Ethical Theory

Managerial Auditing Journal Emerald Article: On ethical theory in auditing Lutz Preuss Article information: To cite this document: Lutz Preuss, (1998),†On ethical theory in auditing†, Managerial Auditing Journal, Vol. 13 Iss: 9 pp. 500 508 Permanent link to this document: http://dx. doi. org/10. 1108/02686909810245910 Downloaded on: 25-11-2012 References: This document contains references to 47 other documents Citations: This document has been cited by 2 other documents To copy this document: [email  protected] com This document has been downloaded 2432 times since 2005. *Users who downloaded this Article also downloaded: * Gary Pflugrath, Nonna Martinov-Bennie, Liang Chen, (2007),†The impact of codes of ethics and experience on auditor judgments†, Managerial Auditing Journal, Vol. 22 Iss: 6 pp. 566 – 589 http://dx. doi. org/10. 1108/02686900710759389 Beverley Jackling, Barry J. Cooper, Philomena Leung, Steven Dellaportas, (2007),†Professional a ccounting bodies' perceptions of ethical issues, causes of ethical failure and ethics education†, Managerial Auditing Journal, Vol. 22 Iss: 9 pp. 928 – 944 http://dx. doi. org/10. 1108/02686900710829426 Douglas E.Ziegenfuss, Anusorn Singhapakdi, (1994),†Professional Values and the Ethical Perceptions of Internal Auditors†, Managerial Auditing Journal, Vol. 9 Iss: 1 pp. 34 – 44 http://dx. doi. org/10. 1108/02686909410050433 Access to this document was granted through an Emerald subscription provided by ASTON UNIVERSITY For Authors: If you would like to write for this, or any other Emerald publication, then please use our Emerald for Authors service. Information about how to choose which publication to write for and submission guidelines are available for all. Please visit www. emeraldinsight. om/authors for more information. About Emerald www. emeraldinsight. com With over forty years' experience, Emerald Group Publishing is a leading independent publi sher of global research with impact in business, society, public policy and education. In total, Emerald publishes over 275 journals and more than 130 book series, as well as an extensive range of online products and services. Emerald is both COUNTER 3 and TRANSFER compliant. The organization is a partner of the Committee on Publication Ethics (COPE) and also works with Portico and the LOCKSS initiative for digital archive preservation. Related content and download information correct at time of download. On ethical theory in auditing Lutz Preuss Lecturer, Heriot-Watt University, Edinburgh, UK This article discusses ways of giving support to auditors in addressing moral dilemmas. Codes of Ethics are very important in this context but are in the ? nal analysis insuf? cient devices, because their necessarily generalised form has to be translated into the speci? c situation and thus requires acceptance rather than merely adherence. Codes have to be complemented with developed ethical r easoning of accountants.Hence, individual ethical principles are discussed which have been applied to accounting in the recent literature, i. e. utilitarianism, deontology, virtue ethics and ethics of care. Unsurprisingly, none emerges as giving completely satisfactory solutions. Yet eclecticism can be avoided by using compound models, which combine individual principles to provide reasonably comprehensive cover of decision-making in a business context. As in any other profession, practitioners of accounting, be they public accountants, management accountants or internal auditors, may face moral dilemmas in their work.In mapping out the ? eld of potentially con? icting interests, the Institute of Chartered Accountants of Scotland (1997, p. i), the oldest professional body in the UK, stipulates: The primary duty of the Institute of Chartered Accountants of Scotland is to the public. This principle determines its status as a professional body . The ICAS (1997, p. vii) requires that: I n addition to the duties owed to the public and to his or her employer, a member of the Institute is bound to observe high standards of conduct, which may sometimes be contrary to his personal self-interest.The author would like to thank Professor Gerald Vinten, editor of Managerial Auditing Journal, and Stephen Morrow, Department of Accountancy and Finance, Heriot-Watt University, Edinburgh, for their helpful suggestions. Managerial Auditing Journal 13/9 [1998] 500–508  © MCB University Press [ISSN 0268-6902] The public accountant is the con? dential agent of the community at large, but the public does not (re)appoint auditors (for speci? c moral dilemmas facing external auditors, see Gunz and McCutcheon, 1991; Moizer 1995; Finn et al. , 1994).Management accountants and internal auditors are employees of the corporation; hence their employment position may collide with their professional values. They too have a responsibility to society – comparable to an engineerâ €™s concern for public safety – and are required by the Institute of Chartered Accountants in England and Wales to observe â€Å"the same standards of behaviour and competence as apply to all other members† (Statement 1. 220, quoted in Maurice, 1996, p. 184). Ethical dilemmas tend to be complex and only hazily de? nable.In accounting, moral agency often becomes a question of causation (Moizer, 1995, p. 425f. ), as the content of an audit report may not have an obvious link with a speci? c result: if an auditor quali? es a company’s accounts, has he actually caused an ensuing bankruptcy? Furthermore, the very subject matter of accounting is such that two equally objective accountants may reach different results. The most visible response by all professional accountancy bodies has been to set up codes of ethics, e. g. by the Institute of Internal Auditors in 1968.Their very existence sets limits for immoral behaviour and offers guidance in ambiguous situations . Studies of members of the Institute of Internal Auditors (Siegel et al. , 1995; Ziegenfuss and Singhapakdi, 1994) found that a clear majority do use the code of ethics in their work. It is seen primarily as an instrument for giving guidance in moral dilemmas (64 per cent) rather than a means to enhance the profession’s public perception (16 per cent). Research by Dittenhofer and colleagues offers an insightful longitudinal perspective nto changing moral beliefs of internal auditors in the light of the IIA Code of Ethics. In 1982 Dittenhofer and Klemm (1983) presented a random sample of IIA members with 20 vignettes, each describing morally contentious issues, and asked respondents to indicate their reaction, ranging from dismissal of the person to doing nothing because no ethical problem is perceived to exist. In 1994 Dittenhofer and Sennetti (1995) repeated the survey, again sampling IIA members. In many cases the authors found signi? cant changes, with IIA members having become more critical and supporting harsher action.So the proportion of respondents who claim they would dismiss an internal audit supervisor who engaged in insider dealing (situation 4) has risen from 50. 1 to 63. 0 per cent, with particularly large increases for trainees/journeymen (+70 per cent), in the category staff status, and insurance (+40 per cent) in the category employer’s activity . However, some cases – arguably less severe ones – have shown little change over the decade or even a decline. In any case, there were no signi? cant differences in terms of gender or age groups.Most astonishing, so Dittenhofer and Sennetti claim, was the fact that for each scenario almost the complete spectrum of attitudes was represented, which indicates that there is no consensus among internal auditors as to what is right or wrong. Codes of ethics are not sufficient to resolve moral dilemmas. For a start, violations of codes have persisted (Finn et al. , 1994; Loeb, 19 71; Pearson, 1987). A distinction has to be made between acceptance of a code and mere adherence to it (Loeb, 1971), which may stem from a strong organisational climate or a fear of being penalised.Furthermore, codes in their universalised form, cannot cover all eventualities. Analysing Dittenhofer and [ 500 ] Lutz Preuss On ethical theory in auditing Managerial Auditing Journal 13/9 [1998] 500–508 Klemm’s (1983) research, Vinten (1996, p. 56) found a two-fold problem: for some scenarios – clearly morally contentious ones – it was unclear exactly which article of the IIA Code applied and also what response the Code required. Swanda (1990) and Vyakarnam et al. , (1996) found that accountants isplay a tendency to revert to technical knowledge when facing ambiguous moral situations. Hence Mintz (1995) argues that accountants need both technical and moral expertise as well as the intention and ability to act against self-interest if morality requires doing so . Vinten (1990, p. 10) sees three types of codes: 1 a regulatory code, such as the Ten Commandments. It establishes a complete overlap between behaviour and code, which is furnished with such a compelling ethical imperative that further discussion is neither necessary nor asked for.It does, however, not recognise shades of grey . 2 an aspirational code, such as the wisdom literature in the Jewish Scriptures. It provides the standard a person should aspire to but recognises that full compliance may rarely be possible. Yet it provides little help in weighing up alternative courses of action. 3 an educational code: this holds rules and regulations to be unhelpful if not damaging and instead stresses the importance of the individual conscience in a professional situation.The unsatisfactory nature of codes of ethics, Vinten suggests, may stem from the predominance of the regulatory model. It emerges that de? ning ethics as â€Å"the rules of conduct recognised in the human life departme nt of the practice of professional accountancy† (Maurice, 1996, p. 9) is too limited. Codes of ethics have to be reinforced with moral development of accountants. Thus researchers have studied the status quo of moral development in the profession (Finn et al. , 1994 of certi? ed public accountants, Ziegenfuss et al. , 1994, of internal auditors and management accountants).Some authors have then discussed possible improvements of moral development within Lawrence Kohlberg’s framework for cognitive moral development (Lovell, 1995, 1997; Sweeney and Roberts, 1997). Others have sought to clarify how far individual ethical theories and principles are applicable to accounting dilemmas, although the discussion has often been limited to utilitarianism and deontology (Maurice, 1996; Moizer, 1995). Only over the last few years have alternative ethical theories been applied to accounting (Mintz, 1995; Francis, 1990; Oakes and Hammond, 1995; Reiter, 1996, 1997).The aim of this arti cle is to draw the discussion of these ethical theories and principles together into a comprehensive system, where the advantages and disadvantages of individual principles are highlighted. Following Hartman (1994) ethical theories shall be understood less as prescription for action than as tools for understanding complex situations. 1. Utilitarian ethics Teleological or consequentialist ethics judges the rightness or wrongness of an act by its consequences. The most elaborate consequentialist theory is that of utilitarianism, as propagated by Jeremy Bentham (1789/1962).In the de? nition of his disciple John Stuart Mill (1861/1962, p. 257): Utility, or the Greatest Happiness Principle holds that actions are right in proportion as they tend to promote happiness, wrong as they tend to produce the reverse of happiness. Utilitarianism is a strongly democratic theory as every individual is to be given as much consideration as anybody else is. It should be pointed out that Bentham saw thi s principle not as a watertight moral theory but as a tool for political decision making. Utilitarianism faces obvious practical problems in its moral arithmetic.Mill introduced a distinction between higher and lower pleasures, yet it is still far from clear how different pleasures of different intensities can be summed up, how a strong immediate desire compares with a life-long moderate one, etc. Agents may lack sufficient time to calculate all the consequences or may overestimate their own sufferings and underestimate somebody else’s happiness. This kind of criticism can partly be averted by applying the principle of utility not to single acts, act-utilitarianism, but to classes of acts, rule-utilitarianism.Under rule-utilitarianism an act is morally obligatory if it falls into a category of acts, which in their collectivity tend to produce more happiness than pain. It is no longer necessary to know all the implications of an action; one can rely on past evidence to get a f airly accurate account of an act’s potential consequences. Rule-utilitarianism is applied in a council recommendation by the Institute of Chartered Accountants of Scotland issued in 1971 (quoted in Moizer, 1995, p. 422), which: †¦ recommends that members †¦ hould not disclose past or intended civil wrongs, crimes †¦ or statutory offences unless they feel the damage to the public likely to arise from non-disclosure is of a very serious nature. [ 501 ] Lutz Preuss On ethical theory in auditing Managerial Auditing Journal 13/9 [1998] 500–508 Rule-utilitarianism, however, can be shown to collapse into act-utilitarianism, if one follows the rule â€Å"in situations of type x, do y or whatever else maximises utility† (Smart, 1967). Also, the difficulty or perhaps impossibility of a moral calculation has not been solved.Since utilitarianism is to consider all consequences of an action, this includes not only consequences for yet unborn generations but al so side effects the agent has not brought about actively, and these may well overshadow intended consequences. Intuitively, one would hesitate to blame a moral agent on this basis. Utilitarianism in accounting Utilitarianism has two advantages over alternative ethical theories for application in business. It links self-interest with moral behaviour, and a company is per de? nition self-interested. Secondly, the calculation of bene? and harm is similar to pro? t and loss accounting and hence more likely to ? nd acceptance with business practitioners than rival ethical theories. By default, utilitarianism is the most in? uential ethical theory in the business context. Most economic and ? nance concepts are implicitly or explicitly built on the assumption that individuals are interested in maximising short-term self-interest. A resulting intellectual parenthood of accounting theory and methods in utilitarianism becomes important in the debate over the neutrality of accounting informati on.Neutralists, like Solomons (1991), argue that it is not the task of accountancy to be an agent of change in society Accountants should merely convey . unbiased information, on which users can then base their decisions. Radical accountants, such as Tinker (1991), have questioned whether accounting information can actually be neutral. As accounting is embedded in social reality – it is neither inexplicably given nor a straight re? ection of social reality – and in social con? ict, its theory and methods inevitably favour one side of the con? ict over another. Thus Lovell (1997) ? ds it problematic that accounting concepts are often presented as neutral or even as morally correct, without pointing out their roots in utilitarian thought. There is evidence (Gray et al. , 1994; Lovell, 1997; Ponemon, 1992) that accounting and other means of organisational control compress moral reasoning within the lowest stages of Lawrence Kohlberg’s (1981) hierarchy of cognitive moral development. Kohlberg sees moral development progressing from an instrumental use of other persons via the acceptance of a social order to abstract principles which, if necessary, over- ride human laws.Where accounting control is assumed to work because people do not want their underperformance to be detected, a stage one motivation exists. Linking acceptable performance to ? nancial bonuses assumes a stage two motivation. At stage three a person performs as required because she wants to win or maintain the respect of colleagues, a stage four motivation shows in a belief that the law, either respective state laws or the organisational â€Å"laws†, are to be obeyed for their own sake (Lovell, 1997, p. 155). On the other hand, inasmuch as it prevents illegal or immoral practices, accounting control does have a moral quality . . Deontological ethics Deontological ethics focuses on duty or moral obligation, deon being the Greek word for duty There are various deontological con. cepts, such as â€Å"Do unto others as thou wouldst have them do unto you†, but the most rigorous version was developed by Immanuel Kant (1785/1898). He sees a sharp difference between self-interest and morality and proposes that an action only has moral value if it is performed from duty Kant proposes his . Categorical Imperative (1785/1898, p. 38). Act only on that maxim whereby thou canst at the same time will that it should become universal law.A different version of the Categorical Imperative reads (1785/1898, p. 47) . So act as to treat humanity, whether in thine own person or in that of any other, in every case as an end, never as means only . A maxim, a rational principle, which underlies an action, has to ful? l two criteria to become universally binding: ? rst it has to be shown that the maxim can be universalised without contradiction. Breaking promises if the disadvantages outweigh the bene? ts is not universalisable, because if it was nobody could rely on an ybody telling the truth anymore.Secondly, one has to show that a rational agent ought to will the maxim, i. e. that it actually creates conditions which are conducive to human life. Some actions, however, are universalisable but nonetheless seem wrong: a religious fundamentalist may reason it necessary to treat opponents in horrendously brutal ways and accept that he would be treated in the very same fashion if he were in the opposing camp. Other cases are not universalisable but do not seem morally wrong; universal contraception would bring humanity to an end but to most people it does not seem wrong in individual cases.Furthermore duties, imposed by [ 502 ] Lutz Preuss On ethical theory in auditing Managerial Auditing Journal 13/9 [1998] 500–508 several categorical imperatives, may clash; here the principle offers no further solution. On the other hand, Kantian morality links with popular conceptions of morality, e. g. that some actions simply are never permissible, whateve r the gain to individuals or society . The link between morality and the will of the agent allows us to praise people for their intentions even if the results fall short of expectations.A deontological approach is also the most important basis for criminal law. Deontology in accounting A deontological perspective underlies much of the self-regulation in accountancy, see for instance the explicit requirement by the Auditing Practices Committee in the US on qualifying a company’s accounts on a going concern basis (Moizer, 1995, p. 424): The auditor should not refrain from qualifying his report if it is otherwise appropriate, merely on the grounds that it may lead to the appointment of a receiver or liquidator. rong, even if it prevented a major disaster; witness the protracted debate on the evaluation of whistleblowing (Vinten, 1994), where accountancy bodies for a long time held that concerned employees may raise their concern to superiors but must not under any circumstances report to outsiders without authorisation. A strong deontological emphasis has the disadvantage that compliance with rules is taken to be moral. â€Å"One feels as an accountant†, said a participant in a study by Vyakarnam et al. (1996, p. 159) that â€Å"there are so many rules and regulations that anything outside is acceptable. A consensus to work just above the required minimum can lead to a situation where the minimum becomes expected and pressure increases to drop standards further. 3. Virtue ethics In contrast to the universal emphasis on moral duty in deontology and on general happiness in utilitarianism, Aristotle emphasises the importance of a person’s character for morality He suggests that the highest . human good is happiness, not in a crude material sense, but in a comprehensive meaning which carries connotations of ? ourishing and well-being.This highest good is closely linked to the function of a human being, which is to obey reason, as this is the mai n characteristic to set humans apart from other living beings. As a good ? autist plays the ? ute well or a good knife cuts well, Aristotle argues, so a good human is good at applying reason. Acting according to good reason is the distinguishing feature of virtuous behaviour. Reason helps to avoid both excess and de? ciency; so the virtue of courage shows the healthy mean between cowardice and rashness. To acquire this kind of virtue, people need practical wisdom, which can only be acquired by experience and habituation.Aristotle (1985, trans. Irwin) de? nes that â€Å"the virtue of a human being will †¦ be the state that makes a human being good and makes him perform his function well† (1106a20-24). Virtue ethics distinguishes between internal and external rewards, a distinction utilitarianism cannot make. Internal goods emerge from speci? c practices. They can only be experienced after a long engagement in the practice and their achievement bene? ts the whole of the co mmunity External goods are . not uniquely related to any practice; they are an individual’s property and are objects of competition.Thus, when Turner revolutionised the painting of sky and clouds, he created an internal good, irrespective of This clear, deontological view is necessitated by the public role of the accountancy profession which requires it to place above any other the public interest in being informed of the auditors doubts about the ability of the company to continue trading. There may be individual cases where an auditor, after considering the consequences of a quali? cation, would want to give the company a clean account; the more since a quali? cation is only a weak indication of business failure.The case of BCCI, Moizer (1995, p. 429) argues, has shown the danger in taking such an act-utilitarian approach. BCCI’s auditors Price Waterhouse considered qualifying the accounts in April 1990 but reasoned that auditors â€Å"owe a duty to shareholders to consider very carefully the possible impact of their report† because a quali? ed report would have more dramatic consequences for a bank than for an industrial concern. BCCI’s accounts were not quali? ed, and the bank was able to trade for another 14 months before it ? nally collapsed in July 1991. Moizer (1995, p. 30) concludes that the profession’s Code, which is based on either deontological or rule-utilitarian approaches, must be followed without regard to the particular situation; auditors ought not to consider the consequences of their actions, â€Å"since they have already been evaluated for the profession as a whole†. Deontological ethics is uncompromising by de? nition, but the complete disregard for circumstances can lead to morally dubious requirements. Few would follow the Kantian notion that telling a lie should always be [ 503 ] Lutz Preuss On ethical theory in auditing Managerial Auditing Journal 13/9 [1998] 500–508 he external goods, such as fame and income, he attained too (MacIntyre, 1985, p. 190f. ). Virtues and accounting Virtue ethics considers both intention and outcome, where duty-based ethics and utilitarianism only see one; it also links morality with self-interest. Principle-based ethics rarely ever gives unambiguous advice; it faces counter-examples, sometimes its conclusion runs counter to moral intuition. Virtue ethics, on the other hand, recognises that there are no easy answers and stresses the importance of practical wisdom in dealing with a moral dilemma.Mintz (1995, p. 259) argues that: virtues enable accounting professionals to resolve con? icting duties and loyalties in a morally appropriate way because they provide the inner strength of character to withstand pressures that might otherwise overwhelm and negatively in? uence professional judgment in a relationship of trust. He sees two virtues as having particular importance for the accounting profession, integrity, which enables the auditor to maintain objectivity under competitive pressure and trustworthiness, which ensures public con? ence in a professional service. The notion of virtuous behaviour in accounting is again linked to the discussion of the neutrality of accounting information. The neutralist perspective (Solomons, 1991) sees a clear distinction between the accountant as accountant and as citizen, and only in the latter capacity can the person legitimately express concern over social issues. Tinker (1991, p. 305), however, sees a social world where roles are inextricably intertwined and con? icting, and where the individual needs to develop a social selfconsciousness for transcending con? cts. The same accounting individual often appears on several sides in the same dispute, and without self-awareness about her role interdependencies, may ultimately contribute to her own repression and exploitation! wisdom, which differs from the routine application of rules, such as those by the professional bodies. Fur thermore, the use of computer-aided decision models does actually remove the possibility of developing virtuous behaviour. The main drawback of virtue ethics lies in its relativism (Hartman, 1994).Aristotle’s de? nition of virtue as a mean between two extremes makes sense to all communities, but only in a formal way The communities decide . what the two extremes are, and by this what the mean is. Hence, there is no neutral objective standpoint from which a good community can be distinguished from bad ones. Most humans belong to a number of communities, the community of their employer, their family, a sports club. There can be diverging de? nitions of community, which lead to diverging conceptions of the required virtues.A management accountant may take her employer to be her community and rate loyalty to it higher than loyalty to the general public. There are also practical problems with virtue ethics. Stressing the importance of character and practical wisdom does not alread y give concrete advice. The focus on character also neglects power distribution in organisations; it may actually disguise power structures and prevent change. The rich language of virtue ethics also lends itself for PR exercises. 4. Ethics of care The ethics of care has been developed as a feminist critique of the traditional moral philosophy on the basis of rights and rules.Traditional ethics is grounded in a view of others as potentially dangerous; thus rights become an important means to underscore claims against others and rules are needed to settle con? ict. The moral responsibility arising from both rights and rules is universal; it binds all moral agents equally (cf. the principle of utilitarianism or Kant’s Categorical Imperative) but makes the individual person replaceable without any loss to the ethical principle. Feminist authors have contended that the ethics of rights is essentially a male perspective and re? cts male dominance in western society and thought. Gi lligan (1982) argues that female morality does not centre on abstract principles but contextualises moral responses by drawing on personal experiences. It focuses on adequate responses to the needs and concerns of close individuals. Feminist philosophers see the self as being determined by its relationship with others Francis (1990) sees three main obstacles for more virtuous accounting. First, the relationship between internal and external rewards is slanted heavily towards the latter. The virtue integrity may be compromised by the bene? s of retaining a client. Secondly, the organisation of accounting into a small number of large private companies may hamper the development of virtuous professional norms. Under the prevailing competitive pressure the profession has demonstrated a lack of solidarity which has manifested itself particularly in auditor switching. Thirdly, virtues require the application of practical [ 504 ] Lutz Preuss On ethical theory in auditing Managerial Auditin g Journal 13/9 [1998] 500–508 (Noddings, 1984). Instead of bargaining between rational agents and rule-based settlement of con? ct, an ethics of care focuses on respect for others; maintaining the relationship is valued higher than scoring a victory or exercising one’s right. Ethics of care and accounting An ethics of care has been applied both directly to the accounting profession and in the concept of a caring organisation. Burton and Dunn (1996, p. 139) suggest that a company could be said to care if â€Å"it exhibits caring behaviour consistent with ? rm policy†. A company is here seen as a secondary caring agent, dependent on the primary caring actions of its organisational members. Liedtka (1996) ? ds that much of the present rhetoric about caring for customers or employees is just â€Å"care-talk† but suggests a caring organisation can be built. Apart from having caring employees, organisational support is crucial, because the organisation largely shapes the person’s role within it and must also provide the resources for caring. Employees would be seen as central, because they are the people who deal directly with customers and thus ultimately determine the success or failure of the business. Reiter (1996) suggests that caring is a valuable trait for employees, especially in the service sector.A caring ? rm could enjoy a competitive advantage in capability-driven markets, because it engenders trust and reduces transaction costs. Reiter (1997) claims that an ethics of care can foster a better understanding of the underlying principles of the accounting profession, such as auditor independence. This independence is to be achieved by a number of detailed rules which determine the relationship between auditor and client, yet true mental separation from the client would require the auditor to work in a social vacuum, and the rule-based approach can be seen as arbitrary The contextual perspective . f the ethics of care, Reit er suggests, provides the alternative metaphor of interdependence. This allows de? ning an appropriate balance of interests, on which users of accounting information could rely more than on the appearance of independence. Such thinking in? uenced the framework approach by the ICAEW (Maurice, 1996, p. 43). In accounting education, an ethics of care provides an important balance to rightsbased approaches. Reiter (1996, p. 48) discusses training material compiled by the American Accounting Association and Arthur Andersen and ? ds a â€Å"tendency of the AAA approach to frame con? icts as moral dilemmas where the choice is between resignation and hopeless compromise of integrity†. A win-win-situation, and above all a positive learning effect for accounting students, may be more likely to come from a care perspective, which â€Å"focusses on understanding others’ situations and points of view and determining what can be done to maintain appropriate relationships between th e self and others† (1996, p. 48). An ethics of care can also impact on accounting research (Oakes and Hammond, 1995), e. g. y challenging the possibility of a disinterested neutral scholar and thus asking how the researcher’s experiences and perspective in? uence the choice of research question. The interconnectedness of accountant and society raises the question of how current accounting practice affects the lives of people, especially the economically disenfranchised. This in turn challenges the assumptions regarding economic behaviour. The contextual approach of an ethics of care introduces a relativist element. This is not merely a practical problem for the accounting profession, in that accounting information may become less comprehensible to outside users.It indicates a deeper philosophical ? aw in the application of an ethics of care to economic life. Burton and Dunn (1996, p. 142) considered the dilemma of a company wanting to dispose of lead acid batteries. Sho wing care for those that are close – its local community – the ? rm decides to recycle the batteries rather than dumping them. The dangers involved in handling the batteries suggest they are better not handled by workers in the US, again the company displays care for close stakeholders. But would recycling in Taiwan be a moral alternative? Burton and Dunn attempt to rescue the model by introducing â€Å"A hybrid approach, ecommending that special attention be given to the least advantaged members of the moral community†, e. g. that the ? rm applies the same employee protection measures in Taiwan as it would in the US, even though this is not mandatory However, exempting the least . advantaged stakeholder shifts the burden onto the second least advantaged, until they turn into the least advantaged and get exempted, etc. The relativist problem leads to a further paradox: if others are all important in the de? nition of the self, then the self does not exist indepe ndently of these others.Wicks et al. , (1994, p. 483) de? ne a company as â€Å"constituted by the network of relationships which it is involved in†, yet as Burton and Dunn (1996) observed, this is close to the view of transaction cost economics. [ 505 ] Lutz Preuss On ethical theory in auditing Managerial Auditing Journal 13/9 [1998] 500–508 5. Compound models of ethics So far ethical theories have been considered in isolation from each other. Unsurprisingly in view of the complex nature of moral issues, none offers a completely satisfactory solution to moral problems in accounting.However, an unsystematic application of ethical principles carries the danger of eclecticism, of a pick-and-mix ethics, where an agent could select the ethical approach most suitable to furthering his own aims. A more secure theoretical foundation can be found in compound models. Brady (1985) suggested a â€Å"Janus-headed† model of ethics, further developed in Brady and Dunn (1995), which combines deontological and utilitarian requirements and proposes that an action should only be carried out if it violates neither.Kantian deontology and utilitarianism should be seen as complementing each other; Moizer (1995) found them to be the two dominant ethical theories in accounting rule making. A particular advantage of this compound model lies in its capability to capture both the universality of a situation, in the universalisability of Kantian deontology, and the particularity, via the utilitarian principle, for which, in order to calculate the greatest good of the greatest number, one must know the particulars of the situation (Brady nd Dunn, 1995, p. 394). Cavanagh et al. (1981) suggest a different version of a compound model, where an act would be unethical if it violated any of the following principles: (a) to optimise utility for all stakeholders involved; (b) to respect the rights of the individuals concerned; and (c) to be consistent with norms of justice. L ater work by the three authors (1995) enlarged this model by asking (d) whether the act arises from an impulse to care, to a utility-rightsjustice-care model.The decision-maker should also take account of clashes between principles, as well as of overwhelming or incapacitating factors, which would â€Å"justify overriding one of the ethical criteria† (1981, p. 370). Cavanagh et al. (1995, p. 399) argue that the advantage of their model lies in its practicality; it does not require business decisionmakers to handle abstract ethical principles but translates them into familiar norms, which then can be applied to concrete situations. Both models have had considerable in? uence in the business ethics debate; they have been applied in case studies and business ethics textbooks.The utility-rights-justice model was even chosen by Arthur Andersen for its business ethics programme. The two teams of authors have spent a great deal of energy arguing the practical applicability or captur ing distinction of their respective models. However, the main problem lies in the legitimation of selecting components. The very existence of competing models and the enlargement of one (the authors also considered the inclusion of a ? fth element of virtue) raises doubt whether any universally acceptable composition of a compound model can be achieved at all. Cavanagh et al. 1981) made important concessions when they allowed for mitigating circumstances for decision-makers who are not fully in control of the situation. Furthermore, decision-makers, who have â€Å"strong and reasonable doubts about the legitimacy of an ethical criterion, can legitimately be excused from adhering to that criterion† (1981, p. 371). None of the two teams offers advice on how to resolve clashes between principles. Cavanagh et al. , (1981, p. 370) note that there are â€Å"no well-de? ned rules† for such clashes, they â€Å"can be resolved only by making a considered judgment concerning wh ich of the con? cting criteria should be accorded the most weight in the given situation. † While the necessity of making such caveats is perfectly understood, nonetheless, a certain amount of arbitrariness may result in the application of compound models. On the other hand, both models secure a reasonably complete â€Å"coverage of the ethical terrain in business decision-making† (Brady and Dunn, 1995, p. 386). Eclecticism in the selection of compound models still guarantees a comprehensive range of ethical principles, which would not be the case for eclecticism in the selection of individual ethical principles.It should also be noted that the utility-rights-justice-care model allows for only one criterion to be set aside, having â€Å"reasonable doubts† about more than one would not longer qualify as legitimate. 6. Conclusion The discussion set out from a realisation that Codes of Ethics in accounting are an important but in the end insufficient device for addr essing moral con? ict, thus stressing the need for moral development of the person. Individual ethical theories and principles have been discussed of which none emerges as an all-round favourite for giving advice in moral dilemmas.Yet between the Scylla of offering no advice at all and the Charybdis of eclecticism in the selection of ethical principles, a safe course can be steered under the guidance of compound models. These are composed of a number of competing principles and the conciliation of these in the [ 506 ] Lutz Preuss On ethical theory in auditing Managerial Auditing Journal 13/9 [1998] 500–508 person of the decision-maker assures a moral quality to the decision. In contrast to a trialand-error approach, the decision-maker would at least have a reliable starting point for the search for ethical guidance.References Aristotle (1985), Nicomachean Ethics, translated by Terence Irwin, Hackett Publishing, Indianapolis, IN. Bentham, J. (1789/1962), â€Å"Introduction to the principles of morals and legislation†, in Warnock, M. (Ed. ) Utilitarianism, Fontana, London. Brady, F. N. (1985), â€Å"A Janus-headed model of ethical theory: Looking two ways at business/society issues†, Academy of Management Review, Vol. 10 No. 3, pp. 568-76. Brady, F. N. and Dunn, C. P. (1995), â€Å"Business metaethics: an analysis of two theories†, Business Ethics Quarterly, Vol. 5 No. 3, pp. 385-98. Burton, B.K. and Dunn, C. P. (1996), â€Å"Feminist ethics as moral grounding for stakeholder theory†, Business Ethics Quarterly, Vol. 6 No. 2, pp. 133-47 Cavanagh, G. F. , Moberg, D. J. and Velasquez, M. (1981), â€Å"The ethics of organizational politics†, Academy of Management Review, Vol. 6, pp. 363-74. Cavanagh, G. F. , Moberg, D. J. and Velasquez, M. (1995), â€Å"Making business ethics practical†, Business Ethics Quarterly, Vol. 5, pp. 399-418. Dittenhofer, M. and Klemm, R. J. (1983), Ethics and the Internal Auditor, Institute of Internal Auditors, Altamonte Springs, FL. Dittenhofer, M. nd Sennetti, J. (1995), â€Å"Ethics and the internal auditor, phase II. A comparison of the 1983 and 1994 surveys†, Managerial Auditing Journal, Vol. 10 No. 4, pp. 35-43. Finn, D. W. , Munter, P. , McCaslin, T. E. (1994), â€Å"Ethical perceptions of CPAs†, Managerial Auditing Journal, Vol. 9 No. 1, pp. 23-28. Francis, J. R. (1990), â€Å"After virtue? 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Ib Pase Paper, Maths Hl

M10/5/MATHL/HP2/ENG/TZ1/XX 22107204 mathematics higher level PaPer 2 Thursday 6 May 2010 (morning) 2 hours iNsTrucTioNs To cANdidATEs ? Write your session number in the boxes above. ? not open this examination paper until instructed to do so. do ? graphic display calculator is required for this paper. A ? section A: answer all of section A in the spaces provided. ? section B: answer all of section B on the answer sheets provided. Write your session number on each answer sheet, and attach them to this examination paper and your cover sheet using the tag provided. At the end of the examination, indicate the number of sheets used in the appropriate box on your cover sheet. ? unless otherwise stated in the question, all numerical answers must be given exactly or correct to three significant figures. 0 0 candidate session number 2210-7204 14 pages  © international Baccalaureate organization 2010 0114 –2– M10/5/MATHL/HP2/ENG/TZ1/XX Full marks are not necessarily awarded for a correct answer with no working. Answers must be supported by working and/or explanations. In particular, solutions found from a graphic display calculator should be supported by suitable working, e. . if graphs are used to find a solution, you should sketch these as part of your answer. Where an answer is incorrect, some marks may be given for a correct method, provided this is shown by written working. You are therefore advised to show all working. Section a Answer all the questions in the spaces provided. Working may be continued below the lines, if necessary. 1. [Maximum mark: 4] The graph below shows y = a cos (bx) + c . y 4 2 x –2 0 –2 –4 2 4 6 Find the value of a , the value of b and the value of c . †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢ € ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦. 2210-7204 0214 –3– 2. [Maximum mark: 5] The system of equations 2 x ? y + 3z = 2 3 x + y + 2 z = ? 2 ? x + 2 y + az = b M10/5/MATHL/HP2/ENG/TZ1/XX is known to have more than one solution. Find the value of a and the value of b . †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦ †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚ ¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. 2210-7204 turn over 0314 –4– 3. [Maximum mark: 6]M10/5/MATHL/HP2/ENG/TZ1/XX In the right circular cone below, O is the centre of the base which has radius 6 cm. The points B and C are on the circumference of the base of the cone. The height AO ? of the cone is 8 cm and the angle BOC is 60? . A diagram not to scale O B ? Calculate the size of the angle BAC . C †¦Ã¢â‚¬ ¦Ã¢â‚ ¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚ ¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. 2210-7204 0414 –5– 4. [Maximum mark: 7] (a) (b)M10/5/MATHL/HP2/ENG/TZ1/XX Solve the equation z 3 = ? 2 + 2i , givin g your answers in modulus-argument form. Hence show that one of the solutions is 1+ i when written in Cartesian form. [6 marks] [1 mark] †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. 2210-7204 turn over 0514 –6– 5. [Maximum mark: 6] M10/5/MATHL/HP2/ENG/TZ1/XX Let A , B and C be non-singular 2 ? 2 matrices, I the 2 ? 2 identity matrix and k a scalar. The following statements are incorrect.For each statement, write down the correct version of the right hand side. (a) (b) (c) ( A + B ) 2 = A2 + 2 AB + B 2 ( A ? kI )3 = A3 ? 3kA2 + 3k 2 A ? k 3 CA = B ? C = B A [2 marks] [2 marks] [2 marks] †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. 2210-7204 0614 –7– 6. [Maximum mark: 5] M10/5/MATHL/HP2/ENG/TZ1/XX Find the sum of all three-digit natural numbers that are not exactly divisible by 3. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦ †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. 2210-7204 turn over 0714 –8– 7. [Maximum mark: 7] M10/5/MATHL/HP2/ENG/TZ1/XX Three Mathematics books, five English books, four Science books and a dictionary are to be placed on a student’s shelf so that the books of each subject remain together. (a) (b) In how many different ways can the books be arranged? In how many of these will the dictionary be next to the Mathematics books? [4 marks] [3 marks] †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦ †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦ †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. 2210-7204 0814 –9– 8. [Maximum mark: 6] M10/5/MATHL/HP2/ENG/TZ1/XX In a factory producing glasses, the weights of glasses are known to have a mean of 160 grams. It is also known that the interquartile range of the weights of glasses is 28 grams. Assuming the weights of glasse s to be normally distributed, find the standard deviation of the weights of glasses. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢ € ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦. 2210-7204 turn over 0914 â €“ 10 – 9. [Maximum mark: 6] Let f ( x) = (a) (b) 4 ? x2 . 4? x M10/5/MATHL/HP2/ENG/TZ1/XX State the largest possible domain for f . Solve the inequality f ( x) ? 1. [2 marks] [4 marks] †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã ¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã ¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. 2210-7204 1014 – 11 – 10. [Maximum mark: 8] The diagram below shows the graphs of y = that all intersect in the same two points. M10/5/MATHL/HP2/ENG/TZ1/XX 3 x ? 3 , y = 3 and a quadratic function, 2 3 x –3 Given that the minimum value of the quadratic function is ? 3 , find an expression for the area of the shaded region in the form a, b, c and t are to be determined. (Note: The integral does not need to be evaluated. ) †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚ ¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã ¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. ? t 0 (ax 2 + bx + c) dx , where the constants 2210-7204 turn over 1114 12 – Section B M10/5/MATHL/HP2/ENG/TZ1/XX Answer all the questions on the answer sheets provided. Please start each question on a new page. 11. [Maximum mark: 20] A plane ? has vector equation r = (? 2i + 3 j ? 2k ) + ? (2i + 3 j + 2k ) +  µ (6i ? 3 j + 2k ) . (a) (b) (c) (d) (e) (f) Show that the Cartesian equation of the plane ? is 3 x + 2 y ? 6 z = 12 . The plane ? meets the x , y and z axes at A, B and C respectively. Find the coordinates of A, B and C. Find the volume of the pyramid OA BC. Find the angle between the plane ? and the x-axis. Hence, or otherwise, find the distance from the origin to the plane ? Using your answers from (c) and (e), find the area of the triangle ABC. [6 marks] [3 marks] [3 marks] [4 marks] [2 marks] [2 marks] 12. [Maximum mark: 15] Casualties arrive at an accident unit with a mean rate of one every 10 minutes. Assume that the number of arrivals can be modelled by a Poisson distribution. (a) (b) (c) Find the probability that there are no arrivals in a given half hour period. A nurse works for a two hour period. Find the probability that there are fewer than ten casualties during this period. Six nurses work consecutive two hour periods between 8am and 8pm.Find the probability that no more than three nurses have to attend to less than ten casualties during their working period. Calculate the time interval during which there is a 95 % chance of there being at least two casualties. [3 marks] [3 marks] [4 marks] [5 marks] (d) 2210-7204 1214 – 13 – 13. [Maximum mark: 11] M10/5/MATHL/HP2/ENG/TZ1/XX Points A, B and C are on the circumference of a circle, centre O and radius r . ? A trapezium OABC is formed such that AB is parallel to OC, and the angle AOC ? is ? , ? ? < ? . 2 B C A r ? O diagram not to scale (a) (b) ? Show that angle BOC is ? ? ? . 3 marks] Show that the area, T , of the trapezium can be expressed as T= 1 2 1 r sin ? ? r 2 sin 2? . 2 2 [3 marks] (c) (i) Show that when the area is maximum, the value of ? satisfies cos ? = 2 cos 2? . (ii) Hence determine the maximum area of the trapezium when r = 1. (Note: It is not required to prove that it is a maximum. ) [5 marks] 2210-7204 turn over 1314 – 14 – 14. [Maximum mark: 14] M10/5/MATHL/HP2/ENG/TZ1/XX A body is moving through a liquid so that its acceleration can be expressed as ? v2 ? ? 32 ? m s ? 2 , ? 200 ? where v m s ? 1 is the velocity of the body at time t seconds.The initial velocity of the body was known to be 40 m s ? 1 . (a) Show that the time taken, T seconds, for the body to slow to V m s ? 1 is given by T = 200 ? (b) (i) 40 V 1 dv . v + 802 2 [4 marks] dv Explain why acceleration can be expressed as v , where s is ds displacement, in metres, of the body at time t seconds. Hence find a similar integral to that shown in part (a) for the distance, S metres, travelled as the body slows to V m s ? 1 . [7 marks] (ii) (c) Hence, using parts (a) and (b), find the distance travelled and the time taken until the body momentarily comes to rest. [3 marks] 2210-7204 1414

Management (Self-Analysis & Professional Development Plan) Essay

Management (Self-Analysis & Professional Development Plan) - Essay Example Before moving into the two sections, listed below are the set of tests that I have undertaken. Also, the results to the test have been attached in the appendix for reference purposes. Before I go into my self analysis, it is essential to get a brief idea about me. I am twenty seven years old and am originally from the Middle East. I have been brought up there and hence English has been my second language. I am a simple and down to earth person with a high zest for life and a learner. Based on all of the results that I obtained from the personality tests, I have got a clearer understanding of myself and I present this as below. My basic character is a good mix of several different aspects like being agreeable, Conscientiousness, and emotional stability. I have scored a total of 13, 12 and 12 for each of the above mentioned respectively. Based on this, it is clear that I have several qualities that will help me in my career. According to the tests, I am a person who is very agreeable, with a calm, trusting, responsible, dependable, imaginative and cooperative nature. I agree to this as by nature I have always been a very quite person, with simple hopes and desires and have always been aware of my responsibilities and my surroundings. I understand the need to be aware of the feeling of others around me. I have been brought up in a large family and hence, these traits have been present from childhood. I have learnt from my mistakes that it is essential to be enthusiastic about things, yet a reasonable and rational decision can only be made with a calm and composed mindset. A saying by George Herbert, reads, à ¢â‚¬ËœBe calm in arguing; for fierceness makes error a fault, and truth discourtesy’1.Hence I pay a lot of attention to being calm, composed yet enthusiastic about life and all its giving. The test to check my Jungian 16 – Type personality also portrays and speaks a lot about me. The results of this test are very accurate and it

Jubilee Debt Campaign Essay Example | Topics and Well Written Essays - 2000 words

Jubilee Debt Campaign - Essay Example There have been claims that these debts have been a continuation of the legacy of colonialism in the way they are given out without looking at the way in which they will be repaid. 1 The jubilee campaign has been a debt calling for eth cancellation of more the $400 billion which the developing nations owe to the G8 countries. It is a campaign that has taken a moral direction on the way these debts have been issued and the legality of these debts in light of the world unequal distribution system of the world. The campaign has been calling for the cancelation of the debts in light of the fact that while the whole nation continue to be paying for the debts, most of them are not used to build the interest of the while nation but rather they are channel to build the interest of the few individuals who are in the leadership of these countries. These debts therefore have a lot of negative impact on women and children, education, climate change, heath sector and the general welfare of the society. The campaign has realized some progress as some debts totaling to $88 billion has been cancelled off but there is a lot that is to be done. The main assertion of the campaign is that most of the debts can be attributed to irresponsible lending decision. Although the developing nations have also been blamed for the increased debt d... The paper look at the case of the helping the poor nation and the moral issues that surrounds. Then it will look at how the developing world has accumulated these debts and the role that developed nation has played. Finally it will set the agenda on what could have been done in order to reverse the situation. Debt as a continuation legacy of colonialism The history of debt in the developing nations has been described as legacy of great siphoning off of their resource by the international financiers. It has been described as unjust process which is used by the rich and developed nation in order to siphon the resource of the developing nations and to ensure that the developing nations live under the shadow of their will. It is a process that has been designed in order to perpetuate itself using a diabolical mechanism where debt replicates in an even greater scale to create cycle that can only be broken by the use of debt relief mechanisms. 2 There have been claims that the process of debt creation in the developing world has been result from unjust transfer to them the debts that they have been accumulated in a colonizing state. To support this argument there is a lot of evidences that shows that a massive $59 billion in form of external debts to the developing world was transferred to the independent state when they were given independence in 1960. The debts have been increasing at a great rate of about 14 percent since then as a result of the unilaterally set rate. This has led to high rate of increase of these debts. Even before the new independent state got time to organize their economies, there were their lenders knocking on the door to get their money back. In order to understand the cycle of debt well,

Reggae Music Essay Example | Topics and Well Written Essays - 1000 words

Reggae Music - Essay Example During the run-up to the elections, he arranged concerts where Eccles’ performed politically loaded numbers. The other Reggae groups that were also supportive of Manley were Dennis Brown, the Chosen Few and The Wailers. Let’s take, for example, Eccles’s â€Å"Rod of Correction†. The title of the song is a reference to the staff given to Michael Manley, as a gift, by Emperor Haile Selassie of Ethiopia. The Emperor was highly revered across the black diasporas and more so in the Caribbean. The biblical symbolism is noted in how the rod commonly came to be known as â€Å"Joshua’s Rod†, whereby making Manley the Joshua – the personification of the â€Å"good† according to the Old Testament. And just as Joshua brought down the walls of Jericho and led the Jews to the land of Canaan, Manley was projected to lead his fellow countrymen to peace and prosperity. There are other symbolic references too. The parable of the destruction of Sodom and Gomorrah and the story of the crossing of the Red Sea by the Israelites are the other references to the Old Testament. Manley is portrayed as Joshua (the Savior) and his incumbent opponent is taken as the Pharoah. Need less to say, such powerful biblical symbolism made a critical impact on the Jamaican society, which was highly religious, and made Michael Manley the Prime Minister. The people will later be disappointed with Manley and disillusioned of his promises of paradise are quite another story (Cooper 1996).

Literature review (2) major project Essay Example | Topics and Well Written Essays - 1500 words

Literature review (2) major project - Essay Example It shall also relate these studies to each other in terms of consistency across a variety of variables. The therapist-driven approach in respiratory therapy as described by the American Association of Respiratory Care (Des Jardins & Burton, 2006, p. 131) focuses first on delivering individualized diagnostic and therapeutic care to patients; second, on assisting the physician in evaluating the needs of the patient; third, on determining the indications for respiratory therapy and what mode would work best in providing quality health services to the patient; and lastly, on empowering practitioners in allocating signs-and-symptoms based algorithms for patient treatment. This type of therapy is based on the individual needs of a patient. The interaction with the therapist would primarily decide the treatment protocols for the patient. Each patient has unique qualities and unique needs that often ultimately influence the choices that would work best for patients. The therapist-driven protocols are also known for being specific respiratory care services which are implemented under the supervision of the respiratory care practitioners (Stoller, 1998, p. 408). It primarily seeks to implement care services for patients more specifically based on their needs, and administered by experts in respiratory care. The development of the care is collaborative and planned with the assistance of the other members of the health care team equipped with the necessary expertise as regards respiratory care. The collaboration takes into consideration all factors pertinent to the needs of the patient. More often than not, the therapist-driven approach extends to the respiratory consult service, also known as the evaluate-and-treat program. This is a program â€Å"in which respiratory care is directed by respiratory care practitioners based on a menu of protocols for individual respiratory care services† (Stoller, 1998,

William Shakespeares Macbeth Essay Example | Topics and Well Written Essays - 1250 words

William Shakespeares Macbeth - Essay Example Thus, there are spectacular evidences all through the characterisation and the plot of the play which suggest the importance of the Elizabethan culture in the representation of 'gender' in Macbeth. Therefore, a paradox of sexual confusion and gender roles is evident at the heart of the play. "'Macbeth, a virile warrior-hero, is at the same time an 'unfinished man' who murders because he has been convinced by his wife that only through violence will he achieve a state of heroism. His manhood, displayed in the utterly 'masculine' form of bloodshed, is not self-determined or innate, but rather infused into him by Lady Macbeth." (Sch'rkhuber, 2007, p. 75). Therefore, this paper analyses Shakespeare's representation of 'gender' in Macbeth and compares it with the representation of 'gender' in the 21st Century referring to current literature, media, and poetry. One of the central concerns of the critics of the play Macbeth has been the author's representation of feminism in the play which is noted for the reversal of gender roles. Thus, one finds that the playwright has been interested in representing females as dominating their counterparts and the major characters and themes of the play depict the difference in the sexual representation of males and females. Whereas both Lady Macbeth and Macbeth are obsessed by the same passion of ambition, the playwright represents its effect differently in males and females. Lady Macbeth, the chief representative of females in the play, evidently violates the natural law concerning gender roles and the character of Macbeth enhances the reversal of gender roles in the play. "Both Lady Macbeth and Macbeth are fired by the same passion of ambition. However, while Macbeth is presented as violating moral law, Lady Macbeth seems to violate natural law: she is unfeminine. By eliminating in herself and her hus band tenderness, pity and vulnerability to feeling - human qualities her culture tends to associate with women - Lady Macbeth becomes half man herself. Through his efforts to be supreme among all men, Macbeth re-enacts the ambivalence of his sexual identity: the more he tries to prove his masculinity, the greater his fusion with Lady Macbeth." (Sch'rkhuber, 2007, p. 75). Therefore, it is palpable to a careful reader of the tragedy that Lady Macbeth makes an essential attempt to assume masculine characteristics which can make herself a stronger person. In addition, she also disparages her male counterpart, Macbeth, by attacking his own masculinity. "Macbeth shows a reversal in gender roles. The play questions what constitutes masculinity and femininity. It is important to Macbeth to be seen as a strong, powerful man. Lady Macbeth taunts him and mocks his sexuality to force him to get the courage to kill Duncan." (Thacker, 2008). In a careful analysis of the representation of feminism and masculinity in the play, one recognises that there is an evident reversal of gender roles by the playwright which can be comprehended as an attempt to identify with the contemporary Elizabethan culture.

Project Management Essay Example | Topics and Well Written Essays - 2500 words - 9

Project Management - Essay Example There exists various ways in which the projects of PFI offer a better time value of money than the original traditional option of purchasing the asset and taking the entire responsibility of maintaining the asset. The operator of the private sector takes the fundamental risks which are involved in designing, operating and financing the asset hence the additional costs and damages that may be incurred thus the local authority is less likely to account for these costs. The local authority indirectly finances the depreciation costs of the asset thus the contractor has the capability of repairing and maintaining the asset in a good condition which is a major advantage over the traditional methods of procurement for it increases the efficiency and performance of the local authority (Hill, 88-126). Since the introduction of the PFIs in the mid-1990s as a tool of funding the private sector it has been very successful and it has most commonly used method to fund health and transportation services. The fundamental principle underlying the operation of the PFIs is the fact that risks and responsibilities that are connected with running the public service are delegated to institutions where they can be well managed. For instance procuring a new building for running a school using the PFI is very different from the traditional procurement method where since the Local Authority enters into an agreement and pays for the accommodation in the school building for a period of time. With reference to the traditional methods, the Local Authority uses funds that have been approved and provided by the central government but the building is designed by architects from the Local Authority however the private sector is limited to the bidding of the contract. The procurement process of the PFI is basically a negotiated process (Allen, 12-62). One of the major

User Centered Design Essay Example for Free

User Centered Design Essay The Liveperson website communicate their purpose very well because the website shows important elements that a website needs to make the website efficient. There are five key elements in order to that may be identified: The Liveperson website shows their company brandmark on the upper-left corner of the website in a normal size very easy to see by the human eye. In the same way the website has in the middle of the page an explicit tagline that reads: â€Å" Liveperson creates meaningful, realtime customer connections that help businesses increase conversions and improve consumers experience,† which summarizes what the company does. Also, the website shows in the middle of the page a photo slider, with their news and more important services to offer. Then, the website shows at the top and at the bottom of the page the main navigation bar, which includes links to other information found on the website. To finalize this point, the home page is different from the other pages on the site but maintains the same font, color palette and design, making the website very consistent. Communicating Information about the company. The second point of this analysis talks about the communicating of information about the company. The website communicate very well their information too. As the first point, the website shows many of the important elements of this point, which makes a website efficient. Among this point we have: The website homepage has a link called â€Å"About Liveperson,† which includes information elements such as: â€Å"our culture, our partner, investor relations, careers and contacts us.† In the same way, at the end of the website we can see the same link with the same information elements. Then, the website has in their main navigation bar a press room link, used for press purposes, showing the most important company headlines. Also, the website, has a privacy policy link at the bottom of the homepage, which include the privacy policy of the company, but does not include internal company information. Content Writing. The writing content of the website is very effective and doesn’t use clever phrases and marketing lingo, making the writing content very readable and easy to understand. Also, all the writing content is written in sentence case and uses a consistent capitalization. Archives and Accessing Past Content. The website doesn’t have an access to past content. Links. The links work perfectly, are different one from each other, and are very scannable. Also the site doesn’t use the word â€Å"link† to indicate links on the page. Navigation. The primary navigation is located in a highly noticeable place, grouping items for the same category link, and doesn’t use made-up words for category navigation choices. Search. The website doesn’t have a search box. Tools and task shortcuts. The website offer direct access to high priority tasks. It includes a chat room, that offers the opportunity to chat with a company representative. Graphic and animations. The website shows the graphic elements in an appropriate way. For example, it shows graphics with real content and displays them ina good size, avoiding watermarks and animation of critical elements of the page such as logo or tag line. Graphic Design. In the graphic design part, the website is very consistent, using the same color palette and two different fonts which works very well. Additionally, the background color and the font color make a highly contrast, making the content very readable and not over loaded for the human eye. The most critical elements of the website are very visible, using their brandmark in a judicious way. The website is presented in vertical scrolling. UI Widgets. The website shows widgets for print and to connect to social network sites such as facebook, twitter, youtube, google, and linkedln. Window Titles. In the windows titles, the website includes a short description of the site, and the windows titles on the website are less than eight words. URLs. The URL is shown in a good way, showing as http://www.liveperson.com. News and Press releases. The News and Press releases section shows the headlines in a succinct and descriptive way. It includes a summary of the headline story, and at the bottom of the story shows a link that reads, â€Å"For more information visit www.websitename.com,† to show the complete story headline. Pop up windows and staging pages. The website only allows one pop up window for the chat room. The chat room pop up window appears once, then users have the opportunity to close or leave open. Advertising. The website only make advertises of their products, making the website more professional. Welcomes. The website doesn’t use welcome messages on the site. Credits. The website doesn’t have a credit link, which makes references to the designers of the website. It only shows the awards that the site has won in the Press releases link. Page Reload and Refresh. The website doesn’t refresh the pages automatically. Customization. The website doesn’t offer customizations. Gathering Customer Data. The website only gather customer data in the chat room section. Fostering community. The website offers a link called customer community, which allows users to post in a blog comments and questions. Dates and Times. The website doesn’t show date and time. Stock Quotes and Displaying Numbers. The website doesn’t show stock quotes and or display numbers. Align decimal points when showing columns of numbers. The website doesn’t show columns of numbers.

Argument for the Use of Consensual Relationship Agreements (Cras) Essay Example for Free

Argument for the Use of Consensual Relationship Agreements (Cras) Essay Argument for the use of consensual relationship agreements (CRAs) in my current (or future) workplace Consensual Relationship Agreement (CRA) is a contract that both employees involved a romantic relationship sign to acknowledge that their relationship is voluntary and that they will abide by the policies of the company regarding anti-harassment and anti-discrimination. (Hellriegel Slocum ,2011). Office romance should be addressed in a more constructive and professional way just like behaviors such as attendance, dress codes, and non-disclosure agreements. If one can argue that it is perfectly acceptable for employees with personal challenges (such as death in the family, childbirth, sick family member) to receive assistance and, support in the form of personal time-off from work then it must be reasonable for other employees who become romantically involved to get some type of support from their employers. The risks of office romance can better be mitigated if the risk factors are known and managed. So, for example, lawsuits from employees who claim they were coerced into an office romance and disadvantaged by it would be avoided because employees signed the CRAs of their own volition. Allegations from other employees of real or imagined favoritism from a supervisor to a subordinate with whom he or she is romantically involved would be averted because HR would have already taken steps to ensure this does not happen. By creating a mechanism which provides specific and detailed guidelines for professional workplace behavior, HR professionals are able to educate and caution employees about appropriate or inappropriate workplace etiquettes. Privacy issues or lack thereof, as related to dating in the workplace would also be appropriately addressed with the employees involved so they cannot claim that were unfairly treated. Another compelling reason why one should argue for CRAs in their workplaces is that in light of the legal ramifications of the rights of employees in the workplace, CRAs become an expedient and prudent way to protect the interest of an organization and simultaneously provide safe and meaningful ways for employees to freely explore and express their sentiments for others who might feel the same way. Organizations spend a lot of time and resources on strategic planning and forecasting. The purpose of such planning is to position the organization to accomplish two major goals: stay abreast with, or ahead of, the competitive marketplace and change current processes and approaches that are not yielding desired, or expected results. These are proactive steps to assure progress and success. CRAs facilitate and enhance such strategic plans by staying ahead of a changing workplace in the modern age and ensuring that valuable human resources will not be lost and thereby jeopardize more important organizational goals and objectives. CRAs provide a framework for responsible behavior by employees, particularly, supervisors. It fosters a win-win work environment for employees who happen to be romantically involved with each other and for employers to provide guidance and direction for appropriate workplace behavioral expectations. By bringing the relationship into the open the employees involved would not feel pressure to go to great lengths to hide their relationship and could focus on being productive. Conversely, these employees would become pre-occupied with finding ways to hide their relationships if the company did not encourage workplace romance. Create a counter argument against the use of CRAs in your current (or future) workplace. Office romance must be forbidden in the workplace because it simply raises too many avoidable issues in the workplace. By its very nature, office romance predisposes employees involved to compromise their rational objective behaviors because their non-objective emotional senses tend to dominate their thoughts and actions. Romantic relationships are the result of spontaneous human behavior; they are not preconceived and therefore cannot be regulated or contained by contracts. Making CRAs a condition for employment is also not a good idea because, over time, the restrictions become unbearable and the parties involved find out that they are better off in different organizations where their actions and whereabouts will no longer be scrutinized by an assigned HR person. CRAs are tantamount to indirect employment agreement in that they indirectly impose restrictions on the individuals involved in the relationship and seek to protect the interest of the organization at the expense of the people who happen to be emotionally attracted to each other and choose to express those sentiments in an open and liberating way. CRAs simply become ineffective tools because the employees who even agree to sign them often find ways to secretly engage in behaviors that result in lapses in judgment because they are emotionally charged. In many respects, CRAs are intrusive and inherently designed to discourage workplace romance. By their very nature they put restrictions on a human behavior that is designed to be spontaneous and natural. Organizations are not equipped to handle issues of the heart and neither should they start now. Organizations need to focus on what they do best, increasing shareholder value and accomplishing other strategic goals and objectives. Discuss the ethical principles involved in the use of CRAs. There are many perspectives and convictions about what is, or is not ethical. However, there are no universally accepted principles and rules for resolving all ethical issues (Hellriegel Slocum ,2011). In a country with such great freedom of expression, one can only imagine the diversity of opinions and positions regarding the factors that drive people’s decisions and formulate their ethics. The range of such factors is as wide as the differences in people and this is reflected in the diversity of ethical principles. However, for the purposes of this paper only a handful of relevant ethics based principles Utilitarian, Professional Standards, Disclosure and Distributive principles will be considered: Utilitarian principle focuses on the magnitude, extent and impact of harm versus good of decisions that are made. The final decision is predicated on the weight of good versus bad – the greatest good for the greatest number (Hellriegel Slocum , 2011). On the basis of this principle, CRAs are believed to generate more good than harm for those who choose to engage in office romance. Disclosure Principle – This principle focuses on how most people in society will react to the details of a decision when it becomes public knowledge. Most organizations take social responsibility very seriously and will do their very best to garner positive public image. Consequently CRAs are viewed favorably by such organizations. Distributive Principle – This principle is predicated on fairness. The thrust of CRAs is a win-win arrangement for both the organization and the individual because most organizations generally perceive CRAs to be fair. Create at least one (1) other option besides CRAs that would address workplace romances. The only other option for workplace romances that may be plausible is for the organization to develop a set of policies based on the organization interest principle which focuses on the basis of what is good for the organization (Hellriegel Slocum , 2011). With this option, employees are expected to disclose any potential or actual conflict interest to the human resource professional within the organization. Conflict of interest covers a broad range of behaviors; however, this paper will focus on personal conflict of interest which stresses zero tolerance for discrimination and sexual harassment. This alternative approach to workplace romances would requires that an employee may not supervise someone with whom they have a close personal relationship such as anyone in their family, household or someone with whom they have or had a romantic relationship or other close personal relationship. Additionally, if one were to supervise someone even indirectly with whom they had one of the relationships described above, one must disclose the relationship promptly. Any acts of discrimination, sexual harassment or other harassment based on race, color, religion, age, gender, sexual orientation would be addressed under a zero tolerance policy.