SEBA Class X English A Letter to God MCQ 2025 HSLC Exam

A Letter to God MCQ 2025 HSLC Exam

1) What was the only hope left in the hearts of Lencho’s family?
(A) compensation from government
(B) help from the farmer’s association
(C) help from God
(D) there was no hope left
Ans: (C) help from God

2) What type of conflict does the chapter highlight?

(A) the conflict between nature and humans
(B) conflict among humans
(C) Conflict between God and nature
(D) both (A) and (B)
Ans: (D) both (A) and (B)

3) What are raindrops like?
(A) Cents
(B) Dollars
(C)Gold coins
(D) Pearls
Ans: (A) Cents

4) Man went out for…
(A) Ploughing
(B) Playing
(D) Buying
(C) Pleasure
Ans: (C) Pleasure

5) What was the reason for Lencho’s satisfaction?
(A) Heavy rain
(B) His children and wife
(C) Money
(D) His house
Ans: (A) Heavy rain

6) What did Lencho hope for?
(A) A good shower of rain for his crop.
(B) A new motorcycle
(C) A tractor
(D) None of the Above
Ans: (A) A good shower of rain for his crop.

7) Why was Lencho satisfied?
(A) On seeing a new tractor in his field.
(B) On seeing a new buffalo.
(C) On seeing the field of ripe corn with flowers.
(D) None of the Above
Ans: (C) On seeing the field of ripe corn with flowers.

8) Who is the author of the lesson ‘A Letter to God?
(A) G.L. Fuentes
(B) J.K. Rowling
(C) William Shakespeare
(D) Roald Dahl
Ans: (A) G.L. Fuentes

9) Where was Lencho’s house situated?
(A) High hill
(C) Low hill
(B) In the village
(D) In the valley
Ans: (C) Low hill

10) What was Lencho expecting from his cornfield?
(A) A good harvest
(B) Lots of money
(C)Treasure
(D) None of the above
Ans: (A) A good harvest

11) What took Lencho to a blow?
(A) Flood
(B) Hurricane
(C)Drought
(D) Hailstones
Ans: (D) Hailstones

12) What do new silver coins mean in the lesson?
(A) Big raindrops
(B) Actual silver coins
(C) Jewels
(D) Corns
Ans: (A) Big raindrops

13) Frozen pearls are
(A) Hails
(B) Flowers
(C) Raindrops
(D) None of the above
Ans: (A) Hails

14) What spoiled Lencho’s dream?
(A) Sunlight
(B) His son
(C) Hailstones
(D) Gold coins
Ans: (C) Hailstones

15) The field looked as if it were ……….covered in
(A) salt
(C) sugar
(B) locusts
(D) ice
Ans: (A) salt

16) What was getting bad in the chapter ‘The Letter to God?
(A) His health
(C) Hailstones
(B) His field
(D) Flowers
Ans: (C) Hailstones

17) Lencho’s field was destroyed.
(A) Partially
(C) Badly
(B) Totally
(D) All of the above
Ans: (B) Totally

18) Lencho’s soul was filled with sadness-why?
(A)Because of the flood
(B) Because corn field was destroyed
(C) Because of his family
(D) Because of money
Ans: (B) Because corn field was destroyed

19) Locusts mean in the chapter ‘The Letter to God’——
(A) Group of flowers
(B) Group of insects
(C) Group of boys
(D) Group of bees
Ans : (B) Group of Insects

20) Lencho needs money for –
(A) Vacation
(B) A new house
(C) Sowing his field
(D) Neighbors
Ans: (C) Sowing his field

21) Lencho’s only hope was
(A) His friend
(C) Villagers
(B) His family
(D) God
Ans: (D) God

22) Which crop was growing on Lencho’s fields?
(A)Corn
(C)Rice
(B) Barley
(D) None of the above
Ans: (A) Corn

23) Where did Lencho expect the downpour to come from?
(A)north
(C)north-west
(B) north-east
(D) south-east
Ans: (B) north-east

24) From the chapter find the words similar to “a strong man”?
(A) Instructed
(B) An ox of a man
(C) At daybreak
(D) conscience
Ans: (B) An ox of a man

25) Where did Lencho carry the letter?
(A) To the village
(B) To town)
(C) To house
(D) To god
Ans: (B) To town

26) What was Lencho’s wish to God?
(A) Money
(B) Gold coins
(C) Fruitful harvest
(D) Strength
Ans: (A) Money boy

27) What Currency of Latin American countries is?
(A) Dollar
(B) Pound
(C) Pesos
(D) Dark
Ans: (C) Pesos

28) What did the postmaster do then?
(A) He decided to collect the money and send it to Lencho.
(B) He throws away the letter
(C) He decided to beat Lencho to teach him a lesson.
(D) He gave it to another postmaster
Ans: (A) He decided to collect the money and send it to Lencho.

29) The postmaster was a fat, amiable man. What is the meaning of amiable?
(A) Rude
(B) Helpful
(C) Friendly
(D) Enthusiastic
Ans: (C) Friendly

30) How much money did Lencho ask for?
(A) 100 pesos
(B) 1000 pesos
(C) 10 pesos
(D) 500 pesos
Ans: (A) 100 pesos

31) What is the irony in this lesson?
(A) Lencho was sad after the hailstorm
even though he was the one waiting for a shower
(B) Postmaster laughed at Lencho but still helped arrange money for him
(C) Lencho blamed the post office employees who in fact helped him
(D) there is no irony
Ans: (C) Lencho blamed the post

32) What did Lencho think of the post-office employees?
(A) bunch of crooks
(B) rude
(C) unhelpful
(D) proud
Ans:(A) bunch of crooks

33) Who read the letter sent by Lencho?
(A) His Wife
(B) The Postmaster
(C) One of the farmer
(D) The God
Ans : (B) The Postmaster

34) Why did the postmaster decide to reply to Lencho’s letter?
(A) He was a good man
(B) He felt empathetic
(C) To preserve Lencho’s faith in God
(D) All of the above
Ans: (C) to preserve Lencho’s faith in God

35) Why was Lencho not surprised on seeing the money in the envelope?
(A) He was too sad to acknowledge it
(B) He had unwavering faith in God
(C) He was an ungrateful man
(D) None of the above
Ans: (B) He had unwavering faith in God

36) How much money was the postmaster able to arrange?
(A) 100 pesos
(B) Nil
(C) 1000 pesos
(D) 70 pesos
Ans: (D) 70 pesos

Oral comprehension check–1 [Page No. 5]

1. What did lencho hope for?
Ans: Lencho hoped for rain for his crops.

2. Why did Lencho say the raindrops were like ‘new coins’?
Ans: He said so because the raindrops would brings coins.

3. How did the rain change? What happened to Lencho’s fields?
Ans: The rain changed into a hailstorm. It completely destroyed Lencho’s fields.

4. What were Lencho’s feelings when the hail stopped?
Ans: Lencho was very sad because the hail left nothing. He thought he would go hungry that year.

Oral comprehension check – 2 [Page No. 6]

1. Who or what did Lencho have faith in? What did he do?
Ans: Lencho had faith in God. He wrote a letter to God.

2. Who read the letter?
Ans: The postmaster read the letter.

3. What did the postmaster do then?
Ans: The Post Master did not want to break Lencho’s faith. He collected money and sent to Lencho.

Oral comprehension check -3 [Page No. 7]

1. Was Lencho surprised to find a letter for him with money in it?
Ans: No, Lencho was not at all surprised to find a letter for him with money in it. Because he had deep faith in God.

2. What made him angry?
Ans: He was angry to find 70 pesos while he asked to send 100 pesos.

Thinking about the Text

1) Who does Lencho have complete faith in? Which sentences in the story tell you this?
Ans: Lencho has complete faith in God as he is instructed that God knows everything and helps us in our problems. There are few sentences which show this

But in the hearts of all who lived in that solitary house in the middle of the valley, there was a single hope help from God.
All through the night, Lencho thought only of his one hope: the help of God, whose eyes, as he had been instructed, see everything, even what is deep in one’s conscience.
“God”, he wrote, “if you don’t help me, my family and I will go hungry this year”.
He wrote ‘To God’ on the envelope, put the letter inside and still troubled, went to town.
God could not have made a mistake, nor could he have denied Lencho what he had requested.

2) Why does the postmaster send money to Lencho? Why does he sign the letter God?
Ans: The postmaster sends money to Lencho in order to keep Lencho’s faith in God alive and firm as he was completely moved by it.
When postmaster reads the letter of Lencho to God, he becomes serious and does not want to shake his faith and decides to answer the letter. He gathers money with the help of his post office employees and friends on behalf of God and signs the letter ‘God’ so that Lencho’s faith does not get shaken.

3) Did Lencho try to find out who had sent the money to him? Why or why not?
Ans: Lencho did not try to find out who had sent the money to him because he never suspected the presence of God and had complete faith in God. He could not believe that it could be – anybody else other than him who would send him the money.
His faith in God was so strong that he believed that he had sent money to him for his help in his problem.

4) Who does Lencho think has taken the rest of the money? What is the irony in the situation? (Remember that the irony of a situation is an unexpected aspect of it. An ironic situation is strange or amusing because it is the opposite of what is expected).
Ans: Lencho thinks that the post office employees have taken the rest of the money as he had demanded a hundred pesos from God and in the letter there was only seventy pesos and God cannot make such a mistake. So, he assumes that they have stolen the money.

The irony in this situation is that Lencho suspects those people who helped him in his problem and tried to keep his faith alive in God.

5) Are there people like Lencho in the real world? What kind of a person would you say he is? You may select appropriate words from the box to answer the question.

Greedy Naive Stupid Ungrateful Selfish Comical Unquestioning

Ans: It is almost impossible to find a person like Lencho as he is an unquestioning and naive kind of person. He is not stupid if he doesn’t know who has sent him money or a letter will reach God without any address. It is Lencho’s faith in God. In real world, people are selfish and greedy and Lencho is totally lovable and different.

6) There are two kinds of conflict in the story between humans and nature and between humans themselves. How are these conflicts illustrated?
Ans: Conflict between Humans and Nature: The conflict between humans and nature is illustrated by the destruction of Lencho’s crop by the hailstorm as Lencho was expecting a good rain to have good harvest as that was the only hope he had for his earning. He worked so hard to feed his family, but nature turned violent and destroyed everything.

Conflict between Humans and Humans: The story also illustrated another conflict, between humans themselves as the postmaster alongwith his friends and staff sent Lencho money that Lencho demanded from God although they didn’t know Lencho. Lencho blamed them for taking away some amount of money. He called them “a bunch of crook”. This shows that man does not have faith in other man, thereby giving rise to this conflict.

Thinking about Language

1. Look at the following sentence from the story.
Suddenly a strong wind began to blow and along with the rain very large hailstones began to fall.

‘Hailstones’ are small balls of ice that fall like rain. A storm in which hailstones fall is a ‘hailstorm’. You know that a storm is bad weather with strong winds, rain, thunder and lightning.

There are different names in different parts of the world’s for storms depending on their nature. Can you match the names in the box with their description below, and fill in the blanks? You may use a dictionary to help you.

Gale, whirlwind, cyclone, Hurricane, tornado, typhoon

1. A violent Tropical Storm in which strong winds move in a circle :——c—-
Ans: Cyclone

2. An extremely strong wind : —a—
Ans: Gale

3. A violent Tropical Storm with very strong winds:——p——
Ans: Typhoon

4. A violent storm whose center is a cloud in the shape of a funnel :——n—–
Ans: Tornado

5. A violent storm with very strong winds, especially in the western Atlantic Ocean :——r—–
Ans: Hurricane

6. A very strong wind that moves very fast in a spinning movement and causes a lot of damage :——l—–
Ans: Whirlwind

7) Match the sentences in column A with the meaning of ‘hope’ in column B.
Ans:

A	B
Will you get the subjects you want to study in college? I hope so.	thinking that this would happen (it may or may not have happened.)
I hope you don’t mind my saying this but 1 don’t like the way you are arguing.	showing concern that what you say should not offend or disturb the other person a way of being polite.
This discovery will give new hope to HIV/AIDS sufferers.	a feeling that something good will probably happen.
We were hoping against hope that the judges would not notice our mistakes.	wishing for something to happen, although this is very unlikely.
I called early in the hope of speaking to her before she went to school.	wanting something to happen (and thinking it quite possible)
Just when everybody had given up hope, the fishermen came back, seven days after the cyclone.	stopped believing that this good thing would happen.

Relative Clauses

Join the sentences given below using who, whom, whose, which, as suggested.

1. I often go to Mumbai. Mumbai is the commercial capital of India. (which)
Ans: I often go to Mumbai which is the commercial capital of India.

2. My mother is going to host a TV show on cooking. She cooks very well. (Who)
Ans: My mother who cooks very well is going to host a TV show on cooking.

3. These sportspersons are going to meet the president. Their performance has been excellent. (Whose)
Ans: The sports persons whose performance has been excellent are going to meet the president.

4. Lencho preyed to God. His eyes see into our minds. (whose)
Ans: Lencho prayed to God whose eyes see into our minds.

5. This man cheated me. I trusted him. (whom)
Ans: This man whom I trusted cheated me.

Using Negatives for Emphasis

Find sentences in the story with negative words, which express the following ideas emphatically.

1. The tress lost all their leaves.
Ans: Not a leaf remained on the trees.

2. The letter was addressed to God himself.
Ans: It was nothing less than a letter to God.

3. The postman saw this address for the first time in his career.
Ans: Never in his career as a postman had he known that address.

NCERT Class X English Chapter The Proposal

The Proposal

By Anton Chekov

BEFORE YOU READ

Activity

1. The word 'proposal' has several meanings. Can you guess what sort of proposal the play is about?

• (i) a suggestion, plan or scheme for doing something
• (ii) an offer for a possible plan or action
• (iii) the act of asking someone's hand in marriage

A Russian Wedding

Do you know anything about a Russian marriage ceremony? Read this article about a Russian wedding.

Preparations for a Russian Wedding: A Russian wedding is very simple. The planning only includes arranging for rings, brides' dress, cars, and a reception. Earlier, the bride's family paid for the reception, but now-a-days brides' and grooms' families usually share expenses. [cite: 14] A Russian wedding lasts for two days; some weddings last as long as a week, and the occasion becomes something to remember for years.

The necessary part of the wedding ceremony is a wedding procession of several cars. The best friends of the groom/bride meet before the wedding a few times, make posters, write speeches and organise contests. When the groom arrives to fetch the bride for the registration, he has to fight to get her! [cite: 18] Russians usually live in apartments in tall buildings, and the groom has to climb several stairs to reach his bride. But at each landing he must answer a question to be allowed to go up.

The bride's friends ask difficult questions (sometimes about the bride, sometimes just difficult riddles), and the groom must answer with the help of his friends. For example, he may be shown a few photos of baby girls and he must say which one his bride is. If he guesses wrong, he must pay cash to move ahead.

After the marriage registration, the newly-married couple leaves the guests for a tour of the city sights. After two or three hours of the city tour the couple arrives at the reception. The couple sits at a specially arranged table with their family, friends and invited guests.

The reception starts with toasts to the couple. A wedding toast is a custom where a close friend or relative of the groom or the bride says a few words to wish the couple, then everyone raises their glass of wine, and drink it up at the same moment. The groom is then asked to kiss the bride. After a few toasts, people start eating and drinking, and generally have fun.

After some time, the bride gets 'stolen'! She disappears, and when the groom starts looking for her, he is asked to pay a fee. Usually it is his friends who 'steal' the bride. Then there are the bride's friends they steal the bride's shoe. The groom must pay money for the shoe too. The guests enjoy watching these tussles, and continue partying.

2. Do you think Indian and Russian weddings have any customs in common? With the help of a partner, fill in the table below.

Wedding Ceremonies in Russia and India

Customs similar to Indian ones	Customs different from Indian ones

The Proposal

"The Proposal" (originally titled "A Marriage Proposal") is a one-act play, a farce, by the Russian short story writer and dramatist Anton Chekhov. It was written in 1888-89.

The play is about the tendency of wealthy families to seek ties with other wealthy families, to increase their estates by encouraging marriages that make good economic sense. Ivan Lomov, a long time wealthy neighbour of Stepan Chubukov, also wealthy, comes to seek the hand of Chubukov's twenty-five-year-old daughter, Natalya. All three are quarrelsome people, and they quarrel over petty issues. The proposal is in danger of being forgotten amidst all this quarrelling. But economic good sense ensures that the proposal is made, after all - although the quarrelling perhaps continues!

Characters

• STEPAN STEPANOVITCH CHUBUKOV: a landowner
• NATALYA STEPANOVNA: his daughter, twenty-five years old
• IVAN VASSILEVITCH LOMOV: a neighbour of Chubukov, a large and hearty, but very suspicious, landowner

A drawing-room in Chubukov's house.

Lomov enters, wearing a dress-jacket and white gloves. Chubukov rises to meet him.

CHUBUKOV:

My dear fellow, whom do I see! Ivan Vassilevitch! I am extremely glad! [Squeezes his hand] Now this is a surprise, my darling... How are you?

LOMOV:

Thank you. And how may you be getting on?

CHUBUKOV:

We just get along somehow, my angel, thanks to your prayers, and so on. Sit down, please do... Now, you know, you shouldn't forget all about your neighbours, my darling. My dear fellow, why are you so formal in your get-up! Evening dress, gloves, and so on. Can you be going anywhere, my treasure?

LOMOV:

No. I've come only to see you, honoured Stepan Stepanovitch.

CHUBUKOV:

Then why are you in evening dress, my precious? As if you're paying a New Year's Eve visit!

LOMOV:

Well, you see, it's like this. [Takes his arm] I've come to you, honoured Stepan Stepanovitch, to trouble you with a request. Not once or twice have I already had the privilege of applying to you for help, and you have always, so to speak... I must ask your pardon, I am getting excited. I shall drink some water, honoured Stepan Stepanovitch. [Drinks.]

CHUBUKOV:

[aside] He's come to borrow money. Shan't give him any! [aloud] What is it, my beauty?

LOMOV:

You see, Honoured Stepanitch... I beg pardon Stepan Honouritch... I mean, I'm awfully excited, as you will please notice... In short, you alone can help me, though I don't deserve it, of course... and haven't any right to count on your assistance...

CHUBUKOV:

Oh, don't go round and round it, darling! Spit it out! Well?

LOMOV:

One moment... this very minute. The fact is I've come to ask the hand of your daughter, Natalya Stepanovna, in marriage.

CHUBUKOV:

[joyfully] By Jove! Ivan Vassilevitch! Say it again - I didn't hear it all! [cite: 78]

LOMOV:

I have the honour to ask...

CHUBUKOV:

[interrupting] My dear fellow... I'm so glad, and so on... Yes, indeed, and all that sort of thing. [Embraces and kisses Lomov] I've been hoping for it for a long time. It's been my continual desire. [Sheds a tear] And I've always loved you, my angel, as if you were my own son. [cite: 83] May God give you both His help and His love and so on, and so much hope... What am I behaving in this idiotic way for? [cite: 84] I'm off my balance with joy, absolutely off my balance! Oh, with all my soul... I'll go and call Natasha, and all that.

LOMOV:

[greatly moved] Honoured Stepan Stepanovitch, do you think I may count on her consent?

CHUBUKOV:

Why, of course, my darling, and... as if she won't consent! She's in love; egad, she's like a lovesick cat, and so on. Shan't be long! [Exit.]

LOMOV:

It's cold... I'm trembling all over, just as if I'd got an examination before me. The great thing is, I must have my mind made up. If I give myself time to think, to hesitate, to talk a lot, to look for an ideal, or for real love, then I'll never get married. Brr... It's cold! Natalya Stepanovna is an excellent housekeeper, not bad-looking, well-educated. What more do I want? But I'm getting a noise in my ears from excitement. [Drinks] And it's impossible for me not to marry. In the first place, I'm already 35 — a critical age, so to speak. In the second place, I ought to lead a quiet and regular life. I suffer from palpitations, I'm excitable and always getting awfully upset; at this very moment my lips are trembling, and there's a twitch in my right eyebrow. But the very worst of all is the way I sleep. I no sooner get into bed and begin to go off, when suddenly something in my left side gives a pull, and I can feel it in my shoulder and head... I jump up like a lunatic, walk about a bit and lie down again, but as soon as I begin to get off to sleep there's another pull! And this may happen twenty times...

[Natalya Stepanovna comes in.]

NATALYA:

Well, there! It's you, and papa said, "Go; there's a merchant come for his goods." How do you do, Ivan Vassilevitch?

LOMOV:

How do you do, honoured Natalya Stepanovna?

NATALYA:

You must excuse my apron and neglige. We're shelling peas for drying. Why haven't you been here for such a long time? Sit down... [They seat themselves.] Won't you have some lunch?

LOMOV:

No, thank you, I've had some already.

NATALYA:

Then smoke. Here are the matches. The weather is splendid now, but yesterday it was so wet that the workmen didn't do anything all day. How much hay have you stacked? Just think, I felt greedy and had a whole field cut, and now I'm not at all pleased about it because I'm afraid my hay may rot. I ought to have waited a bit. But what's this? Why, you're in evening dress! Well, I never! Are you going to a ball or what? Though I must say you look better... Tell me, why are you got up like that?

LOMOV:

[excited] You see, honoured Natalya Stepanovna... the fact is, I've made up my mind to ask you to hear me out... Of course you'll be surprised and perhaps even angry, but a... [aside] It's awfully cold! [cite: 129]

NATALYA:

What's the matter? [pause] Well?

LOMOV:

I shall try to be brief. You must know, honoured Natalya Stepanovna, that I have long, since my childhood, in fact, had the privilege of knowing your family. My late aunt and her husband, from whom, as you know, I inherited my land, always had the greatest respect for your father and your late mother. The Lomovs and the Chubukovs have always had the most friendly, and I might almost say the most affectionate, regard for each other. And, as you know, my land is a near neighbour of yours. You will remember that my Oxen Meadows touch your birchwoods.

NATALYA:

Excuse my interrupting you. You say, "my Oxen Meadows". But are they yours?

LOMOV:

Yes, mine.

NATALYA:

What are you talking about? Oxen Meadows are ours, not yours!

LOMOV:

No, mine, honoured Natalya Stepanovna.

NATALYA:

Well, I never knew that before. How do you make that out?

LOMOV:

How? I'm speaking of those Oxen Meadows which are wedged in between your birchwoods and the Burnt Marsh.

NATALYA:

Yes, yes... they're ours.

LOMOV:

No, you're mistaken, honoured Natalya Stepanovna, they're mine. [cite: 150]

NATALYA:

Just think, Ivan Vassilevitch! How long have they been yours?

LOMOV:

How long? As long as I can remember.

NATALYA:

Really, you won't get me to believe that!

LOMOV:

But you can see from the documents, honoured Natalya Stepanovna. [cite: 159] Oxen Meadows, it's true, were once the subject of dispute, but now everybody knows that they are mine. There's nothing to argue about. You see my aunt's grandmother gave the free use of these Meadows in perpetuity to the peasants of your father's grandfather, in return for which they were to make bricks for her. The peasants belonging to your father's grandfather had the free use of the Meadows for forty years, and had got into the habit of regarding them as their own, when it happened that...

NATALYA:

No, it isn't at all like that! Both grandfather and great-grandfather reckoned that their land extended to Burnt Marsh which means that Oxen Meadows were ours. I don't see what there is to argue about. It's simply silly!

LOMOV:

I'll show you the documents, Natalya Stepanovna!

NATALYA:

No, you're simply joking, or making fun of me. What a surprise! We've had the land for nearly three hundred years, and then we're suddenly told that it isn't ours! Ivan Vassilevitch, I can hardly believe my own ears. These Meadows aren't worth much to me. They only come to five dessiatins, and are worth perhaps 300 roubles, but I can't stand unfairness. Say what you will, I can't stand unfairness.

LOMOV:

Hear me out, I implore you! The peasants of your father's grandfather, as I have already had the honour of explaining to you, used to bake bricks for my aunt's grandmother. Now my aunt's grandmother, wishing to make them a pleasant...

NATALYA:

I can't make head or tail of all this about aunts and grandfathers and grandmothers. The Meadows are ours, that's all.

LOMOV:

Mine.

NATALYA:

Ours! You can go on proving it for two days on end, you can go and put on fifteen dress jackets, but I tell you they're ours, ours, ours! I don't want anything of yours and I don't want to give anything of mine. So there!

LOMOV:

Natalya Stepanovna, I don't want the Meadows, but I am acting on principle. If you like, I'll make you a present of them.

NATALYA:

I can make you a present of them myself, because they're mine! [cite: 189] Your behaviour, Ivan Vassilevitch, is strange, to say the least! Up to this we have always thought of you as a good neighbour, a friend; last year we lent you our threshing-machine, although on that account we had to put off our own threshing till November, but you behave to us as if we were gypsies. Giving me my own land, indeed! No, really, that's not at all neighbourly! In my opinion, it's even impudent, if you want to know.

LOMOV:

Then you make out that I'm a landgrabber? Madam, never in my life have I grabbed anybody else's land and I shan't allow anybody to accuse me of having done so. [Quickly steps to the carafe and drinks more water] Oxen Meadows are mine!

NATALYA:

It's not true, they're ours!

LOMOV:

Mine!

NATALYA:	It's not true! I'll prove it! I'll send my mowers out to the Meadows this very day!
LOMOV:	What?
NATALYA:	My mowers will be there this very day!
LOMOV:	I'll give it to them in the neck!
NATALYA:	You dare!
LOMOV:	[Clutches at his heart] Oxen Meadows are mine! You understand? Mine!
NATALYA:	Please don't shout! You can shout yourself hoarse in your own house but here I must ask you to restrain yourself!
LOMOV:	If it wasn't, madam, for this awful, excruciating palpitation, if my whole inside wasn't upset, I'd talk to you in a different way! [Yells] Oxen Meadows are mine!
NATALYA:	Ours!
LOMOV:	Mine!
NATALYA:	Ours!
LOMOV:	Mine!

[Enter Chubukov]

CHUBUKOV:

What's the matter? What are you shouting for?

NATALYA:

Papa, please tell this gentleman who owns Oxen Meadows, we or he? [cite: 203]

CHUBUKOV:

[to Lomov] Darling, the Meadows are ours!

LOMOV:

But, please, Stepan Stepanovitch, how can they be yours? Do be a reasonable man! My aunt's grandmother gave the Meadows for the temporary and free use of your grandfather's peasants. The peasants used the land for forty years and got accustomed to it as if it was their own, when it happened that...

CHUBUKOV:

Excuse me, my precious. You forget just this, that the peasants didn't pay your grandmother and all that, because the Meadows were in dispute, and so on. And now everybody knows that they're ours. It means that you haven't seen the plan.

LOMOV:

I'll prove to you that they're mine!

CHUBUKOV:

You won't prove it, my darling

LOMOV:

I shall

CHUBUKOV:

Dear one, why yell like that? You won't prove anything just by yelling. I don't want anything of yours, and don't intend to give up what I have. Why should I? And you know, my beloved, that if you propose to go on arguing about it, I'd much sooner give up the Meadows to the peasants than to you. There!

LOMOV:

I don't understand! How have you the right to give away somebody else's property?

CHUBUKOV:

You may take it that I know whether I have the right or not. Because, young man, I'm not used to being spoken to in that tone of voice, and so on. I, young man, am twice your age, and ask you to speak to me without agitating yourself, and all that.

LOMOV:

No, you just think I'm a fool and want to have me on! You call my land yours, and then you want me to talk to you calmly and politely! Good neighbours don't behave like that, Stepan Stepanovitch! You're not a neighbour, you're a grabber!

CHUBUKOV:

What's that? What did you say?

NATALYA:

Papa, send the mowers out to the Meadows at once!

CHUBUKOV:

What did you say, sir?

NATALYA:

Oxen Meadows are ours, and I shan't give them up, shan't give them up, shan't give them up!

LOMOV:

We'll see! I'll have the matter taken to court, and then I'll show you!

CHUBUKOV:

To court? You can take it to court, and all that! You can! I know you; you're just on the look-out for a chance to go to court, and all that. You pettifogger! All your people were like that! All of them! [cite: 235, 239, 240]

LOMOV:

Never mind about my people! The Lomovs have all been honourable people, and not one has ever been tried for embezzlement, like your grandfather!

CHUBUKOV:

You Lomovs have had lunacy in your family, all of you!

NATALYA:

All, all, all!

CHUBUKOV:

Your grandfather was a drunkard, and your younger aunt, Nastasya Mihailovna, ran away with an architect, and so on...

LOMOV:

And your mother was hump-backed. [Clutches at his heart] Something pulling in my side... My head.... Help! Water!

CHUBUKOV:

Your father was a guzzling gambler!

NATALYA:

And there haven't been many backbiters to equal your aunt!

CHUBUKOV:

My left foot has gone to sleep... You're an intriguer....Oh, my heart! And it's an open secret that before the last elections you bri... I can see stars... Where's my hat?

NATALYA:

It's low! It's dishonest! It's mean!

CHUBUKOV:

And you're just a malicious, doublefaced intriguer! Yes!

LOMOV:

Here's my hat. My heart! Which way? Where's the door? Oh I think I'm dying! My foot's quite numb... [Goes to the door.]

CHUBUKOV:

[following him] And don't set foot in my house again!

NATALYA:

Take it to court! We'll see!

[Lomov staggers out.]

CHUBUKOV:

Devil take him! [Walks about in excitement.] [cite: 256, 257]

NATALYA:

What a rascal! What trust can one have in one's neighbours after that!

CHUBUKOV:

The villain! The scarecrow!

NATALYA:

The monster! First he takes our land and then he has the impudence to abuse us.

CHUBUKOV:

And that blind hen, yes, that turnip-ghost has the confounded cheek to make a proposal, and so on! What? A proposal!

NATALYA:

What proposal?

CHUBUKOV:

Why, he came here to propose to you.

NATALYA:

To propose? To me? Why didn't you tell me so before? [cite: 268]

CHUBUKOV:

So he dresses up in evening clothes. The stuffed sausage! The wizen-faced frump!

NATALYA:

To propose to me? Ah! [Falls into an easy-chair and wails] Bring him back! Back! Ah! Bring him here. [cite: 273, 274]

CHUBUKOV:

Bring whom here?

NATALYA:

Quick, quick! I'm ill! Fetch him! [Hysterics.]

CHUBUKOV:

What's that? What's the matter with you? [Clutches at his head] Oh, unhappy man that I am! I'll shoot myself! I'll hang myself! We've done for her! [cite: 280, 281]

NATALYA:

I'm dying! Fetch him!

CHUBUKOV:

Tfoo! At once. Don't yell! [Runs out. A pause.]

NATALYA:

[Natalya Stepanovna wails.] What have they done to me? Fetch him back! Fetch him!

[A pause. Chubukov runs in.]

CHUBUKOV:

He's coming, and so on, devil take him! Ouf! Talk to him yourself; I don't want to...

NATALYA:

[wails] Fetch him!

CHUBUKOV:

[yells] He's coming, I tell you. Oh, what a burden, Lord, to be the father of a grown-up daughter! I'll cut my throat I will, indeed! We cursed him, abused him, drove him out; and it's all you... you! [cite: 303]

NATALYA:

No, it was you!

CHUBUKOV:

I tell you it's not my fault. [Lomov appears at the door] Now you talk to him yourself. [Exit.]

[Lomov enters, exhausted.]

LOMOV:

My heart's palpitating awfully. My foot's gone to sleep. There's something that keeps pulling in my side.... [cite: 307, 308]

NATALYA:

Forgive us, Ivan Vassilevitch, we were all a little heated. I remember now: Oxen Meadows... really are yours. [cite: 309]

LOMOV:

My heart's beating awfully. My Meadows... My eyebrows are both twitching....

NATALYA:

The Meadows are yours, yes, yours. Do sit down. [They sit] We were wrong.

LOMOV:

I did it on principle. My land is worth little to me, but the principle...

NATALYA:

Yes, the principle, just so. Now let's talk of something else.

LOMOV:

The more so as I have evidence. My aunt's grandmother gave the land to your father's grandfather's peasants...

NATALYA:

Yes, yes, let that pass. [aside] I wish I knew how to get him started. [aloud] Are you going to start shooting soon?

LOMOV:

I'm thinking of having a go at the blackcock, honoured Natalya Stepanovna, after the harvest. Oh, have you heard? Just think, what a misfortune I've had! My dog Guess, who you know, has gone lame. [cite: 319, 320]

NATALYA:

What a pity! Why?

LOMOV:

I don't know. Must have got his leg twisted or bitten by some other dog. [sighs] My very best dog, to say nothing of the expense. I gave Mironov 125 roubles for him. [cite: 325, 326]

NATALYA:

It was too much, Ivan Vassilevitch.

LOMOV:

I think it was very cheap. He's a first-rate dog. [cite: 327]

NATALYA:

Papa gave 85 roubles for his Squeezer, and Squeezer is heaps better than Guess! [cite: 328, 332]

LOMOV:

Squeezer better than Guess? What an idea! [laughs] Squeezer better than Guess!

NATALYA:

Of course he's better! Of course, Squeezer is young, he may develop a bit, but on points and pedigree he's better than anything that even Volchanetsky has got.

LOMOV:

Excuse me, Natalya Stepanovna, but you forget that he is overshot, and an overshot always means the dog is a bad hunter!

NATALYA:

Overshot, is he? The first time I hear it!

LOMOV:

I assure you that his lower jaw is shorter than the upper.

NATALYA:

Have you measured?

LOMOV:

Yes. He's all right at following, of course, but if you want to get hold of anything...

NATALYA:

In the first place, our Squeezer is a thoroughbred animal, the son of Harness and Chisels while there's no getting at the pedigree of your dog at all. He's old and as ugly as a worn-out cab-horse. [cite: 343, 344]

LOMOV:

He is old, but I wouldn't take five Squeezers for him. Why, how can you? Guess is a dog; as for Squeezer, well, it's too funny to argue. Anybody you like has a dog as good as Squeezer... you may find them under every bush almost. [cite: 345, 346]

NATALYA:

There's some demon of contradition in you today, Ivan Vassilevitch. First you pretend that the Meadows are yours; now, that Guess is better than Squeezer. I don't like people who don't say what they mean, because you know perfectly well that Squeezer is a hundred times better than your silly Guess. Why do you want to say he isn't? [cite: 348, 349, 350]

LOMOV:

I see, Natalya Stepanovna, that you consider me either blind or a fool. You must realise that Squeezer is overshot! [cite: 356, 357]

NATALYA:

It's not true.

LOMOV:

He is!

NATALYA:

It's not true!

LOMOV:

Why shout madam?

NATALYA:

Why talk rot? It's awful! It's time your Guess was shot, and you compare him with Squeezer!

LOMOV:

Excuse me, I cannot continue this discussion, my heart is palpitating.

NATALYA:

I've noticed that those hunters argue most who know least.

LOMOV:

Madam, please be silent. My heart is going to pieces. [shouts] Shut up!

NATALYA:

I shan't shut up until you acknowledge that Squeezer is a hundred times better than your Guess! [cite: 369, 373]

LOMOV:

A hundred times worse! Be hanged to your Squeezer! His head... eyes... shoulder...

NATALYA:

There's no need to hang your silly Guess; he's half-dead already! [cite: 378]

LOMOV:

[weeps] Shut up! My heart's bursting! [cite: 380, 384]

NATALYA:

I shan't shut up.

[Enter Chubukov.]

CHUBUKOV:

What's the matter now?

NATALYA:

Papa, tell us truly, which is the better dog, our Squeezer or his Guess.

LOMOV:

Stepan Stepanovitch, I implore you to tell me just one thing: is your Squeezer overshot or not? Yes or no? [cite: 388, 389]

CHUBUKOV:

And suppose he is? What does it matter? He's the best dog in the district for all that, and so on.

LOMOV:

But isn't my Guess better? Really, now?

CHUBUKOV:

Don't excite yourself, my precious one. Allow me. Your Guess certainly has his good points. He's purebred, firm on his feet, has well-sprung ribs, and all that. But, my dear man, if you want to know the truth, that dog has two defects: he's old and he's short in the muzzle.

LOMOV:

Excuse me, my heart... Let's take the facts. You will remember that on the Marusinsky hunt my Guess ran neck-and-neck with the Count's dog, while your Squeezer was left a whole verst behind. [cite: 397]

CHUBUKOV:

He got left behind because the Count's whipper-in hit him with his whip.

LOMOV:

And with good reason. The dogs are running after a fox, when Squeezer goes and starts worrying a sheep!

CHUBUKOV:

It's not true! My dear fellow, I'm very liable to lose my temper, and so, just because of that, let's stop arguing. You started because everybody is always jealous of everybody else's dogs. Yes, we're all like that!

LOMOV:

I remember too!

CHUBUKOV:

[teasing him] I remember, too! What do you remember?

LOMOV:

My heart... my foot's gone to sleep. I can't...

NATALYA:

[teasing] My heart! What sort of a hunter are you? You ought to go and lie on the kitchen oven and catch black beetles, not go after foxes! My heart! [cite: 407, 408]

CHUBUKOV:

Yes really, what sort of a hunter are you, anyway? You ought to sit at home with your palpitations, and not go tracking animals. [cite: 409, 410] Let's change the subject in case I lose my temper. You're not a hunter at all, anyway!

LOMOV:

And are you a hunter? You only go hunting to get in with the Count and to intrigue. Oh, my heart! You're an intriguer! [cite: 414, 415]

CHUBUKOV:

What? I am an intriguer? [shouts] Shut up!

LOMOV:

Intriguer!

CHUBUKOV:

Boy! Pup!

LOMOV:

Old rat! Jesuit!

CHUBUKOV:

Shut up or I'll shoot you like a partridge! You fool! [cite: 423]

LOMOV:

Everybody knows that — oh, my heart! — your late wife used to beat you... My feet... temples... sparks... I fall, I fall! [cite: 424, 425, 426, 427, 428]

CHUBUKOV:

And you're under the slipper of your house-keeper!

LOMOV:

There, there, there... my heart's burst! My shoulders come off! Where is my shoulder? I die. [Falls into an armchair] A doctor! [cite: 432, 433]

CHUBUKOV:

Boy! Milksop! Fool! I'm sick! [Drinks water] Sick!

NATALYA:

What sort of a hunter are you? You can't even sit on a horse! [To her father] Papa, what's the matter with him? Papa! Look, Papa! [screams] Ivan Vassilevitch! He's dead! [cite: 435, 436]

CHUBUKOV:

I'm sick! I can't breathe! Air! [cite: 436, 437]

NATALYA:

He's dead. [Pulls Lomov's sleeve] Ivan Vassilevitch! Ivan Vassilevitch! What have you done to me? He's dead. [Falls into an armchair] A doctor, a doctor! [Hysterics.] [cite: 440, 441]

CHUBUKOV:

Oh! What is it? What's the matter?

NATALYA:

[wails] He's dead... dead! [cite: 442]

CHUBUKOV:

Who's dead? [Looks at Lomov] So he is! My word! Water! A doctor! [Lifts a tumbler to Lomov's mouth] Drink this! No, he doesn't drink. It means he's dead, and all that. [cite: 443, 445, 447]

I'm the most unhappy of men! Why don't I put a bullet into my brain? Why haven't I cut my throat yet? What am I waiting for? Give me a knife! Give me a pistol! [Lomov moves] He seems to be coming round. Drink some water! That's right. [cite: 448, 449, 450, 451, 452]

LOMOV:

I see stars... mist... where am I?

CHUBUKOV:

Hurry up and get married and — well, to the devil with you! She's willing! [He puts Lomov's hand into his daughter's] She's willing and all that. I give you my blessing and so on. Only leave me in peace! [cite: 454, 455, 456, 457]

LOMOV:

[getting up] Eh? What? To whom?

CHUBUKOV:

She's willing! Well? Kiss and be damned to you!

NATALYA:

[wails] He's alive... Yes, yes, I'm willing.

CHUBUKOV:

Kiss each other!

LOMOV:

Eh? Kiss whom? [They kiss] Very nice, too. Excuse me, what's it all about? Oh, now I understand... my heart... stars... I'm happy. Natalya Stepanovna... [Kisses her hand] My foot's gone to sleep. [cite: 464, 465, 466]

NATALYA:

I... I'm happy too...

CHUBUKOV:

What a weight off my shoulders, ouf!

NATALYA:

But, still you will admit now that Guess is worse than Squeezer. [cite: 470, 471]

LOMOV:

Better!

NATALYA:

Worse!

CHUBUKOV:

Well, that's a way to start your family bliss! Have some champagne! [cite: 474, 477]

LOMOV:

He's better!

NATALYA:

Worse! Worse! Worse!

CHUBUKOV:

[trying to shout her down] Champagne! Champagne!

---

CURTAIN

Thinking about the Play

1. What does Chubukov at first suspect that Lomov has come for? Is he sincere when he later says "And I've always loved you, my angel, as if you were my own son"? Find reasons for your answer from the play. [cite: 484, 485, 486]
2. Chubukov says of Natalya: "... as if she won't consent! She's in love; egad, she's like a lovesick cat..." Would you agree? Find reasons for your answer. [cite: 487, 488]
3. (i) Find all the words and expressions in the play that the characters use to speak about each other, and the accusations and insults they hurl at each other. (For example, Lomov in the end calls Chubukov an intriguer; but earlier, Chubukov has himself called Lomov a "malicious, doublefaced intriguer." Again, Lomov begins by describing Natalya as "an excellent housekeeper, not bad-looking, well-educated.")
   (ii) Then think of five adjectives or adjectival expressions of your own to describe each character in the play. [cite: 492]
   (iii) Can you now imagine what these characters will quarrel about next?

Thinking about Language

I. 1. This play has been translated into English from the Russian original. Are there any expressions or ways of speaking that strike you as more Russian than English? [cite: 496, 497] For example, would an adult man be addressed by an older man as my darling or my treasure in an English play? Read through the play carefully, and find expressions that you think are not used in contemporary English, and contrast these with idiomatic modern English expressions that also occur in the play. [cite: 499]

2. Look up the following words in a dictionary and find out how to pronounce them. Pay attention to how many syllables there are in each word, and find out which syllable is stressed, or said more forcefully. [cite: 500, 501]

palpitations	interfere	implore	thoroughbred
pedigree	principle	evidence	misfortune
malicious	embezzlement	architect	neighbours
accustomed	temporary	behaviour	documents

3. Look up the following phrases in a dictionary to find out their meaning, and then use each in a sentence of your own. [cite: 505, 506]

• (i) You may take it that
• (ii) He seems to be coming round
• (iii) My foot's gone to sleep

II. Reported Speech

A sentence in reported speech consists of two parts: a reporting clause, which contains the reporting verb, and the reported clause.

(a) "I went to visit my grandma last week," said Mamta.
(b) Mamta said that she had gone to visit her grandma the previous week.

In sentence (a), we have Mamta's exact words. This is an example of direct speech. In sentence (b), someone is reporting what Mamta said. This is called indirect speech or reported speech.

Exercises

You must have noticed that when we report someone's exact words, we have to make some changes in the sentence structure. In the following sentences fill in the blanks to list the changes that have occurred in the above pairs of sentences. One has been done for you. [cite: 540, 541, 542]

1. To report a question, we use the reporting verb asked (as in Sentence Set 1).
2. To report a statement, we use the reporting verb __________. [cite: 545]
3. The adverb of place here changes to __________.
4. When the verb in direct speech is in the present tense, the verb in reported speech is in the __________ tense (as in Sentence Set 3). [cite: 547, 548]
5. If the verb in direct speech is in the present continuous tense, the verb in reported speech changes to __________ tense. For example, __________ changes to was getting. [cite: 548, 549]
6. When the sentence in direct speech contains a word denoting respect, we add the adverb __________ in the reporting clause (as in Sentence Set 1).
7. The pronouns I, me, our and mine, which are used in the first person in direct speech, change according to the subject or object of the reporting verb such as __________ or __________ in reported speech. [cite: 551, 552, 553, 554]

III. Rewrite Activity

Here is an excerpt from an article from the Times of India dated 27 August 2006. Rewrite it, changing the sentences in direct speech into reported speech. Leave the other sentences unchanged. [cite: 555, 556]

"Why do you want to know my age? If people know I am so old, I won't get work!" laughs 90-year-old A. K. Hangal, one of Hindi cinema's most famous character actors. For his age, he is rather energetic. "What's the secret?" we ask. "My intake of everything is in small quantities. And I walk a lot," he replies. "I joined the industry when people retire. I was in my 40s. So I don't miss being called a star. I am still respected and given work, when actors of my age are living in poverty and without work. I don't have any complaints," he says, adding, "but yes, I have always been underpaid." Recipient of the Padma Bhushan, Hangal never hankered after money or materialistic gains. "No doubt I am content today, but money is important. I was a fool not to understand the value of money earlier," he regrets. [cite: 557-563]

Speaking and Writing

1. Anger Management: As adults, one important thing to learn is how to manage our temper. Some of us tend to get angry quickly, while others remain calm. Can you think of three ill effects that result from anger? Note them down. Suggest ways to avoid losing your temper in such situations. Are there any benefits from anger? [cite: 565, 566, 567, 568]

2. In pairs, prepare a script based on the given excerpt from The Home and the World by Rabindranath Tagore. You may write five exchanges between the characters with other directions such as movements on stage and way of speaking, etc. [cite: 569, 570]

One afternoon, when I happened to be specially busy, word came to my office room that Bimala had sent for me. I was startled. "Who did you say had sent for me?" I asked the messenger. "The Rani Mother". "The Bara Rani?" "No, sir, the Chota Rani Mother." The Chota Rani! It seemed a century since I had been sent for by her... (excerpt continues in source text)

3. In groups, discuss the qualities one should look for in a marriage partner. You might consider the following points: Personal qualities, Appearance or looks, Attitudes and beliefs, Sense of humour, Value system, Compassion and kindness, Tolerance, ambition, Attitude to money and wealth, Education and professional background.

4. Are there parts of the play that remind you of film scenes from romantic comedies? Discuss this in groups, and recount to the rest of the class episodes similar to those in the play. [cite: 613, 614]

CONSTITUTION OF INDIA

Part III (Articles 12–35) (Subject to certain conditions, some exceptions and reasonable restrictions) guarantees these Fundamental Rights [cite: 639, 640, 641]

• Right to Equality: before law and equal protection of laws; irrespective of religion, race, caste, sex or place of birth; of opportunity in public employment; by abolition of untouchability and titles.
• Right to Freedom: of expression, assembly, association, movement, residence and profession; of certain protections in respect of conviction for offences; of protection of life and personal liberty; of free and compulsory education for children between the age of six and fourteen years; of protection against arrest and detention in certain cases.
• Right against Exploitation: for prohibition of traffic in human beings and forced labour; for prohibition of employment of children in hazardous jobs.
• Right to Freedom of Religion: freedom of conscience and free profession, practice and propagation of religion; freedom to manage religious affairs; freedom as to payment of taxes for promotion of any particular religion; freedom as to attendance at religious instruction or religious worship in educational institutions wholly maintained by the State.
• Cultural and Educational Rights: for protection of interests of minorities to conserve their language, script and culture; for minorities to establish and administer educational institutions of their choice.
• Right to Constitutional Remedies: by issuance of directions or orders or writs by the Supreme Court and High Courts for enforcement of these Fundamental Rights. [cite: 665, 666]

NCERT Class X Mathematics Examplar Chapter 2

Chapter 2: Polynomials

(A) Main Concepts and Results

Geometrical meaning of zeroes of a polynomial: The zeroes of a polynomial are precisely the x-coordinates of the points where the graph of $y=p(x)$ intersects the x-axis.

Relation between the zeroes and coefficients of a polynomial:

• If $\alpha$ and $\beta$ are the zeroes of a quadratic polynomial $ax^{2}+bx+c$, then $\alpha+\beta=-\frac{b}{a}$ and $\alpha\beta=\frac{c}{a}$.
• If $\alpha$, $\beta$ and $\gamma$ are the zeroes of a cubic polynomial $ax^{3}+bx^{2}+cx+d$, then:
  • $\alpha+\beta+\gamma=-\frac{b}{a}$
  • $\alpha\beta+\beta\gamma+\gamma\alpha=\frac{c}{a}$
  • $\alpha\beta\gamma=\frac{-d}{a}$

The division algorithm: Given any polynomial $p(x)$ and any non-zero polynomial $g(x)$, there are polynomials $q(x)$ and $r(x)$ such that $p(x)=g(x)q(x)+r(x)$ where $r(x)=0$ or degree $r(x) <$ degree $g(x)$.

(B) Multiple Choice Questions

Choose the correct answer from the given four options:

Sample Question 1: If one zero of the quadratic polynomial $x^{2}+3x+k$ is 2, then the value of k is

1. 10
2. -10
3. 5
4. -5

Solution: Answer (B)

Sample Question 2: Given that two of the zeroes of the cubic polynomial $ax^{3}+bx^{2}+cx+d$ are 0, the third zero is

1. $\frac{-b}{a}$
2. $\frac{b}{a}$
3. $\frac{c}{a}$
4. $-\frac{d}{a}$

Solution: Answer (A). [Hint: Because if third zero is $\alpha$, sum of the zeroes $= \alpha+0+0=\frac{-b}{a}$]

EXERCISE 2.1

1. If one of the zeroes of the quadratic polynomial $(k-1)x^{2}+kx+1$ is -3, then the value of k is: (A) $\frac{4}{3}$ (B) $\frac{-4}{3}$ (C) $\frac{2}{3}$ (D) $\frac{-2}{3}$
2. A quadratic polynomial, whose zeroes are -3 and 4, is: (A) $x^{2}-x+12$ (B) $x^{2}+x+12$ (C) $\frac{x^{2}}{2}-\frac{x}{2}-6$ (D) $2x^{2}+2x-24$
3. If the zeroes of the quadratic polynomial $x^2 + (a + 1)x + b$ are 2 and -3, then: (A) $a=-7, b=-1$ (B) $a=5, b=-1$ (C) $a=2, b=-6$ (D) $a=0, b=-6$
4. The number of polynomials having zeroes as -2 and 5 is: (A) 1 (B) 2 (C) 3 (D) more than 3
5. Given that one of the zeroes of the cubic polynomial $ax^{3}+bx^{2}+cx+d$ is zero, the product of the other two zeroes is: (A) $-\frac{c}{a}$ (B) $\frac{c}{a}$ (C) 0 (D) $-\frac{b}{a}$
6. If one of the zeroes of the cubic polynomial $x^{3}+ax^{2}+bx+c$ is -1, then the product of the other two zeroes is: (A) $b-a+1$ (B) $b-a-1$ (C) $a-b+1$ (D) $a-b-1$
7. The zeroes of the quadratic polynomial $x^{2}+99x+127$ are: (A) both positive (B) both negative (C) one positive and one negative (D) both equal
8. The zeroes of the quadratic polynomial $x^{2}+kx+k, k\ne0$: (A) cannot both be positive (B) cannot both be negative (C) are always unequal (D) are always equal
9. If the zeroes of the quadratic polynomial $ax^{2}+bx+c, c\ne0$ are equal, then: (A) c and a have opposite signs (B) c and b have opposite signs (C) c and a have the same sign (D) c and b have the same sign
10. If one of the zeroes of a quadratic polynomial of the form $x^{2}+ax+b$ is the negative of the other, then it: (A) has no linear term and the constant term is negative. (B) has no linear term and the constant term is positive. (C) can have a linear term but the constant term is negative. (D) can have a linear term but the constant term is positive.
11. Which of the following is not the graph of a quadratic polynomial?

(C) Short Answer Questions with Reasoning

Sample Question 1: Can $x-1$ be the remainder on division of a polynomial p(x) by $2x+3$? Justify your answer.

Solution: No, since degree $(x-1)=1=$ degree $(2x+3)$

Sample Question 2: Is the following statement True or False? Justify your answer. If the zeroes of a quadratic polynomial $ax^{2}+bx+c$ are both negative, then a, b and c all have the same sign.

Solution: True, because sum of the zeroes $= -\frac{b}{a} < 0$, so that $\frac{b}{a}> 0$. Also the product of the zeroes $= \frac{c}{a} > 0$.

EXERCISE 2.2

1. Answer the following and justify:

1. Can $x^2-1$ be the quotient on division of $x^{6}+2x^{3}+x-1$ by a polynomial in x of degree 5?
2. What will the quotient and remainder be on division of $ax^{2}+bx+c$ by $px^{3}+qx^{2}+rx+s, p\ne0$?
3. If on division of a polynomial $p(x)$ by a polynomial $g(x)$, the quotient is zero, what is the relation between the degrees of $p(x)$ and $g(x)$?
4. If on division of a non-zero polynomial $p(x)$ by a polynomial $g(x)$, the remainder is zero, what is the relation between the degrees of $p(x)$ and $g(x)$?
5. Can the quadratic polynomial $x^{2}+kx+k$ have equal zeroes for some odd integer $k>1$?

2. Are the following statements 'True' or 'False'? Justify your answers.

1. If the zeroes of a quadratic polynomial $ax^{2}+bx+c$ are both positive, then a, b and c all have the same sign.
2. If the graph of a polynomial intersects the x-axis at only one point, it cannot be a quadratic polynomial.
3. If the graph of a polynomial intersects the x-axis at exactly two points, it need not be a quadratic polynomial.
4. If two of the zeroes of a cubic polynomial are zero, then it does not have linear and constant terms.
5. If all the zeroes of a cubic polynomial are negative, then all the coefficients and the constant term of the polynomial have the same sign.
6. If all three zeroes of a cubic polynomial $x^{3}+ax^{2}-bx+c$ are positive, then at least one of a, b and c is non-negative.
7. The only value of k for which the quadratic polynomial $kx^{2}+x+k$ has equal zeros is $\frac{1}{2}$.

(D) Short Answer Questions

Sample Question 1: Find the zeroes of the polynomial $x^{2}+\frac{1}{6}x-2$, and verify the relation between the coefficients and the zeroes of the polynomial.

Solution: $x^{2}+\frac{1}{6}x-2=\frac{1}{6}(6x^{2}+x-12)=\frac{1}{6}[6x^{2}+9x-8x-12]$

$=\frac{1}{6}[3x(2x+3)-4(2x+3)]=\frac{1}{6}(3x-4)(2x+3)$

Hence, $\frac{4}{3}$ and $-\frac{3}{2}$ are the zeroes of the given polynomial.

The sum of zeroes $=\frac{4}{3}+-\frac{3}{2}=\frac{-1}{6}=-\frac{\text{Coefficient of } x}{\text{Coefficient of } x^{2}}$

and the product of zeroes $=\frac{4}{3}\times\frac{-3}{2}=-2=\frac{\text{Constant term}}{\text{Coefficient of } x^{2}}$

EXERCISE 2.3

Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials:

1. $4x^{2}-3x-1$
2. $3x^{2}+4x-4$
3. $5t^{2}+12t+7$
4. $t^{3}-2t^{2}-15t$
5. $2x^{2}+\frac{7}{2}x+\frac{3}{4}$
6. $4x^{2}+5\sqrt{2}x-3$
7. $2s^{2}-(1+2\sqrt{2})s+\sqrt{2}$
8. $v^{2}+4\sqrt{3}v-15$
9. $y^{2}+\frac{3}{2}\sqrt{5}y-5$
10. $7y^{2}-\frac{11}{3}y-\frac{2}{3}$

(E) Long Answer Questions

Sample Question 1: Find a quadratic polynomial, the sum and product of zeroes are $\sqrt{2}$ and $-\frac{3}{2}$, respectively. Also find its zeroes.

Solution: A quadratic polynomial, the sum and product of whose zeroes are $\sqrt{2}$ and $-\frac{3}{2}$ is $x^{2}-\sqrt{2}x-\frac{3}{2}$

$x^{2}-\sqrt{2}x-\frac{3}{2}=\frac{1}{2}[2x^{2}-2\sqrt{2}x-3]$

$=\frac{1}{2}[2x^{2}+\sqrt{2}x-3\sqrt{2}x-3] = \frac{1}{2}[\sqrt{2}x(\sqrt{2}x+1)-3(\sqrt{2}x+1)]$

$=\frac{1}{2}[\sqrt{2}x+1][\sqrt{2}x-3]$

Hence, the zeroes are $-\frac{1}{\sqrt{2}}$ and $\frac{3}{\sqrt{2}}$

Sample Question 2: If the remainder on division of $x^{3}+2x^{2}+kx+3$ by $x-3$ is 21, find the quotient and the value of k. Hence, find the zeroes of the cubic polynomial $x^{3}+2x^{2}+kx-18$.

Solution: Let $p(x)=x^{3}+2x^{2}+kx+3$

Then, $p(3)=3^{3}+2\times3^{2}+3k+3=21$

i.e., $3k=-27$ which implies $k=-9$

Hence, the given polynomial will become $x^{3}+2x^{2}-9x+3$.

Performing long division: $x^{3}+2x^{2}-9x+3 = (x^{2}+5x+6)(x-3)+21$

Now, $x^{3}+2x^{2}-9x-18=(x-3)(x^{2}+5x+6) = (x-3)(x+2)(x+3)$

So, the zeroes of $x^{3}+2x^{2}+kx-18$ are 3, -2, -3.

EXEMPLAR PROBLEMS

1. For each of the following, find a quadratic polynomial whose sum and product respectively of the zeroes are as given. Also find the zeroes of these polynomials by factorisation.

• (i) $\frac{-8}{3}, \frac{4}{3}$
• (ii) $\frac{21}{8}, \frac{5}{16}$
• (iii) $-2\sqrt{3}, -9$
• (iv) $-\frac{3}{2\sqrt{5}}, -\frac{1}{2}$

2. Given that the zeroes of the cubic polynomial $x^{3}-6x^{2}+3x+10$ are of the form $a, a+b, a+2b$ for some real numbers a and b, find the values of a and b as well as the zeroes of the given polynomial.

3. Given that $\sqrt{2}$ is a zero of the cubic polynomial $6x^{3}+\sqrt{2}x^{2}-10x-4\sqrt{2}$, find its other two zeroes.

4. Find k so that $x^{2}+2x+k$ is a factor of $2x^{4}+x^{3}-14x^{2}+5x+6$. Also find all the zeroes of the two polynomials.

5. Given that $x-\sqrt{5}$ is a factor of the cubic polynomial $x^{3}-3\sqrt{5}x^{2}+13x-3\sqrt{5}$, find all the zeroes of the polynomial.

6. For which values of a and b, are the zeroes of $q(x)=x^{3}+2x^{2}+a$ also the zeroes of the polynomial $p(x)=x^{5}-x^{4}-4x^{3}+3x^{2}+3x+b?$ Which zeroes of $p(x)$ are not the zeroes of $q(x)$?

NCERT Class X Mathematics Examplar Chapter 1

Chapter 1: Real Numbers

(A) Main Concepts and Results

Euclid's Division Lemma: Given two positive integers $a$ and $b$, there exist unique integers $q$ and $r$ satisfying $a=bq+r, 0 \le r < b$.

Euclid's Division Algorithm: To obtain the HCF of two positive integers, say $c$ and $d$ ($c>d$):

1. Step 1: Apply Euclid's division lemma to $c$ and $d$, to find whole numbers $q$ and $r$, such that $c = dq+r, 0 \le r < d$.
2. Step 2: If $r=0$, $d$ is the HCF of $c$ and $d$. If $r \ne 0$ apply the division lemma to $d$ and $r$.
3. Step 3: Continue the process till the remainder is zero. The divisor at this stage will be the required HCF.

Fundamental Theorem of Arithmetic: Every composite number can be expressed as a product of primes, and this expression (factorisation) is unique, apart from the order in which the prime factors occur.

• Let $p$ be a prime number. If $p$ divides $a^2$, then $p$ divides $a$, where $a$ is a positive integer.
• $\sqrt{2}, \sqrt{3}, \sqrt{5}$ are irrational numbers.
• The sum or difference of a rational and an irrational number is irrational.
• The product or quotient of a non-zero rational number and an irrational number is irrational.
• For any two positive integers $a$ and $b$, HCF $(a,b) \times$ LCM $(a,b) = a \times b$.

Let $x = \frac{p}{q}$ be a rational number where $p$ and $q$ are co-prime.

• If the decimal expansion terminates, then the prime factorisation of $q$ is of the form $5^n$; $m, n$ are non-negative integers.
• If the prime factorisation of $q$ is not of the form $2^m 5^n$, then $x$ has a non-terminating repeating decimal expansion.

(B) Multiple Choice Questions

Choose the correct answer from the given four options:

Sample Question 1: The decimal expansion of the rational number $\frac{33}{2^2 \cdot 5}$ will terminate after

• (A) one decimal place
• (B) two decimal places
• (C) three decimal places
• (D) more than 3 decimal places

Solution: Answer (B)

Sample Question 2: Euclid's division lemma states that for two positive integers $a$ and $b$, there exist unique integers $q$ and $r$ such that $a=bq+r$ where $r$ must satisfy

• (A) $1 < r < b$
• (B) $0 < r \le b$
• (C) $0 \le r < b$
• (D) $0 < r < b$

Solution: Answer (C)

EXERCISE 1.1

1. For some integer $m$, every even integer is of the form: (A) $m$ (B) $m+1$ (C) $2m$ (D) $2m+1$
2. For some integer $q$, every odd integer is of the form: (A) $q$ (B) $q+1$ (C) $2q$ (D) $2q+1$
3. $n^2-1$ is divisible by 8, if $n$ is: (A) an integer (B) a natural number (C) an odd integer (D) an even integer
4. If the HCF of 65 and 117 is expressible in the form $65m-117$, then the value of $m$ is: (A) 4 (B) 2 (C) 1 (D) 3
5. The largest number which divides 70 and 125, leaving remainders 5 and 8, respectively, is: (A) 13 (B) 65 (C) 875 (D) 1750
6. If two positive integers $a$ and $b$ are written as $a=x^3y^2$ and $b=xy^3$ ($x, y$ are prime numbers), then HCF ($a, b$) is: (A) $xy$ (B) $xy^2$ (C) $x^3y^3$ (D) $x^2y^2$
7. If two positive integers $p$ and $q$ can be expressed as $p=ab^2$ and $q=a^3b$; $a, b$ being prime numbers, then LCM ($p, q$) is: (A) $ab$ (B) $a^2b^2$ (C) $a^3b^2$ (D) $a^3b^3$
8. The product of a non-zero rational and an irrational number is: (A) always irrational (B) always rational (C) rational or irrational (D) one
9. The least number that is divisible by all the numbers from 1 to 10 (both inclusive) is: (A) 10 (B) 100 (C) 504 (D) 2520
10. The decimal expansion of the rational number $\frac{14587}{1250}$ will terminate after: (A) one decimal place (B) two decimal places (C) three decimal places (D) four decimal places

(C) Short Answer Questions with Reasoning

Sample Question 1: The values of the remainder $r$, when a positive integer $a$ is divided by 3 are 0 and 1 only. Justify your answer.

Solution: No. According to Euclid's division lemma, $a=3q+r$, where $0 \le r < 3$ and $r$ is an integer. Therefore, the values of $r$ can be 0, 1 or 2.

Sample Question 2: Can the number $6^n$, $n$ being a natural number, end with the digit 5? Give reasons.

Solution: No, because $6^n=(2 \times 3)^n = 2^n \times 3^n$, so the only primes in the factorisation of $6^n$ are 2 and 3, and not 5. Hence, it cannot end with the digit 5.

EXERCISE 1.2

1. Write whether every positive integer can be of the form $4q+2$, where $q$ is an integer. Justify your answer.
2. "The product of two consecutive positive integers is divisible by 2". Is this statement true or false? Give reasons.
3. "The product of three consecutive positive integers is divisible by 6". Is this statement true or false? Justify your answer.
4. Write whether the square of any positive integer can be of the form $3m+2$ where $m$ is a natural number. Justify your answer.
5. A positive integer is of the form $3q+1$, $q$ being a natural number. Can you write its square in any form other than $3m+1$, i.e., $3m$ or $3m+2$ for some integer $m$? Justify your answer.
6. The numbers 525 and 3000 are both divisible only by 3, 5, 15, 25 and 75. What is HCF (525, 3000)? Justify your answer.
7. Explain why $3 \times 5 \times 7+7$ is a composite number.
8. Can two numbers have 18 as their HCF and 380 as their LCM? Give reasons.
9. Without actually performing the long division, find if $\frac{987}{10500}$ will have terminating or non-terminating (repeating) decimal expansion. Give reasons for your answer.
10. A rational number in its decimal expansion is 327.7081. What can you say about the prime factors of $q$, when this number is expressed in the form $\frac{p}{q}$? Give reasons.

(D) Short Answer Questions

Sample Question 1: Using Euclid's division algorithm, find which of the following pairs of numbers are co-prime: (i) 231, 396 (ii) 847, 2160

Solution: Let us find the HCF of each pair of numbers.
(i) $396=231 \times 1+165$; $231=165 \times 1+66$; $165=66 \times 2+33$; $66=33 \times 2+0$. Therefore, HCF=33. Hence, numbers are not co-prime.
(ii) $2160=847 \times 2+466$; $847=466 \times 1+381$; $466=381 \times 1+85$; $381=85 \times 4+41$; $85=41 \times 2+3$; $41=3 \times 13+2$; $3=2 \times 1+1$; $2=1 \times 2+0$. Therefore, the HCF=1. Hence, the numbers are co-prime.

Sample Question 2: Show that the square of an odd positive integer is of the form $8m+1$, for some whole number $m$.

Solution: Any positive odd integer is of the form $2q+1$ where $q$ is a whole number.
Therefore, $(2q+1)^2 = 4q^2+4q+1 = 4q(q+1)+1$.
$q(q+1)$ is either 0 or even. So, it is $2m$, where $m$ is a whole number.
Therefore, $(2q+1)^2 = 4 \cdot 2m+1 = 8m+1$.

Sample Question 3: Prove that $\sqrt{2}+\sqrt{3}$ is irrational.

Solution: Let us suppose that $\sqrt{2}+\sqrt{3}$ is rational. Let $\sqrt{2}+\sqrt{3}=a$, where $a$ is rational.
Therefore, $\sqrt{2}=a-\sqrt{3}$.
Squaring on both sides, we get $2=a^2+3-2a\sqrt{3}$. Therefore, $\sqrt{3}=\frac{a^2+1}{2a}$.
Which is a contradiction as the right hand side is a rational number while $\sqrt{3}$ is irrational. Hence, $\sqrt{2}+\sqrt{3}$ is irrational.

EXERCISE 1.3

1. Show that the square of any positive integer is either of the form $4q$ or $4q+1$ for some integer $q$.
2. Show that cube of any positive integer is of the form $4m$, $4m+1$ or $4m+3$, for some integer $m$.
3. Show that the square of any positive integer cannot be of the form $5q+2$ or $5q+3$ for any integer $q$.
4. Show that the square of any positive integer cannot be of the form $6m+2$ or $6m+5$ for any integer $m$.
5. Show that the square of any odd integer is of the form $4q+1$ for some integer $q$.
6. If $n$ is an odd integer, then show that $n^2-1$ is divisible by 8.
7. Prove that if $x$ and $y$ are both odd positive integers, then $x^2+y^2$ is even but not divisible by 4.
8. Use Euclid's division algorithm to find the HCF of 441, 567, 693.
9. Using Euclid's division algorithm, find the largest number that divides 1251, 9377 and 15628 leaving remainders 1, 2 and 3, respectively.
10. Prove that $\sqrt{3}+\sqrt{5}$ is irrational.
11. Show that $12^n$ cannot end with the digit 0 or 5 for any natural number $n$.
12. On a morning walk, three persons step off together and their steps measure 40 cm, 42 cm and 45 cm, respectively. What is the minimum distance each should walk so that each can cover the same distance in complete steps?
13. Write the denominator of the rational number $\frac{257}{5000}$ in the form $2^m \times 5^n$, where $m, n$ are non-negative integers. Hence, write its decimal expansion, without actual division.
14. Prove that $\sqrt{p}+\sqrt{q}$ is irrational, where $p, q$ are primes.

(E) Long Answer Questions

Sample Question 1: Show that the square of an odd positive integer can be of the form $6q+1$ or $6q+3$ for some integer $q$.

Solution: We know that any positive integer can be of the form $6m$, $6m+1$, $6m+2$, $6m+3$, $6m+4$ or $6m+5$, for some integer $m$. Thus, an odd positive integer can be of the form $6m+1$, $6m+3$, or $6m+5$.
Thus we have:
$(6m+1)^2 = 36m^2+12m+1 = 6(6m^2+2m)+1 = 6q+1$, $q$ is an integer.
$(6m+3)^2 = 36m^2+36m+9 = 6(6m^2+6m+1)+3 = 6q+3$, $q$ is an integer.
$(6m+5)^2 = 36m^2+60m+25 = 6(6m^2+10m+4)+1 = 6q+1$, $q$ is an integer.
Thus, the square of an odd positive integer can be of the form $6q+1$ or $6q+3$.

EXERCISE 1.4

1. Show that the cube of a positive integer of the form $6q+r$, $q$ is an integer and $r=0,1,2,3,4,5$ is also of the form $6m+r$.
2. Prove that one and only one out of $n$, $n+2$ and $n+4$ is divisible by 3, where $n$ is any positive integer.
3. Prove that one of any three consecutive positive integers must be divisible by 3.
4. For any positive integer $n$, prove that $n^3-n$ is divisible by 6.
5. Show that one and only one out of $n$, $n+4$, $n+8$, $n+12$ and $n+16$ is divisible by 5, where $n$ is any positive integer. [Hint: Any positive integer can be written in the form $5q, 5q+1, 5q+2, 5q+3, 5q+4$].