Important Questions with Answers

Q. What are some of the positive views on interviews?

Ans. Some people think that in its highest form an interview is a source of truth and an art in its practice. Many people consider it a supremely serviceable medium of communication. Our impressions of the people are created by their interviews. So interviewer holds a position of unprecedented power and influence.

Q. Why do most celebrity writers despise being interviewed?

Ans. Most celebrity writers despise being interviewed because they think it is an intrusion into their lives. They think that the interview somehow diminishes them. V.S. Naipaul feels that some people are wounded by interviews and lose a part of themselves. Rudyard Kipling said that interviewing was immoral. He called it cowardly or vile.

Q. What is the belief in some primitive cultures about being photographed?

Ans. Some primate cultures believed that if one takes a photographic portrait of somebody. Then one is stealing that person’s soul.

Q. What do you understand by the expression “thumbprints on his, windpipe” ?

Ans. The expression “thumbprints on his windpipe’ refers to the feeling of suffocation or strangulation.

Q. Who, in today’s world, is our chief source of information about personalities?

Ans. In today’s world, the interviewer” is our chief source of information about personalities.

Q. Do you think Umberto Eco likes being interviewed? Give reasons for your opinion.

Ans. Yes, we feel that Umberto Eco likes being interviewed. For, in all his answers, he speaks at great length while answering the questions of the interviewer. At no stage, it appears that he wants to get rid of the interviewer. This .is a clear indication that Umberto Eco likes being interviewed.

Q. How does Eco find the time to write so much?

Ans. Eco finds the time to write so much for two reasons. First, as he writes about the same things but in different manners. He has some philosophical interests and he pursues them in his academic work and also in his novels. Even his books for children are about non-violence and peace. The second thing is that he does not let go any time wasted. For example while he was waiting for his interviewer, he had written an article. He calls this time ‘interstice’ and makes full use of it.

Q. What was distinctive about Eco’s academic writing style?

Ans. The distinctive quality of Eco’s academic writing style is his love for narration. Even his dissertation for a doctoral degree was presented as a narrative. This makes even the serious things look interesting.

Q. Did Umberto Eco consider himself a novelist first or an academic scholar?

Ans. Umberto Eco considered himself an academic scholar first. He participates in academic conferences and not in meetings of Pen clubs and writers. He writes novels only when he is not having an academic pursuit at the time. He says “I am a professor who writes novels on Sundays”.

Q. What is the reason for the huge success of the novel. The Name of the Rose?

Ans. The reason for the huge success of the novel ‘The Name of the Rose’ is a mystery. However Eco feels that it might have been because of its timings. Had it been published ten years earlier or ten years after, it might not have succeeded so well, he feels.

Q. What are some of the positive views on interviews?

Ans. Following are some of the Positive views on interviews: (i) The interview has become a commonplam of journalism. (ii) In its highest form, an interview is a source of truth and, in practice, it is an art. (iii) An interview is a supremely serviceable medium of communication.

Q. What do most celebrated writers despise their being interviewed?

Ans. Most celebrated writers despise their being interviewed because of the following reasons: (i) They think that interview is an unjustified intrusion into their private lives. (ii) They feel that interview diminishes thern.

Q. What is the belief in some primitive cultures about being photographed?

Ans. The belief in some primitive cultures is it one takes a photo of somebody, then one is stealing that person’s soul.

Q. What is the chief source of information in today’s world about personalities?

Ans. In today’s world our chief source of information about personalities is interview. We get most vivid impressions of our contemporaries through interviews.

Q. When was the interview invented?

Ans. The interview was invented over 130 years ago.

Q. Why do the opinions of the interview vary considerably?

Ans. Thousands of famous persons have been interviewed over the years. Some of them have been repeated. So the opinions of the interview vary considerably.

Q. What does V.S. Naipaul feel about interviews?

Ans. V.S. Naipaul feels that some people are wounded by interviews and lose a part of themselves.

Q. Who is Lewis Carrol? What did he say about the horror of the interviewer?

Ans. Lewis Carrol is the creator of ‘Alice in Wonderland’. He was horrified of the interviewer and he never consented to be interviewed.

Q. Why did Rudyard Kipling refuse to be interviewed?

Ans. Rudyard Kipling refused to be interviewed because according to him it immoral. It was a crime. It was an offence of assault on his person. It was cowardly and vile. No respectable man would ask it.

Q. What do you understand by the expression ‘The interviewing ordeal’?

Ans. The expression ‘the interviewing ordeal’ implies that when person gives an interview he has to undergo a painful experience.

Q. Is Rudyard Kipling practically right in his assertion for interview?

Ans. No, Rudyard Kipling is not practically right in his assertion for interview. He perpetrated an assault of interview on Mark Twain only a few years before.

Q. Who is Umberto Eco? What does he do on Sundays ?

Ans. Umberto is a professor at the University of Bologna in Italy. He writes novels on Sundays.

Q. What works had already acquired a formidable reputation for Umberto Eco as a scholar before he began to write fiction?

Ans. His ideas on semiotics (the story of signs), literary interpretation, and medieval aesthetics had already acquired a profound reputation for Umberto Eco as a scholar before he began to write fiction.

Q. What do you understand by “The Name of the Rose’? What achievement did it bring to its author?

Ans. ‘The Name of the Rose’ is a novel writteri by Umberto Eco. It acquired the equivalent pr intellectual superstardom for its author.

Q. What did once David Lodge remark?

Ans. David Lodge, an English novelist and academic, once remarked, “I can’t understarid how one man can do all the things he (Eco) does.”

Q. What are interstices, according to Umberto Eco?

Ans. According to Umberto Eco, interstices are the empty spaces in the universe, in all the atoms and in our lives. If we eliminate these empty paces from the universe and from within the atoms, the universe will become as big as his fist.

Q. Which is, according to Mukund, is a marked departure from academic style?

Ans. According to Mukund, Eco’s style is a marked departure from academic style. His scholarly work has a certain playful and personal quality about it which is a marked departure from a regular academic style.

Q. Who was Roland Barthes? Why was he always frustrated?

Ans. Roland Barthes was a friend of Professor Umberto Eco. He was always frustrated that he was an essayist and not a novelist. He wanted to do creative writing one day or the other day.

Q. Why is Umberto Eco not satisfied to be only a novelist?

Ans. Umberto Eco is not satisfied to be only a novelist because he is a university professor. He participates in academic conferences. He identifies himself with the academic community.

Q. What genres are represented in the novel ‘The Name of the Rose’?

Ans. The novel ‘The Name of the Rose’ represents a detective yarn at one level, and metaphysics, theology and medieval history on the other level.

Q. Why are journalists and publishers puzzled?

Ans. Journalists and publishers are puzzled because they believe that people like trash and don’t like difficult reading experiences.

Q. Why was Umberto Eco given an advance for only 3,000 copies of his novel by his American publisher?

Ans. His American publisher gave Umberto Eco an advance for only 3,000 copies of his novel because she didn’t expect to sell more than 3,000 copies of his novel.

Important Question and Answers Poets and Pancakes

Q. What does the writer mean by ‘the fiery misery’ of those subjected to make-up’?

Ans. The make-up room of Gemini Studios looked like a hair cutting salon. It had around half a dozen mirrors with incandescent lights at all angles around them. The artists would feel the heat coming from these lights. Thus, the writer uses the term ‘fiery misery’ to denote the uncomfortable situation of those subjected to make-up.

Q. What is the example of national integration that the author refers to?

Ans. The division of the Gemini Studios’ makeup room was an example of national integration. According to the author, this is so because people from different regions and religions worked together in that department. The department had hierarchy system. The department was headed by a Bengali who was succeeded by a Maharashtrian. The other helpers included a Dharwar Kannadiga, an Andhra, a Madras Indian Christian, an AngloBurmese and the local Tamils.

Q. Why did the author appear to be doing nothing at the studios?

Ans. The job of the author was to cut newspaper clippings and file them. For the other employees, all he seemed to be doing is tearing newspapers, which according to them did not qualify as work. Therefore, they often considered him free and available for their miscellaneous work.

Q. Why was the office boy frustrated? Who did he show his anger on?

Ans. The office boy had joined the studio years ago in the hope of becoming an actor or a screenwriter, or a director, or a lyricist. The fact that he ended up becoming none of these left him frustrated. According to him, “great literary talent was being allowed to go waste in a department fit only for barbers and perverts”. He used to direct his anger at the author even though it was meant for Kothamangalam Subbu.

Q. Why was the legal adviser referred to as the opposite by others?

Ans. A lawyer used to be a part of the Story Department at the Gemini Studios. Though, as a legal adviser, he was supposed to be involved in legal matters, his cagey yet stupid idea led to the end of an actress’s career. Due to this, he was referred to as the illegal advisor, by the people.

Q. Name one example to show that Gemini Studios was influenced by the plays staged by MRA.

Ans. Madras and Tamil drama community included scenes of ‘sunset and sunrise in the manner of Jotham Valley’ in almost all of their plays. This shows how the plays, staged by MRA, influenced Gemini Studios.

Q. What caused the lack of communication between the Englishman and the people at Gemini Studios? Why is the Englishman’s visit referred to as unexplained mystery?

Ans. The accent of Stephen Spender, the Englishman, was the main cause of the lack of communication between him and the people at Gemini Studios. Apart from that, the people did not have any idea about what he was talking. The Englishman’s visit to the Gemini Studios is referred to as an unexplained mystery because no one could decipher his identity, whether he was a poet or an editor. Besides, when he spoke no one at the studio understood what he intended to say as his accent was beyond their comprehension.

Q. What do you understand about the author’s literary inclinations?

Ans. Though the author had a very tedious and unchallenging job at the studios, his interest in literature and writing is apparent in his willingness to participate in the short story contest organised by the British periodical, The Encounter. Moreover, the author appears to be a keen reader visiting libraries and buying books on wide-ranging topics whenever he could afford them. Besides, the narrative also establishes the fact that the author was one of the most knowledgeable persons in Gemini Studios. His idea about how prose writing was not meant for geniuses but for those with patience and perseverance, highlight his reflective and deep thoughts on literature and creative writing.

Indigo Important Questions with Answers

Q. How did Rajkumar Shukla establish that he was resolute? Or Why is Rajkumar Shukla described as being resolute?

Ans. Rajkumar Shukla desired Gandhiji to go with him to his area called Champaran. Gandhiji was engaged at that time. However, Shukla did not leave Gandhiji. He followed him wherever he went. Finally, Gandhiji had to arrange and fix time to go with him. This shows that Shukla was resolute.

Q. Why did Rajkumar Shukla go to meet Gandhiji?

Ans. Rajkumar Shukla went to meet Gandhiji to convince him to visit Champaran to help indigo sharecroppers in their fight against the injustice of the landlord system in Bihar. He went to Calcutta to receive Gandhiji and never left his side. Rajkumar Shukla earnestly wanted that Gandhiji should visit Champaran to solve their problems.

Q. How was Gandhiji treated at Rajendra Prasad’s house? Or Why do you think the servants thought Gandhiji to be another peasant?

Ans. Gandhiji came along with Rajkumar Shukla, who was a peasant, to Rajendra Prasad’s house. He was dressed very simply, so he was treated like an untouchable peasant by not being allowed to drink water from the well.

Q. Why did Gandhiji decide to go to Muzaffarpur before going to Champaran?

Ans. Gandhiji decided to first go to Muzaffarpur because he wanted more information about the conditions in Champaran than Shukla was capable of imparting. It did prove helpful as the lawyers in Muzaffarpur, who frequently represented the peasant groups in the courts, briefed Gandhiji about their cases.

Q. How is Gandhiji critical of the lawyers?

Ans. Gandhiji was critical of the Muzaffarpur lawyers for charging a heavy fee from the sharecroppers, as the peasants were so crushed and fear-stricken that going to the law courts was useless. The real relief for them would be to be free from fear.

Q. How was Gandhiji able to influence lawyers? Give instances.

Ans. Gandhiji asked the lawyers what they would do if he was sentenced to prison. They said that they had come to advise him. If he went to jail, they would go home. Then Gandhiji asked them about the injustice to the sharecroppers. The lawyers held consultations. They came to the conclusion that it would be shameful desertion if they went home. So, they told Gandhiji that they were ready to follow him into jail.

Q. Why did Gandhiji feel that taking the Champaran case to the court was useless?

Ans. Gandhiji went to Champaran to fight the case of peasants. He collected all the information there and reached a conclusion that it was useless taking the Champaran case to the court. He found that the peasants were crushed and fear-stricken. He realised that the making the peasants free from the fear of British landlords was more important than fighting for them.

Q. What made the Lieutenant-Governor drop the case against Gandhiji?

Ans. Thousands of peasants held a spontaneous demonstration in Motihari. The officials felt helpless and the government was baffled. The pressure of the people was mounting. The judge didn’t want to aggravate the situation. He held up the sentence for several days and finally released Gandhiji without bail, thus dropping the case against Gandhiji.

Q. How did the Champaran peasants react when they heard that a Mahatma had come to help them?

Ans. Gandhiji received a summon to appear in court. The next day thousands of peasants had assembled in Motihari. They didn’t know much about Gandhiji. But they knew that he had come there only to take up their cause. Thousands of them held a demonstration.

Q. What were the terms of the indigo contract between the British landlords and the Indian peasants? Or What did the peasants pay to the British landlords as rent?

Ans. The terms of the indigo contract between the British and the peasants were that the peasants were sharecropper tenants, had to plant 15% of the land holding with indigo and surrender the entire indigo harvest to the British landlords as rent.

Q. What made Gandhiji demand 50% refund from the British landlords?

Ans. Gandhiji demanded 50% refund from the British landlords because he knew that the British would negotiate as they had expected full repayment of the money that had illegaly extorted from the sharecroppesrs.Gandhiji wanted them to surrender a part of the money and their prestige also.

Q. Why did Gandhiji agree to a settlement of mere 25 per cent? Or Why did Gandhiji agree to a settlement of25% refund to the farmers?

Ans. For Gandhiji, the amount of the refund was less important than the fact that the landlords had been forced to return part of the money, and with it, part of their prestige too. So, he agreed to settlement of 25 per cent refund to the farmers.

Q. “The battle of Champaran is won.” When and why did Gandhiji exclaim this?

Ans. Gandhiji said that the battle of Champaran is won when the prominent people agreed to go to jail for the course of Champaran. Gandhiji knew that now he would be able to pressurize the government.

Q. How did Gandhiji show that he cared for the cultural and social backwardness of Champaran villages? Or How did Gandhiji help the peasants of Champaran?

Ans. The peasants of Champaran’s villages were culturally and socially backward, besides being crushed and fear-stricken by the British due to the sharecropper agreement. Gandhiji freed them from exploitation by teaching them that they had rights and also supporters of their cause. The backwardness was tackled by opening primary schools, improving the healthcare facilities and teaching the villagers personal cleanliness and community sanitation.

Q. Why do you think Gandhiji considered the Champaran episode to be a turning point in his life?

Ans. Gandhiji considered the Champaran episode to be a turning point in his life because it was the first successful civil disobedience movement for him. Though it began as an ordinary attempt to free the poor peasants from injustice and exploitation, it was important because it wiped out the mortal fear of the Britishers from the hearts of the simple farmers.

Q. Why did Gandhiji oppose when his friend Andrews offered to stay in Champaran and help the peasants?

Ans. C.F. Andrews wanted to stay in Champaran and help the peasants, but Gandhiji objected to it because he wanted to mould ‘a new free Indian’. Although the cause was good and he believed their victory was certain yet they wanted the peasants to be self-reliant.

Q. Why did Rajkumar Shukla invite Gandhiji to Champaran? How did Gandhiji solve the problem of the indigo farmers?

Ans. Rajkumar Shukla invited Gandhiji to Champaran to fight against the injustice meted out to the peasants in Champaran. Gandhiji scolded the lawyers for collecting big fees from the sharecroppers. He telegraphed Dr. Rajendra Prasad to come from Bihar with his friends who conferred with Gandhiji who asked them what they would do if he was sentenced to prison. The senior lawyers replied that they had come to advise and help him. Being a stranger Gandhiji was prepared to go to prison for the sake of the peasants. They also agreed to follow Gandhiji to jail. Gandhiji and the lawyers had written down depositions by about ten thousand peasants and notes made on other evidences. He was served summons but he remained firm. Then he received a written communication from the magistrate that the Lt. Governor of the province had ordered the case to be dropped. Gandhiji agreed to a settlement of 25% refund to the farmers.

Q. Give an account of Gandhiji’s efforts to secure justice for the poor indigo sharecroppers of Champaran? Or Describe the difficulties faced by Gandhiji in Champaran.

Ans. First of all, Gandhiji began by trying to get the facts, for this purpose he visited the secretary of the British Landlord’s Association, but he refused to give any information to an outsider. Next he called upon the British official commissioner of the Tirhut division in which Champaran district lay. The commissioner bullied him and advised him to leave Tirhut. This shows that Gandhiji was a staunch seeker and believer of truth. But Gandhiji did not leave and rather proceeded to Motihari, the capital of Champaran. He mobilized the support of the lawyers and Peasants. He got an official notice to quit Champaran immediately. But he disobeyed the order and was summoned to court. The spontaneous demonstration of thousands of farmers was their liberation of fear of the British. Gandhiji just wanted the civil disobedience movement or Satyagraha in a non-violeht manner. Later on for India’s freedom struggle Satyagraha and non-violence became the pillars of strength.

Q. Why did Gandhiji agree to a settlement of 25% refund to the farmers? How did it influence the peasant-landlord relationship in Champaran?

Ans. Gandhiji fought the case on behalf of the sharecroppers and the evidence that he collected was so overwhelming that the landlords were asked to repay. When Gandhiji asked for 50% repayment, the landlords offered to pay only 25% as they wanted to create a deadlock and thus prolong the dispute. To everybody’s surprise, Gandhiji agreed to a refund of 25%. Gandhiji explained that the amount of refund was not important. What mattered was, that the landlords were obliged to surrender a part of their money and with it, part of their prestige. The peasants saw that they had rights and persons to support them in upholding their rights. They learned courage. Gradually, indigo sharecropping disappeared from the area and the land came back to the poor peasants.

Q. Why is the Champaran episode considered to be the beginning of the Indian struggle for independence?

Ans. The peasants of Champaran were in great fear of the British government because they were forced to plant 15% of their holdings with indigo and surrender the entire produce to the landlord. When synthetic indigo came, the landlords released them after demanding compensation from them. The innocent peasants agreed without realising what they were doing. Later the British hired men to oppose them. When Rajkumar Shukla told Gandhiji about it, Gandhiji visited Champaran and realised that the peasants were greatly in fear of the British. He realised that it was necessary to rid them of their fear. He started Satyagraha movement. The farmers shed their fear and supported Gandhiji by reaching the place of Satyagraha. That is why Champaran episode is considered to be the beginning of the Indian struggle for independence as everyone realised that they can stand against the British who could not order them around in their own country.

Q. The Champaran episode was a turning point in Gandhiji’s life. Explain. Or How did Gandhiji use Satyagraha and non-violence at Champaran to achieve his goal?

Ans. In Champaran the peasants were greatly in fear of the British government. The cause of the problem was indigo and the greed of the landlords. They had forced the tenants to plant 15% of their holdings with indigo and surrender the entire produce to the landlord. When synthetic indigo came, the landlords were ready to release them. They demanded compensation the repercussion of which the peasants did not know and hence agreed to it. Later when the peasants came to know about synthetic indigo they asked for their money the British hired thugs to oppose them. Gandhiji realised that there was no need for lawyers. He realised that it was necessary to release them of their fear which was difficult to achieve as they were uneducated. But with his determination he championed their cause. Soon he led a movement of non-violence and Satyagraha. Many farmers demonstrated around the court room where Gandhiji was summoned, this made the British feel challenged. Sharecroppers from Champaran came barefooted to see Gandhiji. Muzaffarpur lawyers too called on him. He explained that what he had done was an ordinary thing, he had simply told the Britishers that they could not order him in his own country. Gandhiji tried to mould a new free Indian, who could stand on his own feet. This new realisation gave him a direction to lead the freedom struggle and thus proved a turning point in his life. This was the first time Gandhiji realised that India is capable of Mass Movement and it was after this episode that he started the National Struggle for freedom across the country.

Q. Gandhiji’s was not a loyalty to abstractions; it was a loyalty to living, human beings. Why did Gandhiji continue his stay in Champaran even after indigo sharecropping disappeared?

Ans. After the Champaran battle was won and the land reverted to the peasants, Gandhiji continued to stay on in the region. His loyalty was to living human beings and he realised that a lot needed to be done for the upliftment of the peasants in the villages of Champaran. Gandhiji took the initiative and began the work of eradicating their cultural and social backwardness. Primary schools were started so that the ‘poor peasants and their children could be educated. Gandhiji appealed to teachers, and many of his disciples, including his wife and son, volunteered for the work. Health conditions in the area were also miserable. Gandhiji got a doctor to volunteer his services for six months. All this goes to prove that Gandhiji’s loyalty was not to abstractions; his politics was always intertwined with the practical day-to-day problems of the millions.

Q. 7. Describe how, according to Louis Fischer, Gandhiji succeeded in his Champaran campaign.

Ans. The objective of Gandhiji in his Champaran campaign was to mould a new free Indian who could stand on his own feet and thus make India free. He succeeded because, as Rajendra Prasad said, “Gandhiji in this way taught us a lesson in self-reliance”. The peasants did not take the help of any specialist lawyers or any Englishmen like CF Andrews to fight their case. This gave them a new-found confidence in fighting their own battles and they were liberated from fear of the British. The fact that the British planters agreed to refund some of the money paid by the peasants was the crucial point that made the Champaran campaign successful. It showed that both the British and Indians could be treated equally. This ultimately led them in the freedom struggle and gave India its freedom.

Q. “I have come to the conclusion that we should stop going to law courts”, Gandhiji told Muzaffarpur lawyers. How was Gandhiji’s stay in Muzaffarpur?

Ans. Gandhiji decided to go first to Muzaffarpur, which was on the way to Champaran. Gandhiji wanted more complete information. Gandhiji stayed there for two days in the house of Professor Malkani, a teacher in government school. The news of Gandhiji’s arrival and of the nature of his mission spread quickly through Muzaffarpur and to Champaran. Sharecroppers from Champaran began arriving on bare foot and by conveyance to see their champion and defender. Muzaffarpur lawyers, who frequently represented peasant groups in court, called on Gandhiji to brief them. He scolded lawyers for collecting big fees from the sharecroppers. Gandhiji concluded that where the peasants were so crushed and fear-stricken, taking their cases to law courts was useless. He felt that real relief for them was to be free from fear.

The Rattrap Important Question Answers (2 marks) – 30 to 40 words

Q1. How was the peddler received in the cottage?
or
How did the crofter entertain the peddler?

A. The crofter was kind to the peddler. He offered him shelter in his cottage, gave him porridge for supper. Then he cut a big slice from the tobacco roll which was sufficient for both of them and offered it to him. He also confided in the peddler and shared his secrets.

Q2. Why was the crofter so talkative and friendly with the peddler?

A. The crofter lived all alone and wanted someone to talk to. When he saw the peddler, he was happy to get company. He was a generous host and served the peddler porridge, tobacco and also played a card game with him. He even shared his secrets with the peddler.

Q3. In what sense was the world a big rattrap according to the peddler?
Or
Explain the metaphor of the rattrap in context of the story by Selma Lagerlof.

A. The rattrap seller believes that the worldly pleasures are like a bait in the rattrap. They attract us. The one who gets attracted towards the bait, gets trapped. He has to commit all types of misdeeds in order to achieve worldly pleasures. This marks the end for him.

Q4. Why did Edla still entertain the peddler even after she knew the truth about him?
Or
Why did Edla plead with her father not to send the Vagabond away?

A. Edla was a trusting and compassionate person. She wanted the peddler to enjoy himself and remain at peace. As it was Christmas Eve, she did not want to turn away the guest. She reasoned that they had insisted upon the man to visit their home for Christmas Eve. It was her kind gesture that reformed the peddler.

Q5. Why did the ironmaster speak kindly to the peddler and invite him home?
Or
Why was the ironmaster kind to the peddler? Where did he invite him to and why?

A. The ironmaster mistook the peddler for an old comrade. He was happy that he had found his comrade with whom he could spend Christmas eve. He was lonely and so he was desperate to invite the man home. Despite opposition from the peddler, the ironmaster and his daughter convinced him to spend Christmas eve with them. He was eager to take the man home because of the miserable condition which he had found him in.

Q6. Why was the peddler amused at the idea of the world being a rattrap?
A. The peddler was amused when he saw other people getting trapped In the metaphorical rattrap. The thought of other people being trapped, because of the temptations amused him. He was happy because he remained free from the rattrap. He felt joyous to think ill of the world in this way.

Q7. What was the content of the letter written by the peddler to Edla?

A. The peddler wrote that since she had treated him like a captain, he wanted to be nice to her in return. He did not want to embarrass her with the thought that she had invited a thief over for Christmas. He returned the thirty kroners of the crofter and requested her to return the stolen money. The letter read that the rattrap was a Christmas present from a rat who would have been caught in the world’s rattrap if he had not been raised to the status of a captain which motivated him to reform himself.

The Rattrap Important Questions of 5 Marks – 120 to 150 words

Q1. Why did the crofter repose confidence in the peddler?
How did the peddler feel after betraying the crofter?

A. The crofter had been hospitable towards the peddler. Despite this the peddler was ungrateful and betrayed him. The crofter welcomed him in his house when he asked for shelter. He took good care of him, served him supper, shared his tobacco, and played a card game ‘mjolis’ with him until bedtime.

He was also generous with his confidence and told the peddler that he had been a crofter at the Ramjso ironworks and now he had earned 30 kroners by selling the milk of his cow. The crofter did all this because he had been lonely and became happy when he got a company of the peddler. However, the peddler stole the 30 kroners from the crofter’s house. Soon thereafter he felt guilty of cheating the nice man. He felt that the money was bait that had lured him into committing theft.

Q2. How does the story ‘The Rattrap’ highlight the importance of community over isolation? support your rationale with textual evidence.
Or
The Rattrap focuses on human loneliness and the need to bond with others. Comment.

A. Man is a social animal and he cannot live in isolation. In every field of life we interact with humans. In case a person does not socialize and shirks the company of friends, he becomes a recluse. The story ‘The Rattrap’ highlights this fact.

Initially we come to know that the rattrap seller was a lonely man who used to make rattraps and was left to his own meditations. One day when he knocks on a cottage, he meets a crofter who welcomes him and entertains him. The crofter is also a lonely man and is glad to get the company of the peddler.

Then we also find that the ironmaster and his daughter are lonely. They live in a big house but have nobody else for company, so, they insist that the peddler should spend Christmas eve with them.

Q3. What made the peddler finally change his ways?
Or
What miracle did Edla perform in the peddler’s life?
Or
How did Edla bring about a change in the peddler?

A. Edla believed that the peddler was her father’s comrade from the Army. She treated him like a gentleman. She gave him a lot of respect and spoke with kindness. She wanted the peddler to stay with them for Christmas and share the Christmas cheer.

Even when they came to know that the peddler was not the person that Edla’s father had mistaken him to be, Edla’s behaviour towards him remained unchanged. All this moved the peddler. He became a true gentleman and gave up the wrong ways of life. He was thankful to Edla for being so good to him and for transforming him into a true gentleman.

Q4. The peddler believed that the whole world is a rattrap. How did he himself get caught in the same?

A. The peddler gets tempted and trapped when he gets easy access to the crofter’s money. He realizes that he is caught in the trap when he loses his way in the forest, with the few kroners in his pocket. He gets caught in the trap another time when Edla assures him Christmas cheer.

He knows also that the ironmaster is mistaking him for someone else but does not clarify this because he is lured by the attraction of the Christmas cheer. He repeatedly surrenders to the worldly temptations and realizes that the world is a rattrap and the riches, joys, and shelters are worldly temptations.

Q5. How did the seller of rattrap realize that he himself was caught up in a rat trap after he left the crofter’s cottage?

A. The rattrap seller was a habitual thief. Despite the kindness of the crofter, the peddler betrayed him. He stole the money from his house. In order to prevent being caught, he avoided the highway and went into the woods. He got confused and lost his way.
He seemed to move in a circle as the paths seemed to twist back and forth. When he was trapped in the woods he realized that the 30 kroners were bait. He felt that the forest was closing upon him like a prison and he had been trapped.

The Rattrap Subjective Questions 

Q1. From where did the peddler get the idea of the world being a rattrap?

                                       or

Why was the peddler amused by the idea of the world being a rattrap?

Important Point-

– While pondering over the traps made by him

– Comforts of riches, shelter, food, and clothing were baits

– Once trapped everything comes to an end.

– was never treated kindly

– world ensnares all those who succumbed to temptations of riches and joys

Q2. What made the peddler think that he had indeed fallen into a rattrap?
                                  or

Why did the peddler keep to the woods after leaving the crofter’s cottage? how did he feel?

Important Point-

– in order to escape being caught

– being nervous lost his way in the maze 

– by giving in to temptation, had fallen into a rattrap

Q3. Why did Edla still entertain the peddler even after she knew the truth about him?

or

Why did the peddler sign himself as Captain von Stahle?

Important Point-

– knew him to be a homeless tramp

– didn’t feel cheated when my identity revealed

– felt bad for him

– thought it to be wrong to break a promise

– humane and compassionate treatment by Edla

– by returning money, tried to behave as a true captain 

Q4. Give examples from the story, The Rattrap’ to show how the ironmaster is different from his daughter. 

                                   or
Compare and contrast the character of the ironmaster with that of his daughter.

There is a saying ‘ Kindness pays, rudeness never.’ In the story, “ The Rattrap”, Edla’s attitude towards men and matters is different from her father’s attitude. How are the values of concern and compassion brought out in the story?

Important Point-

Ironmaster:

– ambitious and arrogant

– mistook peddler for his old regimental comrade

– had more pride than sympathy and empathy

– called peddler dishonest and threatened to call sheriff 

– worried about his own silverware

Edla:

– kind and compassionate

– sympathetic and considerate towards the peddler right from the beginning

– treated peddler with dignity and respect even after knowing the truth

– changed peddler’s attitude

Important Questions with Answers

Q. What is the “misadventure” that William Douglas speaks about?

The ‘misadventure’ William Douglas speaks about happened in the water-pool of Y.M.C.A. in Yakima. Douglas was about ten or eleven old at the time. He was trying to learn swimming with the help of his waterwings. One day while he was sitting on the side of the pool, a big bruiser of a boy suddenly picked him up and threw him into the nine feet deep water. Douglas used all his mind and might to come out on the surface and paddle to the edge of the pool. However, all his attempts failed. In the end he was so tired that he gave up all efforts to come out and lost consciousness. It was them that someone picked him up and brought him to the shore. He survived somehow but the incident haunted him till many days afterwards. This is the misadventure that William Douglas speaks about.

Q. What were the series of emotions and fears that Douglas experienced when he was thrown into the pool? What plans did he make to come to the surface?

Douglas was frightened when he was thrown into the pool but he had not lost his reason in the beginning. He had a plan to come out of the deep water. However, his repeated attempts to carry out his plan failed. It was then that the sheer stark terror seized him. It was a terror that knew no understanding. Douglas found it difficult to control this emotion of terror. Only those who experience a death-like situation know what it is. Still he made one last effort. This too failed and the terror took a deeper hold on him. He shook and trembled with fear. He tried to call for help, to call for mother. Nothing happened. Then all efforts ceased. He was prepared to die. He crossed into oblivion. The curtain of life seemed to have fallen.

Q. How does Douglas make clear to the reader the sense of panic that gripped him as he almost drowned? Describe the details that have made the description vivid?

Douglas has given a detailed and vivid description of how the sense of panic gripped him as he almost drowned in the pool.

In the beginning Douglas was frightened but not frightened out of wits. On the way down he planned that on touching the ground, he would make a big jump, come to the surface, lie flat on it, and paddle to the edge of the pool. The plan failed and not once but thrice.

It was then that the stark terror seized him. He was shrieking under water slowly, even those screams were frozen. Only his heart, and the pounding in his head, said that he was still alive.

Then all effort ceased. A blackness swept over his brain. It wiped out fear. There was no more panic. He crossed to oblivion, and the curtain of life fell.

It was then, that someone saved him. It took hours before he was able to walk home unsteadily. The panic had gripped him so much that it haunted his heart till many days after the incident.

Q. Why does Douglas as an adult recount a childhood experience of terror and his conquering of it? What larger meaning does he draw from this experience?

Douglas recounts his childhood experience to say that the fear is a most dangerous emotion. To conquer his fear of water he had to work hard for many days but his account shows that he did the right thing.

He draws a larger meaning from his experience. He says that only those who have known stark terror and conquered it can appreciate his emotion. He says that there is terror only in the fear of death. All that we have to fear is fear itself. Since he had experienced both the sensation of dying and the terror that the fear of it can produce, the will to live somehow grew in intensity. ra

Q. Which two frightening experiences did Douglas have in water in his childhood ? Or Which two incidents in Douglas’ early life made him scared of water?

The first incident occurred when he was about four years old at a beach in California. A strong wave knocked Douglas down and he was buried in water. The next incident occurred at the YMCA pool when he was ten or eleven. A big bully of a boy tossed him into the deep end of the pool. He went down to the bottom and almost drowned in the pool.

Q. How did William Douglas’ aversion to water begin?

William Douglas’ aversion to water started when he was three or four years old and his father took him to the beach in California. They stood together in the surf. He hung onto his father, yet the waves knocked him down and swept over him. He was buried in water. His breath was gone and he was frightened. His father laughed, but there was terror in his heart at the overpowering force of the waves.

Q. How did Douglas’ introduction to YMCA pool revive his childhood fear of water?

Unpleasant memories were revived when he went to the YMCA pool for the first time. Childish fears were stirred. In a little while he gathered confidence. He watched other boys paddling on water with their water wings. He tried to learn bY imitating them. He did this two or three times on different days.

Q. Why did Douglas’ mother recommend that he should learn swimming at the YMCA swimming pool? Or Why did Douglas prefer to go to the YMCA swimming pool to learn swimming? Or Why did William Douglas use the YMCA pool and not Yakima river to learn swimming?

Douglas’ mother recommended that he should learn swimming at the YMCA swimming pool because it was safe, being only two or three feet deep at the shallow end with a gradual drop to nine feet at the other end. Also it was close to his residence.

Q. How did his experience at the YMCA swimming pool affect Douglas? Or How did Douglas’ experience at the YMCA pool affect him? Or How did the inicident at the YMCA pool affect Douglas?

The near death experience of drowning had a very strong impact on Douglas’ psychology. He started avoiding venturing Douglas, water and this fear remained with him for many years. It prevented him from fishing, boating and swimming, besides ruining his social life.

Q. What deep meaning did his experience at the YMCA swimming pool have for Douglas?

After his near death experience at the YMCA pool, Douglas started fearing water. He could not enjoy any water sports or go fishing. He decided to overcome his fear and learnt swimming again. He became confident and understood that ‘all that we have to fear is fear itself’.

Q. ‘All we have to Fear is fear itself’. When did Douglas learn this lesson?

These words mean that we fear, fear the most. Those who have undergone this experience of fear can only appreciate its worth. Douglas faced it twice in life. He had a terrible fear of water. He could not go for swimming, Canoeing, boating, rafting, etc. He realized that it would ruin his life since it was following and haunting him wherever he went. Fear is our hard Core enemy.

Q. What did Douglas experience when he went down to the bottom of the pool for the first time?

When Douglas went down the water for the first time, he was afraid but confident with his mind working over a plan that as his feet will touch the floor, he will jump up and pop up like a cork on top and paddle to the edge of the pool.

Q. Why was Douglas determined to get over his fear of water?

Douglas regretted being deprived of enjoying water activities like canoeing, boating, swimming, fishing, etc. The wish to enjoy them and the craving to regain his lost confidence, while being in water, made him try every possible means to get rid of his fear. He was finally able to overcome this mental handicap by getting himself a swimming instructor and further ensuring that no residual fear was left.

Q. How did the instructor turn Douglas into a swimmer?

The instructor adopted a systematic method to turn Douglas into a swimmer.The instructor put a belt around him. A rope attached to the belt went through a pulley then ran on an overhead cable. Thus, he was made to go back and forth across the pool hour after hour. Then he taught Douglas to breathe while swimming, and finally the leg movements and other strokes.

Q. How did Douglas make sure that he conquered the old terror? Or What efforts did Douglas make to get over his fear of water?

Douglas took the help of an instructor. The instructor made Douglas swim five days a week. He was with Douglas for about six months when he was sure Douglas was able to swim alone he left. Finally, Douglas was able to swim the length of the pool up and down but he was not sure he had conquered his fear completely. He went to Wentworth lake and dived off and finally he went to Warm lake, dived and swam across to the other shore and came back.

Q. Why did Douglas go to Lake Wentworth in New Hampshire? How did he make his terror flee?

After taking training in swimming, Douglas went to Lake Wentworth in New Hampshire to conquer his fear of water. He dived off at Triggs Island. He swam across the lake to Stamp Act Island. He swam the crawl, breast stroke, side stroke and back stroke. The terror returned only once but he overcame it. He went up the Tietan to Conrad Meadows and camped there. The next morning he dived into the Warm lake and swam across to the other shore and came back. He shouted with joy as he had conquered his fear of water.

Q. What happened at the YMCA swimming pool which instilled fear of water in Douglas’ mind?

When Douglas was learning at the YMCA pool at the age of ten or eleven, one day while sitting beside the pool waiting for other people to come, a muscular bully picked him up and threw him into the deep end of the pool. As Douglas realised that he might drown, he made three attempts to come up to the water surface, but failed and fell unconscious. Ultimately he was rescued by someone, but this episode in his life reinforced the fear of water in Douglas’ mind which he had first felt when, at the age of three or four, he had been completely swamped by a huge wave at the seaside in California.

His father held on to him at that time to save him from drowning, but at the YMCA pool there was nobody.

Q. How did Douglas’ experience at the YMCA pool affect him? How did he get over this effect?

The experience at the YMCA pool reinforced the fear of water in Douglas’ mind which he had first felt when, at the age of three or four, he had been completely swamped by a huge wave at the seaside in California. The experience at the pool left a haunting fear of water in his heart. He started avoiding water whenever he could, which affected his normal activities as well as his social life.

After a few years of suffering like this, Douglas decided to get an instructor and learn to swim. The instructor systematically taught him how to swim, starting from the basics and taking all the required safety measures, which gave Douglas confidence. After six months of training, Douglas swam across a number of lakes independently thus breaking free from his fear.

Q. Describe the efforts made by Douglas to save himself from, drawing in YMCA swimming pool.

When Douglas was picked and tossed into the the deep end of the pool, he was frightened, but not yet frightened out of his wits. On the way down, he planned that when his feet hit the bottom, he would make a big jump, come to the surface, lie flat on it, and paddle to the edge of the pool.

He imagined he would bob to the surface like a cork. Instead, he came up slowly. He opened his eyes and saw nothing but water. He flailed at the surface of the water, swallowed and choked. He tried to bring his legs up, but they hung as dead weights, paralysed and rigid and he went down again. He struck at the water expanding his strength, remembering that he had to hit the bottom of the floor. But second time too he failed to come up to the surface.

And then sheer, stark terror seized him. The third time he went down he tried to scream but ultimately found himself losing consciousness and started sinking to the bottom of the pool with total silence enveloping him.

Q. “I crossed to oblivion, and the curtain of life fell.” What was the incident which nearly killed Douglas and developed in him a strong aversion to water?

The incident that nearly killed Douglas occurred at the YMCA pool when he was ten or eleven years old. He had decided to learn swimming at the YMCA pool, and thus get rid of his fear of water. One morning, when he was alone at the pool, a big bully of a boy tossed him into the deep end of the pool. Though he had planned a strategy to save himself, his plan did not work. He went down to the bottom and got panicky. Thrice he struggled hard to come to the surface but failed each time. He was almost drowned in the pool. This misadventure developed in him a strong aversion to water.

Q. Douglas fully realised the truth of Roosevelt’s statement ‘All we have to fear is fear itself’. How did this realisation help him brush aside his fear and become an expert swimmer?

Roosevelt said, “All we have to fear is fear itself.” Douglas had experienced both the sensation of dying and the tenor that fear of it can produce. Strong will, hard determination, courage and toil as well as honest labour win over all our terrors and fears. The will to live brushes aside all our fears. This realisation made him resolve to learn swimming by engaging an instructor. This instructor, piece by piece, built Douglas into a swimmer. Then, he went to Lake Wentworth, dived at Triggs Island and swam two miles across the lakes to Stamp Act Island. Finally, he had conquered his fear of water.

Q. How did the instructor make Douglas a good swimmer? Or How did the swimming instructor ‘build a swimmer’ out of Douglas?

Douglas decided to get an instructor to learn swimming. The instructor started working with him five days a Week, an hour each day. He put a belt around him. A rope was attached to the belt went through the pulley that ran on an overhead cable. This, he was made to go back and forth across the pool hour after hour. Then, he taught him to put his head under water and exhale. He taught him to raise his nose and inhale. He taught him all the techniques of swimming in water. He held Douglas on the side of the pool and made him kick with his legs. He made him practice very hard and made a swimmer of Douglas bit by bit. He was with Douglas for about six months, when he was sure Douglas was able to swim alone, he left. Finally, Douglas was able to swim the length of the pool up and down but he was not sure whether he had conquered his fear completely or not.

Q. Do you think the title ‘Deep Water’ is appropriate? Give reasons in support of your.

The title ‘Deep Water’ is quite appropriate. The title is highly suggestive and at once focuses our attention on the main theme — experiencing fear of death under water and the efforts of the author to overcome it. All the details in the excerpt are based on his personal experience and analysis of fear.

The overpowering force of the waves at the California beach stir aversion for water in Douglas. His mother warns him against swimming in the deep waters of the treacherous Yakima River. The nine feet deep water of the swimming pool appears more than ninety to Douglas. However, when he conquers fear, he can dive and swim in the deep water of Lake Wentworth and Warm Lake. Thus, the title is apt and suggestive.

Q. ‘All we have to fear is fear itself’. Courage and optimism are two things that help anyone survive in the period of stress. Comment on the value of being courageous with reference to the chapter ‘Deep Water’.

Everyone in this world in his comes face to face with difficulties at some stage life as life is a mixture of joys and sorrows. These difficulties are like touchstone. They test a person’s will-power as well as his perseverance courage. It is only his optimistic and courageous point of view of life that helps him survive in the period of stress. Douglas also faced the fear of water since his childhood. He never wanted to go near the pool and thus avoided it. But then he decided to get rid of this fear as he had realised that all we have to fear is fear itself. He took the help of an instructor, who guided him into becoming a good swimmer. His aversion to water was gone. The fear of water which had haunted him for years had been overcome by him by his courage and perseverance. He made all his efforts to conquer his fear of water. In the same way, a person must not led any fear overpower him. We should face life boldly and courageously. Courage and determination go hand in hand. If one decides to acheive something, he can very well do it with the help of his positive thinking.

Q. Desire, determination and diligence lead to success. Explain the value of these qualities in the light of Douglas’ experience in ‘Deep Water’.

The terror of water followed Douglas wherever he went. To get rid of it, he made a strong determination. He decided to overcome his fear through his ‘will,’ He engaged an instructor who perfected him in swimming. The instructor gave him hundreds of exercises and taught him to exhale and inhale in water. The practice went on for three months and Douglas was able to counter his terror. Then, after more exercises, the instructor ordered him to dive. He swam across lakes also to gain confidence. He had now completely lost his fear of water. His desire, determination and diligence had succeeded in banishing his fear of water.

Lost Spring – Important Questions

Important Questions with Answers

Q. What does Saheb look for in the garbage dumps? Or What is Saheb looking for in the garbage dumps? Where is he and where has he come from?

In the garbage dumps, Saheb looks for useful items which can be sold for cash. As these items can be traded for money, they are just like ‘gold’ for him. Saheb and his family live in Seemapuri, a slum on the periphery of Delhi. His family had migrated from Bangladesh.

Q. What explanations does the author offer for the children not wearing footwear?

The author has seen many children in the villages as well as cities in India walking barefoot. The general explanation is that it is not lack of money but a tradition to stay barefoot. The author, however, is not sure. She thinks that in many cases it may be a excuse to explain away a perpetual state of poverty.

Q. What makes the city of Firozabad famous?

The city of Firozabad is famous for its bangles. It is the center of India’s glass glowing industry. Here are families who have spent generations working around furnaces, welding glass, and making bangle for all the women of the country.

Q. Why had the ragpickers come to live in Seemapuri? Or To which country did Saheb’s parents originally belong? Why did they come to India? Or Why did Saheb’s parents leave Dhaka and migrate to India?

Once Saheb’s parents lived in Bangladesh, amidst the green fields of Dhaka. There were many storms that swept away their fields and homes. That’s why they migrated to Delhi and settled down in Seemapuri looking for an occupation.

Q. What could be some of the reasons for the migration of people from villages to cities?

Poverty and unemployment are the two most important reasons for the migration of the people from villages to cities. Many people in the villages do not have enough to survive somehow. That is why people travel from as far as Bangladesh to Seemapuri in Delhi. Though the life in the slums of Seemapuri is very bad but they are able to feed themeselves working as rag pickers.

The glare of the city life also attracts many villagers. They are able sell their land at a hefty price and then buy a house in the city. The young love the city life for its crowd, activity and various type of entertainments which it offers.

Q. Describe the irony in Saheb’s name.

Saheb’s name in full is Saheb-e-Alam, which means ‘Lord of the Universe’. But in stark contrast to his name, Saheb is poverty stricken, barefoot, homeless ragpicker who scrounges the garbage dumps of Delhi to take out a livelihood. His name is in total contrast to his very existence and is thus deeply ironical.

Q. What kind of gold did the people of Seemapuri look for in the garbage?

The people of Seemapuri look for items in the garbage which can be traded for money, meaning ‘gold’, as it helps them earn their daily bread and have a roof over their heads. For a child, garbage may mean something wrapped in wonder, whereas for the elders it is a means of survival.

Q. In what sense is garbage gold to the ragpickers? Or What did garbage mean to the children of Seemapuri and to their parents?

Garbage means ‘gold’ to the poor ragpickers because some of it can be sold for cash, thus becoming a means of survival for the children of Seemapuri and for their parents. It is providing them with their daily bread and a roof over their heads.

Q. What forces conspire to keep the workers in bangle industry of Firozabad in poverty?

Many forces conspire to keep the workers in bangle industry of Firozabad in poverty. They are of two types. One is the caste of the people who make bangles. They have the stigma of caste which has been making bangles for many generations. The second factor is the vicious cycle of the sahukaras, middleman, the policeman, the keepers of law, the bureaucrats, and the politicians. The poor need a leader to guide them. Otherwise, the circumstances have imposed the baggage on the child that he can not put down. It needs a lot of courage to go against ones caste and also to fight the vicious circle as well.

Q. Would you agree that promises made to poor children are rarely kept? Why do you think this happens in the incidents narrated in the texts?

It is really true that the promised made to the poor children are seldom kept. In the story this happens because the author was not really serious when he made the promise to Saheb. He had not imagined that Sahib could really be serious about going to school. Even when he made the promise, the author says he was “half-joking”. So How can one keep a promise made as a joke? In fact he never meant to keep the promise.

Q. How in your opinion can Mukesh realize his dream?

Muskesh can realize his dream only if he is determined to do what he wants. It is impossible to find work in a garage. Of course, he will have to face the opposition from his family and he will have to work very hard at least at initial stages. To learn to repair car he will have to work as a labour in the garage. Only after years of hard labour, he will become a good motor mechanic and ultimately be able to drive a car.

Q. Why should child labour be eliminated and how?

Child labour should be eliminated for two important reasons. First, a child should be sent to school because if he gets no education there are no chances of his progress in life. He will be a burden on his family. Second reason is that he is deprived of the childhood which means a time of playing and enjoying. It is necessary of the health of child. If, however, in this age they are sent to work, they work in unhealthy conditions at low wages. After all, they work not because adults are not available but because children are ready to work for really small wages. Their parents send them to work because even a little money is better than no money. So child labor must be eliminated. For this free education alone is not enough. The poor children should be kept in hostels and the society should bear all their expenses.

Q. What does the reference to chappals in ‘Lost Spring’ tell as about the economic condition of the rag pickers?

The ragpickers were extremely poor. They did not have any money to buy chappals. They were poor and impoverished. They lived a hand-to-mouth existence. They were exploited and had no other work to do. They did not have a house to live in too.

Q. How was Saheb’s life at the tea stall? Or What job did Saheb take up? Was he happy? Or Is Saheb happy working at the tea stal? Explain.

No, Saheb does not seem happy working at the tea stall. He is no longer his master and that, relaxed look on his face is also lost. The steel canister seems heavier than the plastic bag that he would carry so lightly over his shoulder. It was because the bag was his and the canister belonged to the man who owned the tea stall.

Q. What is Mukesh’s drem? Do you think he will be able to fulfill is dream? Why? Why not?

Mukesh belongs to bangle-makers of Firozabad where each family is engaged in bangle-making. On asking, Muykesh says, “I will be a motor mechanic.’ Thus, he wants to be his own master. Yes, he has strong determination. He wants to improve his living conditions. However, because he caught up in the vicious cycle created by others, he will not be able to realise his dream and will remain a bangle-maker.

Q. “It is his karam, his destiny.” What is Mukesh’s family’s attitude towards their situation? Or How is Mukesh different from the other bangle-makers of Firozabad? Or How is Mukesh’s attitude to his situation different from that of his family?

Mukesh’s grandmother whose husband became blind with the dust from polishing the glass of bangles accepts the destiny of her husband. She says that her husband was destined to go blind. It was his karam, his destiny. But Mukesh has the courage to dream of becoming a motor mechanic, thus breaking free from destiny.

Q. Mention the hazards of working in the glass bangles industry.

The glass bangles industry has a very hazardous working environment. People work in the glass furnaces with high temperatures, in dingy cells without air and light. Most end up losing their eyesight even before they become adults. Adding to their misery, they are caught in a vicious circle of people who exploit them.

Q. Why could the bangle-makers not organise themselves into a co-operative?

Most of the young bangle-makers have fallen into the trap of the middlemen. They are also afraid of the police. They know that the police will haul them up, beat them and drag to jail for doing something illegal. There is no leader among them to help them see things differently.

Q. Whom does Anees Jung blame for the sorry Plight of the bangle-makers?

Anees Jung blames the vicious circle of the sahukars (moneylenders), middlemen, Policemen, bureaucrats and politicians for the plight of the bangle-makers. They don’t allow the bangle-makers to organise themselves into a cooperative.

Q. In spite of despair and disease pervading the lives of the slum children, they are not devoid of hope. How far do you agree?

In spite of growing up amidst despair and disease, the children who live in slums have the desire to achieve something big in life. This shows that they are not devoid of hope. Saheb, a ragpicker, is eager to go to school and learn. Mukesh, who works in dark, dingy cells making bangles, dreams of becoming a motor mechanic, against his family tradition.

Q. The barefoot ragpickers of Seemapuri live on the periphery of Delhi yet, metaphorically speaking, miles away from it. Comment.

The barefoot ragpickers of Seemapuri live on the periphery of Delhi, yet metaphorically speaking, miles away from it, sums up the true condition of the ragpickers of Seemapuri. Seemapuri is a slum area, which houses approximately 10,000 ragpickers. They live in mud houses with roofs of tin and tarpaulin. There is no sewage, drainage or running water. They came here from Bangladesh in 1971 and have been living here ever since without any identity of their own or permits, but they have ration cards and their names figure in the voter’s list. This is an example of the gross negligence and apathy of the Delhi Government. It has failed to do anything for them. Though Seemapuri is so close to Delhi, almost on its periphery, yet the glitter and glamour, advantages like education, proper facilities for living a clean and decent life are beyond the reach of these slum dwellers of Seemapuri, which is so close to Delhi yet so far.

Q. Give a brief account of life and activities of the people like Saheb-e-Alam settled in Seemapuri. Or “For the children it is wrapped in wonder, for the elders it is a means of survival.” What kind of life do the ragpickers of Seemapuri lead?

Seemapuri is a slum area located on the periphery of Delhi. Most of the residents of Seemapuri consist of people who are refugees from Bangladesh. Saheb’s family is among them. The area consists of mud structures, with roofs of tin and tarpaulin. It has no facilities of sewage, drainage or running water. About 10,000 rag pickers live here.

Their only means of livelihood is finding saleable items from rubbish. Thus, for them, the rubbish is as valuable as gold, for their survival depends on these. These ragpickers have lived here for more than thirty years without any identity. They have no permits but have ration cards, thanks to the selfish whims and wishes of the politicians. With these they can get their name on the voter’s lists and also buy grains for themselves at a subsidised rate.

Q. Describe the difficulties the bangle-makers of Firozabad have to face in their lives. Or Describe the circumstances which keep the workers in the bangle industry in poverty.

The bangle-makers of Firozabad lead a miserable and pathetic life. They live and work in inhuman conditions. They work in dingy cells, without air and light. They have to work in the glass furnaces with high temperatures. Most of them lose their eyesight before they become adults. They live amidst stinking lanes choked with garbage. Their houses have crumbling walls, shaky doors and no windows. They work in the dark hutments in the light of flames of flickering oil lamps. Their eyes are more adjusted to the dark than to the light outside. That is why they often lose their eyesight. They face many difficulties in their lives. They are exploited by the money lenders, the middlemen and the bureaucrats. Even the police does not protect them. They are caught in the web of poverty, burdened by the stigma of caste in which they are born. In fact, no change has taken place in their lives with the passage of time. They are as poor and miserable as they were before.

Q. What circumstances forced Mukesh not to pursue his family business of bangle making? Instead, what did he decide to do?

Mukesh’s family had a family business of making bangles. He did not want to do that because the chances of going blind due to the glass dust was high. The whole family was involved in it yet they were not able to earn much. This made them fall into the vicious circle of the money lenders after which they never come out. The debt would go on for generations. All these reasons made him want to leave his family business of bangle making. Instead, he decided to become a motor mechanic and learn to drive.

Q. How is Mukesh more ambitious in life than Saheb? Give a reasoned answer.

Mukesh is definitely more ambitious than Saheb. Unlike most of his friends in Firozabad, Mukesh did not want to follow the profession of making bangles. No one else could dare to think of breaking the conventional style of living. Mukesh dreamt of becoming a motor mechanic. He had already decided to go to a garage and learn about cars. Though the garage was a long way from his home, he was prepared to walk that distance. He insisted on becoming his own master. Saheb, on the other hand had sacrifised his freedom as a ragpicker to take up a salaried job that would pay him 800 rupees and give him all his meals. Now he was no longer his own master. He had lost his carefree look (which he had when he was a ragpicker). The canister that he carried seemed heavier than the bag he carried as ragpicker, for this job was not to his liking.

Q. How is Mukesh’s attitude towards his situation different from that of Saheb? Why?

Mukesh belongs to a bangle-making family, but he is not content with this profession. He dares to dream of becoming a motor mechanic and driving cars. He has strong will power and wants to achieve what he dreams about, unlike other people in his family. In contrast to this, Saheb is a ragpicker who is content with his life, but becomes unhappy when he gets a job at a tea stall, even though now he is probably earning more and on a regular basis. Saheb is unhappy because he has lost his independence, which he had as a ragpicker. However, Saheb accepts his new situation, whereas Mukesh dares to want to break free from tradition. This is because Mukesh is more courageous and determined than Saheb will ever be.

Q. ‘Lost Spring’ explains the grinding poverty and traditions that condemn thousands of people to a life of abject poverty. Do you agree ? Why/Why not?

‘Lost Spring’ brings home of the dark reality that there are thousands of people who live in a state of abject poverty. The story of both. Saheb in Seemapuri and Mukesh in Firozabad, show the dark reality the lives of so many people. The stories reveal the sheer destitution of the people in the slums and also the life of exploitation that they lead. The children do not go to school. Society as well as their parents neglect them. The stories also tell us that once those people get into the vicious circle of money lenders they are not able to get out of it.

Q. In India we believe in prayers. Whenever we are faced with a problem we pray to God. A son of a priest at Udipi while going to school prayed at the temple for a pair of shoes. Thirty years later we find his son well dressed in a school uniform. What has brought about the change – the father’s prayer or the father having gone to school or both ? Give a reasonable answer.

It was indeed the fact that the father had gone to school and received education because of which his son was wearing a school uniform. Education gave him opportunities to improve the quality of not his life but also of his family and children. Prayers alone cannot help us. We have to put in our efforts to make the things better. The father while going to school not only prayed for school shoes but also must have made efforts to get himself educated. The result was that he was able to get for his son whatever he could not have for himself.

Q. Why should Child labour be eliminated and how?

Child labour is a curse in our society and it must be eliminated. Childhood is a period of mental and spiritual development. This development is only possible if children lead a carefree life and interact with other children, play with them and learn with them. But a child labourer is deprived of these. He has to work for long hours which is not good for them. They are exploited and abused. Their innocent minds do not understand the perils of their working conditions. To eliminate child labour, is a tremendous task. Most child labourers come from poor homes, some of them have lost their parents and have to support their families. Unless the poverty of the family is removed and children get education, training and financial support, child labour cannot be banned or removed. I agree, that the promises made to the poor children are seldom kept. In the text, the author promised to Saheb that she would start a school and he would be able to read there. But she did not mean to build a school. So in the very first place the promise was not meant to be kept. It is an insincere promise. The government has made laws that no child can work in a glass-bangle industry. There are 20,000 poor children working there. They are still there inspite of the law. It is because the government is unable to provide basic necessities to them like employment, free education etc. So, the laws are not enforced. Moreover there is no political will. Poor children are helpless.

Q. ‘Lost Spring’ brings out the condition of some children in India who do not go to school, work in inhuman conditions and live in slums. We, as Indians, have failed in our duty in some way. What values do we need to inculcate among the People to bring back the ‘spring’ in the lives of these poor children?

In ‘Lost Spring’ Mukesh, Savita and Saheb are all victims of child labour. We have not understood their situation adequately. To bring back the ‘spring’ in the lives of these poor children, we must inculcate the values to

1. have a strong will to ensure that all children get basic education. This may be done by helping them join the ‘open school’ system.
2. have a sense of commitment of wanting to help these children; an example can be to find better employment for the adults in their families.
3. say ‘NO’ to child labour in any work related to us during the children’s school hours.
4. feel the need of doing something for such children, exemplifying the saying, ‘where there is a will, there is a way’.
5. create awareness in society about the plight of the underprivileged so that people in power can help them.

Q. ‘Lost Spring’ explains the grinding poverty and tradition that force little children to work at the age of mental and spiritual development. Write an article on ‘Child labour — A Blot On Our Society’.

Child Labour — A Blot on Our Society

Child labour is a curse in our society which condemns children to lead a life of exploitation. It deprives them of schooling and forces them to a perpetual state of poverty. None cares about their dreams which loom like a mirage. Children are forced to work in factories, dhabas, industries, tea stalls for long working hours and less wages. Insincere promises are made to them which are never kept. Child labourers feel helpless under the weight of poverty. Their joys of childhood are snatched away by the circumstances of life. Thus, it is a blot on our society. Children are future of a country. If they remain uneducated, the country will never progress. So child labour must be removed from the society. Free education should be provided to poor children to motivate their parents to send them to school. The government should come forward with some schemes for the upliftment of the poor and unemployed which will take away the burden of earning from the shoulders of poor children. Child labour must be eradicated.

Q. 3. Most of us do not raise our voice against injustice in our society and tend to remain mute spectators. Anees Jung in her story, ‘Lost Spring’ vividly highlights the miserable life of street children and bangle-makers of Firozabad. She wants us to act. Which qualities does she want the children to develop?

Anees Jung want the children to become free from the vicious cycle of poverty into which they have fallen due to the middleman, sahukars, and law enforcement officials. She wants them to be bold enough to raise their voice against their oppressors. She wants them to be fearless and optimistic so that they can dream of taking up other occupations, just like Mukesh, who wants to be a motor mechanic. She wants them to become free from their traditional occupation so that they can realise their lives ambitions. She sees the spark of such quality in Mukesh. Who is willing to go to any lengths to become a motor mechanic. She wants some people to help them develop these qualities so they can be free from injustice and exploitation, taking up other respectable and better paying jobs which will improve their financial condition.

Important Question for Class 12 English Flamingo Book Chapter 1 The Last Lesson

 The Last Lesson 2 Marks Important Questions (30 to 40 words)

Q.- Why did Franz not want to go to school that day?

A. M Hamel had asked the class to revise the grammar topic of Participles for a test. Franz did not know participles and feared the scolding. So, he did not want to go to school.

Q.- How is the mother tongue important to a person? What does M Hamel, the teacher say about it?

A. Mother tongue is the common factor which unites the countrymen. M Hamel made the villagers realize the importance of the mother tongue. He spoke about the beauty of their mother tongue – the French language. He asked the class to guard it because it was the key to their freedom.

Q1.- i. Why did the elders of the village attend the Last Lesson?
ii. Comment on the significance of the villagers sitting at the back in M Hamel’s classroom.

A. Berlin had ordered that French language would no longer be taught across schools in Alsace and Lorraine. The village elders were present in the class which was the last class of the French language. They were there to pay respect to the teacher, M Hamel who had taught there for forty years. They regretted not having attended school in their childhood days.

Q2.- What words did M Hamel write on the blackboard before dismissing the class? What did they mean?

A. Before dismissing the class, M Hamel wrote the following words on the blackboard – “Vive la France”. “Vive la France” means ‘Long live France’. It was a way of showing his love and support towards his mother tongue and his country.

Q3.- What changes did the order from Berlin bring about on the day of the last lesson?

A. The order from Berlin brought all the routine hustle-bustle of the school life to a stand- still. The teacher, M. Hamel, was kind towards his students and taught with more patience. The students became more attentive and concerned about education.

Q4.- How was Mr Hamel dressed differently that day? Why?

A. M Hamel was dressed in his special dress that he wore on a few occasions. It consisted of his beautiful green coat, frilled shirt and a little black silk cap, all embroidered. He wore the special dress because it was the last lesson that he would deliver in the school where he had been teaching for the last forty years.

Q5.- After sitting down at his desk, what unusual thing did Franz observe about M Hamel?

A. Franz observed that M. Hamel was wearing his special dress that he wore on selected occasions only. He was not holding the ruler in his hands. He was calm and kind towards the students. The village elders were present in the class too.

Q6.- Why was Franz not scolded for reaching school late that day?

A. M Hamel was very kind and calm that day because it was the last lesson in French. Berlin had ordered that instead of French, Germany would be taught in the schools of Alsace and Lorraine. So, M Hamel did not scold Franz for being late.

Q7.- How did M Hamel make his last lesson a special one? What did he emphasize on in it?

A. M Hamel made the last lesson by wearing his special dress to the class. He got new copies for the students which had the words “France, Alsace” written beautifully on them.

Q8.- How did Franz react to the declaration that it was their last French lesson?

A. Franz was shocked to know that he could not learn French anymore. He repented that he had not been serious before. He wished he had revised participles and would be able to answer M Hamel’s questions. Franz gained a sudden liking for his teacher and did not want him to leave.

Probability Class 12 Important Questions with Solutions

Question 1.
If P(not A) = 0.7, P(B) = 0.7 and P(B/A) = 0.5, then find P(A/B). (All India 2019)
Answer:

Question 2.
A die marked 1, 2, 3 in red and 4, 5, 6 in green is tossed. Let A be the event ‘number is even’ and B be the event ‘number is marked red’. Find whether the events A and B are independent or not. (Delhi 2019)
Or
A die, whose faces are marked 1,2, 3 in red and 4, 5, 6 in green , is tossed. Let A be the event “number obtained is even” and B be the event “number obtained is red”. Find if A and B are independent events. (All India 2017)
Answer:
When a die is thrown, the sample space is
S = {1, 2, 3, 4, 5, 6}
⇒ n(S) = 6
Also, A: number is even and B: number is red.
∴ A = {2, 4, 6} and B = {1, 2, 3} and A ∩ B = {2}
⇒ n(A) = 3, n(B) = 3 and n(A ∩ B) = 1

Thus, A and B are not independent events.

Question 3.
A black and a red die are rolled together. Find the conditional probability of obtaining the sum 8, given that the red die resulted in a number less than 4. (CBSE 2018)
Answer:
Let us denote the numbers on black die by B₁, B₂, ……., B₆ and the numbers on red die by R₁, R₂, ….., R₆.
Then, we get the following sample space.
s = {(B₁, R₁) ,(B₁, R₂), ……., (B₁, R₆), (B₂, R₂), ………, (B₆, B₁), (B₆, B₂), ……., (B₆,R₆)
Clearly, n(S) = 36
Now, let A be the event that sum of number obtained on the die is 8 and B be the event that red die shows a number less than 4.
Then, A = {(B₂, R₆), (B₆,R₂), (B₃,R₅), (B₅,R₃), (B₄,R₄)}
and B = {(B₁, R₁), (B₁,R₂), (B₁,R₃), (B₂,R₁), (B₂,R₂), (B₂,R₃) ,…….., (B₆,R₁), (B₆,R₂), (B₆,R₃)}
⇒ A ∩ B = {(B₆, R₂), (B₅, R₃)}
Now, required probability,
p(AB) = P(A∩B)P(B)=2361836=218=19

Question 4.
Evaluate P(A ∪ B), if 2P (A) = P(B) = 513 and P(A/ B) = 25. (CBSE 2018C)
Answer:

Question 5.
Prove that if E and F are independent events, then the events E and F’ are also independent. (Delhi 2017)
Answer:
Given, E and F are independent events, therefore
⇒ PE( ∩ F) = P(E) P(F) …….. (i)
Now, we have,
P(E ∩ F’) + P(E ∩ F) = P(E)
P(E ∩ F’) = P(E) – P(E ∩ F)
P(E ∩ F’) = P(E) – P(E ) P(F) [using Eq. (i))
P(E ∩ F’) = P(E) [1 – P(F)]
P (E ∩ F’) = P(E ) P(F’)
∴ E and F ‘are also independent events.
Hence proved.

Question 6.
A and B throw a pair of dice alternately. A wins the game, if he gets a total of 7 and B wins the game, if he gets a total of 10. If A starts the game, then find the probability that B wins. (Delhi 2016)
Answer:
Here, n(S) = 6 × 6 = 36
Let A = Event of getting a sum of 7 in pair of dice = {(1, 6), (2, 5), (3, 4), (6, 1), (5, 2), (4, 3)}
⇒ n(A) = 6
and B = Event of getting a sum of 10 in pair of dice = {(4, 6), (5, 5), (6, 4)} ⇒ n(B) = 3

Now, the probability that if A start the game, then B wins
P(B wins) = P(Ā ∩ B) + P (Ā ∩ B̄ ∩ Ā ∩ B) + P(Ā ∩ B̄ ∩ Ā ∩ B̄ ∩ Ā ∩ B) + …
= P(Ā) P(B) + P(Ā)P(B̄)P(Ā)P(B) + P(Ā)P(B̄)P(Ā)P(B̄)P(Ā) P(B) + ….. (1)
[∵ events are independent]

Question 7.
A and B throw a pair of dice alternately, till one of them gets a total of 10 and wins the game. Find their respective probabilities of winning, if A starts first. (All India 2016)
Answer:
Here, n(s) = 6 × 6 = 36
Let E = Event of getting a total 10
= {(4, 6), (5, 5), (6,4)}
∴ n(E) = 3
∴ P(getting a total of 10) = P(E) = n(E)n(S)=336=112
and P(not getting a total of 10) = P(Ē)
1 – P(E) = 1 – 112 = 111
Thus, P(A getting 10) = 112 and P(B getting 10) and P(A is not getting 10) = P(B is not getting 10) = 1112
Now, P(A winning) = = P(Ā ∩ B) + P (Ā ∩ B̄ ∩ Ā) + P(Ā ∩ B̄ ∩ Ā ∩ B̄ ∩ Ā) + …
= P(Ā) + P(Ā)P(B̄)P(A) + P(Ā)P(B̄)P(Ā)P(B̄)P(A) + ……..

Question 8.
Probabilities of solving a specific problem independently by A and B are 12 and 13, respectively. If both try to solve problem independently, then find the probability that
(i) problem is solved.
(ii) exactly one of them solves the problem. (All India 2015C, Delhi 2011)
Answer:
The problem is solved means atleast one of them solve it. Also, use the concept A and B are independent events, then their complements are also independent.
Let P(A) = Probability that A solves the problem
P(B) = Probability that B solves the problem
P(Ā) = Probability that A does not solve the problem
and P(Ā) = Probability that B does not solve the problem.
According to the question, we have

(i) P (problem is solved)
= P (A ∩ B̄) + P(Ā ∩ B) + P (A ∩ B)
= P(A) ∙ P(B̄) + P(Ā) ∙ P(B) + P(A) ∙ P(B)
[∵ A and B are independent events]
= (12×23)+(12×13)+(12×13)
= 26+16+16=46=23
Hence, probability that the problem is solved, is 23.

(ii) P (exactly one of them solve the problem)
= P (A solve but B do not solve) + P (A do not solve but B solve)
= P(A ∩ B̄) + P(Ā ∩ B)
= P(A) ∙ P(B̄) + P(Ā) ∙ P(B)
= (12×23) = (12×13) = 26+16=36=12

Alternate Method
P (problem is solved)
= 1 – P (none of them solve the problem)
= 1 – P(Ā ∩ B̄)
= l – P(Ā) ∙ P(B̄) = 1 – (12×23)
∵ P(Ā) = 12 and P(B̄) = 23
= 1 – 13 = 23

(ii) P (exactly one of them solve the problem)
= P(A) + P(B) – 2P(A ∩ B)
= P(A) + P(B) – 2P(A) × P(A)
= 12+13−2×12×13=12

Question 9.
A couple has 2 children. Find the probability that both are boys, if it is known that
(i) one of them is a boy.
(ii) the older child is a boy. (Delhi 2014C, All India 2014, 2010)
Answer:
Firstly, write the sample space of given data. Then, use concept of conditional probability
P(A / B) = P(A∩B)P(B) to get the desired result.

Let B and b represent older and younger boy child. Also, let G and g represent older and younger girl child. The sample space of the given question is S = {Bb, Bg, Gg, Gb}
∴ n(S) = 4
Let A be the event that both children are boys.
Then, A = {Bb}
∴ n(A) = 1

(i) Let B : Atleast one of the children is a boy
∴ B = {Bb, Bg, Gb} and n(B) = 3

(ii) Let C : The older child is a boy.
Then, C = {Bb, Bg}
∴ n(C) = 2

Question 10.
Assume that each born child is equally likely to be a boy or a girl. If a family has two children, then what is the conditional probability that both are girls? Given that
(i) the youngest is a girl?
(ii) atleast one is a girl? (Delhi 2014)
Answer:
Let B and b represent elder and younger boy child. Also, G and g represent elder and younger girl child. If a family has two children, then all possible cases are
S = {Bb, Bg, Gg, Gb}
∴ n(S) = 4
Let us define event A : Both children are girls, then A = {Gg} ⇒ n(A) = 1
(i) Let E₁ : The event that youngest child is a girl.
Then, E₁ = {Bg, Gg} and n(E₁) = 2

(ii) Let E₂: The event that atleast one is girl.
Then, E₂ = {Eg, Gg, Gb} ⇒ n(E₂) = 3,

Question 11.
A speaks truth in 75% of the cases, while B in 90% of the cases. In what percent of cases are they likely to contradict each other in stating the same fact? Do you think that statement of B is true? (All India 2013)
Answer:
Let A_T: Event that A speaks truth
and B_T: Event that B speaks truth.
Given, P(A_T) = 75100, then P(Ā_T) = 1 – 75100
[∵ P(Ā) = 1 – P(A)]
= 25100
and P(B_T) = 90100
Then, P(B̄_T) = 1 – 90100 = 10100
Now, P (A and B are contradict to each other)

∴ Percentage of P (A and B are contradict to each other) = 310 × 100 = 30%
Since, B speaks truth in only 90% (i.e. not 100%) of the cases, therefore we think, the statement of B may be false.

Question 12.
P speaks truth in 70% of the cases and Q in 80% of the cases. In what percent of cases are they likely to agree in stating the same fact? Do you think, when they agree, means both are speaking truth? (All India 2013)
Answer:
Let p_T: Event that P speaks truth
and Q_T: Event that Q speaks truth.

No, agree does not mean that they are speaking truth.

Question 13.
A speaks truth in 60% of the cases, while B in 90% of the cases. In what percent of cases are they likely to contradict each other in stating the same fact? In the cases of contradiction do you think, the statement of B will carry more weight as he speaks truth in more number of cases than A? (Delhi 2013)
Answer:
42% Yes

Question 14.
If A and B are two independent events such that P(Ā ∩ B) = 215 and (A ∩ B̄) = 16, then find P (A) and P (B). (Delhi 2015)
Answer:
Given, A and B are two independent events with
P(Ā ∩ B) = 215 and P(A ∩ B) = 16.
We know that, if A and B are independent, then Ā, B and A, B̄ are independent events.

Question 15.
Consider the experiment of tossing a coin. If the coin shows head, toss it again, but if it shows tail, then throw a die. Find the conditional probability of the event that ‘the die shows a number greater than 4’, given that ‘there is atleast one tail’. (Delhi 2014C)
Answer:
The sample space S of the experiment is given as
S = {(H, H), (H, T), (T, 1), (T, 2), (T, 3), (T, 4), (T, 5), (T, 6)}
The probabilities of these elementary events are

The outcomes of the experiment can be represented in the following tree diagram.

Consider the following events:
A = the die shows a number greater than 4 and
B = there is atleast one tail.
We have, A = {(T, 5), (T, 6)},
B = {(H, T), (T, 1), (T, 2), (T, 3), (T, 4), (T, 5), (T, 6)}
and A ∩ B = {(T, 5), (T, 6)}
∴ P(B) = P((H, T)) + P((T, 1)) + P((T, 2)) + P((T, 3)) + P((T, 4)) + P((T, 5)) + P((T, 6))
⇒ P(B) = 14+112+112+112+112+112+112=34
and P(A ∩ B) = P((T, 5)) + P((T, 6)) = 112+112=16
∴ Required probability
= P(AB)=P(A∩B)P(B)=1/63/4=418=29

Topic 2: Baye’s Theorem and Probability Distributions

Question 1.
Find the probability distribution of X, the number of heads in a simultaneous toss of two coins. (All India 2019)
Answer:
When two coins are tossed, there may be 1 head, 2 heads or no head at all. Thus, the possible values of X are 0, 1, 2.
Now, P(X = 0) = P (Getting no head) = P(TT) = 14
P(X = 1) = P (Getting one head) = P(HT or TH) = 24=12
P(X = 2) = P (getting two heads) = P(HH) = 14
Thus, the required probability distribution of X is

Question 2.
The random variable X has a probability distribution P(X) of the following form, where ‘k’ is some number.

Determine the value of ‘k’. (Delhi 2019)
Answer:
Given,

Making it in tabular format, we get the following

Since, sum of all probabilities is equal to 1. (112)
ΣP(X = x) = 1
⇒ P(X = 0) + P(X = 1) + P(X = 2) + 0 + 0 + ……. = 1
⇒ k + 2k + 3k = 1
⇒ 6k = 1 ⇒ k = 16

Question 3.
Suppose a girl throws a the. If she gets 1 or 2, she tosses a coin three times and notes the number of tails. If she gets 3, 4, 5 or 6, she tosses a coin once and notes whether a ‘head’ or ‘tail’ is obtained. If she obtained exactly one ‘tail’, what is the probability that she threw 3, 4, 5 or 6 with the die? (C8SE 2019)
Answer:
Let E₁ be the event that the girl gets 1 or 2.
E₂ be the event that the girl gets 3, 4, 5 or 6
and A be the event that the girl gets exactly a ‘tail’.
Then, P(E₁) = 26=13
and P(E₂) = 46=23
P(AE1) = P (getting exactly one tail when a coin is tossed three times) = 38
P(AE2) = P (getting exactly a tail when a coin is tossed once) = 12
Now, required probability

Question 4.
Two numbers are selected at random (without replacement) from the first five positive integers. Let X denotes the larger of the two numbers obtained. Find the mean and variance of X. (CBSE 2018)
Answer:
Total number of possible outcomes
= ⁵P₂ = 5!3! = 5 × 4 = 20
Here, X denotes the larger of two numbers obtained.
∴ X can take values 2, 3, 4 and 5.
Now, P(X = 2) = P (getting ( 1. 2) or (2, 1)) = 220=110
P(X = 3) = P (getting (1, 3) or (3, 1) or (2, 3) or (3, 2) = 420=210
P(X = 4) = P(getting(1,4) or (4, 1)or (2, 4) or (4, 2) or (3, 4) or (4, 3)) = 620=310
and P(X = 5) = P (getting (1, 5) or (5, 1) or (2, 5) or (5, 2) or (3, 5) or (5, 3) or (4, 5) or (5, 4))
= 820=410
Thus, the probability distribution of X is

Now, mean of X is E(X) = Σ X ∙ P(X)

Question 5.
Two groups are competing for the positions of the Board of Directors of a corporation. The probabilities that the first and second groups will win are 0.6 and 0.4 respectively. Further, if the first group wins, the probability of introducing a new product is 0.7 and the corresponding probability is 0.3 if the second group wins. Find the probability that the new product introduced way by the second group. (CBSE 2018C)
Answer:
Let E₁ and E₂ denote the events that first and second group will win. Then,
P(E₁) = 0.6 and P(E₂) = 0.4
Let E be the event of introducing the new product.
Then, P(EE1) = 0.7 and P(EE2) = 0.3
Now, we have to find the probability that new product is introduced by second event.

Class 12 Maths Chapter 12 Important Extra Questions Linear Programming

Linear Programming Important Extra Questions Very Short Answer Type

Question 1.
Draw the graph of the following LPP:
5x + 2y ≤ 10, x ≥ 0,y ≥ 0.
Solution:
Draw the line AB : 5.v + 2y = 10 …(1),
which meets x-axis at A (2, 0) and y-axis at B (0,5).
Also x = 0 is y-axis and y = 0 is x-axis.
Hence, the graph of the given LPP is as shown (shaded):

Question 2.
Solve the system of linear inequations: x + 2y ≤ 10; 2x + y ≤ 8.
Solution:
Draw the st. lines x + 2y = 10 and 2x + y = 8.
These lines meet at E (2,4).
Hence, the solution of the given linear inequations is shown as shaded in the following figure :

Question 3.
Find the linear constraints for which the shaded area in the figure below is the solution set:

Solution:
From the above shaded portion, the linear constraints are :
2x + y ≥ 2,x – y ≤ 1,
x + 2y ≤ 8, x ≥ 0, y ≥ 0.

Question 4.
A small firm manufactures neclaces and bracelets. The total number of neclaces and bracelets that it can handle per day is at most 24. It takes one hour to make a bracelet and half an hour to make a neclace. The maximum number of hours available per day is 16. If the profit on a neclace is ₹100 and that on a bracelet is ₹300. Formulate an LPP for finding how many of each should be produced daily to maximize the profit ?
It is being given that at least one of each must be produced. (C.B.S.E. 2017)
Solution:
Let ‘x’ neclaces and ‘y’ bracelets be manufactured per day.
Then LPP problem is:
Maximize Z = 100x+300y
Subject to the constraints : x + y ≤ 24,
(1) (x) + 12y ≤ l6,
i.e. 2x + y ≤ 32
and x ≥ 1
and y ≥ 1
i.e. x – 1 ≥ 0
and y – 1 ≥ 0.

Question 5.
Old hens can be bought for ?2.00 each and young ones at ?5.00 each. The old hens lay 3 eggs per week and the young hens lay 5 eggs per week, each egg being worth 30 paise. A hen costs ₹1.00 per week to feed. A man has only ₹80 to spend for hens. Formulate the problem for maximum profit per week, assuming that he cannot house more than 20 hens.
Solution:
Let ‘x’ be the number of old hens and ‘y’ the number of young hens.
Profit = (3x + 5y) 30100 – (x + y) (1)
= 9×10+32yx – y
= y2−x10=5y−x10
∴ LPP problem is:
Maximize Z = 5y−x10 subject to:
x ≥ 0,
y ≥ 0,
x + y ≤ 20 and
2x + 5y ≤ 80.

Linear Programming Important Extra Questions Very Long Answer Type 2

Question 1.
Maximize Z-5x + 3y
subject to the constraints:
3x + 5y ≤ 15, 5x + 2y ≤ 10, x ≥ 0,y ≥ 0. (N.C.E.R.T)
Solution:
The system of constraints is :
3x + 5y ≤ 15 …(1)
5x + 2y ≤ 10 …(2)
and x ≥ 0, y≥ 0 …(3)
The shaded region in the following figure is the feasible region determined by the system of constraints (1) – (3):

It is observed that the feasible region OCEB is bounded. Thus we use Corner Point Method to determine the maximum value of Z, where :
Z = 5x + 3y …(4)

The co-ordinates of O, C, E and B are (0, 0), (2,0), (2019,4519) (Solving 3x + 5y = 15 and 5x + 2y – 10) and (0, 3) respectively.
We evaluate Z at each comer point:

Comer Point	Corresponding Value of Z
O: (0,0)	0
C: (2,0)	10
E((2019,4519))	2019 (Maximum)
B(0.3)	9

Hence’ Z_max = at the Point (2019,4519)

Question 2.
Minimize Z = 3x + 2y subject to the constraints:
x +y ≥ 8, 3x + 5y ≤ 15, x ≥ 0, y ≥ 0. (N.C.E.R.T.)
Solution:
The system of constraints is :
x +y ≥ 8, , x ≥ 0, y ≥ 0…(1)
3x + 5y ≤ 15 …(2)
and x ≥ 0, y ≥ 0 …(3)

It is observed that there is no point, which satisfies all (1) – (3) simultaneously.
Thus there is no feasible region.
Hence, there is no feasible solution.

Question 3.
Determine graphically the minimum value of the objective function :
Z = – 50x + 20y
subject to the constraints:
2x-y ≥ – 5, 3x +y ≥ 3, 2x – 3y ≤ 12, x,y ≥ 0. (N.C.E.R.T.)
Sol. The system of constraints is :
2x-y ≥ – 5 …(1)
3x +y ≥ 3 …(2)
2x – 3y ≤ 12 …(3)
and x,y ≥ 0 …(4)
The shaded region in the following figure is the feasible region determined by the system of constraints (1) – (4).

It is observed that the feasible region is unbounded.
We evaluate Z = – 50x + 20y at the corner points :
A (1, 0), B (6, 0), C (0, 5) and D (0, 3) :

Corner Point	Corresponding Value of Z
A: (1,0)	-50
B : (6, 0)	– 300 (Minimum)
C : (0, 5)	100
D : (0, 3)	60

From the table, we observe that – 300 is the minimum value of Z.
But the feasible region is unbounded.
∴ – 300 may or may not be the minimum value of Z. ”

For this, we draw the graph of the inequality.
– 50x + 20y < – 300
i.e. – 5x + 2y < – 30.
Since the remaining half-plane has common points with the feasible region,
∴ Z = – 50x + 20y has no minimum value.

Question 4.
Minimize and Maximize Z = 5x + 2y subject to the following constraints : x – 2y ≤ 2, 3x + 2y < 12, -3x + 2y ≤ 3, x ≥ 0, y ≥ 0. (A.I.C.B.S.E. 2015)
Solution:
The given system of constraints is :
x – 2y ≤ 2 …(1)
3x + 2y < 12 …(2)
-3x + 2y ≤ 3 …(3)
and x ≥ 0, y ≥ 0.

The shaded region in the above figure is the feasible region determined by the system of constraints (1) – (4). It is observed that the feasible region OAHGF is bounded. Thus we use Corner Point Method to determine the maximum and minimum value of Z, where
Z = 5x + 2y …(5)

The co-ordinates of O, A, H, G and F are :
(0, 0). (2. 0), (72,34) and (32,154), 32)
respectively. [Solving x

2y = 2 and 3x + 2y = 12 for
H and -3x + 2y = 3 and
3x + 2y = 12 for G]
We evaluate Z at each cormer point:

Corner Point	Corresponding value of Z
O: (0,0)	0 (Minimum)
A: (2,0)	10
H(72,34)	19 (Maximum)
G(32,154)	15
F: (0, 32))	3

Hence, Z_max = 19 at (72,34) and
Z_max = 0at (0,0)

Question 5.
A dealer in rural area wishes to purchase a number of sewing machines. He has only ₹5760.00 to invest and has a space for at most 20 items. An electronic sewing machine costs him ₹360.00 and a manually operated sewing machine ₹240.00. He can sell electronic sewing machine at a profit of ₹22.00 and a manually operated sewing machine at a profit of ₹18.00. Assuming that he can sell all the items that he can buy, how should he invest his money in order to maximize his profit. Make it a linear programming problem and solve it graphically. (C.B.S.E. 2014)
Solution:
Let ‘x’ be the number of electronic operated machines and ‘y’ that of manually operated machines be purchased.
Then the LPP problem is as follows :
Maximize:
Z = 22 + 18y
Subject to:
x + y ≤ 20 …(1)
360x + 240y ≤ 5760
i.e. 3x + 2y ≤ 48 …(2)
and x ≥ 0, y ≥ 0 …(3)
The shaded region of the figure represents the feasible region OCEB, which is bounded.

Applying Corner Point Method, we have:

Corner Point	Corresponding value of Z
O: (0,0) C:(16,0) E: (8,12) B: (0,20)	0 352 392 (Maximum) 360

Thus, Z is maximum at E (8, 12).
Hence, the dealer should invest in 8 electronic and 12 manually operated machines.

Question 6.
A manufacturer produces nuts and bolts. It takes 1 hour of work on machine A and 3 hours on machine B to produce a package of nuts. It takes 3 hours on machine A and 1 hour on machine B to produce a package of bolts. He earns a profit of ₹ 35 per package of nuts and ₹ 14 per package of bolts. How many packages of each should be produced each day so as to maximise his profit, if he operates each machine for atmost 12 hours a day? Convert it into an LPP and solve graphically.
Solution:
Let ‘x’ and ‘y’ be the number of packages of nuts and bolts respectively.
We have the following constraints :
x ≥ 0 …(1)
y ≥ 0 …(2)
x + 3y ≤ 12 ….(3)
3x + y ≤ 12 …(4)
Now the profit,P= 35x+ 14y …..(5)
We are to maximize P subject to constraints (1) -(4).
Draw the line AB (x + 3y = 12)
Draw the line CD (3x + y = 12)
These meet at E (3, 3).
The shaded region in the figure represents the feasible region, which is bounded.

Applying Corner Point Method, we have :

Corner Point	P = 35x + 14y
O: (0,0)	0
C : (4, 0)	140
E: (3,3)	147 (Maximum)
B : (0,4)	56

Hence, max. profit is ₹ 147 and it is obtained when 3 packages each of Nuts and Bolts are produced daily.

Question 7.
Two tailors A and B earn ₹ 150 and ₹ 200 per day respectively. A can stitch 6 shirts and 4 pants per day, while B can stitch 10 shirts and 4 pants per day. Form a L.P.P. to minimize the labour cost to produce (stitch) at least 60 shirts and 32 pants and solve it graphically.
Solution:
Let the tailor A work for ‘x’ days and B for ‘y’ days.

Thus, we have the following constraints:
x ≥ 0 …(1)
y ≥ 0 …(2)
6x + 10y ≥ 60
i.e. 3x + 5y ≥ 30 …….(3)
4x + 4y ≥ 32
i.e. x + y ≥ 8 ……(4)
The objective function, or the cost Z is:
Z = 150x + 200 y ……..(5)
For the solution set, we draw the lines:
x = 0, y – 0, 3x + 5y = 30, x + y = 8

The feasible region (shaded) is unbounded. Let us evaluate Z at the comer points:
A (10,0), D(0, 8) and E(5, 3)
[Solving x + y – 8, Sx + 5y – 30; x – 5,y = 3]

Applying Comer Point Method, we have:

Corner Point	Z = 150 x + 200y
A : (10, 0) E: (5,3) D: (0, 8)	1500 1350 (Minimum) 1600

Hence, the tailor A should work for 5 days and B for 3 days.
To Check: Draw 150x + 200y < 1350 i.e. 3x + 4y < 27.
Since there is no region common with feasible region,
∴ Minimum value is ₹ 1350.

Question 8.
A dietician wishes to mix two types of food in such a way that the vitamin contents of the mixture contains at least 8 units of vitamin A and 10 units of vitamin C. Food I contains 2 units/kg of vitamin A and 1 unit/kg of vitamin C. It costs ₹50 per kg to produce food I. Food II contains 1 unit/kg of vitamin A and 2 units/kg of vitamin C and it costs ₹70 per kg to produce food n. Formulate this problem as a LPP to minimise the cost of a mixture that will produce the required diet. Also find the minimum cost
Solution:
Let the quantity of Food I = x kg
and the quantity of Food II = y kg.
Then the LPP problem is as below :
Z = 50x+70y …(1)
Subject to 2x + y ≥ 8 …(2)
x + 2y ≥ 10 ….(3)
and x ≥ 0, y ≥0 ……..(4)
For the solution, we draw the lines :
x = 0,y = 0,2x + y = 8 and x + 2y = 10

The feasible regin is as shown with vertices C(10,0), E(2,4) and B(0,8).
Applying Corner Point Method, we have

Corner Point	Z = 50x + 70y
C: (10,0) E:(2,4) B: (0,8)	500 380 (Minimum) 560

Thus minimum cost is ₹38Q when 2 kg of Food I and 4 kg of Food II are mixed.

Question 9.
A manufacturer makes two types of toys A and B. Three machines are needed for this purpose and the time (in minutes) required for each toy on the machines is given below:

Types of Toys	Machines
I	II	III
A	20	10	10
B	10	20	30

The machines I, II and III are available for a maximum of 3 hours, 2 hours and 2 hours 30 minutes respectively. The profit on each toy of type A is ₹50 and that of type Bis ₹60. Formulate the above problem as a LPP and solve it graphically to maximize profit.
(C.B.S.E. Sample Paper 2018-19)
Solution:
Let ‘x’ and ‘y’ be the number of toys of type A and type B respectively.
Then maximize :
P = 50x + 60y …(1)
Subject to constraints :
20x +10v ≤ 180 …(2)
10x + 20y ≤ 120 …….(3)
10x + 30y ≤ 150 ……(4)
and x ≥ 0, y ≥ 0 …(5)

Applying Corner Point Method we have :

Corner point	P = 50jc + 60y
O: (0,0) F:(0,5) G: (6,3) H: (8,2) A: (9,0)	0 300 480 520 (Maximize) 450

Hence, maximum profit is ₹520 when x = 8 and y = 2. i.e., when 8 toys of type A and 2 toys of type B are made.

Question 10.
A factory manufactures two types of screws A and B, each type requiring the use of two machines, an automatic and a hand- operated. It takes 4 minutes on the automatic and 6 minutes on the hand operated machines to manufacture a packet of screws ‘A’ while it takes 6 minutes on the automatic and 3 minutes on the hand-operated machine to manufacture a packet of screws ‘B\ Each machine is available for at most 4 hours on any day. The manufactures can sell a packet of screws ‘A’ at a profit of 70 paise and screws ‘B’ at a profit of ₹ 1. Assuming that he can sell all the screws he manufactures, how many packets of each type should the factory owner produce in a day in order to maximize his profit? Formulate the above LPP and solve it graphically and find the maximum profit (C.B.S.E. 2018)
Solution:
Let the factory manufacture ‘x’ of type ‘A’ and ‘y’ of type ‘B’
Clearly x ≥ 0 …(1)
and y ≥ 0 …(2)
Since the machines can operate for at the most 4 hours a day,
4x + 6y ≤ 240
i.e., 2x + 3y ≤ 120 …(3)
and 6x + 3y ≤ 240
i.e., 2x + y ≤ 80 …(4)
The objective function or the profit, P, is:
P = 0.7 x + y …(5)

We drawn the lines :
x = 0,y = 0,
2x + 3y = 120
and 2x + y – 80

The feasible region is shown shaded OCPB is bounded, where O is (0, 0), C is (40, 0), B is (0, 40) and P is (30, 20).
[Solving 2x + y – 80 and 2x + 3y = 120; x = 30, y – 20]

Applying Corner Point Method, we have :

Comer point	P = 0.7x + y
O: (0, 0) C : (40, 0) P : (30, 20) B: (0,40)	0 28 41 (Maximum) 40

Hence, in order to maximize profit 30 packets of screw ‘A’ and 20 packets of screw ‘B’ should be manufactured and maximum profit = ₹ 41.

Question 11.
A small firm manufactures chairs and tables. Market demand and available resources indicate that the continued production of chairs and tables should not exceed 50 units per day. It takes 30 minutes to manufacture a chair and 1 hour to manufacture a table. A maximum of 40 man-hours per day are available. The profit on each chair is ₹ 40 and profit on each table is ₹ 60. Determine how many each of chairs and tables should be manufactured per day in order to maximize the profit. What is the maximum profit? Formulate LPPand solve graphically.
Solution:
Let ‘x’ and ‘y’ be the number of chairs and tables respectively.
We have: x ≥ 0 ………..(1)
y ≥ 0 ………(2)
x + y ≤ 50 ……(3)
and x2 + y ≤ 40
⇒ x + 2y < 80 …(4)
The objective function, or the profit, Z is
Z = 40 x + 60 y …(5)

We have to maximize Z subject to (1) – (4).
For solution set, we draw the lines:
x = 0, y – 0, x + y = 50 and x + 2y – 80.
The lines x + y = 50 and x + 2y = 80 meet at E (20, 30).

The shaded portion represents the feasible region, which is bounded.
Applying Corner Point Method, we have:

Corner Point	Z = 40x + 60y
0 =(0,0) A: (50, 0) E: (20,30) B: (0,40)	0 2000 2600 (Maximum) 2400

Hence, the maximum profit is ₹ 2600 when 20 chairs and 30 table are manufactured.

Class 12 Maths Chapter 11 Important Extra Questions Three Dimensional Geometry

Three Dimensional Geometry Important Extra Questions Very Short Answer Type

Question 1.
Find the acute angle which the line with direction-cosines <13√,16√,n> makes with positive direction of z-axis. (C.B.S.E. Sample Paper 2018-19)
Solution:
l² + m² + n² = 1
(13√)2+(16√)2 + n² = 1
⇒ 13+16 + n² = 1
n² = 1 – 12
n² = 12
n = 12√
Thus, cos α = 12√
Hence, α = 45° or π4

Question 2.
Find the direction-cosines of the line.
x−12=−y=z+12 (C.B.S.E. Sample Paper 2018-19)
Solution:
The given line is x−12=y−1=z+12
Its direction-ratios are <2,-1,2>.
Hence, its direction- cosine are:

Question 3.
If α, β, γ are direction-angles of a line, prove that cos 2a + cos 2P + cos 2y +1 = 0. (N.C.E.R.T.)
Solution:
Since α, β, γ are direction-angles of a line,
∴ cos² α + cos² β + cos²γ = 1

⇒ 1 + cos2α + 1 + cos2β + 1 + cos2γ = 2
⇒ cos 2α + cos 2β + cos 2γ + 1 = 0, which is true.

Question 4.
Find the length of the intercept, cut off by the plane 2x + y – z = 5 on the x-axis.   (C.B.S.E. Outside Delhi 2019)
Solution:
The given plane is2x + y – z = 5
⇒ x5/2+y5+z−5=1
Its intercepts are x5/2, 5 and -5.
Hence, the length of the intercept on the x-axis is x5/2

Question 37.
Find the length of the perpendicular drawn from the point P(3, -4,5) on the z-axis.
Solution:
Length of the perpendicular from P(3, -4,5) on the z-axis
= (3)2+(−4)2−−−−−−−−−−√
= 9+16−−−−−√=25−−√ = 5 units.

Question 5.
Find the vector equation of a plane, which is at a distance of 5 units from the origin and whose
normal vector is 2i^−j^+2k^
Solution:
Let n⃗ =2i^−j^+2k^

Question 6.
If a line makes angles 90°, 135°, 45° with the x,y and z-axes respectively, find its direction cosines.
Solution:
Direction cosines of the line are :
< cos 90°, cos 135°, cos 45° >
<0, −12√,12√>

Question 7.
Find the co-ordinates of the point where the line through the points A (3,4,1) and B (5,1, 6) crosses the xy-plane.
The equations of the line through A (3,4,1) and B (5,1,6) are:

Any point on (1) is (3 + 2k,4- 3k, 1 + 5k) …………. (2)
This lies on xy-plane (z = 0).
∴ 1 + 5k = 0 ⇒ k = −15
Putting in (2), [ 3-25, 4 + 35, 1-1)
i.e. (135, 235, 0)
which are the reqd. co-ordinates of the point.

Question 8.
find the vector equation ofthe line which passes through the point (3,4,5) and is parallel to the vector 2i^+2j^−3k^
Solution:
The vector equation of the line is r⃗ =a⃗ +λm⃗ 
i.e., r⃗ =(3i^+4j^+5k^)+λ(2i^+2j^−3k^)

Three Dimensional Geometry Important Extra Questions Short Answer Type

Question 1.
Find the acute angle between the lines whose direction-ratios are:
< 1,1,2 > and <-3, -4,1 >.
Solution:

Question 2.
Find the angle between the following pair of lines:
and
−x+2−2=y−17=z+3−3 and x+2−1=2y−84=z−54
and check whether the lines are parallel or perpendicular. (C.B.S.E. 2011)
Solution:
The given lines can be rewritten as :
−x+2−2=y−17=z+3−3 ………….. (1)
x+2−1=2y−84=z−54 ………..(2)
Here < 2,7, – 3 > and < -1,2,4 > are direction- ratios of lines (1) and (2) respectively.

Hence, the given lines aife perpendicular.

Question 3.
Find the vector equation of the line joining (1.2.3) and (-3,4,3) and show that it is perpendicular to the z-axis. (C.B.S.E. Sample Paper 2018-19)
Solution:
Vector equation of the line passing through
(1.2.3) and(-3,4,3)is r⃗ =a⃗ +λ(b⃗ −a⃗ )
where a⃗ =i^+2j^+3k^ and b⃗ =−3i^+4j^+3k^
⇒ r⃗ =(i^+2j^+3k^)+λ(−4i^+2j^) …(1)
Equation of z-axis is r⃗ =μk^ …(2)
Since (−4i^+2j^)⋅k^=0 = 0,
∴ Line (1) is perpendicular to z-axis.

Question 4.
Find the vector equation of the plane, which is 629√ at a distance of
units from the origin and its normal vector from the origin is 2i^−3j^+4k^ . Also, find its cartesian form. (N.C.E.R.T.)
Solution:

Question 5.
Find the direction-cosines of the unit vector perpendicular to the plane r⃗ ⋅(6i^−3j^−2k^) +1 = 0 through the origin. (N.C.E.R.T.)
Solution:
The given plane is r⃗ ⋅(6i^−3j^−2k^) + 1 = 0
r⃗ ⋅(6i^−3j^−2k^) = 1 ………… (1)
Now |−6i^+3j^+2k^|=36+9+4−−−−−−−−√
=49−−√=7
Dividing (1) by 7,
r⃗ ⋅(−67i^+37j^+27k^)=17
which is the equation of the plane in the form r⃗ ⋅n^=p
Thus, n^=−67i^+37j^+27k^
which is the unit vector perpendicular to the plane through the origin.
Hence, the direction-cosines of n^ are <−67,37,27>

Question 6.
Find the acute angle between the lines
x−43=y+34=z+15 and x−14=y+1−3=z+105
Solution:
Vector in the direction of first line
x−43=y+34=z+15 ,
b⃗ =(3i^+4j^+5k^)

Vector in the direction of second line
x−14=y+1−3=z+105 ,
d⃗ =4i^−3j^+5k^
∴ θ, the angle between two given lines is given by:

Question 7.
Find the angle between the line:
r⃗ =(i^−j^+k^)+λ(2i^−j^+3k^) and the plane r⃗ ⋅(2i^+j^−k^)=4 Also, find whether the line is parallel to the plane or not .
Solution:
The given line is :
r⃗ =(i^−j^+k^)+λ(2i^−j^+3k^)
and the given plane is r⃗ ⋅(2i^+j^−k^) = 4.
Now the line is parallel to 2i^ – j^ + 3k^ and nor¬mal to the plane 2i^ + i^ – k^
If ‘θ’ is the angle between the line and the plane,
then (π2−θ) is the angle between the line and normal to the plane.
Then

Hence, the line is parallel to the plane.

Question 8.
Find the value of ‘λ’, so that the lines:
1−x3=7y−14λ=z−32 and 7−7×3λ=y−51=6−z5 are at right angles. Also, find whether the lines are intersecting or not
Solution:
(i) The given lines are
1−x3=7y−14λ=z−32 ……………….(1)
and 7−7×3λ=y−51=6−z5 ……….. (2)
These are perpendicular if:

Hence λ = 1.

(ii) The direction cosines ofline(1) are <-3,1,2>
The direction cosines of line (2) are < -3,1, -5 >
Clearly, the lines are intersecting.

Question 9.
Find the angle between the line: x−23=y+1−1=z−3−2 and the plane: 3x + 4y + z + 5 = 0.
x-2 y+1 z-3
Sol. The given line is x−23=y+1−1=z−3−2 ………..(1)
and the given plane is :
3x + 4y + z + 5 = 0 …(2)
If the line (1) makes an angle ‘0’ with the plane (2), then the line (1) will make angle (90° – 0) with the normal to the plane (2).
Now direction-ratios of line (1) are:
<3, -1,-2>
and direction-ratios of normal to plane (2) are <3,4,1>.
∴ cos (90° – θ)

Question 10.
State when the line r⃗ =a⃗ +λb⃗  is parallel to the plane r⃗ ⋅n⃗ =d⃗  . Show that the line r⃗ =i^+j^+λ(2^+j^+4k^) is parafiel to the plane r⃗ ⋅(−2i^+k^) = 5. Also, find the distance between the line and the plane.
Solution:
(i) A line is parallel to the plane if it is perpendicular to the normal to the plane.
The given line is r⃗ =a⃗ +λb⃗ 
⇒ b⃗  is parallel to the line.
The given plane is r⃗ ⋅n⃗ =d⃗ 

⇒ n⃗  is normal to the plane.
Thus the line is parallel to the plane when
b⃗ ⋅n⃗  =0.

(ii) Here b⃗ =2i^+j^+4k^ and n⃗ =−2i^+k^
Now b⃗ ⋅n⃗  = (2) (- 2) + (1) (0) + (4) (1)
= -4 + 0 + 4 = 0.
Hence, the given line is parallel to the given plane.

(iii) (1,1,0) is a point on the given line.
Equation of the plane is-2x + z- 5= 0.
∴ Reqd. distance
= ∣∣−2(1)+0−54+0+1√∣∣=75√=75√5units.

Three Dimensional Geometry Important Extra Questions Very Long Answer Type 2

Question 1.
Find the shortest distance between the lines:r⃗ =(4i^−j^)+λ(i^+2j^−3k^) and r⃗ =(i^−j^+2k^)+μ(2i^+4j^−5k^) (C.B.S.E. 2018)
Solution:
Comparing given equations with:

Question 2.
A line makes angles α, β, γ, δ with the four diagonals of a cube, prove that:
cos² α + cos² β + cos² γ + cos² δ= 43. (N.C.E.R.T.)
Solution:
Let O be the origin and OA, OB, OC (each = a) be the axes.

Thus the co-ordinates of the points are :
O (0,0,0), A (a, 0,0), B (0, a, 0), C (0,0, a),
P (a, a, a), L (0, a, a), M (a, 0, a), N (a, a, 0).
Here OP, AL, BM and CN are four diagonals.
Let < l, m, n > be the direction-cosines of the given line.

Now direction-ratios of OP are:
<a-0,a-0,a-0>i.e.<a,a,a>
i.e. < 1,1,1 >,
direction-ratios of AL are:
<0-a, a-0, a-0> i.e. <-a,a,a>
i.e. <-l, 1,1 >,
direction-ratios of BM are:
<a-0,0-a, a-0>
i.e. <a,-a,a> i.e. < 1,-1, 1 >
and direction-ratios of CN are:
<a-0,a-0,0-a> i.e. <a,a,-a>
i.e. < 1,1,-1 >.

Thus the direction-cosines of OP are :
<13√,13√,13√>
the direction-cosines of AL are:
<−13√,13√,13√>
the direction-cosines of BM are :
<13√,−13√,13√>
and the direction-cosines of CN are :
<13√,13√,−13√>
If the given line makes an angle ‘a’ with OP, then :

and cos δ = |l+m−n|3√ ………… (4)
Squaring and adding (1), (2), (3) and (4), we get:
cos² α + cos² β + cos² γ + cos²δ
= 13 [(l + m + n)² + (-l + m + n)²
+ (l-m + n)² + (l + m — n)²]
= 13 [4(l² + m² + n²)] = 13 [4(1)].
Hence,cos² α + cos² β + cos² γ + cos²δ = 43

Question 3.
Find the equation of the plane through the line x−13=y−42=z−4−2 and parallel to the line:
x+12=1−y4=z+21
Hence, find the shortest distance between the lines. (C.B.S.E. Sample Paper 2018-19)
Solution:
The two given lines are:
x−13=y−42=z−4−2 ………… (1)
and x+12=1−y4=z+21 ………….. (2)
Let <a, b, c> be the direction-ratios of the normal to the plane containing line (1).
∴ Equation of the plane is:
a(x- l) + b(y-4) + c(z-4) …(3),
where 3a + 2b – 2c = 0 …(4)
[∵ Reqd. plane contains line (1)] and 2a – 4b + 1.c = 0
[∵ line (1) a parallel to the reqd. plane] Solving (4) and (5),

Putting in (3),
6k(x- 1) + 7k(y – 4) + 16k(z – 4) = 0
= 6(x – 1) + 7(y – 4) + 16(z – 4) =0
[∵k ≠ 0]
⇒ 6x + 7y+ 16z-98 = 0,

which is the required equation of the plane.
Now, S.D. between two lines = perpendicular distance of (-1,1, – 2) from the plane

6(—1) + 7(1) +16(-2) – 98
V(6)2+(7)2+(16)2
-6 + 7-32-98 V36 + 49 + 256

Question 4.
Find the Vector and Cartesian equations of the plane passing through the points (2, 2, -1), (3,4,2) and (7,0,6). Also, find the vector equa¬tion of a plane passing through (4,3,1) and parallel to the plane obtained above. (C.B.S.E. 2019)
Solution:
(i) Cartesian equations
Any plane through (2,2, -1) is :
a(x – 2) + b(y- 2) + c(z + 1) = 0 … (1)
Since the plane passes through the points (3,4,2) and (7,0,6),
∴ a(3 – 2) + b(4 – 2) + c(2 +1) = 0
and a(7 – 2) + b(0 – 2) + c(6 + 1) = 0
⇒ a + 2b + 3c = 0 …(2)
and 5a – 2b + 7c = 0 …(3)
Solving (2) and (3),a14+6=b15−7=c−2−10
⇒ a20=b8=c−12
⇒ a5=b2=c−3 = k (say), value k ≠ 0.
∴ a = 5k,b = 2k and c = -3k,
Putting the values of a, b, c in (1), we get:
5k(x – 2) + 2k(y – 2) – 3k(z + 1) = 0
⇒ 5(x-2) + 2(y-2)-3(z+ 1) =0[∵ k ≠ 0]
=» 5x- 10 + 2y-4-3z-3 = 0
=» 5x + 2y-3z-17 = 0, …(4)
which is the reqd. Cartesian equation.
Its vector equation is r⃗ ⋅(5i^+2j^−3k^) =17.

(ii) Any plane parallel to (4) is
5x + 2y – 3z + λ – 0 … (5)
Since it passes through (4, 3,1),
5(4) + 2(3) – 3(1) + λ = 0
⇒ 20 + 6 — 3 + λ = 0
⇒ λ = -23.
Putting in (5), 5x + 2y – 3z – 23 = 0, which is the reqd. equation.
Its vector equation is r⃗ ⋅(5i^+2j^−3k^) = 23.

Question 5.
Find the co-ordinates of the foot of the perpendicular drawn from the point A (1,8,4) to the line joining B (0, -1,3) and C (2,-3,-1). (A.I.C.B.S.E. 2016)
Solution:
Any point on BC, which divides [BC] in the ratio k: 1,is:

This becomes M, the foot of perp. from A on BC
if AM⊥BC …(2)
But direction-ratios of BC are:
<2-0,- 3 + 1,-1 -3 > i.e. < 2,-2,-4 >
i.e, <1, -1 > -2>
and direction-ratio of AM are:
<2kk+1−1,−3k−1k+1−8,−k+3k+1−4>
i.e. < k- 1,- 11k:-9, -5k— 1 >
∴ Due to (2), (1) (k- 1) + (- 1) (-1 1k-9) + (-2)(-5k- 1) = 0
⇒ k – 1 + 11k + 9 + 10k + 2 = 0
⇒ 22k + 10 = 0
⇒ k = −511
∴ From (1), the co-ordinates of M, the foot of perp. are: