Passage: For achieving inclusive growth there is critical need to rethink the role of the State. The early debate among economists about the size of the Government can be misleading. The need of the hour is to have an enabling Government. India is too large and complex a nation for the State to be able to deliver all that needed. Asking the Government to produce all the essential goods, create all the necessary jobs, and keep a curb on the prices of all goods is to lead to a large cumbersome bureaucracy and widespread corruption. The aim must be to stay with the objective of inclusive growth that was laid down by the founding fathers of the nation and also to take a more modern view of what the State can realistically deliver. This is what leads to the idea of an enabling State, that is, a Government that does not try to directly deliver to the citizens everything that they need. Instead, it (1) creates an enabling ethos for the market so that individual enterprise can flourish and citizens can, for the most part, provide for the needs of one another, and (2) steps in to help those who do not manage to do well for themselves, for there will always be individuals, no matter what the system, who need support and help. Hence we need a Government that, when it comes to the market, sets effective, incentive compatible rules and remains on the sidelines with minimal interference, and, at the same time, plays an important role in directly helping the poor by ensuring that they get basic education and health services and receive adequate nutrition and food.
According to the passage: 1. The objective of inclusive growth was laid down by the founding fathers of the nation. 2. Need of the hour is to have an enabling Government. 3. The Government should engage in maximum interference in market processes. 4. There is a need to change the size of the Government. Which of the statements given above are correct?
- 1 and 2 only
- 2 and 3 only
- 1 and 4 only
- 1, 3 and 4
Explanation: The passage explicitly states that the objective of inclusive growth was laid down by the founding fathers and that the need of the hour is an enabling Government. It argues against maximum interference, advocating minimal interference in the market, and says the debate about the size of Government can be misleading rather than calling for a change in size.
Passage: For achieving inclusive growth there is critical need to rethink the role of the State. The early debate among economists about the size of the Government can be misleading. The need of the hour is to have an enabling Government. India is too large and complex a nation for the State to be able to deliver all that needed. Asking the Government to produce all the essential goods, create all the necessary jobs, and keep a curb on the prices of all goods is to lead to a large cumbersome bureaucracy and widespread corruption. The aim must be to stay with the objective of inclusive growth that was laid down by the founding fathers of the nation and also to take a more modern view of what the State can realistically deliver. This is what leads to the idea of an enabling State, that is, a Government that does not try to directly deliver to the citizens everything that they need. Instead, it (1) creates an enabling ethos for the market so that individual enterprise can flourish and citizens can, for the most part, provide for the needs of one another, and (2) steps in to help those who do not manage to do well for themselves, for there will always be individuals, no matter what the system, who need support and help. Hence we need a Government that, when it comes to the market, sets effective, incentive compatible rules and remains on the sidelines with minimal interference, and, at the same time, plays an important role in directly helping the poor by ensuring that they get basic education and health services and receive adequate nutrition and food.
According to the passage, the strategy of inclusive growth can be effected by focusing on
- Meeting all the needs of every citizen in the country.
- Increasing the regulations over the manufacturing sector.
- Controlling the distribution of manufactured goods.
- Delivery of the basic services to the deprived sections of the society.
Explanation: The passage emphasizes that the Government should play an important role in directly helping the poor by ensuring basic education, health services, nutrition, and food, which corresponds to delivering basic services to deprived sections.
Passage: For achieving inclusive growth there is critical need to rethink the role of the State. The early debate among economists about the size of the Government can be misleading. The need of the hour is to have an enabling Government. India is too large and complex a nation for the State to be able to deliver all that needed. Asking the Government to produce all the essential goods, create all the necessary jobs, and keep a curb on the prices of all goods is to lead to a large cumbersome bureaucracy and widespread corruption. The aim must be to stay with the objective of inclusive growth that was laid down by the founding fathers of the nation and also to take a more modern view of what the State can realistically deliver. This is what leads to the idea of an enabling State, that is, a Government that does not try to directly deliver to the citizens everything that they need. Instead, it (1) creates an enabling ethos for the market so that individual enterprise can flourish and citizens can, for the most part, provide for the needs of one another, and (2) steps in to help those who do not manage to do well for themselves, for there will always be individuals, no matter what the system, who need support and help. Hence we need a Government that, when it comes to the market, sets effective, incentive compatible rules and remains on the sidelines with minimal interference, and, at the same time, plays an important role in directly helping the poor by ensuring that they get basic education and health services and receive adequate nutrition and food.
What constitutes an enabling Government? 1. A large bureaucracy. 2. Implementation of welfare programmes through representatives. 3. Creating an ethos that helps individual enterprise. 4. Providing resources to those who are underprivileged. 5. Offering direct help to the poor regarding basic services. Select the correct answer from the codes given below:
- 1, 2 and 3 only
- 4 and 5 only
- 3, 4 and 5 only
- 1, 2, 3, 4 and 5
Explanation: The passage explicitly mentions creating an enabling ethos for the market, helping those who do not manage well, and directly helping the poor with basic services. It criticizes a large bureaucracy as cumbersome and does not mention implementation through representatives.
Passage: For achieving inclusive growth there is critical need to rethink the role of the State. The early debate among economists about the size of the Government can be misleading. The need of the hour is to have an enabling Government. India is too large and complex a nation for the State to be able to deliver all that needed. Asking the Government to produce all the essential goods, create all the necessary jobs, and keep a curb on the prices of all goods is to lead to a large cumbersome bureaucracy and widespread corruption. The aim must be to stay with the objective of inclusive growth that was laid down by the founding fathers of the nation and also to take a more modern view of what the State can realistically deliver. This is what leads to the idea of an enabling State, that is, a Government that does not try to directly deliver to the citizens everything that they need. Instead, it (1) creates an enabling ethos for the market so that individual enterprise can flourish and citizens can, for the most part, provide for the needs of one another, and (2) steps in to help those who do not manage to do well for themselves, for there will always be individuals, no matter what the system, who need support and help. Hence we need a Government that, when it comes to the market, sets effective, incentive compatible rules and remains on the sidelines with minimal interference, and, at the same time, plays an important role in directly helping the poor by ensuring that they get basic education and health services and receive adequate nutrition and food.
Why is the State unable to deliver 'all that is needed'? 1. It does not have sufficient bureaucracy. 2. It does not promote inclusive growth. Select the correct answer from the codes given below:
- 1 only
- 2 only
- Both 1 and 2
- Neither 1 nor 2
Explanation: The passage attributes the inability to India's large and complex nature, not to insufficient bureaucracy or lack of inclusive growth promotion. In fact, it argues for inclusive growth and warns against excessive bureaucracy.
Passage: For achieving inclusive growth there is critical need to rethink the role of the State. The early debate among economists about the size of the Government can be misleading. The need of the hour is to have an enabling Government. India is too large and complex a nation for the State to be able to deliver all that needed. Asking the Government to produce all the essential goods, create all the necessary jobs, and keep a curb on the prices of all goods is to lead to a large cumbersome bureaucracy and widespread corruption. The aim must be to stay with the objective of inclusive growth that was laid down by the founding fathers of the nation and also to take a more modern view of what the State can realistically deliver. This is what leads to the idea of an enabling State, that is, a Government that does not try to directly deliver to the citizens everything that they need. Instead, it (1) creates an enabling ethos for the market so that individual enterprise can flourish and citizens can, for the most part, provide for the needs of one another, and (2) steps in to help those who do not manage to do well for themselves, for there will always be individuals, no matter what the system, who need support and help. Hence we need a Government that, when it comes to the market, sets effective, incentive compatible rules and remains on the sidelines with minimal interference, and, at the same time, plays an important role in directly helping the poor by ensuring that they get basic education and health services and receive adequate nutrition and food.
What is the essential message being conveyed by the author of the passage?
- The objectives of inclusive growth laid down by the founding fathers of the nation should be remembered.
- The Government needs to make available more schools and health services.
- The Government needs to establish markets and industries to meet the needs of the poor strata of the society.
- There is a need to rethink the role of the State in achieving inclusive growth.
Explanation: The opening and closing of the passage emphasize rethinking the State's role to achieve inclusive growth through an enabling Government rather than direct delivery of everything, making this the central message.
Passage: The concept of 'creative society' refers to a phase of development of a society in which a large number of potential contradictions become articulate and active. This is most evident when oppressed social groups get politically mobilized and demand their rights. The upsurge of the peasants and tribals, the movements for regional autonomy and self-determination, the environmental movements, and the women's movements in the developing countries are signs of emergence of creative society in contemporary times. The forms of social movements and their intensity may vary from country to country and place to place within a country. But the very presence of movements for social transformation in various spheres of a society indicates the emergence of a creative society in a country.
What is the idea of 'creative society'? 1. A society where diverse art forms and literary writings seek incentive. 2. A society where social inequalities are accepted as the norm. 3. A society where a large number of contradictions are recognised. 4. A society where the exploited and the oppressed groups grow conscious of their human rights and upliftment. Select the correct answer using the codes given below:
- 1, 2 and 3 only
- 4 only
- 3 and 4 only
- 2 and 4 only
Explanation: The passage defines a creative society as one where contradictions become active and oppressed groups demand rights, matching statements 3 and 4. It does not mention art forms or accepting inequalities as norms.
Passage: The concept of 'creative society' refers to a phase of development of a society in which a large number of potential contradictions become articulate and active. This is most evident when oppressed social groups get politically mobilized and demand their rights. The upsurge of the peasants and tribals, the movements for regional autonomy and self-determination, the environmental movements, and the women's movements in the developing countries are signs of emergence of creative society in contemporary times. The forms of social movements and their intensity may vary from country to country and place to place within a country. But the very presence of movements for social transformation in various spheres of a society indicates the emergence of a creative society in a country.
What according to the passage are the manifestations of social movements? 1. Aggressiveness and being incendiary. 2. Instigation by external forces. 3. Quest for social equality and individual freedom. 4. Urge for granting privileges and self-respect to disparaged sections of the society. Select the correct answer using the codes given below:
- 1 and 3 only
- 2 and 4 only
- 3 and 4 only
- 1, 2, 3 and 4
Explanation: The passage describes movements where oppressed groups demand rights, indicating a quest for equality and self-respect for disparaged sections. It does not mention aggressiveness or external instigation.
Passage: The concept of 'creative society' refers to a phase of development of a society in which a large number of potential contradictions become articulate and active. This is most evident when oppressed social groups get politically mobilized and demand their rights. The upsurge of the peasants and tribals, the movements for regional autonomy and self-determination, the environmental movements, and the women's movements in the developing countries are signs of emergence of creative society in contemporary times. The forms of social movements and their intensity may vary from country to country and place to place within a country. But the very presence of movements for social transformation in various spheres of a society indicates the emergence of a creative society in a country.
With reference to the passage, consider the following statements: 1. To be a creative society, it is essential to have a variety of social movements. 2. To be a creative society, it is imperative to have potential contradictions and conflicts. Which of the statements given above is/are correct?
- 1 only
- 2 only
- Both 1 and 2
- Neither 1 nor 2
Explanation: The passage links creative society to both the presence of contradictions and the existence of social movements across various spheres, making both statements correct.
Consider the following three statements: 1. Only students can participate in the race. 2. Some participants in the race are girls. 3. All girl participants in the race are invited for coaching. Which one of the following conclusions can be drawn from the above statements?
- All participants in the race are invited for coaching.
- All students are invited for coaching.
- All participants in the race are students.
- None of the above conclusions can be drawn.
Explanation: If only students can participate, then by definition every participant must be a student. The other options introduce elements not supported by the premises.
Examine the following statements: 1. All animals are carnivorous. 2. Some animals are not carnivorous. 3. Animals are not carnivorous. 4. Some animals are carnivorous. Of these four statements, two cannot both be true, but both can be false. Study the statements carefully and identify the two that satisfy the above condition. Select the correct answer using the codes given below:
- 1 and 3
- 1 and 2
- 2 and 3
- 3 and 4
Explanation: Statements 1 and 3 are contrary statements; they cannot both be true but can both be false if some animals are carnivorous and some are not. Statements 1 and 2 are contradictories and cannot both be false.
Examine the following statements: 1. All trains are run by diesel engine. 2. Some trains are run by diesel engine. 3. No train is run by diesel engine. 4. Some trains are not run by diesel engine. Of these four statements, two cannot both be true, but both can be false. Identify the two that satisfy the above condition. Select the correct answer using the codes given below:
- 1 and 2
- 2 and 3
- 1 and 3
- 1 and 4
Explanation: Statements 1 and 3 are universal affirmative and universal negative. They cannot both be true but can both be false if some trains are diesel and some are not.
Passage: Ecosystems provide people with a variety of goods and services; food, clean water, clean air, flood control, soil stabilization, pollination, climate regulation, spiritual fulfillment and aesthetic enjoyment, to name just a few. Most of these benefits either are irreplaceable or the technology necessary to replace them is prohibitively expensive. For example, potable fresh water can be provided by desalinating sea-water, but only at great cost. The rapidly expanding human population and their increased requirements of some of the goods and services, particularly food, fresh water, timber, fibre and fuel. These modifications have contributed substantially to human well being and economic development. The benefits have not been equally distributed. Some people have actually been harmed by these changes. Moreover, short term increases in goods and services have come at the cost of the long term capacity of ecosystems. For example, efforts to increase the production of food and fibre have decreased the ability of ecosystems to provide clean water, regulate flooding and support biodiversity.
With reference to the passage, consider the following statements. Expanding human population has an adverse effect on: 1. Spiritual fulfillment 2. Aesthetic enjoyment 3. Potable fresh water 4. Production of food and fibre 5. Biodiversity. Which of the statements given above are correct?
- 1, 2 and 3 only
- 2, 4 and 5 only
- 3 and 5 only
- 1, 2, 3, 4 and 5
Explanation: The passage explicitly links population-driven production increases to decreased ecosystem capacity for clean water and biodiversity. It does not state adverse effects on spiritual fulfillment, aesthetic enjoyment, or food and fibre production itself.
Passage: Ecosystems provide people with a variety of goods and services; food, clean water, clean air, flood control, soil stabilization, pollination, climate regulation, spiritual fulfillment and aesthetic enjoyment, to name just a few. Most of these benefits either are irreplaceable or the technology necessary to replace them is prohibitively expensive. For example, potable fresh water can be provided by desalinating sea-water, but only at great cost. The rapidly expanding human population and their increased requirements of some of the goods and services, particularly food, fresh water, timber, fibre and fuel. These modifications have contributed substantially to human well being and economic development. The benefits have not been equally distributed. Some people have actually been harmed by these changes. Moreover, short term increases in goods and services have come at the cost of the long term capacity of ecosystems. For example, efforts to increase the production of food and fibre have decreased the ability of ecosystems to provide clean water, regulate flooding and support biodiversity.
The passage mentions that 'some people have actually been harmed by these changes'. What does it imply? 1. The rapid expansion of population has adversely affected some people. 2. Sufficient efforts have not been made to increase the production of food and fibre. 3. In the short term some people may be harmed, but in the long term everyone will benefit from modifications in the Earth's ecosystems. Which of the statement given above is/are correct?
- 1 only
- 2 only
- 1 and 3 only
- None of the statements given above
Explanation: The passage states that benefits have not been equally distributed and some people have been harmed by the changes driven by expanding population. It does not mention insufficient efforts for production or suggest long-term universal benefit.
Passage: Ecosystems provide people with a variety of goods and services; food, clean water, clean air, flood control, soil stabilization, pollination, climate regulation, spiritual fulfillment and aesthetic enjoyment, to name just a few. Most of these benefits either are irreplaceable or the technology necessary to replace them is prohibitively expensive. For example, potable fresh water can be provided by desalinating sea-water, but only at great cost. The rapidly expanding human population and their increased requirements of some of the goods and services, particularly food, fresh water, timber, fibre and fuel. These modifications have contributed substantially to human well being and economic development. The benefits have not been equally distributed. Some people have actually been harmed by these changes. Moreover, short term increases in goods and services have come at the cost of the long term capacity of ecosystems. For example, efforts to increase the production of food and fibre have decreased the ability of ecosystems to provide clean water, regulate flooding and support biodiversity.
With reference to the passage, consider the following statements: 1. It is imperative to modify the Earth's ecosystems for the well being of mankind. 2. Technology can never replace all the goods and services provided by ecosystems. Which of the statements given above is/are correct?
- 1 only
- 2 only
- Both 1 and 2
- Neither 1 nor 2
Explanation: The passage notes that most ecosystem benefits are irreplaceable or replacement technology is prohibitively expensive, supporting statement 2. It does not state that modifying ecosystems is imperative; rather, it warns about costs of such modifications.
Passage: A moral act must be our own act; must spring from our own will. If we act mechanically, there is no moral content in our act. Such action would be moral, if we think it proper to act like a machine and do so. For, in doing so, we use our discrimination. We should bear in mind the distinction between acting mechanically and acting intentionally. It may be moral of a king to pardon a culprit. But the messenger carrying out the order of pardon plays only a mechanical part in the king's moral act. But if the messenger were to carry out king's order considering it to be his duty, his action would be a moral one. How can a man understand morality who does not use his own intelligence and power of thought, but lets himself be swept along like a log of wood by a current? Sometimes a man defies convention and acts on his own with a view to absolute good.
Which of the following statements best describe/describes the thought of the writer? 1. A moral act calls for using our discretion. 2. Man should react to a situation immediately. 3. Man must do his duty. 4. Man should be able to defy convention in order to be moral. Select the correct answer from the codes given below:
- 1 only
- 1 and 3 only
- 2 and 3 only
- 1 and 4 only
Explanation: The writer emphasizes using one's own intelligence, discretion, and power of thought, and mentions defying convention for absolute good. The passage does not advocate immediate reaction or mere duty without discretion.
Passage: A moral act must be our own act; must spring from our own will. If we act mechanically, there is no moral content in our act. Such action would be moral, if we think it proper to act like a machine and do so. For, in doing so, we use our discrimination. We should bear in mind the distinction between acting mechanically and acting intentionally. It may be moral of a king to pardon a culprit. But the messenger carrying out the order of pardon plays only a mechanical part in the king's moral act. But if the messenger were to carry out king's order considering it to be his duty, his action would be a moral one. How can a man understand morality who does not use his own intelligence and power of thought, but lets himself be swept along like a log of wood by a current? Sometimes a man defies convention and acts on his own with a view to absolute good.
Which of the following statements is the nearest definition of moral action, according to the writer?
- It is a mechanical action based on official orders from superiors.
- It is an action based on our sense of discretion.
- It is a clever action based on the clarity of purpose.
- It is religious action based on faith.
Explanation: The passage defines moral action as springing from one's own will and using discrimination and intelligence, not mechanical obedience.
Passage: A moral act must be our own act; must spring from our own will. If we act mechanically, there is no moral content in our act. Such action would be moral, if we think it proper to act like a machine and do so. For, in doing so, we use our discrimination. We should bear in mind the distinction between acting mechanically and acting intentionally. It may be moral of a king to pardon a culprit. But the messenger carrying out the order of pardon plays only a mechanical part in the king's moral act. But if the messenger were to carry out king's order considering it to be his duty, his action would be a moral one. How can a man understand morality who does not use his own intelligence and power of thought, but lets himself be swept along like a log of wood by a current? Sometimes a man defies convention and acts on his own with a view to absolute good.
The passage contains a statement 'lets himself be swept along like a log of wood by a current.' Among the following statements, which is/are nearest in meaning to this? 1. A person does not use his own reason. 2. He is susceptible to influence/pressure. 3. He cannot withstand difficulties/challenges. 4. He is like a log of wood. Select the correct answer using the codes given below:
- 1 only
- 1 and 2 only
- 2 and 3 only
- 1 and 4 only
Explanation: The metaphor describes someone who does not use intelligence or thought and is passively carried by external forces, matching lack of reason and susceptibility to influence. It does not refer to withstanding difficulties or literal woodenness.
Passage: A country under foreign domination seeks escape from the present in dreams of vanished age, and finds consolation in visions of past greatness. That is a foolish and dangerous pastime in which many of us indulge. An equally questionable practice for us in India is to imagine that we are still spiritually great though we have come down in the world in other respects. Spiritual or any other greatness cannot be founded on lack of freedom and opportunity, or on starvation and misery. Many western writers have encouraged that notion that Indians are other-worldly. I suppose the poor and unfortunate in every country become to some extent other-worldly, unless they become revolutionaries, for this world is evidently not meant for them. So also subject peoples. As a man grows to maturity he is not entirely engrossed in, or satisfied with, the external objective world. He seeks also some inner meaning, some psychological and physical satisfactions. So also with peoples and civilizations as they mature and grow adult. Every civilization and every people exhibit these parallel streams of an external life and an internal life. Where they meet or keep close to each other, there is an equilibrium and stability. When they diverge conflict arises and the crises that torture the mind and spirit.
The passage mentions that 'this world is evidently not meant for them'. It refers to people who 1. seek freedom from foreign domination. 2. live in starvation and misery. 3. become revolutionaries. Which of the statements given above is/are correct?
- 1 and 2 only
- 2 only
- 2 and 3 only
- 3 only
Explanation: The phrase immediately follows the mention of the poor and unfortunate who become other-worldly because this world is not meant for them. It does not refer to revolutionaries or those seeking freedom.
Passage: A country under foreign domination seeks escape from the present in dreams of vanished age, and finds consolation in visions of past greatness. That is a foolish and dangerous pastime in which many of us indulge. An equally questionable practice for us in India is to imagine that we are still spiritually great though we have come down in the world in other respects. Spiritual or any other greatness cannot be founded on lack of freedom and opportunity, or on starvation and misery. Many western writers have encouraged that notion that Indians are other-worldly. I suppose the poor and unfortunate in every country become to some extent other-worldly, unless they become revolutionaries, for this world is evidently not meant for them. So also subject peoples. As a man grows to maturity he is not entirely engrossed in, or satisfied with, the external objective world. He seeks also some inner meaning, some psychological and physical satisfactions. So also with peoples and civilizations as they mature and grow adult. Every civilization and every people exhibit these parallel streams of an external life and an internal life. Where they meet or keep close to each other, there is an equilibrium and stability. When they diverge conflict arises and the crises that torture the mind and spirit.
Consider the following assumptions: 1. A country under foreign domination cannot indulge in spiritual pursuit. 2. Poverty is an impediment in the spiritual pursuit. 3. Subject peoples may become other-worldly. With reference to the passage, which of the above assumptions is/are valid?
- 1 and 2 only
- 2 only
- 2 and 3 only
- 1 only
Explanation: The passage states that greatness cannot be founded on starvation and misery, implying poverty impedes spiritual pursuit, and explicitly mentions subject peoples become other-worldly. It does not say they cannot indulge in spiritual pursuit at all.
Passage: A country under foreign domination seeks escape from the present in dreams of vanished age, and finds consolation in visions of past greatness. That is a foolish and dangerous pastime in which many of us indulge. An equally questionable practice for us in India is to imagine that we are still spiritually great though we have come down in the world in other respects. Spiritual or any other greatness cannot be founded on lack of freedom and opportunity, or on starvation and misery. Many western writers have encouraged that notion that Indians are other-worldly. I suppose the poor and unfortunate in every country become to some extent other-worldly, unless they become revolutionaries, for this world is evidently not meant for them. So also subject peoples. As a man grows to maturity he is not entirely engrossed in, or satisfied with, the external objective world. He seeks also some inner meaning, some psychological and physical satisfactions. So also with peoples and civilizations as they mature and grow adult. Every civilization and every people exhibit these parallel streams of an external life and an internal life. Where they meet or keep close to each other, there is an equilibrium and stability. When they diverge conflict arises and the crises that torture the mind and spirit.
The passage thematically centres on
- the state of mind of oppressed people.
- starvation and misery.
- the growth of civilization.
- body mind and spirit of people in general.
Explanation: The passage focuses on how foreign domination, poverty, and subject status lead to other-worldliness, escapism, and internal conflict, which are states of mind.
Passage: A country under foreign domination seeks escape from the present in dreams of vanished age, and finds consolation in visions of past greatness. That is a foolish and dangerous pastime in which many of us indulge. An equally questionable practice for us in India is to imagine that we are still spiritually great though we have come down in the world in other respects. Spiritual or any other greatness cannot be founded on lack of freedom and opportunity, or on starvation and misery. Many western writers have encouraged that notion that Indians are other-worldly. I suppose the poor and unfortunate in every country become to some extent other-worldly, unless they become revolutionaries, for this world is evidently not meant for them. So also subject peoples. As a man grows to maturity he is not entirely engrossed in, or satisfied with, the external objective world. He seeks also some inner meaning, some psychological and physical satisfactions. So also with peoples and civilizations as they mature and grow adult. Every civilization and every people exhibit these parallel streams of an external life and an internal life. Where they meet or keep close to each other, there is an equilibrium and stability. When they diverge conflict arises and the crises that torture the mind and spirit.
According to the passage, the torture of the mind and spirit is caused
- by the impact of foreign domination.
- by the desire to escape from foreign domination and find consolation in visions of past greatness.
- due to lack of equilibrium between an external life and an internal life.
- due to one's inability to be either revolutionary or other-worldly.
Explanation: The passage explicitly states that when external and internal lives diverge, conflict arises and crises torture the mind and spirit.
There are four routes to travel from city A to city B and six routes from city B to city C. How many routes are possible to travel from the city A to city C?
Explanation: Using the fundamental principle of counting, total routes from A to C via B equal 4 multiplied by 6, which equals 24.
A contract on construction job specifies a penalty for delay in completion of the work beyond a certain date is as follows: Rs.200 for the first day, Rs.250 for the second day, Rs.300 for the third day etc., the penalty for each succeeding day being Rs.50 more than that of the preceding day. How much penalty should the contractor pay if he delays the work by 10 days?
- Rs.4950
- Rs.4250
- Rs.3600
- Rs.650
Explanation: The penalties form an arithmetic progression with first term 200, common difference 50, and 10 terms. Sum equals 10/2 multiplied by (2 times 200 plus 9 times 50), which equals 5 times 850, equaling 4250.
A person has only Rs.1 and Rs.2 coins with her. If the total number of coins that she has is 50 and the amount of money with her is Rs.75, then the number of Rs.1 and Rs.2 coins are, respectively.
- 15 and 35
- 35 and 15
- 30 and 20
- 25 and 25
Explanation: Let x be Rs.1 coins and y be Rs.2 coins. Then x plus y equals 50 and x plus 2y equals 75. Solving gives x equals 25 and y equals 25.
Three persons start walking together and their steps measure 40 cm, 42 cm and 45 cm respectively. What is the minimum distance each should walk so that each can cover the same distance in complete steps?
- 25m 20cm
- 50m 40cm
- 75m 60cm
- 100m 80cm
Explanation: The required minimum distance is the LCM of 40, 42, and 45. Prime factorization yields LCM equals 2520 cm, which converts to 25 meters and 20 centimeters.
If a bus travels 160 km in 4 hours and a train travels 320 km in 5 hours at uniform speeds, then what is the ratio of the distances travelled by them in one hour?
Explanation: The bus speed is 160 divided by 4 equals 40 km per hour. The train speed is 320 divided by 5 equals 64 km per hour. The ratio of distances covered in one hour is 40 to 64, which simplifies to 5 to 8.
There are 100 students in a particular class. 60% students play cricket, 30% student play football and 10% students play both the games. What is the number of students who play neither cricket nor football?
Explanation: Students playing cricket equal 60, football equal 30, and both equal 10. Using the principle of inclusion-exclusion, students playing at least one game equal 60 plus 30 minus 10, which equals 80. Therefore, students playing neither equal 100 minus 80, which equals 20.
Passage: A species that exerts an influence out of proportion to its abundance in an ecosystem is called a keystone species. The keystone species may influence both the species richness of communities and the flow of energy and materials through ecosystems. The sea star Pisaster ochraceus, which lives in rocky intertidal ecosystems on the Pacific coast of North America, is also an example of a keystone specie. Its preferred prey is the mussel Mytilus californianus. In the absence of sea stars, these mussels crowd out other competitors in a broad belt of the intertidal zone. By consuming mussels, sea star creates bare spaces that are taken over by a variety of other species. A study at the University of Washington demonstrated the influence of Pisaster on species richness by removing sea stars from selected parts of the intertidal zone repeatedly over a period of five years. Two major changes occurred in the areas from which sea stars were removed. First, the lower edge of the mussel bed extended farther down into the intertidal zone, showing that sea stars are able to eliminate mussels completely where they are covered with water most of the time. Second, and more dramatically 28 species of animals and algae disappeared from the sea star removal zone. Eventually only Mytilus, the dominant competitor, occupied the entire substratum. Through its effect on competitive relationships, predation by Pisaster largely determines which species live in these rocky intertidal ecosystems.
What is the crux of the passage?
- Sea star has a preferred prey.
- A preferred prey determines the survival of a keystone species.
- Keystone species ensures species diversity.
- Sea star is the only keystone species on the Pacific coast of North America.
Explanation: The passage demonstrates that the sea star, as a keystone species, maintains species richness by preventing mussels from monopolizing the habitat. The central idea is that keystone species ensure diversity, not merely that sea stars have a preferred prey.
Passage: A species that exerts an influence out of proportion to its abundance in an ecosystem is called a keystone species. The keystone species may influence both the species richness of communities and the flow of energy and materials through ecosystems. The sea star Pisaster ochraceus, which lives in rocky intertidal ecosystems on the Pacific coast of North America, is also an example of a keystone specie. Its preferred prey is the mussel Mytilus californianus. In the absence of sea stars, these mussels crowd out other competitors in a broad belt of the intertidal zone. By consuming mussels, sea star creates bare spaces that are taken over by a variety of other species. A study at the University of Washington demonstrated the influence of Pisaster on species richness by removing sea stars from selected parts of the intertidal zone repeatedly over a period of five years. Two major changes occurred in the areas from which sea stars were removed. First, the lower edge of the mussel bed extended farther down into the intertidal zone, showing that sea stars are able to eliminate mussels completely where they are covered with water most of the time. Second, and more dramatically 28 species of animals and algae disappeared from the sea star removal zone. Eventually only Mytilus, the dominant competitor, occupied the entire substratum. Through its effect on competitive relationships, predation by Pisaster largely determines which species live in these rocky intertidal ecosystems.
With reference to the passage, consider the following statements: 1. Mussels are generally the dominant species in intertidal ecosystems. 2. The survival of sea stars is generally determined by the abundance of mussels. Which of the statements given above is/are correct?
- 1 only
- 2 only
- Both 1 and 2
- Neither 1 nor 2
Explanation: The passage states mussels dominate only in the absence of sea stars, not generally. It also does not link sea star survival to mussel abundance; rather, sea stars control mussels.
Passage: A species that exerts an influence out of proportion to its abundance in an ecosystem is called a keystone species. The keystone species may influence both the species richness of communities and the flow of energy and materials through ecosystems. The sea star Pisaster ochraceus, which lives in rocky intertidal ecosystems on the Pacific coast of North America, is also an example of a keystone specie. Its preferred prey is the mussel Mytilus californianus. In the absence of sea stars, these mussels crowd out other competitors in a broad belt of the intertidal zone. By consuming mussels, sea star creates bare spaces that are taken over by a variety of other species. A study at the University of Washington demonstrated the influence of Pisaster on species richness by removing sea stars from selected parts of the intertidal zone repeatedly over a period of five years. Two major changes occurred in the areas from which sea stars were removed. First, the lower edge of the mussel bed extended farther down into the intertidal zone, showing that sea stars are able to eliminate mussels completely where they are covered with water most of the time. Second, and more dramatically 28 species of animals and algae disappeared from the sea star removal zone. Eventually only Mytilus, the dominant competitor, occupied the entire substratum. Through its effect on competitive relationships, predation by Pisaster largely determines which species live in these rocky intertidal ecosystems.
Which of the following is/are implied by the passage? 1. Mussels are always bare competitors for sea stars. 2. Sea stars of the Pacific coast have reached the climax of their evolution. 3. Sea stars constitute an important component in the energy flow in intertidal ecosystem. Which of the statements given above is/are correct?
- 1 and 2 only
- 2 only
- 1 and 3 only
- 3 only
Explanation: The passage mentions keystone species influence energy flow and materials, implying sea stars are important components of the ecosystem.
How many prime numbers are there between 1 and 50?
Explanation: The primes between 1 and 50 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47. Total count = 15. Common error: counting 1 as prime gives 16. Missing 2 gives 14.
Which of the following is NOT a prime number?
Explanation: 91 = 7 Ã 13, making it composite. 89, 97, and 101 are all prime. Many students assume 91 is prime because it is not divisible by 2, 3, or 5.
What is the largest prime number less than 100?
Explanation: 97 is prime. 93 = 3Ã31, 99 = 9Ã11, and 91 = 7Ã13 are all composite.
What is the smallest prime number greater than 50?
Explanation: 53 is prime. 51 = 3Ã17, 55 = 5Ã11, and 57 = 3Ã19 are all composite.
How many prime numbers lie between 90 and 100?
Explanation: Only 97 is prime between 90 and 100. 91=7Ã13, 93=3Ã31, 95=5Ã19, 99=9Ã11. Many students guess 2 or 3, assuming odd numbers in this range are prime.
The sum of the first five prime numbers is:
Explanation: First five primes: 2, 3, 5, 7, 11. Sum = 28. Distractor 26 omits 2 (the only even prime). Distractor 30 replaces 11 with 13.
What is the product of the first three prime numbers?
Explanation: First three primes: 2, 3, 5. Product = 30. Distractor 42 uses 2Ã3Ã7 (skipping 5). Distractor 36 uses 2Ã2Ã3Ã3 (using composite 4). Distractor 24 uses 2Ã3Ã4 (using composite 4).
Which of the following statements about prime numbers is correct?
- All prime numbers are odd
- There is no even prime number
- 2 is the only even prime number
- Every even number greater than 2 is prime
Explanation: 2 is the only even prime number. All other even numbers are divisible by 2. 'All primes are odd' misses 2. 'No even prime' is false because of 2. 'Every even number > 2 is prime' is absurd â 4, 6, 8, 10 are all composite.
If p is a prime number greater than 3, then p² â 1 is always divisible by:
Explanation: p²â1 = (pâ1)(p+1). Since p is odd, (pâ1) and (p+1) are consecutive even numbers, giving a factor of 8. One of pâ1, p, p+1 is divisible by 3, and since p > 3 is prime, either pâ1 or p+1 is divisible by 3. Thus 8Ã3 = 24 always divides p²â1.
If p and q are two odd prime numbers, then p² + q² is always:
- Odd
- Even
- Divisible by 3
- Prime
Explanation: The square of any odd number is odd. Odd + Odd = Even. For example, 3² + 5² = 9 + 25 = 34 (even). It is not always divisible by 3 (e.g., 5² + 7² = 74, not divisible by 3). It is never prime for p,q > 2 since it is even and > 2.
Which of the following pairs are twin primes?
- (17, 19)
- (13, 17)
- (23, 29)
- (31, 37)
Explanation: Twin primes are prime pairs with a difference of 2. (17, 19) differ by 2. (13, 17) differ by 4, (23, 29) differ by 6, and (31, 37) differ by 6.
How many twin prime pairs exist below 50?
Explanation: Twin prime pairs below 50: (3,5), (5,7), (11,13), (17,19), (29,31), (41,43). Total = 6. Distractor 5 misses one pair; 4 misses two pairs; 7 incorrectly includes a non-twin pair.
What is the largest gap between consecutive prime numbers below 100?
Explanation: The largest gap below 100 is between 89 and 97 (gap of 8). Other gaps: 7 to 11 (gap 4), 23 to 29 (gap 6), 31 to 37 (gap 6). Distractor 10 would be between 89 and 99 (but 99 is not prime).
What is the longest sequence of consecutive composite numbers below 50?
Explanation: The longest sequence is 24, 25, 26, 27, 28 (5 consecutive composites). Distractor 6 might include 23 or 29 (which are prime). Distractor 7 would require 7 consecutive composites, which does not exist below 50.
Which of the following is a prime triplet?
- (3, 5, 7)
- (5, 7, 11)
- (7, 11, 13)
- (11, 13, 17)
Explanation: A prime triplet is a set of three primes of the form (p, p+2, p+4). The only such triplet is (3, 5, 7) because for any other p, one of p, p+2, p+4 must be divisible by 3. (5, 7, 11) skips 9; (7, 11, 13) has gap 4 between 7 and 11; (11, 13, 17) skips 15.
The number of prime numbers between 20 and 40 is:
Explanation: The primes between 20 and 40 are 23, 29, 31, and 37 â totaling 4. Distractor 3 omits one prime; 5 includes 21 or 27 or 33 or 39 (all composite); 6 includes multiple composites.
If the sum of two prime numbers is 85, then one of them must be:
Explanation: 85 is odd. The sum of two odd numbers is even. Therefore, one prime must be even (i.e., 2) and the other must be 83 (which is prime). 2 + 83 = 85. Any other combination of two odd primes would give an even sum.
Which is the smallest 3-digit prime number?
Explanation: 101 is the smallest 3-digit prime. 100 = 10² (composite), 102 = 2Ã51 (composite). Distractor 103 is the second smallest 3-digit prime.
Which of the following numbers has exactly three distinct prime factors?
Explanation: 30 = 2 à 3 à 5 (three distinct prime factors). 12 = 2² à 3 (only two distinct primes). 8 = 2Âŗ (only one distinct prime). 16 = 2â´ (only one distinct prime).
What is the sum of all prime numbers between 60 and 80?
Explanation: Primes between 60 and 80: 61, 67, 71, 73, 79. Sum = 61+67+71+73+79 = 351. Distractor 340 misses 11 (from 79). Distractor 360 adds a composite number. Distractor 370 is a rough overestimate.
Which of the following numbers is a semiprime (product of exactly two primes, not necessarily distinct)?
Explanation: 49 = 7 Ã 7 (product of exactly two primes, same prime). 27 = 3Âŗ (three prime factors). 16 = 2â´ (four prime factors). 32 = 2âĩ (five prime factors).
Two numbers are said to be co-prime if:
- Their LCM is 1
- Their HCF is 1
- Both numbers are prime
- Their product is 1
Explanation: Co-prime numbers have no common factors other than 1, i.e., HCF = 1. They need not both be prime (e.g., 8 and 9 are co-prime). LCM = 1 only when both numbers are 1. Product = 1 only for 1Ã1.
Which of the following pairs of numbers are NOT co-prime?
- (14, 15)
- (21, 22)
- (18, 24)
- (35, 36)
Explanation: HCF of 18 and 24 is 6 (not 1), so they are not co-prime. HCF(14,15)=1, HCF(21,22)=1, HCF(35,36)=1. Distractor (14,15) looks like consecutive numbers (which are always co-prime).
If two numbers are co-prime, then their LCM is equal to:
- 1
- Their HCF
- Their product
- Their sum
Explanation: For co-prime numbers, LCM Ã HCF = Product. Since HCF = 1, LCM = Product. For example, LCM(8, 9) = 72 = 8Ã9. Distractor 1 is the HCF, not LCM.
How many integers between 1 and 50 are co-prime to 50?
Explanation: Using Euler's Totient Function Ī(50): 50 = 2Ã5², so Ī(50) = 50Ã(1â1/2)Ã(1â1/5) = 50Ã1/2Ã4/5 = 20. These are numbers not divisible by 2 or 5. Distractor 25 assumes half are co-prime; 30 and 40 are overestimates.
Which of the following pairs has an HCF of 1?
- (49, 91)
- (77, 91)
- (65, 78)
- (28, 45)
Explanation: HCF(28, 45) = 1 (co-prime). HCF(49, 91) = 7 (both divisible by 7). HCF(77, 91) = 7 (77=7Ã11, 91=7Ã13). HCF(65, 78) = 13 (65=5Ã13, 78=6Ã13).
If HCF of two co-prime numbers is 1, then their LCM is:
- Always 1
- Equal to their product
- Less than their product
- Greater than their product
Explanation: For co-prime numbers, LCM = Product / HCF = Product / 1 = Product. For example, LCM(7, 11) = 77 = 7Ã11. It cannot be less than the product (since both numbers divide the LCM).
Which of the following statements is TRUE?
- Two prime numbers are always co-prime
- Two composite numbers can never be co-prime
- Two consecutive integers are always co-prime
- All of the above
Explanation: Two consecutive integers are always co-prime (HCF = 1). Two distinct primes are always co-prime, but the statement says 'Two prime numbers' which could include the same prime twice (HCF = p, not 1). Two composites CAN be co-prime (e.g., 8 and 9).
The HCF of two co-prime numbers a and b is 1. If a = 35, which of the following CANNOT be the value of b?
Explanation: HCF(35, 56) = 7 (not 1), so 56 cannot be co-prime with 35. HCF(35, 36) = 1, HCF(35, 48) = 1, HCF(35, 72) = 1. 56 shares factor 7 with 35.
How many pairs of co-prime numbers exist between 10 and 20?
- All pairs are co-prime
- No pairs are co-prime
- Some pairs are co-prime
- Exactly 5 pairs are co-prime
Explanation: Between 10 and 20: (11, 12) are co-prime, but (12, 14) are not (HCF=2). (15, 16) are co-prime, but (15, 18) are not (HCF=3). So some pairs are co-prime and some are not. Distractor 'All pairs' is false; 'No pairs' is false; 'Exactly 5' is arbitrary.
If the product of two co-prime numbers is 143, then their sum is:
Explanation: 143 = 11 Ã 13 (both prime, hence co-prime). Sum = 11 + 13 = 24. Distractor 20 assumes 143 = 1 Ã 143 (sum = 144, not an option). Distractor 22 assumes 143 = 13 Ã 11 but miscalculates. Distractor 26 is close to 24.
What is the total number of factors of 72?
Explanation: 72 = 2Âŗ à 3². Number of factors = (3+1)(2+1) = 4Ã3 = 12. Distractor 10 misses one exponent; 14 adds exponents instead of multiplying; 16 is a guess.
What is the sum of all factors of 28?
Explanation: 28 = 2² à 7. Sum of factors = (2Âŗâ1)/(2â1) à (7²â1)/(7â1) = 7 à 8 = 56. Since 28 is a perfect number, the sum of its proper factors equals 28, and sum of all factors = 2Ã28 = 56. Distractor 54 is close; 58 and 60 are overestimates.
Which of the following is a perfect number?
Explanation: A perfect number equals the sum of its proper divisors. 28 = 1+2+4+7+14 = 28. 24: 1+2+3+4+6+8+12 = 36 â 24. 30: 1+2+3+5+6+10+15 = 42 â 30. 32: 1+2+4+8+16 = 31 â 32.
What is the number of distinct prime factors of 360?
Explanation: 360 = 2Âŗ à 3² à 5š. The distinct prime factors are 2, 3, and 5 â totaling 3. Distractor 4 counts total prime factors with multiplicity (3+2+1=6, not 4). Distractor 2 misses one prime; 5 is a guess.
If N = pâ^a à pâ^b à pâ^c where pâ, pâ, pâ are distinct primes, then the number of factors of N is:
- a + b + c
- a à b à c
- (a+1)(b+1)(c+1)
- (a+b+c+1)
Explanation: Each prime exponent can contribute 0, 1, 2, ..., up to its maximum. So for pâ^a, there are (a+1) choices. Total factors = (a+1)(b+1)(c+1). Distractor a+b+c adds exponents; aÃbÃc multiplies exponents; a+b+c+1 is incorrect.
What is the smallest number that has exactly 6 factors?
Explanation: 12 = 2² à 3š. Number of factors = (2+1)(1+1) = 3Ã2 = 6. 16 = 2â´ has 5 factors. 18 = 2Ã3² has (1+1)(2+1) = 6 factors too, but 12 < 18. 24 = 2ÂŗÃ3 has (3+1)(1+1) = 8 factors.
The prime factorization of 2310 is:
- 2 Ã 3 Ã 5 Ã 7 Ã 11
- 2 Ã 3 Ã 5 Ã 7 Ã 13
- 2 Ã 3 Ã 5 Ã 11 Ã 13
- 2 Ã 3 Ã 7 Ã 11 Ã 13
Explanation: 2310 = 2 Ã 3 Ã 5 Ã 7 Ã 11. Verify: 2Ã3=6, 6Ã5=30, 30Ã7=210, 210Ã11=2310. Distractors replace 11 with 13 or rearrange primes.
Which number has the most factors among the following?
Explanation: 36 = 2²Ã3² has 9 factors. 48 = 2â´Ã3 has 10 factors. 60 = 2²Ã3Ã5 has 12 factors. 120 = 2ÂŗÃ3Ã5 has (3+1)(1+1)(1+1) = 16 factors. More distinct prime factors and higher exponents increase the factor count.
What is the largest prime factor of 182?
Explanation: 182 = 2 Ã 7 Ã 13. The largest prime factor is 13. Distractor 7 is the smaller prime factor. 26 = 2Ã13 is composite. 91 = 7Ã13 is composite.
If a number N has exactly 8 factors, which of the following CANNOT be the form of N?
- p^7
- p^3 Ã q
- p à q à r
- p^2 Ã q^2
Explanation: p^7 has (7+1)=8 factors. pÂŗÃq has (3+1)(1+1)=8 factors. pÃqÃr has (1+1)(1+1)(1+1)=8 factors. p²Ãq² has (2+1)(2+1)=9 factors, not 8.
The sum of the prime factors of 210 is:
Explanation: 210 = 2 Ã 3 Ã 5 Ã 7. Sum of distinct prime factors = 2+3+5+7 = 17. Distractor 10 omits 7; 12 omits 5; 15 is close but incorrect.
Which of the following is a Mersenne prime?
Explanation: A Mersenne prime is a prime of the form 2^n â 1. 31 = 2âĩ â 1 (and 31 is prime). 33 = 3Ã11 (composite). 35 = 5Ã7 (composite). 37 is prime but not of the form 2^nâ1 (2âĩâ1=31, 2âļâ1=63).
According to Goldbach's conjecture, every even integer greater than 2 can be expressed as:
- Product of two primes
- Sum of two primes
- Difference of two primes
- Ratio of two primes
Explanation: Goldbach's conjecture states that every even integer greater than 2 can be expressed as the sum of two primes. For example, 10 = 3+7, 12 = 5+7, 14 = 3+11. It is unproven but verified for very large numbers.
Which even number CANNOT be expressed as the sum of two primes?
Explanation: Goldbach's conjecture applies to even integers GREATER than 2. 2 cannot be expressed as the sum of two primes (the smallest prime is 2, and 2+2=4). 10=3+7, 12=5+7, 14=3+11 all satisfy the conjecture.
If p is a prime number, then p² + p + 1 is always:
- Prime
- Even
- Odd
- Divisible by 3
Explanation: For any prime p > 2, p is odd, so p² is odd, p is odd, and odd+odd+1 = odd+odd+odd = odd. For p=2: 4+2+1=7 (odd). So p²+p+1 is always odd. It is not always prime (e.g., p=7: 49+7+1=57=3Ã19). Not always divisible by 3 (e.g., p=2 gives 7, not divisible by 3).
What is the smallest prime number that is 1 more than a perfect square?
Explanation: 2 = 1² + 1. 5 = 2² + 1. 17 = 4² + 1. 37 = 6² + 1. The smallest is 2. Distractor 5 is the second smallest; 17 and 37 are larger examples.
The product of all prime numbers less than 10 is:
Explanation: Primes less than 10: 2, 3, 5, 7. Product = 2Ã3Ã5Ã7 = 210. Distractor 180 = 2²Ã3²Ã5 (includes composite 4, 6, 9). 240 = 2â´Ã3Ã5. 270 = 2Ã3ÂŗÃ5.
Which of the following statements about prime numbers is FALSE?
- There are infinitely many prime numbers
- Every integer greater than 1 has a prime factor
- The sum of two primes is always even
- There is no largest prime number
Explanation: The sum of two primes is NOT always even: 2 + 3 = 5 (odd). This is false because 2 is the only even prime, so 2 + odd prime = odd. All other statements are true (Euclid's proof, fundamental theorem of arithmetic).
If n is a positive integer such that n, n+2, n+4 are all prime, then n must be:
Explanation: For n > 3, one of n, n+2, n+4 must be divisible by 3 (since they cover all residues mod 3). The only prime triplet (p, p+2, p+4) is (3, 5, 7). For n=5: 5, 7, 9 â but 9 is composite. For n=7: 7, 9, 11 â but 9 is composite.
What is the remainder when the product of the first four prime numbers is divided by 5?
Explanation: First four primes: 2, 3, 5, 7. Product = 2Ã3Ã5Ã7 = 210. 210 Ãˇ 5 = 42 remainder 0. Since 5 is one of the factors, the product is divisible by 5. Distractors assume the product is 2Ã3Ã7=42 (remainder 2) or other miscalculations.
Which of the following is the smallest composite number that is NOT divisible by any prime less than 10?
Explanation: 121 = 11². It is not divisible by 2, 3, 5, or 7. 49 = 7² (divisible by 7). 77 = 7Ã11 (divisible by 7). 91 = 7Ã13 (divisible by 7). Distractor 49 is the smallest composite not divisible by 2, 3, or 5, but it IS divisible by 7.
If Euclid's number E = 2Ã3Ã5Ã7 + 1, then E is:
- Prime
- Composite
- Divisible by 2
- Divisible by 3
Explanation: E = 210 + 1 = 211. Checking: 211 is not divisible by 2, 3, 5, 7, 11, or 13 (sqrt(211) â 14.5). Thus 211 is prime. This demonstrates Euclid's proof that there are infinitely many primes â the product of first n primes + 1 is either prime or has a new prime factor.
According to Wilson's theorem, if p is prime, then (pâ1)! + 1 is divisible by:
Explanation: Wilson's theorem states that for a prime p, (pâ1)! ⥠â1 (mod p), so (pâ1)! + 1 ⥠0 (mod p). Thus (pâ1)! + 1 is divisible by p. Example: p=5, 4!+1 = 25, divisible by 5.
How many prime numbers are there between 100 and 110?
Explanation: Primes between 100 and 110: 101, 103, 107, 109 â totaling 4. Many students miss 101 (assuming all numbers near 100 are composite) or include 105 (divisible by 3, 5, 7).
If p and q are distinct odd primes, then p² â q² is always divisible by:
Explanation: p² â q² = (pâq)(p+q). Since p and q are odd, both pâq and p+q are even. Let p=2k+1, q=2m+1. Then pâq = 2(kâm) and p+q = 2(k+m+1). Since (kâm) and (k+m+1) have opposite parity, one contributes an extra factor of 2. Thus total factors of 2 = 2+2+1 = 3, making p²âq² divisible by 8. Example: 7²â5² = 49â25 = 24, divisible by 8.
The sum of the reciprocals of the first three prime numbers is:
Explanation: First three primes: 2, 3, 5. Sum = 1/2 + 1/3 + 1/5 = 15/30 + 10/30 + 6/30 = 31/30. Distractor 29/30 omits one term; 11/6 = 1/2+1/3+1 (wrong); 5/6 = 1/2+1/3.
Which of the following is an abundant number (sum of proper divisors > number)?
Explanation: 12: proper divisors are 1, 2, 3, 4, 6. Sum = 16 > 12 (abundant). 16: 1+2+4+8 = 15 < 16 (deficient). 28: 1+2+4+7+14 = 28 (perfect). 32: 1+2+4+8+16 = 31 < 32 (deficient).
How many positive integers less than 12 are co-prime to 12?
Explanation: Using Euler's totient Ī(12): numbers co-prime to 12 are 1, 5, 7, 11 â totaling 4. Formula: Ī(12) = 12 Ã (1â1/2) Ã (1â1/3) = 12 Ã 1/2 Ã 2/3 = 4. Distractor 6 counts numbers not divisible by 2 only; 8 is a guess.
Which of the following numbers has exactly 3 factors?
Explanation: Only squares of primes have exactly 3 factors. 4 = 2², factors are 1, 2, 4. 6 = 2Ã3 has 4 factors. 8 = 2Âŗ has 4 factors. 10 = 2Ã5 has 4 factors.