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MP Board Class 6th Maths Solutions Chapter 3 संख्याओं के साथ खेलना Intext Questions

पाठ्य-पुस्तक पृष्ठ संख्या # 51

प्रयास कीजिए

प्रश्न 1.
45, 30 और 36 के सम्भावित गुणनखण्ड कीजिए।
हल :
45 के गुणनखण्ड = 1, 3, 5, 9, 15 और 45
30 के गुणनखण्ड = 1, 2, 3, 5, 6, 10, 15 और 30
36 के गुणनखण्ड = 1, 2, 3, 4, 6, 9, 12, 18 और 36

पाठ्य-पुस्तक पृष्ठ संख्या # 53

प्रश्न 1.
क्या 10 एक सम्पूर्ण संख्या है ?
हल :
10 के सम्भावित गुणनखण्ड = 1, 2, 5 और 10
गुणनखण्डों का योग = 1 + 2 + 5 + 10 = 18
संख्या का दो गुना = 2 x 10 = 20
स्पष्टतः 18 ≠ 20
∴10 एक सम्पूर्ण संख्या नहीं है।

MP Board Class 6th Maths Solutions

MP Board Class 8th Maths Solutions Chapter 10 ठोस आकारों का चित्रण Ex 10.3

प्रश्न 1.
क्या किसी बहुफलक के फलक नीचे दिए अनुसार हो सकते हैं?

1. 3 त्रिभुज
2. 4 त्रिभुज
3. एक वर्ग और चार त्रिभुज।

हल:

1. नहीं, किसी बहुफलक के फलक 3 त्रिभुज नहीं हो सकते।
2. हाँ, किसी बहुफलक के फलक 4 त्रिभुज हो सकते हैं।
3. हाँ, किसी बहुफलक के फलक एक वर्ग और चार त्रिभुज हो सकते हैं।

प्रश्न 2.
क्या ऐसा बहुफलक सम्भव है जिसके फलकों की संख्या कोई भी संख्या हो?
(संकेत: एक पिरामिड के बारे में सोचिए।)
हल:
हाँ, यह सम्भव है जबकि फलकों की संख्या 4 या उससे अधिक हो।

प्रश्न 3.
निम्नलिखित में से कौन-कौन प्रिज्म है?

हल:

(i) और (iii) प्रिज्म नहीं है।
(ii) और (iv) प्रिज्म हैं।

प्रश्न 4.

1. प्रिज्म और बेलन किस प्रकार एक जैसे हैं?
2. पिरामिड और शंकु किस प्रकार एक जैसे हैं?

हल:

1. एक प्रिज्म बेलन का रूप ले लेता है, जब आधार की भुजाओं की संख्या बड़ी तथा और बड़ी होती जाती है।
2. एक पिरामिड शंकु का रूप ले लेता है, जब आधार की भुजाओं की संख्या बड़ी और बड़ी होती जाती है।

प्रश्न 5.
क्या एक वर्ग प्रिज्म और एक घन एक ही होते हैं ? स्पष्ट कीजिए।
हल:
हाँ, एक वर्ग प्रिज्म और एक घन एक ही होते हैं। यह एक घनाभ भी हो सकता है।

प्रश्न 6.
इन ठोसों के लिए ऑयलर सूत्र का सत्यापन कीजिए –

हल:
1. F = 7
V = 10
E = 15
ऑयलर सूत्र – F + V – E = 2
L.H.S. = 7 + 10 – 15 = 17 – 15 = 2 = R.H.S.
अतः ऑयलर सूत्र सत्यापित होता है।

2. F = 9
V = 9
E = 16
L.H.S. = 9 + 9 – 16 = 18 – 16 = 2 = R.H.S.
अतः ऑयलर सूत्र सत्यापित होता है।

प्रश्न 7.
ऑयलर सूत्र का प्रयोग करते हुए, अज्ञात संख्या को ज्ञात कीजिए –

हल:
∵ फलक = किनारे + 2 – शीर्ष
(∵ F + V – E = 2)
∴ फलक = 12 + 2 – 6 = 14 – 6 = 8
∴ शीर्ष = किनारे + 2 – फलक
∴ शीर्ष = 9 + 2 – 5 = 11 – 5 = 6
∴ किनारे = फलक + शीर्ष – 2
∴ किनारे = 20 + 12 – 2 = 32 – 2 = 30

प्रश्न 8.
क्या किसी बहुफलक के 10 फलक, 20 किनारे और 15 शीर्ष हो सकते हैं?
हल:
यहाँ, फलक F = 10
शीर्ष V = 15 तथा
किनारे E = 20
ऑयलर सूत्र से,
F + V – E = 2
L.H.S. = 10 + 15 – 20
= 25 – 20 = 5 + 2
∴ F + V – E #2
अतः किसी बहुफलक के 10 फलक, 20 किनारे और 15 । शीर्ष नहीं हो सकते हैं।

MP Board Class 8th Maths Solutions

MP Board Class 9th Social Science Solutions Chapter 1 Man and Environment

MP Board Class 9th Social Science Chapter 1 Text Book Questions

Choose the Correct Answer:

Question 1.
Unit of measuring noise is:
(a) Centimetre
(b) Decibel
(c) Celsius
(d) Millibar
Answer:
(b) Decibel

Question 2 .
City with the highest intensity of noise in the world –
(a) Mumbai
(b) New York
(c) Rio-de-Janeiro
(d) Tokyo.
Answer:
(c) Rio-de-Janeiro

Question 3.
Where was the ozone hole observed in 1985?
(a) Australia
(b) Antarctica
(c) Western Europe
(d) Alaska.
Answer:
(b) Antarctica

Question 4.
Ozone layer is-
(a) 20-35 km above the earth’s surface
(b) 5-10 km above the earth’s surface
(c) 75-100 km above the earth’s surface
(d) 32-80 km above the earth’s surface
Answer:
(d) 32-80 km above the earth’s surface

Question 5.
Main cause of environmental degradation is –
(a) Increase in tourism
(b) Shifting cultivation
(c) Changing nature of land use
(d) All the above
Answer:
(d) All the above

Question 6.
Population explosion is –
(a) Migration
(b) Equal death rate and birth rate
(c) Crowding
(d) Continuous increase in number of humans.
Answer:
(d) Continuous increase in number of humans.

Question 7.
Cut and Burn is related to –
(a) Shifting cultivation
(b) Tourism and pilgrimage
(c) Mining
(d) Construction of dam.
Answer:
(a) Shifting cultivation

Fill in the blank:

1. Any physical thing or matter, which is valuable to man is called ……………
2. …………… % of the earth is landmass and is covered with water.
3. ……………….. % of the total land area of the world is covered by forest.
4. The term ‘acid rain’ was first coined by a British scientist in ………………

Answer:

1. environment
2. 29, 71
3. 30
4. 1873

MP Board Class 9th Social Science Chapter 1 Very Short Answer Type Questions

Question 1.
What do you mean by environment?
Answer:
All that surrounds us is our environment. Environment consists of’ all living and non – living matters that exist around us. ‘ There is a close relationship between man and environment. Man is born in his special environment and passes his entire life in the same environment.

Question 2.
What is the meaning of cultural environment?
Answer:
This environment develops due to the mutual relationship ‘ between man and natural environment. This includes the economic and social activities created, developed and conducted by man such as farming, industry, rites and rituals, settlement, roads, railways, air services, means of irrigation, government system and science and technology.

Question 3.
Name five polluted rivers of India?
Answer:
The Ganga, Hugli, Damodar, Yamuna, Godawari.

Question 4.
What is global warming?
Answer:
Due to the hole in the ozone layer the wrath of ultraviolet rays on the earth surface is increasing. Carbon-dioxide and other heat resistant gases absorb some part of heat and reverse it back to the earth surface. As a result extra heat is stored in the lower atmosphere and its temperature increases which is called global warming.

MP Board Class 9th Social Science Chapter 1 Short Answer Type Questions

Question 1.
How does air or noise pollution affect human health? Explain.
Answer:
Effects of air pollution on health:
Gases from .industries containing sulphur dioxide, carbon dioxide and phosphorus oxide etc. are harmful suffocating to us. Air pollutants have adverse effect on plant life. The poisonous gases also enter plant tissue of leaves and disturb their physiology. Plants living in polluted conditions suffer poor growth, delayed flowering and maturation of fruits.

Effects of noise pollution on health:
The problem of noise pollution is very acute in urban areas due to sound produced by different vehicles, machines, transistors and radios, drum beats on religious and wedding procession amplified rock music and discos etc.

Constant exposure to this sound produces deafness, increase in nervous tension, blood pressure and causes heart troubles.. It also causes headaches, irritability and affects the sensory and nervous system.

Question 2.
How do radioactive substances spread pollution?
Answer:
During the use of nuclear elements for various purposes the radioactive particles enter the atmosphere and produce radio active pollution. Radioactive pollution have both direct and indirect effects on our body. Radioactive radiation directly enters our body via respiratory track in the form of air and gases. Indirectly it influences our body through food chain. The radioactive substance such as strontium destroys fertility of soil.

Question 3.
Differentiate between pollution and pollutants?
Answer:
1. Pollution:
The unwanted change occurring in the natural formation and balance of the environment is known as environmental pollution.

2. Pollutants:
The unusable substances responsible for pollution which are present in improper quantity are called pollutants.

Question 4.
Explain the problem of ozone depletion?
Answer:
From 30 – 80 km above the earth surface there is a 50 meter thick layer of ozone. This layer absorbs the ultraviolet rays coming from the sun and this saves the living beings on the earth. Due to the excessive use of Chloro Floro Carbon (CFC) in the appliances such as refrigerator and air conditioner, there is a hole in ozone layer.

Due to this hole tire wrath of ultraviolet rays on the earth surface is spreading. As a result the possibility of skin cancer is increasing. It also reduces the immunity of human body. Due to the decreasing rate of photosynthesis growth of flora and fauna is hampered and forests are drying.

Question 5.
what is soil pollution? What are its ill – effects?
Answer:
Any unwanted change in the physical, chemical or biological attributes of the earth which destroys the fertility and utility of land is soil pollution.

Ill – effects of soil pollution:

• Soil pollution and bitter make the visible land dirty.
• Due to the deposition of the waste or coarse material the land becomes unsuitable for other uses.
• There is danger of soil erosion, land sliding and conversion of land into ravines.

Question 6.
What do you understand by population explosion?
Answer:
When the population of country rises at such a fast speed that existing resources of the Country fall short of the necessary requirement of the people, it is called stage of ‘population explosion This has created many problems such as crowd, unemployment, security of resources, pollution, mental and social tension etc.

Question 7.
How does over gradation affect the quality of land?
Answer:
Excessive intake of vegetation by domestic animals is called excessive grazing. As a result cf excessive grazing the vegetation does not grow again very fast in these areas. The layer of vegetation from land is destroyed. Due to soil erosion the danger of desertification arises. In such areas soil absorbs less water and plants do not get adequate water.

Question 8.
Describe the process of shifting agriculture?
Answer:
Shifting agriculture is an old method of cultivation. In this method trees are cut and burnt and then the land is prepared for farming. But this type of land is used for farming for some time and then discarded. But this method of farming has made enough harm to the forests.

Question 9.
What is deforestation? List the reasons of deforestation.
Answer:
Deforestation means cutting forests or plants from some region. Reduction of forest is the result of human effects.

Reasons of deforestation:

• Construction of huge dams.
• Throwing of hydro – electricity projects.
• Distribution of electricity and spread of railways and roads.
• Extension of habitat areas.
• Cutting of trees for fuel.
• Spreading of industries.

Question 10.
Insecticides should not be used in excess amount. Why?
Answer:
It has become a common practice to use plenty of in – pesticides to enhance production. As a result the nutrients of the soil are reduced. The fertility of soil is also hampered. The quality of production is affected.

Question 11.
What are the harmful effects of mining on the environment of a region?
Answer:
Excessive mining means digging of land for extracting minerals and other substances. Trees are being cut from the land. As a result circulation of underground water is hampered, land sliding, deposition of waste material, soil erosion and formation of new’ land forms occur. The ill – effects of excessive mining can be observed in Doon valley formed by Ganga – Yamuna which are surrounded by Himalaya ranges.

Previously this valley area was world famous for the production of Basmati rice, litchi and tea. But now due to uncontrolled mining of lime stone only 12% valley area has greenery. In the Jaipur, Jodhpur and Udaipur districts due to mining of stones the surrounding greenery has been destroyed.

Question 12.
How does urbanization destroy the environment?
Answer:
As a result of urbanization there is increase in population density and transport facilities, roads, railways, hospitals, community centers etc. are on rise. All these increase the pollution. The increasing deposition of waste influences the environment also.

MP Board Class 9th Social Science Chapter 1 Long Answer Type Questions

Question 1.
Explaining the Indian concept of environment describe how man is related to his environment?
Answer:
There has always been a mutual relationship between man and nature. We can find proof in regard in our ancient literature too. Nature is represented in the form of mother in Vedas. In the hope of healthy environment man has maintained passionate relations with the environment in his life-style. His debt towards environment is exposed in the form of sensitivity, honor, passion, and safety.

Man and environment are the creator, parent and savior of each other. Nature is considered as the substitute of life. We have always offered our prayers to water sources such as the Ganga, the Yamuna, the Saraswati, plants like the Banyan, Peepal, Neem, and Tulasi. We have always worshiped the Sun, the Moon and the.

Earth. In turn the mother nature has brought up us like her children. It is believed that the environment receives fragrance as we bum Dhoop. The up nishads assume earth as body of the supernatural, heaven as his brain, the Sun and the Moon as his eyes and skies as his mind. Hence, it is not proper to cut trees and pollute sources of water. Thus, we can say that man and environment are closely related with each other.

Question 2.
What is environment? What are the main elements of environment? How has man affected the environment? Explain.
Answer:
All that surrounds us is our environment. Environment consists of all living and non-living matter that exist around us. There is a close relationship between man and environment. Man is born in his special environment and passes his entire life in the same environment.

Environment may be defined as follows:

• The environment is all that surrounds mankind and directly or indirectly influences man.
• All physical elements and biotic elements together constitute environment.

Main Elements of Environment:.
All the elements which effect environment are called the elements of environment. These elements can be classified into major categories.

1. Elements of natural or physical environment:
This includes all natural elements given by the nature such as location, geomorphology, rocks, climate, vegetation, wild life, minerals, water bodies, ocean, etc.

2. Elements of Sociocultural Environment:
The sociopolitical environment develops due to the mutual relationship between man and natural environment. This includes the economic and social activities created, developed and conducted by man such as farming, industry, rites and rituals, settlement, roads, railways, air, services, means of irrigation, government system and science and technology.

3. How has man affected the environment?
Man is at the top of ecological pyramid. He is omnivorous because he. feeds both on plants and animal life. As a result of senseless and. continuous hunting, some species of birds and animals have become extinct and others are leading to extinction. Cultivation of land has had a serious impact on ecosystem.

The removal of original plants covers and its replacement by single cultivated crop reduces the biological diversity and simplifies the ecosystem  making it venerable to pests and diseases which attack crop. Burning of fossil fuel in large quantities, automobile exhausts, gaseous effluents from factories have led to the pollution of air and water.

Humans have cleared the jungles for their dwellings, for rearing animals and for construction roads and buildings. Too much use of land, forests have caused lot of health problems and ecological problems. As a result of pollution of air and water various species of plants have became extinct because pollution of air and water adversely affects plants.

Question 3.
What do you mean by pollution? Describe differ- it types of pollution.
Answer:
Pollution. The undesirable change in the physical, chemical, and biotic elements of the environment which adversely alters the natural quality of the environment is called environmental pollution. Pollution of the environment may occur naturally without the intervention by human being or may be caused by human activities. Volcanic eruptions, forest fires, space disturbances are the examples of pollution’s caused by nature itself. But pollution is mainly caused by human activities due to over-use, misuse, and mismanagement of resources.

Types of pollution:
1. Air pollution:
Disturbance of balance among various gases in the air or presence of unwanted gases is known as air pollution. The burning of fossil fuels in large quantities during the recent ‘ decades had resulted in gradual increase in carbon – di – oxide content in the atmosphere. The increase in carbon – di – oxide leads to increase in temperature.

Burning of coal and oil adds sulphur dioxide to the atmosphere which causes many body aliments including respiratory diseases. Gaseous effluents from factories also pollute the atmosphere. Air pollution is of global significance unlike water or land pollution which are of local or regional effect. These gases also cause acid rain in industrial regions affecting acquainted ecosystem.

The water so polluted cannot support fish or other life forms in water. Acid rain also damages forests. The ozone layer in the atmosphere serves as a shield protecting the earth from ultraviolet radiation. When the ozone layer is depleted, ultraviolet radiation enters the earth surface and may cause skin cancer.

2. Water Pollution:
The entry of some unwanted substances in the natural water that reduces its purity is known as water pollution. Polluted water is not suitable to men, animals, vegetation, sea, animals and irrigation. The causes of water pollution are soil erosion, leaves of trees, domestic drainage, and sewage, effluents from factories, fertilizers and pesticides and the hot water flowing from nuclear reactors and spreading of mineral oil on sea surface. This contaminates the drinking water and causes water born diseases such as jaundice, dysentery, and typhoid. etc. .

3. Noise Pollution:
The disturbance caused by noise or unwanted sound which interferes with human communication, comfort and health etc. is called Noise Pollution. The problem of noise pollution is very acute in urban areas due to sound produced by different vehicles, machines, transistors and radios, drum beats on religious and wedding procession amplifier rock music and discos etc.

Constant exposure to this sound produces deafness, increase in nervous tension, blood pressure and causes heart troubles. It also causes headaches, irritability and affects the sensory and nervous system.To control noise pollution, improvement in the machines, construction of sound barrier and absorbs, should be done. The ear protective devices should be used.

4. Soil Pollution:
Any unwanted change in the physical, chemical or biological attributes of the earth which destroys the fertility and utility of. land is soil pollution. Excessive use of pesticides and fertilizers, industrialization, urbanization, plastic and polythene, continuous irrigation with salted water, etc. are various causes of soil pollution.

5. Radioactive Pollution:
During the use of nuclear elements for various purposes the radioactive particles enter the atmosphere and produce radioactive pollution. The excessive energy released during nuclear test’ destroys human and animal cells.

6. Thermal Pollution:
An unwanted increase in normal temperature in the world which effects biosphere is thermal pollution. Carbon-dioxide, methane, CFC, Nitro-us oxide, ozone hole, forest fire, nuclear tests etc. increase atmospheric temperature. Drought, flood, low agricultural production due to climate change, ozone depletion etc. are some of the ill-effects of thermal pollution.

Question 4.
What is the meaning of resources? Describe various types of resources with example.
Answer:
All the elements, biological or non – biological, present in nature which are capable of fulfilling human wants are called resources. Land soil, water, air, sun – rays, forests, wild animals, fish, minerals etc. are examples of resources.

All gifts of nature about which man has the knowledge in regard to their use, availability and accessibility are thus called resources. There may be innumerable things hidden in the lap of nature but man may not be knowing about many of them. Thus gifts of nature, about whom man does not know, cannot be called resources.

Various Types of Resources:

• Natural Resources – Renewable (air, agricultural, water, forest), Non – renewable (non-cyclic fossil fuel, as coal, petrol and cyclic-metallic minerals).
• Human Resources – (population, health, education and working efficiency).
• Man – made Resources – (buildings, roads, sociopolitical institution).

1. Renewable Resources:
Resources, which are reproduce-able after the use or can be used again eg. forests, pastures, agricultural land.

2. Non-renewable Resources:
Those resources which once exploited cannot be reproduced or supplied in tire near future such as petroleum, coal.

3. Human Resources:
It means the number and quality of human beings. Education and health increase man’s physical and mental capacity. Man himself is major resource who uses the natural elements as resource on the basis of his knowledge, labor and technical know – how.

4. Man – made Resources:
These are those means of production which are produced by man to utilize the physical matters of environment such as machines, buildings, tools, etc.

5. Land Resources:
It is an important natural resource. Land is used for housing, roads, railways, farming, pasturing and mining, etc.

6. Water Resources:
Water on the earth surface is obtained through rains, rivers, lakes, ponds, glaciers, waterfalls and tube- wells. The water is used for irrigation, industry, domestic supply, fish farming and water transport.

7. Soil Resources:
Good soil is essential for the growth of vegetation.

8. Forest Resources:
Forests are a valuable resources like rivers. Forests provide us many valuable products timber, paper, medicine, firewood, wood pulp, lac, rubber, etc. Forests check soil erosion. Forests prevent floods. Forests provide suitable environment for wild-life. Forests absorb carbon – dioxide and release oxygen to the atmosphere. Forests add water vapor to the atmosphere by the process of transpiration.

Question 5.
How does changing use of land affect the environment? Explain.
Answer:
Man has brought many changes in environment in order to fulfill his ever-growing needs. He has made unnecessary disturbance in natural environment. The major cause of decline of natural environment in the world is changing form of land-use. In India forests are being destroyed for various purposes.

Deforestation and pasturing, construction of big dams for hydro – electricity and irrigation, spread of new roads and railways and development of factories has transformed land – use in India to a great extent. This has disturbed the natural balance.

The continuous interference with environment for the sake of human settlement and industrial and economic development cause environmental deg – radiation. The habitat of various types of animals have been destroyed and thousands species of flora and fauna become extinct.

Question 6.
How does over-population affect human life? Explain.
Answer:
When the population of country rises at such a fast speed that existing resources of the country fall short of the necessity of the people, it is called stage of over – population or population explosion. This has created many problems such as crowd, unemployment, security of resources, pollution, mental and social tension etc.

In the absence of suitable dwelling arrangements people tend to live on fields, streets and lanes. This gives rise to the establishment of jhugg is and subsequent loss to the ecological balance. Over – population has created the problem like overcrowding. It leads to pollution – which leads to many health hazards.

Due to over – population various types of resources are decreasing fast. People cut trees which has created grave situation. People are in race of acquiring more and more material objects. This race has given them mental and social tension. This, over – population has badly affected the human beings. We remain under pressure all the time which may reduce our life span.

Question 7.
How is the construction of huge dams harmful for the environment? Explain.
Answer:
Dams are being constructed on big rivers for purpose of irrigation, hydro – electricity, canals, fishing, water transportation and flood control. About 700 dams have been constructed after independence. But they are not unbeatable from the point of view of environment. Whenever a river valley project is started the construction of houses for the workers, construction of road, railway and underground tunnels become essential.

As a result the greenery of large areas surrounding the construction site disappears. The artificial pond made from the huge dam drowns forests and farm land. The alkalinity of the land increases due to water drained from the canals of the dam and its fertility decreases. Due to the prevalence of continuous water in dam and canal area the nearby land becomes unsuitable for agriculture. Human settlements are displaced from the dam area. Thus construction of huge dams badly affects our environment.

Question 8.
Centralization of industries is very dangerous for the environment. Explain with examples.
Answer:
Industrialization is helpful to man but harmful to our environment. The factories release poisonous gases in the atmo sphere. This disturbs atmospheric balance and increases air pollution. The waste material are left open in surrounding areas. The polluted water is drained in the river which directly influences human health and environment.

Industrialization is the main source of air, water, noise, land, chemical and radioactive pollution. In the Kolkata metropolis of India the water of Damodar and Hugh rivers have been poisoned by the effluents of surrounding steel factories, chemical industries, distilleries and paper and jute mills. Same is the true with the Ganga, Yamuna and Chambal rivers.

Question 9.
What do you mean by water pollution? River pollution is increasing in India. Describe.
Answer:
The entry of some unwanted substance  in the natural water that reduces its purity is known as water – pollution. Water is being polluted by the humans in different ways.

1. When factories, specially, those of paper mills, sugar mills, tanneries etc. discharge their effluents in rivers or allow them stagnate on land.

2. When there is no river nearby the stagnated effluents seep through the land, mix with the underground water and pollute the water. For example, effluents from large number of tanneries in North Arcot district in Tamilnadu have polluted well water in large number of villages.

3. The most widespread source of water pollution is disposal of sewage of urban centers into river. For example, rivers Ganga, Yamuna are polluted by sewage disposal while these rivers also provide drinking water to the people residing nearly.

This contaminates the drinking water and causes water born diseases such as jaundice, dysentery and typhoid etc. Approximately 90% surface water is polluted in India. The Ganga, Hugh, Damodar, Gomti,t Yamuna, Godawari rivers are polluted due to the industrial and the urban wastes and sewages.

MP Board Class 9th Social Science Chapter 1 Other Important Questions

Choose the correct answer:

Question 1.
The religion of Christianity considers nature as a –
(a) God
(b) Goddess
(c) Human being
(d) Nature.
Answer:
(b) Goddess

Question 2.
Lord Buddha got enlightenment under –
(a) The Banyan tree
(b) The Peepal tree
(c) The Neem tree
(d) The Jamun Tree.
Answer:
(a) The Banyan tree

Question 3.
Non – renewable resource is –
(a) Coal
(b) Forest
(e) Pastures
(d) Lands.
Answer:
(a) Coal

Question 4.
The total land area of the world covered by forests is –
(a) 50%
(b) 40%
(c) 30%
(d) 65%
Answer:
(c) 30%

Question 5.
The harmful radioactive substance such as strontium destroys –
(a) Fertility of soil
(b) Growth of plants
(c) Growth of a child
(d) None of the above.
Answer:
(a) Fertility of soil

Fill in the blank:

1. Nature is represent as mother in ………………
2. Under the …………… did Buddha get enligi.. nunent.
3. The unit of noise is …………..
4. Approximately …………… % surface water is poluted in India.
5. ………… and …………. is related to shifting cultivation.

Answer:

1. Vedas
2. Banyan Tree
3. Decibel
4. 90
5. Cut, Burn.

MP Board Class 9th Social Science Chapter 1 Very Short Answer Type Questions

Question 1.
What do you understand by “Man – made environment”?
Answer:
Our cultural environmental is also known as “Man-made environment”. It contains human races, social systems, human habitat and administrative systems. It is different from natural environment but it is effected by natural environment and interact with it.

Question 2.
What are the main causes of air pollution?
Answer:
Main causes of air pollution are following:

• Increase in population
• Deforestation
• Fires
• Industrialization
• Increase in vehicular smoke.

Question 3.
Which are the main components of environment? Write their names.
Answer:
Biotic and abiotic, both are main components of environment.

Question 4.
What is called environment?
Answer:
The surroundings around us is known as our environment. In other words, we can say that environment means the areas and surroundings which cover us and we live in.

Question 5.
Write the main uses of water.
Answer:
Water is used for drinking, cooking, food, irrigation, bathing and washing the clothes. –

Question 6.
How much percentage of surface water is polluted in India.
Answer:
Approximately 90% surface water is polluted in Indian

Question 7.
Which city having highest noise in the world?
Answer:
Rio – de – Janeiro is the city having highest noise in the world, where noise pollution is 120 decibel.

Question 8.
What is the unit of noise?
Answer:
The unit of noise is Decibel.

Question 9.
What is environmental pollution?
Answer:
The undesirable change in the physical, chemical, and biotic elements of the environment which adversely alters the natural quality of the environment is called environmental pollution. Pollution of the environment may occur naturally without the intervention by human being or may be caused by human activities. Volcanic eruptions, forest fires, space disturbances are the examples of pollution’s caused by nature itself. But pollution is mainly caused by human activities due to over-use, misuse, and mismanagement of resources.

MP Board Class 9th Social Science Chapter 1 Short Answer Type Questions

Question 1.
What causes soil pollution?
Answer:
Main causes of soil pollution:

• Artificial chemical fertilizers and insecticides etc. .
• Industrial complexes like pulp, paper mills, oil refineries, rubber producing complexes etc.
• Domestic wastes like broken gadgets, worn out furniture, caves, rags etc.

Question 2.
What will happen if the carbon dioxide content of the air increases?
Answer:
The increase in the carbon dioxide content of the air will result in warming up of the earth. It will cause melting of ice caps and glaciers. This would cause a rise in sea level and violent tidal waves resulting in flooding of the popular coastal plains. It may prove to be an important pollutant in the long run and cause respiratory trouble and direct threat to health.

Question 3.
What are the harmful effects of water pollution?
Answer:
Polluted water causes so many diseases. Jaundice and cholera is caused through contaminated water. As large cities are mainly affected with polluted water so the growth of virus and bacteria etc. get enhanced in these cities and can. the epidemic of jaundice and cholera. Other diseases rela intestinal disorders, liver problems etc.

Question 4.
What is the need of conserving the environment?
Answer:
There is an urgent need for protecting the environment from which man derives his food and other resources. An understanding of the processes which take place in the environment is necessary so that man’s activities, are reorganized in such a manner .so as not to interfere with the environment. Man will have to learn to live in perfect harmony with the physical and biological environment so that the earth continues to be habitable for the future as well.

Question 5.
What is natural environment?
Answer:
Natural Environment:
Natural environment is also called physical environment. It consists of all physical factors bestowed by the nature. Natural environment includes land, climate, minerals and all such endowment without which tire life on earth could have not been possible. It is thus the sum total of all non-living and inorganic matter such as land, air and water. It provides the earth with favorable condition for the existence and growth of life including man.

Question 6.
What is cultural environment?
Answer:
The man – made environment has been included in the cultural environment. The main factors of cultural environment are human races, social systems, human habitat and administrative systems. Through this environment man adapts himself to his physical environment and struggles with the vagaries of nature. His efforts have made him win places which were hitherto untouched by human population. He has succeeded in-exploring the depth of the ocean and height of the mountain.

Question 7.
Define acid rain.
Answer:
The sulphur-dioxide and nitrogen gases coming out of tire factories mix up in the atmosphere, they react with the existing vapors and produce sulphuric acid and nitrite acid respectively. This acid when pours down with the water rains on the earth. It is called the acid ram. The term acid rain was first coined by a British Scientist in 1873.

Question 8.
Write a short note on ‘Urbanization’.
Answer:
The process of extension of cities is urbanization. As a result of urbanization there is increase in population density and transport facilities, roads, railways, hospitals, community centers etc. are on rise. All these increase the pollution.

The increasing deposition of waste influences human health. About 5 crore people in India reside in the four metropolis of Delhi, Mumbai, Kolkata and Chennai. These cities are suffering from drinking water problem, urban pollution, unemployment and lack of peace.

MP Board Class 9th Social Science Chapter 1 Long Answer Type Questions

Question 1.
Describe the factors which are responsible for the changing form of land all over the world.
Answer:

There are many factors which are responsible for the changing form of land all over the world:
1. Population Growth and Reproduction:
Our population is increasing day by day. As a result we are facing the problem of population explosion. Human life-style has improved due to progress in the field of medicine. The fertility and mortality rates have come down. Deaths are on decline. As a result mortality rate has unexpectedly gone down and population has gone up. Due to unlimited growth in population problems like crowd, unemployment, security of resources, pollution, mental and social tension have come up.

2. Deforestation:
It means cutting forests or plants from some region. Reduction of forest is the result of human effects. All methods of shifting cultivation made enough harm to the forests. Other factors which have harmed forests are construction of huge dams, spread of railways and roads, etc.

3. Excessive Grazing:
As a result of excessive grazing the vegetation does not grow again very fast in these areas. The ill- effect is that layer or vegetation from land is destroyed. Due to soil erosion the danger of desertification arises.

4. Harmful Experiments in Agriculture:
In order to fulfill the ever-increasing demand of food crops of more production are grown. It is a common practice to use plenty of chemical fertilizers, pesticides and weed killing chemicals to en-chance production. Salination increases as a result of inadequate drainage of water from the fields. Nutrients of the soil are reduced due to prevalence of excessive water.

5. Urbanization:
It has increased population density, and transport facilities. Roads, railways, hospitals, community centers etc. are on rise. Some other factors are industrialization and extension of slums and construction of huge dams.

MP Board Class 9th Social Science Solutions

MP Board Class 6th Maths Solutions Chapter 2 पूर्ण संख्याएँ Ex 2.3

पाठ्य-पुस्तक पृष्ठ संख्या # 45-46

प्रश्न 1.
निम्नलिखित में से किसमें शून्य निरूपित नहीं होगा?
(a) 1 + 0
(b) 0 × 0
(c) \(\frac { 0 }{ 2 }\)
(d) \(\frac { 10-10 }{ 2 }\)
हल :
(a) 1 + 0 = 1,
(b) 0 x 0 = 0
(c) \(\frac { 0 }{ 2 }\) = 0
(d) \(\frac { 10-10 }{ 2 }\)
= \(\frac { 0 }{ 2 }\)
= 0
अतः (a) 1 + 0 में शून्य निरूपित नहीं होगा।

प्रश्न 2.
यदि दो पूर्ण संख्याओं का गुणनफल शून्य है, तो क्या हम कह सकते हैं कि इनमें से एक या दोनों ही शून्य होने चाहिए ? उदाहरण देकर अपने उत्तर की पुष्टि कीजिए।
हल :
हम जानते हैं कि पूर्ण संख्या और शून्य का गुणनफल शून्य होता है। अर्थात्
0 x 0 = 0,
1 x 0 = 0,
2 x 0 = 0,
0 x 3 = 0 इत्यादि।
हाँ, यदि दो पूर्ण संख्याओं का गुणनफल शून्य है, तो उनमें से एक या दोनों शून्य होनी चाहिए।

प्रश्न 3.
यदि दो पूर्ण संख्याओं का गुणनफल 1 है, तो हम कह सकते हैं कि इनमें से एक या दोनों ही 1 के बराबर होनी चाहिए ? उदाहरण देकर अपने उत्तर की पुष्टि कीजिए।
हल :
हम जानते हैं कि किसी पूर्ण संख्या को 1 से गुणा करने पर वही संख्या प्राप्त होती है।
7 x 1 = 7
119 x 1 = 119
0 x 1 = 0
1 x 1 = 1
23 x 1 = 23
∴1 गुणनफल प्राप्त होने के लिए दोनों संख्याएँ 1 होनी चाहिए।

प्रश्न 4.
वितरण विधि से ज्ञात कीजिए :
(a) 728 x 101
(b) 5437 x 1001
(c) 824 x 25
(d) 4275 x 125
(e) 504 x 35
हल :
(a) 728 x 101
= 728 (100 + 1)
= 728 x 100 + 728 x 1
= 72800 + 728
= 73,528

(b) 5437 x 1001
= 5437 x (1000 + 1)
= 5437 x 1000 + 5437 x 1
= 5437000 + 5437
= 54,42,437

(c) 824 x 25
= 824 x (20 + 5)
= 824 x 20 + 824 x 5
= 16480 + 4120
= 20,600

(d) 4275 x 125
= 4275 x (100 + 20 + 5)
= 4275 x 100 + 4275 x 20 + 4275 x 5
= 427500 + 85500+21375
= 5,34,375

(e) 504 x 35
= (500 + 4) x 35
= 500 x 35 + 4 x 35
= 17500 + 140
= 17.640

प्रश्न 5.
निम्नलिखित प्रतिरूप का अध्ययन कीजिए :
1 x 8 + 1 = 9
12 x 8 + 2 = 98
123 x 8 + 3 = 987
1234 x 8 + 4 = 9876
12345 x 8 + 5 = 98765
अगले दो चरण लिखिए। क्या आप कह सकते हैं कि प्रतिरूप किस प्रकार कार्य करता है ?
(संकेत : 12345 = 11111 + 1111 + 111 + 11 + 1)
हल :
अगले दो चरण होंगे
123456 x 8 + 6 = 987654
और 1234567 x 8 + 7 = 9876543
प्रतिरूप का कार्य
∴ 11 + 1 = 12
111 + 11 + 1 = 123
1111 + 111 + 11 + 1 = 1234
11111 + 1111 + 111 + 11 + 1 = 12345
(1) x 8 + 1 = 9 = (1) x 8 + 1
(12) x 8 + 2 = 98 = (11 + 1) x 8 + 2
(123) x 8 + 3 = 987 = (111 + 11 + 1) x 8 + 3
(1234) x 8 + 4 = 9876 = (1111 + 111 + 11 + 1) x 8 + 4
(12345) x 8 + 5 = 98765 = (11111 + 1111 + 111 + 11 + 1) x 8 + 5
और (123456) x 8 + 6 = 987654 = (111111 + 11111 + 1111 + 111 + 11 + 1) x 8 + 6
(1234567) x 8 + 7 = 9876543 = (1111111 + 111111 + 11111 + 1111 + 111 + 11 + 1) x 8 + 7

MP Board Class 6th Maths Solutions

MP Board Class 6th Social Science Solutions Chapter 7 Latitudes and Longitudes

MP Board Class 6th Social Science Solutions Chapter 7 Text Book Exercise

MP Board Class 6th Social Science Solutions Chapter 7 Short Answer Type Questions

Question 1.
Question (a)
Why latitudes and longitudes have been drawn on a globe map?
Answer:
To locate the exact position of places on earth, latitudes and longitudes lines are drawn on a globe or a map. We can easily locate the geographical position of any village, city, town, country, or any place with the help of these lines.

Question (b)
What are latitudes? Name the main latitude?
Answer:
1. The spherical and horizontal lines drawn parallel to the equator are called latitudes.

2. There are 90 latitudes in the northern hemisphere and 90 latitudes in the southern hemishpere. In all there are 180 latitudes drawn on the globe. The equator is the largest latitude drawn on the globe. The equator is also known as the 0° latitude.

3. The 23\(\frac{1}{2}\)° latitude north, in the northern hemisphere is called the tropic of cancer.

4. Similarly the 23\(\frac{1}{2}\)° latitude south, in the southern hemisphere is called the tropic of capricorn.

5. The Arctic Circle lies at the distances of 66\(\frac{1}{2}\)° N of the equator.

6. The Antarctic Circle 66\(\frac{1}{2}\)° S is similar to the Arctic Circle but lies in the southern hemisphere.

Question (c)
What are longitudes? Name the main longitude of the main meridian?
Answer:

1. The vertical lines drawn from north to south are called longitudes.
2. The longitude passes through Greenwich Observatory near London in England is called 0° longitude, Prime Meridian.
3. The other longitudes are drawn east and west of the Prime Meridian.
4. The Prime Meridian divides the world into Eastern (E) and Western (W) hemisphere.

Question (d)
What are the differences between latitudes and longitudes?
Answer:
Latitudes:

1. The line drawn horizontally parallal to the equator on the globe is latitude.
2. All the horizontally parallel lines are not in equal size.
3. There are 180 latitude lines.

Longitudes:

1. The verticle lines on the globe are longitudes.
2. All the verticle lines are equal in size.
3. There are 360 longitudes lines.

Question (e)
Why the tropic of cancer and the tropic of capricorn has been drawn at 23\(\frac{1}{2}\)°?
Answer:
1. The earth is tilted 23\(\frac{1}{2}\)°on its axis. Therefore, the sun shines straight only once in a year on 23\(\frac{1}{2}\)° northern and southern hemispheres.

2. The sun never shines directly from this latitudes to the northern or southern hemispheres. Therefore, the tropic of cancer and tropic of capricorn has been determined at 23\(\frac{1}{2}\)°.

MP Board Class 6th Social Science Solutions Chapter 7 Long Answer Type Questions

Question 2.
Question (a)
What are latitudes and longitudes. Write the characteristics of these lines?
Answer:
1. Latitudes:
The spherical and horizontal lines drawn parallel to the equator are called latitudes.

Characteristics of the Latitudes:

1. These lines are drawn parallel to the equator from east to west.
2. They are spherical.
3. The distance between two latitudes is the same.
4. The size of the latitudes decrease as we go towards the poles. The poles become just points.
5. The length of the latitudes are not same.
6. The area from north to the equator is called northern hemisphere and that from south to the equator is called southern hemisphere.

2. Longitudes:
The vertical lines drawn from north to south are called longtitudes.

Characteristics of Longitudes:

1. The longtitudes are semi – circular.
2. Their length are the same.
3. The distance between the longitudes is the maximum at the equator, and the distance decreases as we move towards the poles.
4. The longitudes are drawn at 1° interval from the prime meridian. They are 360 in number.

Question (b)
What are equator and prime meridian? In how many hemispheres the world has been divided by these lines?
Answer:
1. Equator:
Equator is an imaginary line which is drawn on the globes (the model of earth) in such a way that at all points it lies exactly half way between the north pole and the south pole.

2. Prime Meridian:

• Like the main latitude (0° equator), there is a main longitude. This longitude passes through Greenwich Observatory near Landon in England. This is called 0° longitude, Prime Meridian.
• The world has been divided by these lines into four hemisphere.

Question 3.

Fill in the blanks:

1. All latitudes are parallel to the ……………….
2. The uppermost point on the north of the world is called ……………. pole and lower most on the south of the world is called ……………….. pole.
3. The main meridian is also called the ……………..
4. The equator divides the world into ……………. and ………………. hemisphere.
5. The total number of longitudes are ……………..

Answer:

1. equator
2. north, south
3. Prime Meridian
4. northern, southern
5. 360.

Question 4.
Match the column

Answer:

(i) (d) Horizontal lines
(ii) (c) Vertical lines
(iii) (a) Nothern Hemisphere
(iv) (b) Southern Hemisphere

Question 5.
Question (a)
The biggest latitude is called?
(i) tropic of cancer
(ii) equator
(iii) tropic of capricorn
(iv) prime meridian
Answer:
(ii) equator.

Question (b)
The distance between the longitudes towards the poles?
(i) increase
(ii) decrease
(iii) parallel
(iv) none of these
Answer:
(ii) decrease.

Project Work

Question 1.
Draw a circle showing latitudes, longitudes and poles in it?
Answer:

MP Board Class 6th Social Science Solutions

MP Board Class 6th Maths Solutions Chapter 2 पूर्ण संख्याएँ Ex 2.2

पाठ्य-पुस्तक पृष्ठ संख्या # 42-43

प्रश्न 1.
उपयुक्त क्रम में लगाकर योग ज्ञात कीजिए :
(a) 837 + 208 + 363
(b) 1962 + 453 + 1538 + 647
हल :
(a) 837 + 208 + 363
= 208 + (363 + 837)
= 280 + (1200)
= 1,408

(b) 1962 + 453 + 1538 + 647
= (1962 + 1538) + (453 + 647)
= (3500) + (1100)
= 4600

प्रश्न 2.
उपयुक्त क्रम में लगाकर गुणनफल ज्ञात कीजिए :
(a) 2 x 1768 x 50
(b)4 x 166 x 25
(c) 8 x 291 x 125
(d) 625 x 279 x 16
(e) 285 x 5 x 60
(f) 125 x 40 x 8 x 25
हल:
(a) 2 x 1768 x 50
= (2 x 50) x 1768
= 100 x 1768
= 1,76,800

(b) 4 x 166 x 25
= (4 x 25) x 166
= 100 x 166
= 16,600

(c) 8 x 291 x 125
= (8 x 125) x 291
= 1000 x 291
= 2,91,000

(d) 625 x 279 x 16
= (625 x 16) x 279
= 10000 x 279
= 27,9,0000

(e) 285 x 5 x 60
= 285 x (5 x 60)
= 285 x 300
= (285 x 3) x 100
= 855 x 100
= 85,500

(f) 125 x 40 x 8 x 25 = (125 x 8) x (40 x 25)
= 1000 x 1000
= 10,00,000

प्रश्न 3.
निम्नलिखित में से प्रत्येक का मान ज्ञात कीजिए :
(a) 297 x 17 + 297 x 3
(b) 54279 x 92 + 8 x 54279
(c) 81265 x 169 – 81265 x 69
(d) 3845 x 5 x 782 + 769 x 25 x 218
हल :
(a) 297 x 17 + 297 x 3 = 297 (17+3)
= 297 x 20
= 5,940

(b) 54279 x 92 + 8 x 54279 = 54279 (92 + 8)
= 54279 x 100
= 54,27,900

(c) 81265 x 169 – 81265 x 69
= 81265 (169 – 69)
= 81265 x 100
= 81,26,500

(d) 3845 x 5 x 782 + 769 x 25 x 218
= 3845 x 5 x 782 + 769 x 5 x 5 x 218
= 3845 x 5 x 782 + (769 x 5) x 5 x 218
= 3845 x 5 x 782 + 3845 x 5 x 218
= 3845 x 5 x (782 + 218)
= 3845 x 5 x (1000)
= 19225 x 1000
= 1,92,25,000

प्रश्न 4.
उपयुक्त गुणों का उपयोग करके गुणनफल ज्ञात कीजिए :
(a) 738 x 103
(b) 854 x 102
(c) 258 x 1008
(d) 1005 x 168
हल :
(a) 738 x 103
= 738 x (100 + 3)
= 738 x 100 + 738 x 3
= 73800 + 2214
= 76,014

(b) 854 x 102
= 854 x (100 + 2)
= 854 x 100 + 854 x 2
= 85400 + 1708
= 87108

(c) 258 x 1008
= 258 x (1000 + 8)
= 258 x 1000 + 258 x 8
= 258000 + 2064
= 2,60,064

(d) 1005 x 168
= (1000 + 5) x 168
= 1000 x 168 + 5 x 168
= 168000 + 840
= 168840

प्रश्न 5.
किसी टैक्सी-ड्राइवर ने अपनी गाड़ी की पेट्रोल टंकी में सोमवार को 40 लीटर पेट्रोल भरवाया। अगले दिन, उसने टंकी में 50 लीटर पेट्रोल भरवाया। यदि पेट्रोल का मूल्य Rs 44 रु. प्रति लीटर था, तो उसने पेट्रोल पर कुल कितना व्यय किया?
हल :
सोमवार तथा मंगलवार को भरवाया गया पेट्रोल
= 40 लीटर + 50 लीटर
= 90 लीटर
पेट्रोल का मूल्य = 44 रुपये प्रति लीटर
पेट्रोल पर कुल व्यय = 44 x 90 रुपये
= 3,960 रुपये

प्रश्न 6.
कोई दूधवाला एक होटल को सुबह 32 लीटर दूध देता है और शाम को 68 लीटर दूध देता है। यदि दूध का मूल्य 15 रुपये प्रति लीटर है, तो दूधवाले को प्रतिदिन कितनी धनराशि प्राप्त होगी? हल :
होटल को सुबह और शाम को दी गई दूध की मात्रा
= 32 लीटर + 68 लीटर
= 100 लीटर
दूध का मूल्य = 15 रुपए प्रति लीटर
∴दूध वाले को प्रतिदिन प्राप्त धनराशि
= Rs 15 x 100
= Rs 1500

प्रश्न 7.
निम्न को सुमेलित (match) कीजिए :
(i) 425 x 136 = 425 x (6 + 30 + 100) (a) गुणन की क्रमविनिमेयता
(ii)2 x 49 x 50 = 2 x 50 x 49 (b) योग की क्रमविनिमेयता
(iii) 80 + 2005 + 20 = 80 + 20 + 2005 (c) योग पर गुणन का वितरण
उत्तर-
(i) → (c),
(ii) → (a),
(iii) → (b)
अब इस सारणी को पूरा कीजिए
हल:

प्रयास कीजिए

प्रश्न 1.
कौन-सी संख्याएँ केवल रेखा के रूप में दर्शाई जा सकती हैं ?
उत्तर- संख्याएँ 2, 5, 7, 11, 13, …… केवल रेखा के रूप में दर्शाई जा सकती हैं।

प्रश्न 2.
कौन-सी संख्याएँ वर्गों के रूप में दर्शाई जा सकती हैं ?
उत्तर-
संख्याएँ 4, 9,16, 25, ….. वर्गों के रूप में दर्शाई जा सकती हैं।

प्रश्न 3.
कौन-सी संख्याएँ आयतों के रूप में दर्शाई जा सकती हैं?
उत्तर-
संख्याएँ 4, 6, 8, 9, 10, 12, …….. आयतों के रूप में दर्शाई जा सकती हैं।

प्रश्न 4.
प्रथम सात त्रिभुजाकार संख्याओं को लिखिए (अर्थात् जिन्हें त्रिभुज के रूप में व्यवस्थित किया जा सकता है) 3,6,……
उत्तर-
प्रथम सात त्रिभुजाकार संख्याएँ हैं. 3, 6, 10, 15, 21, 28 और 36

प्रश्न 5.
कुछ संख्याओं को दो आयतों के रूप में दर्शाया जा सकता है।
उदाहरणार्थ;

इसी प्रकार के क्रम में कम-से-कम पाँच उदाहरण दीजिए।
उत्तर-
6 → 3 x 2; 2 x 3
18 → 3 x 6, 2 x 9
20 → 5 x 4, 10 x 2
24 → 3 x 8, 4 x 6
30 → 5 x 6, 10 x 3

MP Board Class 6th Maths Solutions

MP Board Class 8th Maths Solutions Chapter 10 ठोस आकारों का चित्रण Ex 10.2

प्रश्न 1.
एक नगर के दिए हुए मानचित्र को देखिए। निम्नलिखित प्रश्नों के उत्तर दीजिए –

(a) इस मानचित्र में इस प्रकार रंग भरिए –
नीला – जल; लाल – फायर-स्टेशन; नारंगी – लाइब्रेरी; पीला – स्कूल; हरा – पार्क; गुलाबी – कॉलेज बैंगनी – अस्पताल; भूरा – कब्रिस्तान।
(b) सड़क C और नेहरू रोड के प्रतिच्छेदन पर एक हरा ‘X’ तथा गाँधी रोड़ और सड़क A के प्रतिच्छेदन पर एक हरा ‘Y’ खींचिए।
(c) लाइब्रेरी से बस डिपो तक एक छोटा सड़क मार्ग लाल रंग से खींचिए।
(d) कौन अधिक पूर्व में है-सिटी पार्क या बाज़ार?
(e) कौन अधिक दक्षिण में है – प्राइमरी स्कूल या सीनियर सेकण्डरी स्कूल?

हल:
(a), (b) और (c) के लिए विद्यार्थी अभीष्ट मानचित्र में दिये गये निर्देशानुसार स्वयं रंग भरें। (d) ‘सिटी पार्क’ अधिक पूर्व में है। (e) सीनियर सेकण्डरी स्कूल अधिक दक्षिण में है।

प्रश्न 2.
उचित पैमाने और विभिन्न वस्तुओं के लिए संकेतों का प्रयोग करते हुए, अपनी कक्षा के कमरे का एक मानचित्र खींचिए।
हल:
कक्षा के कमरे का मानचित्र –

प्रश्न 3.
उचित पैमाने और विभिन्न विशेषताओं (वस्तुओ) जैसे खेल का मैदान, मुख्य भवन, बगीचा इत्यादि के लिए संकेतों का प्रयोग करते हुए, अपने विद्यालय परिसर (compound) का एक मानचित्र खींचिए।
हल:
विद्यालय परिसर का मानचित्र –

प्रश्न 4.
अपने मित्र के मार्ग दर्शन के लिए एक मानचित्र खींचिए ताकि वह आपके घर बिना किसी कठिनाई के पहुँच जाए।
हल:
\(\frac{1}{2}\) किमी.

जानकारी के लिए निर्देश –

1. सिटी सेण्टर से अकबर रोड पर आइए।
2. अकबर पर आगे बढ़िए, लाइब्रेरी तक आइए।
3. लाइब्रेरी के सामने नेहरू रोड पर आइए।
4. कुछ कदम आगे आइए, दाहिनी ओर पुलिस थाना आयेगा।
5. पुलिस थाना से आगे आइए, कुछ दूरी चलकर दाहिने मुड़िए।
6. आगे प्राइमरी स्कूल तक आइए।
7. प्राइमरी स्कूल के सामने चलिए।
8. हरी मिष्ठान से आगे बढ़िए और दाहिनी ओर मुड़िए।
9. फायर स्टेशन से आगे सीधी सड़क पर चलिए।
10. लगभग आधा किलोमीटर चलकर आप मेरे घर पर होंगे।

पाठ्य-पुस्तक पृष्ठ संख्या # 173

फलक, किनारे और शीर्ष

पहेली:
मेरा कोई शीर्ष नहीं है। मेरा कोई सपाट फलक नहीं हैं। मैं कौन हूँ?
उत्तर:
किनारा।

पाठ्य-पुस्तक पृष्ठ संख्या # 175

इन्हें कीजिए (क्रमांक 10.4)

प्रश्न 1.
निम्नलिखित बहुफलकों के लिए फलकों (faces), किनारों (edges) और शीर्षों (vertices) की संख्याओं को सारणीबद्ध कीजिए (यहाँ Vशीर्षों की संख्या, F फलकों की संख्या तथा E किनारों की संख्या प्रदर्शित करता है)।

आप अन्तिम दो स्तम्भों से क्या निष्कर्ष निकालते हैं? क्या प्रत्येक स्थिति में आप F + V = E + 2, अर्थात् F + V – E = 2 प्राप्त करते हैं? यह सम्बन्ध ऑयलर सूत्र (Euler’s Formula) कहलाता है। वास्तव में, यह सूत्र प्रत्येक बहुफलक के लिए सत्य है।
हल:
यहाँ, V- शीर्षों की संख्या, F – फलकों की संख्या तथा E किनारों की संख्या है।

अन्तिम दो स्तम्भों से यह निष्कर्ष निकलता है कि F + V = E + 2,
अर्थात् F + V – E = 2
यह सूत्र ऑयलर सूत्र कहलाता है जो प्रत्येक बहुलक के लिए सत्य है।

सोचिए, चर्चा कीजिए और लिखिए

प्रश्न 1.
यदि किसी ठोस में से कोई टुकड़ा काट दिया जाए, तो F, V और E में क्या परिवर्तन होता है ? (प्रारम्भ करने के लिए, एक प्लास्टिसीन का घन लीजिए तथा उसका एक कोना काटकर इसकी खोज कीजिए।)
हल:
माना कि ABCDEFGH एक प्लास्टिसीन का घन है। इस घन में से एक टुकड़ा XYZ एक कोने से काटकर अलग किया गया है। यहाँ, X, Y तथा Z सह किनारों क्रमशः FE, FG तथा FB के बिन्दु हैं।
स्थिति 1:
घन ABCDEFGH से,
फलकों की संख्या F = 6
शीर्षों की संख्या V = 8
किनारों की संख्या E= 12
अब – F + V = 6 + 8 = 14
= 14 + 2 = E + 2
अतः यहाँ ऑयलर सूत्र का सत्यापन होता है।

स्थिति 2:
जब कि घन से एक टुकड़ा समतल XYZ द्वारा काट दिया जाता है –
फलकों की संख्या, F = 7
शीर्षों की संख्या, V = 10
किनारों की संख्या, E= 15
अब, F + V = 7 + 10 = 17
= 15 + 2 = E + 2
अतः, यहाँ ऑयलर सूत्र का सत्यापन होता है।

MP Board Class 8th Maths Solutions

MP Board Class 6th Maths Solutions Chapter 2 पूर्ण संख्याएँ Intext Questions

पाठ्य-पुस्तक पृष्ठ संख्या # 29

प्रयास कीजिए

प्रश्न 1.
19; 1997%; 12000; 49; 100000; 2440701; 100199; और 208090 के पूर्ववर्ती और परवर्ती लिखिए।
हल :
पूर्ववर्ती : 19 – 1 = 18; 1997 – 1 = 1996; 12000 – 1 = 11999; 49 – 1 = 48; 100000 – 1 = 99999; 2440701 – 1 = 2440700; 100199 – 1 = 100198; 208090 – 1 = 208089;
परवर्ती : 19 + 1 = 20; 1997 + 1 = 1998; 12000 + 1 = 12001; 49 + 1 = 50; 100000 + 1 = 100001; 2440701 + 1 = 2440702; 100199 + 1 = 100200; 208090 + 1 = 208091

प्रश्न 2.
क्या कोई ऐसी प्राकृत संख्या है जिसका कोई पूर्ववर्ती नहीं है ?
उत्तर-
हाँ, 1 ऐसी प्राकृत संख्या है जिसका कोई पूर्ववर्ती नहीं है।

प्रश्न 3.
क्या कोई ऐसी प्राकृत संख्या है जिसका कोई परवर्ती नहीं है ? क्या कोई अंतिम प्राकृत संख्या है ?
उत्तर-
नहीं, ऐसी प्राकृत संख्या नहीं है जिसका कोई परवर्ती नहीं।
नहीं, कोई संख्या अंतिम प्राकृत संख्या नहीं है।

पाठ्य-पुस्तक पृष्ठ संख्या # 30

प्रयास कीजिए

प्रश्न 1.
क्या सभी प्राकृत संख्याएँ पूर्ण संख्याएँ भी हैं ?
उत्तर-
हाँ, सभी प्रांकृत संख्याएँ पूर्ण संख्याएँ भी हैं।

प्रश्न 2.
क्या सभी पूर्ण संख्याएँ प्राकृत संख्याएँ भी हैं ?
उत्तर-
नहीं, सभी पूर्ण संख्याएँ प्राकृत संख्याएँ नहीं हैं। क्योंकि शून्य (0) पूर्ण संख्या है लेकिन यह प्राकृत संख्या नहीं है।

प्रश्न 3.
सबसे छोटी पूर्ण संख्या कौन-सी है ?
उत्तर-
सबसे छोटी पूर्ण संख्या शून्य (0) है।

प्रश्न 4.
सबसे बड़ी पूर्ण संख्या कौन-सी है ?
उत्तर-
कोई भी सबसे बड़ी पूर्ण संख्या नहीं है।

पाठ्य-पुस्तक पृष्ठ संख्या # 31

प्रश्न 1.
संख्या रेखा का प्रयोग करके, 4 + 5; 2 + 6; 3 + 5; और 1 + 6 को ज्ञात कीजिए।
हल :
(i) 4 + 5

∴ 4 + 5 = 9

(ii) 2 + 6

∴ 2 + 6 = 8

(iii) 3 + 5

∴ 3 + 5 = 8

(iv) 1 + 6

∴ 1 + 6 = 7

पाठ्य-पुस्तक पृष्ठ संख्या # 32

प्रयास कीजिए

प्रश्न 1.
संख्या रेखा का प्रयोग करके, 8 – 3; 6 – 2; और 9 – 6 ज्ञात कीजिए।
हल :
(i) 8 – 3

∴ 8 – 3 = 5

(ii) 6 – 2

∴ 6 – 2 = 4

(iii) 9 – 6

∴ 9 – 6 = 3

प्रयास कीजिए

प्रश्न 1.
संख्या रेखा का प्रयोग करके, 2 x 6; 3 x 3; और 4 x 2 ज्ञात कीजिए।
हल :
(i) 2 x 6

∴ 2 x 6 = 12

(ii) 3 x 3

∴ 3 x 3 = 9

(iii)4 x 2

∴ 4 x 2 = 8

MP Board Class 6th Maths Solutions

MP Board Class 6th Maths Solutions Chapter 1 अपनी संख्याओं की जानकारी Ex 1.3

प्रश्न 1.
व्यापक नियम का प्रयोग करते हुए, निम्नलिखित में से प्रत्येक का आकलन कीजिए :
(a) 730 + 998
(b) 796 – 314
(c) 12,904 + 2,888
(d) 28,292 – 21,496
जोड़ने, घटाने और उनके परिणामों के आकलन के दस और उदाहरण बनाइए।
हल :
(a) 730 + 998
∵730 → 700 (सौ के सन्निकटित)
998 → 1000 (सौ के सन्निकटित)
∴आकलित योग = 700 + 1000 = 1700

(b) 796 – 314
∵796 → 800 (सौ के सन्निकटित)
314 → 300 (सौ के सन्निकटित)
∴आकलित अन्तर = 800 – 300 = 500

(c) ∵12,904 + 2,888
12,904 → 13,000 (सौ के सन्निकटित)
2,888 → 3,000 (सौ के सन्निकटित)
∴आकलित योग = 13000 + 3000 = 16000

(d) 28,292 – 21,496
∵28292 → 28000 (सौ के सन्निकटित)
21496 → 21000 (सौ के सन्निकटित)
∴आकलित अन्तर = 28000 – 21000 = 7000

10 अन्य उदाहरण
(i) 838 + 895
(ii) 890 – 210
(iii) 740 + 781
(iv) 980 – 688
(v) 13904 + 2868
(vi) 38394 – 31496
(vii) 14909 + 2675
(viii) 28283 – 21529
(ix) 25688 + 4213
(x) 28457 – 23624.
हल :
(i) ∵ 838 → 800; 895 → 900
∴आकलित योग = 800 + 900 = 1700

(ii) ∵ 890 → 900; 210 → 200
∴आकलित अन्तर = 900 – 200 = 700

(iii) ∵ 740 → 700; 781 → 800
∴आकलित योग = 700 + 800 = 1500

(iv) ∵ 980 → 1000; 688 → 700
∴आकलित अन्तर = 1000 – 700 = 300

(v) ∵ 13904 → 14000; 2868 → 3000
∴आकलित योग = 14000 + 3000 = 17000

(vi) ∵ 38394 → 40000; 31496 → 30000
∴आकलित अन्तर = 40000 – 30000 = 10000

(vii) ∵ 14909 → 15000; 2675 → 3000
∴आकलित योग = 15000 + 3000 = 18000

(viii) ∵ 28283 → 30000; 21529 → 20000
∴आकलित अन्तर = 30000 – 20000 = 10000

(ix) ∵ 25688 → 26000; 4213 → 4000
∴आकलित योग = 26000 + 4000 = 30000

(x) ∵ 28457 → 30000; 23624 → 20000
∴आकलित अंतर = 30000 – 20000 = 10000

प्रश्न 2.
एक मोटे तौर पर (Rough) आकलन (सौ तक सन्निकटन) और एक निकटतम आकलन (दस तक सन्निकटन) दीजिए:
(a) 439 + 334 + 4,317
(b) 1,08,734 – 74,599
(c) 8,325 – 491
(d) 4,89,348 – 48,365
ऐसे चार और उदाहरण बनाइए।
हल :
(a) 439 + 334 + 4,317
∵ 439 → 400
334 → 300
(प्रत्येक को सौ के सन्निकटन करने पर)
4317 → 74300
(प्रत्येक को सौ के सन्निकटन करने पर)
∵ रफ आकलन = 400 + 300 + 4300
= 5000
439 → 440
334 → 330
(प्रत्येक को दहाई के सन्निकटन करने पर)
∴4317 → 4320
निकटतम आकलन = 440 + 330 + 4320
= 5090

(b) 1,08,734 – 47,599
108734 → 108700
47599 → 47600 (सौ के सन्निकटन करने पर)
∴रफ आकलन = 108700 – 47600
= 61,100
पुनः 1,08,734 → 1,08,730
47,599 → 47,600
(दहाई के सन्निकटन करने पर)
∴निकटतम आकलन = 108730 – 47600
= 61,130

(c) 8325 – 491
8325 → 8300
491 → 500
(सौ के सन्निकटन करने पर)
∴रफ आकलन = 8300 – 500 = 7800
8325 → 8330
491 → 490
(दहाई के सन्निकटन करने पर)
∴निकटतम आकलन = 8330 – 490
= 7840

(d) 4,89,348 → 48,365
4,89,348 → 4,89,300
48,365 → 48,400
(सौ के सन्निकटन करने पर)
∴रफ आकलन = 489300 – 48400
= 4,40,900
पुनः
489348 → 489350
48365 → 48370
(दहाई के सन्निकटन करने पर)
∴निकटतम आकलन = 489350 – 48370
= 4,40,980

चार और उदाहरण
(i) 593 + 434 + 5317
(ii) 1,09,834 – 48,596
(iii) 7,625 – 591
(iv) 4,79,548 – 47,465
हल :
(i) 593 + 434 + 5317
593 → 600
434 →400
5317 → 5300 (सौ के सन्निकटन करने पर)
∴रफ आकलन = 600 + 400 + 5300 = 6300
पुनः 593 → 590
434 → 430
5317 → 5320
(दहाई के सन्निकटन करने पर)
∴निकटतम आकलन = 590 + 430 + 5320
= 6340

(ii) 1,09,834 – 48,596
1,09,834 → 1,09,800
(सौ के सन्निकटन करने पर)
48,596 → 48,600
∴रफ आकलन = 109800 – 48600
= 61,200
पुनः 109834 → 109830
48596 → 48600
(दहाई के सन्निकटन करने पर)
∴निकटतम आकलन = 109830 – 48600
= 61,230

(iii) 7,625 → 591
7,625 → 7,600
591 → 600 (सौ के सन्निकटन करने पर)
∴रफ आकलन = 7600 – 600
= 7,000
पुनः 7,625 → 7,630
591 → 600 (दहाई के सन्निकटन करने पर)
∴निकटतम आकलन = 7630 – 600
= 7,030

(iv) 4,79,548 – 47,465
4,79,548 → 4,79,500
47,465 → 47,500 (सौ के सन्निकटन करने पर)
∴रफ आकलन = 479500 – 47500
= 4,32,000
पुनः 4,79,548 → 4,79,550
47,465 → 47,470
(दहाई के सन्निकटन करने पर)
निकटतम आकलन = 4,79,550 – 47,470
= 4,32,080

प्रश्न 3.
व्यापक नियम का प्रयोग करते हुए, निम्नलिखित गुणनफलों का आकलन कीजिए :
(a) 578 x 161
(b) 5281 x 3419
(c) 1291 x 592
(d) 9250 x 29
ऐसे चार और उदाहरण बनाइए।
हल :
(a) 578 x 161
578 → 600
161 → 200 (सौ के सन्निकटन)
∴गुणनफल का आकलन = 600 x 200
= 1,20,000

(b) 5281 x 3491
5281 → 5000 (हजार के सन्निकटन)
3491 → 3500 (सौ के सन्निकटन)
∴गुणनफल का आकलन = 5000 x 3500
= 1,75,00,000

(c) 1291 x 592
1291 → 1300
592 + 600 (सौ के सन्निकटन)
∴गुणनफल का आकलन = 1300 x 600
= 7,80,000

(d) 9250 x 29
9250 → 9300 (सौ के सन्निकटन)
29 → 30 (दहाई के सन्निकटन)
∴गुणनफल का आकलन = 9300 x 30
= 2,79,000

चार और उदाहरण
(a) 680 x 271
(b) 2368 x 4483
(c) 7360 x 38
(d) 1487 x 697
हल :
(a) 680 x 271
∵680 → 700
271 → 300
∴गुणनफल का आकलन = 700 x 300
= 2,10,000

(b) 2368 x 4483
∵2368 → 2400
4483 → 4500
∴गुणनफल का आकलन = 2400 x 4500
= 1,08,00,000

(c) 7360 x 38
∵7360 → 7400
38 → 40
∴गुणनफल का आकलन = 7400 x 40
= 2,96,000

(d) 1487 x 697
∵1487 → 1500
697 → 700
∴गुणनफल का आकलन = 1500 x 700
= 10,50,000

पाठ्य-पुस्तक पृष्ठ संख्या # 25

प्रयास कीजिए

प्रश्न 1.
कोष्ठकों का प्रयोग करते हुए निम्नलिखित में मे प्रत्येक के लिए व्यंजक लिखिए :
(a) नौ और दो के योग का चार से गुणा।
(b) अठारह और छः के अन्तर को चार से भाग।
(c) पैंतालीस को तीन और दो के योग के तिगुने से भाग देना।
हल :
(a) 4 x (9 + 2)
(b) (18 – 6) ÷ 4
(c) 45 ÷ [3(3 + 2)]

प्रश्न 2.
(5 + 8) x 6 के लिए तीन विभिन्न स्थितियाँ लिखिए।
(ऐसी एक स्थिति है : सोहनी और रीता ने 6 दिन कार्य किया। सोहनी 5 घण्टे प्रतिदिन कार्य करती है और रीता 8 घण्टे प्रतिदिन कार्य करती है। दोनों ने एक सप्ताह में कुल कितने घण्टे कार्य किया ?
हल :
स्थिति 1-राहुल एक उपन्यास के 5 पृष्ठ प्रतिदिन सुबह और 8 पृष्ठ प्रतिदिन शाम को पढ़ता है। वह 6 दिन में कितने पृष्ठ पढ़ लेगा ?
स्थिति 2-राहुल प्रतिदिन 5 रुपये के अण्डे और 8 रुपये का दूध खरीदता है। वह 6 दिनों में कितने रुपए खर्च करेगा ?
स्थिति 3-मीनाक्षी और अनुपमा 6 दिन कार्य करती हैं। मीनाक्षी प्रतिदिन 5 रुपये और अनुपमा प्रतिदिन 8 रुपये कमाती हैं। दोनों मिलकर 6 दिन में कितना कमाएँगी?

प्रश्न 3.
निम्नलिखित के लिए पाँच स्थितियाँ लिखिए, जहाँ कोष्ठकों का प्रयोग आवश्यक हो :
(a) 7 (8 – 3)
(b) (7 + 2) (10 – 3)
हल :
(a) 7(8 – 3) स्थिति 1 – राम को प्रतिदिन जेब खर्च के लिए 8 रुपये मिलते हैं। उनमें से वह 3 रुपये बचाता है। 7 दिन में वह कितने रुपये खर्च करेगा?
स्थिति 2 – 8 और 3 के अन्तर को 7 से गुणा करने पर क्या प्राप्त होगा?
स्थिति 3 दूध से भरी 7 टंकियाँ हैं। प्रत्येक टंकी में 8 लीटर दूध भरा हुआ है। यदि प्रत्येक टंकी में से 3 लीटर दूध निकाल लें तो टंकियों में कितना दूध रह जाएगा?
स्थिति 4 – सात बालक बाजार गए। प्रत्येक बालक के पास 8 रुपये थे। उनमें से प्रत्येक ने 3 रुपये की गेंद खरीद ली। अब उनके पास कुल कितने रुपये शेष रहे?
स्थिति 5 – माना कि एक ड्राइवर 8 घण्टे कार्य करता है। लेकिन किसी कारणवश उसे प्रतिदिन 3 घण्टे के लिए घर जाना पड़ता है। 7 दिन में उसने कुल कितने घण्टे कार्य किया?

(b) (7 + 2) (10 – 3) स्थिति 1 – 7 और 2 योग को 10 और 3 के अन्तर से गुणा करने पर गुणनफल क्या होगा ?
स्थिति 2 – 7 सेल्स गर्ल्स और 2 मैनेजर की एक टीम एक गारमेण्ट्स की दुकान पर 10 घण्टे प्रतिदिन कार्य करती है।
रविवार को दुकान 3 घण्टे पहले बन्द हो गई। उस टीम ने कुल कितने घण्टे दुकान पर कार्य किया?
स्थिति 3 – 7 व्यक्तियों ने रेलवे स्टेशन जाने के लिए 10 रुपये प्रति व्यक्ति के हिसाब से एक वैन किराये पर ली। वैन चालक ने उसमें 2 सवारी और बैठा ली और उसने 3 रुपए प्रति सवारी किराया कम कर दिया। वैन चालक ने किराये से कुल कितनी धनराशि प्राप्त की?
स्थिति 4 – एक माली को एक महीने में 7 दिन कार्य करने के लिए रखा। प्रतिदिन उसे 10 घण्टे कार्य करना था। लेकिन उसने प्रतिदिन 3 घण्टे कम कार्य किया। लेकिन उसने दो दिन अधिक कार्य किया। उसने कुल कितने घण्टे कार्य किया?
स्थिति 5 – एक रेस्टोरेंट चालक 7 व्यक्ति और 2 बच्चों के लिए एक सोसाइटी के 10 घरों में लंच टिफिन भेजता है। रविवार को 3 घरों को उसने लंच टिफिन नहीं भेजा। रविवार को कितनों घरों में टिफिन भेजा गया?

पाठ्य-पुस्तक पृष्ठ संख्या # 27

प्रयास कीजिए

प्रश्न 1.
रोमन पद्धति में लिखिए :
1.73
2.92
हल :
1. 73 = 50 + 20 + 3
= LXXIII
2. 92 = 90 + 2 = (100 – 10) + 2
= XCII

MP Board Class 6th Maths Solutions

MP Board Class 6th Maths Solutions Chapter 3 संख्याओं के साथ खेलना Ex 3.7

पाठ्य-पुस्तक पृष्ठ संख्या # 72

प्रश्न 1.
रेणु 75 किग्रा और 69 किग्रा भारों वाली दो खाद की बोरियाँ खरीदती है। भार के उस बट्टे का अधिकतम मान ज्ञात कीजिए जो दोनों बोरियों के भारों को पूरा-पूरा माप ले।
हल :
दोनों बोरियों के भारों को पूरा-पूरा मापने के लिए अधिकतम भार म. स. होगा।
∴75 = 3 x 5 x 5
69 = 3 x 23
∴सार्व गुणनखण्ड = 3, अत : म. स. = 3
अतः अधिकतम भार = 3 किग्रा

प्रश्न 2.
तीन लड़के एक ही स्थान से एक साथ कदम उठाकर चलना प्रारम्भ करते हैं। उनके कदमों की माप क्रमश: 63 सेमी, 70 सेमी और 77 सेमी है। इनमें से प्रत्येक कितनी न्यूनतम दूरी तय करे कि वह दूरी पूरे-पूरे कदमों में तय हो जाए?
हल :
प्रत्येक द्वारा तय की गई दूरी उनके कदमों का ल. स. होगी।
अतः

∴ल. स. = 2 x 3 x 3 x 5 x 7 x 11 = 6930
∴अभीष्ट न्यूनतम दूरी = 6930 सेमी

प्रश्न 3.
किसी कमरे की लम्बाई, चौड़ाई और ऊँचाई क्रमश: 825 सेमी, 675 सेमी और 450 सेमी हैं। ऐसा सबसे लम्बा फीता (tape) ज्ञात कीजिए जो कमरे की तीनों विमाओं (dimensions) को पूरा-पूरा माप ले।
हल :
फीते की अधिकतम लम्बाई 825, 675 और 450 का म. स. होगी।
∴825 = 3 x 5 x 5 x 11
675 = 3 x 3 x 3 x 5 x 5
450 = 2 x 3 x 3 x 5 x 5

∴म. स. = 3 x 5 x 5 = 75
अत: फीते की अधिकतम लम्बाई = 75 सेमी

प्रश्न 4.
6,8 और 12 से विभाज्य तीन अंकों की सबसे छोटी संख्या ज्ञात कीजिए।
हल :
तीन अंकों की सबसे छोटी संख्या = 100
तीन अंकों की सबसे छोटी संख्या जो 6, 8 और 12 से पूर्णतः विभाजित हो उनका ल. स. है।

∴ल. स. = 2 x 2 x 2 x 3 = 24
24 के सभी गुणज 6, 8 और 12 से विभाज्य होंगे। लेकिन हमें 3 अंकों का 24 का सबसे छोटा गुणज चाहिए।

अब 100 से बड़ी और 24 से पूर्णतया विभाज्य संख्या = (100 – 4) + 24 = 120
अत: अभीष्ट संख्या = 120

प्रश्न 5.
8, 10 और 12 से विभाज्य तीन अंकों की सबसे बड़ी संख्या ज्ञात कीजिए।
हल :
8, 10 और 12 का ल. स. :

∴ल. स. = 2 x 2 x 2 x 3 x 5 = 120
चूँकि 120 के सभी गुणज 8, 10 और 12 से भी विभाज्य होंगे।
अब 3 अंकों की सबसे बड़ी संख्या = 999

∴999 – 39 = 960 जो कि 120 का गुणज है।
अत : अभीष्ट संख्या = 960

प्रश्न 6.
तीन विभिन्न चौराहों की ट्रैफिक लाइट (traffic lights) क्रमशः प्रत्येक 48 सेकण्ड, 72 सेकण्ड और 108 सेकण्ड बाद बदलती है। यदि वे एक साथ प्रातः 7 बजे बदलें, तो वें पुनः एक साथ कब बदलेंगी?
हल :
अभीष्ट समय 48,72 और 108 का ल. स. होगा।

ल. स. = 2 x 2 x 2 x 2 x 3 x 3 x 3 = 432 सेकण्ड
अत : अभीष्ट न्यूनतम समय जब लाइटें दोबारा अपने आप बदलेंगी = 432 सेकण्ड = 7 मिनट 12 सेकण्ड
इसलिए वे दोबारा 7 बजकर 7 मिनट और 12 सेकण्ड पर बदलेंगी।

प्रश्न 7.
तीन टैंकरों में क्रमशः 403 लीटर, 434 लीटर और 465 लीटर डीजल है। उस बर्तन की अधिकतम धारिता ज्ञात कीजिए जो इन तीनों टैंकरों के डीजल को पूरा-पूरा माप देगा।
हल:
403 = 13 x 31
434 = 2 x 7 x 31
645 = 3 x 5 x 31
∴म. स. = सार्व गुणनखण्ड = 31
अत: बर्तन की अधिकतम अभीष्ट धारिता = 31 लीटर

प्रश्न 8.
वह सबसे छोटी संख्या ज्ञात कीजिए जिसे 6, 15 और 18 से भाग देने पर प्रत्येक दशा में 5 शेष रहे।
हल:

∴ल. स. = 2 x 3 x 3 x 5 = 90
अत : अभीष्ट संख्या = 90 + 5 = 95

प्रश्न 9.
चार अंकों की सबसे छोटी संख्या ज्ञात कीजिए जो 18, 24 और 32 से विभाज्य है।
हल:

∴ल. स. = 2 x 2 x 2 x 2 x 2 x 3 x 3 = 288
चार अंकों की सबसे छोटी संख्या = 1000
∴4 अंकों की सबसे छोटी संख्या जो 288 से विभाज्य हो
= 1000 – 136 + 288
= 1152

अत : अभीष्ट संख्या = 1152

प्रश्न 10.
निम्नलिखित संख्याओं का ल. स. ज्ञात कीजिए जिनमें एक संख्या सदैव 3 का गुणज है :
(a) 9 और 4
(b) 12 और 5
(c) 6 और 5
(d) 15 और 4
प्राप्त ल. स. में एक सामान्य गुण का अवलोकन कीजिए। क्या ल. स. प्रत्येक स्थिति में दोनों संख्याओं का गुणनफल है? क्या हम यह निष्कर्ष निकाल सकते हैं कि दो संख्याओं का ल. स. सदैव 3 का एक गुणज है।
हल :
(a)

∴ल. स. = 2 x 2 x 3 x 3 = 36
9 और 4 का गुणनफल = 9 x 4 = 36
∴4 और 9 का ल. स.= 9 और 4 का गुणनफल

(b)

∴ल. स.= 2 x 2 x 3 x 5 = 60
12 और 5 का गुणनफल = 12 x 5 = 60
∴12 और 5 का ल. स. = 12 और 5 का गुणनफल

(c)

∴ल. स. = 2 x 3 x 5 = 30
6 और 5 का गुणनफल = 6 x 5 = 30
∴6 और 5 का ल. स. = 6 और 5 का गुणनफल

(d)

∴ल. स. = 2 x 2 x 3 x 5 = 60
15 और 4 का गुणनफल = 15 x 4 = 60
∴15 और 4 का ल. स. = 15 और 4 का गुणनफल
हम पाते हैं कि
36 = 3 x 12,
60 = 3 x 20,
30 = 3 x 10
यहाँ प्रत्येक स्थिति ल. स. 3 का गुणज है।
हाँ, प्रत्येक स्थिति में ल. स. = दो संख्याओं का गुणनफल
हम यह निष्कर्ष नहीं निकाल सकते कि दो संख्याओं का ल. स. सदैव 3 का गुणज होता है।

प्रश्न 11.
निम्नलिखित संख्याओं का ल. स. ज्ञात कीजिए जिनमें एक संख्या दूसरी संख्या का एक गुणनखण्ड
(a) 5, 20
(b) 6, 18
(c) 12, 48
(d) 9, 45
प्राप्त परिणामों में आप क्या देखते हैं?
हल :
(a)

∴ल. स. = 2 x 2 x 5 = 20

(b)

∴ल. स. = 2 x 3 x 3 = 18

(c)

∴ल. स. = 2 x 2 x 2 x 2 x 3 = 48

(d)

∴ल. स. = 3 x 3 x 5 = 45
प्राप्त परिणामों से स्पष्ट है कि प्रत्येक स्थिति में दी हुई नंख्याओं का ल. स. उन दोनों में से बड़ी संख्या है।

MP Board Class 6th Maths Solutions