Class 12

Odisha State Board CHSE Odisha Class 12 Economics Solutions Chapter 2 Basic Economic Concepts (Wants, Utility, Goods, Value, Price and Wealth) Questions and Answers.

CHSE Odisha 12th Class Economics Chapter 2 Question Answer Basic Economic Concepts (Wants, Utility, Goods, Value, Price and Wealth)

Group – A

Short type Questions with Answers

I. Answer within Two/Three sentence
Explain the following statements

Question 1.
Free goods have value in use.
Answer:
Free goods are free gifts of nature whose supply is abundant in ralation to its demand. So these goods do not have value-in-exchange but have value in use.

Question 2.
Goods having Value-in-exchange must have value-in-use.
Answer:
Goods having value-in-exchange are capable of satisfying human wants & can be exchanged in the market So goods having value” in exchange must have value in use.

Question 3.
Goods having value-in-use may not have value-in-exchange.
Answer:
There are certain free gifts of nature whose supply is more in relation to its demand & hence are not exchanged. Goods like water of the river, air etc have value in use but due to abundant supply, these goods have no value-in-exchange.

Question 4.
The consumption of public goods is non-excludable.
Answer:
The public goods are supplied by the government for the collective welfare of the people. As it is collectively consumed & the government is the owner of these goods & the consumer can not exclude other from consuming these goods & hence the consumption becomes non-excludable.

Question 5.
Producers goods do not provide direct statistaction
Answer:
Producers goods are used for further production of goods. As such, these goods do not provide direct satisfaction.

Question 6.
Utility is subjective.
Answer:
Utility is psychic entity which resides in the mind of the consumer. The utility derived from a goods is assessed by the consumer concerned & hence it is subjective.

Question 7.
Utility is not same as usefulness
Answer:
There are some goods like wire, opium etc are not useful to human beings. But these are consumed because of their utility

Question 8.
Wants are competitive.
Answer:
On account of unlimited wants and limited resources, a man can not satisfy all of his wants. But because of multiplicity of wants, these compete with each other to be satisfied.

Question 9.
Wants are alternative.
Answer:
There are certain wants Which can be satisfied in an altermative manner. Like hunger which can be satisfied by rice or bread. Hence, wants are altermative.

Question 10.
All capital are wealth; but all wealth are not capital
Answer:
Wealth refers to all the goods having utility, scarcity marketability & external possession. These can satify the human wants directly or it can be used as capital in producing goods. So all capital are wealth; but all wealth are not capital.

Question 11.
A particular want is sutiable
Answer:
Thoughj the wants are unlimited, yet if aman tries its best to satisfy a particular wants by purchasing various units of it, that wants may be satisfied. So, a particular wants is satisfied.

Question 12.
Price of goods may change but value remains constant
Answer:
Price is the exchange value of a goods measured in terms of money. It refers to what a commodity can purchase in terms of other other commodities. So price measures the value in exchange in terms of money. Thus the price may change but value-in-exchange itself remains constant.

Question 13.
Goods satisfy human wants.
Answer:
Goods possesses the wants satisfying power (utlity). So it .can satisfy human wants.

Question 14.
The gifts of nature are free goods.
Answer:
The supply of gift of nature to mankind is abundant in relation to its demand. Hence there goods are available freely & treated as free goods.

Question 15.
Economic goods have value-in-exchange.
Answer:
The supply of goods is scarce in relation to its demand & hence there goods are bought & sold in the market. Thus, the economic goods have value-in-exchange.

Question 16.
Public goods are meant for collective consumption.
Answer:
Public goods are supplied by the government for the collective welfare & also indivisible in nature. So these goods are collectively consumed.

Question 17.
The consumption of public goods is non-excludable. Or, Principle of exclusion is not applicable to public goods.
Answer:
The public goods are meant for joint or collective consumption & hence its consumption is non-rival. So, the principle of exclusion is not applicable to public goods.

Question 18.
Utility is subjective.
Answer:
Utility derived from a goods depend upon the human psychology & it resides in the mind of the consumer. So utility is subjective.

Question 19.
Utility is not same as usefulness.
Answer:
Utility is simple the wants satisfying power of a commodity, but usefulness describes the beneficial effects of a good. So utility is quite different from usefulness.

Question 20.
All the goods have value-in-use but not value-in-exchange.
Answer:
The goods are consumed because of value-in-use. The free goods are of these category which have value in use but not value in exchange.

Question 21.
Economic goods are scarce in supply.
Answer:
Any goods whose supply is scarce in relation to its demand can have value-in-exchange. Economic goods have the attribute like value-in-exchange because of scarce supply & abundant demand.

Question 22.
There can be general rise or fall in price but not value.
Answer:
Value in economics refers to value in exchange i.e. What a commodity can purchase in terms of other commodities. But price is the monetary expression of value in exchange which can be changed.

Question 23.
All the economic goods are wealth.
Answer:
The economic goods possess the characteristics of wealth. So, all the economic goods are wealth.

Question 24.
Health is not treated as wealth in economics.
Answer:
Health does not possess all. the features of wealth like transferability or marketability. Hence, it is not treated as wealth in Economics.

Question 25.
Voice of a singer can not be treated as wealth.
Answer:
Voice of a singer is a personal quality & hence cannot be transferred. So it is not wealth.

Question 26.
H. S. C. certificate is not wealth in Economics.
Answer:
H. S. C. certificate cannot be legally marketed nor transferred. So it is not wealth in economics.

Question 27.
Human wants are unlimited.
Answer:
Human wants have no limit. If one wants is satisfied, another wants takes its place. Satisfaction of wants being a continuous process, the human wants are said to be unlimited.

Question 28.
Wants are recurring.
Answer:
Because of limited resources, one gan not satisfy all of his wants. So the goods compete with each other for which choice is to be made.

Question 29.
Wants are recurring.
Answer:
There are some wants like taking meals or drinks very often occur in a day. So these wants are said to be recurring.

Question 30.
The classification of human wants is relative.
Answer:
The classification of human wants into necessaries, comforts & luxuries is relative in the sense that it varies from person to person, place to place and time to time. The goods like car is said to be a luxurious items for a student whereas it is necessary for an officer.

Question 31.
The consumption of luxuries is not justified.
Answer:
The consumption of luxuries is morally bad & economically wasteful. Hence, its consumption is not justified.

II. Answer within Five/Six sentence :

(A) Write Short Notes on :

Question 1.
Utility:
Answer:
The wants satisfying power of a commodity is known as utility. It is the capacity or the quality of a goods to satisfy wants. Those goods which only possesses utility can satisfy the wants & hence those goods are purchased or consumed. Utility is at the root of consumption. Utility is subjective and it is not same as usefulness. It is different from pleasure and satisfaction, utility can be measured cardinally or ordinally.

Question 2.
Service Utility:
Answer:
The utility derived from various services is called service utility. Like goods, the services extended to a man can satisfy his wants. The service of a doctor, a lecturer or an engineer can satisfy the wants & hence it has utility. In other words, the utility derived from the use or consumption of non-material goods is termed as service utility.

Question 3.
Value:
Answer:
In economics, value means value-in-exchange. Value in exchange means what a commodity can purchase in terms of other commodities. It is the purchasing power of one commodity in terms of other commodities and services. According to Marshall, the value that is exchange value is the amount of second thing which can be got there and then in exchange for the first. Exchange value expresses the relation between two things at particular time and place. Economic goods have got value in-exchange. Free goods have got no value-in-exchange as nobody would give anything in exchange of it.

Question 4.
Price:
Answer:
Prices is the measure of the exchange value of a commodity in terms of money. When we say that the price of sweets is one rupee, it implies that rupee one is equal to the exchange value of that sweets. Price thus expresses the purchasing power of a commodity in terms of money. Value of a commodity means the comparison between any two commodities whereas price is the money value of commodity. Price is highly fluctuating.

Question 5.
Cosmopolitan wealth:
Answer:
Cosmopolitan wealth is the wealth of the whole world. It is the sum total of the wealth of all nations. While estimating all international debts should be deducted. All the nations have collective right on such types of wealth. U.N.O., International Monetary Fund (I.M.F.), International Bank of Reconstruction and Development the oceans of the world, the surface of the moon or mars are the examples of International wealth. There are some who consider scientific.knowledge which crosses the boundary of a particular state are regarded as international wealth.

Question 6.
Economic Goods :
Answer:
The goods having value-in-exchange is called economic goods. These goods are exchanged for money. Price is paid for the possession of these goods. The demand for these goods is more in relation to its supply i.e. supply of these goods are scarce in relation to its demand. Economic problems are created in case of these goods. TV, Radio, watch, food, Dress etc. are the examples of economic goods.

Question 7.
Consumers goods:
Answer:
The goods which can directly satisfy the human wants are called consumers goods. These goods are called goods of first order. These goods may be perishable or durable in nature. The goods which can be used once in the satisfaction of wants is perishable goods & the goods which can be frequently used for the satisfaction of human wants are called durable goods. Food items are the example of perishable goods and T.V., Radio are the durable goods.

Question 8.
Intermediate goods:
Answer:
The goods which are used in the further production are called intermediate goods. These goods can not directly satisfy the human wants. Rather these are used in producing goods. These goods provide indirect satisfaction to the consumer as these are used in the production. Flour for bread is an example of intermediate goods. The value of these goods are excluded while calculating national income.

Question 9.
Private goods:
Answer:
The goods which are owned by the private individuals are called private goods. These are possessed by the private individuals for their own use or consumption. Others do not have any right over these goods. In case of these goods the ‘Principle of exclusion’ is highly applicable. The person who consumes or purchases it can exclude others from the field of consumption. One who pays the price purchases it. There observed competition among the consumers for the purchase of these goods.

Question 10.
Public Goods:
Answer:
Public goods are owned & supplied by the government. These goods are used collectively by the common mass. Principle of exclusion is not applicable to such goods. There observed no rivalry, nor competition amongst the consumers for the possession of these goods. These goods are supplied for collective welfare. Government is the prime supplier of public goods.

Question 11.
Social wealth:
Answer:
The wealth under the possession of the society is called social wealth. The ownership of such goods is vested with the society. All the members of the society have equal right over this wealth. This wealth is otherwise known as collective wealth. These goods are collectively consumed by the members of the society.

Question 12.
Goods and Services :
Answer:
Anything which can satisfy the human wants is called goods. It may be material or non-material. Non-material goods are called services. Goods which usually denotes material goods is visible, tangible and it has got a shape but service is invisible, intangible and it does not have any shape. Goods can directly satisfy the wants of a person who consumes it. But in case of service, one person renders service to other for the satisfaction of his wants. A consumer may feel the utility of a goods for a long-time. But the utility of services perishes at the very moment it is rendered.

Question 13.
Free Goods & Economic Goods:
Answer:

• Free goods are the gift of nature whereas the economic goods are the outcome of human effort.
• Free goods are available freely; but economic goods are available with the payment of price.
• The supply of free goods is abundant in relation to its demand; but the supply of economic goods is scarce in relation to its demand.
• There is no ownership of any individual on free goods; but the individual (s) or any organisation own the economic goods.

Question 14.
Consumers’ goods & Producers’ goods :
Answer:

• Consumers’ goods are those goods which can directly satisfy the human wants whereas the producers’ goods are those goods which can indirectly satisfy human wants.
• Consumers’ goods are exclusively meant for direct consumption but the producers’ goods are used for further production.
• Consumers goods and producers goods may be perishable or durable in nature.
• The distinction between consumers goods & producers goods is relative in nature. One good may be treated as consumers goods & producers goods on the basis of its use.
• The demand for consumers goods is direct demand whereas the demand for producers goods is derived demand.

Question 15.
Private goods & Public goods :
Answer:

• The goods which are under the control & ownership of private individuals (s) are called private goods. But the goods which are owned by the government is called public goods.
• Private goods are used for the satisfaction of the individual wants; but public goods are used for the satisfaction of collective wants.
• The consumption of private goods is rival & excludable but the consumption of public goods is non-rival & non-excludable.
• The consumption of private goods leads to create welfare of the individual but the consumption of public goods create collective welfare for the society as a whole.

Question 16.
Wealth and Title to wealth :
Answer:
The wealth which can satisfy the wants directly or indirectly is known as wealth. The assets claimed to be economic goods are called titles to wealth and can not satisfy the wants directly. Rather, the human wants can be satisfied with these assets. Stocks, shares, Bills of exchange and even money can be called as titles to wealth.

Question 17.
Wealth and Capital:
Answer:

• Wealth refers to all economic goods which can satisfy the human wants directly or indirectly. But the capital is that part of the wealth which can only satisfy the human wants indirectly.
• Wealth covers all the economic goods whether it is consumers’ goods or producers’ goods. But capital is only the producers goods which are used for further production.
• Wealth used for further production or for earning income is treated as capital in Economics.

Question 18.
Wealth and Money :
Answer:

• Wealth refers to economic goods having four features like utility, scarcity, transferability & external possession. But money is a medium of exchange by which goods & services can be possessed. It does not have utility of its own.
• Wealth can satisfy the human directly because of its utility; but money acts a medium to purchase the wealth for the satisfaction of human wants.
• Wealth is demanded for its own sake; but money is demanded for the purchase of wealth.
• Wealth has got both value-in-use and value in exchange; but money has got only the value-in-exchange.

Question 19.
Wealth and Income :
Answer:

• Wealth is a stock of economic goods possessed by a man. Income is a flow which arises when wealth is invested by a way of capital.
• Wealth is a fund but income is a stream.
• Wealth generates income. It is a flow which originates from economic goods.
• The stock of wealth may remain unchanged but the income earned from it goes on changing.

Question 20.
Wealth and Welfare :
Answer:

• The possession of wealth is the root cause of welfare. Greater stock of wealth yields more of welfare.
• Sometimes, huge amount of wealth leads to none of welfare but wealth is one of the conditions of welfare whereas welfare is a multivariate concept.
• Wealth can be externally possessed but welfare is greatly a psychological concept.
• More of wealth does not lead to more of welfare always, it may even impede the welfare.

Question 21.
Desire and human wants
Answer:

• Desire refers to willingness of the individual for having a goods; but human wants refers to effective desire for having a goods.
• Desire of a man only symbolises the willingness of that man for having a goods; but human wants indicates the willingness to buy a goods backed by ability to pay and willingness to spend.
• Desire is one of the conditions of human wants.
• Human wants is a wider concept whereas desire is a narrower concept.
• One man may have desire to have a goods but it can not be called as his wants if he does not have adequate purchasing power or if he is not interested to spend money towards this goods.

Group – B

Long Type Questions With Answers

Question 1.
What is utility ? What are the characteristics of utility ?
Answer:
Utility refers to wants satisfying power of a commodity. The capacity or the quality of the goods by which human wants are satisfied is called utility. If the goods and services directly or indirectly satisfy the wants, it can be asserted that the goods possess the utility. Thus, utility indicates the power of the commodity to satisfy the wants. It is the root-cause of consumption. People purchase or consume goods because of its utility.

FEATURES OF UTILITY:
Utility of a goods being the basic cause of consumption possesses several features. The important features of utility are mentioned below :

(a) Utility is subjective : Utility is a subjective concept. It resides in the minds of a person. It can be adjudged by introspection. It is not objective. The same commodity will provide different utility to different persons. An orange if consumed by an ordinary man gives less utility than a- person who likes orange. So, utility is subjective and also a relative concept. It changes from person to person, time to time and place to place.

(b) Utility is not equal to usefulness : It is highly changeable. Utility is different from usefulness. A thing like opium or wine has no usefulness but possesses utility as they satisfy human wants. Hence, these are useful. Some people may consider this as immoral. But utility as a concept is ethically neutral. Things considered bad or undesirable has got utility like poison or opium.

(c) Utility is different from pleasure : Things may not be pleasant but they have utility. Bitter medicines or arms and ammunitions do not give pleasure but they have utility. So, utility can be found in unpleasant things.

(d) Utility is different from satisfaction : Satisfaction is the result of using a commodity. When we consume a commodity we get satisfaction. We consume it because it has utility. Utility is the cause of consumption and satisfaction is the effect of consumption.

(e) Utility may be cardinal or ordinal: These two terms refer to the measurement of utility. Utility are measured like 1 2 3 or 4. This is called cardinal measurement. Here one is half of two or two is double of one. The otherway to measure it is to say it as 1 st, 2nd, 3rd etc. These measurement is called ordinal numbers. Here, the utility is ranked or ordered. After 1 st comes 2nd but one cannot say 2nd is twice of the first. It may be more than double or less than double of 1 st.

Question 2.
What is Wealth ? What are the features of Wealth ?
Answer:
In economics, wealth refers to all goods having exchange value. Therefore, all economic goods are treated as wealth. According to Lord Keynes, the famous economist of the century, all these goods which have value and the capacity to satisfy human wants are called wealth in economics. Wealth is nothing but economic goods.

Characteristics of Wealth : There are four attributes of wealth such as (i) Utility, (ii) Scarcity, (iii) Transferability, and (iv) External to man.
(i) Wealth must possess Utility : Any goods which can satisfy human want is called wealth. A rotten egg is not wealth because it can not satisfy a human want. On the other hand, wine and opium are wealth because it does satisfy a human want.

(ii) Scarcity : The goods must be limited in supply in comparison to its demand. The commodity can be obtained by a payment. In other words, all economic goods are wealth Air, water, sunshine or moonlight are not wealth because they are available in plenty. Nobody pays anything to get these things. These things have got’great value in use but no value in exchange. This explaines the paradox of value in economics. Diamond which has no value-in-use has got great value-in-exchange because it is scarce in supply. Wealth should be always scarce.

(iii) Marketability or Transferability : A commodity to be wealth must be transferable and marketable. If a thing is marketable, it must be transferable. Transferability and marketability mean both aspects of the same things. Transferability does not mean physical transferability, it may mean transferability of ownership. A man can sell his land or good will of his business to others. This is also transferability. Because wealth will be marketable. Personal qualities like honesty, skill, ability and intelligence are not wealth. These are the source of wealth but not wealth proper because it cannot be sold in the market nor the ownership changed.

(iv) External to man : In order to be wealth a thing must be external to man. If something is an internal quality like the quality of a dancer, singer, painter or actor that cannot be regarded as wealth. The quality of these people are not wealth but their serv ice which can be sold in the market are called wealth. A person’s M.A. degree can not be regarded as wealth, since it cannot be transferred it is not wealth.

The above analysis illustrates that wealth has four attributes. Hence, anything possessing all these attributes can be considered as wealth in economics. As such, to treate anything wealth, all these four features should be present in that thing.

Question 3.
What is wealth ? Make a classification of wealth.
Answer:
In economics wealth refers to all goods having exchange value. Therefore, all economic goods are considered as wealth. According to Lord Keynes, the famous economist of the century, all these goods which have value and the capacity to satisfy human wants are called wealth in economics. Wealth is nothing but economic goods.
Classification of Wealth : Wealth can be classified into different types. The important classification of wealth is described below :

(A) Individual Wealth :

1. It refers to the house building furniture, bond and shares of man. These are transferable material goods. From this, the negative wealth of a person like shares and bonds which he holds to pay others are deducted.
2. Non-material goods like the good- will of a business man, or his professional connections which can be marketed and brings income to him are called individual wealth.

(B) National Wealth: Wealth which belongs to the entire nation is called national wealth. In a broad sense, it includes the following four elements :

1. The sum total of all individual wealth constitute the wealth of the nation. The internal debt due to the one another and external debt and loan of the country should be deducted from the aggregate wealth to arrive at the proper national wealth.

2. Marshall included the free gifts of nature like river, mountains and forests as the wealth of the country. In the strict sense of the term, these are not economic goods and hence not wealth.

3. Public material property like roads, bridges, canal, public parks and the railway of the country are included in the national wealth.

4. Some economists suggest that good scientific knowledge of production and scientific attitudes profoundly influence the national wealth of the country and as such it should be included in the national wealth. German skill or American business talent are the examples of national wealth. But Marshall was of the opinion that scientific discoveries, inventions and enchanting literature belong to the whole of humanity and cannot be regarded as national wealth.

C. Cosmopalitan Wealth: Cosmopolitan wealth is the wealth of the whole world. It is the sum total of the wealth of all nations. While estimating, we should deduct all international debts. All the nations have collective right on such types of wealth. U.N.O., International Monetary Fund (I.M.F.), International Bank for Reconstruction and Development the Oceans of the World, the surface of the Moon or Mars are the examples of International wealth.

D. Potential Wealth: Potential wealth is the wealth which lie unused or unexploited like coal, iron or gold.

E. Negative Wealth: It refers to the borrowing of the country or individual which will be returned after a time.

Question 4.
What do you mean by human wants ? What are the characteristics of human wants ?
Answer:
Human wants is the starting point of all the economic activities. Wants, efforts and satisfaction constitute the subject-matter of economics. Thus, human wants occupy the first position in economics. Human wants refer to the desire for the possession of a commodity. This desire must be effective. The desire is said to be effective if the consumer has willingness to buy, ability to pay and willingness to spend. So, the effective desire for the possesion of a commodity is known as human wants.

Human wants is a vital concept in the study of economics. It possesses various features. The important features or characteristics of human wants are mentioned below.

Characteristics of human wants :
(a) Human wants are unlimited : Human wants have no end to it. These are unlimited in number. According to Marshall, human wants are countless in number and various in kinds. No man is able, to satisfy his wants however rich he may be. If he wants a television and gets it, then he wants a car and the moments he gets a car, he wants a building. If he is a poor man he thinks about food, cloth and shelter. When one of his want is satisfied, another takes its place. Wants multiply with the progress of civilization. Modem man has variety of wants, like the waves of the sea which is countless and numberless.

(b) A Particular Human Wants is Satiable : A particular wants is limited. Each separate wants has a limit. If a man needs a cup of tea, he can saisfy this particular wants by taking few cups of tea. There will be a time when he will refuse to have more of it. So the want of tea being a particular want is limited and satiable.

(c) Wants are Competitive : Wants compete with each other to be satisfied. Wants compete because the wants are limitless in number and the means to satisfy these wants are limited. So a choice is made as to what is more urgent want and less urgent. Wants are graded in order of importance and then the most important one is satisfied. The competition among wants may be closer or distant. Close competition is held between substitutes like tea and coffee or oranges and apples. But distant competition is made between dissimilar things like motor car and cinema show.

(d) Wants are Complementary: There are some goods required in groups for the satisfaction of our want. Motor car will not run without the petrol and mobil or a pen will not write without ink. So the want of motor car also implies want for petrol and mobil. Here wants are complementary because commodities are wanted jointly.

(e) Wants are Alternative : Some wants are alternative because they can be satisfied by alternative things. Thus, hunger can be satisfied by bread or rice or sweets. Thrist can be quenched by water, lassi or Cola.

(f) Wants Recur Again: Some wants recur again and again. They are satisfied in the morning but again arise in the evening. We take a cup of tea in the morning and satisfy our wants but again in the evening we want another cup of tea. There are some persons who feel the want of tea always. Thus, some wants recur.

(g) Some Wants Become Habits : Wants change into habits. At that time it becomes a permanent want. Someone may take tea when he suffers from cold, but if he starts taking it regularly it becomes his habit. Our want for wine, cigarette, tea, opium, coffee or pan are examples of habitual wants.

(h) Wants Vary in Intensity : Wants have different intensities. Some wants are more urgent and some are less. Those wants which are more urgent are more intense. This is nothing but grading wants according to their importance. A students feels the intensity of wants of book and journals than a bicycle.

(i) Wants are both Complementary and Competitive : Some wants are complementary in the beginning but become competitive afterwards. The usual example of such a wants is man and machine. The machine at first runs with the help of a man & it becomes competitive with man afterwards because man and machine do the same work or satisfy the same want. The agitation of workers in some industries in India against automation is due to the competition of man and machine to do the same work.

(j) Present Wants are preferred to Future Wants : Present wants are always felt more important than the future wants. People always satisfy their current wants and postpone their future wants. Future is uncertain and hence people satisfy the current wants first and postpone future wants.

(k) Wants vary with time, place and person:. We do want a glass of cold drinks in summer season but a cup of tea in winter to quench our thirst. Similarly, hot water is not wanted at Cuttack during summer but it is required in Simla. A person living in a town may require pair of trousers whereas a man in rural area will prefer a dhoti. Wants are influenced not only by time, place or person but also by salesmanship and advertisement. People want a thing because of advertisement. Social factors and customs influence the wants of a man.

Question 5.
What do you mean by human wants ? Make a classification of human wants.
Answer:
Human wants is the starting point of all the economic activities. Wants, efforts and satisfaction constitute the subject-matter of economics. Thus, human wants occupy the first position in economics. Human wants refer to the desire for the possession of a commodity. This desire must be effective. The desire is said to be effective if the consumer has willingness to buy, ability to pay & willingness to spend. So, the effective desire for the possesion of a commodity is known as human wants.
(a) Classification of Human Wants : Human wants are classified necessaries such as. Comforts and luxuries. Necessaries are again divided into three categories such as-

(a) Necessaries of existence or Bare necessaries.
(b) Necessaries of Efficiency.
(c) Necessaries of Convention or conventional necessaries.

(i) Necessaries : Necessaries are those things of life without which a man cannot live. Necessaries constitute the urgent wants of life. Necessaries are further sub-divided into necessaries of life, efficiency and convention.

(a) Necessaries of Life: Necessaries of life are those wants without the satisfaction of which a man cannot live. This is otherwise known as necessaries of existence. Food, clothing and shelter are the examples of necessarise of life. These are called absolute necessaries.

(b) Necessaries of Efficiency: Those commodities and services which increase the efficiency of the workers are known as necessaries of efficiency which means the working ability of a person. A pen for a student or a car for a doctor increases their efficiency to work. Similarly, good food like milk, egg and fish increase the efficiency of the worker. A good workshop is favourable for increasing the efficiency.

(c) Conventional Necessaries : These necessaries arise out of social convention and habit. Heavy dowrey and wasteful expenditure on ceremonies are conventional necessaries which arise due to social custom. Habit of smoking, drinking of wine or chewing pan are the examples of conventional necessaries which arise due to habit. If these necessaries are abolished, the efficiency of a person will not be affected. All these necessaries differ from person to person, country to country and climate to climate.

(ii) Comfort: Comforts are those things of life which make life easy and comfortable. The use of those things do not increase our efficiency but the non-use of it decreases our efficiency. An electric fan in summer season, a cycle for a student residing four kilometers away from the college or good food are the examples of comfort. The distinction between necessaries of efficiency and comforts is not clear cut. They differ in degree but not in kind.

(iii) Luxuries : Luxuries are those things which are not necessary for life nor for increasing efficiency. These are things which satisfy superfluous consumption. Motor car for a student is an example of luxury as a diamond necklace for a lady. Luxuries like expensive food or television set or gold ornaments are harmless but wine and gambling are harmful luxuries. They decrease the efficiency of workers. The consumption of such luxuries increase social evil and tension. It tempts a man to practise corruption and he accumulates black money.

Question 6.
What are the luxuries ? Is the consumption of luxuries justified ?
Answer:
Luxuries are those things of human life which satisfy superflous wants. These are intended for display of vanity and aristocracy of a person. There is a controversy among economists whether consumption of luxury is beneficial or harmful. Arguments in favour of consumption of luxuries:

(a) Employment: Production of luxurious articles create employment. But production of necessaries also can create similar or even more employment.

(b) Transfer of Income : Luxuries are consumed by the rich and are produced by the poor. Therefore, there is transfer of income from the rich to the poor. But transfer of wealth from the rich to the poor can be affected by taxing rich and doing public works for poor than misutilising resources of luxurious production.

(c) Consumption of luxuries encourage invention and new method of production. But such inventions are also possible in case of necessaries.

(d) Consumption of luxuries develop a taste for liberal art like painting music, dance, drama and decoration. Nobody objects to have such cultural shows. They are a part of national life. But every body will object if it is only intended for rich people.

Against Consumption of Luxuries :
(i) Luxuries display a sense of vanity and as such gives rise to jealousy and dissatisfation. Luxuries reduce the incentive and efficiency of worker by encouraging consumption of liquor.

(ii) Consumption of luxury diverts resources from production of essential commodities to inessential commodities. It involves waste of resources.

(iii) It encourages social tension and class conflict. Poor people in society revolt against the luxuries of the rich. Therefore, consumption of luxuries is harmful.

Consumption of luxuries in a poor country like India is doubly harmful because we have no resources to produce necessaries. We can get only resources by reducing consumption and increasing saving. If reduction of consumption is desirable reduction of luxurious consumption is doubly desirable.

Group – C

Objective type Questions with Answers
I. Multiple Choice Questions with Answers :

Question 1.
What is meant by utility in Economics ?
(i) Usefulness of the good
(ii) Want satisfying power of a good
(iii) Desirabity of the good
(iv) All of the above
Answer:
(ii) Want satisfying power of a good

Question 2.
Which of the followings is not the feature of utility ?
(i) utility is subjective
(ii) utility may be cardinal or ordinal
(iii) utility is same as usefulness
(iv) utility is different from, pleasure
Answer:
(iii) utility is same as usefulness

Question 3.
Making a chair from the wood creates ____ utility?
(i) form utility
(ii) place utility
(iii) time utility
(iv) service utility
Answer:
(i) form utility

Question 4.
Storing rice in winter and selling it during rainy season creates ____
(i) place utility
(ii) time utility
(iii) form utility
(iv) none of the above
Answer:
(ii) time utility

Question 5.
Which of the following goods does not create utility ?
(i) Orange
(ii) Curd
(iii) Opium
(iv) None of the above
Answer:
(iv) None of the above

Question 6.
Measurability of utility interms of 1, 2, 3 ……….. shows that
(i) Utility is subjective
(ii) Utility is ordinal
(iii) Utility is cardinal
(iv) All of the above
Answer:
(iii) Utility is cardinal

Question 7.
Which is of the followings is not true ?
(i) Utility can be arranged in order
(ii) Utility is not found in unpleasant goods
(iii) Utility differes from usefulness
(iv) None of the above
Answer:
(ii) Utility is not found in unpleasant goods

Question 8.
Service of a teacher creates
(i) Time utility
(ii) Service utility
(iii) Place utility
(iv) No utility is created
Answer:
(ii) Service utility

Question 9.
Utility varies from person to person this notion indicates that
(i) Utility is objective
(ii) Utility can be measured
(iii) Utility is subjective
(iv) None of the above
Answer:
(iii) Utility is subjective

Question 10.
People consume opium because
(i) It has utility
(ii) It is useful
(iii) It is necessary for life
(iv) All of the above
Answer:
(i) It has utility

Question 11.
Selling the goods of rural area in town creates.
(i) From utility
(ii) Time utility
(iii) Place utility
(iv) Service utility
Answer:
(iii) Place utility

Question 12.
Wealth in Economics refers to
(i) All economic goods
(ii) valuable goods
(iii) Money, gold & diamond
(iv) goods having value in use
Answer:
(i) All economic goods

Question 13.
Which of the following is not wealth ?
(i) Bay of Bengal
(ii) Sun or moon
(iii) The Himalayas
(iv) None of the above
Answer:
(iv) None of the above

Question 14.
Which of the following is not the charactersistic of wealth.
(i) High value in use
(ii) High value in exchange
(iii) Marketability
(iv) External possession
Answer:
(i) High value in use

Question 15.
Wood in the jungle is not wealth because
(i) It has no utility
(ii) It is not transferable
(iii) It is not scarce
(iv) All of the above
Answer:
(iii) It is not scarce

Question 16.
Money is not wealth because
(i) It has no utility
(ii) It can not be transferred
(iii) It is not scarce
(iv) All of the above
Answer:
(i) It has no utility

Question 17.
Diamond is wealth but air sunshine are not wealth because
(i) Diamond has high value in use
(ii) Diamond is scarce in supply
(iii) Air sunshine are gift of nature
(iv) Diamond is very costly.
Answer:
(ii) Diamond is scarce in supply

Question 18.
What type of wealth is your Kisan Vikas Patra?
(i) National wealth
(ii) Potential wealth
(iii) Individual wealth
(iv) Negative wealth
Answer:
(iii) Individual wealth

Question 19.
Which of the followings is not wealth?
(i) Your H.S.C. certificate
(ii) Technical skill
(iii) Honesty
(iv) All of the above
Answer:
(iv) All of the above

Question 20.
Voice of a singer is not wealth because.
(i) It is not transferable
(ii) It is not externally possessed
(iii) It is not marketable
(iv) All of the above
Answer:
(iv) All of the above

Question 21.
Which of the followings is an economic goods though it is a free gift of nature?
(i) Land
(ii) Ocean
(iii) River
(iv) Forests
Answer:
(i) Land

Question 22.
Which of the followings is wealth in Economics?
(i) Voice of Kishore Kumar
(ii) Service of a doctor
(iii) Motor Cycle of your father
(iv) Good health of your brother
Answer:
(iii) Motor Cycle of your father

Question 23.
Price is money value of
(i) Utility
(ii) Value in use
(iii) Value in exchange
(iv) All of the above
Answer:
(iii) Value in exchange

Question 24.
Which is not a feature of human wants?
(i) Wants vary from person to person
(ii) Wants are competitive
(iii) Wants are relative in nature
(iv) None of the above
Answer:
(iv) None of the above

Question 25.
Man and machine describes that
(i) Wants are competitive
(ii) Wants are complementary
(iii) Wants are both competitive and complementary
(iv) None of these
Answer:
(i) Wants are competitive

Question 26.
Which of the followings refers to utility?
(i) usefulness of a good
(ii) pleasure dervided from a good
(iii) wants satisfying power of a good
(iv) All of the above
Answer:
(iii) wants satisfying power of a good

Question 27.
Utility is a an
(i) objective & cardinal concept
(ii) subjective & cardinal concept
(iii) subjective & ordinal concept
(iv) objective & ordinal concept
Answer:
(ii) subjective & cardinal concept

Question 28.
Which of the followings has no utility?
(i) Wine
(ii) Cigarette
(iii) Opium
(iv) None of these
Answer:
(iv) None of these

Question 29.
What type of utility is created if a table is made from a log of wood?
(i) place utility
(ii) time utility
(iii) service utility
(iv) form utility
Answer:
(iv) form utility

Question 30.
What type of utility is created when river water is supplied in the town?
(i) Place utility
(ii) form utility
(iii) service utility
(iv) time utility
Answer:
(i) Place utility

Question 31.
What happens to the utility of a sweater if it is stored during summer & sold during winter?
(i) utility remains same
(ii) utility decreases
(iii) utility increases
(iv) utility first increases then decreases
Answer:
(iii) utility increases

Question 32.
Consumption refers to the
(i) creation of utility
(ii) destruction of utility
(iii) increas in utility
(iv) decrease in the utility
Answer:
(ii) destruction of utility

Question 33.
What type of utility is caused on account of export of a goods?
(i) Time utility
(ii) Place utility
(iii) Form utility
(iv) None of the above
Answer:
(ii) Place utility

Question 34.
A table for a student is a
(i) Producers goods
(ii) Consumers good
(iii) Public goods
(iv) Free goods.
Answer:
(ii) Consumers good

Question 35.
What type of good is your club?
(i) Private goods
(ii) Producers goods
(iii) Public goods
(iv) All of the above
Answer:
(iii) Public goods

Question 36.
For which goods principle of exclusion is not applicable.
(i) Consumers goods
(ii) Public goods
(iii) Producers goods
(iv) Private goods
Answer:
(ii) Public goods

Question 37.
‘Principle of exclusion’ is applicable to
(i) Material goods
(ii) Economic goods
(iii) Private goods
(iv) Public goods
Answer:
(iii) Private goods

Question 38.
A log of teak wood collected from the jungle & used in the city is a
(i) Producers goods
(ii) Free goods
(iii) Economic goods
(iv) Consumers goods
Answer:
(i) Producers goods

Question 39.
What type of good is the service of a doctor?
(i) Consumers goods
(ii) Economic goods
(iii) Service
(iv) None of the above
Answer:
(iii) Service

Question 40.
A goods having value in use but not value in exchange is
(i) Economic goods
(ii) Non-material goods
(iii) Free goods
(iv) All of the above
Answer:
(iii) Free goods

Question 41.
Cotton used for preparing cloth is a
(i) Consumers goods
(ii) Producers goods
(iii) Free goods
(iv) Non-material goods
Answer:
(ii) Producers goods

Question 42.
Goods which yields direct satisfaction is
(i) Producers goods
(ii) Free goods
(iii) Consumers goods
(iv) Private goods
Answer:
(iii) Consumers goods

Question 43.
What type of goods is the mid-day meal supplied to the primary student?
(i) Public goods
(ii) Private goods
(iii) Producers goods
(iv) All of the above
Answer:
(i) Public goods

Question 44.
What type of goods is your personal computer?
(i) Private goods
(ii) Public goods
(iii) Consumers goods
(iv) All of the above
Answer:
(i) Private goods

Question 45.
Goods which are visible, tangible & have a shape is called
(i) Consumers goods
(ii) Private goods
(iii) Material goods
(iv) Non-material goods
Answer:
(iii) Material goods

Question 46.
Which of the followings is not free good?
(i) Air
(ii) Sand
(iii) Mid-day meal to children
(iv) Land
Answer:
(iii) Mid-day meal to children

Question 47.
In economics, wealth means
(i) All the costly items
(ii) All producers goods
(iii) Goods having value in use
(iv) Goods having value in exchange
Answer:
(iv) Goods having value in exchange

Question 48.
The scarcity as a characteristics of wealth refers to
(i) Its demand is scarce in relation to its supply
(ii) Both the demand & supply of it are scarce.
(iii) Its supply is scarce in relation to its demand
(iv) All of the above
Answer:
(iii) Its supply is scarce in relation to its demand

Question 49.
In economics, health is not treated as wealth; because
(i) it has no utility
(ii) it is not scarce
(iii) it can not be marketed
(iv) All of the above
Answer:
(iii) it can not be marketed

Question 50.
What type of wealth is Paradeep Port?
(i) National wealth
(ii) Personal wealth
(iii) Cosmopolitan wealth
(iv) None of the above
Answer:
(i) National wealth

Question 51.
What type of wealth is money ?
(i) National wealth
(ii) International wealth
(iii) Personal wealth
(iv) None of the above
Answer:
(iv) None of the above

Question 52.
Why is not money treated as wealth ?
(i) It has no value in use
(ii) It has no value-in-exchange
(iii) It is not transferable
(iv) All of the above
Answer:
(i) It has no value in use

Question 53.
What type up wealth is the cheque drawn on a nation ?
(i) Personal wealth
(ii) National wealth
(iii) International wealth
(iv) Consumption wealth
Answer:
(i) Personal wealth

Question 54.
A milk man bringing milk from the village to the city’ creates
(i) place utility
(ii) time utility
(iii) form utility
(iv) both (i) and (ii)
Answer:
(i) place utility

Question 55.
The value of comodity expressed in terms of money is called
(i) usefulness
(ii) utility
(iii) price
(iv) both (i) and (ii)
Answer:
(iii) price

Question 56.
Sand on the sea shore is not wealth because it lacks
(i) utility
(ii) scarcity
(iii) marketability
(iv) external Possession
Answer:
(ii) scarcity

Question 57.
Air is not treated as wealth because it does not have
(i) utility
(ii) scarcity
(iii) marketability
(iv) both (i) and (ii)
Answer:
(ii) scarcity

Question 58.
In Economics, value of a good refers to
(i) utility
(ii) value-in-use
(iii) value-in-exchange
(iv) All of the above
Answer:
(iii) value-in-exehange

Question 59.
Under what type of wealth can the invention of a scientist of a nation be included ?
(i) Personal wealth
(ii) International wealth
(iii) National wealth
(iv) Not wealth at all
Answer:
(iii) National wealth

Question 60.
The number of goods which can be commanded in exchange of a particular goods shows,
(i) utility
(ii) exchange value
(iii) value-in-use
(iv) both (i) and (ii)
Answer:
(ii) exchange value

Question 61.
One rupee coin is not wealth because it has no
(i) utility
(ii) value-in-use
(iii) value-in-exchange
(iv) Both (i) and (ii)
Answer:
(iv) Both (i) and (ii)

Question 62.
Your H. S. C. certificate cannot be treated as wealth in economics because it has no
(i) utility
(ii) scarcity
(iii) transferability
(iv) All of the above
Answer:
(iii) transferability

Question 63.
The desire for the possession of a commodity is known as
(i) demand
(ii) effective desire
(iii) human wants
(iv) None of the above
Answer:
(iii) human wants

Question 64.
Effective desire indicates
(i) willingness to buy
(ii) ability to pay
(iii) willingness to spend
(iv) All of the above
Answer:
(iv) All of the above

Question 65.
Wants of car can be satisfied with the satisfaction of wants of petrol; it shows the feature that
(i) wants are competitive
(ii) wants are complementary
(iii) wants are both competative
(iv) All of the above
Answer:
(ii) wants are complementary

Question 66.
A hungry man can satisfy his wants either by taking rice or bread ; it shows the feature that
(i) wants are complementary
(ii) wants are alternative
(iii) wants are competitive
(iv) wants are recurring
Answer:
(ii) wants are alternative

Question 67.
The satisfaction of wants like wine is the type of
(i) luxuries
(ii) comforts
(iii) necessaries for efficiency
(iv) conventional necessaries
AnsWER:
(iv) conventional necessaries

Question 68.
A car for a student is a
(i) Necessary for efficiency
(ii) comforts
(iii) luxury
(iv) conventional necessary
Answer:
(iii) luxury

Question 69.
A pen for a student is
(i) Necessary
(ii) comforts
(iii) luxury
(iv) conventional necessary
Answer:
(i) Necessary

Question 70.
The consumption of luxuries is justified because
(i) it increases the efficiency of the consumer
(ii) it increases the employment
(iii) it leads to social welfare
(iv) all of the above
Answer:
(ii) it increases the employment

II. Fill in the blanks :

Question 1.
Human wants are ____
Answer:
Unlimited

Question 2.
All ____ may rise or fall together, but not values.
Answer:
Prices

Question 3.
Air in Puri sea beach is an example of ____ goods.
Answer:
Free

Question 4.
Your Fountain Pen is an example of ____ wealth.
Answer:
Personal

Question 5.
The desire to have a thing is called ____
Answer:
Wants

Question 6.
Man and machine are both complementary and ____
Answer:
Competitive

Question 7.
Chewing pan is an example of ____
Answer:
Conventional necessary

Question 8.
An electric fan in summer season is an example of ____
Answer:
Comfort

Question 9.
____ arise due to scarcity of resources and unlimited nature of human wants.
Answer:
Economic problems

Question 10.
According to ____ Economics is a study of mankind in the ordinary business of life.
Answer:
Marshall

Question 11.
All the human wants are not ____ important.
Answer:
Equally

Question 12.
Resources are of ____ uses.
Answer:
Alternative uses.

Question 13.
Utility is a ____ concepts
Answer:
Subjective

Question 14.
Cuttack Paradip Road is a ____ wealth
Answer:
National

Question 15.
Pasific ocean is an example of ____ wealth
Answer:
International

Question 16.
Wants of Pen and Ink is said to be ____
Answer:
Complementary

Question 17.
Feast in the marriage is ____ necessary
Answer:
Conventional

Question 18.
Your Matriculation certificate is not wealth because it is not ____
Answer:
Transferable

Question 19.
The transportation of goods by a truck driver causes ____ utility
Answer:
Service

Question 20.
The chair made from a log of wood is an example of ____ utility
Answer:
Form Utility

Question 21.
The Ice cream consumed during summer creates ____ utility
Answer:
Time

Question 22.
Selling the vegetables in the town by transporting it from rural areas creates ____ utility
Answer:
Place

Question 23.
According to Marshall Utility is measurable in terms of ____
Answer:
Money

Question 24.
The voice of Lata Mangeskar is not considered to be wealth because it lacks ____
Answer:
External Possession

Question 25.
The treatment of doctor is an example of ____utility
Answer:
Service

III. Correct the Sentences :

Question 1.
Utility is objective
Answer:
Incorrect.
Correct: Utility is subjective

Question 2.
Wine has no utility.
Incorrect.
Correct: Wine has no usefulness

Question 3.
Utility is the effect of consumption and satisfaction is the cause of the consumption.
Answer:
Incorrect.
Correct: Utility is the cause of consumption and satisfaction is the effect of consumption.

Question 4.
Water supplied from a river to a city creates form utility.
Answer:
Incorrect.
Correct: Water supplied from a river to a city creates place utility.

Question 5.
If a good is stored and sold later on it creates form utility.
Answer:
Incorrect.
Correct: If a good is stored and sold later on it creates time utility.

Question 6.
Making of a chair from a log of wood is an example of time utility.
Answer:
Incorrect.
Correct: Making of a chair from a log of wood is an example of form utility.

Question 7.
Treatment made by a doctor is a case of time utility.
Answer:
Incorrect.
Correct: Treatment made by a doctor is a case of service utility.

Question 8.
Services are non-material goods.
Answer:
Correct.

Question 9.
All goods are free gift of nature.
Answer:
Incorrect.
Correct: Only free goods are free gift of nature.

Question 10.
The supply of free goods is scarce in relation to its demand.
Answer:
Incorrect.
Correct: The supply of free goods is abundant in relation to its demand.

Question 11.
Free goods does not have value-in-exchange.
Answer:
Correct

Question 12.
Water supplied in a city is a free goods.
Answer:
Incorrect.
Correct: Water supplied in a city is an economic goods.

Question 13.
The supply of economic goods are scarce.
Answer:
Correct.

Question 14.
Consumers goods are used for further production.
Answer:
Incorrect.
Correct: Producers goods are used for further production.

Question 15.
Producers goods satisfy the human wants directly.
Answer:
Incorrect.
Correct: Consumers goods satisfy the human wants directly.

Question 16.
Intermidiate goods are either consumed or used for resale.
Answer:
Incorrect.
Correct: Final goods are either consumed or used for resale.

Question 17.
Intermediate goods are used for further production.
Answer:
Correct.

Question 18.
The Principle of exclusion is applicable to public goods.
Answer:
Incorrect.
Correct: The Principle of exclusion is applicable to private goods.

Question 19.
The supply of private goods leads to collective welfare.
Answer:
Incorrect.
Correct: The supply of public goods leads to collective welfare.

Question 20.
National Highway is a public goods.
Answer:
Correct.

Question 21.
All the goods are wealth.
Answer:
Incorrect.
Correct: Only the economic goods are wealth.

Question 22.
Wealth has only value-in-use.
Answer:
Incorrect.
Correct: Wealth has value in use and value in exchange.

Question 23.
Money is wealth.
Answer:
Incorrect.
Correct: Money is title to wealth.

Question 24.
Health is wealth.
Answer:
Incorrect.
Correct: Health is not wealth.

Question 25.
Cheque & Draft are wealth.
Answer:
Incorrect.
Correct: Cheque and draft are representative wealth.

Question 26.
Park is an individual wealth.
Answer:
Incorrect.
Correct: Park is a social wealth.

Question 27.
Bay of Bengal is a national wealth.
Answer:
Incorrect.
Correct: Bay of Bengal is an international wealth.

Question 28.
Good will of a business is not wealth.
Answer:
Incorrect.
Correct: Good-will of a business is an individual wealth.

Question 29.
Borrowings of an individual is an individual wealth.
Answer:
Incorrect.
Correct: Borrowing of an individual is a negative wealth.

Question 30.
All wealth is capital.
Answer:
Incorrect.
Correct: All capital is wealth.

Question 31.
Wealth is a flow but income is a stock.
Answer:
Incorrect.
Correct: Wealth is a stock but income is a flow.

Question 32.
Capital is a part of wealth.
Answer:
Correct.

Question 33.
Value in economics refers to value-in-use.
Answer:
Incorrect.
Correct: Value in economics refers to value-in exchange.

Question 34.
Value-in-exchange indicates price.
Answer:
Correct.

Question 35.
Value of all commodities change simultaneously: but price of different goods remain uncharged.
Answer:
Incorrect.
Correct: Price of different commodities change simultaneously; but value of all goods remain unchanged.

Question 36.
Wants have same intensity.
Answer:
Incorrect.
Correct: Wants vary in intensity.

Question 37.
Wants never change with time, place and person.
Answer:
Incorrect.
Correct: Wants vary with time place and person.

Question 38.
News paper for a teacher is comfort.
Answer:
Incorrect.
Correct: News paper for a teacher is necessaries.

Question 39.
Habit of smoking is necessaries for efficiency.
Answer:
Incorrect.
Correct: Habit of smoking is conventional necessaries.

Question 40.
Medicine for patient is necessaries.
Answer:
Correct.

Question 41.
Wine for a drunkard is necessaries for life.
Answer:
Incorrect.
Correct: Wine for a drunkard is conventional necessaries.

Question 42.
Tea is a necessaries for efficiency.
Answer:
Incorrect.
Correct: Tea is a conventional necessaries.

IV. Answer the following questions in one word :

Question 1.
What is utility ?
Answer:
Utility refers to wants satisfying power of a goods.

Question 2.
State a peculiarity of ‘Utility.’
Answer:
Utility is a subjective concept.

Question 3.
Give an example of a goods which has utility but does not have usefulness.
Answer:
Alcohol has utility but does not have usefulness.

Question 4.
What is form utility ?
Answer:
If the utility of a goods increases with the change of its form, it is called form utility.

Question 5.
What is place utility ?
Answer:
If the utility of a goods increases with the change of place, it is called place utility.

Question 6.
What type of utility is created if a log of wood is converted into a table ?
Answer:
Form utility is created if a log of wood is converted into table.

Question 7.
Give an example of place Utility ?
Answer:
If the water collected from the river is supplied in the city, it is called ‘place utility’.

Question 8.
What is service utility ?
Answer:
The service of a person, if satisfies the wants of other it is called service utility.

Question 9.
Give an example of service utility ?
Answer:
Service of a doctor is called ‘service utility.’

Question 10.
What type of utility is created if one goods is stored and sold later on ?
Answer:
Time utility is created if one goods is stored & sold later on.

Question 11.
What is goods ?
Answer:
Anything which satisfies the human wants is called goods.

Question 12.
What is the difference between goods & services ?
Answer:
Goods may be viewed as both the commodities which are visible, tangible & having a shape but services which are invisible, intangible & do not have any shape.

Question 13.
Which goods does not have value-in-exchange ?
Answer:
Free goods does not have value in-exchange.

Question 14.
What is economic goods ?
Answer:
Economic goods are those goods which are possessed with the payment of price & have both value-in-use & value in-exchange.

Question 15.
Give an example of economic goods which is a free gift of nature ?
Answer:
Land is an economic goods which is a free gift of nature.

Question 16.
What is consumers’ goods ?
Answer:
Consumers’ goods are those goods which can satisfy the human wants directly.

Question 17.
What is producers’ goods ?
Answer:
The goods which are used for further production is called producers’ goods.

Question 18.
Give an example of producers goods ?
Answer:
Machine is an example of producers’ goods.

Question 19.
Which goods are used for further production ?
Answer:
Producers’ goods are used for further production.

Question 20.
Which goods are called goods of first order ?
Answer:
Consumers’ goods are called goods of first order.

Question 21.
What is intermediate goods ?
Answer:
The goods which are used for producing goods is called intermediate goods.

Question 22.
What is final goods ?
Answer:
Goods which can be directly used for consumption or for resale is called final goods.

Question 23.
Which are the features of private goods ?
Answer:
Private goods are those goods whose consumption is rival and excludable and also owned by private individuals.

Question 24.
To which goods, the principle of exclusion is applied ?
Answer:
Principle of exclusion is applied to private goods.

Question 25.
What is wealth in economics ?
Answer:
In Economics, wealth refers to all economic goods.

Question 26.
What are the characteristics of wealth ?
Answer:
Utility, scarcity, transferability & external possession are the features of wealth.

Question 27.
Which characteristics of wealth shows its marketability ?
Answer:
The features like transferability shows the marketability of wealth.

Question 28.
Why health is not treated as wealth in economics ?
Answer:
Health does not satisfy the features of wealth like transferability & external possession.

Question 29.
Is money wealth ?
Answer:
Money is not wealth because it does not have utility of its own.

Question 30.
What type of wealth is cheque, shares and bonds ?
Answer:
Cheque, share & bonds are representative wealth.

Question 31.
Give an example of national wealth ?
Answer:
Hirakud Dam is an example of national wealth.

Question 32.
What type of wealth is U.N.O ?
Answer:
U.N.O. is a cosmopolitan wealth.

Question 33.
What type of wealth is Rourkela Steel Plant ?
Answer:
Rourkela Steel plant is a national wealth.

Question 34.
What type of wealth is a club ?
Answer:
A club is a social wealth.

Question 35.
What is value in economics ?
Answer:
Value in economics refers to value in exchange.

Question 36.
What is Price ?
Answer:
Value-in-exchange expressed in terms of money is called price.

Question 37.
What is value in exchange.
Answer:
Value in exchange refers to what a commodity can purchase in terms of other commodity.

Question 38.
What type of wealth is business good-will ?
Answer:
Business good will is an individual wealth.

Question 39.
Why air is not wealth ?
Answer:
As air is not scarce, it is not called wealth.

Question 40.
What type of wealth is river Mahanadi ?
Answer:
River Mahanadi is a national wealth.

Question 41.
What is human wants ?
Answer:
The desire for the possession of a goods is called human wants.

Question 42.
What is complementary goods ?
Answer:
There are some goods which are simultaneously consumed for the satisfaction of a wants are called complementary goods.

Question 43.
What are the necessaries ?
Answer:
Necessaries are those goods which are the basic needs of life.

Question 44.
What is conventional necessaries ?
Answer:
Those necessaries which arise out of the social convention are called conventional necessaries.

Question 45.
What type of wants is car for a doctor ?
Answer:
Car for a doctor is necessaries for efficiency.

Question 46.
What are comforts ?
Answer:
Comforts are those goods which make life comfortable.

Question 47.
What type of wants is wine for a drunkard ?
Answer:
Wine for a drunkard is a conventional necessaries.

Question 48.
What type of wants is a grand feast for marriage ?
Answer:
A Grand feast for marriage is a conventional necessaries.

Question 49.
What type of want is smoking for a smoker ?
Answer:
Conventional necessaries

Question 50.
What type of want is a car for a student ?
Answer:
A car for a student is luxury.

Question 51.
Why an aeroplane is not treated as wants for a poor man ?
Answer:
An aeroplane is not a want for a poor man because he does not have adequate purchasing power for it.

Odisha State Board CHSE Odisha Class 12 Economics Solutions Chapter 1 Definition of Economics and Central Problems of An Economy Questions and Answers.

CHSE Odisha 12th Class Economics Chapter 1 Question Answer Definition of Economics and Central Problems of An Economy

Group – A

Short type Questions with Answers
I. Answer within Two/Three sentence.

Question 1.
What is the meaning of the term ‘Economics’ ?
Answer:
The term “Economics” is originally derived from Greek words “Oikis” which means ! iousehold” & “Nemein” which means “Management”. As such, economics is referred as management, of household.

Question 2.
Write down the wealth definition given by Adam Smith.
Answer:
The first systematic definition of economics is given by Adam Smith , the father of economics in his masterpiece “An Enquiry into the Nature and Causes of wealth of Nations” published in 1776. He defined economics as “Science of Wealth” . It includes the acquisition, accumulation and spending of wealth.

Question 3.
Describe Welfare definition of Alfred Marshall.
Answer:
Alfred Marshall propounded a new definition with different touch in his book “ Principles of Economics” published in 1890. His definition is accepted as “Welfare Definition.”
According to Dr. Marshall “ Economics is a study of mankind in ordinary business of life; it examines that part of individual & social actiuon which is most closely connected with the attainment and with the use of material requisites of well being.”

Question 4.
What is Scarcity definition ?
Answer:
The scarcity definition has been enunciated by Lionel Robbins in his book “Essay on the Nature and Significance of Economic Science” published in the year 1932.
According to Robbins, “Economics is a science which studies the human behaviour as a relationship between ends and scarce means which have alternative uses.”

Question 5.
What is the classical view on Subject matter of economics ?
Answer:
Subject matter of economics is a controversial subject. That is why Mrs. Barbara Wooton said, “Whenever six economists are gathered, there are seven opinions.” The classical economists like Adam Smith, J.S. Mill, David Ricardo, LB. Say regarded economics as a science which studied wealth. They considered only material goods as wealth. And wealth formed the subject matter of economics.

Question 6.
What is the Central Problems of Economics ?
Answer:
the origin of the economic problem is in scarcity of resources Multiplicity of end forces on us the problem of choice among the ends so that the most intense among them are satisfied now.

Question 7.
What is Production Possibility Curve ?
Answer:
A production-possibilities curve shows the various combinations of the goods which an economy can produce with given resources and under the given technology. It is a downwards sloping curve which is concave to the origin.

II. Answer within Five/Six sentence :

Question 1.
Write short notes on Economic Activity.
Answer:

• Economic activity refers to that activity which is concerned with earning of income and spending of income.
• All the economic activities include those activities related to consumption, production and distribution
• Economic activities are undertaken in order to satisfy various human wants.
• The economic activities constitute the ordinary business of life.
• Economic activities are executed by the rational human beings who pursue to maximise the satisfaction with limited resources.

Question 2.
What is Economic problem ?
Answer:

• Economic problem arises becauses of unlimited wants and imited resources.
• Choice in the context of multiplicity of wants and limited resources constitute the basic economic problem.
• Economic problem emerges because of scarce resources having alternative uses for which choice is to be made.
• Problem relating to allocation of resources, production of goods and distribution of goods also constitute economic problem.
• Problem on the attainment of economic growth also forms the component of economic problem.

Question 3.
Describe Wealth Definition.
Answer:

• Adam Smith, the father of economics formulated the wealth definition of economics
• It is considered to be the first systematic definition of economics
• According to Adam Smith, economics is a ‘Science of Wealth’ and gives emphasis on material wealth.
• It deals with the acquisition of wealth, accumulation of wealth and spending of wealth.
• Thus, Wealth definition deals with the consumption, production and distribution.

Question 4.
Describe Welfare definition.
Answer:

• The welfare definition has been enunciated by Alfred Marshall in his book “Principle s of economics” published in 1890.
• According to Marshall, “Economics is a study of mankind in the ordinary business of life; it examines that part of individual and social action which is closely connected with the attainment and the use of material requisites of well being.”
• Marshall gave primary place to man and secondary to wealth.
• According to Marshall, economics deals with the material welfare.
• Marshall’s definition, thus, classifies the economic activities into material welfare and non-material welfare.

Question 5.
Describe Robbins definitio.
Answer:

• Lionel Robbins formulated a definition which is called “Scarcity definition.”
• According to him, “Economics is the science which studies the human behaviour as a relationship between ends and scarce means which have alternative uses.”
• Choice in the context of satisfaction of multiple wants and scarcity of resources form the basis of this definition.
• Robbins definition deals with the unlimited wants, limited resources having altmative uses and choice for the satisfaction of wants in order of intensity.
• Robbins definition is more analytical, comprehensive and treats economics as a positive science.

Question 6.
Central Problems of Economics.
Answer:
The origin of the economic problem is in scarcity of resources. Multiplicity of ends forces on us the problem of choice among the ends so that the most intense among them are satisfied now. If there were only a single end, the problem of how to use the means would be a technological problem. Solution of a technical problem requires knowledge solely of engineering and physical sciences. Solving an economic problem involves value judgements, for such a problem inevitably involves the calculation of how much of one goal has to be sacrificed to attain a particular increment in an other goal. This is known as the Principle of opportunity cost. It tells us the rate at which we have to sacrifice one goal in order to satisfy another goal by a given amount. This principle is very well illustrated by the Production Possibility Curve to study the economic problems.

Question 7.
What do you mean by production possibility curve ?
Answer:
The set of problems facing every economy can be very clearly analysed with the help of what Professor Samuelson called the Production-Possibilities or Boundary Curve. This curve helps us in distinctly showing the relationships among the set of problems/of an economy. The production- possibility curve illustrates three concepts : scarcity, choice and opportunity cost. A production-possibilities curve shows the various combinations of the goods which an economy can produce with given resources and under the given technology.

Group – B

Long Type Questions With Answers

Question 1.
Describe“Wealth Definition” of Economics.
Answer:
Adam Smith is considered as the first economist who has formulated a systematic definition of economics for the first time in his book “An Enquiry into the Nature and Causes of Wealth of Nations” published in 1776. He defined economics “as a Science of Wealth. ” Hence, his definition is universally accepted as. “Wealth definition. ”
According to Adam Smith, all that economics studies is wealth. Economics deals with the acquisition, accumulation and utilisation of wealth. It looks into the process of production and consumption of wealth.
Features : The “Wealth definition of economics as pronounced by Adam Smith contains the following features:

1. Study of Wealth : Adam Smith’s Wealth definition is the study of wealth alone, Hence, it deals with those activities which are related to production, consumption, exchange and distribution.
2. Considers only material commodities : Smith’s definition categorically emphasises on only material commodities. Economics, according to Adam Smith, constitutes only material commodities. These are called wealth according to Adam Smith’s definition. As such, this definition ignores non-matrial goods like services of all types, free goods like air, water etc.
3. Deals with causes of Wealth : In Wealth definition, it is described that economics studies the causes of wealth accumulation. To increase wealth, production of material goods will have to be stepped up.
4. Much emphasis on Wealth: In this definition, wealth is considered to be the sole factors. The main aim of the political economy is to increase the riches (wealth) of the economy.

MERITS :

• Adam Smith’s definition is the first systematic definition of economics which separates economics from politics. This makes economics as an independent subject.
• The Wealth definition of Adam Smith seeks to look into the possible causes which lead to increase the wealth.
• This definition signifies the material goods (material wealth) which are scarce.
• This definition dictates the nature of an economic man who pursues to achieve his needs to the maximum.

DEMERITS :
(i) Gives much emphasis to wealth : Adam Smith’s defintion gives too much emphasis to wealth. Only wealth is treated to be the most significant factor which is even more important than man. Wealth occupies a primary place whereas man occupies secondary place. Thus, definition itself restricts the scope of economics by giving much importance to wealth.

(ii) Provides restricted meaning of wealth: This definition provides a restricted meaning of wealth by considering only material commodities (material wealth). Non-material goods like services of all types are ignored though these services constitute a part of wealth in modern days. Thus, by restricting the wealth to material goods only, this definition has narrowed the scope of economics.

(iii) Ignores wealfare : The concept of welfare which is a significant and long cherished concept has been outrightly ignored by Adam Smith. This definition has not given importance to the economic welfare. It emphasises only on the accumulation of wealth but pays no attention to the equitable distribution of wealth and its uses for the welfare of the society.

(iv) Concept of Economic man : This definition is based on the concept of economic man who works for. Selfish ends alone and not found in real life. But in realism, man’s activities are influenced by moral, social and religious factors.

(v) Ignores problem of Scarcity and Choice : This definition does not make any discussion on the problem of scarcity and choice which are two common concepts to be discussed. Besides, this definition is ambiguous and static in nature Critics like Carlyle, Ruskin and others criticised Adam Smith for his definition and treats economics as a “Dismal Science.” Above all, Wealth definition given by Adam Smith is narrow, controversial and unscientific

Question 2.
“Economics is a Science of Chocie” Discuss this view in the context of Robbins’ definition of Economics.
Answer:
Lionel Robbins, an eminent English economist enunciated a comprehensive definition on economics. The imperfections and inadequacies of previous definitions inspired him to advocate an analytical definition. In his words, economics is the science which studies the human behaviour as a relationship between ends and scarce means which have alternative uses “Robbins raised economics to a dignified status. In his book “Nature and Significance of Economic Science (1932)” he discussed on the several universal facts with following elements.
ELEMENTS:

1. Unlimited wants
2. Limited scarce resources.
3. Alternative uses of resources.
4. Different intensity of wants.
5. Problem of choice

1. Unlimited wants : Robbins calls wants as the ends. These ends or wants are numerous, limitless and numberless. When one wants is satisfied, another wants takes its place. Thus, it contended that the human beings confront with the multiplicity of wants. So it is impossible to satisfy all of the wants of human beings.

2. Limited (scarce) resources : In Robbins definition, the term ‘means’ indicate resources. Resources are those things which can satisfy human wants directly or indirectly. But the resources are scarce in the sense that these goods have limited supply in relation to its demand. So the human beings fail to satisfy all of his wants and are compelled to postpone the satisfaction of less urgent wants. The relative scarcity of the resources poses economic problem. So, economics is termed as a Science of Scarcity.

3. Alternative uses of the resources : Prof. Robbins reveals the alternative uses of the resources in his difinition. It implies that these resources can be put into alternative uses. For example, – coal has several uses. This leads to create an economic problem in the allocation of these limited resources.

4. Different intensity of wants : It is derived from the definition that all the wants are not equally important or urgent. It means wants are of different intensity. Some wants are more urgent than other. Thus, a man is forced to make a choice of wants. So Economics, according to Robbins is a Science of Choice.

5. Problem of Choice : Unlimited wants, limited resources and alternative uses of resources create an economic problem. Every man confronts with scarcity of resources. Hence, he is forced to make a choice of wants in his allocation of resources. This problem is the central problem of economics.

MERITS :
Robbins definition is comprehensive and scientific in out look. This definition is superior to wealth and welfare definition and hence is universally appreciated. The major merits of Robbins definition are as follows :

1. An analytical definition : Robbins definition is an analytical definition. He provides the reasons for the study of economic problems.
2. A universal definition : Robbins definition is universal in nature. It deals with common problems arising out of unlimited wants and limited resources. So it is applicable everywhere.
3. More comprehensive definition: Robbins definition combines human behaviour with the choice between ends and scarce means. So this definition has wider scope than other definition.

DEMERITS :
Though Robbins definition is logical and scientific yet it suffers from several demerits.

(i) Self contradictory : Robbins has contradicted himself by his two views about choice between ends, hi the first place, he contends that Economics is the Science of choice These two contentions are mutually contradictory.

(ii) Concept of Welfare : Robbins’ definition has also hidden concept of welfare. According to Robbins, Economics deals with the choice between ends and means. It implies that there is human welfare to solve this problem. Thus, the idea of welfare is very much in Robbins definition.

(iii) Narrow definition : Another drawback of Robbins definition is that it only deals with the problem of choice. But in modem days, the allocation of resources is not the only problem. Rather, there are other problems like distribution of national income, employment, regional development which are ignored in Robbins definition.
However, Robbins definition is unique and has practical validity. It is a comprehensive definition that touches different aspects of Economics ‘

Question 3.
Explain Marshall’s definition of Economics.
Answer:
Alfred Marshall, an eminent British economist has enunciated a definition of economics in his masterpiece “Principles of Economics” published in 1890. Being dissatisfied with the definition given by Adam Smith, Marshall tries to interprete Economics as a Science of Welfare. Hence, Marshall’s definition is otherwise called “Welfare definition”. In his definition, Marshall emphasised on human welfare than wealth. According to him, wealth is a means to satisty human wants but not an end in itself.

Marshall’s definition reads as ‘‘Economics is a study of mankind in the ordinary business of life. It examines that part of individual and social action which is most closely connected with the attainment and with the use of material requisites of well-being ”

Features :
1. A study of mankind : Economics studies the economic activities of man. Man performs many types of activities. They are social, religious and economics. Economics is the study of economic activities which are concerned with the economic activities of man.

2. Ordinary business of life : Every man works mostly to earn wealth and spends his earning to get maximum satisfaction out of it. This is the activity of an ordinary man. Economics studies only ordinary man not extra ordinary people like Sadhus and Santhas etc.

3. Study of Individual & Social action: Economics studies the personal and social activities of man which are concerned with his material welfare. It is a study of the individuals on the one hand and social organisation of economic activities on the other.

4. Study of material welfare : The main emphasis of this definition is on material welfare. This is the major difference of this definition from the definition given by Adam Smith. One must note that economics is a subject which studies the material welfare of man. The study of non material welfare is ignored in his definition.

5. Normative Scieence : According to this definition, economics is the study of the causes affecting material welfare It is therefore a social science. Economics doesnot only concern with the material means; it studies the related activities which of course concern with wealth.

Merits :
1. A classificatory Definition : Marshall’s definition classifies the economic activities of man into two types: (i) Material welfare, (2) Non-material welfare. Similliarly, men are classified as ordinary and extraordinary. Economic activites are classified as individual and social. Thus Marshall’s definition has served to put economics as a class by itself, distinguished from other sciences.

2. Avoids criticism made against Adam Smith : This definition emphasises man and his welfare.lt mentions wealth later on – Prof. Pigou compared economics with the science of medicine. He regarded it as an instrument of the material welfare of mankind. Thus, economics is no more a dismal science.

3. Clear about the Nature of Economics : This definition tells that economics is a social science. It is not a pure science. It is also not an art. It is one among the social sciences.

4. Clear on the scope of Economics : The definition is also having the merit of laying down the scope of economics clearly. It studies only the material activities of man. It is concerned with the ordinary men not extraordinary men.

Criticism:
1. Study of all types of economic activity of men: Marshall’s definition restricted economics to the study of man in the ordinary business of life. According to Robbins, all men have economic problem. This problem is of limited resources and comparatively much more ends or wants. This problem may be called the problem of scarcity. All men, whether rich or poor, are faced by this problem. Therefore, economics studies all men, whether rich or poor.

2. Restricts the scope of Economics : In Robbin,s view, this definition has limited the scope of economics to the study of material goods only which promote material welfare. But there are non-material services of a singer, a doctor, a teacher or a lawyer which have economic value. Thus, the scope of economics is restricted

3. Economics as pure science : The definition based on material welfare tends to show that economics is a social science. This idea in Robbin’s view, is wrong. Economics is not a social science simply because it studies human beings. It may at best be called a human science. It is a pure science like Physics, Chemistry because it has universally applicable laws.

4. The definition is not analytical: This definition is only classificatory in nature,It doesnot tells us the central problem of economics. In Robbins’ view, the definition of economics must be related to a scientific analyscis of economic activities.

5. Economics is a positive Science: Robbins also criticises the Marshallian definition for its normative character. In Robbins’ view, economics is entirely neutral between ends as every positive science is. The study of ends is outside of its scope. An economist does not study the nature of norms.

6. Impractical: This definition is impractical. The material welfare definition assumes that it is possible to divide a man’s activities in to material and non- material, economic and non-economic. In practice, there is no such clearcut distinction between economic and non economic activities. Therefore, the definition is not practical.

Question 4.
Make comparative analysis between Robbins’ and Marshall’s definitions of Economics.
Answer:
Alfred Marshall and Lionel Robbins propounded two different and separate definitions of economics. Both the formulations bear different aspects of economics. Marshall’s welfare definition and Robbins scarcity definition are enunciated during different periods. But the detailed analysis of the two theme observed certain similarities and dismilarities in their respective contents. So it is not justified to derive a hasty conclusion that these two definitions are completely different from each other.These similarities between the two definitions of economics are discussed below :

Similarities :
1. Wealth and Scarce means : In Marshallian framework, the term wealth is used in form of materal welfare. In Scarcity definition, Lionel Robbins has introduced ‘scarce means’ which simply denotes wealth. It is only the change of words to express and emphasise the wealth in both the definitions. Thus, these two definitions are similar in this respect.

2. Primary place to man : Alfred Marshall’s & Lionel Robbins have given much emphasis on the study of man. Marshall studied wealth for human welfare and Robbins’ described human behaviour as a relationship between ends and scarce means which have alternative uses. Furthermore, Marshall also interpretes that it is on the one side a study of wealth and on the other and more important is the study of a man. Thus, both the definitions aim at the study of human beings.

3. Rational behaviour of man : The comprehensive analysis of both the definitions reveals that both presumes rational man for their study. These two definitions are based on rational behaviour of man. In Marshallian analysis, it tells that man always pursues to maximise his welfare whereas in Robbins language, man tries to maximise his satisfaction. In this context, both the definitions are construed as similar.

Dissimilarities:
In spite of the above mentioned similarities, these two definitions contain certain different concepts. There observed certain fundamental differences between the two definitions.The important distinctions between the two are mentioned below.

1. Distinction between social and human science: According to Marshall and his disciples, economics is viewed as a social science. It studies rational, ordinary social human beings. In Robbins’ view, economics is viewed as human science associated with the economic activities of ordinary and extraordinary men. Every man confronts with economic problem.

2. Distinction between economic and non-economic activities : In Marshallian version, economic activities refer to those activities which are concerned with material commodities promoting material welfare. Robbins’language, all those economic activities the problem of valuation. Thus, Robbins definition is more comprehensive as it includes both the material and non material goods

3. Normative & Positive science: In welfare definition, Marshall clearly describes economics as a normative science. Because it values the welfare of human beings. In Robbins’ version, economics is a positive science.Thus, there lies the difference between the two.

4. Classification and Analytical definition : Marshall’s definition is classificatory in nature because it puts economics as a subject as it is separated from other social sciences. He delimits the subject matter of economics to material activities leading to material welfare. On the otherhand, Robbins submits an analytical definition which concentrates on the basic economic problem.

5. Difference regarding man and his welfare: Marshall’s definition gives much stress on man whereas Robbins’ definition emphasises on the economic problem. From the above analysis, it is presumed that though both the definitions contain some common concepts yet there observed significant differences between the two.

Question 5.
What is the scope of Economics ?
Answer:
By scope of economics, we mean the area of its study or the extent of its study. It is essential to know the boundaries of the study of economics for scientific analysis of the subject In the scope of economics, we discuss its boundaries. Scope of economics answers mainly the following three questions:
1. What is the subject matter of economics ?
2. What is the nature of economics ?
3. What are the limitations of economics ?
Now we will study in detail the answers to these three questions.
Subject Matter of Economics
Subject matter of economics is a controversial subject. That is why Mrs. Barbara Wooton said, “Whenever six economists are gathered, there are seven opinions.” The classical economists like Adam Smith, J.S. Mill, David Ricardo, LB. Say regarded economics as a science which studied wealth. They considered only material goods as wealth. And wealth formed the subject matter of economics. The philosophers of that time criticized this view regarding the subject matter of economics. Marshall removed the defects of the classical view. He regarded economics as a social science studying all those human activities which are related to material welfare. Prof. Robbins found faults with Marshall’s view. So he gave his own opinion and widened the scope of economics, He made economics a science studying all those activities which are related to scarce means in relation to unlimited wants. Thus, according to Prof. Robbins, the problems of valuation and choice are studied in economics.

The scope of economics is very vast. In economics, we study the circular flow of efforts made to satisfy wants and the resulting satisfaction from these efforts.

The economic circle, given on below shows that man has several wants.. In order to satisfy his wants, he makes efforts and thus produces goods and services. From the consumption of these goods and services, he gets satisfaction. Another feature of wants is that when a particular want is satisfied, another want takes its place. So this circular flow goes on as long as a man is alive. It should be borne in mind that wants are of two types: (i) Natural wants, (ii) Artificial wants. Natural wants are those wants which are satisfied by the free gifts of nature like wind, water, heat etc. We do not have to make any effort to get these goods and services. Such wants are not studied in economics.

Artificial wants are satisfied with the man made goods and services like .cloth, food, services of a doctor, etc. Thus we have to make efforts to satisfy them. Artificial wants result from the development of civilisation. They are wants of food, cloth, etc. Only artificial wants form the subject matter of economics.

The study of wants efforts satisfaction is divided into various sections of study. They are : consumption, production, exchange, distribution, public finance and international trade. In consumption, the laws concerning human wants are studied. For example, law of diminishing marginal utility, law of equimarginal utility, etc. In production, we study the means of production and the laws of production. In exchange the price determination through the forces of demand and supply is studied.

We know that production is the result of the combined efforts of the four factors of production which are land, labour, capital, organisation and the entrepreneur. So production is divided among the four factors of production. This division is studied in the distribution section of economics. The last section of the study of economics is public finance and international trade.

According to Chapman, “Economics is that branch of knowledge which studies the consumption, production, exchange and distribution of wealth.”

Peterson said, “Economics is a study of the processes by which goods and services are produced exchanged and consumed.”

Modem economists say that in economics we study the consumer’s equilibrium, producer’s equilibrium, commodity price determination and factor pricing. Both micro and macro types of economic activities are studied in economics. Static as well as dynamic economic activities come under the study of economics.

Now we can conclude that all economic problems, economic policies and economic laws which are concerned with economic activities and human welfare are included in the subject matter of economics. ‘

Question 6.
What are the central economic problems ?
Answer:
The subject matter of economics is concerned with the rational management or optional allocation of scarce economic resources among the alternative uses so that a consumer (individual) can maximize his satisfaction or a producer (entrepreneur) can maximize his profit (output) & economic as a whole can maximize social welfare. It is a fact that economic system is complex as numerous economic agents pursue & prefer to make choices & guided by incentives. In addition to this, the economic activities undertaken by these economic agents are also numerous which & in to all these make the economic system complex.

Every economy has to solve the basic universal economic problem of allocating scarce resources among competing ends. Professor Frank H. Knight pointed out in his book Economic Organisation that the economic problem maybe sub-divided into five interrelated problems. Every society has to devise its methods of solving these five distinct though interrelated problems. These problems are:

1. fixing standards (What to produce ?)
2. organising production (How to produce ?)
3. distributing the product (how to distribute or whom to produce ?)
4. providing for economic maintenance and progress (or how to ensure economic growth?)
5. adjusting consumption to production over short periods (how to ration the limited supplies).

Being confronted with the above fundamental economic problems, the functions of an economy is concentrated on the rational solution of these problem which originate due to scarcity of resources and competing ends. Solving an economic problem value judgements, for such a problem inevitably involves the calculation of one goal which is to be for gone to achieve a particular increment in an other goal. This is known as the “Principle of Opportunity Cost”. Applying this concept, the economy functions to solve the above problems.

1. Determinng What to Produce : Given the economy’s resource endowments, the first function of an economy aims at determining the composition of output. In a free economy the forces determining the composition of output can be classified into two types : (1) the technology (or input-output co-efficients) which determines the relative cost of each product, and (2) consumers ’ tastes and preferences which determine the relative prices of different goods. Since resources at the disposal of every economy are limited, the allocation of given resources has to be done according to the technology available for transforming the resources into the desired goods which, in turn, depends upon the tastes of the consumers. In a free enterprise economy, the composition of output is determined by the equality of the marginal rate of transformation of a good A into good Y with the marginal rate of substitution of the community for the two goods.

2. Determining How to Produce: Once composition of output is determined by consumers tastes and preferences, the organisation of production is taken up by the business firms. The business firms decide on the allocation of resources and the methods of production keeping the relative prices of the resources and technique of production in vient. The firms would try to attain productive efficiency by combining resources to obtain the given output at least cost and selling the produced output in the most profitable way. Thus, a free enterprise economy, depending upon the price mechanism, takes the two decisions of what to produce and how to produce at the same time. In the process, each resource is used according to its relative abundance or scarcity among different uses.

3. Distribution of National Product (Determining whom to produce) : Decision about the distribution of the national product among the members of the community has two aspects. First, the economy has to determine the relative sizes of the shares to be received by each household. Second, the economy has to determine the bundles of goods and services available to each household. The resource- owning households sell their resources for production and, with the incomes so earned, demand the goods and services produced by the producing firms. The resource owners-sell their services at the highest obtainable prices for them. At the same time, the households try to purchase the satisfaction-maximising bundles of goods and services available through spending their incomes. In this way, the decisions about production and distribution are co-ordinated and made consistent.

4. Rationing of Available Supplies : Price mechanism in a free-enterprise economy also decides how the available supplies of consumers goods would be rationed to consumers over the short run. Some commodities may be in shortage for some time to come. For example, sugar or vanaspati ghee supply may be short of demand which leads to rise in their prices. The consumers adjust their demands for these commodities according to their tastes and incomes in view of the high prices of these commodities. As the seasonal supplies of these commodities get depleted, their prices rise further to attain the limited supplies among prospective consumers.

5. Economic Maintenance and Progress (Economic Growth) : This function of an economic system has three aspects :
(a) maintaining the economy’s productive powers in the face of increasing population;
(b) maintaining the production system through replacement of capital goods which are depreciating in the process of production;
(c) improving the technical processes so as to enhance the nation’s productive power and step up the rate of growth of the national product.

According to Frank H. Knight, this function of maintaining the economic system cuts across all the other functions. It implies stabilising the rate of investment to provide for replacement and growth of capital stock on the one hand and improving the productivity of resources within the economy through technological progress. In a free enterprise economy, this function is performed by individual firms, the government providing the needed infrastructure to facilitate their work.

Question 7.
Discuss the concept of Production possibility curve.
Answer:
The production-possibility curve illustrates three concepts : scarcity, choice and opportunity cost. Let us take up the problem of choice betw een Goods X and Goods Y meant for. If we have full employment of resources and we want to produce more X, then we must produce less of all other goods, thereby reducing the quantity of goods (Y) available to satisfy the needs. The country must make a choice of the combination of X and Y. More X mean less for consumption and vice versa. The opportunity cost of more X is shown by the amount of goods Y forgone in greater production of goods X.

Choice & concept of Opportunity cost: The following table shows the combinations of X and Y for a country which has a choice of production between butter and guns. Given the research and technology in our hypothetical economy, we can produce only so much X and so many Y. The table given below gives the combination of Y and X which can be produced assuming that’ only these two outputs are produced. Combination A shows no X, all Y, On the other extreme, combination F shows all X, no Y. Combinations B to E show that in order to have more Y, we must have less of X. This is the principle of opportunity cost. The opportunity cost of getting five lakh quintals of X in combination B, for example, is three Lakh Y which have to be given up in moving from combination A to combination B.

TABLE
Production Possibilities

Combinations	X (lakh quintals)	Y (lakhs)
A	0	40
B	5	37
C	1o	32
D	15	25
E	20	15
F	25	0

The concepts of scarcity, choice and opportunity cost become even more illuminating if we translate the table of production possibilities into a graph, which we call a production-possibilities curve.

A production-possibilities curve shows the various combinations of the goods which an economy can produce with given resources and under the given technology. Figure reveals a Production-Possibility curve from the point A to F. On the horizontal & axis, the quantity of goods X produced is measured while the vertical Y-axis measures the quantity of all other goods, that is, goods Y).

We plot-all those combinations of X goods and Y goods which can be produced if all the resources are fully employed. The points A, E, B, Fare the points on the production possibility curve AF. This curve separates the combinations of the goods obtainable from the use of given resources from those which are not attainable. Points in side the boundary such as C show the combination of and goods xy which are attainable. Points outside the boundary such as D show combinations which are not attainable because there are not enough resources to produce them. Points on the production- possibility boundary such as E and B are just obtainable. These are the combinations which can be produced only if we use all the available resources. The fact that there are combinations which are not attainable in the diagram shows that there is scarcity of resources and we are thereby forced to make a choice between more or less of one type of goods or the other.

The downward slope of the production possibility curve shows that there is an opportunity cost of producing more of goods X or more of goods Y the opportunity cost being measured by the quantity forgone of the other type of commodity. Thus if we go from point E to point B we are reallocating resources out of production of Y and into production of X as a result of which the quantity of production of X rises from OQ to OS while that ofY production of falls from OP to OR. Thus the opportunity cost of getting QS more of goods X produced is PR goods y sacrificed. This illustrates the problem of allocation of resources in the economy.

It is, of course, always possible that actual production in the economy takes place “at some point inside the production-possibility curve. This is possible either because some of its resources are lying idle or because its resources are being used inefficiently in production; Most of the world’s developed “economies are found to operate on or near about the production- possibility curve in normal times. Almost all the world’s less developed countries produce well inside the production- possibility curve simply because they are unable to manage full employment of their resources. The point C in Fig. is one such point which shows considerable unemployment of the economy’s resources.

The higher the proportion of resources unemployed, the closer will the actual point of production be to the origin.

If the economy is at some point inside the boundary such as point C, then it must be enquired as to why the available resources are not being fully utilised. If the reasons are not being fully utilised because of imperfections in the market mechanism, then these imperfections must be removed. On the other hand, if the un-employed resources are idle because of lack of some complementary factors, then these can be imported to employ these resources. Or the structure of production in the economy has to be changed in order to use all the resources in the country and that too efficiently.

Finally, let us deal with the question of economic growth. A country can push its production- possibilities curve outwards by increasing the economy’s capacity to produce goods over a period of time. For example. Fig. shows the shift of the curve PQ to the position P’Q’ through increased productive capacity which is measured as PP’ of goods Y or QQ’ of goods Y. In this case, if the economy remains on the product ion- possibility boundary, it will be possible to increase the production of all goods over time, moving, for example, from point A to point D. It is clear from this analysis that if we want to increase the production of all goods in the economy, it is necessary to take one of the two steps possible :

(1) If the economy is operating at a point inside the production-possibility curve, then the economy can be made to move on to a point on the curve itself, for example, from point C to point B. This can be done by improving the efficiency of production.

(2) If the economy is already operating on the boundary, then it is necessary to take steps which will move the boundary itself outwards so that production can expand. Shift of the production-possibility curve to the right is possible only through economic growth.
The first method consists of a set of policies based on macroeconomics. The second method is based on what has come to be called economics of growth.

Group – C

Objective type Questions with Answers
I. Multiple Choice Questions with Answers :

Question 1.
Who is the father of Economics ?
(i) J.M. Keynes
(ii) Adam Smith
(iii) Alfred Marshal
(iv) Robbins
Answer:
(ii) Adam Smith

Question 2.
Who has propounded the Welfare definition of Economics ?
(i) J.M. Keynes
(ii) Adam Smith
(iii) Alfred Marshal
(iv) Robbins
Answer:
(iii) Alfred Marshal

Question 3.
Who told, “Economics is a Science of Choice” ?
(i) J.M. Keynes
(ii) Adam Smith
(iii) Alfred Marshal
(iv) Lionel Robbins
Answer:
(iv) Lionel Robbins

Question 4.
Who told, “Economics is a Science of Welath” ?
(i) J.M. Keynes
(ii) Adam Smith
(iii) Alfred Marshal
(iv) Lionel Robbins
Answer:
(ii) Adam Smith

Question 5.
Who has termed Economics as a Science of Material Welfare ?
(i) J.M. Keynes
(ii) Adam Smith
(iii) Alfred Marshal
(iv) Lionel Robbins
Answer:
(iii) Alfred Marshal

Question 6.
Who has Popularised the scarcity definition of Economics ?
(i) J.M. Keynes
(ii) Adam Smith
(iii) Alfred Marshal
(iv) Lionel Robbins
Answer:
(iv) Lionel Robbins

Question 7.
What Constitutes the subject matter of Economics ?
(i) Wants
(ii) Efforts
(iii) Satisfaction
(iv) All of the above
Answer:
(iv) All of the above

Question 8.
Which is the basic components of Scarcity definition ?
(i) Wants are unlimited
(ii) Resources are limited
(iii) Resources are alternatively used
(iv) All of the above
Answer:
(iv) All of the above

Question 9.
Scarcity of resources and choice are very much present in the definition of
(i) J.M. Keynes
(ii) Adam Smith
(iii) Alfred Marshal
(iv) Lionel Robbins
Answer:
(iv) Lionel Robbins

Question 10.
Which definition studies the ordinary business of life ?
(i) Welfare definition of Marshall
(ii) Adam Smith’s Wealth definition
(iii) Lionel Robbin’s Scarcity definition
(iv) None of the above
Answer:
(i) Welfare definition of Marshall

II. Fill in the blanks :

1. The term “Economics” is originally derived from Greek words _____
Answer:
“Oikis”

Question 2.
“Oikis” means _____
Answer:
“Household”

Question 3.
The term “economics” was first of all used by _____
Answer:
Dr. Alfred Marshall

Question 4.
The Principle of Economics” published in 1890 was written by _____
Answer:
Alfred Marshall

Question 5.
_____ had given Wealth Definition of economics
Answer:
Adam Smith

Question 6.
_____ is the exponent of Welfare Definition of Economics.
Answer:
Alfred Marshall

Question 7.
The Scarcity Definition of Economics is given by _____
Answer:
Lionel Robbins

Question 8.
The Name of the book written by Adam Smith was _____
Answer:
“An Enquiry into the Nature and Causes of wealth of Nations”

Question 9.
According to Adam Smith Economics is the study of _____
Answer:
Wealth.

Question 10.
According to Marshall Economics is the study of _____
Answer:
Mankind.

Question 11.
Human wants are _____ and Resources are _____
Answer:
unlimited, limited

Question 12.
The scarcity definition has been enunciated by _____
Answer:
Lionel Robbins

Question 13.
_____ arises beacause of unlimited wants and limited resources.
Answer:
Economic Problem

Question 14.
Wants, effort _____ constitute the subject matter of economics
Answer:
Satisfaction

Question 15.
The production possibility curve slopes _____to the Right:
Answer:
Downwards

Question 16.
The production possibility curve is _____ the origin
Answer:
Concave

III. Correct the Sentences :

Question 1.
Economic problem arises because of limited resources & limited wants.
Answer:
Incorrect:
Correct: Economic problem arises becaues of limited resources & unlimited wants.

Question 2.
Economics has been derived from latin words.,
Answer:
Incorrect:
Correct: Economics has been derived from Greek words.

Question 3.
Marshall formulated the first systematic definiton of Economics
Answer:
Incorrect:
Correct: Adam Smith formulated the first systematic definition of economics.

Question 4.
Marshall defined economics as a “Science of Wealth.
Answer:
Incorrect:
Correct: Adam Smith defined economics as a “Science of Wealth.”

Question 5.
Lionel Robbins is the Father of Economics.
Answer:
Incorrect :
Correct: Adam Smith is the Father of Economics.

Question 6.
Adam Smith gave much emphasis to material welfare.
Answer:
Incorrect
Correct Marshall gave much emphasis to material welfare.

Question 7.
Adam Smith formulated “ Welfare definition” of economics.
Answer:
Incorrect
Correct Alfred Marshall formulated “Wealfare defintion of economics.

Question 8.
Alfred Marshall wrote the book “Principles of economics”.
Answer:
Correct

Question 9.
In Marshall’s definition wealth occupies a primary place.
Answer:
Incorrect
Correct In Marshall definition man occupies a primary place.

Question 10.
In Marshall’s definition the term welfare includes only material welfare.
Answer:
Correct

Question 11.
Lionel Robbins’ defined economics a Science of Choice.
Answer:
Correct

Question 12.
Marshall enunciated “Scarcity definition” of economics.
Answer:
Incorrect
Correct Lionel Robbins’ enunciated “Scarcity definition of economics.’

Question 13.
Want are limited but resoures are unlimited.
Answer:
Incorrect
Correct. Wants are unlimited but resoures are limited.

Question 14.
Scarcity means excess of supply over demand.
Answer:
Incorrect
Correct. Scarcity means excess of demand over supply

Question 15.
Resources are of single use.
Answer:
Incorrect.
Correct. Resources are of alternative uses.

Question 16.
Choice in Robbins’ definition refer to choice of resources.
Answer:
Incorrect.
Correct. Choice in Robbins’ definition refer to choice of satisfaction of present wants over future. ‘

Question 17.
The Production possibility curve is upward sloping.
Answer:
Incorrect
Correct: The Production possibility curve is downward sloping

Question 18.
The Production possibility curve is concave to the origin.
Answer:
Correct: The Production possibility curve is concave to the origin.

Question 19.
Scarcity of resources is the starting point of economics.
Answer:
Incorrect
Correct: Human wants is the starting point of economics.

Question 20.
Wants, efforts and Satisfaction constitutes the scope of economics.
Answer:
Incorrect
Correct: Wants, efforts and Satisfaction constitutes the Subject matter of economics.

IV. Answer the following questions in One word/One Sentence :

Question 1.
What is meant by Economics ?
Answer:
Economics studies all the human activities concerning wealth. It studies the production, consumption, exchange and distribution of wealth.

Question 2.
What is the basic economic problem ?
Answer:
The basic problem in Economics is the satisfaction of wants which involves choice. The choice in the context of multiplicity of wants and limited resources poses to be the basic economic problem.

Question 3.
Why do economic problems arise ?
Answer:
Economic problems arise on account of scarcity of resources and unlimited nature of human wants.

Question 4.
From which word the term ‘Economics’ has been derived ?
Answer:
Economics has been derived from two Greek terms like “Oikos” which means household and “Nemein” which means‘management.’

Question 5.
What is Economics ?
Answer:
Economics is a social science which deals with consumption, production, distribution and exchange of wealth.

Question 6.
What is subject matter of Economics ?
Answer:
Wants, efforts and satisfaction constitute the subject matter of economics.

Question 7.
What is basic economic problem ?
Answer:
Unlimited wants, scarcity of resources and choice for satisfaction of wants constitute the basic economic problem.

Question 8.
Who is the first economist to use the term “Economics” ?
Answer:
Alfred Marshall is the first economist who used the term “Economics” in his book “Principle of Economics” in 1890.

Question 9.
What are the economic activities ?
Answer:
Economic activities refer to those activitis which are concerned with earning of income and spending of income.

Question 10.
Who is the “Father of Economics” ?
Answer:
Adam Smith is the “Father of Economics.”

Question 11.
Which book is the first systematic book on Economics ?
Answer:
“An Enquiry into the Nature and Causes of the Wealth of Nations” is the first book written by Adam Smith in a systematic manner.

Question 12.
What is the name of the book written by Adam Smith ?
Answer:
“An Enquiry into thne Nature and the Causes of Wealth of Nations” is wirtten by Adam Smith.

Question 13.
Who formulated the Wealth definition ?
Answer:
Adam Smith formulated Wealth definition.

Question 14.
What is material wealth ?
Answer:
Material wealth refers to all those commodities which are tangible, visible & have exchange value.

Question 15.
What is the name of the books written by Alfred Marshall ?
Answer:
Alfred Marshall wrote a book named “Principles of Economics” in 1890.

Question 16.
Who gave the Welfare definition of Economics ?
Answer:
Alfred Marshall gave the welfare definition of Economics.

Question 17.
Which occupied primary place in Marshall’s definition ?
Answer:
Man occuupies primary place in Marshall’s definition.

Question 18.
Which concept constitutes the sole factor in Marshall’s definition ?
Answer:
Material welfare.

Question 19.
What is material Welfare ?
Answer:
Material welfare refers to acquision and utilisation of material wealth which can promote human welfare.

Question 20.
What is the ordinary business of life ?
Answer:
The ordinary business of life is to earn and to use the material means for the satisfaction of human wants.

Question 21.
What is Robbins’definition ?
Answer:
Robbins’ definition says “Economics is the science which studies the human behaviour as a relationship between ends and scarce means which have alternative uses.”

Question 22.
Who propounded the “Scarcity definition” of Economics” ?
Answer:
Lionel Robbins propounded the “scarcity definition of Economics.”

Question 23.
What do you mean by scarcity ?
Answer:
Scarcity refers to a situation of excess demand in relation to supply.

Question 24.
What do you mean by resources ?
Answer:
Resources are those goods and services which can satisfy human wants directly and indirectly.

Question 25.
What is the nature of resources ?
Answer:
Resources are of alternative uses.

Question 26.
What is the meaning of “Ends” in Robbins’ definition ?
Answer:
In Robbins’definition “ends” means wants.

Question 27.
What do you mean by wants ?
Answer:
The desire for the possession of a commodity is known as wants.

Question 28.
Who said “ Economics is a science of choice” ?
Answer:
Robbins’ said “Economics is a Science of Choice”.

Odisha State Board CHSE Odisha Class 12 Logic Solutions Chapter 1 The Theory of Inference Questions and Answers.

CHSE Odisha 12th Class Logic Chapter 1 Question Answer The Theory of Inference

Group – A

Short type Questions with Answers
I. Answer with in Two/Three sentence.

Question 1.
What is called an inference?
Answer:

1. Inference is a mental fact or a mental process or a mental product.
2. It is an indirect way to get the different types of knowledge.
3. Example : By observing the smoke arising out of the hil if we say there is fire in that hill then this is called the process of inference.

Question 2.
What is called in immediate inference?
Answer:
(i) Immediate inference is a kind of deductive inference where the conclusion comes out of the only one premise.

(ii) Immediate inference is classified into 4 types, such as conversion, obversion, inversion and contraposition.

Question 3.
What is called deductive inference?
Answer:

1. Deductive inference is that inference where the conclusion comes out of the premises by the process of all to all or all to some, known to known and observe to observe.,
2. Deductive inference is two type such as immediate and mediate.
3. In deductive inference, the conclusion is less general than the premises.

Question 4.
What is called an inductive inference?
Answer:
(i) When the conclusion is drawn out of the premises by the process of some to all, known to known observe to unobserved that is called an inductive inference.

(ii) Here the conclusion is more general than the premises.

Question 5.
What are the main classifications of immediate inference?
Answer:
(i) Immediate inference is classified into 4 types.
(ii) These are conversion, obversion, inversion, contraposition.

Question 6.
Define conversion. Give an example of it.
Answer:
(i) Conversion is a kind of deductive immediate inference in which there is a legitimate transposition between the subject and predicate of the given proposition.

(ii) Example : A = All is P
∴ I = Some P is S

Question 7.
Write any two rules of conversion.
Answer:
(i) The subject of convertend will be predicate in converse and predicate of convertend will be subject in converse.
(ii) Quality will be same both in convertend and converse.

Question 8.
What is called convertend?
Answer:
(i) The given premise of conversion is called converted.
(b) Example : Converted (A) = All S is P
∴ Coverse (I) = Some P is S

Question 9.
What is called converse?
Answer:
(i) The conclusion of conversion is called converse.
(ii) Example : Converted (E) = No. dogs are cats.
∴ Converse (E) = No cats are dogs.

Question 10.
What is obverstion?
Answer:
(i) Obverstion is a kind of immediate deductive inference where the subject of the given premise will be same in conclusion and the predicate of the given premise will be contradictory form in the conclusion.

(ii) Example : A = All S is P
∴ E = No S is not P

Question 11.
Write any three rules of obversion.
Answer:
(i) The subject of obvertend will be same in obverse.
(ii) The predicate of obvertend will be contradictory form in obverse.
(iii) Quality will be change and quantity will be same.

Question 12.
What is called obvertend?
Answer:
(i) The given premise of obversion is called obvertend.

(ii) Example : Obvertend (A) = All S is P
∴ Obverse (E) = No S is not P

Question 13.
What is called obverse?
Answer:
(i) The conclusion of obversion is called obverse.
(ii) Example : obverted (E) – No S is P.
Obverse (A) = All S is not P

Question 14.
What is called mediate inference?
Answer:
(i) Mediate inference is that inference where the conclusion comes out of the two premises taken jointly but not separately.
(ii) Example : All men are mortal All students are men.
∴ All students are mortal

II. Answer with in Five/Six sentence :

Question 1.
Simple conversion :
Answer:
Simple conversion is that conversion where the quality and quantity of both convertend and converse are same.
For example;
Convertend (E) = No S is P.
Converse (E) = No P is S.
Converted (I) = Some S is P.
Converse (I) = Some P is S.

Question 2.
Partial conversion/conversion per limitation/conversion per accidence.
Answer:
In a conversion, if the quality of both convertend and converse are same but the quantity is different that is called partial conversion.
For example:
Convertend (A) = All S is P.
Conversion (I) = Some P is S.

Question 3.
State the rules of conversion.
Answer:
(i) The subject of convertend becomes the predicate of converse.
(ii) The predicate of convertend becomes the subject of converse.
(iii) Quality will be same both in convertend and in converse.
(iv) The term which is not distributed in convertend that should not be distributed in converse.

Question 4.
State the rules of obversion.
Answer:
(i) The subject of obvertend becomes the subject of obverse.
(ii) The predicate of obvertend becomes the contradictory form in obverse.
(iii) Quality of the obvertend will be change in obverse.
(iv) Quantity will be same both in obvertend and in obverse.
(v) The terms which is not distributed in obvertend that terms should not be distributed in obverse.

Question 5.
What is material obversion?
Answer:
Material obversion is a fallacious form of obversion in which the meaning of subject and predicate of conclusion are opposite of the subject and predicate of the premise and the quality remains same. This fallacy is given by the logician Bain.
For example;
Knowledge is good.
∴ Ignorance is bad.

Question 6.
Why ‘O’ proposition cannot be converted?
Answer:
If we convert the ‘O’ proposition then the conclusion will be ‘o’ proposition, in which the predicate ‘S’ term will be distribute. But as this term is not distributed in the premise, so it leads the fallacy, which violates the rules of conversion. Hence ‘o’ proposition cannot be converted.

Question 7.
Distinguish between Immediate and mediate inference.
Answer:
(i) Immediate inference is a kind of deductive inference in which the conclusion is drawn out of the only one premise.

(ii) Immediate inference is divided into 4 types, such as conversion, obversion, inversion and contraposition.

(iii) For example;
All men are mortal.
∴ Some mortal beings are men.

(iv) Mediate inference is a kind of deductive inference in which the conclusion is drawn from two premises taken jointly but not separately. It is otherwise called as syllogism.
For example;
All men are mortal.
All students are men.
All students are mortal

Question 8.
What are the different kinds of inferences?
Answer:
Inference is mainly divided into two types, such as deductive inference and inductive inference. Again deductive inference is divided into 2 types, such as, immediate inference and mediate inference. Mediate inference which is called as syllogism can be pure and mixed.

Question 9.
Distinguish between convertend and converse.
Answer:
The given premise of conversion is called convertend. But the conclusion of converse is called converse. For example;
Convertend (E) = No Dogs are Cats.
Converse (E) = No Cats are Dogs.

Question 10.
Define Obversion.
Answer:
Obversion is a kind of immediate deductive inference where the subject of the given premise will be same in the conclusion but the predicate of the given premise will be the contradictory form in the conclusion.
For example:
Obvertend (A) = All S is P.
∴ Obverse (E) = No S is not P.

Group – B

Long Type Questions With Answers

Question 1.
What is meant by inference ? Is immediate inference an inference at all ? Discuss.
Answer:
Inference is a valid source of knowledge. In most cases we depend upon inferential knowledge. For example, where there is smoke, there is fire. By observing the smoke arising out of the hill, we can infer that there is fire in the hill. So it is a process from something known to unknown. The word ‘inference’ has a double meaning. It is used either as a mental process or a mental product.

(1) As a mental process inference means the process of thought by which we pass from something known to something unknown. The known truths are called the premises and the unknown truth which is inferred from the known truths is called conclusion. In other words, inference is the process of thought by which we derive the conclusion on the basis if one or more premises. So it is a form of reasoning.

(2) As a mental product inference means the product or the result of the mental process. The conclusion alone is the product or result of our thinking. The conclusion which is Justified by the premises is valid. An inference requires more than one propositions and when it is expressed in Language that is called an argument. So an argument consists of more than one propositions. The given proposition is called premise and the proposition which we derive is called conclusion. So it is said that Logic is directly concerned with argument but indirectly concerned with inference.

Classification of inference : Inference is mainly divided into two types; such as :
(i) Deductive inference.
(ii) Inductive inference.

(i) Deductive inference : In deductive inference the conclusion is not more general than the premises. It is implied by the premise or premises. So the conclusion adds nothing new to our knowledge.
Example:
All men are mortal.
Mohan is a man.
∴ Mohan is a man.

(ii) Inductive inference : In inductive inference the conclusion is more general than the premises. It asserts more than what is implied in the premises. It adds something new to our knowledge.
Example:
Ram is mortal.
Hari is mortal.
∴ All men are mortal.

Deductive inference again has been subdivided into two classes viz; (1) Immediate and (2) Mediate. An immediate inference is a kind of deductive inference in which the conclusion is drawn from one premise only.
Example :
All crows are black.
∴ Some crows are black.

A Mediate inference is a kind of deductive inference in which the conclusion is drawn from more than one premise.
Example :
All men are honest.
Madhu is a man.
∴ Madhu is a man.

Some Logicians like Bain and Mill are of the opinion that immediate inference is not an inference all. There is only re-arrangement of terms in the conclusion. The conclusion does not tell anything new, Mill says, it is “inference improperly so-called”. Bain says that the conclusion never goes beyond what is asserted in the premise. But such type of objection are not Justified. One cannot simply ignore the usefulness of immediate inference by criticising that they do not state anything new. Welton says, “In immediate inference the conclusion helps in making the meaning explicit what was implicitly contained in the premises’. Hence it is said that immediate inference is a true form of inference.

Question 2.
Define conversion. Explain its rules and determine the converse of A, E, I and C propositions.
Answer:
Definition : Conversion is a kind of immediate deductive inference in which there is a legitimate transposition of the subject and the predicated of the given proposition.
Conversion is an inference, because here we draw a conclusion from a given premise. It is an immediate inference because here the conclusion is drawn from one premise. It is a deductive inference, because here the conclusion is never more general than the premise.
(i) The name of the proposition which is given in conversion is called convertend.
(ii) The conclusion of conversion is called converse.

Rules of Conversion :

1. The subject of the convertend will be the predicate in the converse.
2. The predicate of the convertend will be the subject in the converse.
3. Quality will be remains same.
4. If a term is not distributed in the convertend, it should not be distributed in the converse.

Explanation of the rules :

1. Rule first + second state the defining characteristics of conversion.
2. Rule third states that the premise and the conclusion have exactly the same terms. Only their positions are interchanged. So if they be positively related in the premise, they cannot be negatively related in the conclusion. Therefore, the quality of the premise cannot be changed in the conclusion.
3. As the conversion is deductive in character so the conclusion cannot be wider than the premise.

Converse of Propositions :
Premise = A = All S is P.
The conclusion must be affirmative (A or I). If we make it ‘A’ (All P is S), then the term ‘P’ will be distributed in the conclusion, without being distributed in the premise. So it must be T (Some P is S). Therefore the converse of A is I. Converse of A is I.

Convertend = ‘A’ = All mangoes are fruits.
Converse = T = Some fruits are Mangoes.
Premise = L = No S is P.
Conclusion = E = No P is S.
Here no rule is violated.
Convertend = ‘E’ = No cats are dogs.
Converse = ‘E’ = No dogs are cats.
Premise T = Some S is P.
Conclusion T = Some P is S.
Here no rule is Violated.
Convertend T = Some fruits are sweet.
∴ Converse T =Some sweet things are fruits.
Premise ‘O’ = Some S is not P.
Conclusion = Nothing.

Convertend T = Some students are not intelligent.
Converse = Nothing.
Here out of the premise (O), if we draw any conclusion then it will be ‘O’ proposition in which ‘S’ term will be distributed, which will never be distributed in the convertend or premise. Therefore ‘O’ Proposition cannot be converted.

Out of the above analysis it is concluded that A given I, I gives I, E gives E, ‘O’ does not give any proposition. I = I, E = E are the example of simple conversion because here the quality and quantity if both convertend and converse are equal. But A = I is the example of partial conversion because here the quantity of both convertend and Converse are different from each other but the quality is same.

Question 3.
State the rules of obversion. Apply them A, E, I and O propositions.
Answer:
Obversion is a kind of immediate inference where the predicate of the conclusion is the contradictory of the predicate of the premise, the subject remaining same. The premise of the obversion is called obvertend and the conclusion is called obverse. There are some rules which are to be followed while obverting a proposition. Those rule are discussed below.

• Rule-1: The subject of the obvertend becomes the subject of the obverse. The predicate of the obverse is the contradictory of the predicate of obvertend.
• Rule-2 : The quality change. If the premise is affirmative, the conclusion is negative and if the premise is negative the conclusion is affirmative.
• Rule-3 : The quantity of the obverse remains same as the quantity of the obvertend.
• Rule-4 : The term which is not distributed in the obvertend can not be distributed in the obverse.

These rules of obversion can be applied to different propositions and obversion can be done in the following way.

Obversion of ‘A’ proposition :
A-All swans are white (obvertened)
∴ E – No swans are not-white (obverse)
By the application of the rules of obversion ‘A’ proposition can be validly obverted to ‘E’ proposition. By the rule-I the subject ‘Swans’ remained same in the obverse. But the predicate became contradictory from ‘white’ to not – white. By the rule-2, the quality changed. The premise is affirmative whereas the conclusion is negative. By the 3rd rule the quantity of both is universal. Again no term is distributed in the obverse without being distributed in the obvertend.

Obversion of ‘E’ proposition :
E-No swans are white (obvertend)
∴ A-All swans are not-white (obverse)
As we see here, in obversion ‘E’ becomes ‘A’. Here all the rules of obversion are followed. The subject ‘swan’ is distribute in both the places.

Obversion of ‘I’ proposition :
I-some swans are white (obvertend).
∴ O-some swans are not not-white (obvertend).
Here again all the rules are followed. By the application of the rules we get ‘O’ proposition from T proposition.

Obversion of ‘O’ proposition :
O-some swans are not white, (obvertend).
I-some swans are not – white, (obverse).
By the application of all rules, in obversion ‘O’ proposition gives ‘T’ propositions.

Question 4.
State and explain the rules of contraposition.
Answer:
Contraposition is a logical rule that involves transforming a given proposition to an equivalent form. It is particularly useful in formal logic and is employed in various deductive reasoning processes. The rules of contraposition are applied to categorical propositions, which are statements that assert or deny the inclusion or exclusion of a particular subject within a specified class. These propositions are usually expressed in the form “All S is P,” “No S is P,” “Some S is P,” or “Some S is not P.” The contraposition rule primarily applies to universal affirmative and universal negative propositions. Let’s explore the rules of contraposition in detail.

Universal Affirmative Proposition (A-type):
The contraposition of a universal affirmative proposition “All S is P” is derived by transforming it into its logically equivalent form. The contrapositive statement is “All non-P is non-S.”

For example, if we start with the proposition “All birds are animals,” the contrapositive would be “All non-animals are non-birds.” This transformation maintains the logical equivalence between the original statement and its contrapositive.The reasoning behind this lies in recognizing that if everything belonging to class S is also in class P, then everything outside of class P is also outside of class S.

Universal Negative Proposition (E-type):
The contraposition of a universal negative proposition “No S is P” involves transforming it into the logically equivalent form “No non-P is non-S.”

Consider the proposition “No humans are immortal.” The contrapositive would be “No non-immortals are non-humAnswer:” In this case, the contraposition maintains the logical relationship between the absence of inclusion in class P and the absence of inclusion in class S.The contraposition of a universal negative proposition reflects the idea that if no members of S are in P, then no members outside of P are in S.

Particular Affirmative Proposition (I-type) :
Contraposition is not directly applicable to particular affirmative propositions (“Some S is P”). However, it is essential to note that the contrapositive of a particular affirmative proposition is not necessarily logically equivalent to the original statement. The contrapositive of “Some S is P” would be “Some non-P is non-S,” but this does not necessarily preserve the logical relationship between the classes. Due to this limitation, contraposition is most commonly and effectively applied to universal propositions.

Particular Negative Proposition (O-type) :
Similarly to particular affirmative propositions, contraposition is not directly applicable to particular negative propositions (“Some S is not P”). The contrapositive of “Some S is not P” would be “Some non-P is not non-S,” but this does not maintain a clear logical equivalence.

In practice, contraposition is most confidently applied when dealing with universal propositions, where the transformation retains the logical relationship between the classes involved.

In conclusion, contraposition is a valuable rule in logic, particularly when working with universal propositions. It allows for the transformation of statements while preserving logical equivalence. Universal affirmative propositions are contraposed by stating that everything outside of the predicate class is also outside of the subject class. Similarly, universal negative propositions are contraposed by asserting that nothing outside of the predicate class is inside the subject class. It is important to recognize the limitations of contraposition when dealing with particular propositions, as the contrapositives may not maintain a clear logical relationship.

Question 5.
Distinguish between :
(a) Mediate and immediate inference
(b) Simple conversion and conversion by limitation.
Answer:
(a) Mediate and immediate inference.
In logic, mediate inference and immediate inference are two types of logical reasoning processes that involve drawing conclusions from given propositions. These forms of inference play crucial roles in decretive reasoning and are integral to understanding and constructing logical arguments.

Immediate Inference :
Immediate inference involves drawing conclusions directly from a single proposition without the need for an additional premise. It is an inference where the conclusion follows immediately from the given statement. Immediate inferences are typically based on the conversion, obversion, or contraposition of a given proposition. These processes allow us to manipulate the original proposition to derive an immediate inference.

1. Conversion : Conversion is an immediate inference that involves switching the subject and predicate terms of a proposition while maintaining its quality. There are two types of conversion: simple conversion and conversion by limitation.

1. Simple Conversion: In simple conversion, the subject and predicate terms are switched without any change in quantity or quality. For example, from the proposition “All men are mortal,” we can immediately infer “All mortals are men.”
2. Conversion by Limitation: In conversion by limitation, the original proposition is converted, and a limiting term is added. For instance, from “No birds are mammals,” we can infer “No mammals are birds of any kind.”

2. Obversion: Obversion is another immediate inference that involves negating the predicate term of ^ proposition while maintaining the same subject and quality. Additionally, a new term, called the “term of obversion,” is introduced by negating the original predicate. For example, from the proposition “Some cats are black,” we can immediately infer “Some cats are not non-black.”

3. Contraposition : Contraposition is an immediate inference primarily applied to universal affirmative and universal negative propositions. It involves switching the subject and predicate terms and negating both. For instance, from the proposition “All humans are mortal,” we can infer “All non-mortals are non-humAnswer:”

Immediate inferences are particularly useful for simplifying and clarifying propositions, allowing for the quick derivation of conclusions based on the structure and content of a single statement. These processes provide a direct route from a given proposition to a logically equivalent conclusion.

Mediate Inference :
In contrast to immediate inference, mediate inference involves drawing conclusions by using two or more propositions in a logical sequence. This form of inference relies on the establishment of a relationship between premises and the subsequent derivation of a conclusion. Syllogism, a fundamental structure in mediate inference, consists of three propositions: two premises and a conclusion.

1. Categorical Syllogism : A categorical syllogism is a specific form of mediate inference that involves three categorical propositions. The premises and conclusion are statements that assert or deny the inclusion or exclusion of a particular subject within a specified class. The classic example of a categorical syllogism is :
• Premise 1: All humans are mortal.
• Premise 2: Socrates is a human.
• Conclusion: Therefore, Socrates is mortal.
The conclusion follows logically from the combination of the two premises, demonstrating the process of mediate inference.

2. Hypothetical Syllogism : Hypothetical syllogism involves conditional propositions or “if-then” statements. If one proposition implies another and the second proposition implies a third, then the first proposition implies the third. For example :
• Premise 1 : If it rains, then the streets will be wet.
• Premise 2 : If the streets are wet, then people will use umbrellas.
• Conclusion : Therefore, if it rains, people will use umbrellas.
The conclusion is reached by combining the implications of the two conditional premises

3. Disjunctive Syllogism : Disjunctive syllogism involves a disjunctive proposition (an “either/or” statement). If one of the alternatives is eliminated, the other must be true. For example :
• Premise: Either it is sunny or it is raining.
• Elimination: It is not sunny.
• Conclusion: Therefore, it is raining.
The conclusion is derived by eliminating one of the alternatives presented in the initial disjunctive proposition.

In conclusion, mediate inference involves the use of multiple propositions to establish a logical relationship and draw conclusions. Categorical, hypothetical, and disjunctive syllogisms are common forms of mediate inference, providing a structured approach to reasoning and deduction. Immediate inference, on the other hand, allows for the direct derivation of conclusions from a single proposition through processes like conversion, obversion, and contraposition. Both immediate and mediate inferences are fundamental to understanding and constructing logical arguments in various fields of study.

(b) Simple conversion and conversion by limitation.
In the realm of categorical propositions, conversion is a logical operation that involves interchanging the subject and predicate terms of a given statement. Two main types of conversion are simple conversion and conversion by limitation. These techniques are employed to derive nev. propositions from existing ones, and understanding the distinctions between them is crucial for effective reasoning in formal logic.

Simple Conversion :
Simple conversion is a straightforward process that involves interchanging the subject and predicate terms of a given categorical proposition without altering the quality (affirmative or negative) of the original statement. It is applicable to both universal and particular propositions.

Universal Affirmative (A-type) :
For a universal affirmative proposition like “All S is P,” simple conversion yields “All P is S.” This maintains the original affirmation and switches the subject and predicate terms.

Universal Negative (E-type) :
In the Case of a universal negative proposition such as “No S is P,” simple conversion results in “No P is S.” The negativity of the original statement is preserved, but the subject and predicate terms are interchanged.

Particular Affirmative (I-type):
Simple conversion is not applicable to particular affirmative propositions (“Some S is P”) Attempting to convert a particular affirmative proposition using the simple method may lead to ambiguous or invalid conclusions.

Particular Negative (O-type) :
Similarly, simple conversion is not applicable to particular negative propositions (“Some S is not P”). The attempt to convert a particular negative proposition using simple conversion can result in an ambiguous or invalid statement.

Conversion by Limitation :
Conversion by limitation is a more nuanced form of conversion that involves interchanging the subject and predicate terms of a given proposition while also making adjustments to the quantity (universal or particular) and quality (affirmative or negative) of the original statement. This method is applicable to both universal and particular propositions.

Universal Affirmative (A-type) :
When applying conversion by limitation to a universal affirmative proposition “All S is P,” the result is a particular affirmative proposition, “Some P is S.” This conversion maintains the affirmation but changes the quantity from universal to particular.

Universal Negative (E-type) :
Conversion by limitation applied to a universal negative proposition “No S is P” yields a particular negative proposition, “Some non-P is non-S.” Here, the negativity is preserved, and the quantity changes from universal to particular.

Particular Affirmative (I-type) :
For a particular affirmative proposition “Some S is P,” conversion by limitation results in another particular affirmative proposition, “Some P is S.” The original affirmation is retained, and the quantity remains particular.

Particular Negative (O-type) :
Conversion by limitation applied to a particular negative proposition “Some S is not P” produces another particular negative proposition, “Some non-P is not non-S.” The negativity is preserved, and the quantity remains particular.

Distinctions :

• Quantity and Quality :
  1. Simple conversion maintains the quantity and quality of the original proposition.
  2. Conversion by limitation involves adjusting both the quantity and quality during the conversion process.
• Applicability :
  1. Simple conversion is applicable to universal affirmative and negative propositions.
  2. Conversion by limitation is applicable to both universal and particular propositions, and it allows for a more nuanced transformation.
• Resulting Proposition:
  1. Simple conversion results in a proposition with the same quantity and quality as the original statement.
  2. Conversion by limitation results in a proposition with a modified quantity while preserving the quality of the original statement.

In conclusion, while both simple conversion and conversion by limitation involve interchanging subject and predicate terms, they differ in terms of the adjustments made to quantity and quality. Simple conversion is straightforward and maintains the original quantity and quality, whereas conversion by limitation involves more nuanced adjustments, particularly in changing the quantity of the proposition. Understanding these distinctions is essential for precise and accurate reasoning in formal logic.

Group – C

Objective type Questions with Answers
I. Multiple Choice Questions with Answers :

Question 1.
An immediate inference in which the subject and the predicate are interchanged is called :
(i) Conversion
(ii) Obversion
(iii) Inversion
(iv) Nothing
Answer:
(i) Conversion

Question 2.
A term which is not distributed in the premise :
(i) can be distributed in the conclusion
(ii) cannot be distributed in the conclusion
(iii) may sometimes be distributed in the conclusion
(iv) None of these
Answer:
(ii) cannot be distributed in the conclusion

Question 3.
Which of the following is not true of immediate inference ?
(i) It’s conclusion follows from a single premise
(ii) It is a deductive inference
(iii) It is an inductive inference
(iv) Conversion, obversion, contraposition etc. are it’s types
Answer:
(iii) It is an inductive inference

Question 4.
Which of the following is called an inference?
(i) Inference is a logical phenomena
(ii) Inference is a mental phenomena
(iii) Inference is a philosophical phenomena
(iv) None of these are correct.
Answer:
(ii) Inference is a mental phenomena

Question 5.
When an inference is expressed in Language that is called what ?
(i) Argument
(ii) Proposition
(iii) Judgement
(iv) Term
Answer:
(i) Argument

Question 6.
Deductive inference is divided into which of the following?
(i) Conversion and obversion
(ii) Obversion and contraposition
(iii) Immediate and Mediate
(iv) Direct and indirect
Answer:
(iii) Immediate and Mediate

Question 7.
Which of the following is the main division of immediate inference?
(i) Conversion and obversion
(ii) Conversion, obversion, inversion and contraposition
(iii) Conversion, obversion, syllogism.
(iv) None of these are correct .
Answer:
(ii) Conversion, obversion, inversion, contraposition

Question 8.
In immediate inference, the conclusion is drawn from how many premises ?
(i) One
(ii) Two
(iii) Three
(iv) Four
Answer:
(i) One

Question 9.
Conversion is a what kind of inference?
(i) mediate
(ii) Immediate
(iii) Inductive
(iv) Both (ii) & (iii)
Answer:
(ii) immediate

Question 10.
The given premise of conversion is called vyhat?
(i) Convertend
(ii) Convert
(iii) Obvertend
(iv) Converse
Answer:
(i) Convertend

Question 11.
The conclusion of conversion is called what?
(i) Convertend
(ii) Convert
(iii) Converse
(iv) Obverse
Answer:
(iii) Converse

Question 12.
When a conclusion is drawn from more than one premise that is called what?
(i) Immediate inference
(ii) Mediate inference
(iii) Deductive inference
(iv) Inductive inference
Answer:
(ii) Mediate inference

Question 13.
Which of the following are the main classificatioq of inference?
(i) Mediate and immediate
(ii) Deductive and inductive
(iii) Conversion and obversion
(iv) Direct and indirect
Answer:
(ii) Deductive and inductive

Question 14.
What is the coversion of ‘A’ proposition?
(i) ‘A’
(ii) ‘E’
(iii) ‘T’
(iv) ‘O’
Answer:
(iii) ‘T’

Question 15.
What is the conversion of ‘E’ proposition ?
(i) ‘A’
(ii) ‘E’
(iii) ‘I’
(iv) ‘O’
Answer:
(ii) ‘E’

Question 16.
What is the conversion of T proposition?
(i) ‘A’
(ii) ‘E’
(iii) ‘T’
(iv) ‘O’
Answer:
(iii) ‘T’

Question 17.
What is the conversion of ‘O’ proposition ?
(i) ‘A’
(ii) ‘E’
(iii) ‘I’
(iv) Cannot be converted
Answer:
(iv) Cannot be converted

Question 18.
Which of the following propositions convert simply?
(i) I and O proposition
(ii) A and E proposition
(iii) E and I proposition
(iv) E and O proposition
Answer:
(iii) E and I proposition

Question 19.
Which of the following propositions convert practically ?
(i) ‘A’proposition
(ii) ‘E’Proposition
(iii) T proposition
Answer: (i) ‘A’ proposition

Question 20.
Conversion is mainly divided into
(i) Two types
(ii) Threes types
(iii) Four types
(iv) Five Types
Answer:
(i) Two types

Question 21.
Which proposition cannot be converted?
(i) ‘E’
(ii) T
(iii) ‘O’
(iv) ‘A’
Answer:
(iii) ‘O’

Question 22.
State the conversion of “All men are honest”
(i) Some honest beings are men
(ii) No men are not honest
(iii) Some men are honest
(iv) Some men are not honest
Answer:
(i) Some honest beings are men

Question 23.
State the conversion of “No men are Birds”.
(i) All men are not birds
(ii) No birds are men
(iii) Some men are not birds
(iv) No men are not birds
Answer:
(ii) No birds are men

Question 24.
State the conversion of “Some students are intelligent”.
(i) Some students are not intelligent
(ii) Some intelligent beings are students
(iii) No students are intelligent
(iv) No intelligent beings are students
Answer:
(ii) Some intelligent beings are students

Question 25.
State the conversion of “some students are not intelligent”
(i) Some intelligent beings are not students
(ii) Some students are not intelligents
(iii) All students are intelligent
(iv) None of these
Answer:
(iv) None of these

Question 26.
The given premise of obversion is called What ?
(i) Convertend
(ii) Obvertend
(iii) Obverse
(iv) Ob vert
Answer:
(ii) Obvertend

Question 27.
The conclusion of obversion is called what?
(i) Obverse
(ii) Obvertend
(iii) Obvert
(iv) ‘O’Proposition
Answer:
(i) Obverse

Question 28.
What is the obversion of ‘A’ Proposition?
(i) ‘A’proposition
(ii) ‘E’ proposition
(iii) ‘T’proposition
(iv) ‘O’proposition
Answer:
(ii) ‘E’ proposition

Question 29.
What is the obversion of ‘E’ proposition ?
(i) ‘A’proposition
(ii) ‘E’Proposition
(iii) ‘T’Proposition
(iv) ‘O’proposition
Answer:
(i) ‘A’ proposition ,

Question 30.
What is the obversion of T proposition?
(i) ‘A’Proposition
(ii) ‘E’Proposition
(iii) T Proposition
(iv) ‘O’Proposition
Answer:
(iv) ‘O’ Proposition

Question 31.
If we obvert the proposition ‘O’ then we will get which proposition?
(i) ‘A’Proposition
(ii) ‘E’Proposition
(iii) ‘T’Proposition
(iv) ‘O’Proposition
Answer:
(iii) ‘T’ Proposition

Question 32.
The fallacy of obversion is called what?
(i) Fallacy of material obversion
(ii) Fallacy of Accident
(iii) Fallacy of Accent
(iv) None of these
Answer:
(i) Fallacy of material obversion

Question 33.
When the quality and quantity of both convertend and converse are equal that is called what?
(i) Simple conversion
(ii) Partial conversion
(iii) Conversion per limitation
(iv) Material obversion
Answer:
(i) Simple conversion

Question 34.
When only the quantity of both converted and converse are differ from each other but the quality is remain same that is called what?
(i) Simple conversion
(ii) Partial conversion
(iii) Material obversion
(iv) None of these
Answer:
(ii) Partial conversion

Question 35.
What kind of obversion is the following?
Knowledge is good.
Ignorance is bad
(i) Conversion
(ii) Obversion
(iii) Fallacy of material obversion
(iv) None of these
Answer:
(iii) Fallacy of material obversion

Question 36.
State the obvert of the proposition, Some flowers are fragrance.
(i) Some flowers are fragrance
(ii) Some flowers are not-fragrance
(iii) All flowers are fragrance
(iv) No flowers are fragrance.
Answer:
(ii) Some flowers are not-fragrance

Question 37.
The other name of mediate inference is called what?
(i) Syllogism
(ii) Conversion
(iii) Obversion
(iv) Contraposition
Answer:
(i) Syllogism .

Question 38.
How the predicate of obverse is related to the predicate of obvertend?
(i) Same
(ii) Contradictory
(iii) Contrary
(iv) None of these
Answer:
(ii) Contradictory .

II. Fill in the blanks :

Question 1.
_______ is an indirect way of getting the different of knowledge.
Answer:
immediate, mediate

Question 2.
Inference is a _____ process.
Answer:
mental

Question 3.
When an inference is expressed in language is called _____
Answer:
Argument.

Question 4.
Inference is mainly divided into two types such as _____ and _____.
Answer:
Deductive, inductive

Question 5.
Deductive inference is sub divided into _____ and _____ .
Answer:
immediate, mediate

Question 6.
In an inference if the conclusion is drawn out of the only one premise that is called _____ inference.
Answer:
Immediate

Question 7.
In an inference if the conclusion is drawn from two premises that is called _____ inference.
Answer:
Mediate

Question 8.
Immediate inference is divided into _____ types.
Answer:
four

Question 9.
Immediate inference is divided into four types, such as; _____,_____,_____ and _____.
Answer:
Conversion, obversion, contraposition, inversion

Question 10.
In deductive inference we proceed from _____ .
Answer:
all to some

Question 11.
In _____ inductive inference we proceed from _____.
Answer:
some to all

Question 12.
In _____ inference the conclusion is more general than the premises.
Answer:
Inductive

Question 13.
In _____ inference the conclusion is less general than the premises.
Answer:
Deductive

Question 14.
Conversion is a kind of _____ inference.
Answer:
Immediate

Question 15.
The given premise of conversion is called _____.
Answer:
Convertend

Question 16.
The conclusion of conversion is called _____ .
Answer:
Converse

Question 17.
In conversion the quality is_____.
Answer:
Remain same

Question 18.
If the convertend is affirmative, the converse is_____.
Answer:
Affirmative

Question 19.
If the convertend is negative, the converse is _____.
Answer:
Negative

Question 20.
The converse of ‘A’ is _____.
Answer:
‘I’

Question 21.
The converse of ‘E’ is _____.
Answer:
‘E’

Question 22.
The converse of‘T’ is _____.
Answer:
‘T’

Question 23.
The converse of ‘O’ is _____.
Answer:
Impossible

Question 24.
Conversion is divided into two ways, such as _____ and _____.
Answer:
Simple, partial

Question 25.
In _____ there is a legitimate transposition of the subject and the predicate of a proposition.
Answer:
conversion

Question 26.
There is no change in _____ conversion.
Answer:
quality

Question 27.
_____ proposition cannot be converted.
Answer:
‘O’ proposition

Question 28.
If a term is not distributed in the premise, it should not be _____ in the conclusion.
Answer:
Distributed

Question 29.
Where the qualify and quantity of both convertend and converse are same that is called _____ conversion.
Answer:
Simple.

Question 30.
Where the quantity of both convertend and converse are differ but quality is same that is called _____ conversion.
Answer:
Partial

Question 31.
‘I’ gives ‘I’, ‘E’ gives ,‘E’, are the example of _____ conversion.
Answer:
Simple

Question 32.
Generally ‘A’ proposition is converted by _____
Answer:
Limitation

Question 33.
‘A’ gives ‘I’ is the example of _____ conversion.
Answer:
Partial

Question 34.
Obversion is a kind of _____ inference.
Answer:
Immediate

Question 35.
In obversion, ‘A’ gives _____ .
Answer:
‘E’

Question 36.
The obverse of ‘E’ is _____ .
Answer:
‘A’

Question 37.
The obverse of‘T’ is _____.
Answer:
‘O’

Question 38.
The obverse of ‘O’ is _____.
Answer:
‘I’

Question 39.
The given premise of obversion is called _____.
Answer:
Obvertend

Question 40.
The conclusion of obversion is called _____.
Answer:
obverse

Question 41.
In obversion the quantity is _____ .
Answer:
same

Question 42.
In obversion the quality is _____.
Answer:
change

Question 43.
The kind of obversion based on the facts of experience is called as _____.
Answer:
Material obversion

Question 44.
______ has putforth material obversion.
Answer:
Bain

Question 45.
If we violate the rule of obversion then we commit the fallacy of _______.
Answer:
Material obversion

Question 46.
Knowledge is good
∴ Ignorance is bad.
This is an example of ______.
Answer:
Material obversion

Question 47.
The other name of mediate inference is called ______.
Answer:
syllogism

III. Correct the Sentences:

Question 1.
Logic is directly concerned with inference and indirectly concerned with argument.
Answer:
Logic is directly concerned with argument and indirectly concerned with inference.

Question 2.
In deductive inference, the conclusion speaks something new than the premises.
Answer:
In inductive inference, the conclusion speaks something new than the premises.

Question 3.
In inductive inference, the conclusion does not say anything about the inference.
Answer:
In deductive inference, the conclusion does not say anything about the inference.

Question 4.
In inductive inference the conclusion is less general than the premises.
Answer:
In inductive inference the conclusion is more general than the premises.

Question 5.
In inductive inference the conclusion is less general than the premises?
Answer:
In deductive inference the conclusion is less general than the premises

Question 6.
Deductive inference is divided into two types, such as conversion and obversion.
Answer:
Deductive inference is divided into two, types, such as immediate and mediate inference.

Question 7.
Conversion, observation, inversion and contraposition are.
Answer:
Conversion, obversion, inversion and contraposition are the division of immediate inference.

Question 8.
The premise of conversion is called converse.
Answer:
The premise of conversion is called convertend.

Question 9.
The conclusion of conversion is called convertend.
Answer:
The conclusion of conversion is called converse.

Question 10.
The subject of the convertend becomes the subject of the converse and the predicate of the convertend becomes the predicate of the converse.
Answer:
The subject of the convertend becomes the predicate of the converse and the predicate of the convertend becomes the subject of the converse.

Question 11.
The quality of the convertend is the opposite quality of the converse.
Answer:
The quality of the convertend is same with the quality of the converse.

Question 12.
The term which is not distributed in the convertend is distributed in converse.
Answer:
The term which is not distributed in the convertend should not be distributed in converse.

Question 13.
If the quality of convertend and converse remains the same, it is called simple conversion.
Answer:
If the quality of convertend and converse remains the same, it is called partial conversion.

Question 14.
Conversion of ‘E’ proposition is called conversion per accidens.
Answer:
Conversion of ‘A’ proposition is called conversion per accidens.

Question 15.
The conversion of ‘A’ and ‘O’ propositions are called simple conversion.
Answer:
The conversion of ‘E’ and ‘T’ proposition are called simple conversion.

Question 16.
‘A’proposition convert to‘E’proposition.
Answer:
‘A’ proposition covert to ‘T’ proposition.

Question 17.
‘ E ’ proposition convert to ‘ A’ proposition.
Answer:
‘E’ proposition convert to ‘E’ proposition.

Question 18.
T proposition convert to ‘O’ proposition.
Answer:
T proposition convert to ‘T’ proposition.

Question 19.
‘O’ proposition covert to T proposition.
Answer:
‘O’ proposition cannot be converted.

Question 20.
Obversion is a kind of mediate inference.
Answer:
Obversion is a kind of immediate inference.

Question 21.
The premise of obversion is called obverse.
Answer:
The conclusion of obversion is called obverse.

Question 22.
The conclusion of obversion is called obvertend.
Answer:
The premise of obversion is called obvertend.

Question 23.
The quality of the obverse is the same as the quality of the obvertend.
Answer:
The quality of the obverse is the opposite of the quality of the obvertend.

Question 24.
The quantity of the obverse is the opposite of the quantity of the obvertend.
Answer:
The quantity of the obverse is the same as the quantity of the obvertend.

Question 25.
The term which is not distributed in the obvertend is distributed in the Obverse.
Answer:
The term which is not distributed in the obvertend should not be distributed in the obverse.

Question 26.
‘A’ proposition obvert to‘T’ proposition.
Answer:
‘A’ proposition obvert to ‘E’ proposition.

Question 27.
‘E ’ proposition obvert to ‘E’ proposition.
Answer:
‘E’ proposition obvert to ‘A’ proposition.

Question 28.
‘T’ proposition obvert to ‘O’ proposition.
Answer:
‘T’ proposition obvert to ‘O’ proposition.

Question 29.
‘O’ proposition obvert to ‘O’ proposition.
Answer:
‘O’ proposition obvert to ‘T’ proposition.

Question 30.
Mediate inference is otherwise called a conversion.
Answer:
Mediate inference is otherwise called a Syllogism.

IV. Answer the Following Questions in One Word :

Question 1.
What type of knowledge gives us inference?
Answer:
Indirect

Question 2.
What type of process in inference?
Answer:
Mental

Question 3.
How many kinds of inference are there?
Answer:
Two

Question 4.
In which inference the conclusion is more general than the premises?
Answer:
Inductive

Question 5.
In which inference the conclusion is less general than the premises?
Answer:
Deductive

Question 6.
How many types of classifications are there of deductive inference?
Answer:
Two

Question 7.
How many types of classifications are there of immediate inference?
Answer:
Four types
OD

Question 8.
What is the other name of mediate inference?
Answer:
Syllogism

Question 9.
In which inference the conclusion is drawn from only one premise?
Answer:
Immediate

Question 10.
What is the premise of conversion called?
Answer:
Convertend

Question 11.
What is the conclusion of conversion called?
Answer:
Converse

Question 12.
How many propositions does an immediate inference consist of?
Answer:
Two

Question 13.
How many propositions does the mediate inference consist of?
Answer:
Three

Question 14.
What the quality of convertend and converse?
Answer:
Remains same

Question 15.
‘A’proposition convert to which proposition?
Answer:
‘T’

Question 16.
‘E ’ proposition convert to which proposition?
Answer:
‘E’

Question 17.
‘I’ proposition convert to which proposition?
Answer:
‘T’

Question 18.
‘O’ proposition convert to which proposition?
Answer:
cannot be converted

Question 19.
Which proposition cannot be converted?
Answer:
‘O’ proposition

Question 20.
In which conversion the quantity of convertend and converse remain the same?
Answer:
Simple conversion

Question 21.
In which conversion the quantity of convertend and converse are different from each other?
Answer:
partial

Question 22.
Which proposition convert simply?
Answer:
E & I

Question 23.
Which proposition convert partially?
Answer:
‘A’

Question 24.
What the premise of obversion is called?
Answer:
Obvertend

Question 25.
What the conclusion of obversion is called ?
Answer:
Obverse

Question 26.
What the quality of the obvertend and obverse?
Answer:
Change

Question 27.
What the quantity of the obvertend and obverse?
Answer:
Same

Question 28.
What the obvert of ‘A’ proposition?
Answer:
‘E’

Question 29.
What the obvert of ‘E’ proposition ?
Answer:
‘A’

Question 30.
What the obvert of‘T’ proposition?
Answer:
‘O’

Question 31.
What the obvert of ‘O’ proposition?
Answer:
‘T’

Question 32.
What is the name of the fallacy of Obversion?
Answer:
Material Obversion

Question 33.
Can Material obversion be regarded as a form of obversion?
Answer:
No

CHSE Odisha 12th Class Logic Book Solutions (+ 2 2nd Year)

CHSE Odisha 12th Class Logic Book Solutions in English Medium

• Chapter 1 The Theory of Inference
• Chapter 2 Syllogism
• Chapter 3 Mixed Syllogism
• Chapter 4 Deductive and Inductive Fallacies
• Chapter 5 Symbolic Logic
• Chapter 6 Experimental Methods of Mill
• Chapter 7 Scientific Explanation
• Chapter 8 Naya Theory of Knowledge
• Chapter 9 Doctrine of Karma
• Chapter 10 Gandhi: Truth and Non-Violence

CHSE Odisha 12th Class Logic Book Solutions in Odia Medium

Unit 1 ଅନୁମାନ – ଅନୁମାନର ପ୍ରକାରଭେଦ – ଅବ୍ୟବହିତ ଓ ବ୍ୟବହିତ

• ଅନୁମାନ – ଅନୁମାନର ପ୍ରକାରଭେଦ – ଅବ୍ୟବହିତ ଓ ବ୍ୟବହିତ Objective Questions
• ଅନୁମାନ – ଅନୁମାନର ପ୍ରକାରଭେଦ – ଅବ୍ୟବହିତ ଓ ବ୍ୟବହିତ Short & Long Answer Questions

Unit 2 ବ୍ୟବହିତ ଅନୁମାନ ଓ ମିଶ୍ର ତ୍ରିପଦୀଯୁକ୍ତି

• ବ୍ୟବହିତ ଅନୁମାନ ଓ ମିଶ୍ର ତ୍ରିପଦୀଯୁକ୍ତି Objective Questions
• ବ୍ୟବହିତ ଅନୁମାନ ଓ ମିଶ୍ର ତ୍ରିପଦୀଯୁକ୍ତି Short Answer Questions
• ବ୍ୟବହିତ ଅନୁମାନ ଓ ମିଶ୍ର ତ୍ରିପଦୀଯୁକ୍ତି Long Answer Questions

Unit 3 ତର୍କଦୋଷ ଓ ପ୍ରତୀକାତ୍ମକ ତର୍କଶାସ୍ତ୍ର

• ତର୍କଦୋଷ ଓ ପ୍ରତୀକାତ୍ମକ ତର୍କଶାସ୍ତ୍ର Objective Questions
• ତର୍କଦୋଷ ଓ ପ୍ରତୀକାତ୍ମକ ତର୍କଶାସ୍ତ୍ର Short Answer Questions
• ତର୍କଦୋଷ ଓ ପ୍ରତୀକାତ୍ମକ ତର୍କଶାସ୍ତ୍ର Long Answer Questions

Unit 4 ମିଲ୍‌ଙ୍କ ପରୀକ୍ଷଣ ପଦ୍ଧତି, ବୈଜ୍ଞାନିକ ବ୍ୟାଖ୍ୟାନ

• ମିଲ୍‌ଙ୍କ ପରୀକ୍ଷଣ ପଦ୍ଧତି, ବୈଜ୍ଞାନିକ ବ୍ୟାଖ୍ୟାନ Objective Questions
• ମିଲ୍‌ଙ୍କ ପରୀକ୍ଷଣ ପଦ୍ଧତି, ବୈଜ୍ଞାନିକ ବ୍ୟାଖ୍ୟାନ Short & Long Answer Questions

Unit 5 ନ୍ୟାୟଙ୍କ ଜ୍ଞାନ ସିଦ୍ଧାନ୍ତ ଓ କର୍ମବାଦ

• ନ୍ୟାୟଙ୍କ ଜ୍ଞାନ ସିଦ୍ଧାନ୍ତ ଓ କର୍ମବାଦ Objective Questions
• ନ୍ୟାୟଙ୍କ ଜ୍ଞାନ ସିଦ୍ଧାନ୍ତ ଓ କର୍ମବାଦ Short & Long Answer Questions

CHSE Odisha Class 12 Logic Syllabus (+2 2nd Year) in English

Unit 1
The Theory of Inference: Classification of Inference, Conversion, Obversion, Categorical Syllogism: Structure, Figure, Moods. Rules of syllogism, Determination of valid Moods.

Unit 2
Special rules of Figures, Aristotle’s Dictum, Direct and Indirect Reduction.
Mixed Syllogism: Different forms – Hypothetical categorical, Alternative Categorical, Disjunctive Categorical, Dilemma: Forms, Refutation, Rebuttal of Dilemma.

Unit 3
Fallacy: Deductive Fallacy, Semi-logical Fallacies, Inductive Fallacies: Fallacy of Illicit Generalisation, False Analogy, Ignoratio Elenchi. Propositional Logic: Symbolic Logic and its Characteristics, Propositional Variables, Logical Constants, Propositional Connectives, Truth Functions, Construction of Truth Tables, Testing Validity by direct Truth Table Method.

Unit 4
Methods of Experimental Enquiry: Mill’s Five Experimental Methods.
Scientific Explanation: Nature of Scientific Explanation.

Unit 5
Nyaya Theory of Knowledge :Perception and Inference: Vyapti and its ascertainments.
Doctrine of karma: Niskama Karma of Bhagavad Gita, Gandian Concept of Non Violence.

CHSE Odisha Class 12 Logic Book Syllabus (+2 2nd Year)

Unit 1 ଅନୁମାନ – ଅନୁମାନର ପ୍ରକାରଭେଦ – ଅବ୍ୟବହିତ ଓ ବ୍ୟବହିତ
ଅନୁମାନ – ଅନୁମାନର ପ୍ରକାରଭେଦ – ଅବ୍ୟବହିତ ଓ ବ୍ୟବହିତ ଅନୁମାନ – ଅବ୍ୟବହିତ ଅନୁମାନ – ସମବର୍ତ୍ତନ, ବ୍ୟାବର୍ତ୍ତନ; ବ୍ୟବହିତ ଅନୁମାନ– ତ୍ରିପଦୀଯୁକ୍ତି, ଏହାର ଅବୟବାବଳୀ, ନ୍ୟାୟ-ସଂସ୍ଥାନ, ନ୍ୟାୟରୂପ, ତ୍ରିପଦୀଯୁକ୍ତିର ସାଧାରଣ ନିୟମାବଳୀ, ନ୍ୟାୟରୂପ-ମାନଙ୍କର ନିର୍ଦ୍ଧାରଣ ପ୍ରକ୍ରିୟା ।

Unit 2 ବ୍ୟବହିତ ଅନୁମାନ ଓ ମିଶ୍ର ତ୍ରିପଦୀଯୁକ୍ତି
ପ୍ରତ୍ୟେକ ସଂସ୍ଥାନର ସ୍ବତନ୍ତ୍ର ନିୟମାବଳୀ, ଆରିଷ୍ଟୋଟଲଙ୍କ ମୌଳିକ ସୂତ୍ର, ସାକ୍ଷାତ୍ ଓ ଅସାକ୍ଷାତ୍ ରୂପାନ୍ତରୀକରଣ ।
ମିଶ୍ର ତ୍ରିପଦୀଯୁକ୍ତି – ବିଭିନ୍ନ ପ୍ରକାର : ପ୍ରାକଳ୍ପିକ ନିରପେକ୍ଷ, ବିଯୋଜକ – ନିରପେକ୍ଷ, ବୈକଳ୍ପିକ — ନିରପେକ୍ଷ, ଦ୍ବିଶୃଙ୍ଗକ ନ୍ୟାୟ, ଦ୍ବିଶୃଙ୍ଗକ ଯୁକ୍ତିର ପ୍ରକାରଭେଦ, ଦ୍ବି ଶୃଙ୍ଗକ ଯୁକ୍ତିର ଆକାରଗତ ବୈଧତା, ବସ୍ତୁଗତ ସତ୍ୟାସତ୍ୟ ବିଚାର, ଦ୍ବିଶୃଙ୍ଗକ ଯୁକ୍ତିର ପ୍ରତିରୋଧ ।

Unit 3 ତର୍କଦୋଷ ଓ ପ୍ରତୀକାତ୍ମକ ତର୍କଶାସ୍ତ୍ର
ତର୍କଦୋଷ – ଅବରୋହୀ ତର୍କଦୋଷ, ଅବରୋହୀ – ଅନୁମାନ ସମ୍ପର୍କୀୟ ତର୍କଦୋଷ, ଆପାତଃ ତର୍କଦୋଷ । ଆରୋହୀ ତର୍କଦୋଷ – ଅବୈଧ ସାମାନ୍ୟକରଣ ତର୍କଦୋଷ, ଦୁଷ୍ଟ ଉପମା କିମ୍ବା ଦୁର୍ବଳ ଉପମା ତର୍କଦୋଷ, ବଳକା ତର୍କଦୋଷ, ଅବାନ୍ତର ପ୍ରସଙ୍ଗ ଦୋଷ ।
ପ୍ରତୀକାମୂକ ତର୍କଶାସ୍ତ୍ର ଏବଂ ଏହାର ବୈଶିଷ୍ଟ୍ୟ- ତର୍କବାକ୍ୟମୂଳକ ଚଳ- ତର୍କଶାସ୍ତ୍ରୀୟ ସ୍ଥିରାଙ୍କ- ସତ୍ୟଫଳନ – ସତ୍ୟସାରଣୀ- ସତ୍ୟ ସାରଣୀ ପଦ୍ଧତି । ସାକ୍ଷାତ୍ ସତ୍ୟସାରଣୀ ପଦ୍ଧତି ସାହାଯ୍ୟରେ ବୈଧତା ପରୀକ୍ଷା ।
ପୁନରୁକ୍ତିକ ତର୍କବାକ୍ୟମୂଳକ ସୂତ୍ର- ବିରୁଦ୍ଧ ତର୍କବାକ୍ୟମୂଳକ ସୂତ୍ର- ଆପାତିତ ତର୍କବାକ୍ୟମୂଳକ ସୂତ୍ର । ବିଭିନ୍ନ ଉଦାହରଣମାନଙ୍କର ସାକ୍ଷାତ୍ ସତ୍ୟ ସାରଣୀ ପଦ୍ଧତି ସାହାଯ୍ୟରେ ବୈଧତା ପରୀକ୍ଷା ।

Unit 4 ମିଲ୍ଲଙ୍କ ପରୀକ୍ଷଣ ପଦ୍ଧତି, ବୈଜ୍ଞାନିକ ବ୍ୟାଖ୍ୟାନ
ମିଲ୍‌ଙ୍କ ପରୀକ୍ଷଣ ପଦ୍ଧତି—ମିଲ୍‌ଙ୍କ ପାଞ୍ଚଟି ପରୀକ୍ଷଣ ପଦ୍ଧତି— (୧) ଅନ୍ବୟ ପଦ୍ଧତି, (୨) ବ୍ୟତିରେକ ପଦ୍ଧତି, (୩) ସଂଯୁକ୍ତ ପଦ୍ଧତି, (୪) ସହଚାରୀ ପରିବର୍ତ୍ତନ ପଦ୍ଧତି, (୫) ପରିଶେଷ ପଦ୍ଧତି ।
ବୈଜ୍ଞାନିକ ବ୍ୟାଖ୍ୟାନ– ବୈଜ୍ଞାନିକ ବ୍ୟାଖ୍ୟାନର ଲକ୍ଷଣ ।

Unit 5 ନ୍ୟାୟଙ୍କ ଜ୍ଞାନ ସିଦ୍ଧାନ୍ତ ଓ କର୍ମବାଦ
ନ୍ୟାୟଙ୍କ ଜ୍ଞାନ ସିଦ୍ଧାନ୍ତ – ପ୍ରତ୍ୟକ୍ଷ ଓ ଅନୁମାନ, ବ୍ୟାପ୍ତି ଓ ଏହାର ନିର୍ଦ୍ଧାରଣ ପ୍ରକ୍ରିୟା
କର୍ମବାଦ – ଭଗବତ୍ ଗୀତାର ନିଷ୍କାମ କର୍ମ, ଗାନ୍ଧିଜୀଙ୍କ ଅହିଂସାବାଦ

CHSE Odisha Class 12 Text Book Solutions

CHSE Odisha 12th Class Economics Book Solutions (+ 2 2nd Year)

CHSE Odisha Class 12 Economics Book Solutions in Odia Medium

Chapter 1 ଅର୍ଥଶାସ୍ତ୍ରର ସଂଜ୍ଞା, ପରିସର ଓ ବିଷୟବସ୍ତୁ

• ଅର୍ଥଶାସ୍ତ୍ରର ସଂଜ୍ଞା, ପରିସର ଓ ବିଷୟବସ୍ତୁ Objective & Short Answer Questions
• ଅର୍ଥଶାସ୍ତ୍ରର ସଂଜ୍ଞା, ପରିସର ଓ ବିଷୟବସ୍ତୁ Long Answer Questions

Chapter 2 ଅର୍ଥବ୍ୟବସ୍ଥାର ପରିଚୟ ଏବଂ ଅର୍ଥଶାସ୍ତ୍ରର କେନ୍ଦ୍ରୀୟ ସମସ୍ୟାବଳୀ

• ଅର୍ଥବ୍ୟବସ୍ଥାର ପରିଚୟ ଏବଂ ଅର୍ଥଶାସ୍ତ୍ରର କେନ୍ଦ୍ରୀୟ ସମସ୍ୟାବଳୀ Objective & Short Answer Questions
• ଅର୍ଥବ୍ୟବସ୍ଥାର ପରିଚୟ ଏବଂ ଅର୍ଥଶାସ୍ତ୍ରର କେନ୍ଦ୍ରୀୟ ସମସ୍ୟାବଳୀ Long Answer Questions

Chapter 3 ମୌଳିକ ଧାରଣା (ମାନବୀୟ ଅଭାବ, ଉପଯୋଗିତା, ଦ୍ରବ୍ୟ, ମୂଲ୍ୟ, ଦର ଓ ସମ୍ପଦ)

• ମୌଳିକ ଧାରଣା Objective Questions
• ମୌଳିକ ଧାରଣା Short Answer Questions
• ମୌଳିକ ଧାରଣା Long Answer Questions

Chapter 4 ଉପଭୋଗର ନିୟମ

• ଉପଭୋଗର ନିୟମ Objective Questions
• ଉପଭୋଗର ନିୟମ Short & Long Answer Questions

Chapter 5 ଚାହିଦା

• ଚାହିଦା Objective Questions
• ଚାହିଦା Short Answer Questions
• ଚାହିଦା Long Answer Questions

Chapter 6 ଉତ୍ପାଦନ

• ଉତ୍ପାଦନ Objective Questions
• ଉତ୍ପାଦନ Short & Long Answer Questions

Chapter 7 ପରିବ୍ୟୟ

• ପରିବ୍ୟୟ Objective Questions
• ପରିବ୍ୟୟ Short Answer Questions
• ପରିବ୍ୟୟ Long Answer Questions

Chapter 8 ଆୟ

Chapter 9 ଯୋଗାଣ

• ଯୋଗାଣ Objective & Short Answer Questions
• ଯୋଗାଣ Long Answer Questions

Chapter 10 ବଜାର

• ବଜାର Objective Questions
• ବଜାର Short Answer Questions
• ବଜାର Long Answer Questions

Chapter 11 ସମଷ୍ଟି ଅର୍ଥନୀତି

• ସମଷ୍ଟି ଅର୍ଥନୀତି Objective & Short Answer Questions
• ସମଷ୍ଟି ଅର୍ଥନୀତି Long Answer Questions

Chapter 12 ଜାତୀୟ ଆୟ

• ଜାତୀୟ ଆୟ Objective Questions
• ଜାତୀୟ ଆୟ Short Answer Questions
• Long Answer Questions

Chapter 13 କେନ୍‌ସଙ୍କ ଆୟ ନିର୍ଦ୍ଧାରଣ ତତ୍ତ୍ବ

• କେନ୍‌ସଙ୍କ ଆୟ ନିର୍ଦ୍ଧାରଣ ତତ୍ତ୍ବ Objective Questions
• କେନ୍‌ସଙ୍କ ଆୟ ନିର୍ଦ୍ଧାରଣ ତତ୍ତ୍ବ Short Answer Questions
• କେନ୍‌ସଙ୍କ ଆୟ ନିର୍ଦ୍ଧାରଣ ତତ୍ତ୍ବ Long Answer Questions

Chapter 14 ମୁଦ୍ରା

• ମୁଦ୍ରା Objective Questions
• ମୁଦ୍ରା Short & Long Answer Questions

Chapter 15 ବ୍ୟାଙ୍କ

• ବ୍ୟାଙ୍କ Objective Questions
• ବ୍ୟାଙ୍କ Short Answer Questions
• ବ୍ୟାଙ୍କ Long Answer Questions

Chapter 16 ରାଷ୍ଟ୍ରବିତ୍ତ

• ରାଷ୍ଟ୍ରବିତ୍ତ Objective Questions
• ରାଷ୍ଟ୍ରବିତ୍ତ Short Answer Questions
• ରାଷ୍ଟ୍ରବିତ୍ତ Long Answer Questions

Chapter 17 ବଜେଟ୍

CHSE Odisha Class 12 Economics Book Solutions in English Medium

Part A: Introductory Microeconomics

Unit 1 Introduction

• Chapter 1 Definition of Economics and Central Problems of An Economy
• Chapter 2 Basic Economic Concepts (Wants, Utility, Goods, Value, Price and Wealth)

Unit II Consumption and Demand

• Chapter 3 Laws of Consumption
• Chapter 4 Demand and Price Elasticity of Demand

Unit III Production

• Chapter 5 Production

Unit IV Cost, Revenue and Supply

• Chapter 6 Cost and Revenue
• Chapter 7 Supply

Unit V Market

• Chapter 8 Market

Part B: Introductory Macroeconomics

Unit VI Introduction

• Chapter 9 Meaning of Macroeconomics

Unit VII National Income

• Chapter 10 Aggregates Related to National Incorne
• Chapter 11 Theory of Income Determination

Unit VIII Money, Banking and Public Finance

• Chapter 12 Money and Banking
• Chapter 13 Public Finance

CHSE Odisha Class 12 Economics Syllabus (+2 2nd Year)

Second Year CHSE (2025-2026)
Economics Paper-II
(Elementary Micro and Macro Economics)

Part A: Introductory Micro Economics

Unit I Introduction (10 Periods, 10 Marks)

• Definition, scope, and subject matter of economics.
• Meaning of economy and central problems of an economy – scarcity and choice, what, how, and for whom to produce?
• Basic concepts – wants, utility, goods, value, price, and wealth.

Unit II Consumption and Demand (14 Periods, 15 Marks)

• Laws of consumption – marginal and total utility, law of diminishing marginal utility, the law of equimarginal utility, and conditions of consumer’s equilibrium.
• Demand – meaning and determinants, individual and market demand, demand schedule and demand curve, movement along and shifts in the demand curve.
• Price elasticity of demand – concept, determinants, measurement of price elasticity of demand; percentage and geometric methods (linear demand curve), the relation of price elasticity of demand with total expenditure.

Unit III Production (10 Periods, 10 Marks)

• Meaning of production and production function – short run and long run.
• Total, Average, and Marginal Product.
• Law of variable proportions and returns to a factor.

Unit IV Cost, Revenue, and Supply (12 Periods, 15 Marks)

• Cost – money and real cost, implicit and explicit cost, fixed and variable cost, Total, average, and marginal costs in the short ru,n and their relationship (simple analysis).
• Revenue – Total, average, and marginal revenue and their relationship.
• Supply – meaning and law of supply

Unit V Market (8 Periods, 10 Marks)

• Meaning and forms of market, pure and perfect competition, price determination under perfect competition, and effects of shifts in demand and supply.
• Meaning and features of monopoly, monopolistic competition, and oligopoly.

Part B: Introductory Macroeconomics

Unit VI Introduction (4 Periods, 5 Marks)

• Meaning of macroeconomics, Distinction between macro- and microeconomics, the subject matter of macroeconomics

Unit VII National Income (10 Periods, 15 Marks)

• Meaning and aggregates related to national income – GNP, NNP, GDP, and NDP at market price and factor cost.
• National disposable income (Gross and Net), Private Income, Personal income, Personal disposable income, Nominal and real national income.
• Income determination – Aggregate Demand and Supply and their components, simple Keynesian Theory of Income Determination.

Unit VIII Money, Banking, and Public Finance (12 Periods, 20 Marks)

• Meaning and Functions of Money.
• Meaning and Functions of Commercial Banks.
• Functions of the Central Bank.
• Meaning of Public Finance and Difference between public and private finance.
• Budget – Meaning and objectives, a balanced and unbalanced budget, surplus and deficit budget.

CHSE Odisha Class 12 Text Book Solutions

Odisha State Board Elements of Mathematics Class 12 CHSE Odisha Solutions Chapter 4 Matrices Ex 4(b) Textbook Exercise Questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 4 Matrices Exercise 4(b)

Question 1.
State which of the following matrices are symmetric, skew-symmetric, both or not either:

Solution:
(i) Symmetric
(ii) Neither Symmetric nor skew-symmetric
(iii) Symmetric
(iv) Skew symmetric
(v) Both
(vi) Neither symmetric nor skew-symmetric
(vii) Skew symmetric

Question 2.
State ‘True’ or ‘False’:
(i) If A and B are symmetric matrices of the same order and AB – BA ≠ 0, then AB is not symmetric.
Solution:
True

(ii) For any square matrix A, AA’ is symmetric.
Solution:
True

(iii) If A is any skew-symmetric matrix, then A² is also skew-symmetric.
Solution:
False

(iv) If A is symmetric, then A², A³, …, Aⁿ are all symmetric.
Solution:
True

(v) If A is symmetric then A – A¹ is both symmetric and skew-symmetric.
Solution:
False

(vi) For any square matrix (A – A¹)² is skew-symmetric.
Solution:
True

(vii) A matrix which is not symmetric is skew-symmetric.
Solution:
False

Question 3.
(i) If A and B are symmetric matrices of the same order with AB ≠ BA, final whether AB – BA is symmetric or skew symmetric.
Solution:
A and B are symmetric matrices;
Thus A’ = A and B’ = B
Now (AB – BA)’ = (AB)’ – (BA)’
= B’A’ – A’B’
= BA – AB = – (AB – BA)
∴ AB – BA is skew symmetric.

(ii) If a symmetric/skew-symmetric matrix is expressed as a sum of a symmetric and a skew-symmetric matrix then prove that one of the matrices in the sum must be zero matrix.
Solution:
We know that zero matrix is both symmetric as well as skew-symmetric.
Let A is symmetric.
∴ A = A + O where A is symmetric and O is treated as skew-symmetric. If B is skew-symmetric then we can write B = O + B where O is symmetric and B is skew-symmetric.

Question 4.
A and B are square matrices of the same order, prove that
(i) If A, B and AB are all symmetric, then AB – BA = 0
Solution:
Let A, B and AB are all symmetric.
∴A’ = A, B’ = B and (AB)’ = AB
⇒ B’A’ = AB
⇒ BA = AB
⇒ AB – BA = 0

(ii) If A, B and AB are all skew symmetric then AB + BA = 0
Solution:
Let A, B and AB are all skew symmetric matrices
∴ A’ = -A, B’ = -B and (AB)’ = -AB
Now (AB)’ = -AB
⇒ B’A’ = -AB
⇒ (-B) (-A) = -AB
⇒ BA = -AB
⇒ AB + BA = 0

Question 5.
If A = \(\left[\begin{array}{rrr}
1 & 2 & 0 \\
0 & 1 & 3 \\
-2 & 5 & 3
\end{array}\right]\), then verify that A’ = \(\left[\begin{array}{ccc}
1 & 0 & -2 \\
2 & 1 & 5 \\
0 & 3 & 3
\end{array}\right]\)

(i) A+A’ is symmetric

(ii) A-A’ is skew-symmetric

Question 6.
Prove that a unit matrix is its own inverse. Is the converse true?
IfA = \(\left[\begin{array}{rrr}
0 & 1 & -1 \\
4 & -3 & 4 \\
3 & -3 & 4
\end{array}\right]\) show that A² = I and hence A= A^-1.
Solution:
No the converse is not true for example:

Question 7.
Here A is an involuntary matrix, recall the definition given earlier.
Solution:

Question 8.
Show that \(\left[\begin{array}{ll}
\mathbf{0} & \mathbf{1} \\
\mathbf{1} & \mathbf{0}
\end{array}\right]\) is its own inverse.
Solution:

Question 9.
Express as a sum of a symmetric and a skew symmetric matrix.

Solutions:

Question 10.
What is the inverse of

Question 11.
Find inverse of the following matrices by elementary row/column operation (transformations):
(i) \(\left[\begin{array}{ll}
1 & 2 \\
3 & 5
\end{array}\right]\)
Solution:

(ii) \(\left[\begin{array}{ll}
2 & 5 \\
1 & 3
\end{array}\right]\)
Solution:

(iii) \(\left[\begin{array}{cc}
4 & -2 \\
3 & 1
\end{array}\right]\)
Solution:

(iv) \(\left[\begin{array}{ll}
2 & 5 \\
1 & 3
\end{array}\right]\)
Solution:

(v) \(\left[\begin{array}{cc}
1 & 0 \\
2 & -3
\end{array}\right]\)
Solution:

(vi) \(\left[\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}\right]\)
Solution:

Question 12.
Find the inverse of the following matrices using elementary transformation:
(i) \(\left[\begin{array}{lll}
\mathbf{0} & \mathbf{0} & 2 \\
\mathbf{0} & \mathbf{2} & \mathbf{0} \\
\mathbf{2} & \mathbf{0} & \mathbf{0}
\end{array}\right]\)
Solution:

(ii) \(\left[\begin{array}{lll}
0 & 1 & 2 \\
1 & 2 & 3 \\
3 & 1 & 1
\end{array}\right]\)
Solution:

(iii) \(\left[\begin{array}{ccc}
3 & -2 & 3 \\
2 & 1 & -1 \\
4 & -3 & 2
\end{array}\right]\)
Solution:

(iv) \(\left[\begin{array}{lll}
1 & 1 & 2 \\
0 & 1 & 2 \\
1 & 2 & 1
\end{array}\right]\)
Solution:

(v) \(\left[\begin{array}{lll}
1 & 2 & 3 \\
2 & 1 & 4 \\
1 & 0 & 2
\end{array}\right]\)
Solution:

Odisha State Board Elements of Mathematics Class 12 CHSE Odisha Solutions Chapter 7 Continuity and Differentiability Ex 7(b) Textbook Exercise Questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 7 Continuity and Differentiability Exercise 7(b)

Question 1.
Differentiate from definition
(i) e^3x
Solution:

(ii) 2^x2
Solution:

(iii) In (3x + 1)
Solution:

(iv) logx⁵ (Hint : logx⁵ = \(\frac{\ln 5}{\ln x}\))
Solution:

(v) In sin x
Solution:

(vi) x² a^2x
Solution:

Odisha State Board Elements of Mathematics Class 12 Solutions CHSE Odisha Chapter 5 Determinants Ex 5(b) Textbook Exercise Questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 5 Determinants Exercise 5(b)

Question 1.
Write the number of solutions of the following system of equations.
(i) x – 2y = 0
Solution:
No solution

(ii) x – y = 0 and 2x – 2y = 1
Solution:
Infinite

(iii) 2x + y = 2 and -x – 1/2y = 3
Solution:
No solution

(iv) 3x + 2y = 1 and x + 5y = 6
Solution:
One

(v) 2x + 3y + 1 = 0 and x – 3y – 4 = 0
Solution:
One

(vi) x + y + z = 1
x + y + z = 2
2x + 3y + z = 0
Solution:
No solution

(vii) x + 4y – z = 0
3x – 4y – z = 0
x – 3y + z = 0
Solution:
One

(viii) x + y – z = 0
3x – y + z = 0
x – 3y + z = 0
Solution:
One

(ix) a₁x + b₁y + c₁z = 0
a₂x + b₂y + c₂z = 0
a₃x + b₃y + c₃z = 0
and \(\left|\begin{array}{lll}
a_1 & b_1 & c_1 \\
a_2 & b_2 & c_2 \\
a_3 & b_3 & c_3
\end{array}\right|\) = 0
Solution:
Infinite solutions as Δ = Δ₁ = Δ₂ = Δ₃ = 0

Question 2.
Show that the following system is inconsistent.
(a – b)x + (b – c)y + (c – a)z = 0
(b – c)x + (c – a)y + (a – b)z = 0
(c – a)x + (a – b)y + (b – c)z =1
Solution:

Question 3.
(i) The system of equations
x + 2y + 3z = 4
2x + 3y + 4z = 5
3x + 4y + 5z = 6 has
(a) infinitely many solutions
(b) no solution
(c) a unique solution
(d) none of the three
Solution:
(a) infinitely many solutions

(ii) If the system of equations
2x + 5y + 8z = 0
x + 4y + 7z = 0
6x + 9y – z = 0
has a nontrivial solution, then is equal to
(a) 12
(b) -12
(c) 0
(d) none of the three
Solution:
(b) -12

(iii) The system of linear equations
x + y + z = 2
2x + y – z = 3
3x +2y + kz = 4
has a unique solution if
(a) k ≠ 0
(b) -1 < k < 1
(c) -2 < k < 2
(d) k = 0
Solution:
(a) k ≠ 0

(iv) The equations
x + y + z = 6
x + 2y + 3z = 10
x + 2y + mz = n
give infinite number of values of the triplet (x, y, z) if
(a) m = 3, n ∈ R
(b) m = 3, n ≠ 10
(c) m = 3, n = 10
(d) none of the three
Solution:
(c) m = 3, n = 10

(v) The system of equations
2x – y + z = 0
x – 2y + z = 0
x – y + 2z = 0
has infinite number of nontrivial solutions for
(a) = 1
(b) = 5
(c) = -5
(d) no real value of
Solution:
(c) = -5

(vi) The system of equations
a₁x + b₁y + c₁z = 0
a₂x + b₂y + c₂z = 0
a₃x + b₃y + c₃z =0
with has
(a) more than two solutions
(b) one trivial and one nontrivial solutions
(c) No solution
(d) only trivial solutions
Solution:
(a) more than two solutions

Question 4.
Can the inverses of the following matrices be found?
(i) \(\left[\begin{array}{ll}
0 & 0 \\
0 & 0
\end{array}\right]\)
Solution:
|A| = 0
∴ A^-1 can not be found.

(ii) \(\left[\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right]\)
Solution:
∴ |A| = 4 – 6 = -2 ≠ 0
∴ A^-1 exists.

(iii) \(\left[\begin{array}{ll}
1 & 1 \\
1 & 1
\end{array}\right]\)
Solution:
|A| = \(\left[\begin{array}{ll}
1 & 1 \\
1 & 1
\end{array}\right]\) = 1 – 1 = 0
∴ A^-1 does not exist.

(iv) \(\left[\begin{array}{ll}
1 & 2 \\
2 & 4
\end{array}\right]\)
Solution:
|A| = \(\left[\begin{array}{ll}
1 & 2 \\
2 & 4
\end{array}\right]\) = 4 – 4 = 0
∴ A^-1 does not exist.

(v) \(\left[\begin{array}{lll}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right]\)
Solution:
|A| = \(\left[\begin{array}{lll}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right]\) = 1 ≠ 0
∴ A^-1 exists.

Question 5.
Find the inverse of the following:
(i) \(\left[\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right]\)
Solution:

(ii) \(\left[\begin{array}{cc}
2 & -1 \\
1 & 3
\end{array}\right]\)
Solution:

(iii) \(\left[\begin{array}{cc}
4 & -2 \\
3 & 1
\end{array}\right]\)
Solution:

(iv) \(\left[\begin{array}{ll}
2 & 5 \\
1 & 3
\end{array}\right]\)
Solution:

(v) \(\left[\begin{array}{cc}
1 & 0 \\
2 & -3
\end{array}\right]\)
Solution:

(vi) \(\left[\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}\right]\)
Solution:

(vii) \(\left[\begin{array}{cc}
i & -i \\
i & i
\end{array}\right]\)
Solution:

(viii) \(\left[\begin{array}{ll}
x & -x \\
x & x^2
\end{array}\right]\), x ≠ 0, x ≠ -1
Solution:

Question 6.
Find the adjoint of the following matrices.
(i) \(\left[\begin{array}{ccc}
1 & 1 & -1 \\
2 & -1 & 2 \\
1 & 3 & -2
\end{array}\right]\)
Solution:

(ii) \(\left[\begin{array}{ccc}
-2 & 2 & 3 \\
1 & 4 & 2 \\
-2 & -3 & 1
\end{array}\right]\)
Solution:

(iii) \(\left[\begin{array}{lll}
2 & 1 & 2 \\
2 & 2 & 1 \\
1 & 2 & 2
\end{array}\right]\)
Solution:

(iv) \(\left[\begin{array}{ccc}
1 & 3 & 0 \\
2 & -1 & 6 \\
5 & -3 & 1
\end{array}\right]\)
Solution:

Question 7.
Which of the following matrices are invertible?
(i) \(\left[\begin{array}{ccc}
1 & 0 & 0 \\
1 & 1 & 1 \\
2 & -1 & 1
\end{array}\right]\)
Solution:

(ii) \(\left[\begin{array}{ccc}
2 & 1 & -2 \\
1 & 2 & 1 \\
3 & 6 & 4
\end{array}\right]\)
Solution:

(iii) \(\left[\begin{array}{ccc}
-1 & -2 & 3 \\
2 & 1 & -4 \\
-1 & 0 & 2
\end{array}\right]\)
Solution:

(iv) \(\left[\begin{array}{ccc}
1 & 0 & 1 \\
2 & -2 & 1 \\
3 & 2 & 4
\end{array}\right]\)
Solution:

Question 8.
Examining consistency and solvability, solve the following equations by matrix method.
(i) x – y + z = 4
2x + y – 3z = 0
x + y + z = 2
Solution:

(ii) x + 2y – 3z = 4
2x + 4y – 5z = 12
3x – y + z = 3
Solution:
Let

(iii) 2x – y + z = 4
x + 3y + 2z = 12
3x + 2y + 3z = 16
Solution:

(iv) x + y + z = 4
2x + 5y – 2x = 3
x + 7y – 7z = 5
Solution:

(v) x + y + z = 4
2x – y + 3z = 1
3x + 2y – z = 1
Solution:

(vi) x + y – z = 6
2x – 3y + z = 1
2x – 4y + 2z = 1
Solution:

(vii) x – 2y = 3
3x + 4y – z = -2
5x – 3z = -1
Solution:

(viii) x + 2y + 3z = 14
2x – y + 5z = 15
2y + 4z – 3x = 13
Solution:

(ix) 2x + 3y +z = 11
x + y + z = 6
5x – y + 10z = 34
Solution:

Question 9.
Given the matrices
A = \(\left[\begin{array}{ccc}
1 & 2 & 3 \\
3 & -2 & 1 \\
4 & 2 & 1
\end{array}\right]\), X = \(\left[\begin{array}{l}
x \\
y \\
z
\end{array}\right]\) and C = \(\left[\begin{array}{l}
1 \\
2 \\
3
\end{array}\right]\)
write down the linear equations given by AX = C and solve it for x, y, z by matrix method.
Solution:

Question 10.
Find X, if \(\left[\begin{array}{ccc}
1 & 1 & 1 \\
1 & 1 & -1 \\
2 & 1 & -1
\end{array}\right]\) X = \(\left[\begin{array}{l}
6 \\
0 \\
1
\end{array}\right]\) where X = \(\left[\begin{array}{l}
x_1 \\
x_2 \\
x_3
\end{array}\right]\)
Solution:

Question 11.
Answer the following:
(i) If every element of a third order matrix is multiplied by 5, then how many times its determinant value becomes?
Solution:
125

(ii) What is the value of x if \(\left|\begin{array}{ll}
4 & 1 \\
2 & 1
\end{array}\right|^2=,\left|\begin{array}{ll}
3 & 2 \\
1 & x
\end{array}\right|-\left|\begin{array}{cc}
x & 3 \\
-2 & 1
\end{array}\right|\) ?
Solution:
x = 6

(iii) What are the values of x and y if \(\left|\begin{array}{ll}
x & y \\
1 & 1
\end{array}\right|=2,\left|\begin{array}{ll}
x & 3 \\
y & 2
\end{array}\right|=1\) ?
Solution:
x = 5, y = 3

(iv) What is the value of x if \(\left|\begin{array}{ccc}
x+1 & 1 & 1 \\
1 & 1 & -1 \\
-1 & 1 & 1
\end{array}\right|\) = 4?
Solution:
x = 0

(v) What is the value of \(\left|\begin{array}{ccc}
\mathbf{o} & -\mathbf{h} & -\mathbf{g} \\
\mathbf{h} & \mathbf{0} & -\mathbf{f} \\
\mathbf{g} & \mathbf{f} & \mathbf{0}
\end{array}\right|\)?
Solution:
0

(vi) What is the value of \(\left|\begin{array}{l}
\frac{1}{a} 1 \mathrm{bc} \\
\frac{1}{b} 1 c a \\
\frac{1}{c} 1 a b
\end{array}\right|\)
Solution:
0

(vii) What is the co-factor of 4 in the determinant \(\left|\begin{array}{rrr}
1 & 2 & -3 \\
4 & 5 & 0 \\
2 & 0 & 1
\end{array}\right|\)
Solution:
-2

(viii)In which interval does the determinant \(\left|\begin{array}{ccc}
1 & \sin \theta & 1 \\
-\sin \theta & 1 & \sin \theta \\
-1 & -\sin \theta & 1
\end{array}\right|\) lie?
Solution:
[2, 4]

(ix) Ifx + y + z = n, what is the value of Δ = \(\left|\begin{array}{ccc}
\sin (x+y+z) & \sin B & \cos C \\
-\sin B & 0 & \tan A \\
\cos (A+B) & -\tan A & 0
\end{array}\right|\) Where A, B, C are the angles of triangle.
Solution:
0

Question 12.
Evaluate the following determinants:
(i) \(\left|\begin{array}{ccc}
14 & 3 & 28 \\
17 & 9 & 34 \\
25 & 9 & 50
\end{array}\right|\)
Solution:
\(\left|\begin{array}{ccc}
14 & 3 & 28 \\
17 & 9 & 34 \\
25 & 9 & 50
\end{array}\right|\)
= 2\(\left|\begin{array}{ccc}
14 & 3 & 28 \\
17 & 9 & 34 \\
25 & 9 & 50
\end{array}\right|\) = 0
(∵ C₁ = C₃)

(ii) \(\left|\begin{array}{ccc}
16 & 19 & 13 \\
15 & 18 & 12 \\
14 & 17 & 11
\end{array}\right|\)
Solution:
\(\left|\begin{array}{ccc}
16 & 19 & 13 \\
15 & 18 & 12 \\
14 & 17 & 11
\end{array}\right|\) = \(\left|\begin{array}{ccc}
1 & 1 & 1 \\
1 & 1 & 1 \\
14 & 17 & 11
\end{array}\right|\)
( R₁ = R₁ – R₂, R₂ = R₂ – R₃)
= 0 (∵ R₁ = R₂)

(iii) \(\left|\begin{array}{ccc}
224 & 777 & 32 \\
735 & 888 & 105 \\
812 & 999 & 116
\end{array}\right|\)
Solution:
\(\left|\begin{array}{ccc}
224 & 777 & 32 \\
735 & 888 & 105 \\
812 & 999 & 116
\end{array}\right|\)
= 7\(\left|\begin{array}{ccc}
32 & 777 & 32 \\
105 & 888 & 105 \\
116 & 999 & 116
\end{array}\right|\) = 0
(∵ C₁ = C₂)

(iv) \(\left|\begin{array}{lll}
1 & 1 & 1 \\
2 & 3 & 4 \\
3 & 4 & 6
\end{array}\right|\)
Solution:

(v) \(\left|\begin{array}{ccc}
1 & 2 & 3 \\
3 & 5 & 7 \\
8 & 14 & 20
\end{array}\right|\)
Solution:

(vi) \(\left|\begin{array}{ccc}
1^2 & 2^2 & 3^2 \\
2^2 & 3^2 & 4^2 \\
3^2 & 4^2 & 5^2
\end{array}\right|\)
Solution:

= 225 – 256 – 4(100 – 144) + 9(64 – 81)
= -31 – 4(-44) + 9(-17)
= -31 + 176 – 153 = -184 + 176
= -8

(vii) \(\left|\begin{array}{ccc}
1 & 0 & -5863 \\
-7361 & 2 & 7361 \\
1 & 0 & 4137
\end{array}\right|\)
Solution:
\(\left|\begin{array}{ccc}
1 & 0 & -5863 \\
-7361 & 2 & 7361 \\
1 & 0 & 4137
\end{array}\right|\)
= 2\(\left|\begin{array}{cc}
1 & -5863 \\
1 & 4137
\end{array}\right|\)
(expanding along 2nd column)
= 2(4137 + 5863)
= 2 × 10000 = 20000

(viii) \(\left|\begin{array}{lll}
265 & 240 & 219 \\
240 & 225 & 198 \\
219 & 198 & 181
\end{array}\right|\)
Solution:

(ix) \(\left|\begin{array}{ccc}
0 & a^2 & b \\
b^2 & 0 & a^2 \\
a & b^2 & 0
\end{array}\right|\)
Solution:

= -a² (0 –  a²) + b (b⁴ –  0) = a⁵ + b⁵

(x) \(\left|\begin{array}{ccc}
a-b & b-c & c-a \\
\boldsymbol{x}-\boldsymbol{y} & \boldsymbol{y}-\boldsymbol{z} & z-\boldsymbol{x} \\
\boldsymbol{p}-\boldsymbol{q} & \boldsymbol{q}-\boldsymbol{r} & \boldsymbol{r}-\boldsymbol{p}
\end{array}\right|\)
Solution:
\(\left|\begin{array}{lll}
a-b & b-c & c-a \\
x-y & y-z & z-x \\
p-q & q-r & r-p
\end{array}\right|\)
= \(\left|\begin{array}{lll}
0 & b-c & c-a \\
0 & y-z & z-x \\
0 & q-r & r-p
\end{array}\right|\) (C₁ = C₁ + C₂ + C₃)
= 0 ( ∵ C₁ = 0)

(xi) \(\left|\begin{array}{lll}
a-b & b-c & c-a \\
b-c & c-a & a-b \\
c-a & a-b & b-c
\end{array}\right|\)
Solution:
\(\left|\begin{array}{lll}
a-b & b-c & c-a \\
b-c & c-a & a-b \\
c-a & a-b & b-c
\end{array}\right|\)
= \(\left|\begin{array}{lll}
0 & b-c & c-a \\
0 & c-a & a-b \\
0 & a-b & b-c
\end{array}\right|\) (C₁ = C₁ + C₂ + C₃)
= 0

(xii) \(\left|\begin{array}{ccc}
-\cos ^2 \theta & \sec ^2 \theta & -0.2 \\
\cot ^2 \theta & -\tan ^2 \theta & 1.2 \\
-1 & 1 & 1
\end{array}\right|\)
Solution:

(Expanding along 3rd row)
= (-cos² θ + sec² θ) (-tan² θ – 1.2) – (sec² θ + 0.2) (cot² θ – tan² θ)
= sin² θ – 1.2 cos² θ – sec² θ tan² θ – 1.2 sec² θ – cosec² θ +  sec² θ tan² θ – 0.2 cot² θ + 0.2 tan² θ
= sin² θ – cosec² θ + 1.2 (cos² θ – sec² θ) + 0.2 (tan² θ – cot² θ) ≠ 0
The question seems to be wrong.

Question 13.
If \(\left|\begin{array}{ccc}
1 & 1 & 1 \\
1 & 1+x & 1 \\
1 & 1 & 1+y
\end{array}\right|\) = 0 what are x and y?
Solution:

or, xy – 0 = 0 ⇒ xy = 0, ⇒ x = 0, or y = 0

Question 14.
For what value of x \(\left|\begin{array}{ccc}
2 x & 0 & 0 \\
0 & 1 & 2 \\
-1 & 2 & 0
\end{array}\right|\) = \(\left|\begin{array}{lll}
1 & 0 & 0 \\
2 & 3 & 4 \\
0 & 3 & 5
\end{array}\right|\)?
Solution:

Question 15.
Solve \(\left|\begin{array}{ccc}
x+a & 0 & 0 \\
a & x+b & 0 \\
a & 0 & x+c
\end{array}\right|\) = 0
Solution:
\(\left|\begin{array}{ccc}
x+a & 0 & 0 \\
a & x+b & 0 \\
a & 0 & x+c
\end{array}\right|\) = 0
or, (x – a) \(\left|\begin{array}{cc}
x+b & 0 \\
0 & x+c
\end{array}\right|\) = 0
or, (x + a) (x + b) (x + c) = 0
x = -a, x = -b, x = -c

Question 16.
Solve \(\left|\begin{array}{lll}
a+x & a-x & a-x \\
a-x & a+x & a-x \\
a-x & a-x & a+x
\end{array}\right|\) = 0
Solution:

Question 17.
Solve \(\left|\begin{array}{ccc}
x+a & b & c \\
a & x+b & c \\
a & b & x+c
\end{array}\right|\) = 0
Solution:

Question 18.
Show that x = 2 is a root of \(\left|\begin{array}{ccc}x & -6 & -1 \\ 2 & -3 x & x-3 \\ -3 & 2 x & x+2\end{array}\right|\) = 0 Solve this completely.
Solution:
Putting x = 2,

= (x – 1) (-15x + 30 – 5x² + 10x)
= (x – 1) (-5x² – 5x + 30)
= -5(x – 1) (x² + x – 6)
= -5(x – 1) (x + 3) (x – 2) = 0
⇒ x = 1 or, -3 or 2.

Question 19.
Evaluate \(\left|\begin{array}{ccc}
1 & a & b c \\
1 & b & c a \\
1 & c & a b
\end{array}\right|\) – \(\left|\begin{array}{lll}
1 & a & a^2 \\
1 & b & b^2 \\
1 & c & c^2
\end{array}\right|\)
Solution:

= (a – b) (b – c) [(-a + c) – (b + c – a – b)]
= (a – b) (b – c) (-a + c – c + a) = 0

Question 20.
\(\left|\begin{array}{lll}
a & a^2-b c & 1 \\
b & b^2-a c & 1 \\
c & c^2-a b & 1
\end{array}\right|\)
Solution:

Question21.
For what value of X the system of equations
x + y + z = 6, 4x + λy – λz = 0, 3x + 2y – 4z = -5 does not possess a solution?
Solution:

= 24 – 6λ – 2λ = 24 – 8λ
when Δ = 0
We have 24 – 8λ, = 0 or, λ = 3
The system of equations does not posses solution for λ = 3.

Question 22.
If A is a 3 × 3 matrix and |A| = 2, then which matrix is represented by A × adj A?
Solution:

Question 23.
If A = \(\left[\begin{array}{cc}
0 & -\tan \frac{\alpha}{2} \\
\tan \frac{\alpha}{2} & 0
\end{array}\right]\)
show that (I + A) (I – A)^-1 = \(\left[\begin{array}{cc}
\cos \alpha & -\sin \alpha \\
\sin \alpha & \cos \alpha
\end{array}\right]\) where I = \(\left[\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right]\)
Solution:

Question 24.
Prove the following:
(i) \(\left|\begin{array}{ccc}
a^2+1 & a b & a c \\
a b & b^2+1 & b c \\
a c & b c & c^2+1
\end{array}\right|\) = 1 + a² + b² + c²
Solution:

(ii) \(\left|\begin{array}{ccc}
1 & 1 & 1 \\
a & b & c \\
a^3 & b^3 & c^3
\end{array}\right|\) = (b – c) (c – a) (a – b) (a + b + c)
Solution:

= (a – b) (b – c) (b² + bc + c² – a² – ab – b²)
= (a – b) (b- c) (c² – a² + bc – ab)
= (a – b) (b – c) {(c – a) (c + a) + b(c – a)}
= (a – b) (b – c) (c – a) (a + b + c) = R.H.S.
(Proved)

(iii) \(\left|\begin{array}{lll}
\boldsymbol{a} & \boldsymbol{b} & \boldsymbol{c} \\
\boldsymbol{b} & \boldsymbol{c} & \boldsymbol{a} \\
\boldsymbol{c} & \boldsymbol{a} & \boldsymbol{b}
\end{array}\right|\) = 3abc – a³ – b³ – c³
Solution:

= (a + b + c) {(b – c) (a – b) – (c – a)²}
= (a + b + c) (a + b + c) (ab – b² – ca + bc – c² – a² + 2ca)
= (a + b + c) (-a² – b² – c² + ab + bc + ca)
= -(a + b + c) (a² + b² + c² – ab – bc – ca)
=- (a³ + b³ + c³ – 3abc)
= 3abc – a³ – b³ – c³

(iv) \(\left|\begin{array}{lll}
b^2-a b & b-c & b c-a c \\
a b-a^2 & a-b & b^2-a b \\
b c-a c & c-a & a b-a^2
\end{array}\right|\) = 0
Solution:

= (b² – a² + bc – ac) (a – b) {(-a + b) (c – a) – (bc – ac – ab + a²)}
= (b² – a² + bc – ac) (a – b) (- ca + a² + bc – ab – bc + ac + ab – a²)
= (b² – a² + bc – ac) (a – b) × 0 = 0
= R.H.S.
(Proved)

(v) \(\left|\begin{array}{ccc}
-a^2 & a b & a c \\
a b & -b^2 & b c \\
a c & b c & -c^2
\end{array}\right|\) = 4a²b²c²
Solution:

(vi) \(\left|\begin{array}{lll}
(b+c)^2 & a^2 & b c \\
(c+a)^2 & b^2 & c a \\
(a+b)^2 & c^2 & a b
\end{array}\right|\) = (a² + b² + c² ) (a + b + c) (b – c) (c – a) (a – b)
Solution:

= (a – b) (b – c) (a² + b² + c²) (-a² – ab + bc + c²)
= (a – b) (b – c) (a² + b² + c²) {(c² – a²) + b(c – a)}
= (a² + b² + c²) (a – b) (b – c) (c – a) (c + a + b)

(vii) \(\left|\begin{array}{lll}
b+c & a+b & a \\
c+a & b+c & b \\
a+b & c+a & c
\end{array}\right|\) = a³ + b³ + c³ – 3abc
Solution:

= (a + b +c) {(a – b) (a – c) – (c – b) (b – c)}
= (a + b + c) (a² – ac – ab + bc – bc + c² + b² – bc)
= (a + b + c) (a² + b² + c² – ab – bc – ca)
= (a³ + b³ + c³ – 3abc)

(viii) \(\left|\begin{array}{ccc}
a+b+c & -c & -b \\
-c & a+b+c & -a \\
-b & -a & a+b+c
\end{array}\right|\) = 2(b + c) (c + a) (a + b)
Solution:

= -2(a + b) (b + c) (-a – b – c + b)
= 2(a + b) (b + c) (c + a)

(ix) \(\left|\begin{array}{ccc}
a x-b y-c z & a y+b x & a z+c x \\
b x+a y & b y-c z-a x & b z+c y \\
c x+a z & a y+b z & c z-a x-b y
\end{array}\right|\) = (a² + b² + c²) (ax + by + cz) (x² + y² + z²)
Solution:

Question 25.
If 2s = a + b + c show that \(\left|\begin{array}{ccc}
a^2 & (s-a)^2 & (s-a)^2 \\
(s-b)^2 & b^2 & (s-b)^2 \\
(s-c)^2 & (s-c)^2 & c^2
\end{array}\right|\) = 2s³ (s – a) (s – b) (s – c)
Solution:
Let s – a = A, s – b = B, s – c = C
A + B + C = 3s – (a + b + c)
= 3s – 2s = s
Also B + C = s – b + s – c = 2s – (b + c)
= (a + b + c) – b + c = a
Similarly C + A = b, A + B = c

= 2 ABC (A + B + C)²
[Refer Q.No.9 (xii) of Exercise 5(a)]
= 2(s – a) (s – b)(s – c) s³

Question 26.
if \(\left|\begin{array}{ccc}
x & x^2 & x^3-1 \\
y & y^2 & y^3-1 \\
z & z^2 & z^3-1
\end{array}\right|\) = 0 then prove that xyz =1 when x, y, z are non zero and unequal.
Solution:

= (x – y) (y – z) (z – x) (xyz – 1)
It is given that
(x – y) (y – z) (z – x) (xyz – 1) = 0
⇒ xyz – 1 (as x ≠ y ≠ z)

Question 27.
Without expanding show that the following determinant is equal to Ax + B where A and B are determinants of order 3 not involving x.
\(\left|\begin{array}{ccc}
x^2+x & x+1 & x-2 \\
2 x^2+3 x-1 & 3 x & 3 x-3 \\
x^2+2 x+3 & 2 x-1 & 2 x-1
\end{array}\right|\)
Solution:

Question 28.
If x, y, z are positive and are the pth, qth and rth terms of a G.P. then prove that \(\left|\begin{array}{lll}
\log x & p & 1 \\
\log y & q & 1 \\
\log z & r & 1
\end{array}\right|\) = 0
Solution:
Let the G.P. be
a, aR, aR², aR³ …..aR^n-1
p th term = aR^p-1
q th term = aR^q-1
r th term = aR^r-1
x = aR^p-1, y= aR^q-1, z = aR^r-1
log x = log a + (p – 1) log R,
log y = log a + (q – 1) log R,
log z = log a + (r – 1) log R

Question 29.
If D_j = \(\left|\begin{array}{ccc}
j & a & n(n+2) / 2 \\
j^2 & b & n(n+1)(2 n+1) / 6 \\
j^3 & c & n^2(n+1)^2 / 4
\end{array}\right|\) then prove that \(\sum_{j=1}^n\)D_j = 0.
Solution:

Question 30.
Ifa₁, a₂,……a_n are in G.P. and a_i > 0 for every i, then find the value of
\(\left|\begin{array}{ccc}
\log a_n & \log a_{n+1} & \log a_{n+2} \\
\log a_{n+1} & \log a_{n+2} & \log a_{n+3} \\
\log a_{n+2} & \log a_{n+3} & \log a_{n+4}
\end{array}\right|\)
Solution:

Question 31.
If f(x)= \(\left|\begin{array}{ccc}
1+\sin ^2 x & \cos ^2 x & 4 \sin ^2 x \\
\sin ^2 x & 1+\cos ^2 x & 4 \sin 2 x \\
\sin ^2 x & \cos ^2 x & 1+4 \sin ^2 x
\end{array}\right|\) what is the least value of f(x)?
Solution:

As minimum value of sin 2x is 0. So the minimum value of above function f(x) is 2.

Question 32.
If f_r(x), g_r(x), h_r(x), r = 1, 2, 3 are polynomials in x such that f_r(a) = g_r(a) = h_r(a) and
F(x) = \(\left[\begin{array}{lll}
f_1(x) & f_2(x) & f_3(x) \\
g_1(x) & g_2(x) & g_3(x) \\
h_1(x) & h_2(x) & h_3(x)
\end{array}\right]\) find F'(x) at x = a.
Solution:
We have

[Since f₁a) = g₁(a) = h₁(a), f₂(a) = g₂(a) = h₂(a) and f₃(a) = g₃(a) = h₃(a) So that each determinant is zero due to presence of two identical rows.]

Question 33.
If f(x) = \(\left[\begin{array}{ccc}
\cos x & \sin x & \cos x \\
\cos 2 x & \sin 2 x & 2 \cos 2 x \\
\cos 3 x & \sin 3 x & 3 \cos 3 x
\end{array}\right]\) find f'(\(\frac{\pi}{2}\)).
Solution:

Odisha State Board Elements of Mathematics Class 12 CHSE Odisha Solutions Chapter 4 Matrices Ex 4(a) Textbook Exercise Questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 4 Matrices Exercise 4(a)

Question 1.
State the order of the following matrices.
(i) [abc]
(ii) \(\left[\begin{array}{l}
1 \\
2
\end{array}\right]\)
(iii) \(\left[\begin{array}{ll}
x & y \\
y & z \\
z & x
\end{array}\right]\)
(iv) \(\left[\begin{array}{cccc}
1 & 0 & 1 & 4 \\
2 & 1 & 3 & 0 \\
-3 & 2 & 1 & 3
\end{array}\right]\)
Solution:
(i) (1 x 3)
(ii) (2 x 1)
(iii) (3 x 2)
(iv) (3 x 4)

Question 2.
How many entries are there in a
(i) 3 x 3 matrix
(ii) 3 x 4 matrix
(iii) p x q matrix
(iv) a sqare matrix of order p?
Solution:
(i) 9
(ii) 12
(iii) pq
(iv) p²

Question 3.
Give an example of
(i) 3 x 1 matrix
(ii) 2 x 2 matrix
(iii) 4 x 2 matrix
(iv) 1 x 3 matrix
Solution:
(i) \(\left(\begin{array}{l}
a \\
b \\
c
\end{array}\right)\)
(ii) \(\left(\begin{array}{ll}
a & b \\
c & d
\end{array}\right)\)
(iii) \(\left(\begin{array}{ll}
a & b \\
c & d \\
e & f \\
g & h
\end{array}\right)\)
(iv) (1, 2, 3)

Question 4.
Let A = \(\left[\begin{array}{lllll}
1 & 2 & 3 & 4 & 1 \\
4 & 5 & 6 & 1 & 2 \\
3 & 9 & 1 & 1 & 6
\end{array}\right]\)
(i) What is the order of A?
(ii) Write down the entries a₃₁, a₂₅, a₂₃
(iii) Write down A^T.
(iv) What is the order of A^T?
Solution:
A = \(\left[\begin{array}{lllll}
1 & 2 & 3 & 4 & 1 \\
4 & 5 & 6 & 1 & 2 \\
3 & 9 & 1 & 1 & 6
\end{array}\right]\)
(i) Order of A is (3 x 5)
(ii) a₃₁ = 3, a₂₅= 2, a₂₃ = 6
(iii) A^T = \(\left[\begin{array}{lll}
1 & 4 & 3 \\
2 & 5 & 9 \\
3 & 6 & 1 \\
4 & 1 & 1 \\
1 & 2 & 6
\end{array}\right]\)
(iv) Order of A^T is (5 x 3).

Question 5.
Matrices A and B are given below. Find A + B, B + A, A – B and B – A. Verify that A + B = B + A and B – A = -(A – B)
(i) A = \(\left[\begin{array}{l}
7 \\
1
\end{array}\right]\), B = \(\left[\begin{array}{c}
-6 \\
9
\end{array}\right]\)
Solution:

(ii) A = \(\left[\begin{array}{cc}
1 & 2 \\
3 & -1
\end{array}\right]\), B = \(\left[\begin{array}{cc}
4 & 1 \\
-3 & 2
\end{array}\right]\)
Solution:

(iii) A = \(\left[\begin{array}{ll}
\frac{1}{2} & \frac{1}{4} \\
\frac{1}{3} & \frac{1}{5}
\end{array}\right]\), B = \(\left[\begin{array}{ll}
\frac{1}{3} & \frac{1}{2} \\
\frac{1}{2} & \frac{4}{5}
\end{array}\right]\)
Solution:

(iv) A = \(\left[\begin{array}{cc}
1 & a-b \\
a+b & -3
\end{array}\right]\), B = \(\left[\begin{array}{cc}
1 & b \\
-a & 5
\end{array}\right]\)
Solution:

(v) \(\left[\begin{array}{rrr}
1 & -2 & 5 \\
-1 & 4 & 3 \\
1 & 2 & -3
\end{array}\right]\), B = \(\left[\begin{array}{rrr}
-1 & 2 & -5 \\
1 & -3 & -3 \\
1 & 2 & 4
\end{array}\right]\)
Solution:

Question 6.
(i) Find the 2×2 matrix X
if X + \(\left[\begin{array}{ll}
0 & 1 \\
1 & 0
\end{array}\right]\) = \(\left[\begin{array}{ll}
2 & 0 \\
0 & 2
\end{array}\right]\)
Solution:

(ii) Given
[x y z] – [-4 3 1] = [-5 1 0] derermine x, y, z.
Solution:
[x y z] – [-4 3 1] = [-5 1 0]
∴ (x y z) = (-4 3 1) + (-5 1 0) = (-9 4 1)
∴ x = -9, y = 4, z = 1

(iii) If \(\left[\begin{array}{ll}
x_1 & x_2 \\
y_1 & y_2
\end{array}\right]\) – \(\left[\begin{array}{ll}
2 & 3 \\
0 & 1
\end{array}\right]\) = \(\left[\begin{array}{ll}
3 & 5 \\
1 & 2
\end{array}\right]\) determine x₁, x₂, y₁, y₂.
Solution:

(iv) Find a matrix which when added to \(\left[\begin{array}{cc}
2 & -3 \\
-4 & 7
\end{array}\right]\) gives \(\left[\begin{array}{ll}
4 & 1 \\
3 & 2
\end{array}\right]\)
Solution:

Question 7.
Calculate whenever possible, the following products.
(i) \(\left[\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right]\left[\begin{array}{l}
2 \\
3
\end{array}\right]\)
Solution:

(ii) \(\left[\begin{array}{l}
2 \\
3
\end{array}\right]\left[\begin{array}{ll}
1 & 2 \\
4 & 3
\end{array}\right]\)
Solution:
\(\left[\begin{array}{l}
2 \\
3
\end{array}\right]\left[\begin{array}{ll}
1 & 2 \\
4 & 3
\end{array}\right]\) is impossible because number of columns of 1st ≠ number of rows of second.

(iii) \(\left[\begin{array}{ll}
1 & 2 \\
2 & 1
\end{array}\right]\left[\begin{array}{ll}
3 & 1 \\
1 & 1
\end{array}\right]\)
Solution:

(iv) \(\left[\begin{array}{cc}
1 & -2 \\
-2 & 3
\end{array}\right]\left[\begin{array}{lll}
1 & 2 & 3 \\
2 & 3 & 4
\end{array}\right]\)
Solution:

Question 8.
If A = \(\left[\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right]\), B = \(\left[\begin{array}{ll}
3 & 2 \\
1 & 4
\end{array}\right]\), C = \(\left[\begin{array}{ll}
2 & 2 \\
1 & 3
\end{array}\right]\)
Calculate (i) AB (ii) BA (iii) BC (iv) CB (v) AC (vi) CA
Solution:

Question 9.
Find the following products.
(i) \(\left[\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right]\left[\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right]\)
Solution:

(ii) \(\left[\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right]\left[\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right]\)
Solution:

(iii) \(\left[\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right]\left[\begin{array}{ll}
1 & 3 \\
1 & 4
\end{array}\right]\)
Solution:

(iv) \(\left[\begin{array}{ll}
1 & 3 \\
1 & 4
\end{array}\right]\left[\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right]\)
Solution:

(v) \(\left[\begin{array}{cc}
1 & i \\
i & -1
\end{array}\right]^2\) where i = √-1
Solution:

(vi) \(\left[\begin{array}{ll}
\mathbf{0} & \mathbf{1} \\
\mathbf{1} & \mathbf{0}
\end{array}\right]\left[\begin{array}{ll}
a & b \\
c & d
\end{array}\right]\)
Solution:

(vii) \(\left[\begin{array}{ll}
0 & k \\
1 & 0
\end{array}\right]\left[\begin{array}{ll}
a & b \\
c & d
\end{array}\right]\)
Solution:

(viii) \(\left[\begin{array}{ll}
a & b \\
c & d
\end{array}\right]\left[\begin{array}{ll}
0 & 1 \\
1 & 0
\end{array}\right]\)
Solution:

(ix) \(\left[\begin{array}{ll}
1 & 0 \\
0 & k
\end{array}\right]\left[\begin{array}{ll}
a & b \\
c & d
\end{array}\right]\)
Solution:

(x) \(\left[\begin{array}{lll}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{array}\right]\left[\begin{array}{lll}
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{array}\right]\)
Solution:
\(\left[\begin{array}{lll}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{array}\right]\left[\begin{array}{lll}
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{array}\right]\) = \(\left[\begin{array}{lll}
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{array}\right]\)

Question 10.
Write true or false in the following cases:
(i) The sum of a 3 x 4 matrix with a 3 x 4 matrix is a 3 x 3 matrix.
Solution:
False

(ii) k[0] = 0, k ∈ R
Solution:
False

(iii) A – B = B – A, if one of A and B is zero and A and B are of the same order.
Solution:
False

(iv) A + B = B + A, if A and B are matrices of the same order.
Solution:
True

(v) \(\left[\begin{array}{cc}
1 & 0 \\
-2 & 0
\end{array}\right]\) + \(\left[\begin{array}{cc}
-1 & 0 \\
2 & 0
\end{array}\right]\) = 0
Solution:
True

(vi) \(\left[\begin{array}{ll}
3 & 1 \\
6 & 2
\end{array}\right]\) = 3 \(\left[\begin{array}{ll}
1 & 1 \\
2 & 2
\end{array}\right]\)
Solution:
False

(vii) With five elements a matrix can not be constructed.
Solution:
False

(viii)The unit matrix is its own transpose.
Solution:
True

Question 11.
If A = \(\left[\begin{array}{cc}
2 & 4 \\
3 & 13
\end{array}\right]\) and I = \(\left[\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right]\) find A – α I, α ∈ R.
Solution:

Question 12.
Find x and y in the following.
(i) \(\left[\begin{array}{cc}
x & -2 y \\
0 & -2
\end{array}\right]=\left[\begin{array}{cc}
1 & -8 \\
0 & -2
\end{array}\right]\)
Solution:

(ii) \(\left[\begin{array}{c}
x+3 \\
2-y
\end{array}\right]=\left[\begin{array}{c}
1 \\
-3
\end{array}\right]\)
Solution:

(iii) \(\left[\begin{array}{c}
2 x-y \\
x+y
\end{array}\right]=\left[\begin{array}{c}
3 \\
-9
\end{array}\right]\)
Solution:

(iv) \(\left[\begin{array}{l}
x \\
y
\end{array}\right]+\left[\begin{array}{l}
3 \\
4
\end{array}\right]=\left[\begin{array}{c}
2 \\
-1
\end{array}\right]\)
Solution:

(v) [2x -y] + [y 3x] = 5 [1 0]
Solution:

Question 13.
The element of ith row and ith column of the following matrix is i +j. Complete the matrix.

Question 14.
Write down the matrix

Question 15.
Construct a 2 x 3 matrix having elements given by
(i) a_ij = i + j
(ii) a_ij = i – j
(iii) a_ij = i × j
(iv) a_ij = i / j
Solution:

Question 16.
If \(\left[\begin{array}{cc}
2 x & y \\
1 & 3
\end{array}\right]+\left[\begin{array}{cc}
4 & 2 \\
0 & -1
\end{array}\right]=\left[\begin{array}{ll}
8 & 3 \\
1 & 2
\end{array}\right]\)
Solution:

Question 17.
Find A such that
\(\left[\begin{array}{ccc}
2 & 3 & 4 \\
1 & 0 & -2 \\
3 & 1 & -1
\end{array}\right]+A=\left[\begin{array}{ccc}
1 & 2 & -1 \\
2 & -1 & 0 \\
1 & 3 & 2
\end{array}\right]\)
Solution:

Question 18.
If

Question 19.
What is the order of the matrix B if [3 4 2] B = [2 1 0 3 6]
Solution:
(3 4 2) B = (2 1 0 3 6)
Let A = (3 4 2), C = (2 1 0 3 6)
∴ Order of A = (1 x 3)
Order of C = (1 x 5)
∴ Order of B = (3 x 5)

Question 20.
Find A if \(\left[\begin{array}{l}
4 \\
1 \\
3
\end{array}\right]\) A = \(\left[\begin{array}{rrr}
-4 & 8 & 4 \\
-1 & 2 & 1 \\
-3 & 6 & 3
\end{array}\right]\)
Solution:

Question 21.
Find B if B² = \(\left[\begin{array}{cc}
17 & 8 \\
8 & 17
\end{array}\right]\)
Solution:

∴ a² + bc = 17, ab + bd= 8
ca + cd = 8, bc + d² = 17
∴ a² + bc = bc + d²
or, a² + d² or, a = d
or, ca + cd = ab + bd
or, cd + cd – bd + bd
or, 2cd = 2bd = 8
or, b = c and bd = 4 = cd
∴ ab + bd= 8
or, ab + 4 = 8
or, ab = 4
Again, a² + bc = 17
or, a² + b . b = 17 (∵ b = c)
or, a² + b² = 17
Also (a + b)² = a² + b² + 2ab
∴ (a + b)² = 17 + 8 = 25
or, a + b = 5
And (a – b)² = 17 – 8 = 9
or, a – b = 3
∴ a = 4, b = 1, So d = 4, c = 1
∴ B = \(\left[\begin{array}{ll}
4 & 1 \\
1 & 4
\end{array}\right]\)

Question 22.
Find x and y when

Question 23.
Find AB and BA given that:

Question 24.
Evaluate

Question 25.
If

Show that AB = AC though B ≠ C. Verify that
(i) A + (B + C) = (A + B) + C
(ii) A(B + C) = AB + AC
(iii) A(BC) = (AB)C
Solution:

Question 26.
Find A and B where

Question 27.
If A = \(\left[\begin{array}{cc}
4 & 2 \\
-1 & 1
\end{array}\right]\) and I be the 2 × 2 unit matrix find (A – 2I) (A – 3I)
Solution:

Question 28.
Verify that [AB]^T = B^TA^T where

Question 29.
Verify that A = \(\left[\begin{array}{ll}
a & b \\
c & d
\end{array}\right]\) satisfies the equation x² – (a + d)x + (ad – bc)I = 0 where I is the 2 x 2 matrix.
Solution:

Question 30.
If A = \(\left[\begin{array}{rrr}
1 & 2 & 3 \\
3 & -2 & 1 \\
4 & 2 & 1
\end{array}\right]\), show that A³ – 23 A – 40 I = 0
Solution:

Question 31.

Question 32.
If A and B are matrices of the same order and AB = BA, then prove that
(i) A² – B² = (A – B) (A + B)
(ii) A² + 2AB + B² = (A + B)²
(iii) A² – 2AB + B² = (A – B)²
Solution:
(i) (A – B) (A + B)
= A² + AB – BA – B²
= A² + AB – AB- B²(∵ AB = BA)
= A² – B²
(ii) (A + B)² = (A + B) (A + B)
= A² + AB + BA + B²
= A² + AB + AB + B² (∵ AB = BA)
= A² + 2AB + B²
(iii) (A – B)² = (A – B) (A – B)
= A² – AB – BA + B²
= A² – AB – AB + B² (∵ AB = BA)
= A² – 2AB + B²

Question 33.
If α and β are scalars and A is a square matrix then prove that
(A – αI) . (A – βI) = A² – (α + β) A + αβI, where I is a unit matrix of same order as A.
Solution:
(A – αI) (A – βI)
= A² – AβI – αIA + αβI²
= A² – βAI – αA + αβI
(∵ IA = A, I² = I)
= A² – βA – αA + αβI) (∵ AI = A)
= A² – (α + β) A + αβI

Question 34.
If α and β are scalars such that A = αβ + βI, where A, B and the unit matrix I are of the same order, then prove that AB = BA.
Solution:
We have A = αβ + βI
AB (αβ + βI) B
= α βB + βI B
= α βB + βB = (α + I) βB
= βB (α + 1)
(∵ Scalar mltiβlication is associative)
= Bβ (α + 1)
= Bβα + Bβ = Bαβ + BIβ
(∵ BI = B)
= B (αβ + βi) = BA
AB = BA
(proved)

Question 35.

Question 36.

Question 37.

Question 38.

Question 39.

Question 40.

Question 41.

Question 42.

Question 43.

	Men	Women	Children
Family A →	4	6	2
Family B →	2	2	4

	Family B
	Calory	Proteins
Men	2400	45
Women	1900	55
Children	1800	33

Solution:
The given informations can be written in matrix form as

∴ Calory requirements for families A and B are 24600 and 15800 respectively and protein requirements are 576 gm and 332 gm respectively.

Question 44.
Let the investment in first fund = ₹x and in the second fund is ₹(50000-x)
Investment matrix A=[x  50000-x]

⇒ 300000 – x = 278000
⇒ x = 22000
∴ He invests ₹22000 in first bond and ₹28000 in the second bond.

Odisha State Board CHSE Odisha Class 12 Math Solutions Chapter 3 Linear Programming Ex 3(b) Textbook Exercise Questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 3 Linear Programming Exercise 3(b)

Question 1.
Maximize Z = 5x₁+ 6x₂
Subject to: 2x₁ + 3x₂ ≤ 6
x₁, x₂ ≥ 0
Solution:
Step – 1 Treating the constraint as equation, we get 2x₁ + 3x₂ = 6
Step – 2 Let us draw the graph

x1	3	0
x2	0	0

Step – 3 Clearly (0,0) statisfies 2x₁ + 3x₂ ≤ 6
The shaded region is the feasible region with vertices 0(0,0), A(3,0), B(0,2).
Step – 4

Corner point	Z = 5x1+ 6x2
0(0.0)	0
A(3,0)	15 → maximum
B(0,2)	12

Z is maximum at A (3,0)
∴ The solution of LPP is x₁ = 3, x₂ = 0
Z_max = 15

Question 2.
Minimize: Z = 6x₁ + 7x₂
Subject to: x₁ + 2x₂ ≥ 4
x₁, x₂ ≥ 0
Solution:
Step – 1 Treating the constraint as equation we get x₁ + 2x₂ = 0
Step – 2 Let us draw the graph of x₁ + 2x₂ = 4

x1	0	4
x2	2	0

Step – 3 Clearly 0(0,0) does not satisfy
x₁ + 2x₂ > 4, x₁ > 0, x₂ > 0 is the first quadrant.
The feasible region is the shaded region with vertices A(4, 0), B(0, 2).
Step – 4 Z (4, 0) = 24
Z (0, 2) = 14 → minimum
Step – 5 As the feasible region is unbounded we cannot immediately decide Z is minimum at B (0, 2).
Let us draw the half-plane 6x₁ + 7x₂ < 14

x1	0	3.5
x2	2	-1

As this half-plane has no point common with the feasible region, we have Z is minimum for x₁= 0, x₂ = 2 and the minimum value of Z = 14.

Question 3.
Maximize Z = 20x₁+ 40x₂
Subject to: x₁ + x₂ ≤ 1
6x₁ + 2x₂ ≤ 3
x₁, x₂ ≥ 0.
Solution:
Step – 1 Treating the constraints as equations
x₁ + x₂ = 1    …. (1)
6x₁ + 2x₂ = 3   …. (2)
x₁, x₂ ≥ 0
Step – 2 Let us draw the graph:
Table – 1

x1	0	1
x2	1	0

Table – 2

x1	0	0.5
x2	1.5	0

Step – 3 As (0, 0) satisfies both the inequations the shaded region is the feasible region.
Step – 4 Solving
x₁ + x₂ = 1
6x₁ + 2x₂ = 3
we have x₁ = ¼ x₂ = ¾
The vertices are O(0, 0), A(0.5, 0), B(0,1) and C(¼, ¾)
Now Z(O) = 0
Z(A) = 10
Z(B) = 40
Z(C) = 20 × ¼ + 40 × ¾ = 35
∴ Z attains maximum at B for x₁= 0, x₂ = 1
Z_max = 40

Question 4.
Minimize: Z = 30x₁ + 45x₂
Subject to: 2x₁ + 6x₂ ≥ 4
5x₁ + 2x₂ ≥ 5
x₁, x₂ ≥ 0
Solution:
Step – 1 Consider the constraints as equations
2x₁ + 6x₂ = 4
5x₁ + 2x₂ = 5
Step – 2
Table – 1

x1	2	-1
x2	0	1

Table – 2

x1	1	0
x2	0	2.5

Step – 3 Clearly 0(0,0) does not satisfy 2x₁ + 6x₂ ≥ 4 and 5x₁ + 2x₂ ≥ 5.
Thus the shaded region is the feasible region.
Solving the equations we get
x₁ = \(\frac{11}{13}\), x₂ = \(\frac{5}{13}\).
∴ The vertices are A(2, 0)
B(\(\frac{11}{13}\), \(\frac{5}{13}\)) and C(0, \(\frac{5}{2}\)).
Step – 4 Z(A) = 60
Z(B) = \(\frac{555}{13}\) → minimum
Z(C) = \(\frac{225}{2}\)
Step – 5 As the feasible region is unbounded we cannot immediately decide Z is minimum at B(\(\frac{11}{13}\), \(\frac{5}{13}\))
Let us draw the half plane
30x₁ + 45x₂ < \(\frac{555}{13}\)

x1	\(\frac{11}{13}\)	0
x2	\(\frac{5}{13}\)	\(\frac{27}{39}\)

As this half plane and the feasible region has no point in common we have Z is minimum for x₁ = \(\frac{11}{13}\), x₂ = \(\frac{5}{13}\), and Z_min = \(\frac{555}{13}\)

Question 5.
Maximize: Z = 3x₁+ 2x₂
Subject to: -2x₁ + x₂ ≤ 1
x₁ ≤ 2
x₁+ x₂ ≤ 3
x₁, x₂ ≥ 0
Solution:
Step – 1 Treating the constraints as equations
-2x₁ + x₂ = 1        …..(1)
x₁ = 2                   …..(2)
x₁+ x₂ = 3            …..(3)
Step – 2 Let us draw the lines.
Table – 1

x1	0	-1
x2	1	-1

Table – 2

x1	2	2
x2	0	1

Table – 3

x1	0	3
x2	3	0

Step – 3 (0, 0) satisfies all the constraints and x₁, x₂ > 0 is the 1st quadrant the shaded region is the feasible region.

Step – 4 Solving -2x₁ + x₂ = 1
x₁+ x₂ = 3
we have 3x₁ = 2
⇒ x₁ = \(\frac{2}{3}\), x₂ = 3 – \(\frac{2}{3}\) = \(\frac{7}{3}\)
From x₁+ x₂ = 3 and x₁ = 2 we have x₁ = 2, x₂ = 1
∴ The vertices are 0(0, 0), A(2, 0), B(2, 1), C(\(\frac{2}{3}\), \(\frac{7}{3}\)), D(0, 1)
Z(0) = 0, Z(A) = 6, Z(B) = 8, Z(C) = 3.\(\frac{2}{3}\) + 2.\(\frac{7}{3}\) = \(\frac{20}{3}\), Z(D) = 2
Z is maximum at B.
∴ The solution of given LPP is x₁ = 2, x₂ = 1, Z_(max) = 8.

Question 6.
Maximize: Z = 50x₁+ 60x₂
Subject to: x₁ + x₂ ≤ 5
x₁+ 2x₂ ≤ 4
x₁, x₂ ≥ 0
Solution:
Step – 1 Treating the constraints as equations we get
x₁ + x₂ = 5     ….(1)
x₁+ 2x₂ = 4    ….(2)
Step – 2 Let us draw the graph
Table – 1

x1	5	5
x2	0	0

Table – 2

x1	4	0
x2	0	2

Step – 3 0(0,0) satisfies x₁ + x₂ ≤ 5 and does not satisfy x₁+ 2x₂ ≤ 4
Thus the shaded region is the feasible region.
Step – 4 The corner points are A(4,0), B(5,0), C(0,5) , D(0,2)
∴

Corner point	z = 50x1+ 60x2
A(4,0)	200
B (5,0)	250 → maximum
C(0,5)	300
D(0,2)	120

Z is maximum for x₁ = 0, x₂ = 5, Z_(max) = 300.

Question 7.
Maximize: Z = 5x₁+ 7x₂
Subject to: x₁ + x₂ ≤ 4
5x₁+ 8x₂ ≤ 30
10x₁+ 7x₂ ≤ 35
x₁, x₂ ≥ 0
Solution:
Step – 1 Treating the constraints as equations we get,
x₁ + x₂ = 4           …. (1)
5x₁+ 8x₂ = 30      …. (2)
10x₁+ 7x₂ = 35    …. (3)
Step – 2 Let us draw the graph
Table – 1

x1	4	0
x2	0	4

Table – 2

x1	6	2
x2	0	2.5

Table – 3

x1	0	3.5
x2	5	0

Step – 3 0(0,0) satisfies all the constraints.
Thus the shaded region is the feasible region.
From (1) and (2) we get (\(\frac{2}{3}\), \(\frac{10}{3}\))
From (1) and (3) we get
x₁ = \(\frac{7}{3}\), x₁ = \(\frac{5}{3}\)
∴ The corner points are 0(0,0), A(\(\frac{7}{2}\), 0), B(\(\frac{7}{3}\), \(\frac{5}{3}\)), C(\(\frac{2}{3}\), \(\frac{10}{3}\)), D(0, \(\frac{15}{4}\))
Step – 4

Corner point	z = 5x1+ 7x2
0(0,0)	0
A(\(\frac{7}{2}\), 0)	\(\frac{35}{2}\)
B(\(\frac{7}{3}\), \(\frac{5}{3}\))	\(\frac{70}{3}\)
C(\(\frac{2}{3}\), \(\frac{10}{3}\))	\(\frac{80}{3}\)
D(0, \(\frac{15}{4}\))	\(\frac{105}{4}\)

Z attains its maximum value \(\frac{80}{3}\) for x₁ = \(\frac{2}{3}\) and x₂ = \(\frac{10}{3}\).

Question 8.
Maximize: Z = 14x₁ – 4x₂
Subject to: x₁ + 12x₂ ≤ 65
7x₁ – 2x₂ ≤ 25
2x₁+ 3x₂ ≤ 10
x₁, x₂ ≥ 0
Also find two other points which maximize Z.
Solution:
Step – 1 Treating the constraints as equations we get
x₁ + 12x₂ = 65   …. (1)
7x₁ – 2x₂ = 25    …. (2)
2x₁ + 3x₂ = 10   …. (3)
Step – 2 Let us draw the graph
Table – 1

x1	65	5
x2	0	5

Table – 2

x1	5	10
x2	5	22.5

Table – 3

x1	5	2
x2	0	2

Step – 3 Clearly 0(0,0) satisfies x₁ + 12x₂ ≤ 65 and 7x₁ – 2x₂ ≤ 25 but does not satisfy 2x₁+ 3x₂ ≤ 10. Thus shaded region is the feasible region.
Equation (1) and (2) meet at (5, 5).
From (2) and (3)

∴ The corner points of the feasible region are A(0, \(\frac{10}{3}\)), B(\(\frac{19}{5}\), \(\frac{4}{5}\)), C(5, 5), D(0, \(\frac{65}{12}\)).
Step – 4

Corner point	z = 14x1 – 4x2
A(0, \(\frac{10}{3}\))	\(\frac{-40}{3}\)
B(\(\frac{19}{5}\), \(\frac{4}{5}\))	50 → maximum
C(5, 5)	50 → maximum
D(0, \(\frac{65}{12}\))	\(\frac{65}{3}\)

Z is maximum for x₁ = \(\frac{19}{5}\), x₂ = \(\frac{4}{5}\) or x₁ = 5, x₂ = 5 and Z_max = 50
There is no other point that maximizes Z.

Question 9.
Maximize: Z = 10x₁ + 12x₂ + 8x₃
Subject to: x₁ + 2x₂ ≤ 30
5x₁ – 7x₃ ≤ 12
x₁ + x₂ + x₃ = 20
x₁, x₂ ≥ 0
[Hints: Eliminate x₃ from all expressions using the given equation in the set of constraints, so that it becomes an LPP in two variables]
Solution:
Eliminating x₃ this LPP can be written as Maximize Z = 2x₁ + 4x₂ + 160
Subject to: x₁ + 2x₂ ≤ 30
5x₁ – 7x₃ ≤ 12
x₁, x₂ ≥ 0
Step – 1 Treating the consraints as equations we get
x₁ + 2x₂ = 30    …..(1)
5x₁ – 7x₃ = 12   …..(2)
Step – 2 Let us draw the graph
Table – 1

x1	30	0
x2	0	15

Table – 2

x1	8	1
x2	8	20

Step – 3 Clearly 0(0,0) satisfies x₁ + 2x₂ ≤ 30 and does not satisfy 12x₁ + 7x₂ ≤ 152
∴ The shaded region is the feasible region.

Step – 4

Z is maximum for x₁ = 30, x₂ = 0 and Z_max = 220

Question 10.
Maximize: Z = 20x₁ + 10x₂
Subject to: x₁ + 2x₂ ≤ 40
3x₁ + x₂ ≥ 30
4x₁+ 3x₂ ≥ 60
x₁, x₂ ≥ 0
Solution:
Step – 1 Treating the constraints as equalities we have:
x₁ + 2x₂ = 40   ….(1)
3x₁ + x₂ = 30   ….(2)
4x₁+ 3x₂ = 60  ….(3)
Step – 2 Let us draw the graph

Step – 3 (0, 0) satisfies x₁ + 2x₂ ≤ 40 and does not satisfy 3x₁ + x₂ ≥ 30 and 4x₁+ 3x₂ ≥ 60, x₁, x₂ ≥ 0 is the first quadrant.
∴ The shaded region is the feasible region.
Step – 4 x₁ + 2x₂ = 40 and 3x₁ + x₂ = 30

∴ The vetices are A(15, 0), B(10, 0), C(4, 18) and D(6, 12)
Z(A) = 300, Z(B) = 800
Z (C) = 20 x 4 + 10 x 18 = 260
Z (D) = 120 + 120 = 240
Z attains minimum at D(6 ,12).
∴ The required solution x₁ = 6, x₂ =12 and Z_min = 240

Question 11.
Maximize: Z = 4x₁ + 3x₂
Subject to: x₁ + x₂ ≤ 50
x₁ + 2x₂ ≥ 80
2x₁+ x₂ ≥ 20
x₁, x₂ ≥ 0
Solution:
Step – 1 Treating the constraints as equations
x₁ + x₂ ≤ 50    ….(1)
x₁ + 2x₂ ≥ 80  ….(2)
2x₁+ x₂ ≥ 20   ….(3)
Step – 2 Let us draw the graph

Step – 3 (0, 0) satisfies x₁ + x₂ < 50, x₁ + 2x₂ < 80 but does not satisfy
2x₁ + x₂ > 20, x₁ > 0, x₂ > 0 is the 1st quadrant.
Hence the shaded region is the feasible region.
Step – 4 x₁ + x₂ = 50
x₁ + 2x₂ = 80
=> x₂ = 30, x₁ = 20
The vertices of feasible region are
A(10, 0), B(50, 0), C(20, 30), D (0, 40) and E (0, 20)

Point	Z = 4x1 + 3x2
A(10,0)	40
5(50,0)	200
C(20,30)	170
D(0,40)	120
E(0,120)	60

Question 12.
Optimize: Z = 5x₁ + 25x₂
Subject to: -0.5x₁ + x₂ ≤ 2
x₁ + x₂ ≥ 2
-x₁+ 5x₂ ≥ 5
x₁, x₂ ≥ 0
Solution:
Step – 1 Treating the constraints as equations
-0.5x₁ + x₂ = 2   ….(1)
x₁ + x₂ = 2         ….(2)
-x₁+ 5x₂ = 5      ….(3)
Step – 2 Let us draw the graph.

Step – 3 (0, 0) satisfies -0.5x₁ + x₂ ≤ 2, but does not satisfy x₁ + x₂ ≥ 2 and -x₁+ 5x₂ ≥ 5, x₁ > 0, x₂ > 0 is the 1st quadrant.
The shaded region is the feasible region with vertices A(\(\frac{5}{6}\), \(\frac{7}{6}\)) and B(0, 2).
Step – 4 Z can be made arbitrarily large.
∴ Problem has no maximum.
But Z(A) = \(\frac{100}{3}\), Z(B) = 50
Z is minimum at A(\(\frac{5}{6}\), \(\frac{7}{6}\)).
But the feasible region is unbounded.
Hence we cannot immediately decide, Z is minimum at A.
Let us draw the half plane
5x₁ + 25x₂ < \(\frac{100}{3}\)
⇒ 3x₁ + 15x₂ < 20
As there is no point common to this half plane and the feasible region.
we have Z is minimum for x₁ = \(\frac{5}{6}\), x₂ = \(\frac{7}{6}\) and the minimum value = \(\frac{100}{3}\)

Question 13.
Optimize: Z = 5x₁ + 2x₂
Subject to: -0.5x₁ + x₂ ≤ 2
x₁ + x₂ ≥ 2
-x₁+ 5x₂ ≥ 5
x₁, x₂ ≥ 0
Solution:
Step – 1 Treating the constraints as equations
-0.5x₁ + x₂ = 2   ….(1)
x₁ + x₂ = 2         ….(2)
-x₁+ 5x₂ = 5      ….(3)
Step – 2 Let us draw the graph.

Step – 3 The shaded regian is feasible region which is unbounded, thus Z does not have any maximum.

As Z can be made arbitrarily large, the given LPP has no maximum.
Z is minimum at B (0, 2). But we cannot immediately decide, Z is minimum at B.
Let us draw the half plane 5x₁ + 2x₂ < 4

x1	0	4/5
x2	2	0

As there is no point common to this half plane and the feasible region,
we have Z is minimum for x₁ = 0, x₂ = 2 and the minimum value of Z = 4.

Question 14.
Optimize: Z = -10x₁ + 2x₂
Subject to: -x₁ + x₂ ≥ -1
x₁ + x₂ ≤ 6
x₂ ≤ 5
x₁, x₂ ≥ 0
Solution:
Step – 1 Treating the constraints as equations
-x₁ + x₂ = -1     ….(1)
x₁ + x₂ = 6        ….(2)
x₂ = 5                ….(3)
Step – 2 Let us draw the graph
Table – 1

x1	1	0
x2	0	-1

Table – 2

x1	6	0
x2	0	1

Step – 3 Clearly 0(0,0) satisfies all the constraints.
Thus the shaded region is the feasible region.
The vertices are 0(0,0) , A(1,0), B(\(\frac{7}{2}\), \(\frac{5}{2}\)) ,C(1, 5) and D (0, 5)
Step – 4 Z(O) = 0
Z(A) = -10
Z(B) = – 30
Z(C) = 0
Z(D) = 10
∴ Z is maximum for x₁= 0, x, = 5 and Z_(max) = 10
Z is minimum for x₁ = \(\frac{7}{2}\)  x₂ = \(\frac{5}{2}\) and Z_(min) = -30

Question 15.
Solve the L.P.P.s obtained in Exercise 3(a) Q.1 to Q. 9 by graphical method.
(1) Maximise: Z = 1500x + 2000y
Subject to: x + y < 20
x + 2y < 24
x, y ≥ 0
Solution:
Step – 1 Treating the constraints as equations we get
x + y = 20
x + 2y = 24
Step – 2 Let us draw of graph.

Step – 3 Clearly 0(0,0) satisfies all the constraints.
Thus the shaded region is the feasible region.
From (1) and (2) we get
y = 14
x = 16
With vertices 0(0, 0), A(20, 0), B(16, 4), C(0, 12).
Step – 4 Z(0) = 0
Z(A) = 30,000
Z(B) = 32,000 → Maximum
Z(C) = 24000
Z is maximum for x = 16, y = 4 with Z = 32000
To get maximum profit he must keep 16 sets of model X and 4 sets of model Y.
Maximum profit = 1500 × 16 + 2000 × 4 = ₹32,000

(2) Maximize: 15x + 10y
Subject: x + 3y ≤ 600
2x + y ≤ 480
x, y ≥ 0
Solution:
Step – 1 Treating the constraints as equations we get
2x +3y = 600
2a + y = 480
Step – 2 Let us draw the graph

Step – 3 Clearly 0(0,0) satisfies all the constraints.
The corner point are 0(0, 0), A (240, 0) B(210, 60),C(0, 200)
Step – 4 Z(0) = 6
Z(A) = 3600
Z(B) = 3150 + 600
= 3750 → maximum
Z(C) = 2000
Thus Z is maximum for x = 210 and y = 60
and Z_(max) = 3750

(3) Maximize: Z = 20x + 30y
Subject to: x + 2y ≤ 10
x + y ≤ 6
x ≤ 4
x, y ≥ 0.
Solution:
Step – 1 Treating the constraints as equations we get
x + 2y = 10       …(1)
x + y = 6           …(2)
x = 4
Step – 2 Let us draw the graph

Step – 3 As 0(0,0) satisfies all the constraints the shaded region is the feasible region.
Solving (1) and (2) we get x = 2, y = 4.
The vertices and 0(0, 0) , A(4, 0), B(4, 2), C(2, 4), D (0, 5).
Step – 4 Z(0) =0
Z(A) = 80
Z (B) =140
Z(C) = 1 60 → maximum
Z (D) = 150
∴ Z is Maximum when x = 2, y = 4 and Z_(max) = 160

(4) Maximize: Z = 15x + 17y
Subject to: 4x + 7y ≤ 150
x + y ≤ 30
15x + 17y > 300
x, y ≥ 0
Solution:
Step – 1 Treating the constraints as equations we get
4x + 7y = 150      ….(1)
x + y = 30            ….(2)
15x + 17y = 300  ….(3)
Step – 2 Let us draw the graph

Step – 3 Clearly 0(0,0) satisfies all the constraints.
4x + 7y ≤ 150, x + y ≤ 30, but does not satisfy 15x + 17y ≥ 300.
∴ The shaded region is the feasible region.
From (1) and (2) we get

∴ Z is maximum for x = 20. y = 10 and Z_(max) = 470

(5) Maximize: Z = 2x + 4y
Subject to: 3x + 2y ≤ 10
2x + 5y ≤ 15
5x + 6y ≤ 21
x, y ≥ 0
Solution:
Step – 1 Treating the constraints as equations we get
3x + 2y = 10  …(1)
2x + 5y = 15  …(2)
5x + 6y = 21  …(3)
Step – 2 Let us draw the graph

Step – 3 As 0(0,0) satisfies all the constraints the shaded region is the feasible region.
From (1) and (3) we get

From (2) and (3) we get

Step-4 Z(O) = 0

(6) Maximize: Z = 1000x + 800y
Subject to: x + y ≤ 5
2x + y ≤ 9
x, y ≥ 0
Solution:
Step – 1 Treating the constraints as equations we get
x + y = 5    ….(1)
2x + y = 9  ….(2)
Step – 2 Let us draw the graph

Step – 3 Clearly 0(0,0) satisfies all the constraints.
∴ Thus the shaded region is the feasible region.
From (1) and (2) we get x = 4, y = 1.
∴ The vertices are A(0, 0), A(4.5, 0), B(4, 1) and C(0, 5).
Step – 4 Z(0) =0
Z (A) = 4500
Z (B) = 4800 → Maximum
Z (C) = 4000
Z is maximum for x = 4 and y = 1, Z_(max) = 4800

(7) Minimize: Z = 4960 – 70x – 130y
Subject to: x + y ≤ 12
x + y ≥ 6
x ≤ 8
y ≤ 8
x, y ≥ 0
Solution:
Step – 1 Treating the constraints as equations we get
x + y = 12   ….(1)
x + y = 6     ….(2)
x = 8           ….(3)
y = 4           ….(4)
Step – 2 Let us draw the graph

Step – 3 Clearly 0(0,0) satisfies all the constraints except x + y > 6.
The shaded region is the feasible region.
The vertices are A(6, 0), B(8, 0), C(8, 4), D(4, 8), E(0, 8) and F(0, 6).
Step – 4 Z (A) = 4540
Z (B) = 4400
Z (C) = 3880
Z (D) = 3640 → Minimum
Z (E) = 3920
Z (F) = 4180
∴ Z is maximum for x = 4 and y = 8 and Z_(min) = 3640.

(8) Minimize: Z = 16x + 20y
Subject to x + 2y ≥ 10
x + y ≥ 6
3x + y ≥ 8
x, y ≥ 0
Solution:
Step – 1 Treating the constraints as equations we get
x + 2y = 10  ….(1)
x + y = 6      …(2)
3x + y = 8    …(3)
Step – 2 Let us draw the graph

Step – 3 Clearly 0(0,0) satisfies all the constraints. Thus the shaded region is the feasible region.
From (1) and (2) we get y = 4, x = 2.
From (2) and (3) we get x = 1, y = 5.
The vertices are A(10, 0), B(2, 4), C(1, 5), D(0, 8).
Step – 4 Z (A) = 160
Z (B) = 112 → Minimum
Z (C) =116
Z (D) = 160
As the region is unbounded, let us draw the half plane Z < Z_(min)
⇒ 16x + 20y < 112
⇒ 4x + 5y < 28

x1	7	0
x2	0	5.6

There is no point common to the shaded region and the half plane 4x + 5y ≤ 28 other than B(2, 4).
∴ Z is minimum for x = 2, y = 4 and Z_(min) = 112.

(9) Minimize: Z = (512.5)x + 800y
Subject to: 5x + 4y = 40
x ≤ 7
x ≤ 3
x, y ≥ 0
Solution:
Step – 1 Let us draw the graph of
5x + 4y = 40
x = 7, y = 3

x1	8	0
x2	0	10

Step – 1 Let us draw the graph of
5x + 4y = 40
x = 7, y = 3
Step – 2 The line segment AB is the feasible region.
Step – 3 Z (A) = 3587.5 + 1000 = 4587.5
Z (B) = 2870 + 2400 = 5270
Clearly Z is minimum for
x = 7, y = \(\frac{5}{4}\) and Z_(min) = 4587.5

Odisha State Board Elements of Mathematics Class 12 Solutions CHSE Odisha Chapter 2 Inverse Trigonometric Functions Ex 2 Textbook Exercise Questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 2 Inverse Trigonometric Functions Exercise 2

Question 1.
Fill in the blanks choosing correct answer from the brackets:
(i) If A = tan^-1 x, then the value of sin 2A = ________. (\(\frac{2 x}{1-x^2}\), \(\frac{2 x}{\sqrt{1-x^2}}\), \(\frac{2 x}{1+x^2}\))
Solution:
\(\frac{2 x}{1+x^2}\)

(ii) If the value of sin^-1 x = \(\frac{\pi}{5}\) for some x ∈ (-1, 1) then the value of cos^-1 x is ________. (\(\frac{3 \pi}{10}\), \(\frac{5 \pi}{10}\),\(\frac{3 \pi}{10}\))
Solution:
\(\frac{3 \pi}{10}\)

(iii) The value of tan^-1 x (2cos\(\frac{\pi}{3}\)) is ________. (1, \(\frac{\pi}{4}\), \(\frac{\pi}{3}\))
Solution:
\(\frac{\pi}{4}\)

(iv) If x + y = 4, xy = 1, then tan^-1 x + tan^-1 y = ________. (\(\frac{3 \pi}{4}\), \(\frac{\pi}{4}\), \(\frac{\pi}{3}\))
Solution:
\(\frac{\pi}{2}\)

(v) The value of cot^-1 2 + tan^-1 \(\frac{1}{3}\) = ________. (\(\frac{\pi}{4}\), 1, \(\frac{\pi}{2}\))
Solution:
\(\frac{\pi}{4}\)

(vi) The principal value of sin^-1 (sin \(\frac{2 \pi}{3}\)) is ________. (\(\frac{2 \pi}{3}\), \(\frac{\pi}{3}\), \(\frac{4 \pi}{3}\))
Solution:
\(\frac{\pi}{3}\)

(vii) If sin^-1 \(\frac{x}{5}\) + cosec^-1 \(\frac{5}{4}\) = \(\frac{\pi}{2}\), then the value of x = ________. (2, 3, 4)
Solution:
x = 3

(viii) The value of sin (tan^-1 x + tan^-1 \(\frac{1}{x}\)), x > 0 = ________. (0, 1, 1/2)
Solution:
1

(ix) cot^-1 \(\left[\frac{\sqrt{1-\sin x}+\sqrt{1+\sin x}}{\sqrt{1-\sin x}-\sqrt{1+\sin x}}\right]\) = ________. (2π – \(\frac{x}{2}\), \(\frac{x}{2}\), π – \(\frac{x}{2}\))
Solution:
π – \(\frac{x}{2}\)

(x) 2sin^-1 \(\frac{4}{5}\) + sin^-1 \(\frac{24}{25}\) = ________. (π, -π, 0)
Solution:
π

(xi) if Θ = cos^-1 x + sin^-1 x – tan^-1 x, x ≥ 0, then the smallest interval in which Θ lies is ________. [(\(\frac{\pi}{2}\), \(\frac{3 \pi}{2}\)), [0, \(\frac{\pi}{2}\)), (0, \(\frac{\pi}{2}\)])
Solution:
(0, \(\frac{\pi}{2}\)]

(xii) sec² (tan^-1 2) + cosec² (cot^-1 3) = ________. (16, 14, 15)
Solution:
15

Question 2.
Write whether the following statements are true or false.
(i) sin^-1 \(\frac{1}{x}\) cosec^-1 x = 1
Solution:
False

(ii) cos^-1 \(\frac{4}{5}\) + tan^-1 \(\frac{2}{3}\) = tan^-1 \(\frac{17}{6}\)
Solution:
True

(iii) tan^-1 \(\frac{4}{3}\) + cot^-1 (\(\frac{-3}{4}\)) = π
Solution:
True

(iv) sec^-1 \(\frac{1}{2}\) + cosec^-1 \(\frac{1}{2}\) = \(\frac{\pi}{2}\)
Solution:
False

(v) sec^-1 (-\(\frac{7}{5}\)) = π – cos^-1 \(\frac{5}{7}\)
Solution:
True

(vi) tan^-1 (tan 3) = 3
Solution:
False

(vii) The principal value of tan^-1 (tan \(\frac{3 \pi}{4}\)) is \(\frac{3 \pi}{4}\)
Solution:
False

(viii) cot^-1 (-√3) is in the second quadrant.
Solution:
True

(ix) 3 tan^-1 3 = tan^-1 \(\frac{9}{13}\)
Solution:
False

(x) tan^-1 2 + tan^-1 3 = – \(\frac{\pi}{4}\)
Solution:
False

(xi) 2 sin^-1 \(\frac{4}{5}\) = sin^-1 \(\frac{24}{25}\)
Solution:
False

(xii) The equation tan^-1 (cotx) = 2x has exactly two real solutions.
Solution:
True

Question 3.
Express the value of the foilowing in simplest form.
(i) sin (2 sin^-1 0.6)
Solution:
sin (2 sin^-1 0.6)

(ii) tan (\(\frac{\pi}{4}\) + 2 cot^-1 3)
Solution:

(iii) cos (2 sin^-1 x)
Solution:

(iv) tan (cos^-1 x)
Solution:

(v) tan^-1 (\(\frac{x}{y}\)) – tan^-1 \(\frac{x-y}{x+y}\)
Solution:

(vi) cosec (cos^-1 \(\frac{3}{5}\) + cos^-1 \(\frac{4}{5}\))
Solution:

(vii) sin^-1 \(\frac{1}{\sqrt{5}}\) + cos^-1 \(\frac{3}{\sqrt{10}}\)
Solution:

(viii) sin cos^-1 tan sec √2
Solution:
sin cos^-1 tan sec √2
= sin cos^-1 tan sec \(\frac{\pi}{4}\)
= sin cos^-1 1 = sin 0 = 0

(ix) sin (2 tan^-1 \(\sqrt{\frac{1-x}{1+x}}\))
Solution:

(x) tan \(\left\{\frac{1}{2} \sin ^{-1} \frac{2 x}{1+x^2}+\frac{1}{2} \cos ^{-1} \frac{1-y^2}{1+y^2}\right\}\)
Solution:

(xi) sin cot^-1 cos tan^-1 x.
Solution:

(xii) tan^-1 \(\left(x+\sqrt{1+x^2}\right)\)
Solution:

Question 4.
Prove the following statements:
(i) sin^-1 \(\frac{3}{5}\) + sin^-1 \(\frac{8}{17}\) = cos^-1 \(\frac{36}{85}\)
Solution:

(ii) sin^-1 \(\frac{3}{5}\) + cos^-1 \(\frac{12}{13}\) = cos^-1 \(\frac{33}{65}\)
Solution:

(iii) tan^-1 \(\frac{1}{7}\) + tan^-1 \(\frac{1}{13}\) = tan^-1 \(\frac{2}{9}\)
Solution:
L.H.S = tan^-1 \(\frac{1}{7}\) + tan^-1 \(\frac{1}{13}\)

(iv) tan^-1 \(\frac{1}{2}\) + tan^-1 \(\frac{1}{5}\) + tan^-1 \(\frac{1}{8}\) = \(\frac{\pi}{4}\)
Solution:

(v) tan ( 2tan^-1 \(\frac{1}{5}\) – \(\frac{\pi}{4}\) ) + \(\frac{7}{17}\) = 0
Solution:

Question 5.
Prove the following statements:
(i) cot^-1 9 + cosec^-1 \(\frac{\sqrt{41}}{4}\) = \(\frac{\pi}{4}\)
Solution:

(ii) sin^-1 \(\frac{4}{5}\) + 2 tan^-1 \(\frac{1}{3}\) = \(\frac{\pi}{2}\)
Solution:

(iii) 4 tan^-1 \(\frac{1}{5}\) – tan^-1 \(\frac{1}{70}\) + tan^-1 \(\frac{1}{99}\) = \(\frac{\pi}{4}\)
Solution:

(iv) 2 tan^-1 \(\frac{1}{5}\) + sec^-1 \(\frac{5 \sqrt{2}}{7}\) + 2 tan^-1 \(\frac{1}{8}\) = \(\frac{\pi}{4}\)
Solution:

(v) cos^-1 \(\frac{12}{13}\) + 2 cos^-1 \(\sqrt{\frac{64}{65}}\) + cos^-1 \(\sqrt{\frac{49}{50}}\) = cos^-1 \(\frac{1}{\sqrt{2}}\)
Solution:


(vi) tan² cos^-1 \(\frac{1}{\sqrt{3}}\) + cot² sin^-1 \(\frac{1}{\sqrt{5}}\) = 6
Solution:
tan² cos^-1 \(\frac{1}{\sqrt{3}}\) + cot² sin^-1 \(\frac{1}{\sqrt{5}}\)
= tan² tan^-1 √2 + cot² cot^-1 (2)
= 2 + 4 = 6

(vii) cos tan^-1 cot sin^-1 x = x.
Solution.

Question 6.
Prove the following statements:
(i) cot^-1 (tan 2x) + cot^-1 (- tan 2x) = π
Solution:

(ii) tan^-1 x + cot^-1 (x + 1) = tan^-1 (x² + x + 1)
Solution:

(iii) tan^-1 (\(\frac{a-b}{1+a b}\)) + tan^-1 (\(\frac{b-c}{1+b c}\)) = tan^-1 a – tan^-1 c.
Solution:
tan^-1 (\(\frac{a-b}{1+a b}\)) + tan^-1 (\(\frac{b-c}{1+b c}\))
= tan^-1 a – tan^-1 b + tan^-1 b – tan^-1 c
= tan^-1 a – tan^-1 c.

(iv) cot^-1 \(\frac{p q+1}{p-q}\) + cot^-1 \(\frac{q r+1}{q-r}\) + cot^-1 \(\frac{r p+1}{r-p}\) = 0
Solution:

(v)

Solution:

Question 7.
Prove the following statements:
(i) tan^-1 \(\frac{2 a-b}{b \sqrt{3}}\) + tan^-1 \(\frac{2 b-a}{a \sqrt{3}}\) = \(\frac{\pi}{3}\)
Solution:

(ii) tan^-1 \(\frac{1}{x+y}\) + tan^-1 \(\frac{y}{x^2+x y+1}\) = tan^-1 \(\frac{1}{x}\)
Solution:

(iii) sin^-1 \(\sqrt{\frac{x-q}{p-q}}\) = cos^-1 \(\sqrt{\frac{p-x}{p-q}}\) = cot^-1 \(\sqrt{\frac{p-x}{x-q}}\)
Solution:

(iv) sin² (sin^-1 x + sin^-1 y + sin^-1 z) = cos² (cos^-1 x + cos^-1 y + cos^-1 z)
Solution:

(v) tan (tan^-1 x + tan^-1 y + tan^-1 z) = cot (cot^-1 x + cot^-1 y + cot^-1 z)
Solution:

Question 8.
(i) If sin^-1 x + sin^-1 y + sin^-1 z = π, show that x\(\sqrt{1-x^2}\) + x\(\sqrt{1-y^2}\) + x\(\sqrt{1-z^2}\) = 2xyz
Solution:
Let sin^-1 x = α, sin^-1 y = β, sin^-1 z = γ
∴ α + β + γ = π
∴ x = sin α, y = sin β, z = sin γ
or, α + β = π – γ
or, sin(α + β) = sin(π – γ) = sin γ
and cos(α + β) = cos(π – γ) = – cos γ

(ii) tan^-1 x + tan^-1 y + tan^-1 z = π show that x + y + z = xyz.
Solution:

(iii) tan^-1 x + tan^-1 y + tan^-1 z = \(\frac{\pi}{2}\). Show that xy + yz + zx = 1
Solution:

or, 1 – xy – yz – zx = 0
⇒ xy + yz + zx = 1

(iv) If r² = x² +y² + z², Prove that tan^-1 \(\frac{y z}{x r}\) + tan^-1 \(\frac{z x}{y r}\) + tan^-1 \(\frac{x y}{z r}\) = \(\frac{\pi}{2}\)
Solution:

(v) In a triangle ABC if m∠A = 90°, prove that tan^-1 \(\frac{b}{a+c}\) + tan^-1 \(\frac{c}{a+b}\) = \(\frac{\pi}{4}\). where a, b, and c are sides of the triangle.
Solution:
L.H.S. tan^-1 \(\frac{b}{a+c}\) + tan^-1 \(\frac{c}{a+b}\)

Question 9.
Solve
(i) cos (2 sin^-1 x) = 1/9
Solution:

(ii) sin^-1 x + sin^-1 (1 – x) = \(\frac{\pi}{2}\)
Solution:
sin^-1 x + sin^-1 (1 – x) = \(\frac{\pi}{2}\)
or, sin^-1 (1 – x) = \(\frac{\pi}{2}\) – sin^-1 x = cos^-1 x
or, sin^-1 (1 – x) = sin^-1 \(\sqrt{1-x^2}\)
or, 1 – x = \(\sqrt{1-x^2}\)
or, 1 + x² – 2x = 1 – x²
or, 2x² – 2x  = 0
or, 2x (x – 1) = 0
∴ x = 0 or, 1

(iii) sin^-1 (1 – x) – 2 sin^-1 x = \(\frac{\pi}{2}\)
Solution:
sin^-1 (1 – x) – 2 sin^-1 x = \(\frac{\pi}{2}\)
⇒ – 2 sin^-1 x = \(\frac{\pi}{2}\) – sin^-1 (1 – x)
⇒ cos^-1 (1 – x)
⇒ cos (– 2 sin^-1 x) = 1 – x      ….. (1)
Let sin^-1 Θ ⇒ sin Θ
Now cos (– 2 sin^-1 x) = cos (-2Θ)
= cos 2Θ = 1 – 2 sin² Θ = 1 – 2x²
Using in (1) we get
1 – 2x² = 1 – x
⇒ 2x² – x = 0 ⇒ x (2x – 1) = 0
⇒ x = 0, ½, But x = ½ does not
Satisfy the given equation, Thus x = 0.

(iv) cos^-1 x + sin^-1 \(\frac{x}{2}\) = \(\frac{\pi}{6}\)
Solution:

(v) tan^-1 \(\frac{x-1}{x-2}\) + tan^-1 \(\frac{x+1}{x+2}\) = \(\frac{\pi}{4}\)
Solution:

(vi) tan^-1 \(\frac{1}{2 x+1}\) + tan^-1 \(\frac{1}{4 x+1}\) = tan^-1 \(\frac{2}{x^2}\)
Solution:

(vii) 3 sin^-1 \(\frac{2 x}{1+x^2}\) – 4 cos^-1 \(\frac{1-x^2}{1+x^2}\) + 2 tan^-1 \(\frac{2 x}{1-x^2}\) = \(\frac{\pi}{3}\)
Solution:

(viii) cot^-1 \(\frac{1}{x-1}\) + cot^-1 \(\frac{1}{x}\) + cot^-1 \(\frac{1}{x+1}\) = cot^-1 \(\frac{1}{3x}\)
Solution:

(ix) cot^-1 \(\frac{1-x^2}{2 x}\) =  cosec^-1 \(\frac{1+a^2}{2 a}\) – sec^-1 \(\frac{1+b^2}{1-b^2}\)
Solution:

(x) sin^-1 \(\left(\frac{2 a}{1+a^2}\right)\) + sin^-1 \(\left(\frac{2 b}{1+b^2}\right)\) = 2 tan^-1 x
Solution:

(xi) sin^-1 y – cos^-1 x = cos^-1 \(\frac{\sqrt{3}}{2}\)
Solution:

(xii) sin^-1 2x + sin^-1 x = \(\frac{\pi}{3}\)
Solution:

Question 10.
Rectify the error ifany in the following:
sin^-1 \(\frac{4}{5}\) + sin^-1 \(\frac{12}{13}\) + sin^-1 \(\frac{33}{65}\)
Solution:

Question 11.
Prove that:
(i) cos^-1 \(\left(\frac{b+a \cos x}{a+b \cos x}\right)\) = 2 tan^-1 \(\left(\sqrt{\frac{a-b}{a+b}} \tan \frac{x}{2}\right)\)
Solution:

(ii) tan \(\left(\frac{\pi}{4}+\frac{1}{2} \cos ^{-1} \frac{a}{b}\right)\) + tan \(\left(\frac{\pi}{4}-\frac{1}{2} \cos ^{-1} \frac{a}{b}\right)\)
Solution:

(iii) tan^-1 \(\sqrt{\frac{x r}{y z}}\) + tan^-1 \(\sqrt{\frac{y r}{y x}}\) + tan^-1 \(\sqrt{\frac{z r}{x y}}\) = π where r = x + y +z.
Solution:

Question 12.
(i) If cos^-1 (\(\frac{x}{a}\)) + cos^-1 (\(\frac{y}{b}\)) = Θ, prove that \(\frac{x^2}{a^2}\) – \(\frac{2 x}{a b}\) cos Θ + \(\frac{y^2}{b^2}\) = sin² Θ.
Solution:

(ii) If cos^-1 (\(\frac{x}{y}\)) + cos^-1 (\(\frac{y}{3}\)) = Θ, prove that 9x² – 12xy cos Θ + 4y² = 36 sin² Θ.
Solution:

(iii) If sin^-1 (\(\frac{x}{a}\)) + sin^-1 (\(\frac{y}{b}\)) = sin^-1 (\(\frac{c^2}{a b}\)) prove that b²x² + 2xy \(\sqrt{a^2 b^2-c^4}\) a²y² = c²
Solution:

(iv) If sin^-1 (\(\frac{x}{a}\)) + sin^-1 (\(\frac{y}{b}\)) = α prove that \(\frac{x^2}{a^2}\) + \(\frac{2 x y}{a b}\) cos α + \(\frac{y^2}{b^2}\) = sin² α
Solution:

(v) If sin^-1 x + sin^-1 y + sin^-1 z = π prove that x² + y² + z² + 4x²y²z² = 2 ( x²y² + y²z² + z²x² )
Solution:

Question 13.
Solve the following equations:
(i) tan^-1 \(\frac{x-1}{x+1}\) + tan^-1 \(\frac{2 x-1}{2 x+1}\) = tan^-1 \(\frac{23}{36}\)
Solution:

(ii) tan^-1 \(\frac{1}{3}\) + tan^-1 \(\frac{1}{5}\) + tan^-1 \(\frac{1}{7}\) + tan^-1 x = \(\frac{\pi}{4}\)
Solution:

(iii) cos^-1 \(\left(x+\frac{1}{2}\right)\) + cos^-1 x+ cos^-1 \(\left(x-\frac{1}{2}\right)\) = \(\frac{3 \pi}{2}\)
Solution:

(iv) 3tan^-1 \(\frac{1}{2+\sqrt{3}}\) – tan^-1 \(\frac{1}{x}\) = tan^-1 \(\frac{1}{3}\)
Solution: