Author name: Prasanna

NCERT Solutions for Class 10 English First Flight Poem 5 The Ball Poem

NCERT Solutions for Class 10 English

The Ball Poem NCERT Solutions for Class 10 English First Flight Poem 5

The Ball Poem NCERT Text Book Questions and Answers

The Ball Poem Thinking about the Poem

Question 1.
Why does the poet say, “I would not intrude on him”? Why doesn’t he offer him money to buy another ball?
Answer:
The poet decides not to intrude on him. He does not want to intervene in the natural process of learning. He does not offer him money to buy another ball because he wants the child to learn his lesson of responsibility.

Question 2.
“staring down/All his young days into the harbour where/ His ball went.” Do you think the boy has had the ball for a long time? Is it linked to the memories of days when he played with it?
Answer:
Yes, the child has a long association with the ball. The ball is linked to the memories of days when he played with it. That is the reason why the child is upset at the loss of the ball.

Question 3.
What does “In the world of possessions” mean?
Answer:
It means that the world is full of materialistic things and people give maximum importance to these things. This is the reason why they are upset at the loss of things.

NCERT Solutions for Class 10 English First Flight Poem 5 The Ball Poem

Question 4.
Do you think the boy has lost anything earlier? Pick out the words that suggest the answer.
Answer:
No, the boy has never lost anything earlier. It was his first loss. The line “He senses first responsibility” suggests it.

Question 5.
What does the poet say the boy is learning from the loss of the ball? Tty to explain this in your own words.
Answer:
According to the poet the child has learnt his first lesson. He has learnt the sense of responsibility. It is a learning process. He has learnt how to face the loss of something very important to us.

Question 6.
Have you ever lost something you liked very much? Write a paragraph describing how you felt then and saying whether —and how — you got over your loss?
Answer:
Classroom Activity.

The Ball Poem Extra Questions and Answers

The Ball Poem Reference-to-Context Questions

Read the stanza given below and answer the questions that follow:

Question 1.
An ultimate shaking grief fixes the boy
As he stands rigid, trembling, staring down
All his young days into the harbour where
His ball went.

(a) The boy in the above stanza seems to be in a
Answer:
sad

(b) He stands stiff and trembling while staring at his
Answer:
ball

(c) The boy feels that with his ball that has fallen into a harbour, his childhood memories have also been washed off. (True/False)
Answer:
True

(d) The word that means same as ‘final’ is
Answer:
ultimate

NCERT Solutions for Class 10 English First Flight Poem 5 The Ball Poem

Question 2.
I would not intrude on him;
A dime, another ball, is worthless.
Now He senses first responsibility
In a world of possessions.

(a) The poet does not want to ………. the boy’s thoughts.
Answer:
intrude

(b) According to the poet, from the loss of the ball, the boy would learn what it means to lose something in a ………..
Answer:
world of possessions

(c) The poet wants to give monetary help to the boy for buying a new ball. (True/False)
Answer:
False

(d) The word in the stanza means same as ‘encroach upon’.
Answer:
intrude

NCERT Solutions for Class 10 English First Flight Poem 5 The Ball Poem

Question 3.
Money is external.
He is learning, well behind his desperate eyes,
The epistemology of loss, how to stand up
Knowing what every man must one day know
And most know many days, how to stand up.

(a) Money is external as it cannot buy
Answer:
memories

(b) The boy is learning how amidst losses.
Answer:
to stand up

(c) This episode will surely teach the boy the true meaning of life and nature of loss. (True/False)
Answer:
True

(d) ……………. in the stanza means ‘the study of the nature of knowledge it self.
Answer:
Epistemology

The Ball Poem Short Answer Question

Question 1.
‘He senses first responsibility’—What responsibility is referred to here? [2018]
Answer:
The ‘responsibility’ referred to here is relates to learning what it is like to experience grief at the loss of a much loved possession.

NCERT Solutions for Class 10 English First Flight Poem 5 The Ball Poem

The Ball Poem Long Answer Question

Question 1.
‘Possession in nine-tenths of the law’ How far does the contents of the poem, ‘The Ball Poem’, illustrate this idiom?
Answer:
The boy in the poem has lost his ball as it went rolling down the street and into the water. The loss of the ball is a great educator about the value of possession and the responsibility of keeping one’s possessions safely. The boy’s personal life is shattered as his personal possession has slipped away and lies irretrievable, and encompassing all his consciousness.

Though he is consoled by others with the offer of a substitute ball, or a dime to buy a ball, these prove worthless, and the loss awakens in him a sense of responsibility. The boy learns to stand up for his rightful possessions, besides learning to look after them by striving to be a responsible guardian.

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NCERT Solutions for Class 8 English It So Happened Chapter 1 How the Camel Got His Hump

NCERT Solutions for Class 8 English

How the Camel Got His Hump NCERT Solutions for Class 8 English It So Happened Chapter 1

How the Camel Got His Hump NCERT Text Book Questions and Answers

How the Camel Got His Hump Comprehension check – I

Question 1.
What tasks, do you think, were assigned to the dog and the ox?
Answer:
The dog carried the task of fetching and carrying things, and the ox helped man plough fields.

Question 2.
Why did the camel live in the middle of the desert?
Answer:
The camel lived in the middle of the desert because he did not want to work, and did not want to be disturbed by anyone.”

NCERT Solutions for Class 8 English It So Happened Chapter 1 How the Camel Got His Hump

Question 3.
What made the dog, the horse and the ox very angry?
Answer:
The camel did not do his share of work. So, the man made the dog, the horse and the ox do double the amount of work they used to do. This made them furious.

Question 4.
How did the Djinn know the horse was complaining against the camel?
Answer:
The Djinn knew that the horse was complaining against the camel because it was his camel. Moreover, the horse was ranting against “a thing in the middle of the Desert with a long neck and long legs”. The Djinn understood that the horse was talking about his camel.

How the Camel Got His Hump Comprehension check – II

Question 1.
The camel was looking at this own reflection in the pool. What does it suggest to you about the camel?
Answer:
It suggests that the camel was narcissistic, who loved himself more than anything else in the world. His love for himself and idleness characterized him.

NCERT Solutions for Class 8 English It So Happened Chapter 1 How the Camel Got His Hump

Question 2.
The camel said, “Humph ” repeatedly. How did it affect him?
Answer:
When the Djinn came to know that the camel says “Humph” all the time, he gave him a real hump on his back.

Question 3.
What, according to the Djinn, was the use of the “humph ’’?
Answer:
The “humph” would help the camel feel full in his stomach for three days so that he can work without food and make up for lost time.

Question 4.
“…he has never yet learnt to behave ”. In the light of this, what is the writer’s opinion about the camel?
Answer:
The writer believes that the camel is a lazy animal, and does not do his fair share of the work. He finds that all the other animals are very hard-working, whereas the camel sits around all day eating thorns and prickles. According to the writer, the camel spends most of his time in idleness.

How the Camel Got His Hump Exercise Questions and Answers

Discuss the following topics in groups.

Encourage the students to discuss the given questions in groups.

Question 1.
Can this story be factually true?
Answer:
No, this story cannot be factually true. The author is making up a story to entertain the readers. In reality, the camel is a hard-working animal that spends his days helping people who live in deserts. He carries heavy loads and makes their life comfortable. He carries them from one place to another.The hump on the camel’s back stores fat so that he can survive in the hot desert environment of the desert where food is scarce and limited.

NCERT Solutions for Class 8 English It So Happened Chapter 1 How the Camel Got His Hump

Question 2.
What, according to you, is the story about?
Answer:
The story is about how animals began to help human beings in doing work such as ploughing fields, carrying things and going from one place to another. It shows how we are dependent on animals and how each living being on this earth is allotted their share of work and contributes to the world in the best way they can. This is why we should all take our work seriously and do our fair share. It would be unjust to be lazy when there is so much work to be done around us.

The story demonstrates this way of thought by saying that the camel got his hump because he was not willing to do his fair share of the work and to help humans. So, the Djinn gave him a hump so that he did not have to eat for three days and do his work easily. The writer uses the sound that camel makes ‘humph’ and relates it to its hump, saying that there is a connection between the two. However, it is not true, and is merely said for the sake of humour.

Question 3.
What did you do over the weekend? Were you generally active or idle?
Answer:
I met with all my friends over the weekend to plan a getaway on my upcoming birthday. All of us met at a popular restaurant in the city and then headed for my home. There, we all sat and chatted over fresh lime sodas and cucumber sandwiches. We discussed our school and college days. It was a lot of fun. I wish all my weekends could be spent this way.

NCERT Solutions for Class 8 English It So Happened Chapter 1 How the Camel Got His Hump

Question 4.
There are broadly two categories of workers—those who prefer to do today what they can do tomorrow, and those who prefer to do tomorrow what they can do today. Where do you belong?
Answer:
Unfortunately, I belong to those who prefer to do tomorrow what they can do today. I believe that I do most of my things on the last day, as I tend to procrastinate. I am trying to break this habit, but it is not an easy one to get out of. I often get into a lot of trouble for this. I wish I belonged to the group who do their things beforehand.

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NCERT Solutions for Class 10 English First Flight Chapter 4 From the Diary of Anne Frank

NCERT Solutions for Class 10 English

From the Diary of Anne Frank NCERT Solutions for Class 10 English First Flight Chapter 4

From the Diary of Anne Frank NCERT Text Book Questions and Answers

From the Diary of Anne Frank Oral Comprehension Check

Question 1.
What makes writing in a diary a strange experience for Anne Frank?
Answer:
First, she had never written anything like this before and secondly, she thought that nobody was going to read or would be interested in her diary.

Question 2.
Why does Anne want to keep a diary?
Answer:
Because she doesn’t have a friend with whom she can share her feelings.

Question 3.
Why did Anne think she could confide more in her diary than in other people?
Answer:
She could confide in a close friend but she didn’t have one, the friends she had were there to have more fun and good times rather than the ones in whom she could confide.

NCERT Solutions for Class 10 English First Flight Chapter 4 From the Diary of Anne Frank

Question 4.
Why does Anne provide a brief sketch of her life?
Answer:
She thought that nobody would understand a single word without her giving the brief sketch of her life.

Question 5.
What tells you that Anne loved her grandmother?
Answer:
She often thought of her grandmother and still loved her.

Question 6.
Why was Mr Keesing annoyed with Anne? What did he ask her to do?
Answer:
Mr Keesing was annoyed with her because Anne talked very much in class. After several warnings he assigned her extra homework.

Question 7.
How did Anne justify her being a chatterbox in her essay?
Answer:
She gave two arguments to justify that she was a chatterbox, one that chatting was a student’s trait, and the other reason that nothing could be done about the inherited traits.

Question 8.
Do ybu think Mr Keesing was a strict teacher?
Answer:
Yes, he was a strict teacher because he wanted to set right Anne for talking too much in class.

Question 9.
What made Mr Keesing allow Anne to talk in class?
Answer:
When Anne had written her assignment – a long beautiful verse, Mr Keesing liked it. He read the poem to the class, added his comments to it and then he allowed her to talk.

From the Diary of Anne Frank Thinking about the Text

Question 1.
Was Anne right when she said that the world would not be interested in the musings of a thirteen- year-old girl?
Answer:
No, Anne was not right when she said that the world would not be interested in the musings of a thirteen-year-old girl.

Question 2.
There are some examples of diary or journal entries in the ‘Before You Read’ section. Compare these with what Anne writes in her diary? What language was the diary originally written in? In what way was Anne’s diary different?
Answer:
Anne started with a diary format but it became a memoir later on.

NCERT Solutions for Class 10 English First Flight Chapter 4 From the Diary of Anne Frank

Question 3.
Why does Anne need to give a brief sketch about her family? Does she treat ‘Kitty’ as an insider or outsider?
Answer:
She feels she is very lonely and this feeling is with her despite the fact that she has a family. That is why she wants to give a brief sketch of her family. She treats Kitty as an insider.

Question 4.
How does Anne feel about her father, her grandmother, Mrs Kuperus and Mr Keesing? What do these tell you about her?
Answer:
Anne Frank loved her father too much. She described him as the most adorable father she had ever seen. She was deeply attached to her grandmother. She felt extremely lonely after her death and she even lit a candle for her on her next birthday. Anne got attached to her headmistress, Mrs Kuperous and became emotional when bidding farewell. Mr Keesing, her maths teacher was very strict and she got pretty well with him.

Question 5.
What does Anne write in her first essay?
Answer:
She wants to write convincing arguments on the essay A chatterbox’. She writes that it is a student’s trait and that she will try to keep it under control. But she says she wouldn’t be able to cure it as the trait has been inherited from her mother.

Question 6.
Anne says teachers are most unpredictable. Was Mr Keesing unpredictable? How?
Answer:
Yes, what she says is right. She couldn’t predict the reactions of Mr Keesing on her assignments.

Question 7.
What do these statements tell you about Anne Frank as a person?
(i) We don’t seem to be able to get any closer, and that’s the problem. Maybe it’s my fault that we don’t confide in each other.
(ii) I don’t want to jot down the facts in this diary the way most people would do, but I want the diary to be my friend.
(iii) Margot went to Holland in December, and I followed in February, when I was plunked down on the table as a birthday present for Margot.
(iv) If you ask me, there are so many dummies that about a quarter of the class should be kept back, but teachers are the most unpredictable creatures on earth.
(v) Anyone could ramble on and leave big spaces between the words, but the trick was to come up with convincing arguments to prove the necessity of talking.
Answer:
(i) Friendly and warm
(ii) Wants to be different and original
(iii) Witty
(iv) Humorous
(v) Intelligent

NCERT Solutions for Class 10 English First Flight Chapter 4 From the Diary of Anne Frank

From the Diary of Anne Frank Extra Questions and Answers

From the Diary of Anne Frank Reference-to-Context Questions

Read the following extract carefully and answer the questions that follow.

Question 1.
Writing in a diary is a really strange experience for someone like me. Not only because I’ve never written anything before, but also because it seems to me that later on neither I nor anyone else will be interested in the musings of a thirteen-year-old school girl. Oh well, it doesn’t matter. I feel like writing, and I have an even greater need to get all kinds of things off my chest.

(a) Anne found her diary the best to share her thoughts and feelings with.
Answer:
companion

(b) The narrator thought that no one would be interested in the of a 13-year-old school girl.
Answer:
musings

(c) Writing in a diary was really a strange experience for a new hand. (True/False)
Answer:
True

(d) Give a synonymous word for ‘reflection of one’s thoughts’ from the extract.
Answer:
‘musings’

NCERT Solutions for Class 10 English First Flight Chapter 4 From the Diary of Anne Frank

Question 2.
‘Paper has more patience than people.’ I thought of this saying on one of those days when I was
feeling a little depressed and was sitting at home with my chin in my hands, bored and listless, wondering whether to stay in or go out. I finally stayed where I was, brooding: Yes, paper does have more patience, and since I’m not planning to let anyone else read this stiff-backed notebook grandly referred to as a‘diary’, unless I should ever find a real friend, it probably won’t make a bit of difference.

(a) Paper has more than people.
Answer:
patience

(b) Anne was feeling a little bored and listless that time.
Answer:
Depressed

(c) Anne preferred to write her musings on the blackboard. (True/False)
Answer:
False

(d) Find a synonymous word for ‘thinking a lot about something’ from the extract.
Answer:
Brooding.

NCERT Solutions for Class 10 English First Flight Chapter 4 From the Diary of Anne Frank

Question 3.
Now I’m back to the point that prompted me to keep a diary in the first place: I don’t have a friend. Let me put it more clearly, since no one will believe that a thirteen-year-old girl is completely alone in the world. And I’m not. I have loving parents and a sixteen-year-old sister, and there are about thirty people I can call friends. I have a family, loving aunts and a good home. No, on the surface I seem to have everything, except my one true friend.

(a) Anne had loving friends and a sister.
Answer:
16 year-old

(b) Anne seemed to have everything, except her one
Answer:
True friend

(c) Anne Frank was all alone in the world. (True/False)
Answer:
False

(d) Find a synonymous word for ‘motivate’ from the extract.
Answer:
‘prompted’.

Question 4.
All I think about when I’m with friends is having a good time. I can’t bring myself to talk about anything but ordinary everyday things. We don’t seem to be able to get any closer, and that’s the problem. Maybe it’s my fault that we don’t confide in each other. In any case, that’s just how things are, and unfortunately they’re not liable to change. This is why I’ve started the diary.

(a) When Anne was with her friends, she used to talk about everyday things.
Answer:
Ordinary

(b) Although Anne spent her time with her friends, she could not in them.
Answer: confide

(c) Anne confided in her dear friends, so she started writing her diary. (True/False)
Answer: False

(d) Find the synonym of ‘confess’ from the extract.
Answer:
Confide

NCERT Solutions for Class 10 English First Flight Chapter 4 From the Diary of Anne Frank

Question 5.
To enhance the image of this long-awaited friend in my imagination, I don’t want to jot down the facts in this diary the way most people would do, but I want the diary to be my friend, and I’m going to call this friend ‘Kitty’. Since no one would understand a word of my stories to Kitty if I were to plunge right in, I’d better provide a brief sketch of my life, much as I dislike doing so.

(a) Kitty was the name of diary.
Answer:
Anne’s

(b) Anne decided to provide a brief of her life before sharing her thoughts with Kitty.
Answer:
Sketch

(c) Anne never treated her diary Kitty as her friend. (True/False)
Answer:
False

(d) Give the antonym of ‘emerge/rise’ from the extract.
Answer:
Plunge

NCERT Solutions for Class 10 English First Flight Chapter 4 From the Diary of Anne Frank

Question 6.
My father, the most adorable father I’ve ever seen,didn’t marry my mother until he was thirty-six and she was twenty-five. My sister, Margot, was born in Frankfurt in Germany in 1926. I was born on 12 June 1929. I lived in Frankfurt until I was four. My father emigrated to Holland in 1933. My mother, Edith Hollander Frank, went with him to Holland in September, while Margot and I were sent to Aachen to stay with our grandmother. Margot went to Holland in December, and I followed in February, when I was plunked down on the table as a birthday present for Margot.

(a) Anne Frank was born on in Frankfurt in Germany.
Answer:
12 June 1929

(b) Anne’s mother was
Answer:
Edith Hollander Frank

(c) Anne’s mother was senior to her father by ten years. (True/False)
Answer:
False

(d) Find the synonym of ‘lovable’ from the extract.
Answer:
‘Adorable’

NCERT Solutions for Class 10 English First Flight Chapter 4 From the Diary of Anne Frank

Question 7.
I started right away at the Montessori nursery school. I stayed there until I was six, at which time I started in the first form. In the sixth form my teacher was Mrs Kuperus, the headmistress. At the end of the year we were both in tears as we said a heartbreaking farewell.
In the summer of 1941 Grandma fell ill and had to have an operation, so my birthday passed with little celebration. Grandma died in January 1942. No one knows how often I think of her and still love her. This birthday celebration in 1942 was intended to make up for the other, and Grandma’s candle was lit along with the rest.The four of us are still doing well, and that brings me to the present date of 20 June 1942, and the solemn dedication of my diary.

(a) Anne stayed at the until she was six.
Answer:
Montessori nursery school

(b) Mrs. Kuperus was the of the Montessori nursery school.
Answer:
Headmistress

(c) Anne’s birthday was celebrated in 1932. (True/False)
Answer:
False

(d) Give the antonym of ‘jovial/joyous’ from the extract.
Answer:
Solemn.

Question 8.
Our entire class is quaking in its boots. The reason, of course, is the forthcoming meeting in which the teachers decide who’ll move up to the next form and who’ll be kept back. Half the class is making bets. G.N. and I laugh ourselves silly at the two boys behind us, C.N. and Jacques, who have staked their entire holiday savings on their bet. From morning to night, it’s “You’re going to pass”, “No, I’m not”, “Yes, you are”, “No, I’m not”. Even G.’s pleading glances and my angry outbursts can’t calm them down. If you ask me, there are so many dummies that about a quarter of the class should be kept back, but teachers are the most unpredictable creatures on earth.

(a) The entire class was quaking in its boots because at the meeting the teachers would declare result.
Answer:
Forthcoming

(b) Half of the class was making who will pass and who will not.
Answer:
Bets

(c) According to Anne Frank, her teachers were the most predictable creatures on the earth. (True/False)
Answer:
False

(d) Give a synonym of ‘requested’ from the extract.
Answer:
Pleaded.

NCERT Solutions for Class 10 English First Flight Chapter 4 From the Diary of Anne Frank

Question 9.
The only subject I’m not sure about is maths. Anyway, all we can do is wept. Until then, we keep telling each other not to lose heart. I get along pretty well with all my teachers. There are nine of them, seven men and two women. Mr Keesing, the old foget who teaches maths, was annoyed with me for ages because I talked so much. After several warnings, he assigned me extra homework. An essay on the subject, A Chatterbox’. A chatterbox what can you write about that? I’d worry about that later, I decided. I jotted down the title in my notebook, tucked it in my bag and tried to keep quiet.

(a) Anne’s performance was not good in …………..
Answer:
Maths

(b) Anne got pretty well with all her teachers except …………, her maths teacher.
Answer:
Mr Keesing

(c) As a punishment, Mr. Keesing asked Anne to write an essay on the subject, ‘A Chatterbox’.(True/False)
Answer:
True

(d) Give an antonym of ‘relieved’ from the extract.
Answer:
Annoyed

Question 10.
That evening, after I’d finished the rest of my homework, the note about the essay caught my eye. I began thinking about the subject while chewing the tip of my fountain pen. Anyone could ramble on and leave big spaces between the words, but the trick was to come up with convincing arguments to. prove the necessity of talking. I thought and thought, and suddenly I had an idea. I wrote the three pages Mr Keesing had assigned me and was satisfied. I argued that talking is a student’s trait and that I would do my best to keep it under control, but that I would never be able to cure myself of the habit since my mother talked as much as I did if not more, and that there’s not much you can do about inherited traits.

(a) Anne argued that talking is a student’s
Answer:
Trait

(b) Anne wrote pages on the subject, ‘A Chatterbox’.
Answer:
Three

(c) Anne told her teacher that her habit of talking too much was an inherited trait from her mother. (True/False)
Answer:
True

(d) Give a synonym of ‘attributed’ from the extract.
Answer:
Trait

NCERT Solutions for Class 10 English First Flight Chapter 4 From the Diary of Anne Frank

Question 11.
Mr Keesing had a good laugh at my arguments, but when I proceeded to talk my way through the next lesson, he assigned me a second essay. This time it was supposed to be on An Incorrigible Chatterbox’. I handed it in, and Mr Keesing had nothing to complain about for two whole lessons. However, during the third lesson he’d finally had enough. “Anne Frank, as punishment for talking in class, write an essay entitled — ‘Quack, Quack, Quack, Said Mistress Chatterbox’. The class roared. I had to laugh too, though I’d nearly exhausted my ingenuity on the topic of chatterboxes. It was time to come up with something else, something original. My friend, Sanne, who’s good at poetry, offered to help me write the essay from beginning to end in verse and I jumped for joy. Mr Keesing was trying to play a joke on me with this ridiculous subject, but I’d made sure the joke was on him.

(a) How did Mr Keesing react to her essay?
Answer:
Mr. Keesing had a good laugh at her arguments.

(b) Why did Mr. Keesing assign her another essay? What was it all about?
Answer:
She could not control over her habit of chatting in the class. As a punishment he gave her another essay an Incorrigible Chatterbox.

(c) What did he ask Anne to do as third punishment?
Answer:
Mr. Keesing, as punishment for talking in class, asked her to write an essay entitled ‘Quack, Quack, Quack, Said Mistress Chatterbox’.

(d) What was Mr Keesing trying to do this time by giving that subject?
Answer:
Mr. Keesing was trying to play a joke on her with that ridiculous subject.

Question 12.
I finished my poem, and it was beautiful! It was about a mother duck and a father swan with three baby ducklings who were bitten to death by the father because they quacked too much. Luckily, Mr Keesing took the joke the right way. He read the poem to the class, adding his own comments, and to several other classes as well. Since then I’ve been allowed to talk and haven’t been assigned any extra homework. On the contrary, Mr Keesing’s always making jokes these days.

(a) Who is I here? What did she do?
Answer:
I here is- Anne Frank. She wrote an essay in poetic form. It was an assignment given to her by her teacher, Mr Keesing.

(b) What was it all about?
Answer:
It was about a mother duck and a father swan with three baby ducklings who were bitten to death by the father because they quacked too much.

(c) How did Mr Keesing take it?
Answer:
Mr Keesing took the joke in the right way. He added his own comments in it and read the poem to the class.

NCERT Solutions for Class 10 English First Flight Chapter 4 From the Diary of Anne Frank

(d) How did this change the whole situation?
Answer:
Mr Keesing allowed Anne to talk in the class and never assigned any extra work. He himself started making jokes.

From the Diary of Anne Frank Short Answer Questions

Question 1.
What prompted Anne to maintain a diary?
Answer:
After the death of her grandmother, Anne felt quite lonely. She started writing a diary to share her sorrows and joys.

Question 2.
Who became Anne’s friend, and what was the friend’s name?
Answer:
Anne’s diary became her friend. She named it ‘Kitty’.

Question 3.
For whom was Anne “a birthday present” and why?
Answer:
Anne was ‘a birthday present’ to her sister, Margot, because she came to England on Margot’s birthday and was plunked down on the table as the birthday present.

Question 4.
Why did Anne think that she was alone? Give reasons.
Answer:
Anne felt very lonely though she had loving parents and many other friends because she could not share her feelings with any one of them.

Question 5.
How do you know that Anne was close to her grandmother?
Answer:
Anne spent her early childhood with her grandmother. After her death she missed her a lot and often thought of her. She even lit a candle as a tribute to her deceased grandmother.

Question 6.
(i) Where did Anne stay before going to Holland?
(ii) Why was she in tears when she left the Montessori School?
Answer:
(i) Anne stayed at Aachen where her grandmother stayed.
(ii) When she parted with her loving teacher, Mrs Kuperus, she wept piteously.

Question 7.
Why was the entire class quaking in their boots?
Answer:
The entire class was quaking in the boots as they knew that their fate was going to be decided in a meeting. The teachers would decide who would pass and who would fail. This worried them a lot.

From the Diary of Anne Frank Long Answer Questions

Question 1.
What idea do you form of Mr Keesing as a teacher? What is that you like the most about him?
Answer:
Mr Keesing, the maths teacher, was very strict. He got annoyed with Anne as she talked too much. He warned Anne several times and after that he assigned her extra homework. When she completed it she was assigned one more essay by Mr Keesing. He found all the essays correct and laughed at her arguments. This shows his liking for Anne. At last, he tried to play a joke on Anne by giving her a ridiculous topic Quack, Quack, Quack, said mistress Chatterbox.

The poem written by Anne, completely transformed Mr Keesing. Now, he had started having fun with students and even allowed . them to talk. Mr Keesing was a good teacher. He was a very disciplined and concerned teacher. He wanted his students to be serious in his classes. However, he was a short tempered teacher who punished Anne without understanding her stand. When Anne cracked a joke on him, he took it in a positive way. This trait of his character is very impressive.

NCERT Solutions for Class 10 English First Flight Chapter 4 From the Diary of Anne Frank

Question 2.
How does Anne feel about her father, her grandmother, Mrs Kuperus and Mr Keesing? How does Anne’s description of these characters reflect her own character? Is she fair, critical or biased about them?
Answer:
Anne Frank loved her father too much. She described him as the most adorable father she had ever – seen. She was deeply attached to her grandmother. She felt extremely lonely after her death and she even lit a candle for her on her next birthday. Anne got attached to her headmistress, Mrs Kuperous and became emotional when bidding farewell. Mr Keesing, her maths teacher was very strict and she got pretty well with him. Anne’s description of these characters shows that Anne herself is a good human being. She has respect for all. She does not hesitate in making her teacher realise that he is wrong. She is fearless but talkative. She is fair and critical in her approach. She is not biased.

Question 3.
Write a brief character sketch of Anne. How does she impress you? What will you learn from her?
Answer:
Young Anne was a very intelligent girl and had a flair for writing essays. She could write essays and convinced her teacher that the talkativeness was her birthright and that she had no control over her talkativeness as she had inherited the art from her mother. She outwitted her teacher by writing ‘ the essay. But when the teacher punished her again and asked her to write another essay, “The Incorrigible Chatterbox”, she composed a poem and gave a message through it to the teacher.

The teacher was so impressed by her little poem that he decided not to punish her. Thus, we see that she is capable of writing good essays and win the heart of Mr Keesing and make him realise his mistake. Her fearlessness, critical thinking, humility and unbiased approach are some of the values reflected in her personality. I like her creativity and humorous approach to deal with her strict maths teacher, Mr Keesing.

Question 4.
Do you think Keesing was justified in punishing Anne? Would you support such a punishment in your class? Why/Why not?
Answer:
Mr Keesing was a very strict teacher who could not tolerate Anne’s talkativeness and would punish her. He even did not try to find out why the girl was always talking in his class. He was not justified in punishing her because he should have tried to make his teaching more impressive and interesting. Anne was weak in Mathematics; naturally, she was not interested in learning in his class.

First, he insulted her and asked her to write an essay on a chatterbox. As a teacher he should have been careful enough not to insult a young girl in front of the class. He further punished her to write another essay. The children are loving young ones who should not be punished at all but treated affectionately. No, I will not support any kind of punishment. A teacher should not use punishment to control the class. He/she should understand the level of students and modify his/her method of teaching.

Question 5.
Anne was very much attached to her grandmother. What should be our attitude towards our elders? What do you learn from Anne?
Answer:
Anne Frank was a thirteen year old girl. She was born at Frankfurt in Germany. She lived there until she was four. Her parents emigrated to Holland and she was sent to stay with her grandmother. Anne was very close to her grandmother. She found her a lovely lady. She loved her the most.

After her death, Anne missed her very much. She remarks, “No one knows how often I think of her and still love her.” It shows that Anne was very much attached to her grandmother. We should be respectful and sympathetic towards our elders. We get love and wishes from our elders if we treat them respectfully. It is our duty to take a good care of our elders. We learn from Anne that if we love our elders we get love in return.

NCERT Solutions for Class 10 English First Flight Chapter 4 From the Diary of Anne Frank Read More »

NCERT Solutions for Class 8 English It So Happened Chapter 10 The Comet 2

NCERT Solutions for Class 8 English

The Comet 2 NCERT Solutions for Class 8 English It So Happened Chapter 10

The Comet 2 NCERT Text Book Questions and Answers

The Comet 2 Comprehension check – I

Question 1.
“For a moment James wondered if he had done his sums right. ” Why was James doubtful about his sums and calculations?
Answer:
James was doubtful about his sums and calculations because if they were correct, it would mean that the Earth was going to be hit by a comet. He could not believe that this was possible, especially when the sky appeared quite peaceful as it did that day.

NCERT Solutions for Class 8 English It So Happened Chapter 10 The Comet 2

Question 2.
What did the scientists at the conference say about James s “sums ”?
Answer:
The scientists checked and rechecked James’ calculations with the latest observations of Comet Dutta. They later agreed that his predictions were absolutely correct.

Question 3.
Immediate action was needed, the scientists decided. Give one example each of ‘defensive ’ and “offensive ” action mentioned in the text.
Answer:
The “defensive” action suggested by the scientists was that all earthlings should go and live in underground bunkers. The “offensive” action that was decided upon was to deflect Comet Dutta from its path by giving it a push. The scientists were going to place the nuclear payload in a spaceship, and send it to intercept the approaching comet. Then they would detonate it by remote control.

NCERT Solutions for Class 8 English It So Happened Chapter 10 The Comet 2

Question 4.
“I am not buying any Christmas presents till December 15. ” What did Sir John mean by that?
Answer:
December 15 was predicted to be the date when Comet Dutta would hit the Earth and destroy it. Sir John was not confident that the Earth and earthlings would survive the collision. So, he did not want to buy Christmas presents before the Earth survived that day.

The Comet 2 Comprehension check – II

Question 1.
What is Duttada expected to do on his return from London?
Answer:
Duttada is expected to perform a yajna on his return from London so that the comet that he has discovered does not cause any ill effects on Earth.

Question 2.
What is his reaction to the proposal?
Answer:
He is full of anger that his family could be so superstitious. He says that it was futile for him to explain anything about comets to them, as they had never even read the elementary books on science.

Question 3.
i. What does “Project Light Brigade” refer to?
Answer:
“Project Light Brigade” refers to the operation of preventing Comet Dutta from colliding with the Earth.

NCERT Solutions for Class 8 English It So Happened Chapter 10 The Comet 2

ii. What does Sir John say about the Project in his letter to Duttada in October?
Answer:
Sir John includes a secret message about the Project in his letter to Duttada in October. He tells him in his secret code that the project had begun.

Question 4.
Did Sir John buy Christmas presents on December 15? How didDuttada get to know about it?
Answer:
Yes, Sir John bought Christmas presents on December 15. Duttada got to know about it through an urgent telex message sent by Sir John and brought to him by a special messenger on a scooter from the British Council. It was Sir John’s message that he was confident about buying Christmas presents on December 15.

Question 5.
Why, according to Indrani Debi, had the comet not been disastrous? Do you agree with her?
Answer:
According to Indrani Debi, the comet had not been disastrous because Duttada’s grandson, Khoka, had performed the yajna on his behalf to avert the side-effects of the comet.

Encourage the students to share their own opinions. No, I do not agree with her. The disaster was averted because of the efforts put in by the scientists, and not because of the yajna.

NCERT Solutions for Class 8 English It So Happened Chapter 10 The Comet 2

Question 6.
Is Duttada’s general outlook
i. rational?
ii. moral?
iii. traditional?

Choose the right word. Say why you think it right.
Duttada’s general outlook is rational because he does not believe that performing the yajna had stopped the disastrous effects of the comet.

The Comet 2 Exercise Question and Answer

Discuss the following topics in small groups. Write your answers afterwards.

Encourage the students to use their creativity to formulate their individual answers.

Question 1.
Should a scientist s findings be suppressed if they seem disturbing? Give reasons for and against the topic.
Answer:
A scientist’s findings should be repressed for the larger good and to protect the larger interests of mankind. For example, if a scientific finding can create panic among the masses, it is best to suppress such a finding. However, if the information safeguards the interests of political parties, lobbyists and only some sections of the society, it is best to bring the issue to light. The findings ought to be published if they serve mankind in any way and pave a better future for us.

NCERT Solutions for Class 8 English It So Happened Chapter 10 The Comet 2

Question 2.
Do you think ours is a traditional society? What are some of the things we do to be called traditional? Do you find these things useless or useful?
Answer:
Yes, India is a traditional society. We worship idols and even the moon. Our forms of worship resonate with an ancient belief system. Yajnas and poojas are still performed to pacify deities. Many people still believe in astrology and think that stars in the sky affect our behaviours and personalities. A majority of these things do not have any logic behind them, and are remnants of a superstitious society that was largely uneducated. So, most of them are highly useless.

Question 3.
Give two or three examples to show how science has been useful to us.
Answer:
Science has given us many gifts and has made our lives easier. The advancement of science and medicine has helped us to find cures to diseases that were thought to be incurable such as jaundice, tuberculosis and even cancer. Science has given us many miracles.

Also, it is science that has made transport and communication easier. Today it is possible to travel large distances in a very short time. We have so many modes of travel today to choose from. For example, superfast trains, metros and aeroplanes are all gifts of science.

NCERT Solutions for Class 8 English It So Happened Chapter 10 The Comet 2

Question 4.
Give one example to show that science has been misused, and as a result been harmful to us.
Answer:
The invention of nuclear bombs is one way that science has proved to be harmful to mankind. It has destroyed whole cities and can be blamed to have taken away countless human lives. The atomic bombing of Hiroshima and Nagasaki during the World War is one example of a negative effect of scientific progress.

Similarly, it has become very easy to make bombs nowadays. For this reason, terrorism is on the rise and has entered the common parlance. It is a curse that has befallen on mankind, and science has made it possible.

NCERT Solutions for Class 8 English It So Happened Chapter 10 The Comet 2 Read More »

NCERT Solutions for Class 9 Maths Chapter 2 Polynomials (Hindi Medium)

NCERT Solutions for Class 9 Maths Chapter 2 Polynomials (Hindi Medium)

NCERT Solutions for Class 9 Maths Chapter 2 Polynomials (बहुपद) (Hindi Medium)

These Solutions are part of NCERT Solutions for Class 9 Maths in Hindi Medium. Here we have given NCERT Solutions for Class 9 Maths Chapter 2 Polynomials

प्रश्नावली 2.1 

Ex 2.1 Class 9 गणित Q1. निम्नलिखित व्यंजकों में कौन-कौन एक चर में बहुपद हैं  और कौन-कौन नहीं हैं ? कारण के साथ उत्तर दीजिए :
(i) 4x2 – 3x + 7 

(ii) y2 + √2
(iii)3√t + t√2
(iv) y + \(\frac { 2 }{ y }\)
(v) x10 + y3 + t50
हल:
(i) 4x2 – 3x + 7
यह एक चर में बहुपद है क्योंकि चर घात एक प्राकृत संख्या है |

(ii) y2 + √2
यह एक चर में बहुपद है क्योंकि चर घात एक प्राकृत संख्या है |

(iii)3√t + t√2
यह एक चर में बहुपद नहीं है क्योंकि चर का घात एक भिन्नात्मक संख्या है कोई प्राकृत संख्या नहीं है |

(iv) y + \(\frac { 2 }{ y }\)
यह एक चर में बहुपद नहीं है |

(v) x10 + y3 + t​50
यह एक चर में बहुपद नहीं है | बल्कि यह तीन चर में बहुपद है |

Ex 2.1 Class 9 गणित Q2. निम्नलिखित में से प्रत्येक में x2 का गुणांक लिखिए |
(i) 2 + x2 + x 
(ii) 2 – x2 + x3
iii) \(\frac { \pi }{ 2 }\) x2 + x
(iv) √2x −1
हल: 
(i) 2 + x2 + x
x2 का गुणांक = 1

(ii) 2 – x2 + x3
x2 का गुणांक = –1 ​

(iii) \(\frac { \pi }{ 2 }\) x2 + x
x2 का गुणांक = \(\frac { \pi }{ 2 }\)

(iv) √2x −1
x2 का गुणांक = 0 [क्योंकि यहाँ xनहीं है इसलिए इसका गुणांक शून्य होगा |]

Ex 2.1 Class 9 गणित Q3. 35 घात के द्विपद का और 100 घात के एकपदी का एक-एक उदाहरण दीजिए|
हल:
35 घात का एक द्विपदी
⇒ 2x35 + 5y ​
Note: द्विपदी का अर्थ दो पदों वाला व्यंजक जैसे –  x + 5, 3a – 2b, 3t + 7 आदि.
100 घात का एक एकपदी
⇒ 3y100
Note: एकपदी का अर्थ एक पद वाला व्यंजक जैसे- 3x, 5t, y, 3xy आदि.

Ex 2.1 Class 9 गणित Q4. निम्नलिखित बहुपदों में से प्रत्येक के घात लिखिए:
(i) 5x3 + 4x2 + 7x

(ii) 4 – y2
(iii) 5t – √7
(iv) 3
हल:
(i) 5x3 + 4x2 + 7x
उत्तर: बहुपद का घात = 3
[नोट: बहुत का घात ज्ञात करने के लिए सभी घातों में से सबसे बड़ी घात को चुना जाता है |]

(ii) 4 – y​2
उत्तर: बहुपद का घात = 2

(iii) 5t – √7
उत्तर: बहुपद का घात = 1

(iv) 3
उत्तर: बहुपद का घात = 0
[नोट: चूँकि यहाँ कोई चर नहीं है इसलिए बहुपद का घात शून्य (0) है |]

Ex 2.1 Class 9 गणित Q5. निम्नलिखित को रैखिक, द्विघात और त्रिघात बहुपद में वर्गीकृत कीजिए:
(i) x​2 + x 
(ii) x – x3 
(iii) y + y2 + 4
(iv) 1 + x
(v) 3t
(vi) r2 
(vii) 7x2
हल:
(i) x​2 + x
उत्तर: द्विघात बहुपद

(ii) x – x3
उत्तर: त्रिघात बहुपद

(iii) y + y2 + 4
उत्तर: द्विघात बहुपद

(iv) 1 + x
उत्तर: रैखिक बहुपद

(v) 3t
उत्तर: रैखिक बहुपद

(vi) r2
उत्तर: द्विघात बहुपद

(vii) 7x2
उत्तर: त्रिघात बहुपद

प्रश्नावली 2.2

Ex 2.2 Class 9 गणित प्र1. निम्नलिखित पर बहुपद 5x – 4x2 + 3 के मान ज्ञात कीजिए :
(i) x = 0
(ii) x = –1
(iii) x = 2

हल:
(i) p(x) = 5x – 4x+ 3
बहुपद p(x) में x = 0 रखने पर
P(0) = 5(0) – 4(0)+ 3 = 0 – 0 + 3 = 3
अत: बहुपद का मान 3 है |

(ii) p(x) = 5x – 4x+ 3
बहुपद p(x) में x = -1 रखने पर
P(1) = 5(-1) – 4(-1)+ 3 = – 5 – 4 + 3 = – 9 + 3 = – 6
अत: बहुपद का मान – 6 है |

(iii) p(x) = 5x – 4x+ 3
बहुपद p(x) में x = 2 रखने पर
P(2) = 5(2) – 4(2)+ 3 = 10 -16 + 3 = – 3
अत: बहुपद का मान – 3 है |

Ex 2.2 Class 9 गणित Q2. निम्नलिखित बहुपदों में से प्रत्येक के लिए p(0), p(1) और p(2) ज्ञात कीजिए| 
(i)  p(y) = y2 – y + 1
(ii) p(t) = 2 + t + 2t 2 – t 3
(iii) p(x) = x3
(iv) p(x) = (x – 1) (x + 1)
हल:
(i) p(y) = y– y + 1
P(0) के लिए
P(0) = (0)2– 0 + 1 = 1
P(1) के लिए
P(1) = (1)2– 1 + 1
= 1 – 1 + 1 = 1
P(2) के लिए
P(2) = (2)2– 2 + 1
= 4 – 2 + 1 = 3

(ii) p(t) = 2 + t + 2t2– t3
P(0) के लिए
P(0) = 2 + 0 + 2(0)2– (0)3 = 2
P(1) के लिए
P(1) = 2 + 1 + 2(1)– (1)3 = 4

P(2) के लिए
P(2) = 2 + 2 + 2(2)2– (2)3
= 4 + 8 – 8 = 4

(iii) p(x) = x3
P(0) के लिए
P(0) = (0)3 = 0
P(1) के लिए
P(1) = (1)= 1
P(2) के लिए
P(2) = (2)= 8

(iv) P(x) = (x – 1) (x + 1)
P(0) के लिए
P(0) = (0 – 1) (0 + 1) = (-1) (1) = -1
P(1) के लिए
P(1) = (1 – 1) (1 + 1) = 0 (1) = 0
P(2) के लिए
P(2) = (2 – 1) (2 + 1) = 1(3) = 3

Ex 2.2 Class 9 गणित Q3. सत्यापित कीजिए कि दिखाए गए मान निम्नलिखित स्थितियों में संगत बहुपद के शुन्यक हैं :
NCERT Solutions For Class 9 Maths Polynomials Hindi Medium 2.2 3
हल:  
(i) P(x) = 3x + 1
Maths NCERT Solutions Class 9 Polynomials Hindi Medium 2.2 3.1
p(x) = 0, अत: दिया गया x का मान बहुपद का शुन्यक है |
(ii) P(x) = 5x – π
NCERT Solutions for Class 9 Maths Chapter 2 (Hindi Medium) 2.2 3.2
= 5 – π
∵ ​P(x) ≠ 0
∴ x के लिए दिया गया मान P(x) का शुन्यक नहीं है|

(iii) P(x) = x– 1
Class 9 Maths NCERT Polynomials Solutions Hindi Medium 2.2 3.3
NCERT Maths Solutions For Class 9 Polynomials Hindi Medium 2.2 3.4
NCERT Class 9 Maths Hindi Medium Polynomials Solutions 2.2 3.5
NCERT Maths Class 9 Hindi Medium Polynomials Solutions 2.2 3.6
NCERT Solutions For Maths Class 9 Polynomials Hindi Medium 2.2 3.7

Ex 2.2 Class 9 गणित Q4. निम्नलिखित स्थितियों में से प्रत्येक स्थिति मेंबहुपद का शुन्यक ज्ञात कीजिए :
(i) P(x) = x + 5
(ii) P(x) = x – 5
(iii) Px) = 2x + 5
(iv) P(x) = 3x – 2
(v) P(x) = 3x
(vi) P(x) = ax, a ≠ 0
हल (i) :
(i)   P(x) = x + 5
⇒ x + 5 = 0
⇒ x = – 5
बहुपद का शुन्यक – 5 हैं |

 हल (ii) :
(ii) P(x) = x – 5
⇒ x – 5  = 0
⇒ x = 5
बहुपद का शुन्यक 5 है|

Maths NCERT Solutions Class 9 Polynomials Hindi Medium 2.2 4
बहुपद का शुन्यक \(\frac { -5 }{ 2 }\) है |

(iv)  P(x) = 3x – 2
3x – 2 = 0 ≠
NCERT Solutions for Class 9 Maths Chapter 2 (Hindi Medium) 2.2 4.1
बहुपद का शुन्यक \(\frac { 2 }{ 3 }\) है |
Maths NCERT Class 9 Solutions Polynomials Hindi Medium 2.2 4.2
Maths Class 9 NCERT Solutions Hindi Medium 2.2 4.3

प्रश्नावली 2.3

Ex 2.3 Class 9 गणित Q1. x3 + 3x2 + 3x + 1 को निम्नलिखित से भाग देने पर शेषफल ज्ञात कीजिए : 
(i) x + 1 
(ii) x – \(\frac { 1 }{ 2 }\)
(iii) x 
(iv) x + θ
(v) 5 + 2x 
हल : (i) x3 + 3x2 + 3x + 1 को x + 1 से भाग देने पर 
Polynomials Maths Solutions For Class 9 NCERT Hindi Medium 2.3 1
अत: भाग देने पर शेषफल 0 है|

Polynomials Solutions For Maths NCERT Class 9 Hindi Medium 2.3 1.1

हल : (iii)  x3 + 3x2 + 3x + 1 को x से भाग देने पर       
Class 9 NCERT Maths Polynomials Solutions Hindi Medium 2.3 1.2
अत: भाग देने पर शेषफल 1 है|

हल : (iv) x3 + 3x2 + 3x + 1 को x + π से भाग देने पर 
NCERT Solutions For Class 9 Maths Polynomials PDF Hindi Medium 2.3 1.3
अत: भाग देने पर शेषफल – π3 + 3π2 – 3π + 1 है|
हल : (v) x3 + 3x2 + 3x + 1 को 5 + 2x से भाग देने पर
NCERT Solutions for Class 9 Maths Chapter 2 (Hindi Medium) 2.3 1.4
Class 9th Maths NCERT Polynomials Solutions Hindi Medium 2.3 1.5
NCERT Maths Book Class 9 Polynomials Solutions Hindi Medium 2.3 1.6

Ex 2.3 Class 9 गणित Q2. x3  ax2 + 6x  a  को x  a से भाग देने पर शेषफल ज्ञात कीजिए |
हल : p(x) = x3 – ax2 + 6x – a  और g(x) = x – a है |
g(x) = x – a का शुन्यक
अत:  x – a = 0
x = a
अत: शेषफल प्रमेय से
p(x) को x – a से भाग देने पर शेषफल प्रमेय द्वारा शेषफल p(a) प्राप्त होगा |
इसलिए, p(a) = (a)3 – a(a)2 + 6(a) – a
= a3 – a3 + 6a – a = 5a
अत: शेषफल 5a है |

Class 9 NCERT Solutions Maths Polynomials Hindi Medium 2.3 3

प्रश्नावली 2.4

Ex 2.4 Class 9 गणित Q1. बताइए कि निम्नलिखित बहुपदों में से किस बहुपद का एक गुणनखंड x + 1 है|
(i) x3 + x2 + x + 1
(ii) x4 + x3 + x2 + x + 1
(iii) x4 + 3x3 + 3x2 + x + 1
(iv) x3 – x3 – (2 + √2)x + √2
हल : (i) p(x) = x3 + x2 + x + 1
माना g(x) = x + 1 = 0
⇒ x = – 1
अब गुणनखण्ड प्रमेय के प्रयोग से
p(x) = 0 यदि x = -1 p(x) का शुन्यक है |
अत: p(x) में x = -1 रखने पर
p(x) = x3 + x2 + x + 1
p(-1) = (-1)3 + (-1)2 + (-1) + 1
= – 1 + 1 – 1 + 1 = 0
चूँकि p(-1) = 0 इसलिए -1 p(x) का शुन्यक है और x + 1 p(x) का एक गुणनखंड है |

हल : (ii) p(x) = x4 + x3 + x2 + x + 1
माना g(x) = x + 1 = 0
⇒ x = – 1
अब गुणनखण्ड प्रमेय के प्रयोग से
p(x) = 0 यदि x = -1 p(x) का शुन्यक है |
अत: p(x) में x = -1 रखने पर
p(x) = x4 + x3 + x2 + x + 1
p(-1) = (-1)4 + (-1)3 + (-1)2 + (-1) + 1
= 1 – 1 + 1 – 1 + 1 = 1
चूँकि p(-1) = 1 इसलिए -1 p(x) का शुन्यक नहीं है इसलिए गुणनखंड प्रमेय से x + 1 p(x) का एक गुणनखंड नहीं है |

हल : (iii) p(x) = x4 + 3x3 + 3x2 + x + 1
माना g(x) = x + 1 = 0
⇒ x = – 1
अब गुणनखण्ड प्रमेय के प्रयोग से
p(x) = 0 यदि x = -1 p(x) का शुन्यक है |
अत: p(x) में x = -1 रखने पर
p(x) = x4 + 3x3 + 3x2 + x + 1
p(-1) = (-1)4 + 3(-1)3 + 3(-1)2 + (-1) + 1
= 1 – 3 + 3 – 1 + 1 = 1
चूँकि p(-1) = 1 इसलिए -1 p(x) का शुन्यक नहीं है अत: गुणनखंड प्रमेय से x + 1 p(x) का एक गुणनखंड नहीं है |
माना g(x) = x + 1 = 0
⇒ x = – 1
अब गुणनखण्ड प्रमेय के प्रयोग से
p(x) = 0 यदि x = -1 p(x) का शुन्यक है |
अत: p(x) में x = -1 रखने पर
इसलिए -1 p(x) का शुन्यक नहीं है अत: गुणनखंड प्रमेय से x + 1 p(x) का एक गुणनखंड नहीं है |
NCERT Solutions for Class 9 Maths Chapter 2 (Hindi Medium) 2.4 1
9th Class Maths NCERT Polynomials Hindi Medium Solutions 2.4 1.1

Ex 2.4 Class 9 गणित Q2. गुणनखंड प्रमेय लागु करके बताइए कि निम्नलिखित स्थितियों में से प्रत्येक स्थिति में g(x), p(x) का एक गुणनखंड है या नहीं :
(i) p(x) = 2x3 + x2 – 2x – 1, g(x) = x + 1
(ii) p(x) = x3 + 3x2 + 3x + 1, g(x) = x + 2
(iii) p(x) = x3 – 4x2 + x + 6, g(x) = x – 3
हल : (i) p(x) = 2x3 + x2 – 2x – 1, g(x) = x + 1
g(x) का शुन्यक
⇒  x + 1 = 0
अत: x = – 1
गुणनखंड प्रमेय लागु करने पर यदि p(-1) = 0, तो गुणनखंड है अथवा नहीं |
अत: p(x) = 2x3 + x2 – 2x – 1  दिया है |
अब, p(-1) = 2(-1)3 + (-1)2 – 2(-1) – 1
= 2 (-1) + 1 + 2 – 1 = – 2 + 1 + 2 – 1 = 0
चूँकि p(-1) = 0 है इसलिए -1 p(x) का एक शुन्यक है अत: गुणनखंड प्रमेय से x + 1 p(x) का एक गुणनखंड है |

हल : (ii) p(x) = x3 + 3x2 + 3x + 1, g(x) = x + 2
g(x) का शुन्यक
⇒ x + 2 = 0
अत: x = – 2
गुणनखंड प्रमेय लागु करने पर यदि p(-2) = 0, तो गुणनखंड है अथवा नहीं |
अत: p(x) = x3 + 3x2 + 3x + 1  दिया है |
अब, p(-2) = (-2)3 + 3(-2)2 + 3(-2) + 1
= -8 + 12 – 6 + 1 = 13 – 14 = – 1
चूँकि p(-2) = – 1 है इसलिए -2 p(x) का एक शुन्यक नहीं है अत: गुणनखंड प्रमेय से x + 2 p(x) का एक गुणनखंड भी नहीं है |

हल : (iii) p(x) = x3 – 4x2 + x + 6, g(x) = x – 3
g(x) का शुन्यक
⇒ x – 3 = 0
अत: x = 3
गुणनखंड प्रमेय लागु करने पर यदि p(3) = 0, तो गुणनखंड है अथवा नहीं |
अत: p(x) = x3 – 4x2 + x + 6  दिया है |
अब, p(3) = (3)3 – 4(3)2 + 3 + 6
= 27 – 36 + 3 + 6 = 36 – 36 = 0
चूँकि p(3) = 0 है इसलिए 3 p(x) का एक शुन्यक है अत: गुणनखंड प्रमेय से x – 3 p(x) का एक गुणनखंड है |

Ex 2.4 Class 9 गणित Q3. k का मान ज्ञात कीजिए जबकि निम्नलिखित स्थितियों में से प्रत्येक स्थिति में (x – 1), p(x) का एक गुणनखंड हो :
(i) p(x) = x2 + x + k
(ii) p(x) = 2x2 + kx + √2
(iii) p(x) = kx2 – √2x + 1
(iv) p(x) = kx2 – 3x + k
हल : (i) p(x) = x2 + x + k
x – 1 p(x) का एक गुणनखंड है |
इसलिए x – 1 = 0 => x = 1
अत: 1 p(x) का शुन्यक है |
इसलिए p(1) = 0
अब p(x) = x2 + x + k = 0
p(1) = (1)2 + (1) + k = 0
1 + 1 + k = 0
2 + k = 0
k = – 2

हल : (ii) p(x) = 2x2 + kx + √2
चूँकि x – 1 p(x) का एक गुणनखंड है|
इसलिए x – 1 = 0
⇒ x = 1
अत: 1 p(x) का शुन्यक है |
इसलिए p(1) = 0
अब p(x) = 2x2 + kx + √2 = 0
p(1) = 2(1)2 + k(1) + √2  = 0
2 + k + √2 = 0
k = – 2 – √2
k = – (2 + √2)

हल : (iii) p(x) = kx2 – √2x + 1
चूँकि x – 1 p(x) का एक गुणनखंड है |
इसलिए x – 1 = 0 => x = 1
अत: 1 p(x) का शुन्यक है |
इसलिए p(1) = 0
अब p(x) = kx2 – √2x + 1 = 0
p(1) = k(1)2 – √2(1) + 1 = 0
k – √2 + 1 = 0
k = √2 – 1

हल : (iv) p(x) = kx2 – 3x + k
चूँकि x – 1 p(x) का एक गुणनखंड है |
इसलिए x – 1 = 0
⇒ x = 1
अत: 1 p(x) का शुन्यक है |
इसलिए p(1) = 0
अब p(x) = kx2 – 3x + k = 0
p(1) = k(1)2 – 3(1) + k = 0
k – 3 + k = 0
2k – 3 = 0
2k = 3
k = 3/2

Ex 2.4 Class 9 गणित Q4. गुणनखंड ज्ञात कीजिए :
(i) 12x2 – 7x + 1
(ii) 2x2 + 7x + 3
(iii) 6x2 + 5x – 6
(iv) 3x2 – x – 4
हल : (i) 12x2 – 7x + 1
⇒ 12x2 – 3x – 4x + 1
⇒ 3x(4x – 1) – 1(4x – 1)
⇒ (4x – 1) (3x – 1)

हल : (ii) 2x2 + 7x + 3
⇒ 2x2 + 6x + x + 3
⇒ 2x(x + 3) + 1(x + 3)
⇒ (x + 3) (2x + 1)

हल :  (iii) 6x2 + 5x – 6
⇒ 6x2 + 9x – 4x – 6
⇒ 3x(2x + 3) – 2(2x + 3)
⇒ (2x + 3) (3x – 2)

हल : (iv) 3x2 – x – 4
⇒ 3x2 – 4x + 3x – 4
⇒ x(3x – 4) + 1(3x – 4)
⇒ (3x – 4) (x + 1)

Ex 2.4 Class 9 गणित Q5. गुणनखंड ज्ञात कीजिए :
(i) x3 – 2x2 – x + 2
(ii) x3 – 3x2 – 9x – 5
(iii) x3 + 13x2 + 32x + 20
(iv) 2y3 + y2 – 2y – 1
हल : (i) x3 – 2x2 – x + 2
बहुपद का संभावित शुन्यक हैं – ±1 और ±2
अत: बहुपद x3 – 2x2 – x + 2 में x = 1 रखने पर
p(x) = (1)3 – 2(1)2 – (1) + 2
=  1 – 2 – 1 + 2 =  0
चूँकि p(x) = 0 है, अत: 1 p(x) का शुन्यक है इसलिए x – 1 p(x) का एक गुणनखंड है |

पहली विधि : x – 1 से x3 – 2x2 – x + 2 में भाग देने पर
CBSE Class 9 Maths Polynomials Hindi Medium Solutions 2.4 5
अत: x3 – 2x2 – x + 2 = (x – 1) (x2 – x – 2) [चूँकि p(x) = g(x) × q(x) ]
= (x – 1) (x– 2x + x – 2)
= (x – 1) [x(x – 2) + 1(x – 2)]
= (x – 1) (x – 2) (x + 1)

नोट: चूँकि यह त्रिघात बहुपद है इसलिए इसके तीन शुन्यक होंगे और तीन गुणनखंड होंगे |

दूसरी विधि : हम यहाँ पर x – 1 से भाग की लंबी प्रक्रिया न अपनाकर गुणनखंड विधि से अन्य गुणनखंड प्राप्त कर सकते हैं | चूँकि एक गुणनखंड x – 1 प्राप्त है|
x3 – 2x2 – x + 2 = x2(x -1) – x2 – x + 2
= x2(x -1) – x(x – 1) – 2x + 2
= x2(x -1) – x(x – 1) – 2(x – 1)
= (x – 1) (x2 – x – 2)
= (x – 1) (x– 2x + x – 2)
= (x – 1) [x(x – 2) + 1(x – 2)]
= (x – 1) (x – 2) (x + 1)

तीसरी विधि : हमें बहुपद का संभावित शुन्यक ±1 और ±2 ज्ञात है :
p(x) में x = 1, – 1, 2 और – 2 रखने पर
p(1) = 0 है | अत: x – 1 एक गुणनखंड है |

अब p(-1) = x3 – 2x2 – x + 2
= (-1)3 – 2(-1)2 -(-1) + 2
= -1 – 2 + 1 + 2 = 0
अत: p(-1) = 0 है अत: x + 1 एक गुणनखंड है |

अब p(2) = x3 – 2x2 – x + 2
= (2)3 – 2(2)2 -(2) + 2
= 8 – 8 – 2 + 2
= 0
p(2) = 0 है अत: x – 2 p(x) का एक गुणनखंड है |

अब p(-2) = x3 – 2x2 – x + 2
= (-2)3 – 2(-2)2 -(-2) + 2
= -8 – 8 + 2 + 2
= -16 + 4 = -12
p(-2) ≠ 0 अत: – 2 p(x) का शुन्यक नहीं है |
अत:  x3 – 2x2 – x + 2 के गुणनखंड है (x – 1) (x + 1) (x – 2)

हल : (ii) x3 – 3x2 – 9x – 5
बहुपद का संभावित शुन्यक ± 1 और ±5 है |
बहुपद में x = -1 रखने पर
p(-1) = x3 – 3x2 – 9x – 5
= (-1)3 – 3(-1)2 – 9(-1) – 5
= -1 – 3 + 9 – 5 = 9 – 9 = 0
अत: x = -1 p(x) का शुन्यक है इसलिए x + 1 एक गुणनखंड है |
x3 – 3x2 – 9x – 5 = x2(x + 1) – 4x2 – 9x – 5
= x2(x + 1) – 4x(x + 1) – 5x – 5
= x2(x + 1) – 4x(x + 1) – 5(x + 1)
= (x + 1) (x2 – 4x – 5)
= (x + 1) (x2 – 5x + x – 5)
= (x + 1) [x(x – 5) +1(x – 5)]
= (x + 1) (x – 5) (x + 1)
अत: त्रिघात बहुपद के गुणनखंड (x + 1), (x – 5) और (x + 1) है |

हल : (iii) x3 + 13x2 + 32x + 20
बहुपद का संभावित शुन्यक ±1, ±2, ±4, ±5, ±10 और ±20 हैं |
बहुपद में x = – 1 रखने पर
p(x) = x3 + 13x2 + 32x + 20
= (-1)3 + 13(-1)2 + 32(-1) + 20
= -1 + 13 – 32 + 20 = 33 – 33 = 0
चूँकि p(-1) = 0 है अत: x + 1 बहुपद का एक गुणनखंड है |
x3 + 13x2 + 32x + 20 = x2(x + 1) + 12x2 + 32x + 20
= x2(x + 1) + 12x(x + 1) + 20x + 20
= x2(x + 1) + 12x(x + 1) + 20(x + 1)
= (x + 1) (x2 + 12x + 20)
= (x + 1) (x2 + 10x + 2x + 20)
= (x + 1) [(x(x + 10) + 2(x + 10)]
= (x + 1) (x + 10) (x + 2)
अत: त्रिघात बहुपद के गुणनखंड (x + 1), (x + 10) और (x + 2) है|

हल : (iv) 2y3 + y2 – 2y – 1
= y2(2y + 1) -1(2y + 1)
= (y2 – 1) (2y + 1)
= (y + 1) ( y – 1) (2y + 1)
बहुपद के गुणनखंड (y + 1), ( y – 1) और (2y + 1)हैं |

उपयोगी बीजगणितीय सर्वसमिकाएँ: 

  1. (x + y)2 = x2 + 2xy + y2
  2. (x – y)2 = x2 – 2xy + y2
  3. x2 – y2 = (x + y) (x – y)
  4. (x + a) (x + b) = x2 + (a + b)x + ab
  5. (x + y)3 = x3 + 3x2y + 3xy2 + y3
  6. (x – y)3 = x3 – 3x2y + 3xy2 – y3
  7. x3 + y3 = (x + y) (x2 – xy + y2)
  8. x3 – y3 = (x – y) (x2 + xy + y2)
  9. (x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2zx
  10. x3 + y3 + z– 3xyz = ( x + y + z) (x2 + y2 + z2 – xy – yz – zx)

प्रश्नावली 2.5

Ex 2.5 Class 9 गणित Q1. उपयुक्त सर्वसमिकाओं को प्रयोग करके निम्नलिखित गुणनफल ज्ञात कीजिए:
(i) (x + 4) (x + 10)
(ii) (x + 8) (x – 10)
(iii) (3x + 4) (3x – 5)
NCERT Solutions for Class 9 Maths Chapter 2 (Hindi Medium) 2.5 1
(v) (3 – 2x) (3 + 2x)
हल: 
(i) (x + 4) (x + 10) 
सर्वसमिका  (x + a) (x + b) = x2 + (a + b)x + ab का प्रयोग करने पर
(x + 4) (x + 10) = x2 + (4 + 10)x + (4)(10)
= x2 + 14x + 40

(ii) (x + 8) (x – 10) 
सर्वसमिका  (x + a) (x + b) = x2 + (a + b)x + ab का प्रयोग करने पर
(x + 8) (x – 10) = x2 + [8 + (-10)]x + (8)(-10)
= x2 – 2x – 80

(iii) (3x + 4) (3x – 5)
सर्वसमिका  (x + a) (x + b) = x2 + (a + b)x + ab का प्रयोग करने पर
(3x + 4) (3x – 5) = (3x)2 + [4 + (-5)]3x + (4)(-5)
= 9x2 – 3x – 20
NCERT Solutions for Class 9 Maths Chapter 2 (Hindi Medium) 2.5 1.1
सर्वसमिका (x + y) (x – y) = x2 – y2 का प्रयोग करने पर
NCERT Solutions for Class 9 Maths Chapter 2 (Hindi Medium) 2.5 1.2

(v) (3 – 2x) (3 + 2x)
सर्वसमिका (x + y) (x – y) = x2 – y2 का प्रयोग करने पर  ​
(3 – 2x) (3 + 2x) = (3)2 – (2x)2
= 9 – 4x2

Ex 2.5 Class 9 गणित Q2. सीधे गुना किये बिना निम्नलिखित गुणनफलों के मान ज्ञात कीजिए :
(i) 103 × 107
(ii) 95 × 96
(iii) 104 × 96
हल:
(i) 103 × 107 = (100 + 3) (100 + 7)
सर्वसमिका  (x + a) (x + b) = x2 + (a + b)x + ab का प्रयोग करने पर
(100 + 3) (100 + 7) = (100)2​ + (3 + 7)100 + 3×7
=10000 + 1000 + 21 = 11021

(ii) 95 × 96 = (90 + 5) (90 + 6)
सर्वसमिका  (x + a) (x + b) = x2 + (a + b)x + ab का प्रयोग करने पर
(90 + 5) (90 + 6) = (90)2​ + (5 + 6)90 + 5×6
= 8100 + 990 + 30 = 9120

(iii)  104 × 96 = (100 + 4) (100 – 4)
सर्वसमिका (x + y) (x – y) = x2 – y2 का प्रयोग करने पर  ​
(100)2 – (4)2
= 10000 – 16 = 9984

Ex 2.5 Class 9 गणित 3. उपयुक्त सर्वसमिकाओं को प्रयोग करके निम्नलिखित का गुणनखंड कीजिए:
(i) 9x2 + 6xy + y2 
(ii) 4y2 – 4y + 1
NCERT Solutions for Class 9 Maths Chapter 2 (Hindi Medium) 2.5 3.1
हल:
(i) 9x2 + 6xy + y2 
= (3x)2 + 2.3x.y + (y)2     [ ∵ x2 + 2xy + y2 = (x + y)2]
∴ = (3x + y)2
=  (3x + y)  (3x + y)

(ii) 4y2 – 4y + 1 
= (2y)2 – 2.2y.1 + (1)2     [ ∵ x2 – 2xy + y2 = (x – y)2]
∴ = (2y – 1)2
=  (2y – 1)  (2y – 1)
NCERT Solutions for Class 9 Maths Chapter 2 (Hindi Medium) 2.5 3
NCERT Solutions For Class 9 Maths Polynomials Hindi Medium 2.5 3.2
[ ∵ x2 – y2 = (x + y) (x – y) ​]

Ex 2.5 Class 9 गणित Q4. उपयुक्त सर्वसमिकाओं को प्रयोग करके निम्नलिखित में से प्रत्येक का प्रसार कीजिए:
(i) (x + 2y + 4z)2 
(ii) (2x – y + z)2 
(iii) (–2x + 3y + 2z)2
(iv) (3a – 7b – c)2 
(v) (–2x + 5y – 3z)2
हल:
(i) (x + 2y + 4z)2  
यहाँ माना कि a = x, b = 2y, c = 4z और a, b तथा c का मान सर्वसमिका
(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca में रखने पर
∴ (x + 2y + 4z)2 = (x)2 + (2y)2 + (4z)2 + 2(x)(2y) + 2(2y)(4z) + 2(4z)(x)
= x2 + 4y2 + 16z2 + 4xy + 16yz + 8zx

(ii) (2x – y + z)2 
यहाँ माना कि a = 2x, b = – y, c = z और a, b तथा c का मान सर्वसमिका
(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca में रखने पर
∴ (2x – y + z)2 = (2x)2 + (- y)2 + (z)2 + 2(2x)(- y) + 2(- y)(z) + 2(z)(2x)
= 4x2 + y2 + z2 – 4xy – 2yz + 4zx

(iii) (–2x + 3y + 2z)2
यहाँ माना कि a = – 2x, b = 3y, c = 2z और a, b तथा c का मान सर्वसमिका
(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca में रखने पर
∴ (-2x + 3y + 2z)2
= (-2x)2 + (3y)2 + (2z)2 + 2(-2x)(3y) + 2(3y)(2z) + 2(2z)(-2x)
= 4x2 + 9y2 + 4z2 – 12xy  + 12yz – 8zx

(iv) (3a – 7b – c)2 
यहाँ माना कि x = 3a, y = -7b, z = -c और x, y तथा z का मान सर्वसमिका
(x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2zxमें रखने पर
∴ (3a – 7b – c)2
= (3a)2 + (-7b)2 + (-c)2 + 2(3a)(-7b) + 2(-7b)(-c) + 2(-c)(3a)
= 9a2 + 49b2 + c– 42ab  + 14bc – 6ac

(v) (-2x + 5y – 3z)2
यहाँ माना कि a = – 2x, b = 5y, c = -3z और a, b तथा c का मान सर्वसमिका
(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca में रखने पर
∴ (-2x + 5y – 3z)2
= (-2x)2 + (5y)2 + (-3z)2 + 2(-2x)(5y) + 2(5y)(-3z) + 2(-3z)(-2x)
= 4x2 + 25y2 + 9z2 – 20xy  – 30yz + 12zx

Ex 2.5 Class 9 गणित Q5. गुणनखंड कीजिए:
(i) 4x2 + 9y2 + 16z2 + 12xy – 24yz – 16xz

हल:
(i) 4x2 + 9y2 + 16z2 + 12xy – 24yz – 16xz
= (2x)2 + (3y)2 + (4z)2 + 2(2x)(3y) + 2(3y)(4z) + 2(4z)(2x)
[∵ a2 + b2 + c2 + 2ab + 2bc + 2ca = (a + b + c)2 ]
= (2x + 3y + 4z)2
= (2x + 3y + 4z) (2x + 3y + 4z)
NCERT Solutions for Class 9 Maths Chapter 2 (Hindi Medium) 2.5 5
Maths NCERT Solutions Class 9 Polynomials Hindi Medium 2.5 5.1

Ex 2.5 Class 9 गणित Q6. निम्नलिखित घनों को विस्तारित रूप में लिखिए :
(i) (2x + 1)3 

(ii) (2a – 3b)3
Class 9 Maths NCERT Polynomials Solutions Hindi Medium 2.5 6
NCERT Maths Solutions For Class 9 Polynomials Hindi Medium 2.5 6.1
हल:
(i) (2x + 1)3 
[सर्वसमिका के प्रयोग से (a + b)3 = a3 + 3a2b + 3ab2 + b3]
(2x + 1)3 = (2x)3 + 3 (2x)(1) + 3 (2x) (1)2 + (1)3
= 8x3 + 12x2 + 6x + 1

(ii) (2a – 3b)3
[सर्वसमिका के प्रयोग से (x – y)3 = x3 – 3x2y + 3xy2 – y3]
(2a – 3b)3 = (2a)3 – 3 (2a)(3b) + 3(2a) (3b)2 – (3b)3
= 8a3 – 36a2b + 54ab2 – 27b3
NCERT Solutions for Class 9 Maths Chapter 2 (Hindi Medium) 2.5 6.2
[सर्वसमिका के प्रयोग से (a + b)3 = a3 + 3a2b + 3ab2 + b3]
NCERT Class 9 Maths Hindi Medium Polynomials Solutions 2.5 6.3
NCERT Maths Class 9 Hindi Medium Polynomials Solutions 2.5 6.4
[सर्वसमिका के प्रयोग से (a – b)3 = a3 – 3a2b + 3ab2 – b3]
NCERT Solutions For Maths Class 9 Polynomials Hindi Medium 2.5 6.5

Ex 2.5 Class 9 गणित Q7. उपयुक्त सर्वसमिका का प्रयोग कर निम्नलिखित का मान ज्ञात कीजिए :
(i) (99)3 
(ii) (102)3 
(iii) (998)3
हल : 
(i) (99)3 
= (100 – 1)3
[सर्वसमिका के प्रयोग से (a – b)3 = a3 – 3a2b + 3ab2 – b3]
(100 – 1)= (100)3 – 3(100)2(1) + 3(100)(1)2 – (1)3
= 1000000 – 30000 + 300 – 1 = 1000300 – 30001 = 970299

(ii) (102)3 
= (100 + 2)3
[सर्वसमिका के प्रयोग से (a + b)3 = a3 + 3a2b + 3ab2 + b3]
(100 + 2)= (100)3 + 3 (100)(2)+ 3 (100) (2)2 + (2)3
= 1000000 + 60000 + 1200 + 8 = 1061208

(iii) (998)3
= (1000 – 2)3
[सर्वसमिका के प्रयोग से (a – b)3 = a3 – 3a2b + 3ab2 – b3]
(1000 – 2)= (1000)3 – 3 (1000)(2)+ 3(1000) (2)2 – (2)3
= 1000000000 – 6000000 + 12000 – 8
= 1000012000 – 6000008
= 994011992

Ex 2.5 Class 9 गणित Q8. निम्नलिखित का गुणनखंड कीजिए :
(i) 8a3 + b3 + 12a2b + 6ab2
(ii) 8a2 – b2 – 12a2b + 6ab2
(iii) 27 – 125a3 – 135a + 225a2
(iv) 64a3 – 27b3 – 144a2b + 108ab2
NCERT Solutions for Class 9 Maths Chapter 2 (Hindi Medium) 2.5 8.1
हल:
(i) 8a3 + b3 + 12a2b + 6ab2 
= (2a)3 +(b)3 + 3(2a)2(b) + 3(2a)(b)2
[सर्वसमिका के प्रयोग से  x3  + y+ 3x2y + 3xy2 = (x + y)3 ]
= (2a)3 +(b)3 + 3(2a)2(b) + 3(2a)(b)2 = (2a + b)3
= (2a + b)(2a + b)(2a + b)

(ii) 8a2 – b2 – 12a2b + 6ab2
= (2a)3 – (b)3 – 3(2a)2(b) + 3(2a)(b)2
[सर्वसमिका के प्रयोग से  x3 – y– 3x2y + 3xy2 = (x – y)3 ]
= (2a)3 – (b)3 – 3(2a)2(b) + 3(2a)(b)2 = (2a – b)3
= (2a – b)(2a – b)(2a – b)

(iii) 27 – 125a3 – 135a + 225a2
= (3)3 – (5a)3 – 3(3)2(5a) + 3(3)(5a)2
[सर्वसमिका के प्रयोग से  x3 – y– 3x2y + 3xy2 = (x – y)3 ]
= (3)3 – (5a)3 – 3(3)2(5a) + 3(3)(5a)2= (3 – 5a)3
= (3 – 5a)(3 – 5a)(3 – 5a)

(iv) 64a– 27b3 – 144a2b + 108ab2  
= (4a)3 – (3b)3 – 3(4a)2(3b) + 3(4a)(3b)2
[सर्वसमिका के प्रयोग से x3 – y– 3x2y + 3xy2 = (x – y)3 ]
= (4a)3 – (3b)3 – 3(4a)2(3b) + 3(4a)(3b)2 = (4a – 3b)3
= (4a – 3b)(4a – 3b)(4a – 3b)
[सर्वसमिका के प्रयोग से x3 – y– 3x2y + 3xy2 = (x – y)3 ]
Maths NCERT Solutions Class 9 Polynomials Hindi Medium 2.5 8.2

Ex 2.5 Class 9 गणित Q9. सत्यापित कीजिए :
(i) x3 + y3 = (x + y) (x2 – xy + y2)
हल :
RHS = (x + y) (x2 – xy + y2)
= x(x2 – xy + y2) + y (x2 – xy + y2)
= x3 – x2y + xy2 + x2y – xy2 + y3
Maths NCERT Class 9 Solutions Polynomials Hindi Medium 2.5 8.2
= x3 + y3
∵ LHS = RHS सत्यापित

(ii) x3 – y3 = (x – y) (x2 + xy + y2)
हल :
RHS = (x – y) (x2 + xy + y2)
x(x2 + xy + y2) – y (x2 + xy + y2)
= x3 + x2y + xy2 – x2y – xy2 – y3
Maths Class 9 NCERT Solutions Hindi Medium 2.5 8.4
= x3 – y3
∵ LHS = RHS सत्यापित |

Ex 2.5 Class 9 गणित Q10. निम्नलिखित में से प्रत्येक का गुणनखंड ज्ञात कीजिए:
(i) 27y3 + 125z3 
(ii) 64m3 – 343n3
हल : 
(i) 27y3 + 125z3 
= (3y)3 + (5z)3
[सर्वसमिका के प्रयोग से x3 + y3 = (x + y) (x2 – xy + y2) ]
(3y)3 + (5z)3​ = (3y + 5y) [(3y)2 – (3y)(5z) + (5z)2]
= (3y + 5y) (9y2 – 15yz + 25z2)

(ii) 64m3 – 343n3
हल : 
(ii) 64m3 – 343n3
= (4m)3 – (7n)3
[सर्वसमिका के प्रयोग से x3 – y3 = (x – y) (x2 + xy + y2) ]
(4m)3 – (7n)3​ = (4m – 7n) [(4m)2 + (4m)(7n) + (7n)2]
= (4m – 7n) (16m2 + 28mn + 49n​2)

Ex 2.5 Class 9 गणित Q11. गुणनखण्ड ज्ञात कीजिए : 27x3 + y3 + z3 – 9xyz
हल : 
= (3x)3 + (y)3 + (z)– 9xyz
∵ x+ y3 + z3 – 3xyz =  (x + y + z) (x2 + y2 + z​2 – xy – yz – zx)
सर्वसमिका के प्रयोग से :
= (3x + y + z) ((3x)2 + (y)2 + (z)2 – (3x)(y) – (y)(z) – (z)(3x))
= (3x + y + z) (9x2 + y2 + z2 – 3xy – yz – 3zx)

Ex 2.5 Class 9 गणित Q12. सत्यापित कीजिए: 
x+ y3 + z3 – 3xyz = \(\frac { 1 }{ 2 }\) (x + y + z) [(x – y)2 + (y – z)2 + (z – x)2]
हल : 
LHS =  \(\frac { 1 }{ 2 }\) (x + y + z) [x2 – 2xy + y2 + y2 – 2yz + z2 + z2 – 2xz + x2]
= \(\frac { 1 }{ 2 }\) (x + y + z) (2x+ 2y2 + 2z2 – 2xy – 2yz – 2xz)
= \(\frac { 1 }{ 2 }\) × 2(x + y + z) (x+ y2 + z2 – xy – yz – xz)
= (x + y + z)(x+ y2 + z2 – xy – yz – xz)
= x+ y3 + z3 – 3xyz [सर्वसमिका के प्रयोग से ]
LHS = RHS

Ex 2.5 Class 9 गणित Q13. यदि x + y + z = 0 हो, तो दिखाइए कि x3 + y3 + z3 = 3xyz है | ​
हल : x + y + z = 0 दिया है |
x3 + y3 + z3 – 3xyz = ( x + y + z) (x2 + y2 + z2 – xy – yz – zx)
= (0) (x2 + y2 + z2 – xy – yz – zx) = 0
अत: x3 + y3 + z3 – 3xyz = 0
या x3 + y3 + z3 = 3xyz सत्यापित

Ex 2.5 Class 9 गणित Q14. वास्तव में घनों का परिकलन किए बिना निम्नलिखित में से प्रत्येक का मान ज्ञात कीजिए : 
(i) (-12)3 + (7)3 + (5)3
(ii) (28)3 + (-15)3 + (-13)3
हल : (i) (-12)3 + (7)3 + (5)3

प्रश्न 13. में हमने एक सर्वसमीका प्राप्त किया था कि यदि x + y + z = 0 हो तो
x3 + y3 + z3 = 3xyz है |
अत: इस सर्वसमिका में x = -12, y = 7 और z = 5 रखने पर
चूँकि – 12 + 7 + 5 => -12 + 12 = 0
अत: x + y + z = 0 है |
अब, x3 + y3 + z3 = 3xyz  [x, y, और z का मान रखने पर ]
=> (-12)3 + (7)3 + (5)3 = 3 × (-12) × 7 × 5
= – 1260

हल : (ii) (28)3 + (–15)3 + (–13)3
28 + (-15) + (-13) = 28 – 28 = 0
चूँकि x + y + z = 0 है |
इसलिए x3 + y3 + z3 = 3xyz
अब, (28)3 + (–15)3 + (–13)3​ = 3 × 28 × (-15) × (-13)
= 133380

Ex 2.5 Class 9 गणित Q15. नीचे दिए गए आयातों, जिनमें उनके क्षेत्रफल दिए गए है, में से प्रत्येक की लंबाई और चौड़ाई के लिए संभव व्यंजक दीजिये | 
(i) क्षेत्रफल : 25a2 – 35a + 12 
(ii) क्षेत्रफल : 35y2 + 13y – 12 
हल : (i) क्षेत्रफल : 25a2 – 35a + 12
क्षेत्रफल = लंबाई × चौड़ाई
अत: 25a2 – 35a + 12 के दो गुणनखंड होंगे जिसमें एक लंबाई होगा और दूसरा चौड़ाई होगा |
गुणनखंड करने पर :
25a2 – 35a + 12 = 25a2 + 15a + 20a + 12
= 5a(5a + 3) + 4(5a + 3)
= (5a + 3) (5a + 4)
चूँकि (5a + 3) < (5a + 4) है |
अत: लंबाई = 5a + 4 और चौड़ाई = 5a + 3

हल : (ii) क्षेत्रफल : 35y2 + 13y – 12
गुणनखंड करने पर
35y2 + 13y – 12 ​= 35y2​ + 28y – 15y – 12
= 7y(5y + 4) – 3(5y + 4)
= (5y + 4) (7y – 3)
अत: लंबाई = 5y + 4 और चौड़ाई = 7y – 3

Ex 2.5 Class 9 गणित Q16. घनाभों (cuboids), जिनके आयतन नीचे दिए गए हैं कि, विमाओं के लिए संभव व्यंजक क्या हैं ? 
(i) आयतन : 3x3 – 12x 
(ii) आयतन : 12ky2 + 8ky – 20k 
हल : (i) आयतन : 3x3 – 12x
गुणनखंड करने पर
आयतन = 3x3 – 12x = 3x(x – 4)
चूँकि आयतन = L × B × H
अत: L = 3, B = x और H = x – 4

हल : (ii) आयतन : 12ky2 + 8ky – 20k
आयतन = 12ky2 + 8ky – 20k
= 4k (3y2 + 2y – 5)
= 4k (3y2 + 5y – 3y – 5)
= 4k [y (3y + 5) – 1(3y + 5)]
= 4k (3y + 5) (y – 1)
चूँकि आयतन = L × B × H
अत: L = 4k, B = (3y + 5) और H = (y – 1)

Hope given NCERT Solutions for Class 9 Maths Chapter 2 are helpful to complete your homework.

NCERT Solutions for Class 9 Maths Chapter 2 Polynomials (Hindi Medium) Read More »

NCERT Solutions for Class 10 English Footprints Without Feet Chapter 4 A Question of Trust

In this article, we have created the most comprehensive NCERT Solutions for Class 10 English Footprints Without Feet Chapter 4 A Question of Trust.

A Question of Trust NCERT Solutions for Class 10 English Footprints Without Feet Chapter 4

A Question of Trust NCERT Text Book Questions and Answers

A Question of Trust Read and Find out

Question 1.
What does Horace Danby like to collect?
Answer:
Horace Danby is a voracious reader. He loves to collect rare and expensive books.

Question 2.
Why does he steal every year?
Answer:
He steals every year to be able to make his both ends meet and to be able to purchase expensive books.

Question 3.
Who is speaking to Horace Danby?
Answer:
The smart woman in red dress is speaking to Horace Danby.

NCERT Solutions for Class 10 English Footprints Without Feet Chapter 4 A Question of Trust

Question 4.
Who is the real culprit in the story?
Answer:
The woman in the red is the real culprit in the story as she succeeds in befooling Horace Danby.

A Question of Trust Extra Questions and Answers

A Question of Trust Short Answer Questions

Question 1.
How were flowers hindering Horace in his work?
Answer:
Horace suffered from hay fever and was allergic to some flowers. On smelling the flowers, he started sneezing and was caught red-handed by another thief.

Question 2.
Why was it not difficult for Horace to open the safe?
Answer:
It was not difficult for Horace to open the safe because he was a perfect and experienced locksmith. He had collected all the information about the safe before entering the house.

Question 3.
What advice did the lady give Horace regarding his hay fever? Was she really interested in his health?
Answer:
The lady advised Horace that he could find a cure to the hay fever by trying to find which plant gave the disease. She was not interested in his disease or its cure, but she was rather making fun of him.

NCERT Solutions for Class 10 English Footprints Without Feet Chapter 4 A Question of Trust

Question 4.
Why did Horace Danby feel sure of his success in that year’s robbery?
Answer:
Horace Danby felt sure of his success in that year’s robbery too, because he had planned his work carefully. He studied every detail of the house. He had chosen an appropriate place and time for the robbery.

Question 5.
How did Horace manage the small dog when he attempted to rob the house at Shotover Grange?
Answer:
Horace Danby was an expert thief who planned his mission without any fault. When he tried to rob the house in Grange, he encountered a dog. But Horace Danby calmed the dog by calling him by his name.

Question 6.
What story did the lady tell Horace to get the jewels?
Answer:
The lady told Horace an interesting story. She told that her jewels were lying in the safe which she, needed at once. She also told that she had forgotten the numbers to open the safe.

Question 7.
Did Horace get the jewels from the Grange safe? If not, why did the police arrest him?
Answer:
Horace was not able to get any jewels though he stole them. The young lady in red befooled him. But the police arrested him due to his fingerprints on the Grange safe.

Question 8.
How can you say that Horace Danby was good and respectable but not completely honest? [Delhi 2019]
Answer:
Horace Danby was not a typical thief. He used to rob every year enough money to last for twelve months to buy books which he loved to read. He is described as a good and respectable person but not completely honest because he could not curb his habit of stealing a safe every year.

Question 9.
Why did Horace rob every year? Was he a typical thief? If so, why? In what way could Horace’s arrest have helped the lady?
Answer:
He robbed every year enough money to last for twelve months to buy books which he loved to read. No, he was not a typical thief because he used to steal only to buy interesting books.

NCERT Solutions for Class 10 English Footprints Without Feet Chapter 4 A Question of Trust

Question 10.
Did the young lady expect Horace to be caught after the theft?
Answer:
Yes, the young lady knew that Horace would be caught. As he forgot to put on his gloves. Naturally his finger prints would lead the police towards him. Horace’s arrest would not let anyone think that she was the thief. So she was to be benefitted by his arrest.

Question 11.
Was Horace a typical thief? Why/Why not?
Answer:
No, Horace was not a typical thief. He robbed only once in a year to have enough money to last for twelve months. He was fond of expensive books which he used to buy from the stolen money. Otherwise, he was considered as an honest and respectable person.

Question 12.
What do you think is the meaning of the phrase ‘honour among thieves’? Who lacked it?
Answer:
The phrase ‘honour among thieves’ means that even the thieves have some principles and they do not cheat each other. They trust other thieves and are honest in their dealings with each other. Obviously, it is the young lady in the Red, who lacked honour as she cheated and befooled another thief. She procured the booty whereas Horace went to the jail.

Question 13.
How did Horace get entry into the house?
Answer:
He was very keen in his observation. He was able to see that the housekeepef hung the keys to the kitchen door on a hook. He picked up the keys from there and opened the door.

Question 14.
How did Horace know about the safe behind the painting?
Answer:
A magazine article had described this house giving a plan of all the rooms and a picture of this room. Even the fact that the safe was hidden behind a picture was given there.

A Question of Trust Long Answer Questions

Question 1.
Horace was a thief who planned his work carefully. He was in a way a sucesssful thief. Should we call him a successful thief and appreciate his work? Why/Why not?
Answer:
Horace Danby was a good, honest citizen of about 50 years. He planned his work carefully. He was a meticulous planner. He used to observe and supervise the house to be burgled. He never acted in haste. He studied the map and other minute details of the house at Shotover Grange.

He had the details of electric wiring, dogs and servants of the house. He knew when it was the right time to strike. He did his work so well that there was no cause of his arrest. No doubt he was a successful thief but his act of theft can not be appreciated. Stealing is a vice which can not be appreciated. To fulfil our needs we should not resort to theft.

NCERT Solutions for Class 10 English Footprints Without Feet Chapter 4 A Question of Trust

Question 2.
Do you think Horace Danby was unfairly punished or that he deserved what he got? Do you agree that honesty in wrong acts is not desirable?
Answer:
The story, A Question of Trust’ is a story of distrust. The lady in the red is the real culprit. She was a very clever lady and was successful in befooling Horace. She made him believe that she was the mistress of the house and told him a story with conviction. Horace had taken off his gloves, because he thought that the wife of the owner was with him. That was the biggest mistake of his life.

Horace Danby left his fingerprints and was arrested. He was not punished unfairly. He was not innocent as he entered the house with the intention to rob the house. Honesty in wrong acts can not be justified. It is not desirable at all.

Question 3.
Do intentions justify actions? Would you, like Horace Danby, do something wrong if you thought your ends justify the means? Do you think that there are situations in which it is excusable to act less than honest?
Answer:
Yes, intentions justify the actions. If any wrong act is committed unintentionally, it can be excused. But if the wrong act is done intentionally it is not excusable. Horace Danby had the intention to rob the house. This is an intentional crime. He helped the house lady by opening the safe, he had good intentions but that too, for his own motive of being free. I would not indulge in any wrong act even if it is justified. Mere justifcation of the wrong act does not serve the purpose. Yes, there are situations in which it is excusable to act less than honest. But the case of Horace Danby does not fall in that category.

Question 4.
Did you begin to suspect, before the end of the story, that the lady was not the person Horace Danby took her to be? If so, at what point did you realise this, and how? How will you appreciate the act of the lady?
Answer:
Yes, it is very natural to suspect that the lady was not the owner of the house as she did not express surprise on seeing a burglar in her house rather she promised him that she would not hand him over to the police. Secondly, she even did not know the number of the safe. But Horace was too nervous to notice all these things.

The lady in red was very clever. She was successful in befooling Horace Danby. She was a good actress and acted so smartly that Horace Danby was trapped in her scheme. She lacked honour. Her act cannot be appreciated as she trapped another person for her greed and bad intentions.

NCERT Solutions for Class 10 English Footprints Without Feet Chapter 4 A Question of Trust

Question 5.
What are the subtle ways in which the lady manages to deceive Horace Danby into thinking she is the lady of the house? Why doesn’t Horace suspect that something is wrong? What are the negative aspects of her character?
Answer:
The lady is very smart and clever. She succeeds in befooling Horace Danby that she is the owner of the house. She is a well dressed, well planned and organised thief who drafted her trick so meticulously that a brilliant thief like Horace could not suspect her. The way she enters and talks to Horace Danby, he is unable to doubt her integrity. Even the dog did not bark on seeing her. The lady in red had some negative traits in her personality. She was not honest. She was a thief and befooled another thief. She lacked honesty, integrity and honour.

Question 6.
“Horace Danby was good and respectable – but not completely honest.” Why do you think this description is apt for Horace? Why can’t he be categorised as a typical thief? Should we call him a good human being?
Answer:
The statement is an apt statement for Danby. He was a respectable person, but he was a thief who in order to fulfil his desire used to rob once in a year. He had adopted a dishonest way to fulfil his desire. So he cannot be called an honest person.

Horace Danby is not a typical thief because he steals mainly to buy rare and expensive books. He planned his theft in such a manner that he could not be arrested so far. Being an introvert, he did not blurt about his theft to anybody. No, we can’t call him a good human being. He was a victim of the trick played on him. He is not completely honest. He entered the house with bad intention. He was a thief and a thief can not be called a good human being.

Question 7.
“Horace Danby was a meticulous planner but still he faltered.” Where did he go wrong and why? Negative values never pay in long run: Do you agree?
Answer:
Though Horace Danby was a brilliant thief, he was caught in the end. He faltered because he readily handed over jewellery to the so-called owner of the house. Horace Danby was befooled by the lady in red. She pretended to be the owner of the house and made him open the safe without gloves. Horace left his fingerprints. He failed in his plan and was caught for a crime that he did not commit.

Negative values never pay in the long run. Every criminal has his punishment. Sooner or later one is caught and punished. Horace Danby was no doubt a meticulous planner but he was on a wrong path. His intention had never been good. He planned to rob others and was ultimately paid for it.

Question 8.
The lady in red dress was a more professional thief than Horace Danby. Do you agree? Elaborate.
Answer:
The lady in red not only outsmarted Horace Danby rather she went one step forward to ensure that all the evidence went against Horace to establish him as the real culprit and the lady walked out freely and untraceably. The lady combined her female arrogance and confidence to prepare a perfect recipe to befool Horace as he could not suspect that she was not the real lady of house. The lady tricked Horace Danby to utilise his years of experience in theft for her own benefit and Horace on the basis of his fingerprints was put behind the bars.

NCERT Solutions for Class 10 English Footprints Without Feet Chapter 4 A Question of Trust

Question 9.
How did Horace Danby plan his robbery of shot over Grange?
Answer:
Horace Danby was not a typical thief. He planned his act of burglary meticulously. He always studied his target carefully. He planned his robbery of Shotover Grange carefully. He studied the situation . of rooms, electric fittings and other aspects well in advance. He collected details from different magazines and articles. He was aware about the safe which had jewels worth fifteen thousand pounds.

He knew the family was on vacation in London. He also knew that all the servants had gone for a movie. He knew the place of the keys too. When the right time came he struck his plan but outwitted by the lady in red. His plan failed and he was arrested

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