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CBSE Class 12 Maths Notes: Matrices – Solving Linear Equations using Matrices

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SYSTEM OF SIMULTANEOUS LINEAR EQUATIONS

Consider the following system of n linear equations in n unknowns:

Example -: With the help of matrices, solve the equations; 3x + y + 2z = 3, 2x – 3y – z = –3, x + 2y + z = 4.

Solution:       We can write the given equations as

AX = B                                                …(1)

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NCERT Class VII English Chapter 1 Three Questions

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NCERT Class VII English Chapter 1 Three Questions

National Council of Educational Research and Training (NCERT) Book for Class VII
Subject: English
Chapter: Chapter 1 – Three Questions

Class VII NCERT English Text Book Chapter 1 Three Questions is given below

Before you read

A king has three questions and he is seeking answers to them. What are the questions? Does the king get what he wants?

 

Three Questions

I

 he thought came to a certain king that he would never fail if he knew three things. These three things were: What is the right time to begin something? Which people should he listen to? What is the most important thing for
him to do?

The king, therefore, sent messengers throughout his kingdom, promising a large sum of money to anyone who would answer these three questions.

Many wise men came to the king, but they all answered his questions differently.

In reply to the first question, some said the king must prepare a timetable, and then follow it strictly. Only in this way, they said, could he do everything at its proper time. Others said that it was impossible to decide in advance the right time for doing something. The king should notice all that was going on, avoid foolish pleasures, and

always do whatever seemed necessary at that time. Yet others said that the king needed a council of wise men who would help him act at the proper time. This was because one man would find it impossible to decide correctly, without help from others, the right time for every action.

But then others said that there were some things which could be urgent. These things could not wait for the decision of the council. In order to decide the right time for doing something, it is necessary to look into the future. And only magicians could do that. The king, therefore, would have to go to magicians.

In their answers to the second question, some said that the people most necessary to the king were his councillors; others said, the priests. A few others chose the doctors. And yet others said that his soldiers were the most necessary. To the third question, some said science. Others chose fighting, and yet others religious worship.

As the answers to his questions were so different, the king was not satisfied and gave no reward. Instead, he decided to seek the advice of a certain hermit, who was widely known for his wisdom.

The hermit lived in a wood which he never left. He saw no one but simple people, and so the king put on ordinary clothes. Before he reached the hermit’s hut the king left his horse with his bodyguard, and went on alone.

As the king came near the hermit’s hut, he saw the hermit digging the ground in front of his

 hut. He greeted the king and continued digging. The hermit was old and weak, and as he worked, he breathed heavily.

The king went up to the hermit and said, “I have come to you, wise hermit, to ask you to answer three questions: How can I learn to do the right thing at the right time? Who are the people I need most? And what affairs are the most important?”

The hermit listened to the king, but did not speak. He went on digging. “You are tired,” said the king. “Let me take the spade and work in your place.”

“Thanks,” said the hermit, giving the king his spade. Then he sat down on the ground.

 

 

 

 

 

 

 


When the king had dug two beds, he stopped and repeated his questions. The hermit gave no answer, but stood up, stretching out his hand for the spade, and said, “Now you rest, and let me work.” But the king did not give him the spade and continued to dig.

One hour passed, then another. The sun went down behind the trees, and at last the king stuck the spade into the ground and said, “I came to you, wise man, for an answer to my questions. If you can give me no answer, tell me so and I will return home.”

“Here comes someone running,” said the hermit.

1. Why did the king want to know answers to three questions?
2. Messengers were sent throughout the kingdom

(i) to fetch wise men.
(ii) to find answers to the questions.
(iii) to look for the wise hermit.
(iv) to announce a reward for those who could answer the questions.

Mark your choice.

II

The king turned round and saw a bearded man running towards them. His hands were pressed against his stomach, from which blood was flowing. When he reached the king he fainted and fell to the ground. The king and the hermit removed the man’s clothing and found a large

wound in his stomach. The king washed and covered it with his handkerchief, but the blood would not stop flowing. The king re-dressed the wound until at last the bleeding stopped.

The man felt better and asked for something to drink. The king brought fresh water and gave it to him. By this time the sun had set and the air was cool. The king with the hermit’s help carried the wounded man into the hut and laid him on the bed. The man closed his eyes and lay quiet. The king, tired by his walk and the work he had done, lay down on the floor and slept through the night. When he awoke, it was several minutes before he could remember

 

 

 

 

 

 

 

 



where he was or who the strange bearded man lying on the bed was.

“Forgive me!” said the bearded man in a weak voice, when he saw that the king was awake.

“I do not know you and have nothing to forgive you for,” said the king.

“You do not know me, but 1 know you. 1 am that enemy of yours who swore revenge on you, because you put my brother to death and seized my property. I knew you had gone alone to see that hermit, and I made up my mind to kill you on your way home. But the day passed and you did not return. So I left my hiding-place, and I came upon your bodyguard, who recognised me and wounded me. I escaped from him but I should have died if you had not dressed my wounds. I wished to kill you, and you have saved my life. Now, if I live, I will serve you as your most faithful servant and will order my sons to do the same. Forgive me!”

The king was very happy to have made peace with his enemy so easily, and to have won him over as a friend. He not only forgave him but said he would send his servants and his own doctor to look after him, and he promised to give back the man his property.

Leaving the wounded man, the king went out of the hut and looked round for the hermit. Before going away he wished once more to get answers to his questions. The hermit was on his knees sowing seeds in the beds that had been dug the day before. The king went up to the hermit and said, “For the last time I beg you to answer my questions, wise man.”


“You have already been answered!” said the hermit still bending down to the ground and looking up at the king as he stood before him.

“How have I been answered? What do you mean?”

“Do you not see?” replied the hermit. “If you had not pitied my weakness yesterday and had not dug these beds for me, you would have gone away. Then that man would have attacked you and you would have wished you had stayed with me. So the most important time was when you were digging the beds. And I was the most important man, and to do me good was your most important business. Afterwards, when the man ran to us, the most important time was when you were caring for him, because if you had not dressed his wounds he would have died without having made peace with you. So he was the most important man, and what you did for him was your most important business.

“Remember then, there is only one time that is important and that time is ‘Now’. It is the most important time because it is the only time we have any power to act.

“The most necessary person is the person you are with at a particular moment, for no one knows what will happen in the future and whether we will meet anyone else. The most important business is to do that person good, because we were sent into this world for that purpose alone.”

LEO TOLSTOY
[retold]

 

 

 

 

 

 

 

 

 

 

 

 

 

 


Complete the following sentences by adding the appropriate parts of the sentences given in the box.
1. Many wise men answered the king’s questions,___________________________________

2. Someone suggested that there should be a council of wise men________________________________

3. Someone else suggested that the king should have a timetable__________________________________

4. The king requested the hermit_________________________________________

5. The king washed and dressed the bearded man’s wound,_____________________________________

  • but the bleeding would not stop.
  • to answer three questions.
  • but their answers were so varied that the king was not satisfied.
  • and follow it strictly.
  • to help the king act at the right time.

Working with the Text

Answer the following questions.

1. Why was the king advised to go to magicians?
2. In answer to the second question, whose advice did the people say would be important to the king?
3. What suggestions were made in answer to the third question?
4. Did the wise men win the reward? If not, why not?
5. How did the king and the hermit help the wounded man?
6. (i) Who was the bearded man?
(ii) Why did he ask for the king’s forgiveness?
7. The king forgave the bearded man. What did he do to show his forgiveness?
8. What were the hermit’s answers to the three questions?
Write each answer separately. Which answer do you like most, and why?

Working with Language

1. Match items in List A with their meanings in List B.

2. Each of the following sentences has two blanks. Fill in the blanks with appropriate forms of the word given in brackets.

He has__________________________ to help me. Do you think he will remember his ________________? (promise)

He has promised to help me. Do you think he will_________________ remember his promise?

(i) The _______________said that only fresh evidence would make him change his _____________.(judge)
(ii) I didn’t notice any serious ____________________of opinion among the debaters, although they______________________
from one another over small points. (differ)
(iii) It’s a fairly simple question to ____________________,but will you accept my as final? (answer)
(iv) It isn’t ____________________that should________________ always be the mother of invention. (necessary)
(v) Hermits are __________________men. How they acquire their _______________no one can tell. (wise)
(vi) The committee has________________ to make Jagdish captain of the team. The __________________is likely to please everyone. (decide)
(vii) Asking for is as noble as willingness to_____________ . (forgive)

Speaking and Writing

1. Imagine you are the king. Narrate the incident of your meeting the hermit. Begin like this:

The wise men answered my questions, but I was not satisfied with their answers. One day I decided to go and meet the hermit…

2. Imagine you are the hermit. Write briefly the incident of your meeting the king. Begin like this: One day I was digging in my garden. A man in ordinary clothes came to see me. I knew it was the king…

Do you know…
Does an ostrich really stick its head in the sand to hide from an enemy?

The Squirrel

You may have seen a squirrel sitting on the ground eating a nut. What did it look like? Here is a poet’s description of just such a squirrel.

He wore a question mark for tail,  An overcoat of gray, He sat up straight to eat a nut. He liked to tease and play, And if we ran around his tree, He went the other way.

MILDRED BOWERS ARMSTRONG

 

 

 

 

 

 

 

 

Working with the Poem

1. Why does the poet say the squirrel “wore a question mark for tail”? Draw a squirrel, or find a picture of a squirrel sitting on the ground. How would you describe its tail?
2. Do we usually say that an animal ‘wears’ a tail? What do we say? (Think: Does an animal wear a coat? Consult a dictionary if you like, and find out how ‘wear’ is used in different ways.)
3. “He liked to tease and play”. Who is teasing whom? How ?

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NCERT Class VII Science Chapter 17 Our Lifeline

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NCERT Class VII Science Chapter 17 Our Lifeline

National Council of Educational Research and Training (NCERT) Book for Class VII
Subject: Science
Chapter: Chapter 17 – Our Lifeline

Class VII NCERT Science Text Book Chapter 17 Our Lifeline is given below

One evening Boojho entered the park with an elderly person. He introduced him to his friends. Prof Ahmad was a scientist working in the university. The children started playing while Prof Ahmad sat on a bench in the corner. He was tired as he had participated in the golden jubilee celebrations of the town. After a while, the children also came and sat around him. They wanted to know about the celebrations. Prof Ahmad told them that after the cultural programme, the senior people discussed the town’s unemployment problem. A plan was proposed to put up a factory by clearing an area of the forest just outside the town. This would give the increasing population of the town a chance to get jobs. The children were very surprised when Prof Ahmad told them that many people had objected to this idea.

“This is because the forests serve as green lungs and water purifying systems in nature”, Prof Ahmad explained. The children were confused. Prof Ahmad realised that the children had not visited a forest. The children also wanted to know more about the forest, so they decided to visit it with Prof Ahmad.

17.1 VISIT TO A FOREST

One Sunday morning, the children packed a few things like a knife, a hand lens, a stick, a notebook and walked together through a forest trail near a village. On their way, they met Tibu, a young boy of their age group, of nearby village, who was taking cattle for grazing along with his aunt. He was very agile, running here and there to keep the herd together. When he saw the children, Tibu also started walking along with them, while his aunt went on a different path. As soon as they entered the forest Tibu raised his hand and signalled them to keep quiet because noise could disturb the animals living in the forest. Tibu then took them to a place at a height to show them the broad view of the forest. Children were surprised because they could not see any land (Fig. 17.1). The different treetops had formed green cover over the land. However, the cover was not uniformly green. The

environment was peaceful and a cool breeze was blowing. This made children quite fresh and happy.

While coming down, they got excited on hearing a sudden sound of birds and some noise from the top branches of the trees. Tibu told them to relax since it was a normal phenomenon here.

Because of the children’s presence, some monkeys had climbed higher up on the trees where they disturbed the birds. Animals often give this type of warning call to alert other animals. Tibu also told that many other animals like boar, bison, jackals, porcupine, elephants live in the deeper areas of the forest (Fig. 17.2). Prof Ahmad cautioned children that they should not go deep into the forest.

Boojho and Paheli remembered that they have studied about forests as an example of a habitat in Class VI (Fig. 17.3). They could see now how the forest provides a home for many animals and plants.


The land where the children were walking was uneven and covered with many trees. Tibu helped them to identify sal, teak, semal, sheesham, neem, palash, fig, khair, amla, bamboo, kachnar (Fig. 17.4). Prof Ahmad pointed out that there are several other trees, shrubs, herbs and grasses in the forest. The trees were also covered with different types of creepers and climbers. The sun was barely  visible through the leaves of the trees, making it quite dark inside the forest.

Activity 17.1

Observe the various things in your home and make a list of those which are made from material which may have been obtained from the forest. You might have many wooden items on your list like plywood, fuel wood, boxes, paper, matchsticks, and furniture. Do you know that gum, oils, spices, fodder for animals and medicinal plants are also some of the products which we get from the forest (Fig. 17.5).

Sheila wondered who would have planted these trees. Prof Ahmad replied that in nature trees produce enough seeds. The forest floor provides favourable conditions for them to germinate and develop into seedlings

and saplings. Some grow up into trees. He added that branchy part of a tree above the stem is known as the crown of the tree (Fig. 17.6).

Prof Ahmad asked children to look up and observe how the branches of the tall trees look like a roof over the other plants in the forest. He told them that this is called a canopy (Fig. 17.7).

Activity 17.2

Visit a forest or a park in your neighbourhood. Observe the trees and try to identify them. You can take the help of some elders or books on trees. List the characteristics of the trees that you observe, such as the height, shape of leaves, crown, flowers, and fruits. Also draw the crowns of some trees.

Prof Ahmad pointed out that trees had crowns of different types and sizes. These had created different horizontal layers in the forest. These are known as understoreys (Fig. 17.7). Giant and tall trees constituted the top layer followed by shrubs and tall grasses, and herbs formed the lowest layer.

“Would we see similar kind of trees in every forest?”—asked Boojho. Prof Ahmad said, “No, due to different climatic conditions there are variations in the types of trees and other plants. The types of animals also differ from forest to forest.”

A few children were busy watching beautiful butterflies fluttering here and there on the flowers of shrubs and herbs. They had a close look at the bushes. While doing that their hair and clothes had seeds and shrubs clinging to them.

They came across numerous insects, spiders, squirrels, ants and various

other small animals on the bark of the trees, plant leaves and on decaying leaves on the forest floor (Fig. 17.8). They started making sketches of these creatures. The forest floor seemed dark coloured and was covered with a layer of dead and decaying leaves, fruits, seeds, twigs and small herbs. The decaying matter was moist and warm. Children picked up various seeds and leaves for their collection. Walking over the dead leaf layer on the forest floor was like walking over a spongy carpet! Is the decaying matter always warm? Prof Ahmad suggested that the children could perform an activity to get an answer to this question.

Activity 17.3

Dig a small pit. Put vegetable waste and leaves in it. Cover them with soil. Add some water. After three days, remove the

upper layer of the soil. Does the pit feel warm inside? Paheli asked, “There are so many trees here. Also, there are many forest like this. What difference will it make if we cut some trees for a factory?” Prof Ahmad said, “You have read about autotrophs, heterotrophs and saprotrophs. You have learnt how green plants produce food.

All animals, whether herbivores or carnivores, depend ultimately on plants for food. Organisms which feed on plants often get eaten by other organisms, and so on. For example, grass is eaten by insects, which in turn, is taken by the frog. The frog is consumed by snakes. This is said to form a food chain: Grass→ insects→ frog→ snake→ eagle. Many food chains can be found in the forest. All food chains are linked. If any one food chain is disturbed, it affects other food chains. Every part of the forest is dependent on the other parts. If we remove one component, say trees, all other components would be affected.” Prof Ahmad asked children to pick up leaves from the forest floor and observe them under a hand lens. They found tiny mushrooms over the

decaying leaves. They also saw an army of tiny insects, millipedes, ants and beetle on them. They were wondering how these organisms live there. Prof Ahmad explained that apart from these animals which are easily seen, there are several organisms and micro-organisms that live in the soil. Paheli wondered what mushroom and other micro-organisms eat. Prof Ahmad replied that they feed upon the dead plant and animal tissues and convert them into a dark coloured substance called humus.

You have learnt about humus in Chapter 9. In which layer of the soil would you find humus? What is its importance to the soil? The micro-organisms which convert the dead plants and animals to humus are known as decomposers. These micro-organisms play an important role in the forest. Soon, Paheli removed some dead leaves and discovered under them a layer of humus on forest floor. The presence of humus ensures that the nutrients of the dead plants and animals are released into the soil. From there, these nutrients are again absorbed by the roots of the living plants. “What happens if an animal dies in the forest?” Sheila asked. Tibu replied the dead animals become food for vultures, crows, jackals and insects.” In this way, the nutrients are cycled. So, nothing goes waste in a forest (Fig. 17.9). Paheli reminded Prof Ahmad that he had not explained why forests are called green lungs. Prof Ahmad explained that

plants release oxygen through the process of photosynthesis. The plants help to provide oxygen for animal respiration. They also maintain the balance of oxygen and carbon dioxide in the atmosphere (Fig. 17.10). That is why forests are called lungs.

The children saw clouds forming in the sky. Boojho recalled what he had learnt about the water cycle in Class VI. Trees take in water from their roots and release water vapour into the air through evaporation.

If there were fewer trees, how will the water cycle be affected? Tibu told them that the forest is not just home to plants and animals. Many people also live in the forest. Some of them may belong to different tribes. Tibu explained that these people depend mostly on the forests. The forest provides them with food, shelter, water and medicines. They have traditional knowledge about many medicinal plants in the forest.

While Boojho was drinking water from a small stream, he saw some deer crossing the stream (Fig. 17.11). They disappeared into the bushes. The dense bushes and the tall grass provide animals with the food and shelter. They also protect them from carnivores that live in the forest.

Tibu then started looking closely at the forest floor. Soon he called and showed the children droppings of some animals, and explained the difference between various types of droppings. Prof Ahmad informed them that the forest officers could recognise the presence of some animals in the forest by their droppings and footprints. Boojho called every one and showed them a large, decaying heap of animal dropping. Several beetles and grubs were feeding on the heap and a bunch of seedlings was sprouting. “These seedlings are of the herbs and shrubs. The animals also disperse the seeds of certain plants and help the forest to grow

and regenerate. The decaying animal dung also provides nutrients to the seedlings to grow”, said Prof Ahmad. After listening to this, Boojho noted in his notebook, “By harbouring greater variety of plants, the forest provides greater opportunities for food and habitat for the herbivores. Larger number of herbivores means increased availability of food for a variety of carnivores. The wide variety of animals

helps the forest to regenerate and grow. Decomposers help in maintaining the supply of nutrients to the growing plants in the forest. Therefore, the forest is a ‘dynamic living entity’ — full of life and vitality.”

It was about afternoon and the children wanted to go back. Tibu suggested another route for going back. While they were going back, it started raining. However, surprisingly, they saw that the raindrops were not hitting the forest floor directly. The uppermost layer of the forest canopy intercepted the flow of raindrops, and most of the water was coming down through the branches and the stems of the trees. From the leaves it was dripping slowly over branches of the shrubs and herbs (Fig. 17.13). They found that the ground was still dry. After about half an hour, the rain stopped. They noticed that the layer of dead leaves over the forest floor appeared wet now. But water did not stagnate in the forest.

Boojho thought that if it had rained so heavily in his town, it would have flooded the drains and roads. What would happen if it rains heavily in your town?

Prof Ahmad told them that the forest also acts as a natural absorber of rainwater and allows it to seep. It helps maintain the water table throughout the year. Forests not only help in controlling floods but also help maintain the flow of water in the streams so that we get a steady supply of water. On the other hand, if trees are not present, rain hits the ground directly and may flood the area around it. Heavy rain may also damages the soil. Roots of trees normally bind the soil together, but in their absence the soil is washed away or eroded.

The children spent an hour at Tibu’s village on their way back. The weather of the village was quite pleasant. Villagers told them that due to the surrounding forest, they receive good rainfall. The air also remained cool. Noise pollution, too is less because the forest absorbs the noise of the nearby highway. The children learnt about the history of the village.

It surprised them that the villages and the agricultural fields of that area were created after clearing the forest about sixty years ago. Tibu’s grandfather told them that when he was young, the village was not as large as it was now. It was also surrounded by forests. Construction of roads, buildings, industrial development and increasing demand of wood created pressure on the forests and it started vanishing. He was not happy that the forest adjoining their village is not regenerating and is on the verge of disappearing due to overgrazing of animals and indiscriminate felling of trees. Prof Ahmad said that if we did things wisely we could preserve forests and environment as well as have development. Children prepared a few pictures to show the consequences of such an event.

At the end of the visit, Prof Ahmad asked children to sum up the importance of forests. The children wrote: Forests provide us with  oxygen. They protect soil and provide habitat to a large number of animals. Forests help in bringing good rainfall in  neighbouring areas.  They are a source of medicinal plants, timber and many other useful products. We must preserve our forests.


Keywords

Canopy Seed dispersal Deforestation
Crown Soil erosion Humus
Decomposers Understorey Regeneration

What you have learnt

  • We get various products from the forests surrounding us. Forest is a system comprising various plants, animals and micro- organisms.
  • In a forest, trees form the uppermost layer, followed by shrubs.
  • The herbs form the lowest layer of vegetation. Different layers of vegetation provide food and shelter for animals, birds and insects.
  • The various components of the forest are interdependent on one another.
  • The forest keeps on growing and changing, and can regenerate.
  • In the forest, there is interaction between soil, water, air and living organisms.
  • Forests protect the soil from erosion.
  • Soil helps forests to grow and regenerate.
  • Forests are the lifeline for the forest-dwelling communities.
  • Forests influence climate, water cycle and air quality.

Exercises

1. Explain how animals dwelling in the forest help it grow and regenerate.

2. Explain how forests prevent floods.

3. What are decomposers? Name any two of them. What do they do in the forest?

4. Explain the role of forest in maintaining the balance between oxygen and carbon dioxide in the atmosphere.

5. Explain why there is no waste in a forest.

6. List five products we get from forests?

7. Fill in the blank:

(a) The insects, butterflies, honeybees and birds help flowering plants in .

(b) A forest is a purifier of and .

(c) Herbs form the layer in the forest.

(d) The decaying leaves and animal droppings in a forest enrich the

8. Why should we worry about the conditions and issues related to forests far from us?

9. Explain why there is a need of variety of animals and plants in a forest.

10. In Fig. 17.15, the artist has forgotten to put the labels and directions on the arrows. Mark the directions on the arrows and label the diagram using the following labels:

clouds, rain, atmosphere, carbon dioxide, oxygen, plants, animals, soil, roots, water table.

11. Which of the following is not a forest product?

(i) Gum

(ii) Plywood

(iii) Sealing wax

(iv) Kerosene

12. Which of the following statements is not correct?

(i) Forests protect the soil from erosion.

(ii) Plants and animals in a forest are not depedent on one another.

(iii) Forests influence the climate and water cycle.

(iv) Soil helps forests to grow and regenerate.

13. Micro-organisms act upon the dead plants to produce

(i) sand (ii) mushrooms (iii) humus (iv) wood

Extended Learning — Activities and Projects

1. The Department of Environment is to decide whether some portion of a forest in your area could be cleared for a housing complex. Write a letter to the department explaining your point of view as a concerned citizen.

2. Visit a forest. Here is a list of points that would make your visit more fruitful.

(a) Make sure that you have permission to go into the forest.

(b) Make sure that you can find your way around. Get a map and go along with some one who is familiar with the area.

(c) Keep a record of the things you see and do. Observations make the visit interesting. Sketches and photographs are useful.

(d) You may record bird calls.

(e) Collect different kinds of seeds or hard fruits like nuts.

( f ) Try to recognise various types of trees, shrubs, herbs, etc. Make lists of plants from different places in the forest and of different layers. You may not be able to name all the plants, but it is worth recording and seeing where they grow. Make a record of approximate heights of plants, crown shape, bark texture, leaf size, and flower colour.

(g) Learn to recognise the animal’s droppings.

(h) Interview the forest officials and the people of surrounding vil lages and other visitors.

You must never collect birds’ eggs, and their nests should never be disturbed.

You can read more on the following website:

www.wild-india.com

Did you know?

In India the area under forest cover is about 21% of the total area. It had steadily been falling since independence. But people now seem to have realised the importance of the forest cover. Reports suggest that the area under forest cover has slightly increased in recent years.

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NCERT Class VII Maths Chapter 14 Symmetry

NCERT Class VIII English Chapter 10 The Great Stone Face II

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NCERT Class VIII English Chapter 10 The Great Stone Face II

National Council of Educational Research and Training (NCERT) Book for Class VIII
Subject: English
Chapter: Chapter 10 – The Great Stone Face II

Class VIII NCERT English Text Book Chapter 10 The Great Stone Face II is given below.


Before you read

Many years passed. Ernest was now a man of middle age. To his neighbours, who never suspected that he was  anything more than an ordinary and familiar face, he was only a humble and hardworking, though thoughtful,  person. But what about the old prophecy? Was it ever fulfilled? Let us read and find out.

The years hurried on, and brought white hairs upon the head of Ernest, and made wrinkles  across his forehead and furrows in his cheeks. He was an old man. But not in vain had he  grown old; more numerous than the white hairs on his head were the wise thoughts in 

his mind. And Ernest had ceased to be obscure. Unsought for, undesired, had come the fame  which so many seek. He had become famous beyond the limits of the valley. College  professors, and even the active men of cities, came from far to see and converse with Ernest,  nd he received them with gentle sincerity, and spoke freely with them of whatever came uppermost, or lay deepest in his heart or their own. While they talked together, his face would brighten, unawares, and shine upon them, as with a mild evening light.

While Ernest had been growing old, God had granted a new poet to this earth. He, too, was a  native of the valley, but had spent the greater part of his life in distant cities, pouring out his  sweet music everywhere. Neither was the Great Stone Face forgotten, for the poet had celebrated it in a poem. The songs of this poet found  their way to Ernest. He read them after  is customary toil, seated on the bench before his cottage door. As he read he lifted his eyes to the mountain.

“O Great Stone Face,” he said, “is not this man worthy to be your likeness?” The face seemed  to smile, but did not answer. Now it happened that the poet, though he lived so far away, had  not only heard of Ernest but had thought much about his character and wished to meet this  man whose wisdom walked hand in hand with the noble simplicity of his life. One summer  day, therefore, he arrived at Ernest’s door, where he found the good old man holding a book in his hand, which  he read and, then, with a finger between the leaves, looked lovingly at the Great Stone Face.

“Good evening,” said the poet. “Can you give a traveller a night’s shelter?” “Gladly,”  answered Ernest; and then he added, smiling, “I think I never saw the Great Stone Face look
so hospitably at a stranger.”

The poet sat down beside him, and he and Ernest talked together. Never  before had the poet  talked with a man like  Ernest, so wise, and  gentle, and kind. Ernest, on the other hand, was   moved by the living  images flung out of the  poet’s mind.

As Ernest listened to the poet, he imagined that the Great Stone Face was  bending forward to listen   too. He gazed into the  poet’s eyes. “Who are  you, my gifted guest?” he   asked.

The poet laid his finger on the book that Ernest had been reading. “You have read these poems,” said he. “You  know me, then, for I wrote them.” Again and again, Ernest examined the poet’s features; he turned towards the  Great Stone Face then back. He shook his head and sighed. “Why are you sad?” inquired the poet. “Because,”  replied Ernest, “all through life I have awaited the fulfillment of a prophecy, and when I read these poems, I  hoped that it might be fulfilled in you.” “You hoped,” answered the poet, faintly smiling, “to find in me the  likeness of the Great Stone Face. I am not worthy to be its likeness.”

“And why not?” asked Ernest. He pointed to the book. “Are not those thoughts worthy?”  “You can hear in them the distant voice of a heavenly song. But my life, dear Ernest, has  not corresponded with my thoughts. I have had grand dreams, but they have been only  dreams. Sometimes I lack faith in my own thoughts. Why, then, pure seeker of the good  and true, should you hope to find me in the face of the mountain?”

The poet spoke sadly and his eyes were wet with tears. So, too, were those of Ernest. At the  hour of sunset, as had long been his custom, Ernest was to speak to a group of neighbours in  the open air. Together he and the poet went to the meeting place, arm in arm. From there  could be seen the Great
Stone Face.

Ernest threw a look of familiar kindness around upon his audience. He began to speak to  the people what was in his heart and mind. His words had power, because they agreed with  his thoughts; and his thoughts had reality and depth, because they harmonised with the life which he had always lived. It was not mere breath  that the preacher uttered; they  were the words of life. A life of good deeds and selfless love was melted into them. The  poet, as he listened, felt that the life and character of Ernest were a nobler strain of poetry  than he had ever written. His eyes filled with tears and he said to himself that never was  there so worthy a sage as that mild, sweet, thoughtful face, with the glory of white hair diffused about it.

At a distance, but clearly to be seen, high up in the golden light of the setting sun,  appeared the Great Stone Face, with white mists around it, like the white hairs around the  brow of Ernest. At that moment, Ernest’s face took on an expression so grand that the  poet was moved to throw his arms up and shout. “Behold! Behold! Ernest is himself the  likeness of the Great Stone Face!”

Then all the people looked, and saw that what the poet said was true. The prophecy was fulfilled. But Ernest,  having finished what he had to say, took the poet’s arm, and walked slowly homeward, still hoping that some  wiser and better man than himself would by and by appear, bearing a resemblance to the Great  Stone Face.

NATHANIEL HAWTHORNE
(abridged)

Comprehension Check

Write ‘True’ or ‘False’ against each of the following statements.

1. Ernest’s words reminded people of the wise old sayings. ______

2. Total strangers from far away, who visited Ernest in the valley, found his face familiar. ______

3. The Great Stone Face confirmed Ernest’s view that the poet could be worthy of its likeness. ______

4. When Ernest and the poet met, they respected and admired each other equally. ______

5. The poet along with Ernest addressed the inhabitants of the valley. ______

6. The poet realised that Ernest’s thoughts were far nobler than his own verses. ______

Working with the Text

Answer the following questions.

1. How was Ernest different from others in the valley?

2. Why did Ernest think the poet was like the Stone Face?

3. What did the poet himself say about his thoughts and poems?

4. What made the poet proclaim Ernest was the Stone Face?

5. Write ‘Ernest’ or ‘Poet’, against each statement below.

(i) There was a gap between his life and his words.

(ii) His words had the power of truth as they agreed with his thoughts.

(iii) His words were as soothing as a heavenly song but only as useful as a vague dream.

(iv) His thoughts were worthy.

(v) Whatever he said was truth itself.

(vi) His poems were noble.

(vii) His life was nobler than all the poems.

(viii) He lacked faith in his own thoughts.

(ix) His thoughts had power as they agreed with the life he lived.

(x) Greatness lies in truth. Truth is best expressed in one’s actions. He was truthful, therefore he was great.

6. (i) Who, by common consent, turned out to be like the Great Stone Face?

(ii) Did Ernest believe that the old prophecy had come true? What did he say about it?

Working with Language

1. Mark the meaning that best fits the word or a phrase in the story.

(i) (sun) going down

(a) becoming smaller
(b) weakening
(c) setting

(ii) brightening

(a) making (it) look bright and cheerful
(b) lending (it) a special glow
(c) causing (it) to appear hopeful

(iii) spacious

(a) lonely and wild
(b) big and wide
(c) special and important

(iv) prophecy

(a) proverb
(b) prediction
(c) rumour

(v) marvellous

(a) wonderful
(b) surprising
(c) shocking

(vi) proclaim

(a) reveal
(b) declare
(c) shout

(vii) cease

(a) happen
(b) stop
(c) remain

(viii) (a night’s) shelter

(a) stay
(b) safety
(c) hospitality

(ix) gazed

(a) wandered about
(b) stared at
(c) thought of

(x) took on (an expression)

(a) challenged
(b) resembled
(c) assumed

2. (i) Read the following sentences.

(a) I do hope I’ll live to see him.

(b) He will come! Fear not, Ernest; the man will come.

(c) Gathergold is arriving tomorrow, people said.

(d) Blood-and-Thunder starts his journey back to the valley next week, everyone proclaimed.

(e) The great man is going to spend his old age in his native town.

Notice that in the above sentences, verbs in bold type are in four different forms, denoting four important ways  of expressing future time. None of these can be said to be exclusively used to show future time, though each is  used to refer to some action in future.

(ii) Which form of the verb is more natural in these sentences? Encircle your choice.

(a) I’m not free this evening. I will work/am working on a project.

(b) Have you decided where you will go for your higher secondary? Yes, I have. I will go/am going to the  Kendriya Vidyalaya.

(c) Don’t worry about the dog. It won’t hurt/isn’t hurting you.

(d) The weatherman has predicted that it will snow/is snowing in Ranikhet tonight.

(e) Swapna can’t go out this evening. Her father will come/is coming to see her.

3. (i) Complete these pieces of conversation using will or going to with the verbs given.

(a) Rani : Why are you turning on the radio?

Ravi : I ___________ (listen) to the news.

(b) Rani : Oh, I can’t buy this book. I have no money.

Ravi : Don’t worry. I ___________ (lend) you some.

(c) Rani : Look at those dark clouds.

Ravi : I think it ___________ (rain).

(d) Rani : What shall we have for dinner?

Ravi : I can’t decide.

Rani : Make up your mind.

Ravi : All right, then. We ___________ (have) fried rice and dry beans.

(e) Rani : Why are you filling the kettle with water?

Ravi : I ___________ (make) coffee.

(f) Rani : We need some bread and butter for breakfast.

Ravi : All right. I ___________ (go) to the bakery and get some. (Before he goes out, Ravi talks to their father.)

Ravi : I ___________ (get) some bread and butter. Do you want any thing from the bakery?

Father : Yes, I want some salt biscuits.

Ravi : Fine, I ___________ (get) you a packet.

(ii) Let pairs of children take turns to speak aloud the dialogues.

Speaking and Writing

1. Each of the following words has the sound/f/ as in feel. The words on the left have it initially. Those on the right have it finally. Speak each word clearly.

flail life fact tough
Philip puff fail laugh
flowed deaf fast stiff

2. Underline the letter or letters representing/f/in each of the following words.

file slough faint lift
cough defence afford enough
photograph staff tough aloof
affront philosophy sophistry

3. Imagine that you are the poet. You have come to your native valley to meet a famous preacher called Ernest.  Narrate the incident of your first meeting with him.

4. (i) Put each of the following in the correct order to construct sentences.

  • a resident of Noida near Delhi,/is visually impaired/George Abraham, ______________________________________________

________________________________________________

  • confidence and competitive spirit/and infuses discipline among the participants/It provides ____________________________________________

______________________________________________

  • he has helped/The brain behind the World Cup Cricket,/the disabled to dream _____________________________________________________

________________________________________________

  • to the blind school in Delhi/It was a chance visit/that changed his life ___________________________________________________

__________________________________________________

  • sport is a powerful tool/the disabled/He believes that/for rehabilitation of  ___________________________________________________

______________________________________________________

(ii) Now rearrange the sentences above to construct a paragraph.

George Abraham, _______________________________________________
_________________________________________________________________
_________________________________________________________________
_________________________________________________________________
_________________________________________________________________

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16th NSO 2013 – 2014 Answer Keys

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16th NSO 2013 – 2014 Answer Keys

16th NSO 2013 – 2014 Answer Keys are given below.

SET A

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CBSE Class 12 Maths Notes: Matrices – Algebra of Matrices

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ALGEBRA OF MATRICES

Addition and Subtraction of Matrices:

Any two matrices can be added if they are of the same order and the resulting matrix is of the same order.  If two matrices A and B are of the same order, they are said to be conformable for addition.

Properties:

All the laws of ordinary algebra hold for the addition or subtraction of matrices and their multiplication by scalars.

(i).        If A and B be two matrices of the same order and if k be a scalar, then

k(A + B) = kA + kB

(ii).       If k1 and k2 are two scalars and if A is a matrix, then

(k1 + k2)A = k1A + k2A and k1(k2A) = k2(k1A)

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CBSE Class 12 Maths Notes: Matrices – Classification of Matrices

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Row Matrix:

A matrix having a single row is called a row matrix. e. g. [1 3 5 7]

Square Matrix:

An m x n matrix A is said to be a square matrix if m = n i.e. number of rows = number of columns.

Note:

  • The diagonal from left hand side upper corner to right hand side lower side lower corner is known as leading diagonal or principal diagonal. In the above example square matrix containing the elements 1, 3, 5 is called the leading or principal diagonal.

Diagonal Matrix:

A square matrix all of whose elements except those in the leading diagonal, are zero is called a diagonal matrix. For a square matrix A = [aij]nxn to be a diagonal matrix, aij = 0, whenever i not equal to  j.

Scalar Matrix:

A diagonal matrix whose all the leading diagonal elements are equal is called a scalar matrix.

Triangular Matrix:

A square matrix in which all the elements below the diagonal elements are zero is called Upper Triangular matrix and a square matrix in which all the elements above diagonal elements are zero is called Lower Triangular matrix.

Given a square matrix A = [aij]n´n,

For upper triangular matrix, aij = 0,            i > j

and for lower triangular matrix, aij = 0,     i < j

Notes:

  • Diagonal matrix is both upper and lower triangular
  • A triangular matrix A = [aij]n´n is called strictly triangular if aii = 0 for  1 ≤ i ≤ n.


16th NSO 2013 – 2014 Answer Keys – Class 4

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16th NSO 2013 – 2014 Answer Keys – Class 7

16th NSO 2013 – 2014 Answer Keys – Class 8

16th NSO 2013 – 2014 Answer Keys – Class 9

16th NSO 2013 – 2014 Answer Keys – Class 10


CBSE Class 11 Maths Notes: Sequences & Series – Arithmetic Mean

Rajasthan Board Class 12 Model Question Papers

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Rajasthan Board Books

MP State Board Instruction regarding O.M.R. form 2013 for Class X & XII

CBSE Board Exam 2014 Class 10 SA2 Sample Question Paper – Mathematics for Blind Candidates

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Central Board of Secondary Education SA2 Exam 2014 Sample Question Paper
Year: 2013 – 2014
Examination: SA2 Board Exam for Secondary (Class 10)
Class: X
Subject: Mathematics for Blind Candidates

Central Board of Secondary Education (CBSE) Sample Question Paper for Class 10th SA2 Board Exam 2014 is given below for reference and preparation.

CBSE Board Exam 2014 Class 10 SA2 Sample Question Paper – Mathematics for Blind Candidates
CBSE Board Exam 2014 Class 10 SA2 Sample Question Paper – Mathematics for Blind Candidates

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