This is am african history class, you can choose the topic Research Paper

This is am african history class, you can choose the topic - Research Paper Example They are responsible for leading scientific and intellectual breakthroughs in history. The world, in general, owes the Islam community a debt. In terms of industrialisation, the Islam played an intricate role in the process. They also had a central position to play in the evolution of the philosophical enterprise in the world. This paper will explain how these revolutions shaped the world. In the past, it was easy for scholars, mathematicians, economists, and researchers to draw their inspiration from Islam (Huff 73). Islam worked to open up the minds and hearts of everyone in the world. In thought and study, many scholars can attribute their success to the Islam faith and its growth and spread. It is believed that the Islam community in the late seventeenth century drew their philosophical strength from the Greek. This means that, the introduction of many inventions, for example, the telescope led to the growth of the Western curiosity on the influence of Greco-Islam revolutions. Th e West had a number of inventions and scientific breakthroughs. However, they did not draw their inspiration from anywhere as many civilisations did. Many individuals around the world believe that there was no scientific or intellectual breakthrough in the Islamic world. This is because, most of them believe that all that is Islam developed as a result of the Greek heritage. Without it, many believe that Islam would not have an avenue to exist on its own. Others also believe that, the Arabs cold not identify what the ancient hieroglyphics in their presence meant. This is not until the coming of a European who explored the ancient language and drew conclusions from his research (Abu-Rabi 145). The introduction of foreigners makes the issue of Muslims being intellectually capable of making strides in the world debatable. What these critics do not seem to comprehend is the fact that, every civilisation brings forth diverse worldviews on diverse matters. In the field of optics, the Musl im and Arab community have made significant strides (Huff 79). The most revered thing about the Muslim faith is the belief they exhibit the necessary aspects of knowledge and wisdom in life. This makes them refuse the influence brought on by many education and religious influences from around the world. The fear of foreign influences pushes them to create their own understanding on issues, which pertain to many aspects of life. Islam focuses on regulating the philosophical attitude of the world. This is by overarching the philosophy of embracing mankind regardless of their faith and cultural background. No faith in the world claims that taking another life in the name of religion is justified, and Islam is not the first. Islam embraced the power of science, even when Christianity shunned and condemned the likes of Galileo for the assumptions he made about the earth. This laid the basis for the growth and development of philosophy and science among the Islam community (Huff 81). In A frica, intellectual revolution opened doors for the much needed economic change. Islam can be the perfect ground to lay this intellectual foundation. With the political changes, it is possible to have many channels created by the Western world in the third world countries. In Africa, for instance, the introduction of madrasa ensured that children learned the importance of the different fields in the world. The European and African enlightenment was brought on by the light that Islam shone

Educational management and lifelong learning Essay Example for Free

Educational management and lifelong learning Essay The models of theory which have been introduced into the educational sector are many, and while they have distinct names and functions, several have the tendency of overlapping with others. Six major models are presented here, in the orders of formal, collegial, political, subjective, ambiguity and cultural (Bush, 2003). These models complex constructs that are characterized and grouped into these clusters based on their relationships to different factors within the institution in which they are identified. Classification of these models is based on their agreement in such areas as the goals of the organization, their relationship to the dynamic of structure, as well as their orientation regarding an institution’s interaction with its environment (2003). Adult or lifelong learners are proliferating within today’s educational system and the posture of an institution’s management has the capacity to influence the experience of these types of students. Lifelong learners have needs and concerns that diverge from that of traditional students and the ability of teachers, administrators and principals (or deans) to cater to these non-traditional needs depends not only on their willingness to do so, but also on the type of managerial model that is present within the educational institution. The following paper discusses the principles of each of these managerial theories of education and relates each to the experience of the lifelong learner. Formal Models The formal models of theories of educational leadership and management emphasize the elements of educational institutions that might be considered official or structural (Bush, 2003). These models are based on the hierarchical envisioning of organizations and on the idea that the objectives pursued and achieved by managers are done using rational means. The form or structure of the organizations endows the organizational or departmental heads with their authority based on the positions to which they have been assigned. The formal model comprises five basic models, all of which to some degree contain the following seven features. The first feature is the tendency toward a systemic view of organizations, with each element of the system possessing an identifiable and demonstrable link with the other. Within an educational organization, this is represented by the different departments and elements (professors, secretaries, teachers and students) that make up these departments. The prominence of a given structure within the organization is another feature that defines such formal models. The flow of this structure is usually identifiable through organized patterns, and the usual pattern within educational organizations—that of hierarchy—defines the third feature. This hierarchy of the different departments as well as the order of authority possessed by the members within each defines this structure. Teachers are, for example, subordinate to heads of department, who are in turn subordinate to principals, and so forth (Bush, 2003). The formal models also render schools as organizations that seek out goals, and these official objectives are usually adopted by those who subscribe to the organization (Beare, Caldwell, Millikan, 1989). The rationality of the processes that define managerial decisions within schools is also a feature adopted by the models within the locus of formal theories. Such decisions often involve the consideration of all alternatives before selecting the most feasible and profitable. Within these formal models, the authority granted to leaders and managers is considered to derive from the positions they hold. Therefore, their authority is only tenable as long as they remain within their position. Finally, formal models emphasize the role of the body that sponsors the organization (sponsoring body). This necessitates that the educational institution be accountable to this sponsoring body, which usually takes the form of governments. Formal Models and the Lifelong Learner Within the arena of lifelong learning, the normative and prescriptive nature of these theories possesses strengths and weaknesses. While the systemic and hierarchical nature of the models allow for adult learners to understand and follow the chain of command, the fact that students are often placed at the bottom of this chain proves problematic for the adult learner. A more collaborative and egalitarian setting would be more appropriate for an adult learner, as he or she is likely to possess authority in other areas of life and would be uncomfortable merely being submissive within this educational setting. Furthermore, since adult learners are more apt to take responsibility for their learning, their inclusion in the decision-making processes would be curtailed in a very rigidly structured model of education. The problems identified with the formal model address some of the issues that involve lifelong learners (Bush, 2003). The power distribution and structure denies the important role of the individual in making and influencing decisions. Adult learners are often consulted about programs that they are interested in pursuing, and often chart their own paths in the achievement of goals. Therefore, the goal-oriented nature of the formal model, while technically substantiated where lifelong learners are concerned, gives little acknowledgement to the relative autonomy of this type of student in formulating and achieving these goals. Much of the lifelong learning and continuing studies which takes place within this age does so via the internet. Within such an educational environment, the systemic and hierarchical nature of educational management is vastly altered. Classes are no longer taught by teachers in a teacher-student hierarchy. Rather, students involved in teaching themselves via materials merely uploaded by â€Å"teachers† or professors. Therefore, the idea that the power available within this organization resides at the top of the pyramid again is faulted. Also, the structure of the organization does appear to change into a more egalitarian one in order to accommodate the more responsible adult learner.

The United States of Emergency :: September 11 Terrorism Essays

The United States of Emergency We are living in an ongoing crisis, a long-term state of emergency. We are at war all over the world (U.S. government officials tell us that Al-Qaeda operates in or from 60 countries). We have partial martial law within our country, most obviously in airports. Escalation has taken over. Killing has escalated. Fear has escalated. This gives diversity consciousness a new resonance, a new bottom line. It gives people a renewed purpose. Diversity is the key to survival. So the logic is clear: diversity must be escalated. "They don't think like we do," said a U.S. terrorism expert. "I could never imagine myself diving a plane into a building and killing people." The response of some people shows their cognitive/imaginative threshold. Some people were conceptually blocked. "I can't imagine how they could do that," said many people after Sept. 11. When we define some people as deviant, we can't approach them in our usual way, it puts them beyond our reach, or else we would have to admit that their deviance is within our repertoire as well. The more different from our norm someone is, the more we are challenged to bridge the distance. When we live with great overlap and similarity, we don't practice making contact with "the farther reaches of human nature" (the title of a book by Abraham Maslow), our own or someone else's. We talked about outsiders and outcasts in connection with the School shootings of the past few years. When I was teaching "Lord of the Flies" to a high school English class, students were quick to dismiss a character as nuts. They accepted only a narrow band of normal. We write off, avoid, demonize, pathologize, those who seem different. We homogenize our mental/emotional/political environment. We reinforce our norms and convince ourselves we are the most normal AND the most special people in existence. Feminists are terrorists to a male-dominant world order. Someone who is considered deviant is by definition not accessible in the usual mode and is seen as a threat to one's stability and security. "A physician has to enter the consciousness and subjectivity of the patient or he's not a good physician," says Dr. Oliver Sacks. Dr. Jerrold Post, George Washington University psychiatrist, has been giving interviews in which he says the terrorists are not psychotic.

Physchological Effects of Alcholism

The Psychological Effects of Alcoholism There are many people who live a life that is painful and which they want to escape from. Due to this people will often turn to alcohol and abuse the privilege. When people think of the effects of alcohol, they think about the physical effects it has on the body. What is discussed less frequently, but in some cases equally damaging, is the effect alcohol takes on the mind. Alcoholism will take you on a one way road to failure, which leads to losing a career, family, and most importantly one’s self. Initially, alcohol will increase a person’s confidence. At social gatherings, a person will initiate more conversations, tell funny jokes and feel a sense of acceptance. Unfortunately this is all in their mind; what’s funny to them might really be rude and obnoxious to a sober person. An intoxicated person won’t see how their acting because the alcohol triggers the short release of dopamine. Consequently, the brain will rely on the alcohol to release dopamine for feelings of pleasure and happiness. After continued use of alcohol a person will more often make poor judgment. During their impaired state under the influence, they are likely to lie to people they care about, steal or engage in other destructive mental behavior. While intoxicated, your mind will trick you into believing that your actions are not that bad, leading to a carefree attitude. As a result, a person will have to pay the consequences in jail, the hospital or in a grave. Another downside of an alcoholic is the increased aggression. Violent, aggressive behavior is common among male and sometimes females. The loss of inhibitions, along with the destruction of brain cells caused by drinking can create abusive behavior in alcoholics. As a result, spousal abuse and street fighting are often triggered. There are more shelters opening up for women who are victims of spousal abuse. Often times the abuser will only strike their loved one when intoxicated. Most importantly, alcoholism affects your family and loved ones; the alcoholic is not the only one that gets emotionally scared. An alcoholic doesn’t realize the hurtful words they say, stay in the minds of their children and family. A child’s mind absorbs every detail and growing up in a verbally abusive environment can stunt their development. They have a good chance of growing up with learning and anger management issues.

Pure Hatred Essay

Introduction It is so depressing to say that hate, the most powerful of human emotions is still rampant in today’s world. Despite decades of struggles for civil rights, sad stories of hatred are still being told. A lot of individuals have to walk the streets of cities, the halls of schools and offices, and even the rooms of their own houses in fear. Around this world people are still being attacked because of their race, their sex, or their religion. In this new millennium, is it going to be possible to create a safer environment for all people? Can each country become the â€Å"Land of the Free†? Sadly, individuals and groups that espouse hate are still active in the country. The horrific events of September 11, 2001, and the terrorism that has followed in its wake have made it even more important now than in the past to understand the nature of hate. Given the overwhelming displays of hate currently being displayed in the world, we have a responsibility to seek an understanding of hate, its causes, and its consequences and how to combat it and achieve a culture of peace (Brenes & Du Nann Winter, 201; Brenes & Wessells, 124). Typical Definitions of Hate The typical formulations of hate, those by Aristotle, Descartes, Spinoza, Hume, and Darwin are notable for their contradictions. For Descartes (1694/1989), hate was an awareness of an object as something bad and an urge to withdraw from it. For Spinoza (1677/1985), it was a case of pain (sadness) accompanied by a perception of some external cause. For Aristotle (trans. 1954), the distinguishing phenomenological fact about hate was that it is pain-free (in addition to being incurable by time and striving for the annihilation of its object). Hume (1739-1740/1980) argued that neither love nor hate can be defined at all, because both are irreducible feelings with the introspective immediacy of sensory impressions. Darwin (1872/1998) also saw hate as a special feeling, one that lacks a distinct facial sign and manifests itself as rage. Hatred is causes of bitter sorrow. We find ourselves in repugnance and anger in the presence of one we hate. The joy of hate is being caused by the suffering, loss of power and reputation of the hated person. Shand (192) described hate as a syndrome, or a bundle of episodic dispositions united by a common emotional object or a common category of such objects. The key feature of such a syndrome is that a person may be legitimately characterized as having it without being imputed any corresponding episodic state. Modern Conceptions of Hate Sternberg (123) recently proposed that both disgust and contempt are special kinds of hate, â€Å"cold hate† and â€Å"cool hate,† respectively (see also Oatley & Johnson- Laird, 87, for a claim that hate is a derivative of disgust). Steinberg’s proposal is part of a broad theoretical typology based on the principle that, like love, hate can be characterized in terms of three action-feelings components: (a) intimacy (more precisely, the negation thereof), (b) passion, and (c) commitment. The feelings and actions associated with the first (negation of intimacy) component include revulsion-disgust and distancing, respectively. Fight-or-flight is the action pattern, and anger-fear are the feelings attending the passion element. The last (commitment) component involves an attempt to devalue the target of hatred through contempt. On the basis of this triangular structure, Sternberg posited a variety of hates. There is, for example, the already mentioned â€Å"cool hate,† composed solely of disgust, and â€Å"hot hate,† composed solely of the anger-fear combination. There are also â€Å"cold hate† (devaluation through contempt alone), â€Å"boiling hate† (disgust + anger-fear), â€Å"simmering hate† (disgust + contempt), â€Å"seething hate† (passion + commitment; also called â€Å"revilement†), and, finally, â€Å"burning hate,† which includes all three action-feelings components. True hate, he argued, is an emotion of intimacy, respect, and strength—†There can be no hatred in weakness† (Solomon, 326); he saw this equality of power as part of hate’s special mythology, ensuring that the antagonism involves an element of â€Å"mutual respect.† Though Solomon referred to hate as an emotion, the general affective construct that appears to fit best his own characterization of hate dynamics is that of a syndrome. Types of Hate Hate as an Emotion The hate as an emotion occurs based on the individual emotional experience. It is an emotion where people have to experience that affect the way they live. People come to hate other people whom have mistreated them. Hate that we learn as an Idea It is a long-standing hatred even of people they have never met, simply on the basis of belonging to groups in conflict or as an idea. Prejudice and Discrimination Prejudice is a negative attitude toward an entire category of people, often an ethnic or racial minority. People who have an obvious difference make prejudice easier. If you resent your roommate because he or she is sloppy, you are not necessary guilty of prejudice. However, if you immediately stereotype your roommate on the basis of such characteristics as race, ethnicity, or religion, that is a form of prejudice. Prejudice tends to perpetuate false definitions of individuals and groups. One important and widespread form of prejudice is racism, the belief that one race is supreme and all others are innately inferior. When racism prevails in a society, members of subordinate groups generally experience prejudice, discrimination, and exploitation. In 1990, as concern mounted about racist attacks in the United States, Congress passed the Hate Crimes Statistics Act. This law directs the Department of Justice to gather data on crimes motivated by the victim’s race, religion, ethnicity, or sexual orientation. In 2000 alone, more than 8,063 hate crimes were reported to authorities. Some 54 percent of these crimes against persons involved racial bias, whereas another 18 percent involved religious bias, 16 percent sexual orientation bias, and 11 percent ethnic bias (Department of Justice 2001a). A particularly horrifying hate crime made the front pages in 1998: In Jasper, Texas, three White men with possible ties to race-hate groups tied up a Black man, beat him with chains, and then dragged him behind their truck until his body was dismembered. Numerous groups in the United States have been victims of hate crimes as well as generalized prejudice. In the wake of the terrorist attacks of September 11, 2001, hate crimes against Asian Americans and Muslim Americans escalated rapidly. Prejudice is also happening against Arab Americans and Muslims who live in the United States (226). The activity of organized hate groups appears to be increasing, both in reality and in virtual reality. Although only a few hundred such groups may exist, there were at least 2,000 websites advocating racial hatred on the Internet in 1999. Particularly troubling were sites disguised as video games for young people, or as â€Å"educational sites† about crusaders against prejudice, like Martin Luther King, Jr. The technology of the Internet has allowed race-hate groups to expand far beyond their traditional southern base to reach millions (Sandberg, 105). Hate causes Violence Hate is the most powerful human emotion exists that causes violence. It is a disease like tuberculosis. It may infect others, but it inevitably destroys the hater, diminishing his humanity and perverting the purpose and promise of life itself.   A special case of ostensive formulation might be found in the concept of the so-called hate crime. Hate crimes can be defined as criminal offenses in which the defendant’s conduct was motivated by hatred, bias, or prejudice, based on the actual or perceived race, color, religion, national origin, ethnicity, gender, or sexual orientation of another individual or group of individuals. A more extensive definition can be found in the California Penal Code, which says that: â€Å"Hate crimes . . . means any act of intimidation, harassment, physical force, or the threat of physical force directed against any person, or family, or their property or advocate, motivated either in whole or in part by the hostility to the real or perceived eth nic background, national origin, religious belief, gender, age, disability, or sexual orientation, with the intention of causing fear and intimidation.† Hate crimes are not separate offenses, however, and it is important to realize that many types of felonies can be prosecuted as hate crimes. Hate crime laws, which have developed during the past decade or two, simply enhance or increase the penalties associated with serious offenses that fall into the â€Å"hate crimes† category. At the 1994 is typical of such legislation. The act provides for enhanced sentences where a federal offense is determined to be a hate crime. The federal Hate Crime Statistics Act, signed into law by then-President Bush in April 1990, mandates an annual statistical tally of hate crimes throughout the country. Data collection under the law began in January 1991. Yearly statistics show approximately 10,000 reported instances of hate crimes, including about a dozen murders. Most hate crimes (approximately 65 percent) appear to be motivated by racial bias, while religious hatred (15 percent) and sexual orientation (12 percent) account for most of the remainder. Many hate crimes that are reported fall into the category of â€Å"intimidation,† although vandalism, simple assault, and aggravated assault also account for a fair number of hate crime offenses. Notable in recent years has been a spate of church burnings throughout the south where congregations have been predominantly African-American. A few robberies and rapes are also classified under the hate crime umbrella in any given year. Hate crimes are sometimes also called bias crimes. One form of bias crime that bears special mention is homophobic homicide. Homophobic homicide is a term that refers to the murder of homosexuals by those opposed to their lifestyles.   Some hate crimes are committed by organized hate groups. According to the Intelligence Project of the Southern Poverty Law Center (457) organized hate groups operated in the United States in 1999. Another so-called â€Å"patriot† organizations, many with separatist leanings based on race or ethnicity, existed throughout the country. Some hate crime laws have not passed constitutional muster, often because they have run afoul of First Amendment concerns over free speech. In 1992, for example, in the case of R.A.V. v. City of St. Paul, the U.S. Supreme Court invalidated a St. Paul, Minnesota, city ordinance designed to prevent the bias-motivated display of symbols or objects, such as Nazi swastikas or burning crosses. Also in 1992, in the case of Forsyth County, Ga. v. Nationalist Movement, the Court held that a county requirement regulating parades was unconstitutional because it also regulated freedom of speech—in this case a plan by an affiliate of the Ku Klux Klan to parade in opposition to a Martin Luther King birthday celebration. Some writers have noted that statutes intended to control hate crimes may contravene constitutional guarantees if they: (1) are too vague, (2) criminalize thought more than action, (3) attempt to control what would otherwise be free speech, and deny equal protection of the laws to those who wish to express their personal biases. Examples of effective hate crime legislation can be found in a Wisconsin law that increases penalties for most crimes when the offender â€Å"Intentionally selects the person against whom the crime . . . is committed or selects the property that is damaged or otherwise affected by the crime . . . in whole or in part because of the actor’s belief or perception regarding the race, religion, color, disability, sexual orientation, national origin or ancestry of that person or the owner or occupant of that property, whether or not the actor’s belief or perception was correct.†Wisconsin’s penalty enhancement statute was upheld in the 1993 case of Wisconsin v. Mitchell. In that case, the United States Supreme Court held that Mitchell, a black man whose severe beating of a white boy was racially motivated, could be punished with additional severity as permitted by Wisconsin law because he acted out of â€Å"race hatred.† The Court called the assault â€Å"conduct unprotected by the First Amendment† and upheld the Wisconsin statute saying, â€Å"[since] the statute has no ‘chilling effect’ on free speech, it is not unconstitutionally overbroad.† In 2000, however, the Supreme Court, in the case of Apprendi v. New Jersey,struck down a New Jersey law that allowed judges to sentence offenders to longer prison terms for crimes motivated by racism or other bias. The law did not require that prosecutors prove to a jury that an offense was a â€Å"hate crime† under state law. Are there Any Cures for Hate? There is no magic bullet cure for hate. There are several possible steps, however. Indeed, Staub (240, 124) devised a program for intervening in cases of mass killings and violence (see also Veale & Dona, 147). At the very least, one can start by modifying negative stereotypes, which can be done with some success (Blair & Banaji, 219; Mackie, Allison, Worth, & Asuncion, 156). In general, people need to: †¢ understand the triangular nature of hate and its escalation with successive triangular components so that one can recognize its often subtle presence; †¢ understand how hate is fomented through stories, often by way of propaganda; †¢ understand how hate can lead to massacres and genocide through the translation of feeling triangles into action triangles; †¢ combat feelings of impotence with constructive rather than destructive responses, and act against hate and its consequences rather than stand by as passive observers, as the world so often has done; †¢ realize that passive observation and often attempts at reason enacted in the hope that hate-based massacres and genocides will go away are perceived as weaknesses and tend to encourage rather than to discourage violence; and †¢ combat hate with wisdom. There is no complete cure for hate. Cognitive comprehension of a destructive psychological process does not insulate people from experiencing it. But given the destruction hate has caused over time and geography, there is a need to understand it, its consequences, and ways to at least try to combat it through understanding and especially through action. Indeed, there are few areas of psychology for which it equally can be said that action speaks louder than words. Many of the ways of combating hate are the same that one would use in resolving conflict situations and achieving peace (Christie, Wagner, & Du Nann Winter, 238), including creation of win-win situations, building trust between groups, sharing information, each side asking questions of the other, generating multiple alternative options, and seeking understanding of groups to which one does not belong (Boardman, 149; Isenhart & Spangle, 259). Sometimes when a group communicates to the other the story of what its members have experienced, they can come to an understanding of each other that is not possible when people stay silent and fail to communicate (Albeck, Adwan, & Bar-On, 162). When wrongs have been committed, no solution may be possible unless both sides are willing to forgive (Azar& Mullet, 95). Building tolerance and creating a culture of peace and a society in which people share equally in rights and in participation in the society can go a long way toward resolving problems of violence and hate (Christie & Dawes, 2001; Miall, Ramsbotham, & Woodhouse, 199; Montiel & Wessells, 221). The question is whether people have sufficient good will to achieve this goal. Combating hate requires, first and foremost, taking responsibility for it, its perpetrators, and its consequences. Ultimately, the best way to combat hate may be through wisdom (Steinberg, 198). Intelligent people may hate; wise people do not. People like Mohandas Gandhi, Martin Luther King, Mother Theresa, and Nelson Mandela had the same human passions as any of us, but in their wisdom, they moved beyond hate to embrace love and peace. The balance theory of wisdom (Sternberg, 198) defines wisdom as the application of intelligence, creativity, and experience toward a common good by balancing one’s own interests with others’ interests and institutional interests over the long and short terms. By definition, wise people do not hate others because they care about the individual’s (or group’s) wellbeing as well as their own or that of their group. They seek solutions that embrace the legitimate interests of others as well as of themselves. Someone who cares about another’s interests and well-being cannot hate that person, in part because he or she cannot dehumanize that other. Schools typically teach children knowledge and to think intelligently. But they rarely teach for wisdom. Indeed, in many schools across the globe, they teach hate for one group or another. Ultimately, if society wishes to combat hate, its schools and institutions need to teach students to think wisely. They then will realize that hate is not the solution to any legitimate life problem. Indeed, it foments rather than solves problems. But to teach for wisdom requires wisdom, and so far, the possession of that wisdom is a challenge that many fail to meet, not because we cannot meet it, but rather, because we choose not to. It is to be hoped that, in the future, people will make the better choice—for wisdom rather than for foolishness and the hate that can arise from it. Conclusion To sum up, despite much recent attention to hate as a topic of discussion and intervention, there currently exists no generally accepted definition and cure of hate. More grievously, there is nothing approaching a consensus on how to delimit the domain within which such a definition would fall. Meanings of hate differ both across and within contexts. Thus, it remains unclear if different authors are indeed discussing or intervening against the same thing. The situation raises a number of questions: Why this cornucopia of meaning? How are psychologists to characterize the underlying disagreements? How they to decide which disagreements are are substantive and which are purely semantic? How are people to decide who is right and who is wrong? What would it mean to be right or wrong in this context? These are just some trying questions about hate, to which the answers are still unclear. But one thing is clear, definitely hate is not the answer and we have to control ourselves emotionally and change our minds for the better. WORKS CITED Albeck, J. H., Adwan, S., & Bar-On, D. Dialogue groups: TRT’s guidelines for working through intractable conflicts by personal storytelling. Peace and Conflict: journal of Peace Psychology, 8, 301-322, 2002. Aristotle. The rhetoric and the poetics o fAristotk (W. R. Roberts, Trans.). New York: Modern Library, 1954. (Original work written ca. 340 B.C.) Azar, F., & Mullet, E. Willingness to forgive: A study of Muslim and Christian Lebanese. Peace and Conflict: Journal of Peace Psychology, 8, 17-30, 2002. Blair, I. V., & Banaji, M. R. Automatic and controlled processes in stereotype priming. Journal of Personality and Social Psychology, 70, 1142-1163, 1996. Boardman, S. K. Resolving conflict: Theory and practice. Peace andConftict: Journal of Peace Psychology, 8, 157-160, 2002. Brenes, A., &. Du Nann Winter, D. Earthly dimensions of peace: The Earth charter. Peace and Conflict: Journal of Peace Psychology, 7, 157-171, 2001. Brenes, A., & Wessells, M. Psychological contributions to building cultures of peace. Peace and Conflict: Journal of Peace Psychology, 7, 99-107, 2001. Christie, D. J., & Dawes, A. Tolerance and solidarity. Peace and Conflict: Journal of Peace Psychology, 7, 131-142, 2001. Christie, D.J, R. V. Wagner, R.V. & Winter, D.D. 2001, Peace, Conflict and Violence: Peace Psychology for the 21st Century Upper Saddle River, NJ: Prentice Hall.,106, 2001. Darwin, C. (1998). The expression of the emotions in man and animals. New York: Oxford University Press, 1998. (Original work published 1872) Department of Justice. Hate Crime Statistics, 2000. Washington, D C: U.S. Government Printing Office, 2001a (Accessed October 16, 2002). Descartes, R. On the passions of the soul (S. Voss., Trans.). Indianapolis, IN, 1989. (Original work published 1694) Hume, D. A treatise of human nature. Oxford, England: Oxford University Press, 1980 (Original work published 1739-1740). Isenhart, M., & Spangle, M. Collaborative approaches for resolving conflict. Thousand Oaks, CA: Sage, 2000. Mackie, D. M., Allison, S. T., Worth, L. T., & Asuncion, A. G. (1992). The generalization of outcome-biased counter-stereotypic inferences, journal of Experimental Social Psychology, 28, 43-64, 1992. Miall, H., Ramsbotham, O., & Woodhouse, T. Contemporary conflict resolution. Cambridge, MA: Polity Press, 1999. Montiel, C. }., & Wessells, M. (2001). Democratization, psychology, and the construction of cultures of peace. Peace and Conflict: Journal of Peace Psychology, 7,119-129, 2001. Shand, A. F. The foundations of character (2nd ed.). London: Macmillan, 1920. Solomon, R. The passions. New York: Anchor Books, 1977. Spinoza, B.Ethics. In E. Curley (Ed.), The collected works of Spinoza (Vol. 1, pp. 408-617). Princeton, NJ: Princeton University Press, 1985. (Original work published 1677) Staub, E. (1989). The roots of evil: The origins of genocide and other group violence. New York: Cambridge University Press, 1989. Sternberg, R. J. Why schools should teach for wisdom: The balance theory of wisdom in educational settings. Educational Psychologist, 36, 227-245, 2001. Sternberg, R. J. A duplex theory of hate and its development and its application to terrorism, massacres, and genocides. Review of General Psychology, 7, 299-328, 2003. Veale, A., & Dona, G. Psychosocial interventions and children’s rights: Beyond clinical discourse. Peace and Conflict: Journal of Peace Psychology, 8,47-61, 2002.

The Secret River Shows Essay Example

The Secret River Shows Essay Example The Secret River Shows Essay The Secret River Shows Essay the Secret River shows that even people who are essentially good can do bad deeds. Discuss. The Secret River by Kate Greenville is a novel set in the 19th century, where William Thrill who is from the slums of London, is convicted to New South Wales, Australia for the term of his natural life. With his wife and children he experiences a harsh land and want for more than he ever thought he would be capable of having. However this land comes with the trouble of the unknown Aboriginals. Through this novel we are continually presented with the issue that good people can perhaps do ad deeds. But what defines the good people and the deeds to be wrong? Who are the characters that portray this? And are there any who rise above from this condemning sentence? Good people can do bad deeds is more of a fact to humanity then a question. We see this occur each day, with the presidents, prime ministers, supposedly good celebrities turning to things that are considered in our lives as wrong. Therefore it is only logical that it be incorporated in novels, which mimic life. However we must first ask ourselves what is good? And what are evil deeds? Such broad questions, and yet hen given scenarios, people quickly identify for themselves the difference between good and evil. Yet, the context of novel changes the readers views. Despite the fact that the main character, William Thrill, is a continuous stealer, Greenville presents his act of stealing as a physical necessity, and therefore he earns the sympathy of the reader. Later on when in Australia and he is once again picking up the bad habit of stealing, which our society would Judge as terrible, the reader is able to easily brush it off, and not see it as a horrid deed at all. Our views of the good people are manipulated and shaped by the author throughout the entire novel, and we find ourselves putting the view of society bad deeds being brushed off when our author presents them in good light. The minor characters were explored by the author, and then further by the reader in this area. Sarah Thrill, known as Sal for most of the novel, is Williams wife who is portrayed throughout the novel as a strong, great women. However because we only see the novel from the eyes of William Thrill, we do not know her as anything so terrible, for William dollied Sal from the beginning of the novel. However although she is so beautifully presented by the author, even when she steals for the first time, we see her commit an ugly deed whilst in Australia. This deed is committed through fear. We see her clip her child Johnny on the ear, and what occurred was that fear could slip unnoticed into anger, as if they were one and the same. (IPPP) This is one of the only incidents where we see Sal do something out of her good character that is a bad deed. The Aboriginals are a different case altogether. For although the main character does not always shed them in good light, they are reasonably good people o. We see that William begins to understand this after they burn some of the land, which later the grass grows up and the kangaroos approach. A very clever method to obtain meat. We read how they look after the land they call theirs, even though they dont have any buildings or fences, and that they are generally peaceful people. However because they have spears to protect them , may I remind you that the civilians had guns, and they resort to violence when they are not understood clearly enough by the civilians, they are yet another example of good people, doing bad deeds. However it is their bad deeds that are outweighed as worse than seems by the racial prejudice of characters in the novel. Yet it is because of the people that try to understand the aboriginals in this novel that how we are also able to sympathize over them. Another interesting character who shows this theme of good and evil is Willie Thrill, the first son of Sal and Will. As a child he is of course seen as innocent, or because he lacks understanding, neither good nor bad. However when the novel develops we see this rough boy who holds a racial prejudice towards the aboriginals, a boy who says that they should get the guns and shoot them now. A boy who we begin to be unsure of because of these cruel words coming from this mouth of a child. Despite that he isnt a very developed character through the book, he is yet another diverse example of the good doing bad, and perhaps even becoming bad. For we are our actions, are we not? The most observable character of this theme if of course, William Thrill. As previously mentioned he commits acts that would be considered bad, however he is continuously painted as a good character. The reader likes the way he finds the research in London unjust. The reader enjoys learning of his dream to become a respectable man, like Mr. Middleton, Gals father. Most of all, he utterly contradicts any convict stereotype; of these terrible men that did terrible deeds. When brought to Australia we still see him in good light, he continually reminds himself that he does not want to kill the aboriginals, or harm them at all. He begins to understand their ways and sees that they are not terrible as primarily presumed. However despite all of this, he commits deeds that are seen as corrupt. His character is also changed and developed. His complete and utter desire for the land, takes over him. His need to climb up the ladder of society, where each man [was] higher than one, lower than another (pop) leads him to commits deeds he never wanted to commit. Although he understands that he is in fact taking a land that is not empty, and belongs to the Aboriginals, he continues to be greedy. When he is given a small amount of power, such as when he is given two convicts to rule over, he treats them the way he would never want to be treated. He acts different around the blacks, with more superiority. When the blacks invade his hard worked corn patch, he turns to lenience, hitting the women and children that are stealing his corn. All this anger towards the blacks; for being in the land he desperately needed to call his own, for stealing the things he had worked so hard on, for making Sal want to leave the place even faster, is what pushes him to do the worst act in the whole novel. The massacre. In which Thrill is mostly bewildered, yet he still murders Whisker Harry, an elder aboriginal. After this incident we must evaluate whether we still see Thrill as good or not. Because if he is Just a bad person doing bad deeds, then there is

What young India wants Essays

What young India wants Essays What young India wants Essay What young India wants Essay Essay Topic: Clueless Ask not what your country can do for you; ask what you can do for your country. Distinguished Judges of deference, esteemed dignitaries, teachers and all my friends, warm greetings to you all. Here I am to share what I think What young India wants? The India is going through an upheaval and there are great uncertainties ahead. But it is a moot point whether Indian is affected Indias structural and political problems or mismanagement. Is this the young India? No and we must be strong enough to tell it bold and brave because we are the young India. The young, according to Pearl S. Buck, do not know enough to be prudent, and therefore they attempt the impossible, and achieve it, generation after generation. What is it that mfoung India actually wants? Before telling that let me tell you what are all the problems she currently faces. Anybody would simple tell illiteracy, poverty, income inequality, corruption and so on. But its time we stop talking about this material problem and really find out what is the mess. A clueless society, none to follow, none to lead, no political knowledge, politics deriding youth and a lot more. Arent these enough for a catastrophe of problems? k enough of problems. Come on lets think and then get on to it. At present if no role models then lets create role models. Nothing ever changes in seconds. Rome was not built in a day. It takes time. We are tomorrows officers, teachers, army, council and cabinet. Why we are the tomorrow. We are the young India. We know what we want and let us not wait for it. Let us create it. Why hate politics and leave it behind? Somebody has got to change the system and let it be us. Why not follow ourselves or why not lead? Or better why dont we move together? Just like Humpty Dumptys un-birthdays in Alice and Wonderland, which re celebrated throughout the year except on the day of the birthday, we want young people to participate in democracy all year around, not Just on the day of the elections and create a participative democracy rather than a representative one. A year or a two then we are a part of it. This is the ancient land where wisdom made its home before it went into any other country. It is the same India which has withstood the shocks of centuries, of hundreds of foreign invasions of hundreds of upheavals of manners and customs. It is the same land which stands firmer than any rock in the world, with its undying vigour, indestructible life. We are the children of such a country. I am here to speak to you today about some practical things. The problems in India are more complicated, more momentous, than the problems in any other country. Race, religion, language, government † all these together make a nation. The elements which compose the nations of the world are indeed very few, taking race after race, compared to this country. Here have been the Aryan, the Dravidian, the Tartar, the Turk, the Mogul, the European † all the nations of the world, as it were, pouring their blood into this land. Of languages the most wonderful onglomeration is here; of manners and customs there is more difference between two Indian races than between the European and the Eastern races. We are taught to service, to be brave and to work for victory. Well we know what young India wants, we know why and we know what to do. Then why wait? Today itself my dear friends swear in and work for a prosperous young India. Lets not forget our pledge we take power or better or even the best. Older people believe, for reasons I cant fathom, that youth equals stupidity, when youth is really an attribute to intellect, not a detriment and let us make believe in a greater young India.

How to Pronounce the French X

How to Pronounce the French X Generally, the letter x is used in one of two ways in French: by itself or in combination with c (xc). X: Soft Pronunciation The x is pronounced like the x in the English word fix [ks] in two locations: 1) in front of a  consonant  or 2) at the end of a word or syllable.  Listen   Examples: the letter xune experience experience, experimentla taxe tax Exception:  The x at the end of French  numbers such as  six  and  dix  is pronounced like an S. X: Hard Pronunciation Between two vowels in two different syllables, the x is pronounced [gz] as in exit. ListenException:  The x in  xà ©rà ¨s  is pronounced [gz]. Examples:   le xylophone xylophoneexact exact, correct Liaison  or  Enchaà ®nement In  pronunciations that involve  liaison  or  enchaà ®nement, x is pronounced like a z. XC: Soft Pronunciation In front of a  soft vowel xc is pronounced [ks], like the soft x  pronunciation in which the c is essentially silent. Listen to  excentrà ©. Examples: excellent excellentexcentrà © outlying XC: Hard Pronunciation In front of a hard vowel or a  consonant,  xc is pronounced [ksk], like the soft x plus a hard c. Essentially, they are pronounced individually according to their traditional rules. Listen to  express.   Examples:   excusez-moi excuse meexclamation exclamationexpress espresso

Analysis of theoretical explanations of the relationship between Essay

Analysis of theoretical explanations of the relationship between technology and society - Essay Example 2.1 Technological determinism is a reductionist theory which holds the assumption that a society’s technology plays a crucial role in driving the development of its social structure as well as the cultural values (McLoughlin 1999, 32). The social structures evolve as a result of adapting to the technological change. The theory also suggests that technology moves on its own course which is normally independent of the human direction. It is therefore regarded to as an autonomous system which ultimately permeates all other sub systems of a society. Technological determinism contributes to the conclusion that emergence of automated technology inevitably raises the level of skill as well as the autonomy of the work force. According to Webster (2002, 34), technological determinism is also a clear explanation of the relationship between technology and society which asserts that technology plays an important role in determining the social existence within a society. In most instances technological changes act as catalysts for societal change. The changes are either seen as literally outside the context of the society or metaphorically outside the society. In addition, the technological developments take place as a result of natural logic which is not socially or culturally determined. As a result, the developments enact change and social adaptation (McLouglin 1999, 39). 2.3 The reason as to why I think they are perfect examples is due to the impact they have on the society and their autonomy nature. Bearing in mind that technological determinism refers to technology as an autonomous system, the quotes support the idea behind that. For example, robots are slowly being assimilated into the systems of our homes and lives whereby they will play a central role later in the century just like both the internet and combustion do now. In reference to technological determinism, the revolution of

Market Model Patterns of Change Research Paper Example | Topics and Well Written Essays - 1000 words

Market Model Patterns of Change - Research Paper Example The business expansion that has characterized many companies in the industry has led the expansion in the size of large companies and eventual attainment of a competitive advantage over smaller companies. Currently, the industry is experiencing enormous changes as a result of changing customer needs, emerging markets, and entrance of new competitors, technological advancement, and the strict environment regulations due to substance emissions (Blum, 2009). The changes have affected the demand, value chains and markets that characterize the industry. This has led to the transformation of the market into a more mobile model that is more competitive. The industry players and the new entrants have been forced to apply new competitive strategies that will enable sustainable success in the competitive industry. Pattern of Change Both the emerging and mature markets in the automobile industry are characterized by risk and opportunities. The global automobile industries have engaged in price wars crating frequent capacity overlap and rivalry. Therefore, the oligopolistic market model has been transformed into a competitive model. The competitive pattern of change is evident in its production, international trade changes, and supply chain and pricing. ... However, the lowering of the trade barriers and advancements in globalization has changed the tradition whereby the production location decisions are influenced by the international competitiveness. The international competitiveness is a product of the level of different tasks within the industry due to the vertical disintegration of global production. This is a change from the traditional business norms whereby the global competitiveness was determined by the level of the industry. Variation in Production Table 1: Top 20 Motor Vehicle Producing Countries   China 18,418,876 United States 8,653,560   Japan 8,398,654   Germany 6,311,318   South Korea 4,657,094   India 3,936,448   Brazil 3,406,150   Mexico 2,680,037   Spain 2,353,682   France 2,294,889   Canada 2,134,893   Russia 1,988,036   Ã‚  Iran 1,648,505   Thailand 1,478,460   UK 1,463,999   Czech Republic 1,199,834 Turkey 1,189,131   Indonesia 837,948 Poland 837,132 Argentina 828,771   Italy 790, 348 Source: http://oica.net/category/production-statistics/ Table 2: Annual Change in Production in Automobile Industry Year Production Change 1997 54,434,000    1998 52,987,000 -2.7% 1999 56,258,892 6.2% 2000 58,374,162 3.8% 2001 56,304,925 -3.5% 2002 58,994,318 4.8% 2003 60,663,225 2.8% 2004 64,496,220 6.3% 2005 66,482,439 3.1% 2006 69,222,975 4.1% 2007 73,266,061 5.8% 2008 70,520,493 -3.7% 2009 61,791,868 -12.4% 2010 77,857,705 26.0% Source: http://oica.net/category/production-statistics/1999-statistics/ International Trade The international trade is characterized by the dominance from a small number of companies. However, the industry faces segmentation that has made it more regional than global. The industry is also characterized by cost pressures, overcapacity and low profitability

Analysis of the Value at Risk (VaR) of a Portfolio of 4 Shares Essay

Analysis of the Value at Risk (VaR) of a Portfolio of 4 Shares - Essay Example This research will begin with the introduction of Value-at-Risk (VaR) as an established method for measuring market risk is an element of the advancement of risk management. The relevance of VaR has been extensive from its early use in security houses to profit-making banks and business and from marketplace risk to credit risk. Subsequent to the foreword in October 1994 by the Risk metrics by JP Morgan, the VaR is an assessment of the worst estimated failure that a firm may bear over a stage of time that has been particular by user, under standard market circumstances and a specific level of assurance. This evaluation may be attained in various ways, by means of a numerical model or by Computer calculated models. VaR is a calculation of market risk. It is the highest loss which can happen by incurring N % confidence above the property period of n days. VaR is the predictable loss of a portfolio over a particular time stage for a lay down level of probability. For instance, if every d ay VaR is declared as  £100,000 to a 95% level of confidence and throughout the day there is simply a 5% probability, then the next day loss is better than  £100,000. VaR dealings the potential failure in market value of a portfolio by means of expected instability and correlation. The â€Å"correlation† is considered as the correlation that is present between the market value of diverse appliance in a bank’s portfolio. VaR is considered inside a given confidence gap, typically 95% or 99%; it seeks to compute the probable losses from a place or portfolio under various normal situations. The description of regularity is vital and is fundamentally a statistical conception that varies by the organization and by risk management system. Considering merely, the most frequently used VaR models suppose that the price of resources in the financial markets go behind a standard distribution. To execute VaR, all of a firm’s situations data must be meet into one centrali zed database. Once this is absolute, the general risk has to be designed by combined risks from specified instruments within the whole portfolio. The possible shift in each gadget (that is the single risk factor) has to be incidental from past every day price movements above a given examination period. For dictatorial purpose, this stage is at least one year. Hence, the data where the VaR estimates are supported must confine all appropriate daily market shifts over the preceding year.   VaR is simply a measure of a bank’s risk experience; it an instrument for computing market risk experience. There is no one VaR integer for a single portfolio, as diverse methodologies used for scheming VaR produced dissimilar results. The VaR number confines only those risks that can be calculated in quantitative terms; it does not confine risk exposure such as prepared risk, liquidity risk, regulatory risk or autonomous risk. Assumption of Normality: An allocation is explained as usual, if there is greatest probability that any examination of the populace sample will have an importance that is

Variable pay strategy and program for an online retailer Research Paper

Variable pay strategy and program for an online retailer - Research Paper Example Variable pay can at times be long termed (more than a year) while at times it can be short termed (less than a year). Incentives refer to pre determined standards or criteria used in award determination. Bonuses on the other hand are wards delivered to the people concerned at the end of a certain period. The period on the case of bonuses is determined by the subject judgment of the quality of the performance while a reward is warranted (Variable Pay and Performance Linked Incentives, 2010). Strategy An effective variable pay program is as result of a strategic variable pay strategy, this strategy acts as a guideline for a quality design and program administration. A strategic program has the following characteristics: the clearly fit in the organizational structure and are steers of the organizations success. For a strategy to contribute to the success any organization, it must be in line with the mission and vision of the organization concerned. The speed of a strategy does not brin g about and importance because most likely it may be speeding to the wrong direction. A quality and strategic program should target to meet the objectives of the organization by lying down the require plans which if properly and rightfully administered, it would take the firm to another level. The strategy of the program should be also sound enough to bring about sustainability of the firm putting in consideration all the core strengths of the organization. It is also critical for a strategy to be in a specific context. It must bring out feasibility where it suits with the culture of the organization and that of the human resource, this must consider the organizational environment and realities that exists. Integration must also prevail where the structure of the program suits in the structure of the specific organization including the human resource’s structure. The program must also be flexible enough to respond to appropriateness and changes within an organization (Variabl e Pay Plan Development, 2011). There are differences between other compensations and variable pay, this differences occur on the bases of benefits and base pay. Variable pay program does not fit in entitlement cultures, this is because it’s not career annuity where base pay increase and must be compensated trough earnings over the given period. In reference to financial perspective, variable pay is a variable cost but the benefits and base pay are naturally fixed. There are difficulties in administering variable pay, competitive measurements is on of these difficulties. Levels of variable pay vary dramatically unlike levels of base pays, this helps in transforming average of the prevailing market into a broad-range, other than a specific point. One of the major characteristic of these variable pays is that it can be combined with other components to evaluate the performance of different levels in the organization both long term and long term (Variable Pay and Performance Link ed Incentives, 2010). The most significant benefits associated with variable pay compared to other types of compensations it that cost related to planning can be combined with performance/revenues and therefore vary economically. In Alaska, there is an advantage of using this kind of system because when the world’s economy is open to large variation, it carries with itself competitive advantage because the situation disadvantages them

Measurement and Instrumentation LVDT sheet Lab Report

Measurement and Instrumentation LVDT sheet - Lab Report Example The device consists of a primary winding (P) and two secondary windings named S1 and S2. Both of them are wound on one cylindrical former, side by side, and they have equal number of turns. Their arrangement is such that they maintain symmetry with either side of the primary winding (P). A movable soft iron core is placed parallel to the axis of the cylindrical former. An arm is connected to the other end of the soft iron core and it moves according to the displacement produced. The LVDT is also used as a secondary transducer in various measurement systems. A primary transducer is used to convert the measurand into a displacement (Scholey et al.,1995). The LVDT is then used to measure that displacement. Examples are: The graph above shows the plot between the resulting voltage or voltage difference and displacement. The graph clearly shows that a linear function is obtained between the output voltage and core movement from the null position within a limited range of 4 millimeter. The results showed that there is linear relationship between displacement and voltage and this relationship continues till the maximum displacement of 40 mm, after that output voltage was decreased drastically towards negative side. The results revealed that there is very strong repeatability in LVTD as show in fig. (2). There is almost no difference in output voltage when the experiment was replicated second time. These results are supported by the findings of other researchers who reported the similar results. (Tariq et al., 2002 ; Wu et al., 2008 and Drumea et al., 2006). It is evident from the data that the instrument has high sensitivity, resolution and repeatability. Maintains a linear relationship between the voltage difference output and displacement from each position of the core for a displacement of up to 40 millimeter. Scholey, G. K., Frost, J. D., Lo Presti, D. C. F., & Jamiolkowski, M. (1995). A review of instrumentation

Leadership + Economics of Regulation + Ethics and Organizational Essay

Leadership + Economics of Regulation + Ethics and Organizational Structure - Essay Example a. Employees typically resist a change that they believe will take away something of value. A proposed change in technology, job design or structure may lead to a real or perceived loss of pay, power, prestige or company benefits. (Daft 381-383) The fear of personal loss felt by employees is perhaps the greatest obstacle to organizational change and since larger organizations have a greater number of employees who feel endangered the resistance to change is far greater than the one usually experienced in smaller organizations. b. Lack of trust and understanding also has a magnified impact in larger organizations. Employees very often do not understand the intended reason behind a change and very often mistrust the intentions behind it (Daft 381-383). Having a greater number of people in this case magnifies the air of mistrust and creates further roadblocks to change. c. The uncertainty that every change brings with itself also plays a major role in the resistance to change displayed by employees of a large organization. For employees change means uncertainty and that bring out in them a fear of the unknown. Employees worry about whether they will be able to meet the demands of a new technology or a new procedure (Daft 381-383). The greater the number of uncertain employees, the greater will be the resistance to change. Leadership plays a very important role in the design and the regular updating of an organization’s organizational architecture. It is the leadership style of the leader that determines the organizational logic and the whole purpose of an organizational structure update is to ensure that the organizational chart of the organization accurate reflects how the stake holders in the organization actually work together. So if the leadership style of the leader is autocratic the organizational architecture should reflect it and if the leadership style of the leader is democratic then the organizational architecture should reflect it as

Measurement and Instrumentation LVDT sheet Lab Report

Measurement and Instrumentation LVDT sheet - Lab Report Example The device consists of a primary winding (P) and two secondary windings named S1 and S2. Both of them are wound on one cylindrical former, side by side, and they have equal number of turns. Their arrangement is such that they maintain symmetry with either side of the primary winding (P). A movable soft iron core is placed parallel to the axis of the cylindrical former. An arm is connected to the other end of the soft iron core and it moves according to the displacement produced. The LVDT is also used as a secondary transducer in various measurement systems. A primary transducer is used to convert the measurand into a displacement (Scholey et al.,1995). The LVDT is then used to measure that displacement. Examples are: The graph above shows the plot between the resulting voltage or voltage difference and displacement. The graph clearly shows that a linear function is obtained between the output voltage and core movement from the null position within a limited range of 4 millimeter. The results showed that there is linear relationship between displacement and voltage and this relationship continues till the maximum displacement of 40 mm, after that output voltage was decreased drastically towards negative side. The results revealed that there is very strong repeatability in LVTD as show in fig. (2). There is almost no difference in output voltage when the experiment was replicated second time. These results are supported by the findings of other researchers who reported the similar results. (Tariq et al., 2002 ; Wu et al., 2008 and Drumea et al., 2006). It is evident from the data that the instrument has high sensitivity, resolution and repeatability. Maintains a linear relationship between the voltage difference output and displacement from each position of the core for a displacement of up to 40 millimeter. Scholey, G. K., Frost, J. D., Lo Presti, D. C. F., & Jamiolkowski, M. (1995). A review of instrumentation

Understanding developments Essay Example | Topics and Well Written Essays - 1250 words

Understanding developments - Essay Example Development can only realize its full potential through motivating the people, involving them in the projects and organisation at the grassroots levels. Citizens and the agencies have seen the benefits of including the people in the various projects to achieve success hence made it almost mandatory to seek participation in any project. Participation of the people was embraced due to the concerns that development was not only capital centered but also needed the approval of the local people for it to succeed. Participation was aimed at improving the distribution of growth and to reach out to even the lowest income groups of people and emphasise the importance of development to the entire populations. In the recent decades, participation has taken the form of democratisation where citizens are allowed to vote for what they want, and any project that they feel is best for them (Oakley, 1995). The easiest form of participation is in the political arena where everyone is given the equal right to vote for whoever they want and thus increasing the sense of responsibility. When the communities on the grounds are allowed to participate in the various projects, they are more likely to be successful. The idea of participation started as a way of reducing poverty among the people and increasing their influence over factors meant to benefit them. The increasing levels of poverty were a major concern, and it was believed that people were poor since they did not have any influence over the factors that determined are directly affected them (Dom, 2012). Therefore participation would enable them to have some control over the factors that affect them including access to resources that would help them to improve their standards of living. Decisions on development projects were in the past made by people who did not understand the needs of the poor and thus he projects would at most times be irrelevant to the people.

Peter Skrynecki Belonging Paper Essay Example for Free

Peter Skrynecki Belonging Paper Essay For Peter Skrznecki it is like being part of two cultures but at the same time not able to belong and connect to either one. In Migrant Hostel Peter reflects the unpleasant memories, he remembers at 4 years old, when he and his parents emigrated to Australia from Poland. He uses language techniques such as similes to create a sense of uncertainty and not belonging, because similes are not as certain as metaphors. He uses motifs of birds and pigeons, which is seen in my visual representation. For example, Nationalities sought each other out instinctively like homing pigeons this emphasises the instinctive need to find someone or something familiar in a foreign place, to feel a sense of comfort and belonging. At times he felt unity with his family but this is juxtaposed with the sense of wondering of which culture he belongs to, Poland or Australia? This is represented visually in my collage, with images showing not belonging to groups and being an outcast, there is only one image in the corner representing unity and belonging. This is contrasted by all the other images portraying not belonging and alienation. Which relates to how Peter feels about being a man of two different cultures. In the poem Feliks Skrynecki, Peter describes his father with emotive language on how he tries and fit in to a new culture My gentle father, kept paces only with the Joneses (idiom is used.) But his father still has a strong connection to his culture through his garden, which Peter describes as he loved his garden, like an only child the garden is symbolic and represents his home land and culture. This relates to the film Kite runner, when Amirs father takes soil from Afghanistan and puts it in a case before he crosses the border, the soil becomes symbolic for the fathers culture and treasures it till his death bed. In both texts land is used as a symbol of connection that both fathers show to their culture and homeland. Which relates to my visual representation with an image of a man holding soil from the land. Choices are a large part of belonging to a culture. Peter chooses to be disconnected and he doesnt try and learn about his own culture, which makes him not belong culturally. For example in the poem Feliks Skrzynecki Peter says I forgot my first Polish word but his father repeated it so he would never forget it but Peter choose not to remember it. He also doesnt try to fit in with a new culture, he chooses to be the black sheep which creates a barrier to belong. In kite runner an important quote was this is the real Afghanistan you have always been a tourist in your own country. This shows when Amir lived in Afghanistan as a child his father always protected him and made all of his decisions, which caused him not to know who he really was and never experience the real Afghanistan. But when Amir went to America he starting making his own choices and found out who he was and felt like he belonged there. Both Amir and Peter shows how significant decisions and choices are, because it is your own personal choice to belong to a culture.

Monter Inc. Movie Essay -- essays research papers

Reverse Monsterfication   Ã‚  Ã‚  Ã‚  Ã‚  Throughout the length of the movie, I was taken back to my childhood when there were monsters in my own closet. Over the years, the monsters have all died and been replaced by just as scary skeletons, so my closet is still full. However, to a young child monsters are still lurking in the shadows, and they still make the floor creak. The approach taken by the writers of this film is one of uniqueness and of originality. By successfully juxtaposing the situation between monster and child, the writers were able to confront a touchy subject head on. This was reached through a combination of differences represented by the monsters in the movie and between the child’s impression of monsters in real life. We all know that in real life, to a child, monsters can be very real, intimidating, and extremely terrifying. The monsters in the movie are shown to be emotional creatures, with feelings and concerns. Appropriately enough, the largest monster named Sullivan is be st friends with one of the smallest monsters named Mike. Sullivan the bear, being the largest and scariest of monsters, turns out to be the one with the largest heart and concern for the well being of a small human girl he names Boo. The movie also shows us that the monsters are actually at work, earning money. Not only are they at work, but they also have a society, relationships, and an apparent chain of leadership and authority. To top it off, the monsters are more s...

Lorraine Hansberrys A Raisin In The Sun Essay -- essays research pap

Conflict in A Raisin in the Sun 	In the play A Raisin in the Sun, the playwright Lorraine Hansberry depicts the life of an impoverished African American family living on the south side of Chicago. The Youngers, living in a small apartment and having dreams larger than the world in which the live, often use verbal abuse as a way to vent their problems. Many times, this verbal abuse leads to unnecessary conflict within the family. The most frequently depicted conflict is that between Walter and his sister Beneatha. Walter wants nothing more than to be a wealthy entrepreneur that can provide for his family, while Beneatha plans to go to medical school and become a doctor. Both characters are opposed to the others’ dreams. This opposition creates serious conflict within the Younger household, and specifically among Walter, Beneatha, and Mama. 	During the course of the play, conflicts between Beneath and her brother Walter are revealed. Walter thinks that his sister should be a mainstream woman and not have great dreams and ambitions for her life. "Who the hell told you you had to be a doctor? If you so crazy ‘bout messing ‘round with sick people - then go be a nurse like other women - or just get married an be quiet" (38). This passage shows that Walter is clearly a chauvinist, and does not believe in his sister’s desire to be a doctor. Similarly, Beneatha does not believe in Walters aspirations of becoming a rich entrepreneur, and thinks he is rather...

Project Management E-Mail Essay

Our team was chosen by Mr. Wendell Deirelein, VP, to analyze three projects and give a recommendation of what company, Piper Industries Corp., should invest in. Our recommendation will include our selected project, a description of the five phases of the project and the key deliverables. Piper Industries Corp needs a completed project that would generate revenue within 12 months for the Project Management Office’s. After much review and evaluation we have selected the Project Code name: Stargazer as the project for investment next week by Piper Industries Corp., The Stargazer project has a massive research, and development on their new widgets, the company has spent $450,000 on this product and an estimated to bring in $575,000. Stargazer will be implemented by using the feasibility study. The feasibility study shows that this project has been evaluated and analysis of the potential of proposed project. Although, this product is not in production yet an extensive investigation and research to support the process of decision- making. The strengths and weaknesses of this proposed venture, opportunities and threats to carry out this project for its success. The feasibility is the cost required and the values attain to be successful. The risk of completion this project on time is high which will generate revenue within the 12 months. Our team feels that Stargazer will be an innovative money maker for Piper Industries Corp. The five phases of project management need to be fulfilled for a successful project. We have reviewed each phase and found that the following in regards to the Stargazer  project. Project initiation was completed by the feasibility studies, this project was named and defined, sponsors and stakeholders are deciding if they want to participate in this project. During the project planning phase, Stargazer has developed a plan for cost, scope, time, quality, communication, resources, and risk. This project is this first of its kind in the industry, therefore Stargazer will be seen as a leader in the marketplace. In the Project Execution phase, the project deliverable is developed and completed. This phase along with the project monitoring and control phase go hand in hand and are performed simultaneously. In this phase status meetings, project development updates, status reports, performance reports and human resource development are completed. The stargazer project product life is forecasted to be seven years for this product. Measuring Stargazer’s project performance and progression, calculating key performance indicators for cost and time to measure the degree of variation if any and in which case corrective measures are done in the project monitoring and control phase. Stargazer project is forecasted to have an ROI of $300,000 for the first year; $550,000 the second year and $750,000 the third year. The final phase of the project management is the project closure, in this phase tasks such as making the delivery, relieving resources, reward and recognitions of all the team members and the termination of contractors that were involve in the project. Piper Industries Corp. will have a closure ceremony for the project we selected and recognize all team members that took part in the planning of Stargaze, Piper Industries Corp. will host a launch party to announce that we will invest in Stargazer’s new widget product to be introduced into the marketplace in June, 2015.

Getting Older Essay

Introduction Math and numbers are used in every aspect and in every part of our daily lives. Everything is numbers. This is no different with people. Humans start as newborns and, if they are blessed, become elders. Numerical order is also used in determining our ages. The longer you live the older you become and the more things change. Aging is a part of life that should be embraced gracefully. There are three parts of old age. There are young old, middle aged old, and elderly old ages. Each part of being a senior citizen has its advantages and disadvantages. We still have to acknowledge the fact that the increase in living elders is making a major impact on the political aspects of being an old person. It seems to be and so we will explore why that the more seniors that survive each year seems to correspond with the lowering amount of funds available to them and more. We will attempt to cover the mark this senior citizen boom is making in the world for elders and those others it may affect. â€Å"Aging—a process that begins at birth and ends at death—has implications not only for our own lives but also for every aspect of society.† There are several problems being brought on by the indefinite increase in the aging population in America as well as other places. Not only does it affect the capital that supports many of our elders but it also weighs on what happens with their retirement and if and how they should continue to work. Now more than ever elders find themselves having to work out of obligation to stay financi ally stable and be able to be taken care of by their own finances. Elders from a while back worked because they had the ability and strength to. They were not as out of shape or out of tune with life. They did not  consume as much and many processed foods as our generations have either which would explain the longevity. Depending on what your beliefs are the reason for the ever so shortening or longing of years could have something to do with spirituality as well as environmental conditions. The demographic transition theory shows how the population is changed throughout time.It is based on the study and documentation of changes in birth/death rates. These significant transitions Result from low-high birth/death rates to high-low birth/death rates. When the aging population begins to rapidly increase, it usually is because of less new-birth but also less death. Low birth rates are as a result of access to contraception, preference to smaller families, urbanization and education of women. What a person prefers is often their business and left to their discretion, but with less youth, there are less younger adults that are able to take care of the older generation that may be needing help. Again we discuss nursing homes and health care. This is to say†¦ Ultimately, there is nothing wrong with the elders surviving. However, we should remember that we have to keep an attitude and a balance of the population and the resources available to supply the population.. There are upsides to the increase in the aging population but there are also downsides. The pros could include the fact that elders need to be attended to more could open the door for young adults to find jobs, also being guided by successful elders who hold secrets to prospering is something no one can deny is very helpful, also elders hold to a sense of familiarity meaning that just when all the fads and trends calm down, elders are there to stabilize the air. As for the downsides of the aging population, the economy is not as able to grow properly because the work field is going down, also health care has increased costs as it relates to treatments and medical care that is always needed for old people, and to add with more machinated tasks being carried out by machines as we ll as younger workers those skilled to work in a trade or a craft are not needed. Elders are having to make a choice between personal care from loved ones or the impersonal care that comes from nursing homes, home health nurses, or extended care facilities. Sadly, because of the lack of wanting to burden family members and the like, elders usually choose care from the list of impersonal care and internalize the neglect felt from having to do so. Advantages and disadvantages of implications of an aging population must be examined so that plans of implementation can be properly executed and adopted to fix or balance the situation.  The impact the older population is making does not have to be the be-all end-all factor of what is to happen as predicted by many. Increased dependency is believed to be one of the heaviest burdens that will cause the elderly to cripple the economy but it does not have to be so. We will also use the aforementioned to discuss the actions that must be taken to address the issue of the aging population. As well as having to rely on the government to make sure that this problem that seems to have arisen is taken care of there are things that we can do to be effective as well. We can definitely make sure our elders don’t feel like burdens. When I was younger we were able to adopt a grandparent. They could have kids or grandkids but many did not have any that would visit often or at all. We  would be allowed to spend time with them every weekend and it would just brighten their day to hear about what we are allowed to do and what we would discuss with them. Something as simple as this wakes the elders mind up in the nursing homes, it gets them excited again about life and their mind off where they are. Of course more financial and secure measures need to be taken but with just a simple economical start we could very well being on the road to the discovery of what works. References Demography Is Not Destiny: The Challenges and Opportunities of Global Population Aging. Full Text Available Academic Journal Uhlenberg, Peter; Generations, 2013 Spring; 37 (1): 12-8. (journal article – case study) ISSN: 0738-7806, Database: CINAHL with Full Text Subjects: Demography; Aging; World Health; Forecasting Academic Journal By: Thomas, Kali S.; Mor, Vincent. Health Services Research. Jun2013, Vol. 48 Issue 3, p1215-1226. 12p. 2 Charts, 1 Graph. DOI: 10.1111/1475-6773.12015. The demographic transition: causes and consequences. Detail Only Available Academic Journal By: Galor, Oded. Cliometrica. Jan2012, Vol. 6 Issue 1, p1-28. 28p. DOI: 10.1007/s11698-011-0062-7. , Database: Business Source Elite Subjects: STAGNATION (Economics); ECONOMIC development; HUMAN capital; DEMAND (Economic theory); DEMOGRAPHIC transition; FERTILITY; MORTALITY The Implications of Increased Survivorship for Mortality Variation in Aging Populations Full Text Available Engelman, Michal; Canudas-Romo, Vladimir; Agree, Emily M.; Population and Development Review, September 2010, v. 36, iss. 3, pp. 511-39, Database: EconLit with Full Text Subjects: Health Production ; Demographic Trends, Macroeconomic Effects, and Forecasts ; Economics of the Elderly; Economics of the Handicapped; Non-labor Market Discrimination . The Relationship between Older Americans Act Title III State Expenditures and Prevalence of Low-Care Nursing Home Residents. Full Text Available Academic Journal By: Thomas, Kali S.; Mor, Vincent. Health Services Research. Jun2013, Vol. 48 Issue 3, p1215-1226. 12p. 2 Charts, 1 Graph. DOI: 10.1111/1475-6773.12015. Subjects: NURSING home patients; NURSING care facilities — Finance; MEDICAID — Finance; STATE Agencies On Aging; CAREGIVERS; UNITED States. Older Americans Act of 1965; Administration of Human Resource Programs (except Education, Public Health, and Veterans’ Affairs Programs); Nursing Care Facilities (Skilled Nursing Facilities); Community care facilities for the elderly Markson, E. W. & Stein, P. J. (2012). Social gerontology: Issues & prospects. San Diego, CA: Bridgepoint Education, Inc.

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