History chapter 1 The Rise of Nationalism in Europe Notes

Chapter  1  The Rise of Nationalism in Europe                                      

Frederic  Sorrieu , French Artist’s Painting

• In 1848, Frédéric Sorrieu, a French artist, prepared a series of four prints visualising his dream of a world made up of ‘democratic and social Republics’, as he called them.
•  It’s shows the peoples of Europe and America – men and women of all ages and social classes – marching in a long train

• Offering homage to the statue of Liberty

•   Statue of Liberty having torch of Enlightenment she bears in one hand and the Charter of the Rights of Man in the other

• On the earth in the foreground of the image lie the shattered remains of the symbols of absolutist institutions

•   Leading the procession, way past the statue of Liberty, are the United States and Switzerland

•   She is followed by the peoples of Germany, bearing the black, red and gold flag.

• From the heavens above, Christ, saints and angels gaze upon the scene

• This chapter will deal with many of the issues visualised by Sorrieu

• During the nineteenth century, nationalism emerged as a force which brought about sweeping changes in the political and mental world of Europe.

French Revolution and the idea of Nation

French Revolution

• Political and contitonal changes led to the transfer of sovereignty from monarchy to French citizens

• French revolutionaries introduced measures that could create a sense of collective identity

•  A new French flag was chosen

• Estates General renamed as National Assembly

•  A centralised administrative system formulated uniform laws for all citizens 

•  Uniform system of weights and measures adopted

• French became the common language of the nation

• Revolutionaries further made a mission liberate people of Europe for despotism

• Helping and guiding peoples of  Europe to become nations

Spread of nationalism

•  Standard and education middle classes in Europe began setting up Jacobin clubs

•  Their activities and campaigns prepared the way for French

• French armies moved into Holland, Belgium, Switzerland and Italy

•  With the outbreak of the revolutionary wars, French armies began to carry the of  Nationalism

Resources and Development Nots and ncert basedbook solutons

Resources and Development

Resource: Anything which can be used for satisfying the human needs is called a resource.

Types of Resources:

Resources can be classified on different bases; into following types:
On the basis of origin – Biotic and Abiotic:-
On the basis of exhaustibility – Renewable and Non-renewable
On the basis of ownership – Individual, community, national and international
On the basis of status of development – Potential, Developed, Stock and Reserves

Classification of Resources :- On the basis of Origin

Biotic Resources – All living organisms in our environment are called biotic resources. For example, trees, animals, insects, etc.
Abiotic Resources – All non-living things present in our environment are termed as abiotic resources. For example – earth, air, water, metals, rocks, etc.

Classification of Resources :- On the basis of Exhaustibility:

Renewable Resources: Resources that can be replenished after a short period of time are called Renewable Resources. For example – agricultural crops, water, forest, wildlife, etc.
Non-renewable Resources – Resources which takes million years of time to replenish are called non-renewable resources. For example – fossil fuel.

Classification of Resources - On the Basis of ownership:

Individual – Resources owned by individuals are called Individual Resources. For example – land owned by farmers, house, etc.
Community – Resources owned by community or society are called Community Owned Resources. For example – Graveyard, grazing land, ponds, burial grounds, park, etc.
National Resources – Resources owned by Individual Nations are called National Resources. For example – Government land, Roads, canals, railway, etc.
International Resources – Resources regulate by International bodies are called International Resources. For example – Ocean and sea beyond 200 km of the Exclusive Economic Zone and is called open sea or ocean. No individual country can utilize these resources without the permission of International bodies.

Classification of Resources - On the basis of Status of Development:

Potential Resources: Resources which are found in a particular region, but not yet used properly. For example – Rajasthan and Gujarat receive plenty of solar energy and have plenty of wind energy, but use of these resources so far has not been developed properly.
Developed Resources: Resources which are developed and surveyed for utilization and are being used in present time are known as Developed Resources.
Stock – Resources that are available, but we do not have proper technology to used them are called Stock. For example – water is made of oxygen and hydrogen, which can be used as fuel, but because of lack of proper technology these are not being used.
Reserves – Resources which are available and the knowhow to use them is also present but they are yet to be used are called Reserves. For example – river water which is not used to generate electricity.

Discriminate use of Resources

Resources are vital for human beings. But indiscriminate use of resources is creating many problems.

Example –

Accumulation of resources in only few hands leaves others unsatisfied.
Indiscriminate use of resources is creating many problems around the world, such as global warming, ecological crisis, disturbance in ozone layer, etc.
Thus, equal distribution of resources becomes necessary for sustainable development.
Sustainable Development: Development which takes place without damaging the environment and compromising with needs of future is called sustainable development.
Keeping the view of justified use of resources and sustainable development, the Earth Summit was organized in 1992 in Rio de Janeiro in which more than 100 state heads agreed with Agenda 21 for the sustainable development and proper use of resources.
Agenda 21 is an agenda to combat environmental damage, poverty, disease, etc. through global co-operation on common interests, mutual needs and shared responsibilities.

Resource Planning:

Resource planning is the judicious use of resources. Resource planning becomes more important in a country like India, where resources are not distributed properly. For example; many states are rich in mineral and deficient in other resources, such as Jharkhand is rich in minerals, but there is problem of drinking water and other facilities, Arunachal Pradesh has plenty of water but lack of other development because of lack of resources.
These types of discriminations can be reduced or completely vanished with proper planning of judicious use of resources.

Resource Planning in India:

Resources can contribute in proper development only with a good planning keeping the technology, skills and institution in mind.
Resource Planning in India is one of the most important goals right from its first Five Years Plan. Following are the main points of Resource planning.
(a) Making of inventory of resources after their region-wise identification across the country.
(b) Making of the planning structure with appropriate technology, skill and institutions.
(c) Matching of resource plan with development plan, etc.

Conservation of Resources:

Overuse of resources creates many socio-economic problems. Many leaders and thinkers have been advocating for the judicious use and conservation of resources. Gandhiji told “There is enough for everybody’s need and not for any body’s greed.” He thought that exploitative nature of modern technology is the root cause for depletion at global level. He believed in the production by masses and not in the mass production.
Thus, conservation of resources at various levels becomes most important. Resources can be conserved only with their judicious use.

Land Resources:-

Land is one of the most important natural resources. Land supports our life system. Thus, careful planning of use of land resource is necessary.
India comprises of many types of land. These are mountains, plateau, plains and islands.
Mountains – About 30% of land area in India is in the form of mountain. Mountain supports the perennial flow of rivers, which carry fertile soils, facilitate irrigation and provide drinking water.
Plain – About 43% of land area in India is in the form of plains. Plains provide facilities for agriculture, building of industries and houses, etc.
Plateau – About 27% of land in India is in the form of plateau which provides many types of minerals, fossil fuels and forest.
Land Utilisaiton: Patterns of use of Land Resources

1. Forests

2. Land not available for cultivation: There are two types of land which are not used for agriculture purpose. These are:
(a) Barren and waste land
(b) Lands used for buildings, roads, factories, etc. i.e for non-agriculture purpose.

3. Other uncultivated land (excluding fallow land)

(a) Permanent pastures and grazing land,
(b) Land under miscellaneous tree crops groves (not included in net sown area),
(c) Culturable waste land (left uncultivated for more than 5 agricultural years).

4. Fallow lands

(a) Current fallow-(left without cultivation for one or less than one agricultural year),
(b) Other than current fallow-(left uncultivated for the past 1 to 5 agricultural years).
5. Net sown area: Area which is sown at least once in a year is called net sown area.
6. Gross cropped area: Area sown more than once in an agricultural year plus net sown area is known as gross cropped area.

Land Use Pattern in India:-

Pattern of use of land depends upon physical and human factors both. Climate, topography, type of soil, etc. are considered as physical factors while population, technology, skill, population density, tradition, capability, etc. are considered as human factors.
India has total 3.28 million square kilometer land used data. But only 93% of land of total geographical area is available. This is because land used data has not been collected for the north eastern states except Assam and the land occupied by Pakistan and China has not been surveyed because of many unavoidable reasons.
The land under permanent pasture is decreasing, this will create the problem for grazing. The total net sown are (NSA) is not more than 54% including land other than fallow land. Land other than fallow land is either of poor quality or too costly to cultivate, these lands are cultivated only once or twice in two or three consecutive years.
The pattern of net sown area varies widely from state to state. Where net sown area is 80% in state like Punjab, it is only 10% in the state of Arunachal Pradesh, Mizoram, Manipur and Andaman and Nicobar Island. Such difference is creating lot of discrimination.
According to National Forest Policy (1952), the forest should be 33% of total geographical area, which is essential to maintain ecological balance. But the forest area in India is far less than desired measures. This is because of illegal deforestation and development which cannot be overlooked, such as construction of roads and building, etc. On the other hand, a large population which is dwelling at the fringe of forest depends upon the forest and its produce, resulting in the reduction of forest area.
Moreover, continuous use of land over a long period without taking measures to conserve and manage, degrade them. This has resulted in repercussion in society and creating serious problem to environment.

Land Degradation

Our past generation left land for us without exploiting them too much and it is expected from us too. We fulfill most of our needs from land, such as food, clothing, shelter, drinking water, etc. But in past few decades the quality of land is degrading fiercely because of human activity. Many human activities aggravated the natural forces which are, in turn, degrading the land resources also.
Presently, about 130 million hectare of land is reported under degraded land in India, in which about 28% of land belongs to forest and about 28% is water eroded area. Rest degraded land is because of over deposition of salinity and alkalinity. Overgrazing, mining, deforestation, division of lands in small area because of family feuds, etc. are some of the major causes of degradation of land.
Because of mining in the states of Jharkhand, Chhattisgarh, Orissa and Madhya Pradesh; lands are left abandoned without proper treatment after the completion of mining works. This has resulted in the form of deep scars and traces. Along with mining, deforestation in these states has degraded the land very fast.
In the states of Uttar Pradesh, Punjab, Haryana, over irrigation causes water shortage and increase in salinity and alkalinity due to water logging.
In Bihar, Assam, Arunachal Pradesh, land is degraded because of flood.
States, in which minerals processing, such as grinding of lime stone, manufacturing of cement, etc. produce huge quantity of dust. These dusts prevent the percolation of water because of deposition on the ground and it is responsible for degradation of land.
The degradation of land creates many problems, such as flood, decrease in yield, etc. which leads to decrease in GDP and country has to face economic problems.

Measures to Conserve the Land Resources:-

Degradation of land can be prevented by taking following measures:

• Afforestation
• Proper management of grazing.
• Stabilisation of sand dunes by plantation of thorny bushes.
• Proper management of waste lands.
• By proper irrigation.
• By proper harvesting.
• Control over mining activities.
• Proper management of land after completion of mining work.
• Discharge of industrial waste and effluents only after proper treatment.
• Plantation of trees along the road sides.
• By preventing the deforestation.

Soil as a natural resource:-

Soil is one of the most important natural resources. Soil supports the growth of plants. Soil is the natural home of many living organism, such as ants, rats, snakes, and many insects.
It takes thousands of years to form even 1 cm of soil. Soil is formed by the weathering process of the rocks. Various natural forces, such as temperature, running water, wind, etc. along with many physical and chemical changes are equally important in the formation of soil formation.

Classification of Soil:-

Soil is categorized in many types on the basis of texture, colour, age, chemical properties, etc. India is a vast country which comprises of many types of land. Thus, many types of soils are found in different regions in India.

Alluvial Soil

Alluvial soil is found near the river or plains formed by rivers. Alluvial soil is considered relatively younger in age. In India, alluvial soil is found in the north eastern plain where Ganga, Yamuna, and Brahmaputra flow. Alluvial soil is deposited by river system. Entire northern plain is made of alluvial soil.
Alluvial soil is also found in eastern coastal plains near the Mahanadi, the Krishna, the Godavari and the Kaveri rivers.
Alluvial soil is very fertile, thus plains of Ganga, Brahmaputra, Yamuna, etc. are densely populated. Alluvial soil is the mixture of various proportions of silt, sand and clay. Alluvial soils differ in the size of their particles at the area of break of slope.
Apart from the size of particles of grains, soils are also classified on the basis of their age. The older alluvial soil is known as Bangar and new alluvial soil or Khadar. New alluvial soil has more concentration of coarse grains while Old alluvial soil has plenty of fine particles.
Alluvial soil is rich in potash, phosphoric acid and lime. Because of presence of these chemicals alluvial soil is good for the growth of sugarcane, paddy, wheat, maize, and pulses.

Black Soil

Because of black colour, this type of soil is called black soil. It is also known as Regur Soil. Black soil is found in the north west deccan plateau. It is found in the plateau of Maharashtra, Saurashtra, Malwa, Madhya Pradesh and Chattisgarh and extent along with the valley of Krishna and Godavari Rivers.
Black soil has high concentration of fine particles and thus can hold moisture for long time. It contains calcium, potassium, magnesium and lime. Black soil is suitable for the growth of cotton, but many other crop are grown in the area of black soil.

Red and Yellow Soil

The soil looks red due to presence of iron in crystalline or metamorphic rocks. When the soil look yellow when it occurs in dehydrated form. Red soil is present in the eastern and southern parts of the Deccan Plateau. Red soil is also found in Orissa, Chhattisgarh, on the southern part of the Gangetic plains and along the piedomont zone of the Western Ghats.

Laterite Soil

Laterite soil is formed in regions which get high rainfall with high temperature. This causes leaching of the soil and microorganisms are killed during the process. Due to this, laterite soil does not contain humus or contains very low amount of humus. This soil is mainly found in Kerala, Karnataka, Tamil Nadu, Madhya Pradesh and in hilly areas of Orissa and Assam. This soil can be made cultivable with heavy dose of manures.

Arid Soil

Arid soil is found in those areas which receive scanty rainfall. Due to high temperature, evaporation is faster in these regions. The soil has a high content of salt. Arid soil can be made cultivable with proper treatment. Arid soil is present in Rajasthan and Gujarat.

Forest Soil

The forest soil is found in hilly areas. The soil in upper parts is highly acidic because of denudation. The soil in the lower part is highly fertile.

Soil Erosion and Soil Conservation

Removal of top soil is called soil erosion. Intense farming, grazing, construction activities and other human activities; along with deforestation have led to soil erosion. Soil erosion; if not checked in time; can even lead to desertification.
Soil conservation is important to prevent soil erosion. Soil conservation can be done by many methods. Afforestation is the main method because trees hold the topsoil in place. Terrace farming and shelter belt planting also help in soil conservation.

NCERT Solution

Question – 1 - Which one of the following type of resource is iron ore?

1. Renewable
2. Flow
3. Biotic
4. Non-renewable

Answer: (d) Non-renewable

Question – 2 - Under which of the following type of resource can tidal energy be put?

1. Replenishable
2. Abiotic
3. Human-made
4. Non-recyclable

Answer: (a) Replenishable

Question – 3 - Which one of the following is the main cause of land degradation in Punjab?

1. Intensive cultivation
2. Over irrigation
3. Deforestation
4. Overgrazing

Answer: (b) Over irrigation

Question – 4 - In which one of the following states is terrace cultivation practised?

1. Punjab
2. Haryana
3. Plains of Uttar Pradesh
4. Uttarakhand

Answer: (d) Uttarakhand

Question – 5 - In which of the following states is black soil found?

1. Jammu and Kashmir
2. Rajasthan
3. Gujarat
4. Jharkhand

Answer: (c) Gujarat

Question – 6 - Name three states having black soil and the crop which is mainly grown in it.

Answer: Maharashtra, Madhya Pradesh and Chhattisgarh are some states where black soil is found and cotton is the main crop grown on black soil.

Question – 7 - What type of soil is found in the river deltas of the eastern coast? Give three main features of this type of soil.

Answer: Alluvial soil is found in the river deltas of the eastern coast. Alluvial soil is rich in potash, phosphoric acid and lime. It has a high water retention capacity and it is highly fertile soil.

Question – 8 - What steps can be taken to control soil erosion in the hilly areas?

Answer: Terrace farming and shelter belt plantation can be done to prevent soil erosion in hilly areas.

Question – 9 - What are the biotic and abiotic resources? Give some examples.

Answer: Biotic Resources – All living organisms in our environment are called biotic resources. For example, trees, animals, insects, etc.

Abiotic Resources – All non-living things present in our environment are termed as abiotic resources. For example – earth, air, water, metals, rocks, etc.
Question – 10 - Explain land use pattern in India and why has the land under forest not increased much since 1960-61?

Answer: About 45% of land is used as net sown area, i.e. for farming. About 22% of the land is under forest and the rest of the land is used for various purposes; like housing, recreation and industrial activities. Increasing population and subsequent increase in demand for resources is the main reason that forested land has not increased much during this period.

Question – 11 - How have technical and economic development led to more consumption of resources?

Answer: Economic development creates demand for various resources and technical development gives the knowhow to exploit those resources. Thus, technical and economic development; together lead to more consumption of resources.

Meera Bai

Real Numbers notes and ncert based solution

Real Numbers

Theorem (Euclid’s Division Lemma):

For a pair of given positive integers ‘a’ and ‘b’, there exist unique integers ‘q’ and ‘r’ such that
Explanation:
Thus, for any pair of two positive integers a and b; the relation

Example:-

(a) 20, 8

Let 20 = a and 8 = b

(b) 17, 5

Let 17 = a and 5 = b

Exercise 1.1 (NCERT Book)

Question - 1: Use Euclid’s division algorithm to find the HCF of:

(i) 135 and 225 (ii) 196 and 38220 (iii) 867 and 255

Solution: (i) 135 and 225

In the given problem, let 225 = a, and 135 = b
Since, r (remainder) is not equal to zero (0). Thus, by applying the Euclid’s division algorithm, by taking 135 = a, and 90 = b we get
Since, in this step also, r is not equal to zero(0). Thus by continuing the Euclid’s division algorithm, by taking this time, 90 = a, and 45 = b we get
Therefore, 45 is the HCF of given pair 225 and 135
Thus, Answer: 45

Solution: (ii) 196 and 38220

In the given pair, 38220 > 196, thus let 38220 = a and 196 = b
Now by applying the Euclid’s division algorithm, we get
Since, in the above equation we get, r = 0, therefore, 196 is the HCF of the given pair 196 and 38220.
Answer: 196

Solution:- (iii) 867 and 255

Let a = 867 and b = 255, thus after applying the Euclid’s division algorithm we get
867 = 255 X 3 + 102 ( Where r = 102)
Since, r ≠ 0, therefore, by taking 255 and 102 as a and b respectively we get
255 = 102 X 2 + 51
Similarly, 102 = 51 X 2 + 0 (Where, r = 0)
Since, in this term r = 0, thus HCF of the given pair 867 and 255 is equal to 51
Answer: 51

Question - 2 – Show that any positive odd integer is of the form 6q + 1 or, 6q +3 or, 6q + 5, where q is some integer.

Solution:
Let ‘a’ be any positive odd integer and ‘b = 6’.
Therefore,
Now, by placing r = 0, we get, a = 6q + 0 = 6q
By placing r = 1, we get, a = 6q +1
By placing, r = 2, we get, a = 6q + 2
By placing, r = 3, we get, a = 6q + 3
By placing, r = 4, we get, a = 6q + 4
By placing, r = 5, we get, a = 6q +5
Thus, a = 6q or, 6q +1 or, 6q + 2 or, 6q + 3 or, 6q + 4 or, 6q +5
But here, 6q, 6q + 2, 6q +4 are the even integers
Therefore, 6q + 1 or, 6q + 3 or, 6q + 5 are the forms of any positive odd integers.
Alternate Method: Assume any value for q; like 1, 2, 3, ………….n
If q = 1;
Then;
If q = 2; Then;
Similarly, we can go on substitiuting different values for q and the result would always be a positive odd integer.

Another rationale for this can be as follows:

Since 6 is an even number and product of any number with an even number is always an even number. Moreover, if an odd number is added to any even number, we get an odd number as result.

Question: 3 – An army contingent of 616 members is to march behind an army band of 32 members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march?

Solution: The required number of column is obtained by the HCF of 616 and 32
Let, a = 616 and b = 32, therefore, by applying Euclid’s division algorithm, we get
616 = 32 X 19 + 8, since, here r = 8 and ≠ 0, thus, by continuing the process, we get
32 = 8 X 2 + 0, here r = 0
Thus, the HCF of 616 and 32 is equal to 8
Thus, the required maximum number of column = 8

Question: 4 – Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3m or 3m +1 for some integer m.

[Hint: Let x be any positive integer then it is of the form 3q, 3q +1 or 3q + 2. Now square each of these and show that they can rewritten in the form 3m or 3m +1]
Solution: Let ‘n’ is any positive integer, then it is of the form of 3q or, 3q + 1 or, 3q + 2
Therefore, square of any positive integer is either of the form of 3m or 3m + 1

Alternate method:

Let us start with the smallest square number, i.e. 4
Let us take the next square number, i.e. 9
Let us take the next square number, i.e. 16
Therefore, square of any positive integer is either of the form of 3m or 3m + 1

Question: 5 – Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8.

Solution:
Let ‘a’ is a positive integer and b = 4
Therefore, according Euclid’s division lemma

Thus, cube of any positive integer is in the form of 9m, 9m +1 or 9m +8

Alternate method:

Let us start with the smallest cube number, i.e. 8
Let us take the next cube number, i.e. 27
Let us take the next cube number, i.e. 64
Thus, cube of any positive integer is in the form of 9m, 9m +1 or 9m +8

Fundamental Theorem of Arithmetic:

Every composite number can be expressed (factorised ) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.
This theorem also says that the prime factorisation of a natural number is unique, except for the order of its factors.
For example 20 can be expressed as 2 × 2 × 5
Using this theorem the LCM and HCF of the given pair of positive integers can be calculated.

LCM = Product of the greatest power of each prime factor, involved in the numbers.

HCF = Product of the smallest power of each common prime factor in the numbers.

Exercise 1.2 (NCERT Book)

Question: 1 - Express each number as a product of its prime factors:

(i) 140 (ii) 156 (iii) 3825 (iv) 5005 (v) 7429

Solution:
Question: 2 - Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers.
(i) 26 and 91 (ii) 510 and 92 (iii) 336 and 54
Solution:-

(i) 26 and 91

The prime factors of 26 = 2 × 13
The prime factors of 91 = 7 × 13
Therefore, LCM = 2 × 7 × 13 = 182
And, HCF = 13
Now, LCM × HCF = 182 × 13 = 2366
Product of given numbers = 26 × 91 = 2366
Therefore, LCM × HCF = Product of the given two numbers

(ii) 510 and 92

The prime factors of 510 = 2 × 3 × 5 × 17
The prime factors of 92 = 2 × 2 × 23
Therefore, LCM = 2 × 2 × 3 × 5 × 17 × 23 = 23460
And, HCF = 2
Now,
LCM × HCF = 23460 × 2 = 46920
Product of given two Numbers = 510 × 92 = 46920
Therefore, LCM × HCF = Product of given two numbers

(iii) 336 and 54

The prime factors of 336 = 2 × 2 × 2 × 2 × 3 × 7 = 2⁴ × 3 × 7
The prime factors of 54 = 2 × 3 × 3 × 3 = 2 × 3³
Therefore, LCM of 336 and 54 = 2⁴ × 3³ × 7 = 3024
And, HCF = 2 × 3 = 6
Now, LCM × HCF = 3024 × 6 = 18144
And the product of given numbers = 336 × 54 = 18144
Therefore, LCM × HCF = Product of given numbers

Question: 3. Find the LCM and HCF of the following integers by applying the prime factorization method.

(i) 12, 15 and 21 (ii) 17, 23 and 29 (iii) 8, 9 and 25

Solution:-

(i) 12, 15 and 21

Prime factors of 12 = 2 × 2 × 3 = 2² × 3
Prime factors of 15 = 3 × 5
Prime factors of 21 = 3 × 7
Therefore, LCM = 2 × 2 × 3 × 5 × 7 = 420
HCF = 3

(ii) 17, 23 and 29

Prime factors of 17 = 17 × 1
Prime factors of 23 = 23 × 1
Prime factors of 29 = 29 × 1
Therefore, LCM = 17 × 23 × 29 = 11339
And HCF = 1

(iii) 8, 9 and 25

Prime factors of 8 = 2 × 2 × 2 = 2³
Prime factors of 9 = 3 × 3 = 3²
Prime factors of 25 = 5 × 5= 5²
Therefore, LCM = 2³ × 3² × 5² = 8 × 9 × 25 = 1800
Since, there is no common factors among the prime factors of given three numbers,
Thus HCF = 1

Question: 4. Given that HCF (306, 657) = 9, find LCM (306, 657).

Solution:-

Question: 5. Check whether 6ⁿ can end with the digit 0 for any natural number n.

Solution:-
Numbers that ends with zero are divisible by 5 and 10. Simultaneously they are divisible by 2 and 5 both.
If the number 6ⁿ will be divisible by 2 and 5, then, it will end with the digit 0 otherwise not.
Prime factor of 6n = (2 × 3) ⁿ
Since, 5 is not a prime factor of 6ⁿ
Therefore, for any value of n, 6ⁿ will not be divisible by 5.
Therefore, 6ⁿ cannot end with the digit 0 for any natural number n.
Note: 6ⁿ always has 6 at unit’s place. Following examples illustrate this:
6¹ = 6
6² = 36
6³ = 216
6⁴ = 1296
Hence, 6ⁿ can never end with the digit zero for any natural number n.

Question: 6. Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers.

Solution:

Numbers which have at least one factor other than 1 and number itself are called composite numbers.

Given, first expression:
7 × 11 × 13 + 13
= 13 × (7 × 11 + 1)
= 13 × (77 + 1)
= 13 × 78
Since, given expression 7 × 11 × 13 + 13 has two prime factors other than 1, thus it is a composite number.
Second expression:
7 × 6 × 5 × 4 × 3 × 2 × 1 + 5
= 5 ( 7 × 6 × 4 × 3 × 2 × 1 + 1)
= 5 (1008 + 1)
= 5 × 1009
Since, the given expression has two factors other than 1, thus it is a composite number.

Question: 7. There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. After how many minutes will they meet again at the starting point?

Solution:
Since, they will take different times to complete one round of the sports field. Thus, time after which they will meet again at the starting point will be given by the LCM of time taken to complete one round for each of them.
Time taken by Sonia to complete one round = 18 minute
Time taken by Ravi to complete one round = 12 minute
Prime factors of 18 = 2 × 3 × 3 = 2 × 3²
Prime factors of 12 = 2 × 2 × 3 = 2² × 3
Therefore, LCM = 2² × 3² = 4 × 9 = 36
Thus, after starting simultaneously Ravi and Sonia will meet at starting point after 36 minute.

NCERT Solution - Exercise - 1.3

Question – 1 - Prove that is irrational.

Answer: Let us assume the contrary, i.e. is irrational.

Thus, there can be two integers a and b (b≠0) and a and b are coprime so that;

Squaring on both sides, we get;

This means that a² is divisible by 5 and hence a is also divisible by 5.

This contradicts our earlier assumption that a and b are coprime, because we have found 5 as at least one common factor of a and b.

This also contradicts our earlier assumption that is irrational.

Question – 2 - Prove that is irrational.

Answer: Let us assume to the contrary, i.e. is irrational.

Thus, there can be two integers a and b (b≠0) and a and b are coprime so that;

Since a and b are rational, so  is rational and hence, is rational.

But this contradicts the fact that is irrational.

This happened because of our faulty assumption.

Question – 3 - Prove that following are irrationals:

Answer: Let us assume to the contrary, i.e. is rational.

Thus, there can be two integers a and b (b≠0) and a and b are coprime so that;

Squaring on both sides, we get;

This means that b² is divisible by 2 and hence a is also divisible by 2.

This contradicts our earlier assumption that a and b are co-prime, because 2 is at least one common factor of a and b.

This also contradicts our earlier assumption that is rational.

Hence, is irrational proved.

Answer: Let us assume to the contrary, i.e. is rational.

There can be two integers a and b (b≠0) and a and b are coprime, so that;

Squaring on both sides, we get;

This means that a² is divisible by 245; which means that a is also divisible by 245.

This contradicts our earlier assumption that a and b are coprime, because 245 is at least one common factor of a and b.

This happened because of our faulty assumption and hence, is irrational proved.

Answer: Let us assume to the contrary, i.e. is rational.

Thus, there can be two integers a and b (b≠0) and a and b are coprime so that;

Since a and b are rational, so is rational and hence, is rational.

But this contradicts the fact that is irrational.

This happened because of our faulty assumption.

Hence, is irrational proved.