的圆的方程为：x 2+y 2=1， 此时与椭圆交点为(0,1)，(0,−1)；②当直线l 的斜率为0时，直线l 的方程为y =m ，此时以AB 为直径的圆的方程为：x 2+(y −m)2=2(1−m 2)， 当定点为(0,1)时，解得m =−13，当定点为(0,−1)时，解得m =13∉(−1,0)，∴m =−13，在椭圆上可能存在定点T(0,1)满足条件；③斜率存在时，设直线l 的方程为：y =kx −13，A(x 1,y 1)，B(x 2,y 2)， 与椭圆方程联立，可得(1+2k 2)x 2−43kx −169=0，∴x 1+x 2=4k3(1+2k 2)，x 1x 2=−169(1+2k 2)，TA ⃗⃗⃗⃗⃗ ⋅TB⃗⃗⃗⃗⃗ =(k 2+1)x 1x 2−43k(x 1+x 2)+169=(k 2+1)(−169(1+2k 2))−43k ⋅4k 3(1+2k 2)+169=0，∴过M(0,−13)的直线l 斜率存在时，以AB 为直径的圆过定点T(0,1)，综上所述，m =−13时，过M(0,−13)的直线无论如何转动，以AB 为直径的圆过定点T(0,1)．解析：(1)由题意，b =1，ca =√22，求出a ，b ，即可求椭圆C 的方程；(2)讨论直线l 的斜率不存在，设出直线l 的方程，求得圆的方程，求得定点T ，讨论直线l 的斜率存在，设出直线方程，联立椭圆方程，运用韦达定理和圆的性质，结合向量垂直的条件，即可得到存在定点T ．本题考查椭圆的方程和性质，主要考查椭圆方程的运用，联立直线方程和椭圆方程，运用韦达定理，同时考查直线和圆的位置关系，考查运算能力，属于中档题．18.答案：解：(1)①在RTΔCDO中，∠ACO=θ，所以，所以，在RTΔAGC中，所以，其中，②设AC=y，则在RTΔAGC中CG=√y2−x2，由RTΔCDO与RTΔAGC相似得，COAC =ODAG，即√y2−x2−20y=20x，即x√y2−x2−20x=20y，即x√y2−x2=20(x+y)，即x√y−x=20√x+y即x2(y−x)=400(x+y)，化简得CA=y=x3+400xx2−400，L(x)=2CA=2x3+800xx2−400其中x∈(20,+∞);(2)选择(1)中的第一个函数关系式，研究．，令L′(θ)=0，得令，当θ∈(0,θ0)时，L′(θ)<0，所以L(θ)递减；当时，L′(θ)>0，所以L(θ)递增，所以当时，L(θ)取得最小值，新建道路何时造价也最少．解析：本题主要考查函数模型的应用和导数的实际应用，考查实际应用能力和计算能力，属于中档题．(1)根据题意求函数的解析式即可；(2)利用导数求函数的最值即可．19.答案：解：(1)a=1时，f(x)=4x+x2−23x3，f′(x)=4+2x−2x2，令f′(x)>0，解得：−1<x<2，令f′(x)<0，解得：x>2或x<−1，故f(x)在(−∞,−1)，(2,+∞)上单调递减，在(−1,2)上单调递增；(2)f′(x)=4+2ax−2x2，若f(x)在[1,+∞)递减，则2ax≤2x2−4即a≤x−2x在[1,+∞)恒成立，令g(x)=x−2x，x∈[1,+∞)，则g′(x)=1+2x2>0，则g(x)在[1,+∞)递增，g(x)≥g(1)=−1，故a≤−1．即a的取值范围是(−∞,−1].解析：本题考查了函数的单调性、最值问题，考查导数的应用以及转化思想，是一道中档题．(1)求出函数的导数，解关于导函数的不等式，求出函数的单调区间即可；(2)求出函数的导数，问题转化为a≤x−2x 在[1,+∞)恒成立令g(x)=x−2x，x∈[1,+∞)，根据函数的单调性求出a的范围即可．20.答案：解：(1)a1=2，a2=2+c，a3=2+3c，因为a1，a2，a3成等比数列，所以(2+c)2=2(2+3c)，解得c=0或c=2．当c=0时，a1=a2=a3，不符合题意舍去，故c=2．(2)当n≥2时，由于a2−a1=c，a3−a2=2c，a n−a n−1=(n−1)c，所以a n−a1=[1+2++(n−1)]c=n(n−1)2c．又a1=2，c=2，故a n=2+n(n−1)=n2−n+2(n=2,3，···).当n=1时，上式也成立，所以a n=n2−n+2(n=1,2，···)解析：本题考查数列的性质和应用，解题时要注意计算能力的培养．(1)由题意知(2+c)2=2(2+3c)，解得c =0或c =2.再由当c =0时，a 1=a 2=a 3，不符合题意舍去，知c =2．(2)由题意知a n −a n−1=(n −1)c ，所以a n −a 1=n(n−1)2c.由此可知a n ．21.答案：解：∵点P(x,5)在矩阵M =[1234]对应的变换下得到点Q(y −2,y)， ∴依题意，[1234][5x]=[y y−2]，即{x +10=y −23x +20=y 解得{x =−4y =8 由逆矩阵公式知，矩阵M =[1234]的逆矩阵M −1=[−2132−12]， ∴M −1[y x]=[−2132−12][8−4]=[−1016]．解析：由题意得到{x +10=y −23x +20=y ，从而求出x ，y ，再由逆矩阵公式求出矩阵M 的逆矩阵，由此能求出M −1[y x]．本题考查矩阵变换的应用，是中档题，解题时要认真审题，注意逆矩阵公式的合理运用．22.答案：解：(1)由C 1：x +y =4，得直线C 1的极坐标方程为ρcosθ+ρsinθ=4，由C 2：{x =1+cosθy =sinθ，得(x −1)2+y 2=1，即x 2+y 2=2x ，∴曲线C 2的极坐标方程为ρ=2cosθ； (2)设A(ρ1,α)，B(ρ2,α)，−π4<α<π2， 则ρ1=4cosα+sinα，ρ2=2cosα，|OB||OA|=ρ2ρ1=14⋅2cosα(cosα+sinα)=14(cos2α+sin2α+1)=14[√2cos(2α−π4)+1]，∵−π4<α<π2，∴−√22<cos(2α−π4)≤1，∴0<14[√2cos(2α−π4)+1]≤14(√2+1)， ∴|OB||OA|的取值范围是(0,14(√2+1)].解析：本题考查简单曲线的极坐标方程，考查三角函数最值的求法，是中档题．(1)直接化C 1的普通方程为极坐标方程，化C 2的参数方程为普通方程，进一步化为极坐标方程； (2)设A(ρ1,α)，B(ρ2,α)，−π4<α<π2，则ρ1=4cosα+sinα，ρ2=2cosα，代入|OB||OA|，整理后利用三角函数的最值得答案．23.答案：解：(1)f(x)=|x|2−|x|−2≥0⇒(|x|−2)(|x|+1)≥0，所以|x|≥2，解得x ∈(−∞,−2]∪[2,+∞)．(2)由题意得|a −b +1|≤1，|3a −b +9|≤1，故|f(2)|=|2a −b +4|=12|(a −b +1)+(3a −b +9)−2|≤12(|f(1)|+|f(3)|+2)≤2， 取等条件为{a −b +1=−1,3a −b +9=−1⇒a =−4,b =−2．故|f(2)|的最大值为2．解析：本题考查绝对值不等式． (1)利用分解因式，得|x|≥2求解即可;(2)由已知得a −b +1|≤1，|3a −b +9|≤1，然后将f(2)用f(1)和f(2)表示绝对值不等式的三角不等式求解即可．24.答案：解：(1)由f(1)=(1+a)n =a 0+a 1+a 2+⋯+a n =∑a i n i=0=(23)n ．得a =−13；(2)[f(x)]2=b 0+b 1x 2+b 2x +⋯+b 2n x 2n =[(1−13x)n ]2=(1−13x)2n ，∴b k =C 2n k (−13)k ，令b k 3k =C 2n k (−1)k ，k =1，2，3…，2n ，首先考虑1C 2n+1k+1C 2n+1k+1=k!(2n+1−k)!(2n+1)!+(k+1)!(2n−k)!(2n+1)!=k!(2n −k)!(2n +1−k +k +1)(2n +1)!=k!(2n−k)!(2n+2)(2n+1)!=2n+2(2n+1)C 2nk ，则1C 2nk =2n+12n+2(1C 2n+1k+1C 2n+1k+1)，因此1C 2nk −1C 2nk+1=2n+12n+2(1C 2n+1k−1C 2n+1k+2).故13b 1+132b 2+133b 3+⋯+132n b 2n =−C 2n 1+C 2n 2−C 2n 3+⋯+(−1)k C 2n k +⋯+C 2n 2n=−2n +12n +2(1C 2n+11−1C 2n+13+1C 2n+13−1C 2n+15+⋯+1C 2n+12n−1−1C 2n+12n+1) =−2n +12n +2(1C 2n+11−1C 2n+12n+1)=2n +12n +2(12n +1−1) =n n+1．解析：(1)在已知等式中取x =1，结合∑a i n i=0=(23)n 即可求得a 值；(2)由已知结合(1)可得b k =C 2n k (−13)k ，令b k 3k =C 2n k (−1)k ，k =1，2，3…，2n ，得到1C 2n k =2n+12n+2(1C 2n+1k +1C 2n+1k+1)，因此1C 2n k −1C 2n k+1=2n+12n+2(1C 2n+1k −1C 2n+1k+2)，代入得答案． 本题考查二项式定理的应用，考查组合数公式的性质，考查计算能力，是中档题． 25.答案：解：(1)由题意可知：P =(34)3×14=27256;(2)X 的所有可能值为0，1，2，3，4.则P(A k )=34(k =1,2,3,4)，且A 1，A 2，A 3，A 4相互独立．故P(X =0)=P(A 1)=14，P(X =1)=P(A 1·A 2)=34×14=316，P(X =2)=P(A 1A 2A 3)=(34)2×14=964， P(X =3)=P(A 1A 2A 3A 4)=(34)3×14=27256， P(X =4)=P(A 1A 2A 3A 4)=(34)4=81256，从而X 的分布列为所以E(X)=0×14+1×316+2×964+3×27256+4×81256=525256．解析：(1)由相互独立事件的概率计算公式计算即可；(2)先得出X的可能取值，求出对应概率，列出分布列求出数学期望即可．。
2020年新GRE数学模拟考试试题及答案
1. A,B,C,D,E,F排在1,2,3,4,5,6六个位置上,问A不在1, B不在2, C不在3的情况下,共有多少种排法?
(A) 720
(B) 450
(C) 180
(D) 216
(E) 320
2. 一直线L过点A(5,0), B(0,2), 坐标原点为O, 点P(X,Y)为三角形OAB中一点, 问：Y
(A) 1/4
(B) 3/8
(C) 1/2
(D) 5/8
(E) 3/4
3. In an insurance company, each policy has a paper record and an electric record. For those policies having incorrect paper record, 60% also having incorrect electric record; For policies having incorrect electric record, 75% also having incorrect paper record. 3% of all policies have both incorrect paper and incorrect electric records. If we randomly pick out one policy,what’s the probability that the one having both correct paper and correct electric records?
(A) 0.80
(B) 0.94
(C) 0.75
(D) 0.88
(E) 0.92
4. If Bob can do a job in 20 days and Jane can do the job in 30 days, they work together to do this job and in this period, Bob stop work for 2.5 days and Jane stop work for x days, and the job be finished for 14 days, what is x?
(A) 1.6
(B) 3.2
(C) 1.5
(D) 1.25
(E) 1.15
5. The probability of A is 60% and the probability of B is 50%, what is the most possible probability that neither A nor B would happen?
(A) 0.80
(B) 0.40
(C) 0.75
(D) 0.55
(E) 0.68
6. There are 1200 respondents to a poll, each favoring
their preference for candidates A,B, and C. 54% favored A, 48% favored B, and 42% favored C, and there is 30% favored both A and B. what’s the largest possible number of respondents favoring C, but not C&B, nor C&A?
(A) 25%
(B) 30%
(C) 28%
(D) 38%
(E) 40%
参考答案：
1.解：首先考虑总的可能性为，再考虑A在1，B在2，C在3的
可能性分别为，中重复计算了三者交集，分别为AB在1，2，AC在1，3，BC在2，3，所需将三种情况加回，即，但考虑这三种加回的交集
又重复计算了ABC在1，2，3的情况，所以应减去P33
2.解：在平面直角坐标系中，T
3.解：设总数为x，设incorrect paper record有y, incorrect electric record有z，则：
x·y·60%=3%·x y=5%
x·z·75%=3%·x z=4%
则，两者至少有一个错误的百分比为5%+4%-3%=6%，所准确答案为94%
4.解：1/20(14-2.5)+1/30(14-x)=1，得出x=1.25
5.解：划出图表来能够一目了然：A， B均不发生的概率为40%，最小概率为0.
6.解：A和B的并集为：54%+48%-30%=72%，所C为28%.。
