Class 11

Odisha State Board CHSE Odisha Class 11 Math Notes Chapter 8 Permutations And Combinations will enable students to study smartly.

CHSE Odisha 11th Class Math Notes Chapter 8 Permutations And Combinations

Fundamental Principle Of Counting:
(a) Fundamental principle of Multiplication:
If we choose an element from set A with m element and then one element from set B  with n elements, then are total number of ways we can make a choice is exactly mn.

OR

If an event can occur in m different ways, following which another event can occur in n different ways, then the total number of ways in which both the events can occur in succession in mn ways.

(b) Fundamental Principle of addition: If there are two events such that they can be performed independently in m and n different ways respectively, then either of two events can be performed in (m + n) ways.

Note:
(a) Use the multiplication principle if by doing one part of the job, the job remains incomplete.
(b) Use the addition principle if by doing one part of the job, the job is completed.

Factorial Notation:
If n ∈ N then the factorial of n, denoted by n! or ∠n is defined as
n! = n (n – 1). (n – 2) … 3.2.1.

Note:
0! = 1

Properties of Factorial:
(1) Factorial of negative integers is not defined
(2) n! = n(n – 1)!
= n(n – 1) (n – 2)!
= n(n – 1) (n – 2) (n – 3)!
(3) \(\frac{n !}{r !}\) = n(n – 1) (n – 2) ….. (r + 1)
(4) Exponent of a prime number p in n! denoted by
\(\mathrm{E}_p(n !)=\left[\frac{n}{p}\right]+\left[\frac{n}{p^2}\right]+\ldots \ldots\)

Permutation:
Each of the arrangements which can be made by taking some or all objects or things at a time is called a permutation.

(a) Permutation of n different objects:

• Number of permutations of n different objects have taken all at a time = \({ }^n \mathrm{P}_n\) = n!.
• Number of permutations of n different objects taken none at a time = \({ }^n \mathrm{P}_0\) = 1
• Number of permutations of n different objects taken r at a time = \({ }^n \mathrm{P}_r\) = P(n, r) = \(\frac{n !}{(n-r) !}\)

(b) Permutation ofnon-distinct objects:
(1) Number of permutations of n objects taken all at a time of which p objects are of same kind and others are distinct = \(\frac{n !}{p !}\)
(2) Number of permutations of n objects taken all at a time of which p objects are of one kind, q objects are of a second kind and other are distinct = \(\frac{n !}{p ! q !}\)
(3) Number of permutations of n objects taken all at a time in which p₁ objects are of one kind, p₂ are of second kind, p₃ are 3rd kind ….. and
p_n are of nth kind and other are distinct. = \(\frac{n !}{p_{1} ! p_{2} ! \cdots p_{n} !}\)

(c) Restricted permutations:

• Permutation of distinct objects with repetition: The number of permutations of n different things taken r at a time when each thing may be repeated any number of times = n^r
• Number of permutations of n different things taken r at a time when a particular thing is to be always included in each arrangement = r. ^n-1P_r-1.
• Number of permutations of n different things, taken r at the time when p particular are to be always included in each arrangement = P(r – (p – 1) ^n-pP_r-p.
• Number of permutations of n different things taken r at a time, when a particular thing is never taken in each arrangement = ^n-1P_r.
• Number of permutations of n different things taken r at a time, when p particular things never taken in each arrangement = ^n-pP_r.

(d) Circular permutation:
(1) When we do an arrangement of objects along a closed curve we call it the circular permutation.
(2) Number of circular permutations of n distinct objects taken all at a time = (n – 1)!, where clockwise and anti-clockwise orders are taken as different, as arrangements round a table.
(3) Number of circular permutations of n distinct objects taken all at a time, where clockwise and anti-clockwise orders make no difference as beads or flowers in a necklace or garland.
= \(\frac{(n-1) !}{2}\)
(4) Number of circular permutations of n different things taken r at a time where clockwise and anti-clockwise orders are different = \(\frac{\left({ }^n \mathrm{P}_r\right)}{r}\)
(5) Number of circular permutations of n different things taken r at a time where clockwise and anti-clockwise orders make no difference = \(\frac{\left({ }^n \mathrm{P}_r\right)}{2 r}\)

(e) Some more restricted permutations:

• Number of permutations of n different things taken all at a time, when m specified things come together = m!(n – m + 1)!.
• Number of permutations of n different things taken all at a time when m specified things never come together = n!  – m!(n – m + 1)!.

Combinations:
Each of the different selections made by taking some or all objects at a time irrespective of any order is called a combination.

(a) Difference between permutation and combination:

• A combination is a selection but a permutation is not a selection but an arrangement.
• In combination the order of appearance of objects is immaterial, whereas in a permutation the ordering is essential.
• Practically to find permutations of n different objects taken r at a time, we first select objects then we arrange them.
• One combination corresponds to many permutations.

(b) Combinations of n different things taken r at a time:
The number of combinations of n different things have taken r at a time ⁿc_r = C(n, r) = \(\left(\begin{array}{l}
n \\
r
\end{array}\right)=\frac{n !}{r !(n-r) !}\)

(c) Properties of ⁿc_r :
(1) ⁿc_r = ⁿC₀ = 1, ⁿC₁ = n
(2) ⁿC_r = ⁿC_n-r
(3) ⁿC_r + ⁿC_r-1 = ⁿ⁺¹C_r (Euler’s formula)
(4) ⁿC_x = ⁿC_y ⇒ x = y or x + y = n
(5) n. ^n-1C_r-1 = (n – r + 1) ⁿC_r-1
(6) ⁿC_r = \(\frac{n}{r}{ }^{n-1} \mathrm{C}_{r-1}\)
(7) \(\frac{{ }^n \mathrm{C}_r}{{ }^n \mathrm{C}_{r-1}}=\frac{n-r+1}{r}\)
(8) If n is even then the greatest value of ⁿC_r is ⁿC_n/2.
⇒ If n is odd then the greatest value of ⁿC_r is \({ }^n \mathrm{C}_{\left(\frac{n+1}{2}\right)} \text { or }{ }^n \mathrm{C}_{\left(\frac{n-1}{2}\right)}\)

(d) Number of combinations of n different things taken r at a time, when k particular things always occur = ^n-kC_r-k

(e) The number of combinations of n different things, taken r at a time where k particular things never occur = ^n-kC_r

(f) The total number of combinations of n different things taken one or more at a time (or the number of ways of n different things selecting at least one of them) = ⁿC₁ + ⁿC₂ + ⁿC₃ + ….. + ⁿC_n = 2ⁿ -1

(g) The number of combinations of n identical things taken r at a time = 1.

(h) Number of ways of selecting r things out of n alike things where r = 0, 1, 2, 3 ….. n is (n+ 1).

(i) Division into groups:

• The number of ways in which (m + n) different things can be divided into two groups which contain m and n things respectively = \(\frac{(m+n) !}{m ! n !}\) for m ≠ n.
• If m-n then the groups are of equal size. Thus, division can be done in two ways as:
  ⇒ If order of groups is not important: In this case the number of ways = \(\frac{(2 n) !}{2 !(n !)^2}\)
  ⇒ If order of groups is important: In this case the number of ways = \(\frac{(2 n) !}{(n !)^2}\)

(j) Arrangement in groups:

1. The number of ways in which n different things can be arranged into r different groups = ^n+r-1P_n or n! ^n-1C_r-1
2. The number of ways in which n different things can be distributed into r different groups = rⁿ – ^rC₁(r – 1)ⁿ + ^rC₂(r – 2)ⁿ ….. + (-1)^r-1 . ^rC_r-1. (Blank groups are not allowed)
3. The number of ways in which n identical things can be distributed into r different groups where blank groups are allowed
   = ^(n+r-1)C_(r-1)
   = ^(n+r-1)C_n
4. Number of ways in which n identical things can be distributed into r different groups where blank groups are not allowed (each group receives at least one item) = ^n-1C_r-1

(k) Number of divisors:

Odisha State Board CHSE Odisha Class 12 Invitation to English 4 Solutions Grammar Additional Questions Textbook Activity Questions and Answers.

CHSE Odisha 11th Class English Grammar Additional Questions

A. Rewrite the passages after correcting all grammatical errors in it.

(1) There is great excitement in the planet of Venus this week. For the first time, Venusia scientists manage to land an unmanned spacecraft in the planet Venus, and it is sending back signals, as well as photographs, ever since. The craft directed into an area known as Gonebay.
Answer:
There was great excitement on the planet of Venus this week. For the first time, Venusian scientists managed to land an unmanned spacecraft on the planet Venus, and it has been sending back signals, as well as photographs ever since. The craft was directed into an area known as Gonebay.
(2) How did birds know when to flew south for winter? How long do a bear sleep in winter? Do a porcupine really shoot its quills at an enemy? How do squirrels know where he buries nuts?
Answer:
How do birds know when to fly south for the winter? How long does a bear sleep in winter? Does a porcupine really shoot his quills at an enemy? How does a squirrel know where he buried a nut?

B. Correct the errors.

(a) He is an European.
(b) I met the concerned clerk.
(c) It is high time you get up early.
(d) It has been five years since I last met you.
(e) I congratulate you for winning the prize.
Answer:
(a) He is a European.
(b) I met the clerk concerned.
(c) It is high time you got up early.
(d) It is five years since I last met you.
(e) I congratulate you on winning the prize.

C. Supply the correct tense of the verbs given in brackets.

1. Water always (freeze) at 0 degrees centigrade.
2. Students frequently (make) mistakes of tense usage when they do this exercise.
3. I (have) my hair cut whenever it gets too long.
4. I (take) my dog for a walk every evening before it died.
5. He (come) to my office whenever he needed money.
6. Last year she (wear) the same dress at every party.
7. Whenever I climb a hill, my ear (boil).
8. She (sing) very beautifully before she was married, but nowadays she (not sing) anymore.
9. I seldom (see) him at concerts these days. He (go) to them regularly before the war.
10. She cooks very well but her sister (cook) much better when I knew her.
11. Every time he opens his mouth, he (say) something foolish.
12. He occasionally makes a big effort, but usually he (not bother).
13. Whenever I (go) to see him, he was out.
14. In the past men frequently (fight) duels. Nowadays they seldom (do).
15. How often you (go) to the theatre when you were in London?
16. You (play) with dolls when you were a little girl?
17. The ancient Egyptians (build) pyramids as tombs for their kings.
18. When I was young, my father always (give) me some money on Saturdays.
19. If he is wise, a pianist (practise) four hours a day.
20. His parents don’t know what to do with their child. He (lie) habitually.
21. My aunt Jane (hate) girls who made up.
22. We all (study) Latin when we were at school.
23. Wood always (float).

Answer:
1. Water always freezes at 0 degrees centigrade.
2. Students frequently make mistakes of tense usage when they do this exercise.
3. I have my hair cut whenever it gets too long.
4. I took my dog for a walk every evening before it died.
5. He came to my office whenever he needed money.
6. Last year she wore the same dress at every party.
7. Whenever I climb a hill, my ear boils.
8. She sang very beautifully before she was married, but nowadays she does not sing anymore.
9. I seldom see him at concerts these days. He went to them regularly before the war.
10. She cooks very well but her sister cooked much better when I knew her.
11. Every time he opens his mouth, he says something foolish.
12. He occasionally makes a big effort, but usually, he does not bother.
13. Whenever I went to see him, he was out.
14. In the past men frequently fought duels. Nowadays they seldom do.
15. How often you go to. the theatre when you were in London?
16. Did you play with dolls when you were a little girl?
17. The ancient Egyptians built pyramids as tombs for their kings.
18. When I was young, my father always gave me some money on Saturdays.
19. If he is wise, a pianist practices four hours a day.
20. His parents don’t know what to do with their child. He lies habitually.
21. My aunt Jane hated girls who made up.
22. We all studied Latin when we were at school.
23. Wood always floats.

D. Put the verbs in brackets into the correct Present Tense, Continuous, or Simple.

1. Buses usually (run) along this street. but today they (not run) because it is under repair.
2. John (pass) the post office on his way to work every day.
3. She usually (sit) at the back of the class, but today she (sit) in the front row.
4. I rarely (carry) an umbrella, but I (carry) one now because it is raining.
5. What you generally (do) for a living?
6. You (enjoy) your English class today?
7. You (enjoy) washing dishes as a rule?
8. We nearly always (spend) our holidays at the seaside, but this year we are going to France.
9. Mr. Jones usually (sell) only newspapers, but this week he (sell) magazines as well.
10. You (wash) your hands before every meals.
11. Mary generally (begin) cooking at 11, but today she came home early and (cook) now, although il is only 10.30.
12. I’m sorry you can’t see her. She (sleep) still. She usually (wake) much earlier.
13. Why you (wear) a coat this morning? I never (wear) one till October.
14. Joan still (do) her homework. Her sister, who always (work) quicker, (play) already in the garden.
15. These builders generally (build) very rapidly. They (work) at present on two separate contracts.
16. What (do) you at this moment? If you (not do) anything, please help me.
17. John, who (study) medicine at present, hopes to go abroad after graduation.
18. He generally (come) to my office every clay, but today he (visit) his parents in the country.
19. You (watch) television often? The electrician (install) ours at this moment.
20. Mary usually (wear) a hat to go shopping. but today, as the sun (shine) she (not wear) one.

Answer:
1. Buses usually run along this Street, but today they are not running because it is under repair.
2. John passes the post office on his way to work every day.
3. She usually sits at the back of the class, but today she is sitting in the front row.
4. I rarely carry an umbrella, but I am carrying one now because it is raining.
5. What do you generally do for a living?
6. Are you enjoying your English class today?
7. Do you enjoy washing dishes as a rule?
8. We nearly always spend our holidays at the seaside, but this year we are going to France.
9. Mr. Jones usually sells only newspapers, but this week he is selling magazines as well.
10. Do you wash your hands before every meal?
11. Mary generally begins cooking at 11, but today she came home early and is cooking now, although it is only 10.30.
12. I’m sorry you can’t see her. She is still sleeping. She usually wakes much earlier.
13. Why are you wearing a coat this morning? I never wear one till October.
14. Joan is still doing her homework. Her sister, who always works quicker, is already playing already in the garden.
15. These builders generally build very rapidly. They are working at present on two separate contracts.
16. What are you doing at this moment? If you are not doing anything, please help me.
17. John, who is studying medicine at present, hopes to go abroad after graduation.
18. He generally comes to my office every day, but today he (is visiting) his parents in the country.
19. Do you often watch television? The electrician is installing ours at this moment.
20. Mary usually wears a hat to go shopping, but today, as the sun is shining she is not wearing one.

E. Put the verbs in brackets into the correct tense, Continuous or Simple.

1. You (see) the house on the corner? That is where I was born.
2. You (listen) to what I am saying You (understand) me?
3. I (notice) Mary (wear) a new hat today.
4. She (not understand) what you (mean).
5. I (need) a new suit. They (offer) special prices at the tailor this week.
6. You (smell) gas? I (think) the new stove is leaking.
7. Look at Mary! She (drink) up her medicine, but I can say that she (hate) it.
8. John (seem) rather tired today.
9. it still (rain), but it (look) as if it will soon stop.
10. You (mind) helping me a moment? I (try) to mend this table.
11. Ask him what he (want).
12. You (remember) the name of that girl who (walk) on the other side of the street?
13. ‘Will you have some tea?’ ‘I (prefer) coffee, please.’
14. I (suppose) I must go now. My wife (wait) for me at home.
15. You (see) this box? It (contain) matches.
16. These twins, who (resemble) one another so strongly, (study) art at present.
17. After what has happened, you really (mean) to say that you still (believe) him?
18. You (suppose) the children still (sleep)?
19. ‘The train still (stand) in the station. You (think) we can just catch it?
20. I (notice) you (possess) a copy of Waugh’s latest book. Will you lend it me?

Answer:
1. Do you see the house on the corner? That is where I was born.
2. Are you listening to what I am saying? Do you understand me?
3. I notice Mary is wearing a new hat today.
4. She does no: understand what you mean.
5. I need a new suit. They are offering special prices at the tailor this week.
6. Do you smell gás? I think the new stove is leaking.
7. Look at Mary! She is drinking up her medicine, but I can say that she hates it.
8. John seems rather tired today.
9. It is still raining, but it looks as if it will soon stop.
10. Do you mind helping me a moment? I urn Irving to mend this table.
11. Ask him what he wants.
12. Do you remember the name of that girl who is walking on the other side of the street?
13. ‘Will you have some tea?’ ‘I prefer coffee, please.’
14. 1 suppose I must go now. My wife is waiting for me at home.
15. Do you see this box? It contains matches.
16. These twins, who resemble one another so strongly, are studying art at present.
17. After what has happened, do you really mean to say that you still believe him?
18. Do you suppose the children still sleeping?
19. ‘The train is still standing in the station. Do you think we can just catch it
20. I notice you possess a copy of Waugh’s latest book. Will you lend it me?

F. Supply the correct form of the Present Perfect tense, Continuous or Simple in the place of the verbs in brackets.

1. They just (arrive) from New York.
2. They still (not succeed) in reaching the summit.
3. I this very minutes (receive) a telegram from my brother in India.
4. We already (have) breakfast.
5. I now (study) your proposals and regret I cannot accept them.
6. They (live) here since January.
7. We (wait) on the platform since three o’ clock.
8. She already (ring) the bell twice.
9. I see you just (have) your hair cut.
10. She (write) letters all morning, but I (not start) to write any yet.
11. The children (sleep) all this afternoon.
12. How long you (stay) in that old hotel?
13. They (work) in the same factory for twenty years now.
14. Since when you (have) that new car?
15. I (knock) on the door for ten minutes now without an answer.
16. They (build) that bridge for over a year and still it isn’t finished.
17. I (try) three times and (be) successful only once.
18. How many times you (be) to the cinema this week?
19. He (go) to the dentist off and on for six months.
20. He (take) the exam. three times and (fail) every time.
21. William (marry) the eldest Jones girl at last.
22. I (try) to get in touch with you for several days now.
23. She just (spend) three weeks at her grandmother’s.
24. He (work) hard on his book for some time and (finish) it at last.
25. You ever (read) ‘War and Peace’?

Answer:
1. They have just arrived from New York.
2. They have still not succeeded in reaching the summit.
3. I have this very minutes received a telegram from my brother in India.
4. We have already had breakfast.
5. I have now studied your proposals and regret I cannot accept them.
6. They have been living here since January.
7. We have been waiting on the platform since three o’ clock.
8. She has already rung the bell twice.
9. I see you have just had your hair cut.
10. She has been writing letters all morning, but I have no: starred to write any yet.
11. The children have been sleeping all this afternoon.
12. How long have you been staying in that old hotel?
13. They have been working in the same factory for twenty years now.
14. Since when have you had that new car?
15. I have been knocking on the door for ten minutes now without an answer.
16. They have been building that bridge for over a year and still it isn’t finished.
17. I have tried three times and have been successful only once.
18. How many times have you been to the cinema this week?
19. He has been going to the dentist off and on for six months.
20. He has taken the exam. three times and has failed every time.
21. William has married the eldest Jones girl at last.
22. I have been trying to get in touch with you for several days now.
23. She has just spent three weeks at her grandmother’s.
24. He has been working hard on his book for some time and has just finished it at last.
25. Have you ever read ‘War and Peace’?

G. Supply the missing prepositions using only to or at.

1. He is quite blind _____ her faults.
Answer:
to

2. He is extraordinarily clever _____ mimicking others.
Answer:
at

3. She, on the other hand, is very efficient _____ her work.
Answer:
at

4. He is an expert making himself understood in foreign languages.
Answer:
at

5. Contrary my expectations, I quite enjoyed myself at the party.
Answer:
to

6. She was standing too close _____ the fire and got burned.
Answer:
to

7. That fellow’s no good games at all.
Answer:
at

8. He carried out the project which had always been dear _____ his heart.
Answer:
to

9. We are all very indignant _____ the injustice done to him.
Answer:
at

10. I’m sorry! I’m very bad _____ explaining myself.
Answer:
at

11. He’s the one who’s so very lucky _____ cards.
Answer:
at

12. That device is entirely new _____ me.
Answer:
to

13 He’s not equal _____ the job they’ve given him.
Answer:
to

14. He remained faithful _____ his principles in spite of great pressure.
Answer:
to

15. The delay proved fatal _____ our plans.
Answer:
to

16. She was overjoyed _____ the prospect of meeting him again.
Answer:
at

17. His activities are very harmful _____ my interests.
Answer:
to

18. The government showed itself hostile _____ any progress.
Answer:
to

19. She’s no terribly cruel _____ her dog.
Answer:
to

20. I engaged him because he was so prompt in understanding my instructions.
Answer:
at

21. He was heartbroken _____ her indifference to him.
Answer:
at

22. He doesn’t like me although I’ve always been kind _____ him.
Answer:
to

23. Yes, he’s the kind of person who is always quick figures.
Answer:
at

24. This is much inferior _____ the one I bought last week.
Answer:
to

25. You will be liable _____ a heavy fine if you do that.
Answer:
to

26. This flower is not native _____ England.
Answer:
to

27. Naturally she was sad _____ the death of her parrot.
Answer:
at

28. He does his work carefully but he’s terribly slow ______ it.
Answer:
at

29. Who wouldn’t be triumphant _____ their success in the examination?
Answer:
at

30. A dutiful daughter and obedient _____ her parents.
Answer:
to

31. You shouldn’t be surprised _____ a thing like that.
Answer:
at

32. It should be obvious _____ the meanest intelligence.
Answer:
to

33. The seeds are peculiar ______ this genus of plant.
Answer:
to

34. Can’t you manage to be a little more polite your aunt?
Answer:
to

35. He works in a factory. Previous _____ that he was in a laundry.
Answer:
to

36. I have been truly astonished _____ the number of people who believe it.
Answer:
at

37. This is quite irrelevant ______ the matter we are discussing.
Answer:
to

38. They were shocked _____ his apparent lack of appreciation.
Answer:
at

39. He was so rude _____ her that she never spoke to him again.
Answer:
to

40. Sacred _____ the memory of Mary Jones.
Answer:
to

41. They are very sensitive _____ people’s opinion of them.
Answer:
to

42. I’ve got one similar _____ yours.
Answer:
to

43. Subject _____ the exigencies of the service.
Answer:
to

44. The people of this country are very skilful ______ making dolls.
Answer:
at

45. It’s useful _____ me to have him about the house.
Answer:
to

46. It’s vital _____ a proper understanding of the problem.
Answer:
to

H. Supply the missing preposition using with, for or of.

1. Don’t be afraid _____ the dog! It won’t bite you.
Answer:
of

2. I can’t be angry _____ him now that he’s apologized _____ what he has done.
Answer:
with, for

3. He’s far ahead _____ the others in arithmetic.
Answer:
of

4. You ought to be ashamed _____ yourself.
Answer:
of

5. Are you aware _____ the fact that it is half past ten.
Answer:
of

6. I’m sorry. They are simply not capable doing it.
Answer:
of

7. Don’t disturb him! He’s busy _____ his accounts.
Answer:
with

8. He’s ambitious and eager _____ honours.
Answer:
for

9. For goodness sake! Do be careful _____ that vase. You could easily drop it.
Answer:
with

10. Children must be taught to be careful _____ traffic.
Answer:
of

11. I am not at all certain _____ the date of his arrival.
Answer:
of

12. His explanation was not consistent _____ the facts.
Answer:
with

13. Monsieur X was famous _____ his collection of pictures.
Answer:
for

14. I was conscious _____ a feeling of uneasiness.
Answer:
of

15. Kingston lies due west _____ London.
Answer:
of

16. I’m sorry. I’m not content _____ your explanations.
Answer:
with

17. The soldier was pronounced fit _____ service.
Answer:
for

18. Mary was terribly envious _____ Joan’s new hat.
Answer:
of

19. We are all very fond _____ going to the theatre.
Answer:
of

20. John is very discontented _____ his salary.
Answer:
with

21. I shall be grateful any advice you can give me.
Answer:
for

22. This exercise is full ______ the most terrible mistakes.
Answer:
of

23. Are you familiar _____ the works of Milton?
Answer:
with

24. The manager is well qualified _____ his position.
Answer:
for

25. That is something I am profoundly glad.
Answer:
of

26. This chair is not identical _____ the one I bought last year.
Answer:
with

27. He’s a sporting fellow, and always ready for anything.
Answer:
for

28. That student is ignorant _____ the first rules of grammar.
Answer:
of

29. Your explanation is incompatible _____ the story I heard.
Answer:
with

30. John, you will be responsible for providing the drinks.
Answer:
for

31. Judges must be independent of

political pressure.
Answer:
of

32. I know he’s a difficult child, but you must be patient _____ him.
Answer:
with

33. I am extremely sorry _____ the delay, but I was held up.
Answer:
for

34. He was jealous _____ his brother’s good fortune.
Answer:
of

35. Comedians are always popular holiday crowds.
Answer:
with

36. I can’t bake a cake as we are short _____ eggs this week.
Answer:
of

37. His income is sufficient _____ his needs.
Answer:
for

38. He’s always very shy approaching his chief.
Answer:
of

39. Let us be thankful _____ small mercies.
Answer:
for

40. It is wise to be sure _____ your facts before you speak.
Answer:
of

41. One is generally tolerant _____ small faults.
Answer:
of

42. She is, unfortunately, devoid _____ a sense of humor.
Answer:
of

43. They gave him a visa valid _____ all countries in Europe.
Answer:
for

44. I’m tired _____ arguing with you.
Answer:
of

45. They proved themselves unworthy _____ the trust which was placed in them.
Answer:
of

K. Supply the missing prepositions, from, about, on or in.

1. Keep away _____ the machine while it is running.
Answer:
from

2. He wsa singularly fortunate _____ his choice of wallpaper.
Answer:
in

3. He is intent _____ attending the football match on Saturday.
Answer:
on

4. I am very dubious _____ your chances of passing the examination.
Answer:
about

5. They are proficient _____ the use of their fists.
Answer:
in

6. This is quite different _____ what I expected.
Answer:
from

7. The diet here is deficient _____ vitamins.
Answer:
in

8. That young man is very keen _____ cycling.
Answer:
on

9. We are all very enthusiastic ________ our next holiday.
Answer:
about

10. It was far ______ my intention to suggest that he was unintelligent.
Answer:
from

11. He was perfectly honest _____ his intentions to win the prize at all costs.
Answer:
about

12. The secretary was not well qualified _____ shorthand.
Answer:
in

13. Some people appear completely immune _____ this disease.
Answer:
from

14. I am very reluctant _____ asking him to do this.
Answer:
about

15. Our plans must remain dependent _____ the weather.
Answer:
on

16. Everybody was very uneasy _____ the outcome of the negotiations.
Answer:
about

17. Of course, you are quite right _____ that.
Answer:
about

18. He was involved in an accident, resulting _____ the slippery condition of the road.
Answer:
from

19. The enemy is weak _____ artillery.
Answer:
in

20. Entomologists are still curious _____ the life-cycle of that month.
Answer:
about

21. I am not very interested _____ the story of your life.
Answer:
in

22. Put that cake back in the cupboard. where it will be safe _____ the cat.
Answer:
from

23. I’m afraid he is quite wrong _____ the date of the invasion.
Answer:
about

24. His father was very sad _____ his son’s failure in his final exams.
Answer:
about

25. I am extremely doubtful _____ the wisdom of pursuing that course of action.
Answer:
about

L. Supply the prepositions at or to as appropriate.

1. What time did you arrive _____ your home?
Answer:
at

2. He finds it difficult to accustom himself _____ the climate.
Answer:
to

3. AIl the visitors exclaimed _____ the beauty of the place.
Answer:
at

4. Just glance _____ this for me, would you?
Answer:
at

5. His debts amount _____ a considerable sum.
Answer:
to

6 I can only guess _____ the extent of the damage.
Answer:
at

7. It is useless to appeal _____ his better nature.
Answer:
to

8. She hinted darkly _____ all sorts of wild actions in his youth.
Answer:
at

9. If you want permission. you must apply _____ the caretaker.
Answer:
to

10. He was attached _____ the French Army during the war.
Answer:
to

11. He is too sick to attend _____ his duties.
Answer:
to

12. _____ what do you attribute your success in life?
Answer:
to

13. I’m sure this one doesn’t belong _____ me.
Answer:
to

14. He challenged him _____ a game of chess.
Answer:
to

15. It is very unkind to joke _____ the expense of the disabled.
Answer:
at

16. Shall I compare thee _____ a summer’s day?
Answer:
to

17. If you want them to hear, you’ll have to knock a good deal harder _____ the door.
Answer:
at

18. The prisoner was condemned _____ penal servitude for life.
Answer:
to

19. The fire was confined _____ the kitchen regions.
Answer:
to

20. I will never consent _____ her marrying that man.
Answer:
to

21. Just have a look _____ this for me, would you?
Answer:
at

22. I have been entirely converted _____ the use of an electric razor.
Answer:
to

23. Employees who have twenty-five years’ service become entitled _____ a pension.
Answer:
to

24. You can safely entrust your little son _____ her care.
Answer:
to

25. People who heard her voice marvelled _____ it.
Answer:
at

26. Let’s invite them all _____ dinner.
Answer:
to

27. Listen _____ me!
Answer:
to

28. Do you object my smoking?
Answer:
to

29. He peered _____ the exhibit on account of his short-sightedness.
Answer:
at

30. Such an idea would never occur _____ me!
Answer:
to

31. The patient is reacting very unsatisfactorily _____ the drug.
Answer:
to

32. The children peeped _____ the guests as they were arriving.
Answer:
at

33. I have been reduced ______ using oil for lack of fat.
Answer:
to

34. The children are playing _____ Red Indians again.
Answer:
at

35. She has been forced to resort _____ all sorts of devices to avoid him.
Answer:
to

36. The patient has not responded _____ treatment.
Answer:
to

37. It is very rude to point people in the street.
Answer:
at

38. If you’ll bring the drinks, I’ll see _____ the food.
Answer:
to

39. They have been subjected _____ all sorts of indignities.
Answer:
to

40. I refuse to submit ______ that sort of treatment.
Answer:
to

41. Can you wonder _____ it, if they are reduced _____ begging.
Answer:
at, to

42. I’m sorry he finally succumbed _____ the temptation of stealing.
Answer:
to

43. If you want to pass your examination, you il have to work very hard _____ your Latin.
Answer:
at

44. We shall never surrender _____ that enemy.
Answer:
to

45. I don’t know him, but he has been starting _____ me for ten minutes.
Answer:
at

46. I have never subscribed _____ the general opinion of him.
Answer:
to

47. Turn _____ page 22 and start reading!
Answer:
to

48. The government has again yielded _____ the pressure from outside.
Answer:
to

49. She always trusts _____ her neighbors to help her.
Answer:
to

50. Would you please reply my question?
Answer:
to

M. Turn the following into passive.

1. The government has called out troops.
Answer:
Troops have been called out.

2. Fog held up the trains. (agent required)
Answer:
Trains were held up by fog.

3. You are to leave this here. Someone will call for it later on.
Answer:
This is to be left here. k will be called for.

4. We called in the police.
Answer:
Police were called in.

5. They didn’t look after the children properly.
Answer:
Children were not properly looked after.

6. They are flying in reinforcements.
Answer:
Reinforcements are being flown in.

7. Then they called up men of 28.
Answer:
Men of 28 were called up.

8. Everyone looked up to him. (agent required)
Answer:
He was looked up to by everyone.

9. All the ministers will see him off at the airport. (agent required)
Answer:
He will be seen off at the airport by all the ministers.

10. He hasn’t slept in his bed.
Answer:
Bed hasn’t been slept in.

11. We can build on more rooms.
Answer:
More rooms can be built on.

12. They threw him out.
Answer:
He was thrown out.

13. They will have to adopt a different attitude.
Answer:
Different attitude will have to be adopted.

14. He’s a dangerous maniac. They ought to lock him up.
Answer:
He ought to be locked up.

15. Her story didn’t take them in. (agent required)
Answer:
They weren’t taken in by her story.

16. Burglars broke into the house.
Answer:
House was broken into.

17. The manufacturers are giving away small plastic toys with each packet of cereal.
Answer:
Small plastic toys are being given away.

18. They took down the notice.
Answer:
Notice was taken down.

19. They frown on smoking here.
Answer:
Smoking is frown on.

20. After the government had spent a million pounds on the scheme they decided that it was impracticable and gave it up. (Make only the first and last verbs passive)
Answer:
After a million pounds had been spent, the scheme was given up.

21. When I returned I found that they had towed my car away. I asked why they had done this and they told me that it was because I had parked it under a No Parking sign. (four passives)
Answer:
My car had been towed away. I asked why this had been done and was told that it had been parked.

22. People must hand in their weapons.
Answer:
Weapons must be handed in.

23. The crowd shouted him down.
Answer:
He was shouted down.

24. People often take him for his brother.
Answer:
He is often taken for his brother.

25. No one has taken cut the cork.
Answer:
The cork hasn’t been taken out.

26. The film company were to have used the pool for aquatic displays, but now they have changed their minds about it and are filling it in. (Make the first and last verbs passive)
Answer:
Pool was to have been used it is being filled in.

27. This college is already full. We ne turning away students the whole time.
Answer:
Students are being turned away.

28. You will have to pull down this skyscraper as you have not complied with the town planning regulations.
Answer:
Skyscraper will have to be pulled down as the town planning regulations have not been complied with.

N. Put the following sentences into passive, using infinitive construction where possible.

1. We added up the money and found that it was correct.
Answer:
Money was added up and found to be correct.

2. I’m employing a man to tile the bathroom.
Answer:
I am having the bathroom tiled.

3. Someone seems to have made a terrible mistake.
Answer:
A terrible mistake seem to have been made.

4. It is your duty to make tea at eleven o’ clock. (Use suppose.)
Answer:
You are supposed to make tea.

5. People know that he is armed.
Answer:
He is known to be armed.

6. Someone saw him pick up the gun.
Answer:
He was seen to pick up?

7. We know that you were in town on the night of the crime.
Answer:
You are known to have been.

8. We believe that he has special knowledge which may be useful to the police. (one passive)
Answer:
He is believed to have special knowledge.

9. You needn’t have done this.
Answer:
This needn’t have been done.

10. It’s a little too loose: you had better ask your tailor to take it in. (one passive)
Answer:
You had better have it taken in.

11. He likes people to call him — ‘sir’.
Answer:
He likes to be called sir’.

12. Don’t touch this switch.
Answer:
This switch isn’t to be/mustn’t be touched.

13. You will have to get someone to see to it.
Answer:
You will have to have/get it seen to.
(Or) It will have to be seen to.

14. It is impossible to do this. (Use can’t)
Answer:
This can’t be done.

15. Someone is following us.
Answer:
We are being followed.

Odisha State Board CHSE Odisha Class 11 Math Notes Chapter 5 Principles of Mathematical Induction will enable students to study smartly.

CHSE Odisha 11th Class Math Notes Chapter 5 Principles of Mathematical Induction

Principles Of Mathematical Induction

(i) Principle – 1
Let P(n) is a statement , n ∈ Z

Step – 1: Verification step:
verify that P(1) is true.

Step – 2: Induction step – 1:
Assume that P(k) is true for any arbitrary k ∈ N.

Step – 3: Induction step – 2:
prove that P_(k+1) is true using step – 1 and step – 2

Step – 4: Conclusion Step:
If P_(k+1) is true then take a conclusion that by Principle of mathematical induction P(n) is true for all n ∈ N.

(ii) Principle – 2
Let P(n) be a statement, n ∈ N.

Step – 1: Verification step:
verify the P(1) is true

Step – 2: Induction step – 1:
Assume that P(2), P(3),….. P(k) is true.

Step – 3: Induction step – 2:
Using step – 1 and step – 2 prove that P(k + 1) is true.

Step – 4: Conclusion step:
If P(k + 1) is true, then take a conclusion that by Principle of mathematical induction P(n) is true for all n ∈ N.

Odisha State Board CHSE Odisha Class 11 Math Notes Chapter 4 Trigonometric Functions will enable students to study smartly.

CHSE Odisha 11th Class Math Notes Chapter 4 Trigonometric Functions

Angle:
If A, B, and C are three non-collinear points, then ∠ABC = \(\overrightarrow{\mathrm{BA}} \cup \overrightarrow{\mathrm{BC}}\)

\(\overrightarrow{\mathrm{BA}}\) is the initial side, \(\overrightarrow{\mathrm{BC}}\) is called the terminal side and B is called the vertex of the angle.

Positive and negative angles:
If the direction of rotation is anti-clockwise then the angle is positive and if the direction of rotation is clockwise then the angle is negative.

Measure of an angle:
(a) Sexagesimal system or English System (Degree measure):
1 degree = 1° = \(\left(\frac{1}{360}\right) \text { th }\) of revolution from initial side to terminal side.

• One revolution = 360°
• 1° = 60′ (sixty minute)
• 1′ = 60” (sixty seconds)

(b) Circular system(Radian measure):
One radian = 1^c = The angle at the centre of the circle by an arc where the arc length equals to

Note:
(i) θ = \(\frac{l}{r}=\frac{\text { arc }}{\text { radius }}\) where θ is an radian.
(ii) θ in radian is created as a real number.

(c) Relation between Degree and radian measure:

• 2π radians = 360°
  ⇒ π radian = 180°
• We can convert radian to degree or degree to radian by using the identity.
  \(\frac{\mathrm{D}}{180}=\frac{\mathrm{R}}{\pi}\) where D is the degree measure and R is the radian measure of an angle.

Trigonometry Functions:

(i) Sign of trigonometry functions:

(ii) \(\begin{array}{cccc}
\text { Add } & \text { Sugar } & \text { To } & \text { Coffee } \\
\downarrow & \downarrow & \downarrow & \downarrow \\
\text { all }+ & \sin + & \tan + & \cos +
\end{array}\)

(iii) Periodicity of trigonometry functions:

Trigonometric function	Period
sin x	2π
cos x	2π
tan x	π
cot x	π
sec x	2π
cosec x	2π
sin2 x or cos2 x	π
|sin x| or |cos x|	π

Trigonometric functions of some standard angles

Fundamental trigonometric identities:
(a) sin θ = \(\frac{1}{{cosec} \theta}\)
(b) cos θ = \(\frac{1}{{sec} \theta}\)
(c) tan θ = \(\frac{1}{{cot} \theta}\)
(d) sin² θ + cos² θ = 1
(e) sec² θ – tan² θ = 1
(f) cosec² θ – cot² θ = 1
(g) sin (-θ) = -sin (θ)
(h) cos (-θ) = -cos (θ)

Trigonometric functions of allied angles:
(a) sin \(\left((2 n+1) \frac{\pi}{2} \pm \theta\right)\) = (±) cos θ choose + or – in (±) by using ASTC rule

(b) cos \(\left((2 n+1) \frac{\pi}{2} \pm \theta\right)\) = (±) sin θ choose + or – in (±) by using ASIC rule
Similar technique can be used for other trigonometric functions.

(c) sin (nπ ± θ) = (±) sin θ
cos (nπ ± θ) = (±) cos θ
tan (nπ ± θ) = (±) tan θ
choose + or – in (±) by using ASTC rule.

Sum And Difference Formulae:
(a) sin(A + B) = sin A . cos b + cos A . sin B

(b) sin(A – B) = sin A . cos B – cos A . sin B

(c) cos(A + B) = sin A . cos B – cos A . sin B

(d) cos(A – B) = sin A . cos B + cos A . sin B

(e) tan(A + B) = \(\frac{\tan A+\tan B}{1-\tan A \cdot \tan B}\)

(f) tan(A – B) = \(\frac{\tan A-\tan B}{1+\tan A \cdot \tan B}\)

(g) cot(A + B) = \(\frac{\cot A \cdot \cot B-1}{\cot A+\cot B}\)

(h) cot(A – B) = \(\frac{\cot A \cdot \cot B+1}{\cot A-\cot B}\)

(i) sin(A + B) + sin (A – B) = 2 sin A . cos B

(j) sin (A + B) – sin(A – B) = 2 cos A . sin B

(k) cos(A + B) + cos(A – B) = 2 cos A . cos B

(l) cos(A + B) – cos(A – B) = -2sin A . sin B

(m) sin(A + B) sin(A – B) = sin² A – sin² B = cos² B – cos² A

(n) cos(A + B) cos(A – B) = cos² A – sin² B = cos² B – sin² A

(o) sin 2A = 2 sin A cos A = \(\frac{2 \tan \mathrm{A}}{1+\tan ^2 \mathrm{~A}}\)

(p) cos 2A = cos² A – sin² A
= 2 cos² A – 1
= 1 – 2 sin² A
= \(\frac{1-\tan ^2 \mathrm{~A}}{1+\tan ^2 \mathrm{~A}}\)

(q) tan 2A = \(\frac{2 \tan A}{1-\tan ^2 A}\)

(r) tan(A + B + C) = \(\frac{\tan A+\tan B+\tan C-\tan n A \cdot \tan B \cdot \tan C}{1-\tan A \cdot \tan B-\tan B \cdot \tan C-\tan C \cdot \tan A}\)

(s) sin 3A = 3 sin A – 4 sin³ A
= 4 sin A sin(\(\frac{\pi}{3}\) – A) sin(\(\frac{\pi}{3}\) + A)

(t) cos 3A = 4 cos³ A – 3 cos A
= 4 cos A cos(\(\frac{\pi}{3}\) – A) cos(\(\frac{\pi}{3}\) + A)

(u) tan 3A = \(\frac{3 \tan A-\tan ^3 A}{1-3 \tan ^2 A}\)
= tan A. tan(\(\frac{\pi}{3}\) – A) tan(\(\frac{\pi}{3}\) + A)

Sum or Difference → Product:
(a) sin A + sin B = 2 sin(\(\frac{A+B}{2}\)) cos(\(\frac{A-B}{2}\))
(b) sin A – sin B = 2 cos(\(\frac{A+B}{2}\)) cos(\(\frac{A-B}{2}\))
(c) cos A + cos B = 2 cos(\(\frac{A+B}{2}\)) cos(\(\frac{A-B}{2}\))
(d) cos A – cos B = -2 sin(\(\frac{A+B}{2}\)) sin(\(\frac{A-B}{2}\))

Submultiple Arguments:
(a)

(b) 2 sin² \(\frac{\theta}{2}\) = 1 – cos θ
2 cos² \(\frac{\theta}{2}\) = 1 + cos θ

(c) tan \(\frac{\theta}{2}\) = \(\frac{\sin \theta}{1+\cos \theta}=\frac{1-\cos \theta}{\sin \theta}\)

(d) sin θ = 3 sin \(\frac{\theta}{2}\) – 4 sin³ \(\frac{\theta}{2}\)
cos θ = 4 cos³ \(\frac{\theta}{2}\) – 3 cos \(\frac{\theta}{2}\)

(e) tan θ = \(\frac{3 \tan \frac{\theta}{2}-\tan ^3 \frac{\theta}{2}}{1-3 \tan ^2 \frac{\theta}{2}}\)

Trigonometric Equations:
(a) Equation involving trigonometric equations of unknown angles are called trigonometric function.
(b) Principle solution: The solution ‘x’ of a trigonometric equation is said to be a principle solution if x ∈ (0, 2π)
(c) The solution considered over the entire set R are called the general solution.
(d) General solution of some standard trigonometric equations.

• sin x = 0 ⇒ x = nπ, n ∈ Z
• cos x = 0 ⇒ x = (2n + 1) \(\frac{\pi}{2}\), n ∈ Z
• tan x = 0 ⇒ x = nπ, n ∈ Z
• sin x = sin α ⇒ x = nπ + (-1)ⁿ, n ∈ Z
• cos x = cos α ⇒ x = 2nπ ± α, n ∈ Z
• tan x = tan α ⇒ x = nπ + α, n ∈ Z
• \(\left.\begin{array}{l}
  \sin ^2 x=\sin ^2 \alpha \\
  \cos ^2 x=\cos ^2 \alpha \\
  \tan ^2 x=\tan ^2 \alpha
  \end{array}\right]\) ⇒ x = nπ ± α
• \(\left.\begin{array}{l}
  \cos x=\cos \alpha \\
  \text { and } \sin x=\sin \alpha
  \end{array}\right]\) ⇒ x = nπ ± α, n ∈ Z

Sine Formula:
In any Δ ABC, \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\) or, \(\frac{\sin \mathrm{A}}{a}=\frac{\sin \mathrm{B}}{b}=\frac{\sin \mathrm{C}}{c}\) = 2R
∴ a = 2R sin A, b = 2R sin B and c = 2R sin C
Also, sin A = \(\frac{a}{2R}\), sin B = \(\frac{b}{2R}\) and sin c = \(\frac{c}{2R}\)

Cosine fromulae:
In any Δ ABC,
(i) a² = b² + c² – 2bc cos A
(ii) b² = c² + a² – 2ca cos B
(iii) c² = a² + b² – 2ab cos C
or, → cos A = \(\frac{b^2+c^2-a^2}{2 b c}\)
→ cos B = \(\frac{c^2+a^2-a^2}{2 c a}\)
→ cos C = \(\frac{a^2+b^2-c^2}{2 a b}\)

Projection formulae:
In any Δ ABC,
(i) a = b sin C + c sin B
(ii) b = c cos A + a cos C
(iii) c = a cos B + b cos A

Tangent formulae (Napier’s Analogy);
In any Δ ABC

Area of Triangle (Heron’s formulae):
(i) Area of triangle ABC

(ii) Heron’s formulae:
In any Δ ABC Let 2S = a + b + c
Area of Δ ABC = Δ = \(\sqrt{s(s-a)(s-b)(s-c)}\)
Δ = \(\frac{1}{2}\) bc sin A = \(\frac{1}{2}\) ca sin B
= \(\frac{1}{2}\) ab sin C, Δ = \(\frac{abc}{4R}\)

Semi-Angle Formulae:

Odisha State Board CHSE Odisha Class 11 Math Notes Chapter 1 Mathematical Reasoning will enable students to study smartly.

CHSE Odisha 11th Class Math Notes Chapter 1 Mathematical Reasoning

Proposition: (Mathematically Acceptable)
A proposition (or mathematically acceptable statement) is a declarative sentence that is either true or false but not both.
(1) Thus a sentence will be a statement if

• It is declarative
• It has a truth value (either true (T) or false (F).

(2) A sentence cannot be a statement if it is
(i) A question
(ii) An order
(iii) An exclamation
(iv) A wish
(v) Advice or it involves

• variable time like ‘today’, ‘tomorrow’, ‘yesterday’ etc.
• Variable place like ‘here’, ‘there’, etc.
• pronouns like ‘he’, ‘she’, ‘they’ etc.
• Relative words/adjectives / undefined words like ‘good’, ‘bad’, ‘beautiful’, ‘wise’ etc
• Variable x, y, z, u, v….etc

(3) We denote statements by same letters are p, q, r, s, etc.

Negative (~): Denial of a statement is its negation.
Axiom of negation:
For any proposition p, if p is true, then ~p (Negation of p) is false and if p is false, then ~p is true,
Truth table of Negation:

p	~p
T	F
F	T

Logical Connectives:

• Two statements can be combined together by using the words like or, and, if, only if, if and only if etc. These are known as logical connectives.
• A proposition in which one or more connectives appear is known as a composite or compound proposition.

Conjunction (∧), (and):
Axiom: A conjunction p ∧ q is true if both ‘p’ and ‘q’ are both true and false otherwise.
Truth table:

p	q	p ∧ q
T	T	T
T	F	F
F	T	F
F	F	F

Disjunction (∨) (or):
Axiom: A disjunction p ∨ q is false only when both ‘p’ and ‘q’ are false and is true otherwise.
Truth table:

p	q	p ∨ q
T	T	T
T	F	T
F	T	T
F	F	F

• Inclusive and exclusive sense of ‘OR’

→ An employee either goes on leave or attends to his duty. (Exclusive)
→ In this restaurant you can order veg or non-veg items. (Inclusive)

Conditional (→)(if … then):
Axiom: A conditional p → q is false only when ‘p’ is true and ‘q’ is false in all other cases it is true.
Truth table:

p	q	p → q
T	T	T
T	F	F
F	T	T
F	F	T

Converse, Inverse and Contrapositive:

• Converse of p → q is q → p
• Inverse of p → q is ~p → ~q
• Contra positive of p → q is ~q → ~ p

Biconditional (p ↔ q)(p if and only if q):
Axiom: A biconditional p ↔ q is true if both ‘p’ and ‘q’ have same truth value and is false otherwise.
Truth table:

p	q	p ↔ q
T	T	T
T	F	F
F	T	F
F	F	T

Equivalent statements:
Two statements ‘p’ and ‘q’ are said to be equivalent statements if they have same truth values.

Tautology:
A statement is a tautology if it is always true:

Implication and double implication:

• If a conditional p → q is a tautology then we say p implies q and we write P ⇒ q
• If a biconditional p ↔ q is a tautology and we write p ⇒ q.

Contradiction:
A contradiction we mean a proposition that is false for all possible assignments of truth values to its prime components.

Logical Quantifiers:
Logical quantifiers are the words that associate a quantity to it. There are two types of logical quantifiers.
(i) Existential (There exists)
(ii) Universal (For all, for every).

Validity Of Statements
A statement is said to be valid if it is true.
Techniques to check the validity of a statement:

Validity Of Statements With ‘And’
To prove p ∧ q is true we follow the following steps :
Step – 1: Show that ‘p’ is true.
Step – 2: Show that ‘q’ is true.

Validity Of Statements With ‘ OR’
To prove p ∧ q is true we have to consider the following cases :
Case – 1: By assuming p is false, prove that q is true.
Case – 2: By assuming q is false, prove that p is true.

Validity Of Statements With ‘if … then’
To prove if ‘p’ then ‘q’ is true we can adopt any one of the following methods.

• Method – 1 (Direct Method):
  Assume ‘p’ is true and prove that ‘q’ is true (i.e. p ⇒ q)
• Method – 2 (Contrapositive Method):
  Assume ‘q’ is false and prove that ‘p’ is false. (i.e. ~ q ⇒ ~ p)
• Method – 3 (Contradiction Method):
  → Assume that p → q is false, i.e. p is true and q is false
  → Obtain an absurd result
  → This is due to our false assumption.
  → Thus by the method of contradiction p → q is true. i.e., the statement is valid.
• Method – 4 (By giving a counter-example):
  To prove a statement is false we give a single example where it is false.

Validity Of Statement With ‘if and only if’.
To prove ‘p’ if and only if ‘q’ is true we have to follow the following steps.
Step – 1: Take ‘p’ is true and prove that ‘q’ is true.
Step – 2: Take q is true and prove that ‘p’ is true.

Odisha State Board CHSE Odisha Class 11 Math Notes Chapter 3 Relations And Function will enable students to study smartly.

CHSE Odisha 11th Class Math Notes Chapter 3 Relations And Function

Order Pairs
An ordered pair consists of a pair of objects, or elements or numbers or functions in order.
We denote order pairs as (a, b)

• An order pair is not a set of two objects.
• (a, b) = (c, d) ⇒ a = c and b = d
• (a, b) × (b, a)

Cartesian Product Of Sets:
If A and B are non-empty sets, then their Cartesian product, denoted by A × B and defined by A × B = {(a, b): a ∈ A, b ∈ B} = Set of all ordered pairs (a,b) where a ∈ A and b ∈ B
Note:
1. For finite sets A and B |A × B| = |A| . |B|
2. A × B = Φ ⇔ A = Φ or B = Φ
3. A² = A × A

Properties of Cartesian product:
1. A × B ≠ B × A (Cartesian product is non-commutative)
2. A × (B ∪ C) = (A × B) ∪ (A × C)
3. A × (B ∩ C) = (A × B) ∩ (A × C)
4. A × B = B × A ⇔ A = B
5. A × (B – C) = ( A × B) – (A × C)
6. A ⊂ B ⇒ A × A ⊂ (A × B) ∩ (B × A)
7. A ⊂ B ⇒ A × C ⊂ B × C
8. A ⊂ Band C ⊂ D ⇒ A × C ⊂ B × D
9. (A × B) ∩ (C × D) = ( A ∩ C) × (B ∩ D)

Relation
Let A and B be two arbitrary sets. A binary relation from A to B is a subset of A × B.
OR f is a relation from A to B if f ⊆ A × B
Note:

• If a of A is related to b of B by relation ‘f’ then we write (a,b) ∈ f or a f b
• As Φ ⊂ A × B we have Φ is a relation from A to B. This relation is known as a null of void or empty relation.
• As A × B ⊆ A × B, A × B is also a relation from A to B. This relation is known as universal relation.
• If |A| = m and |B| = n then number of relations from A to B is 2^mn

Domain, co-domain, and Range of a relation:
Let f is a relation from A to B. Domain of f = Dom (f) or D_f
={x ∈ A : (x, y) ∈ f for some y ∈ B) Co-domain of f = B
Range of ‘f’ = Rng (f) or R_f = {y ∈ B : (x, y) ∈ f for some x ∈ A}

Types Of Relation:
(a) One-many relation: A relation f from A to B is one many if (a, b) and (a, b’) ∈ f ⇒ b ≠ b’
(b) Many-one relations: A relation f from A to B is many-one if (a, b) and (a’, b) ∈ f ⇒ a ≠ a’
(c) One-one relation: A relation f from A to B is one-one if (a, b), (a, b’) ∈ f ⇒ b = b’ and (a, b), (a’, b) ∈ f ⇒ a = a’

Inverse of a relation: Let f is a relation from A to B. The inverse of f is denoted by f^-1 is a relation from B to A defined as f^-1 = {(b, a): (a, b) ∈ f}

Function:
A relation ‘f’ from X to Y is called a function if:
(a) D_f = Dom (f) = X and
(b) (x, y) and (x, z) ∈ f ⇒ y = z or A relation from A to B is a function
if ⇒ Domain of f = X i.e All elements of X is engaged in the relation and
⇒ f is not one many.

Note:
(1) If a relation f from X to Y becomes a function then we write f: X → Y.
(2) If f is a function from A to B i.e. f: X → Y and (x, y) ∈ f then we write y = f(x)
(3) Mapping, map, transformation, transform, operator, and correspondence are different synonym terms of function.
(4) If f: X → Y is defined as y = f(x), then

• y is called the value of the function at x or the image of x under f or the dependent variable.
• x is called the independent variable or pre-image of y under f.

Domain, Co-domain or Range of a function:
Let f: X → Y defined as y = f(x)
(a) Domain of ‘f’ = Dom f = D_f = {x ∈ X: y = f(x)}
(b) Range of f = Rng f = R_f = f(A) = {f(x) ∈ Y: x ∈ A } Clearly f(A) ⊆ y
(c) If |A| = m, |B| = n then number of functions from A to B = n^m

Real valued function :
A function f: A → B is a real-valued function if B ⊆ R.
→ f is a real function if A ⊆ R and B ⊆ R

Techniques to find Domain and Range of a Real function:
(a) Techniques to find Domain: Let the function is defined as y = f(x).
Step -1: Check the values of x for which f(x) is well defined.
Step -2: The set of all values obtained from step -1 is the domain of ‘f.

(b) Techniques to find range: Let the function is y = f(x)

• Method-1 (By inspection):
  → Step -1: Get values of y for all values x ∈ Dom f.
  → Step -2: Set of all these values of y = Rng f
• Method-2:
  → Step -1: Write x in terms of y
  → Step -2: Get values of y for which x is well defined in Dom f.
  → Step -3: Rng (f) = The set of all y obtained from step 2.

Some Real Functions:
(a) Constant function: A function f: A → R defined as f(x) = k, for some k ∈ R is called a constant function.

(b) Identity function: Let A ⊆ R. The function f: A → A defined as f(x) = x, x ∈ A is called the identity function on A. We denote it by I_A

(c) Polynomial function: A function f: A → R defined by f(x) = f(x) = a₀ + a₁x + a₂x² + a_nxⁿ where a₀, a₁, a₂, ….., a_n are real numbers and a_n ≠ 0 is called a polynomial function (polynomial) of degree n.

(d) Rational function: A function of form f(x) = \(\frac{\mathrm{P}(x)}{\mathrm{Q}(x)}\) where P(x) and Q(x) are polynomial functions of x is known as a rational function.

(e) Absolute value function OR modulus function: The function f: R → R defined as  f(x) = |x| = \(\begin{cases}x, & x \geq 0 \\ -x, & x<0\end{cases}\) is called as the modulus function.
→ Rng f = [ 0, ∞] = R⁺U {0}

Properties Of Modulus Function:
1. For any real number x, we have \(\sqrt{x^2}=|x|\)
2. If a and b are positive real numbers then

• x² ≤ a² ⇔ |x| ≤ a
• x² ≥ a² ⇔ |x| ≥ a
• a² ≤ x² ≤ b² ⇔ a ≤ |x| ≤ b ⇔ x ∈ [-b, – a] ∪ [a, b]

(f) Signum function: The function f: R → R defined as f(x) = \(\begin{cases}\frac{x}{|x|}, & x \neq 0 \\ 0, & x=0\end{cases}\) is called signum function.
→ We denote a signum function as f(x) = sgn(x)
→ Range of a signum function = {-1, 0, 1}

(g) Greatest integer function: The function f: R → R defined by f(x) = [x] is called the greatest integer function. [x] = The greatest among all integers ≤ x. OR [x] = n for n ≤ x < n + 1

Properties of the greatest integer function :
Let n is an integer and x is a real number between n and n + 1
(i) [-n]= -[n]
(ii) [x + k] = [x] + k (for an integer ‘k’)
(iii) [-x] = -[x] – 1
(iv) [x] + [-x] = \(\begin{cases}-1, & \mathrm{x} \notin \mathrm{Z} \\ 0, & \mathrm{x} \in \mathrm{Z}\end{cases}\)
(v) [x] – [-x] = \(\begin{cases}2[\mathrm{x}]+1, & \mathrm{x} \notin \mathrm{Z} \\ 2[\mathrm{x}], & \mathrm{x} \in \mathrm{Z}\end{cases}\)
(vi) [x] ≥ k ⇒ x ≥ k for k ∈ Z
(vii) [x] ≤ k ⇒ x < k +1 for k ∈ Z
(viii) [x] > k ⇒ x > k + 1 for k ∈ Z
(ix) [x] < k ⇒ x < k for k ∈ Z

(h) Exponential Function: A function f: R → R defined as f(x) = a^x where a > 0 and a ≠ 1 is called the exponential function.

Properties Of Exponential Function:
1. a^x+y =  a^x . a^y
2. (a^x)^y = a^xy
3. a^x = 1 if x = 0
4. If a > 1, a^x > a^y ⇒ x > y
5. If a < 1, a^x > a^y ⇒ x < y

Logarithmic Function:
Let a ≠ 1 is a positive real number. The function f: (0, ∞) → R defined by f(x) = log_a^x is called the logarithmic function, where y = log_a^x  ⇔ a^y = x
→ Domain of a logarithmic function = (0, ∞) and Range = R

Properties of logarithmic function:
1. log_a (xy) = log_ax + log_ay
2. log_a (x/y) = log_ax – log_ay
3. log_aa = 1
4. log_a(x)^y = y log_ax
5. log_a x = 0 ⇔ x = 1
6. log_ax = \(\frac{1}{\log _a{ }_a}\) , x ≠ 1
7. log_ab = \(\frac{\log _a b}{\log _c a}\)
8. \(\log _{a^n}\left(x^m\right)\) = \(\frac{m}{n}\) log_a|x|

Different Categories of function:
(a) Algebraic Function: A function that can be generated by a variable by a finite number of algebraic operations such as addition, subtraction, multiplication, division, square root, etc. is called an algebraic function.

(b) Transcendental function: A non-algebraic function is a transcendental function.
⇒ Trigonometric, trigonometric, Exponential, and logarithmic functions are transcendental functions.

Even And Odd Functions:
A function ‘f’ is an even function  if f(-x) = x and is an odd function
if f(-x) = x and is an odd function: if f(-x) = -f(x)
Note:
1. If ‘f’ is any function f(x) + f(-x) is always an even function and f(x) – f(-x) is an odd function.
2. Every function f(x) can be expressed as the sum of an even and an odd function as f(x) = g(x) + h(x), where
g(x) = \(\frac{f(x)+f(-x)}{2}\)
h(x) = \(\frac{f(x)-f(-x)}{2}\)

Periodic Function:
A function is called a periodic function with period k if f(x + k) = f(x) for some constant k ≠ 0. The least positive value of k for which f(x + k) = f(x) holds is called the fundamental period of f.

Properties of periodic function :
(1) If k is the period of f then any non-zero integral multiples of k is also a period of f.
(2) If k is the period of f(x) then f(ax + b) is also periodic with period \(\frac{k}{a}\)
(3) If f₁(x) + f₂(x) and f₃(x) are periodic functions with periods k₁, k₂, k₃, respectively then the function a₁f₁(x) + a₂f₂(x) + a₃f₃(x) is also a periodic function with period, LCM (k₁, k₂, k₃)

Algebra Of Real functions:
(a) Equality of two functions: Two functions f and g are equal iff ‘
(i) Dom f = Dom g
(ii) Co-Dom f = Co-Dom g
(iii) f(x) = g(x) for all x belonging to their common domain.

(b) Addition of two functions: Let f: D₁ → R and g: D₂ → R be two real functions.
The sum function f + g is defined by f + g: D₁ ∩ D₂ → R and (f + g)(x) = f(x) + g(x) ∀ x ∈ D₁ n D₂

(c) Subtraction of two functions: Let f: D₁ → R and g: D₂ → R. The difference function (f – g) is f – g: D₁ ∩ D₂) → R defined by (f – g) (x) = f(x) – g(x) ∀ x ∈ D₁ ∩ D₂

(d) Scalar multiplication: Let f: D → R and c is any scalar. The scalar multiple of f by the scalar c is cf: D → R defined as (cf)(x) = c. f(x) ∀ x ∈ D₁.

(e) Multiplication of two functions: Let f: D₁ → R and g: D₂ → R are two real functions. The product function (fg) is (fg): D₁ ∩ D₂ → R defined as (fg)(x) = f(x)g(x) ∀ ∈ D₁ ∩ D₂

(f) The quotient of two functions: Let f: D₁ → R and g: D₂ → R are two real functions. the quotient function (\(\frac{f}{g}\)) i,e,. \(\frac{f}{g}\): D₁ ∩ D₂ → R, defined by (\(\frac{f}{g}\))(x) = \(\frac{f(x)}{g(x)}\), ∀ x ∈ D₁ ∩ D₂

Odisha State Board CHSE Odisha Class 11 Math Notes Chapter 2 Sets will enable students to study smartly.

CHSE Odisha 11th Class Math Notes Chapter 2 Sets

Set:
Set is an undefined term in mathematics. But we understand set as “a collection of well-defined objects”.

• Set is a collection.
• The objects (called elements) in a set must be well-defined.

Set Notation:
We denote set as capital alphabets like A, B, C, D…..and the elements by the small alphabets like x, y, z ….

• If x is an element of set A we say “x belongs to A” and write ‘x ∈ A’.
• If x is not an element of set A we say “x does not belong to A” and we write ‘x ∉ A’.

Set Representation:
(a) Extension or tabular or Roster Method: In this method, we describe a set listing the elements, separated by commas within curly brackets.
Note: While listing out the elements the repetition of an object has no effect. Thus, we don’t do this.

(b) Intention or set builder or set selector method: In this method: a set is described by a characterizing property p(x) of element x. In this case, the set is described as {x : p(x) holds}

Types Of Set:
(a) Empty of full or void set: It is a set with no element.

• We denote empty set by ‘Φ’
• There is only one empty set.

(b) Singleton set: It is a set with only one element.

(c) Finite set: A set is finite if it has a finite number of elements.

(d) Infinite Set: A set that is not finite is called an infinite set.

(e) Equal sets: Two sets A and B are equal if they have the same elements. Two sets A and B are equal if all elements of A are also elements of B and all elements of B are also elements of A.

(f) Equivalent set: Two finite sets A and B are equivalent if they have the same number of elements.

Subsets: Let A and B be two sets. If every element of A is an element of B then A is called a. subset of B (we write A ⊂ B) and B is called a superset of A (We write B ⊃ A)
Thus A ⊂ B is x ∈ A ⇒ x ∈ B
Note.
(i) A set is a subset of itself.
(ii) Empty set Φ is a subset of every set.
(iii) A is called a proper subset of B if B contains at least one element that is not in A
(iv) If A has n elements then total number of subsets of A = 2ⁿ.

Universal set:
A set ‘U’ that contains all sets in a given context is called the universal set.

Power set:
Let A is any set. The collection (or set) of all subsets of A is called the power set of A. We denote it as P(A)
P(A) = { S: S ⊂ A }

Set Operations:
(a) Union of sets :
The union of two sets A and B is the set of all elements of A or B or both.
∴ A ∪ B = {x ∈ A or x ∈ B}

(b) Intersection of sets:
Intersection of two sets A and B is the set of all those elements that belong to both A and B . (or all common elements of A and B)
∴ A ∩ B = {x: x ∈ A and x ∈ B }
Two sets A and B are disjoint if A ∩ B = Φ. Otherwise, A and B are intersecting or overlapping sets.

(c) Difference of sets: The difference of two sets ‘A and B’ is the set of all elements of A which do not belong to B.
∴ A- B = {x: x ∈ A and x ∈ B)

(d) Symmetric difference of two sets: Symmetric difference of two sets A and B is the set (A – B) ∪ (B – A)
∴ A Δ B = (A – B) ∪ (B – A) = (A ∪ B) – (A ∩ B)

(e) Complement of a set: Let the complement of a set A (denoted as A’ or A^c) be defined as U – A

• A’ = {x ∈ U) : x ∉ A)
• x ∈ A’ ⇔ x ∉ A

Laws Of Set Algebra:
(a) Idempotent law: For any set A we have
(i) A ∪ A = A
(ii) A ∩ A = A

(b) Identity laws: For any set A we have
(i) A ∪ Φ = A and
(ii) A ∩ U = A

(c) Commutative laws: For any three sets A, B, and C
(i) A ∪ B = B ∪ A
(ii) A ∩ B = B ∩ A

(d) Associative laws: For any three sets A, B, and C
(i) A ∪ (B ∪ C) = (A ∪ B) ∪ C
(ii) A ∩ (B ∩ C) = (A ∩ B) ∩ C

(e) Distributive laws: For any three sets A, B, and C
(i) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
(ii) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

(f) De-morgans laws: For any two sets A and B
(i) (A ∪ B)’ = A’ ∩ B’
(ii) (A ∩ B)’ = A’ ∪ B’

Some more properties of sets: For any three sets A, B, and C
(a) A ⊂ (A ∪ B) and (A ∩ B) ⊂ A
(b) A ∪ B = B ⇔ A ⊂ B
(c) A ∩ B = A ⇔ A ⊂ B
(d) B ⊂ A and C ⊂ A ⇒ (B ∪ C ) ⊂ A and A ⊂ B, A ⊂ C ⇒ A ⊂ (B ∩ C)
(e) B ⊂ C ⇒ A ∪ B ⊂ A ∪ C and A ∩ B ⊂ A ∩ C
(f) A – B = A ∩ B’
(g) A – B = A ⇔ A ∩ B = Φ
(h) (A – B) ∪ B = A ∪ B and (A – B ) ∩ B = Φ
(i) A ⊆ B ⇔  B’ ⊆ A’
(j) A Δ B = B Δ A

Cardinality or order of a finite set: The cardinality or order of a finite set A (denoted as |A| or O(A) or n (A)) is the number of elements in ‘A’.

Some important results on the cardinality of finite sets and applications of set theory:
If A, B, and C are finite sets and ‘U’ is the finite universal set then a number of elements belonging to at least one of A or B.
(a) |A ∪ B| = |A| + |B| – |A ∩ B|
(b) |A ∪ B| = |A| + |B| for A ∩ B = Φ i.e. for two disjoint sets A and B
(c) Number of elements belonging to at least one of A, B, or C
= |A ∪ B ∪ C|
= |A| + |B| + |C| – |A ∩ B| – |B ∩ C| – |C ∩ A| + |A ∩ B ∩ C|
(d) Number of elements belonging to exactly two of the three sets A, B, and C = |A ∩ B| + |B ∩ C| + |C ∩ A| – 3 |A ∩ B ∩ C|
(e) Number of elements belonging to exactly one of the three sets A, B, and C = |A| + |B| + |C| – 2 |A ∩ B| -2 |B ∩ C| – 2 |C ∩ A| + 3 |A ∩ B ∩ C|
(f) Number of elements belonging to A but not B = |A – B| = |A| – |A ∩ B|
∴ |A| = |A – B| + |A ∩ B|
(g) Number of elements belonging to exactly one of A or B
= |A Δ B| = |A – B| + |B – A|
= |A| + |B| – 2 |A ∩ B|
(h) |A’ ∪ B’| = |U| – |A ∩ B|
(i) |A’ ∩ B’| = |U| – |A ∪ B| = Number of elements belonging to neither A nor B.

Odisha State Board CHSE Odisha Class 11 Invitation to English 4 Solutions Grammar Prepositions Textbook Activity Questions and Answers.

CHSE Odisha 11th Class English Grammar Prepositions

Look at the following examples :
1. He went to Puri.
2. He went Puri.
In the sentences written above, Sentence 1 is grammatical and acceptable but Sentence 2 is not grammatical and acceptable. Sentence 2 i.e., He went Puri, is a non-standard and unusual sentence. It is not written or spoken by educated native speakers. The small word to is left out in Sentence 2. The absence of the small word ‘to’ gives no meaning to the construction. So the sentence is not acceptable and recommendable.

In Sentence – 1, the small word ‘to’ is more attracted/associated towards the Noun Phrase‘ Puri’ than the verb went. So we call the small word a preposition.  Prepositions are words or word groups that are usually used before a noun phrase. Their normal position in a sentence is in front of a noun phrase with its determiner if any. They (prepositions) sometimes are used in other positions than the one in front of a ‘noun phrase’. We can see different types of prepositions like time, place, movement, etc.

Activity – 1

Use the prepositions on, in, at with the time expressions given below.

(a) February                               (b) midnight               (c) eight o’clock (d) the afternoon                       (e) night                      (f) Christmas day (g) the eighteenth century         (h) lunchtime             (i) 1999 (j) Wednesday                            (k) the morning         (l) March 15 (m) Saturday night                      (n) Easter

In the box below, write ‘on’ ‘in’ or ‘at’, whichever is appropriate.

a

b

c

d

e

f

g

h

i

j

k

l

m

n

Answer:

a

b

c

d

e

f

g

h

i

j

k

l

m

n

in

at

at

in

at

on

in

at

in

on

in

on

at

on

Activity – 2

Fill in the blank spaces with on, in, or at :
1. ___________ the daytime the streets are crowded but ___________night they are quite deserted.
2. He’s always in a bad temper ___________ breakfast time.
3. We couldn’t offer him a room in our flat, because ___________ that time our guest was staying with us.
4. Children get presents ___________ Christmas and ___________ their birthdays.
5. We want three seats for ‘Boothnaath’ ___________ Friday night.
6. I will reach Bhubaneswar ___________ 3 o’clock ___________ the morning but the offices start working only ___________ 10 a.m.
7. Millions of Indian soldiers were killed ___________ the Second World War.
8. It was ___________ the spring of 1985 that my brother and sister-in-law met for the first time. However, they got married only ___________ 1995.
9. ___________ 1950 the cost of living was only a fraction of what it was ___________ 1990.
10. They get up ___________ dawn and reached the summit noon.
11. We didn’t want anything to go wrong ___________ our sister’s wedding day.
12. I’ll meet you ___________ 2.30 ___________ Friday, August 2001.
13. I’ll see you ___________ six days’ time.
14. Birds don’t find much to eat ___________ winter.
15. What will you be doing ___________ the holidays?

Answer:
1. In the daytime the streets are crowded but at night they are quite deserted.
2. He’s always in a bad temper in breakfast time.
3. We couldn’t offer him a room in our flat, because at that time our guest was staying with us.
4. Children get presents at Christmas and on their birthdays.
5. We want three seats for ‘Bhootnaath’ on Friday night.
6. I will reach Bhubaneswar at 3 o’clock in the morning, but the offices start at working only at 10 a.m.
7. Millions of Indian soldiers were killed in the Second World War.
8. It was in the spring of 1985 that my brother and sister-in-law met for the first time. However, they got married only in 1995.
9. In 1950 the cost of living was only a fraction of what it was in 1990.
10. They got up at dawn and reached the summit at noon.
11. We didn’t want anything to go wrong on our sister’s wedding day.
12. I’ll meet you at 2.30 on Friday, August 2001.
13. I”ll see you in six day’s time.
14. Birds don’t find much to eat in winter.
15. What will you be doing on the holidays?

Activity – 3

Insert suitable prepositions in the blank spaces from the following list, (during, since, for, from ___________ to ___________.)
1. They’ve lived in this town _____________ five years.
2. The farmers have been working in the field _____________10 o’clock.
3. The employees stop working _____________ the night.
4. The players practised in the field _____________ 4 p.m. _____________6 p.m.
5. He looked as if he hadn’t slept _____________weeks.
6. The carpenters worked _____________9 a.m. _____________5 p.m. with an hour for lunch.
Can you explain, in your own words, what the rules are for the use of these prepositions?

Answer:
1. They’ve lived in this town for five years.
2. The farmers have been working in the field since 10 o’clock.
3. The employees stop working during the night.
4. The players practiced in the field from 4 p.m. to 6 p.m.
5. He looked as if he hadn’t slept for weeks.
6. The carpenters worked from 9 a.m. to 5 p.m. with an hour for lunch.

The following are the rules for the use of the prepositions such as ‘from, to, for, since, during’.
For is used to talk about a period of time continuing up to the present. It is used to say how long something has lasted. It can be used to talk about the past, present, or future.
‘Since’ is used to say when the action/event started (the starting point of the actions and situations). Both ‘since’ and ‘for’ are usually used in the present perfect tense.
‘The use of ‘from’ in a sentence says when the action/situation starts and it finishes.
The idea is normally expressed with from….to or from…. till/until…………
‘During’ is used to say when something happened but ‘for’ is used to say how long it took

Activity – 4

Fill in the blanks using appropriate prepositions.
1. All the students are busy _____________ the moment.
2. Sunita and Mahima finished their homework _____________ the same time.
3. They are getting married _____________six months time.
4. Hemanta is in class 8. He’ll be leaving school _____________ two years time.
5. It was a very interesting book. I read it _____________ a day.
6. Hurry up! We’ve got to go _____________ten minutes.
7. My sister is a doctor but she is out of work _____________ present.
8. He’s busy just now but he’ll be with you a _____________ moment.
9. A: I” ll meet you at 10.30.
B: OK, but please be _____________ time.
10. A child ran across in front of the car, but the driver managed to stop just _____________ time.
11. At first I didn’t like getting up early, but _____________ the end I got used to it.
12. Workers normally get paid _____________the end of the month.

Answer:
1. All the students are busy at the moment.
2. Sunita and Mahima finished their homework at the same time.
3. They are getting married in six months time.
4. Hemanta is in class 8. He’ll be leaving school in two years time.
5. It was a very interesting book. I read it in a day.
6. Hurry up! We’ve got to go in ten minutes.
7. My sister is a doctor but she is out of work at present.
8. He’s busy just now but he’ll be with you in a moment.
9. A: I” ll meet you at 10.30.
B: OK, but please be in time.
10. A child ran across in front of the car, but the driver managed to stop just on time.
11. At first I didn’t like getting up early, but in the end, I got used to it.
12. Workers normally get paid at the end of the month.

Activity – 5

Match the expressions in A with their meanings in B.

Answer:
on time — punctual, not late
in time — soon enough
at the end — at the time when something ends
in the end — finally

SECTION – 2

(Place)
Activity – 6

Complete the sentences using the prepositions at, in, or on.
1. There’s some sugar ___________ the shelf ___________the cupboard.
2. Is Seema ___________the kitchen ?
3. Sambalpur is ___________ the west of Orissa, ___________the River Mahanadi.
4. There’s a grocery shop ___________the comer ___________the end of the street.
5. There are three books ___________ the table.
6. He is ___________ the roof.
7. The old man is standing ___________ the gate.
8. He put the money ___________ his pocket.
9. They’ll meet the players ___________ the dining table.
10. I waited ___________ the bus stop for two hours.

Answer:
1. There’s some sugar on the shelf in the cupboard.
2. Is Seema in the kitchen?
3. Sambalpur is in the west of Orissa, on the River Mahanadi.
4. There’s a grocery shop in the comer at the end of the street.
5. There are three books on the table.
6. He is on the roof.
7. The old man is standing at the gate.
8. He put the money in his pocket.
9. They’ll meet the players at the dining table.
10. I waited at the bus stop for two hours.

Now complete the following activity by matching the prepositions with their respective meanings.

Answer:

Activity – 7

Complete the sentences using at, in or on.
1. Our teacher’s flat is ___________the second floor.
2. The boy was holding a ball ___________ his hand.
3. The children spent an afternoon ___________ the zoo.
4. He traveled from Calcutta to Delhi. He is ___________ Delhi now.
5. They have gone to a wedding. They are probably ___________ the wedding.
6. He was taken to hospital. He is ___________ hospital now.
7. The baby was playing ___________ the floor.
8. There’s ink ___________ your shirt.
9. Hari met his friend ___________ the crossroad.
10. Gauhati is ___________ Assam.

Answer:
1. Our teacher’s flat is on the second floor.
2. The boy was holding a ball in his hand.
3. The children spent an afternoon in the zoo.
4. He traveled from Calcutta to Delhi. He is in Delhi now.
5. They have gone to a wedding. They are probably in the wedding.
6. He was taken to the hospital. He is in hospital now.
7. The baby was playing on the floor.
8. There’s ink in your shirt.
9. Hari met his friend at the crossroad.
10. Gauhati is in Assam.

Activity – 8

Complete the sentences using above, across, along, below, down, over, past, through, under, and up.
1. Jitu ran ___________ the road to meet his friend.
2. They took a shorter route ___________ the forest to save time.
3. He walked ___________ the road slowly, examining the shop windows.
4. They saw the car going ___________the house and shouted to the driver to turn back.
5. The woman climbed ___________ the stairs to the first floor.
6. We saw the new bridge which has been built ___________the river.
7. Madhu lives on the third floor. He came ___________ the stairs to greet his friends.
8. The temperature is ___________ zero in the poles.
9. The plane went up quickly. Soon it was ___________ the clouds.
10. We live on the earth, ___________ the sky.

Answer:
1. Jitu ran across the road to meet his friend.
2. They took a shortcut through the forest to save time.
3. He walked alone the road slowly, examining the shop windows.
4. They saw the car going past the house and shouted to the driver to turn back.
5. The woman climbed up the stairs to the first floor.
6. We saw the new bridge which has been built over the river.
7. Madhu lives on the third floor. He came down the stairs to greet his friends.
8. The temperature is below zero in the poles.
9. The plane went up quickly. Soon it was above the clouds.
10. We live on the earth, under the sky.

Activity – 9

Complete the sentences using the prepositions at, in, or on. (More than one answer is possible)
1. There’s chemist’s ___________ the comer ___________ the end of the street.
2. They had breakfast ___________ Khan’s Cafe ___________ the main road their way home.
3. Calcutta is ___________ the eastern part of India ___________ the River Ganga.
4. They were waiting ___________ the station.
5. He’d spend the whole day sitting ___________ a desk.
6. If you walk further, you’ll see a small shop ___________ the comer.
7. Is Mother ___________ the kitchen? No, she’s ___________ the back of the house.

Answer:
1. There’s chemists in the comer at the end of the street.
2. They had breakfast at Khan’s Cafe on the main road on their way home.
3. Calcutta is in the eastern part of India on the River Ganga.
4. They were waiting at the station.
5. He’d spend the whole day sitting at a desk.
6. If you walk further, you’ll see a small shop at the comer.
7. Is Mother in the kitchen? No, she’s at the back of the house.

SECTION – 3

Study the use of in, on, and at in the following sentences.
1. (a) They are not in town. They are on holiday in Darjiling.
(b) They heard the news on the radio.
(c) He has put on weight. He’ll have to go on a diet.
Some useful expressions with on are the following.

on business, on a tour, on a cruise, on television, on the phone, on strike, on fire, on the whole, on purpose.
2. (a) The old woman did not like the sun. She preferred to sit in the shade, (b) My friend always writes in pencil.
Some more expressions within are given below.
in the rain, in the sun, in the dark, in bad weather, in ink, in words, in figures, in block letters, in cash

3. He left school at the age of 14.
Some more expressions with at are given below.
at a speed of, at a temperature of

Activity – 10

Complete the sentence using on, in, or at.
1. When you write a cheque, you should write the amount ____ words as well as figures.
2. Water boils ____ 100 degrees Celsius.
3. He avoided meeting him ____ purpose.
4. My brother is going ____ a tour tomorrow.
5. The old man likes to keep warm, so he does not go out ____ cold weather.
6. Look! The train is ____ fire!
7. The workers are ____ strike.
8. He left home ____ the age of 10.
9. We pay for things ____ cash.
10. I watch the morning news ____ television.

Answer:
1. When you write a cheque, you should write the amount in words as well as in figures.
2. Water boils at 100 degree Celsius.
3. He avoided meeting him on purpose.
4. My brother is going on a tour tomorrow.
5. The old man likes to keep warm, so he does not go out in cold weather.
6. Look! The train is on fire!
7. The workers are on strike.
8. He left home at the age of 10.
9. We pay for things in cash.
10. I watch the morning news on television.

Activity – 11

Fill in the blanks using the prepositions in, on, or at.
We live ___________ a house ___________ Janpath. We live ___________ 85, Bapuji Nagar. Our house is ___________ a convenient location, just ___________ the main road. It is ___________ the intersection of Cuttack Road and Lewis Road.
Answer:
We live in a house on Janpath. We live on 85 Bapuji Nagar. Our house is in a convenient location, just on the main road. It is in the intersection of Cuttack Road and Lewis Road.

SECTION – 4

Study the following sentences.
(a) He called the stranger by mistake.
(b) My father always makes payments by cheque.
(c) My sister goes to college by bus.
(d) My friend did not use his car. He came in a taxi.

In (a) and (b) by is used with mistake and cheque. The following are some of the expressions that usually go with by.
by chance, design, accident
by letter, post, hand, cable, telegram
by heart
by day, night
In (c), by is used with bus. By is generally used to refer to some means of transport. Here are some more examples.
by land, by air, by road, by sea, by coach, by train, etc.
But by is replaced by in, on, etc. when a noun such as a ship, car, bus, etc. is used along with a determiner such as my, a, the. For example: in my car, on my bicycle, etc.

Activity – 12

Complete the sentences using by, on, or in.
1. A bat sleeps __________ day and flies __________ night.
2. The tourists have decided to travel to Australia __________ sea rather than __________ air.
3. Usha usually goes to college __________ her scooter, but sometimes she goes __________ bus.
4. It was only __________ chance that he passed the examination. He hadn’t worked at all.
5. The officer did not come to work __________ his car on Monday. His car had broken down and he had to come __________ taxi.
6. We decided not to go __________ bus. We went __________ my bike instead.
7. The journey takes 20 minutes __________bus and about 50 minutes __________foot.
8. They went for a ride __________ a motorbike.
9. I traveled to Delhi __________ train.
10. How long does it take to get to Sri Lanka __________ boat?

Answer:
1. A bat sleeps by day and flies by night.
2. The tourists have decided to travel to Australia by sea rather than by air.
3. Usha usually goes to college in her scooter, but sometimes she goes by bus.
4. It was only by chance that he passed the examination. He hadn’t worked at all.
5. The officer did not come to work in his car on Monday. His car had broken down and he had to come by taxi.
6. We decided not to go by bus. We went in my bike instead.
7. The journey takes 20 minutes by bus and about 50 minutes on foot.
8. They went for a ride on/in a motorbike.
9. I traveled to Delhi by train.
10. How long does it take to get to Sri Lanka by boat?

(Adjective + Preposition) Combinations
Certain adjectives are always used in combination with particular prepositions. Here are some common examples of these adjective + preposition combinations.

of
1. Children are afraid of snakes.
2. We are proud of being Indians.
3. Cats are fond of milk.
Some more examples :

frightened of           jealous of             conscious of         tolerant of scared of                 envious of             capable of            independent of full of                       suspicious of        short of                typical of ashamed of              aware of               critical of              aware of tolerant of

about/with
1. We are excited about our sister’s wedding tomorrow.
2. My younger sister was delighted with the present I gave her.
Some more examples:
worried/upset/nervous/happy/annoyed/furious etc. + about.
pleased/satisfied/disappointed/bored/angry etc. with
at/by/with
1. My friend is very good at mathematics.
2. Everybody was shocked at/by what they heard.

Some more examples.
bad at        brilliant at
clever at     surprised at/by
hopeless    at astonished at/by
excellent    at amazed at/by

of/to
1. It is so nice / kind of you to visit us.
2. My mother is very nice / kind to the poor.
Some more examples.
silly/stupid/generous/good + of somebody (to do something)
cruel/friendly/polite/generous/good + to somebody.

Activity – 13

Fill in the blanks with suitable prepositions.
1. The boy’s quite capable __________ solving the problem.
2. Are you worried __________ your examination?
3. The children were very disappointed __________ the magic show.
4. Minu is still upset __________ what you said to her on last Sunday.
5. One should not be cruel __________ animals.
6. We were amused __________ the way he spoke.
7. My elder brother is not aware __________ his responsibility.
8. The beggar is not ashamed __________ what other people think of them.

Answer:
1. The boy’s quite capable of solving the problem.
2. Are you worried about your examination?
3. The children were very disappointed with the magic show.
4. Minu is still upset about what you said to her on last Sunday.
5. One should not be cruel to animals.
6. We were amused at the way he spoke.
7. My elder brother is not aware of his responsibility.
8. The beggar is not ashamed of what other people think of them.

Activity – 14

Write sentences using the adjectives in brackets followed by suitable prepositions. The first two have been done for you.
1. Can I help you with your luggage?
(kind) It’s very kind of you.
2. I went out in the rain without an umbrella.
(silly) It was silly of me to go out in the rain without an umbrella.
3. My friend didn’t thank me for the present.
(not polite) ____________________________
4. The couple had an argument and now they refuse to speak to one another.
(childish) ____________________________
5. Uma offered to carry the luggage to the station.
(nice) ____________________________
6. Mohan shouted at his friends in his birthday party.
(not nice) ____________________________
7. I needed money, and Shyam gave me some.
(generous) ____________________________

Answer:
3. My friend did not thank me for the present.
(not polite) It’s not polite of him.
4. The couple had an argument and now they refuse to speak to one another.
(childish) It’s childish of them.
5. Uma offered to carry the luggage to the station.
(nice) It’s nice of her.
6. Mohan shouted at his friends in his birthday party.
(not nice) It’s not nice of Mohan.
7. I needed money and Shyam gave me some.
(generous) It’s generous of Shyam.

Activity – 15

Put each of the adjectives followed by an appropriate preposition in the right blank.
[rude, brilliant, annoyed, typical, bored, furious]
1. Why do you always get so ___________ little things.
2. He isn’t happy at school. He says he’s ___________ the courses he’s doing.
3. The teacher was ___________ us for making so much noise in the class.
4. Why are you always so ___________ your friends? Can’t you be ___________ them?
5. We’re not surprised he changed his mind at the last moment. That’s ___________ him.
6. He is ___________ telling jokes.

Answer:
1. Why do you always get so worried about little things.
2. He isn’t happy at school. He says he’s annoyed with the courses he’s doing.
3. The teacher was furious at us for making so much noise in the class.
4. Why are you always so rude to your friends? Can’t you be nice to them?
5. We’re not surprised he changed his mind at the last moment. That’s typical of him.
6. He is bored with telling jokes.

SECTION – 6

(Verb + Preposition) Combinations
Certain verbs are followed by particular prepositions. Here are some common examples
of these verb + preposition combinations.
at
1. (a) Don’t point that knife at the child. It’s dangerous.
(b) He spoke in such a manner that everyone laughed at him.
Some more verbs used with ‘at’

aim at              jump at             stare at            glance at wonder at       hint at               peck at              jeer at

for
2. (a) The students are waiting for their teacher to come.
(b) One should always hope for the best.
Some more verbs used with ‘for’

apologize for         cry for                send for beg for                   feel for               strive for care for                   forgive for         sue for charge for               long for             wish for compensate for      mourn for          search for

from
3. (a) The speaker digressed from the topic and spoke a lot of nonsense.
(b) Students will benefit from the new book.
Some more verbs used with ‘from’

hide from              derive from           escape from          save from abstain from         desist from            exclude from         separate from borrow from         differ from             prevent from         subtract from emerge from        digress from           recover from         refrain from

in
4. (a) Most people believe in astrology.
(b) My friend did well in English but failed in Mathematics.
Some more verbs with ‘in’

abound in            excel in            involved in           succeed in admit of               boast of           dispose of             suspect of approve of           smell of           dream of               think of

 

of
5. (a) The students are tired of doing the same task every day.
(b) Many smokers die of lung cancer.
Some more verbs used with ‘of’

accuse of                beware of              complain of              deprive of admit of                 boast of                  dispose of                 suspect of approve of             smell of                  dream of                   think of

on
6. (a) Success in life depends on hard work.
(b) My brother spends a lot on books.
Some more verbs used with ‘on’

base on               impose on            reflect on              comment on insist on              rely on                  congratulate on    intrude on resolve on          decide on              operate on            feast on

to
7. (a) None listened to the Chief Guests lecture.
(b) Students contributed a lot to the Indian Freedom Struggle.
Some more verbs with ‘to’

add to          attend to             conform to          submit to adhere to    belong to             consent to           surrender to amount to   commit to            introduce to         yield to aspire to      confine to            object to              speak to

with
8. (a) Our neighbor always quarrels with children.
(b) Gandhi never compromised with falsehood.
Some more verbs with ‘with’

coincide with      fill with               part with               unite with comply with        grapple with      supply with           deal with cope with            interfere with    sympathize with    overwhelm with disagree with      meddle with      threaten with         reconcile with

Activity – 16

Complete the sentences using appropriate prepositions.
1. The old man accused the boy ___________ trying to steal his bag.
2. My friend invited me ___________ his sister’s marriage.
3. The local people have warned the tourists ___________ swimming in that part of the river.
4. Will you please remind us ___________ the party next Monday?
5. He always borrows money ___________ his friends and never cares to return it.
6. Our teacher congratulated us ___________ our good performance in the examination.
7. Pradip blames other people ___________ his own mistakes.
8. Our Independence Day coincides ___________ the birthday of Sri Aurobindo.
9. The principal approved ___________ the stand taken by the students on the dowry system.
10. We are involved ___________ a project on the uplift of the poor.

Answer:
1. The old man accused the boy of trying to steal his bag.
2. My friend invited me to his sister’s marriage.
3. The local people have warned the tourists of swimming in that part of the river.
4. Will you please remind us of the party next Monday?
5. He always borrows money from his friends and never cares to return it.
6. Our teacher congratulated us on our good performance in the examination.
7. Pradip blames other people for his own mistakes.
8. Our Independence Day coincides with the birthday of Sri Aurobindo.
9. The principal approved of the stand taken by the students on the dowry system.
10. We are involved in a project on the uplift of the poor.

Activity – 17

Complete the sentences, using one of the following verbs in the correct form with appropriate prepositions.
write, speak, glance, listen, talk, explain, ask, shout
1. Please ___________ him. ! He may have something interesting to tell you.
2. He didn’t have his watch. He ___________ my watch to see what the time was.
3. I had an argument with Sikha and now we’re not ___________ each another.
4. Please don’t ___________ the children. Be nice to them.
5. That old woman’s a bit lonely. She needs somebody to ___________.
6. Can you ___________ me how this machine works?
7. Don’t forget to ___________ me while you’re away.
8. His son is always ___________ him ___________ money.

Answer:
1. Please listen to him. ! He may have something interesting to tell you.
2. He didn’t have his watch. He glanced at my watch to see what the time was.
3. I had an argument with Sikha and now we’re not talking to each another.
4. Please don’t shout at the children. Be nice to them.
5. That old woman’s a bit lonely. She needs somebody to speak to.
6. Can you explain to me how this machine works?
7. Don’t forget to write to me while you’re away.
8. His son is always asking him for money.

Activity – 18

Insert the correct prepositions.
1. The parents searched everywhere ___________ their baby but couldn’t find it.
2. ‘Are you going to Calcutta tomorrow ?’
‘I hope so. It depends ___________ the weather.’
3. I don’t mind buying this book. But who is going to pay ___________ it?
4. These two brothers are suspected ___________ stealing a car.
5. You can rely ___________ your grandfather, who has never refused you anything.
6. That handbag belongs ___________ the old man standing in the comer.
7. Warm clothes protect us ___________ cold.
8. Mother asked me to fill the bucket ___________ water.

Answer:
1. The parents searched everywhere for their baby but couldn’t find it.
2. ‘Are you going to Calcutta tomorrow ?’
‘I hope so. It depends on the weather.’
3. I don’t mind buying this book. But who is going to pay for it?
4. These two brothers are suspected of stealing a car.
5. You can rely on your grandfather, who has never refused you anything.
6. That handbag belongs to the old man standing in the comer.
7. Warm clothes protect us from cold.
8. Mother asked me to fill the bucket with water.

Activity – 19

Look at this paragraph from a letter and put in these verbs with suitable prepositions.
applied, agree, care, ask, caring, decided, concentrate suffering, pay

I’m working at a private factory now. I ___________ a caterer’s job last August and started in November. I don’t earn much money, and I even had to___________ my uniform out of my own money. Perhaps I should ___________ a pay rise. But I don’t really ___________ the money. The work is the important thing. Of course, it’s very hard work ___________ the patients, and at the moment I’m ___________ backache. But I knew it would be like this even when I ___________ a career in a hospital. I just try to forget all the problems and ___________ the job. I think it’s a worthwhile thing to do: I hope you ___________ me.

Answer:
I’m working at a private factory now. I decided about a caterer’s job last August and started in November. I don’t earn much money, and I even had to pay for my uniform out of my own money. Perhaps I should ask for a pay rise. But I don’t really care for the money. The work is the important thing. Of course, it’s very hard work caring for the patients, and at the moment I’m suffering from backache. But I knew it would be like this even when I applied for a career in a hospital. I just try to forget all the problems and concentrate on the job. I think it’s a worthwhile thing to do; I hope you agree with me.

Odisha State Board CHSE Odisha Class 11 Invitation to English 4 Solutions Grammar The Imperative Textbook Activity Questions and Answers.

CHSE Odisha 11th Class English Grammar The Imperative

Look at the sentences below.
1. (a) Hurry up!
(b) Stop!
2. (a) Be careful!
(b) Mind your language!
3. Borrow the book from your teacher, read the chapter and make notes on the main points.
All the sentences above are imperative. Sentences 1 and 2 are used for orders, advice, and warnings but Sentence 3 is used to give direction. There is no special form of the verb imperative in English.

Look at the following sentences.
4. Come in, make yourselves at home.
5. Please start, don’t wait for me.

These above Sentences 4 and 5 are used for ‘invitations’.
6. Push.
7. Keep off the grass.

These sentences are used for signboards and notices.
To suggest doing something together we use Let’s / Let us.
8. Let’s hire a taxi.
9. Let us go out.

Activity – 1

Match the sentences in Column A with the imperatives in Column B.

Answer:

Activity – 2

Here is a recipe for making an omelette. Put the following verbs in the correct spaces.
( break, beat, heat, add, turn, make, put, pour)
1. First, two eggs and _____________ them in a bowl.
2. _____________ them up and _____________ salt and pepper to taste.
3. _____________ a little butter or oil in a frying pan.
4. Then _____________ the mixture in, and _____________ sure that the omelette doesn’t burn.
5. _____________ it over when the omelette is fried.
6. Serve it hot.
Answer:
1. First, two eggs and break them in a bowl.
2. Heat them up and add salt and pepper to taste.
3. Pour a little butter or oil in a frying pan.
4. Then pour the mixture in and make sure that the omelette doesn’t burn.
5. Beat it over when the omelette is fried.
6. Serve it hot.

Activity – 3

Match each of the imperatives in column A with their functions in column B.

Answer:
1. Enjoy yourself! —Making a friendly remark.
2. Come in and make yourself at home. — Inviting.
3. Get a Number 7 bus, that’s direct. — Making a suggestion.
4. Turn right at the traffic lights. — Giving direction.
5. Open your book at page 20. — Giving an instruction.
6. Mind the floor, it’s slippery. — Warning.

Activity – 4

Write suggestions that will match the statements below. Use clues to form sentences beginning with Let.
1. Tomorrow is my sister’s birthday. (buy / her / present)
___________________________
2. 1 can’t wait for the train. (take / taxi)
___________________________
3. I want to see a film. (go / cinema)
___________________________
4. It’s hard to believe. (forget / it)
___________________________
5. It’s very cold in here. (light/fire)
___________________________
Answer:
1. Tomorrow is my sister’s birthday.
Let’s buy her a present.
2. I can’t wait for the train.
Let’s take a taxi.
3. I want to see a film.
Let’s go to see the cinema.
4. It’s hard to believe.
Let’s forget it.
5. It’s very cold in here.
Let’s light a fire.

Activity – 5

Fill in each blank choosing the right verb from the following list.
(turn, ask, go, cross, walk, continue, take.)
Stranger: Excuse me, could you tell me how to get to Lewis Road, please?
Young boy: Yes, certainly. _________ along this road till you come to the traffic lights. Then, _________ over and _________ going till you come to the roundabout. _________ right there, then _________ as far as the next roundabout. _________ the first turning to the left and then the first turning to the right. Lewis Road is the second one on the left. _________ someone if you get lost, but it’s really quite easy to find.
Stranger: Thank you very much.
Answer:
Stranger: Excuse me, could you tell me how to get to Lewis Road, please?
Young boy: Yes, certainly. Go along this road till you come to the traffic lights. Then, take over and continue going till you come to the roundabout. Turn right there, then walk as far as the next roundabout. Cross the first turning to the left and then the first turning to the right. Lewis Road is the second one on the left. Ask someone if you get lost, but it’s really quite easy to find.
Stranger: Thank you very much.

Odisha State Board CHSE Odisha Class 11 Invitation to English 3 Solutions Notice Writing Textbook Activity Questions and Answers.

CHSE Odisha 11th Class English Notice Writing

Writing Notices

A notice is a written or printed news, announcement or information. It is usually displayed publicly on a school/college notice-board. A notice can also be given for insertion in a newspaper like an advertisement. But there is one main difference between an advertisement and a notice. An advertisement is chiefly commercial (or matrimonial) in nature; but a notice is a general piece of information for a particular group. A notice should be in complete sentences or even in the form of a short paragraph, whereas an advertisement can be in the form of merely catching phrase and slogans.

Activity 8

The Dramatic and Cultural Association of your college plans to organise an Oriya Debate Competition for +2 students. As the Secretary of the Association, you have put up the following notice:

A well-written notice will tell its readers :
what is about to happen
when it will happen
where it will happen
who can take part
who to contact/apply to
how to contact/apply
where to apply to
etc.

Read the notice given above and check if it contains all these points.
Answer:
This notice does not contain all these points such as, ‘how to conduct/apply/ where to apply to’.

Activity 9

ere is a notice put up by the Youth Adventure Club of Bombay. The sentences in the notice are not in the right order. Re-write the notice, arranging the sentences in the correct order.
1. The voyage is fully sponsored by the Club.
2. The Club plans to take young people on a one-year round-the-world voyage of EXPLORATION AND DISCOVERY, on SS VARSHA, starting 1st January 2010.
3. In addition, a knowledge of cooking and nursing is desirable.
4. The Youth Adventure Club invites young people of both sexes to apply to take part in the ADVENTURE OF A LIFETIME.
5. If interested, please contact the Secretary, Youth Adventure Club, 7 Marine Drive, Bombay on or before 24 October.
6. Those interested should be aged 16-24, enthusiastic, and physically fit.
7. They should have experience of sailing and swimming.

Answer:

Youth Adventure Club, Bombay
NOTICE

10 June, 20_

The Youth Adventure Club invites young people of both sexes to apply to take part in the ADVENTURE OF A LIFE-TIME. The Club plans to take young people on a one year, round-the-world voyage of EXPLORATION AND DISCOVERY, on SS VARSH, starting 1st January, 20. The voyage is fully sponsored by the Club. Those interested should be aged 16-24, enthusiastic and physically fit. They should have experience of sailing and swimming. In addition, a knowledge of cooking and nursing preferable. If interested, please contact the Secretary, Youth Adventure Club, 7, Marine Drive, Bombay on or before 24 October.

Activity 10

The following is a notice put up by the Cultural Secretary of the college. Through this notice, he wants to inform 1 and II year +2 students about an Inter-College One Act Play Competition. Read the notice carefully and try to improve upon it.

Answer:

+2 Cultural Association
B.J.B. College, Bhubaneswar
NOTICE

1 March, 20______

An Inter-College One-Act Play Competition for I and II year students will be held in our college on 9.3.20_. Willing participants are required to submit their names to the Cultural Secretary by 11 a.m. on 3.3.20_.

Pranab Mishra
Secretary

Activity 11

(a) Your college has arranged to present an episode on Door Darshan in the popular programme “Meet the Young Artists”. Write a notice in not more than 50 words for the college notice board, inviting talented students to appear for trials for different items on 20 April. You have invited a famous Door Darshan artist to be present at the selection. You are the Secretary of the College Cultural Association.
(b) You are the Secretary of the Day Scholars’ Association. You have planned an excursion to South India. It is partly subsidised. Write a notice, inviting +2 students to take part in the excursion. Your notice should include all the relevant details.
The Youth Adventure Club invites young people of both sexes to

Answer:
(a)

J.K.B.K. College, Cultural Association
NOTICE

13.4.20

J.K.B.K. Cultural Association invites students (singers, musicians, dancers, monoactors etc.) of all classes to display their talents in a trial test before Mr. Prafulla Mohanty, AIR artist on 20th April 20 at the College Auditorium. Selected students will have the opportunity to appear in Doordarshan’s “Meet the Young Artists” programme. Details can be had from the undersigned.

Pranab Kumar Das
Secretary
J.K.B.K. Cultural Association

(b)

Day Scholars’ Association
S.B. Women’s College, Cuttack .
NOTICE

25 April 20

The Day Scholars’ Association invites +2 students for a heavily subsidised South India tour starting 1st May from Bhubaneswar. The tour will cover entire South India in 15 days with stopovers at Madras, Bangalore, Ooty, Madurai, Pondicherry. Interested students can immediately contact the undersigned with Rs. 2000/- to be paid towards boarding, lodging, train-fare and local transport.
Selection on first come first serve basis.

Rita Mishra
Secretary
Day Scholars’ Association

Odisha State Board CHSE Odisha Class 11 Invitation to English 4 Solutions Grammar Countable and Uncountable Nouns Textbook Activity Questions and Answers.

CHSE Odisha 11th Class English Grammar Countable and Uncountable Nouns

SECTION – 1

Study the following sentences :

• Red is a color.
• Pradeep is a man of ability.
• Nothing succeeds like success.

The underlined words, namely, red, ability, and success are Nouns.
Here, the definition of old grammar does not account for identifying a Noun in a sentence, i.e. name of a person or place or thing.
Let’s see for ourselves what a Noun in English is and whether there is any precise way of knowing (identifying) it in a sentence.
Look at the different natures and functions of a Noun.

• Birds fly.
• The tall dark red-haired Russian scientist will give a talk tomorrow morning.

The above-underlined words, such as, ‘Birds and scientist’ function as the ‘headword’ in the subject part of the sentences.
We can distinguish a ‘Noun ’ from other word classes if they satisfy the following criteria :

(1) These are open-class words to which new items are constantly added.
Example :
cosmonaut, astronaut, videotape, flyover, etc.

(2) Noun can function as the subject, object, complement, or adjunct in a sentence.
Example :
The man died yesterday, (subj.)
Grammarians are nasty creatures, (comp.)
I am writing this lesson now. (obj.)
The student looked up the word (obj.) in the dictionary. (adjunct)

(3) Most nouns form plurals by taking -s or ~es. e.g. -birds, books, watches, etc.
(4) Nouns can be preceded by determiners like a, an, the, one, my, two, some, a few, this, that, these, those or can go without any determiner.
(5) Many nouns can go with genitives such as -’s, -s’.
Example: A bird’s nest is destroyed.
There will be a writers’ camp soon.
(6) Nouns can be preceded by prepositions. e.g. – It is meant for birds only.
(7) Nouns can also be used directly before other nouns as modifiers, e.g. tea- stall, Art College, etc.
(8) Words quoted with plural marker -s are also sometimes used as nouns e.g., ifs and buts, ups and downs, etc.

The above properties provide a clear-cut basis for identifying words as nouns.
Countable nouns refer to ‘things’ (nouns) that can be counted (numbered) and they have got two forms, such as singular form and plural forms. For example, a pen [singular countable], and two pens [plural countable]. On the other hand, uncountable nouns do not take a plural form. These are normally used in the singular form. For example, hair, sugar, and water. [However, in exceptional cases, we, under the circumstances, use the plural form of uncountable nouns, like – water – waters, hair – hairs, etc.]

Now mark the following conversation/talking.
Mother: We have run out of rice, flour, butter, and a few other things.
Father: All right. I’ll make a shopping list. Let me get a piece of paper and a pen.

Now you need some sugar, tea, and milk, right? We also need ten kilos of rice and half a liter of cooking oil. Anything else?
Remember the criteria or markers we discussed earlier and try out yourself first to find out nouns in the above conversation/dialogue. Then check your answers with the answers given below.
Answer:
rice, flour, butter, things, list, piece, paper, pens, sugar, tea, milk, kilos, oil, etc.

Having discussed the various properties/characteristics of nouns in general, let us now try to classify the nouns. The nouns like pen, piece, kilo, etc. can be counted. So they are called Countable Nouns and nouns like rice, paper, oil, water, etc. cannot usually be counted. So these nouns are called Uncountable Nouns. So there are two groups or categories of nouns, such as Countable Nouns and Uncountable Nouns.

Countable Nouns have the following characteristics.
(i) They have two number forms, such as pen-pens, book-books, etc. (singular – plural forms).
(ii) The singular countable nouns take modifiers like an, a, the, three, fourth, etc. (called determiners) before them.
(We cannot say: There is a book on the table.)
(iii) The plural form may or may not take modifiers like the few, and many before them. (Modifiers are optional here.)
(iv) Nouns like sheep and deer are count nouns, but they have the same form for singular and plural. We call them one-form count nouns.

Uncountable Nouns have the following characteristics.
Examine the characteristics of rice the noun :
– Rice is our chief food.
– The rice in his shop is fine.
– Much rice is grown in the plains of Orissa.
– Little rice is grown in hilly areas.
The noun rice (uncountable) has the following characteristics.
(i) It has one form. It does not take -s or -es form.
(ii) It can be used without modifiers.
(iii) It can be used with the modifiers like much, and little but not with a few, or many.
(iv) It cannot be used with numerals like one, two, third, fourth, etc.
Such nouns like ‘rice’ are called uncountables.

Activity – 1:

Say whether the nouns in the list below are countables or uncountables.

1. car        2. Music     3. Cloud       4. information       5. bottle 6. chair     7. Advice    8. Loaf          9. video               10. pot 11. fruit   12. Hair      13. Girl          14. water               15. weather 16. taxi    17. Traffic  18. leather     19. furniture           20. apple

Answer:
1. countable
2. uncountable
3. countable
4. uncountable
5. countable
6. countable
7. uncountable
8. countable
9. countable/uncountable
10. countable
11. uncountable/countable
12. uncountable/countable
13. countable
14. uncountable
15. uncountable
16. countable
17. uncountable
18. uncountable
19. uncountable
20. countable

Activity — 2

Choose the correct alternatives in the sentences below.
1. There is/are usually a lot of traffic/traffics in the city during working hours.
2. The young man is looking for a work /job at the moment.
3. I want to make a list of candidates attending the meeting. Have you got a paper/ some paper?
4. I don’t want to have a bread/bread for my breakfast.
5. The girl with a blonde (golden or pale colored) hair/hairs lives next door.
6. Good accommodation/accommodations is/are not available in this city.
7. We need some meat/meats for the dinner tonight.
8. The old man could not carry the luggage/luggages.
Answer:
1. There is usually a lot of traffic in the city during working hours.
2. The young man is looking for a job at the moment.
3. I want to make a list of candidates attending the meeting. Have you got a paper
4. I don’t want to have bread for my breakfast.
5. The girl with blonde hair lives next door.
6. Good accommodation is not available in this city.
7. We need some meat for the dinner tonight.
8. The old man could not carry the luggage.

Activity – 3

Use a, an, the, somewhere required, In the blank spaces below. If no word is required, leave the space blank.
1. _________health is more precious than _________ wealth. To keep healthy, we need _________ good food, _________ sleep, and _________ exercise.
2. _________ travel gives us _________ opportunity of seeing how_________ other people live. When we go on _________ journey, we should take _________ note-book with us to make _________ notes of the names of _________ people we meet.
Answer:
1. Health is more precious than wealth. To keep healthy, we need some good food, sleep, and exercise.
2. Travel gives us an opportunity of seeing how other people live. When we go on a journey, we should take a notebook with us to make a note of the names of people we meet.

Activity – 4

There are three countable nouns in the list below. Can you find them?

gold	match	rubbish	love
jam	fun	equipment	knowledge
happiness	wood	homework	food
snow	progress	cheese	bottle
cream	help	cupboard	wind

Answer:
Countable
match, cupboard, bottle

SECTION – 2

Read the dialogue below.
Wife: What did you buy in the market?
Husband: I bought a bottle of milk, a packet of washing powder, and a tube of toothpaste.
Wife: What about the bar of chocolate I asked you to buy?
Husband: Oh dear. I completely forgot.

Do mark now the uncountable nouns used as countable in the above dialogue between wife and husband. We made countable quantities with uncountable nouns such as milk, washing powder, and toothpaste using a bottle of milk, and a packet of washing powder. and a tube of toothpaste.

An uncountable noun does not take words like a, an or one, two before it and does not have a plural form; but we can use expressions like a piece of a drop of a bar of, etc. before it to make it countable. They are used to refer to ‘units’ of such uncountables.
For example — a ball of string, a bar of chocolate, a kilo of rice, etc.

Activity – 5

Match the items in Column ‘A’ with suitable items in Column-‘B’.

Answer:

Activity – 6

Imagine that you ‘went with a friend, on a week-long camping trip. You took some supplies (food, matches, candles, etc.) with you, but you find that most of the things have been used up by the third day. Ask your friend to tell you what supplies are left with him/her, and then tell your friend what things are left with you.
The following is the list of supplies that you took on the trip. You will have to decide which of these supplies have been completely used up and which are still left, and the quantities that still remain.
Follow this pattern :
A: Is there any sugar left?
B: Yes, we still have about a kilogram of sugar.
A: What about soap?
B: I’m afraid there’s no soap left.
(or)
B: Yes, we have four cakes of soap left.
List of supplies taken on the camping trip.

rice	salt	matchbox	chewing-gum
bread	sugar	milk	condensed milk
soap	torch	chocolate	toothpaste
tea	bandages	batteries	antiseptic ointment

Answer:
A: Is there any rice left?
B: No, there isn’t any rice left.
A: What about bread?
B: No, we don’t have any loaf of bread.
A: What about soap?
B: Yes, we have a cake of soap left.
A: Is there any salt left?
B: Yes, we have some pinches of salt left.
(Or)
B: I’m afraid there is no grain of salt left.
A: Is there any chewing-gum left?
B: No, there is no chewing-gum left.
A: Is there any condensed milk left?
B: No, there is no condensed milk left.
A: What about your torch?
B: Yes, we have a torch. It is still working well.
A: Is there any antiseptic ointment left?
B: I’m afraid there’s no antiseptic ointment left.
A: Is there any matches left?
B: Yes, there are five boxes of matches left.
A: What about toothpaste?
B: Yes, we have five tubes of toothpaste left.
A: Is there any chocolate left?
B: Yes, there is still a bar of chocolate left.
A: What about milk?
B: Yes, we have some pints of milk left.
A: Is there any tea left?
B: Yes, we have a packet/pound of tea left.
(Or)
B: We have fifty grams of tea left.
A: What about sugar?
B: Yes, we have two kilograms of sugar left.
A: What about batteries?
B: Yes, we still have five (pieces of) batteries left.
A: Is there any bandage left?
B: Yes, we still have four (strips of) bandages left.

SECTION – 3

Nouns that can be countable as well as uncountable.
Look at the examples mindfully.
1.
(a) Our science teacher wrote a paper on the effects of the cyclone, (a paper means here an article (essay): countable)
(b) Books are printed on paper. (uncountable)
(c) every day I read a paper, (newspaper: countable)
(d) The geography paper was difficult, (subject: countable)

2.
(a) Yesterday I bought an iron. (countable: a thing for ironing dress/clothes)
(b) Iron is stronger than wood, (metal: uncountable)
(c) He is a man of iron, (physical strength: uncountable)
In the above examples paper and iron have different meanings in their use. They are used both as countable and uncountable.

Uncountable		Countable
1.	There is a good deal of noise.	1.	Do not make a noise.
2.	Bread is our staple food.	2.	What breads have you got today? (types of)
3.	Do you have much difficulty with your English?	3.	We have had very few difficulties so far.
4.	The temple is made of stone.	4.	The boys are pelting stones at the frogs.
5.	There is some egg on your chin, (food)	5.	There are four eggs in, the basket.
6.	Eat a little more fish, (food)	6.	There are fish/fishes in the pond.
7.	I don’t like tea.	7.	We want three teas and two coffees.
8.	I don’t like your talk, (gossip)	8.	He gave a nice talk on science.
9.	The table is made of wood.	9.	The poet loves woods. Ismail forest!
10.	Beauty is to be admired.	10.	Lara Dutt is a beauty.
11.	He ate a whole chicken.	11.	I Would like to have some more chicken.
12.	Could I have a glass of water?	12.	Glass is brittle.
13.	Tea is grown in Assam.	13.	This shop sells teas from different parts of India.

SECTION – 4

More on Quantifiers.
1. much, many, a lot of, (a) little, (a) few
Much and many are used in negatives and questions/interrogatives generally. Much is used before uncountable nouns and many with plural countable.

• A lot of and lots of cans be used with countable and uncountables in positive (remarks) sentences.
• A little and little can be used with uncountable nouns only.
• Few and a few can be used with plural countable.

Activity – 7

Use much, many, a lot of, lots of in the blank spaces, where required in the sentences below. (In some blank spaces more than one alternative is possible.)
(a) I can’t come with you. I’ve got _________ work to do.
(b) He’s not got _________money, so he can’t buy that house.
(c) He is very quiet person. He doesn’t speak _________.
(d) I’m hoping to get a ticket for the match. But there aren’t _________ seats left, I hear.
(e) That car is very old. It uses _________ petrol.
(f) I haven’t got _________ time for watching sport at the moment.
(g) ______ people go me to the public meeting addressed by the Chief Minister.
(h) We didn’t visit _________ places when we were on holiday.
(i) We heard the cheapest washing machine costs 10,000 rupees. That is _______, in my opinion.
(j) My father drinks _________ water, – ten liters a day.
(k) The players haven’t won _________ medals.
(l) I take photographs but not as _________ as I used to.
Answer:
(a) much a lot of
(b) much
(c) much
(d) many
(e) a lot of
(f) much/a lot of
(g) lots of
(h) many
(i) lots of
(j) lots of
(k) many
(l) many

Activity – 8

Complete these mini dialogues with much, many, a lot of, and lots of
(a) A: Too ________ students fail in English every year.
B: Yes, and the schools aren’t doing ________ about it.
(b) A: We didn’t have ________ time to spare at the railway station.
B: No, we didn’t have ________ either.
(c) A: There were ________ people at the annual function, weren’t there?
B: Yes, we weren’t expecting so ________.
(d) A: We haven’t bad ________ rain this year, have we?
B: No, there haven’t been ________ rainy days.
(e) A: I don’t think my daughter knows ________ about people!
B: I don’t think ________ children know ________ about people!
Answer:
(a) A: many     B: much
(b) A: much     B: much
(c) A: a lot of   B: many
(d) A: much     B: many
(e) A: much     B: many, much

2. few, a few, little, a little.
We have discussed few, a few, little, and a little in Section 4.1. Do mark that a few, few are used with plural countables. They are not generally used with uncountables.
Example:
He is a man of few words, (negative meaning)
A few students passed in English last year, (positive meaning)
There is little water in the glass, (negative meaning)
I need a little help to move the box. (positive meaning)

Now, look at the following examples.
(a) He isn’t very fond of books. He has only a few books at home.
(b) This is a difficult book to read. I’ve had to look up quite a few words in the dictionary.
(c) If what you say is true, there is little we can do about it.
(d) I can’t give you an opinion now. I need a little time to -think.
In the above examples few’ means ‘not many’, ‘a few’ means ‘a small number’ and ‘little’ means ‘not much’, and ‘a little’ means ‘a small amount’.

Activity – 9

Choose the correct alternative in the sentences below.
(a) They could speak few/a few words of Assamese, but they weren’t very fluent.
(b) This is a boring little town; there’s little/a little to do here.
(c) A: Would you like some pepper in your soup?
B: Yes, please, little / a little.
(d) The mud was quite deep. They had little / a little hope of getting out.
(e) Would you like a little/little more tea? There’s still a little/little left in the pot.
(f) I don’t think Ranjan can become a scientist. He’s got little / a little intelligence.
(g) A: Have you ever been to Koraput?
B: Yes, we’ve been there few / a few times.
(h) Father will be away for / a few days next week.
(i) My brother has got a few /few friends in Delhi and he is very happy there.
(j) They won’t take much time to reach the station. There’s a little/little traffic on the road at this time of the day.
Answer:
(a) a few
(b) little
(c) a little
(d) a little
(e) a little, a little
(f) little
(g) a few
(h) a few
(i) a few
(j) a little

Activity – 10

Use a little, a lot of, few, a few, fewer, many, and much where required, in the sentences below:
I moved to this neighborhood two years ago. There seemed to be _________people in this area who were without telephones, so I expected to get a new phone quickly. I applied for one as soon as I moved into the new house. “We aren’t supplying _________ new phones in your area”, an engineer told me.” _________ people want new phones at present and the company is employing _________ engineers than last year so as to save money. A new phone won’t cost _________ money, but it will take _________ time. We can’t do anything for you before December. You need _________ patience if you are waiting for a new phone and you should have _________ friends whose phones you can use when necessary.” Fortunately, I had both. December came and went, but there was no sign of a phone. I went to the office of the telephone company to protest. They told me I would have a phone by December. I protested. “Which year ?“ the clerk asked.
Answer:
I moved to this neighborhood two years ago. There seemed to be a lot of people in this area who were without telephones, so I expected to get a new phone quickly. I applied for one as soon as I moved into the new house. “We aren’t supplying many new phones in your area”, an engineer told me. “A lot of people want new phones at present and the company is employing fewer engineers than last year so as to save money. A new phone won’t cost much money, but it will take a lot of time. We can’t do anything for you before December. You need a little patience if you are waiting for a new phone and you should have a few friends whose phones you can use when necessary.” Fortunately, I had both. December came and went, but there was no sign of a phone. I went to the office of the telephone company to protest. They told me I would have a phone by December. I protested. “Which year ?“ the clerk asked.

SECTION – 5

The articles: a/an and the
Study the following sentences.
(a) I met a beggar and an orphan. I didn’t like the beggar much, but the orphan was very nice.
(b) My brother wrote a novel and a play. I found the novel very interesting, but the play was boring.

A / An is used before singular countable nouns when the speaker or the writer does not know the person or thing. It means that when he/she wants to say about a thing or a person for the first time, he/she uses a/an. A/An is used for indefinite things or persons. So, a and an are known as indefinite articles.

When the speaker or writer speaks or writes about a person or thing (not for the first time) for the second, third time, and so on, he/she uses article the. It gives the definite or particular meaning of the noun. It is used for countable and uncountable nouns. So it is called the ‘definite article’.

Activity – 11

Use a/an or the, where required, in the blank spaces below.
(a) My uncle lives in _________ small house in _________ remote village. There is _________ beautiful garden behind _________ house. _________ garden has many rare plants.
(b) There are two bags on the table: _________ white one and _________ yellow one. _________ white one belongs to my friend but I don’t know who _________ owner of _________ yellow one is.
(c) My friend witnessed _________ accident this morning. _________ truck crashed into _________ lamp post. _________ driver of _________ truck wasn’t hurt but _________ truck was badly damaged.
(d) _________ taxi drove up to our house. _________ taxi stopped outside our house and _________ woman got out of the taxi. _________ man who was carrying _________ case in his hand also got out. With _________ case in his hand, _________ man looked like _________ salesman.
Answer:
(a) My uncle lives in a small house in a remote village. There is a beautiful garden behind the house. The garden has many rare plants.
(b) There are two bags on the table: a white one and a yellow one. The white one belongs to my friend but I don’t know who the owner of the yellow one is.
(c) My friend witnessed an accident this morning. A truck crashed into a lamp post. The driver of the truck wasn’t hurt but the truck was badly damaged.
(d) A taxi drove up to our house. The taxi stopped outside our house and a woman got out of the taxi. A man who was carrying a case in his hand also got out. With the case in his hand, the man looked like a salesman.

Activity – 12

Complete the story below by using a, an, or the where required.
_________ man walked into _________ bank in America and handed _________ note to one of the cashiers, _________ young woman. _________ woman read _________ note, which told her to give _________ man some money. Afraid that he might have _________ gun, she followed _________ instruction. _________ man then walked out of _________ building, leaving _________ note behind. However, it was not _________ successful crime. _________ man had no time to spend _________ money because he was arrested _________ same day. He had made _________ stupid mistake. He had written _________ note on _________ back of _________ envelope. On _________ other side of _________ envelope was _________ man’s name and address. This information was enough for the police to arrest the man.
Answer:
A man walked into a bank in America and handed a note to one of the cashiers, a young woman. The woman read the note, which told her to give the man some money. Afraid that he might have a gun, she followed the instruction. The man then walked out of the building, leaving the note behind. However, it was not a successful crime. The man had no time to spend the money because he was arrested the same day. He had made a stupid mistake. He had written a note on the back of the envelope. On the other side of the envelope was the man’s name and address. This information was enough for the police to arrest the man.

SECTION – 6

More on the indefinite article (a/an).
Look at the following sentences.
My friend is a doctor. He was born in America but now he lives in a small village in a tribal district of Orissa. He works in a hospital attached to a Catholic Church.
A/An is used before singular count nouns.
Example: doctor, village, hospital, etc. Here we refer to persons, people, and things. With the use of a/an article, we usually get an indefinite meaning.

Activity – 13

Complete the sentences below. (You will have to use a/an in the first blank space in each sentence.)
(a) A place where bread is made is called _________.
(b) A shelter for a horse is called _________.
(c) The home of a lion is called _________.
(d) A place where birds are kept is called _________.
(e) A person who mends water pipes is called _________.
(f) A person who tells the future by the stars is called _________.
(g) A person who treats diseases by performing operations is called _________.
(h) A book in which the events of the day are recorded is called _________.
(i) A play with a happy ending is called _________.
(j) A list of the items to be served at a meal is called _________.
Answer:
(a) a bakery
(b) a stable
(c) a den
(d) a cage
(e) a plumber
(f) an astrologer
(g) a surgeon
(h) a diary
(i) a comedy
(j) a menu

Activity – 14

Can you expand the newspaper headlines below into sentences? You will have to use a/an as well as the other words which are missing. The first one has been done for you.
(a) Blast in Billiar town. ➨ There was a blast in a town in Bihar.
(b) Bomb scare delays train. ➨ Bomb scare delays an Express train.
(c) Snake found in a fruit basket at the hotel. ➨ A snake was found in a fruit basket at a hotel.
(d) Indian ship sinks off Abu Dhabi. ➨ An Indian ship sinks off Abu Dhabi.
(e) Ten injured in the clash. ➨ Ten people were injured in a clash.
(f) Bag snatched. ➨ A bag full of gold was snatched by a robber.

Activity – 15

Who were these people?
1. Kalidas
Answer:
Kalidas was an Indian poet.

2. Newton
Answer:
Newton was a great scientist.

3. Charlie Chaplin
Answer:
Charlie Chaplin was an artist.

4. Nargis
Answer:
Nargis wasa him-star.

5. Tansen
Answer:
Tansen was a great musician.

SECTION – 7

The definite article: the
(a) My uncle bought the house next to the post office.
(b) We are not keen on meeting the people next door.
(c) He works in the garage opposite our house.
(d) The boy I met at the station last night is going to America on a scholarship.
In the above examples look at the words in italics. They tell us something about the noun that comes just before them. With the use of those situations/contexts, we will
be able to know that they identify the persons, things, or places. They speak about a definite/particular person, thing or place. So ‘the’ is usually used before them.

Activity – 16

Rewrite the sentences below, inserting the definite article the at the places.
(a) Cottage by the river has been deserted for many years.
Answer:
The cottage by the river has been deserted for many years.

(b) I bought the book you referred to yesterday.
Answer:
I bought the book you referred to yesterday.

(c) They wanted to meet old man living in the yellow house.
Answer:
They wanted to meet the old man living in the yellow house.

(d) I like tea grown in gardens of Assam.
Answer:
I like the tea grown in gardens of Assam.

(e) This is house that my father built.
Answer:
This is the house that my father built.

SECTION – 8

The definite article used for ‘unique reference’.
Mark the use of ‘the’ in the following sentences.
(a) The moon goes round the earth and the earth goes round the sun.
(b) What is the longest bridge in the world?
(c) No one can predict the end of the universe.
The above nouns such as moon, earth, sun, world, and universe are the unique nouns. Their number is one. They don’t usually have plural form. So the definite article ‘the’ is used before them. The use of a/an is possible before them.
Look at the following example.

• The sky got dark.
• There is only one sky in the universe.

Activity – 17

Use the in the blanks, wherever necessary.
(a) _________Prime Minister met _________ President yesterday.
(b) _________ sky is blue.
(c) I am studying History at _________ university.
(d) It’s difficult to live near _________ North Pole.
(e) _________ manufacturing industry is using _________ computers more and more.
(f) Sailors used to spend weeks crossing _________ Atlantic Sea.
(g) _________ equator runs through Africa.
(h) _________ grass is green but _________ grass in my neighbor’s garden is greener.
(i) There are hundreds of small islands in _________ Pacific Ocean.
Answer:
(a) The Prime Minister met the President yesterday.
(b) The sky is blue.
(c) I am studying at university.
(d) It’s difficult to live near the North Pole.
(e) The manufacturing industry is using computers more and more.
(f) Sailors used to spend weeks crossing the Atlantic Sea.
(g) The equator runs through Africa.
(h) Grass is green but the grass in my neighbor’s garden is greener.
(i) There are hundreds of small islands in the Pacific Ocean.

SECTION – 9

Look at the use of the in the passage below.
When we got home, everyone seemed to be busy. Brother was working in the garden. Father was reading the newspaper in the living room. Mother was making tea in the kitchen. Sister was playing with her friend on the terrace.
The use of the nouns garden, living room, kitchen, etc. is for a definite or particular thing or place. That is why the is used in the above situations.

Activity – 18

Use the where necessary.
(a) _______ teacher entered _______ classroom, went to _______ table, held _______ duster in his left hand and _______ book in his right hand. Then he cleaned _______ blackboard and started writing a passage from _______ book.
(b) Ravi went to the post office and talked to _______ man at _______ counter. He asked Ravi to meet _______ postmaster. _______ postmaster asked him to sign _______ register. Then he handed _______ parcel over to Ravi.
Answer:
(a) The teacher entered the classroom, went to the table, held the duster in his left hand and the book in his right hand. Then he cleaned the blackboard and started writing a passage from the book.
(b) Ravi went to the post office and talked to the man at the counter. He asked Ravi to meet the postmaster. The postmaster asked him to sign the register. Then he handed the parcel over to Ravi.

Activity – 19

Supply ‘the’ where necessary.
(a) A: Who’s at ______ door?
B: It is ______ postman.
(b) Will you please go to ______ market and get some butter for ______ cake I am baking?
(c) They prefer to spend their holidays in ______ country, among ______ mountains, or by ______ sea.
(d) This is ______ front room. ______ ceiling and ______ walls need repair but ______ floor is in good condition.
(e) A: Where is your sister?
B: She is in ______ kitchen at ______ moment.
Answer:
(a) A: Who’s at the door?
B: It is the postman.
(b) Will you please go to (the) market and get some butter for the cake I am baking?
(c) They prefer to spend their holidays in the country, among the mountains or by the Sea.
(d) This is the front room. The ceiling and the walls need repair but the floor is in good condition.
(e) A: Where is your sister?
B: She is in the kitchen at the moment.

SECTION – 10

Place names with and without ‘the’.
We normally use ‘the’ with the following.
(a) Seas and oceans: The Pacific Sea, the Mediterranean.
(b) Rivers : The Mahanadi, the Baitarani, the Kathajori etc.
(c) Canals: The Suez Canal, the Panama Canal, etc.
(d) Deserts The Sahara, the Thar.
(e) Island groups: The West Indies, the Canaries.
(f) Hotels, Cinemas: The Grand Hotel, The Metro Cinema.
(g) Museums, Clubs: The Salarjung Museum (Hyderabad), the Saturday Club, etc.
(h) Restaurants, Pubs: The Sultan Cafe, the Swan (Pub)

Names of ships, trains, newspapers, etc. very often have ‘the’.

• the Queen Elizabeth (ship)
• the Falakanama Express
• the Times of India
• the harmonium (any musical instrument)

We do not normally use the following place names.
(a) Continents: Europe, Asia, Africa
(b) Countries, States, Departments: India, China, Orissa
These are the single names of the Countries or States. So they do not go with the definite article ‘the’.
(c) Cities, towns, and villages: Delhi, Bhubaneswar, Astarang.
(d) Individual Islands: Long Island
(e) Lakes : Lake Chilika, Lake Ansupa
(f) Individual mountains: Mount Everest
(g) Streets, Roads: College Street, Lewis Road
(h) Names of games: Football, Cricket, Volley Ball.

We use the with place names when they include a count noun
e.g. Union, Republic, State, Kingdom, Isle, etc.
We use ‘the’ before place names containing of phrases with of:
The University of Delhi, The State of Liberty
The Bank of India, The History of India, etc.

We use ‘the’ when we refer to the parts of the body.
→ She pulled me by the hair.
→ I shook her by the hand.
We use ‘the’ before ordinal or cardinal numbers.
→ Bakul is in the sixth class/class six.
→ Today is the fifteenth of August.
→ The book was published in the 1970s / in the seventies.
We use the + adjective to talk about a whole group of people, a class of people e.g. the poor, the young, the old, the weak, the injured etc.

Activity – 20

Fill in the blanks with ‘the’ where necessary.
(a) _______ Mount Everest is the highest peak in _______ Himalayas.
(b) _______ India celebrated the fiftieth anniversary of its independence in _______ year 1997.
(c) _______ United Nations is an international organization.
(d) _______ United States of America is a republic.
(e) We visited _______ Nehru Park when we were in Hyderabad.
(f) Is _______ Pacific Ocean larger than _______ Indian Ocean?
(g) We read _______ Statesman every day and _______ India Today every week.
(h) _______ University of Utkal is older than Berhampur University.
(i) _______ Grand Trunk Road is the oldest road in India.
(j) _______ Oberoi is _______ only 5-star hotel in _______ Bhubaneswar.
(k) _______ Thar is in _______ Rajasthan.
(l) _______ President visits _______ Pun tomorrow.
Answer:
(a) Mount Everest is the highest peak in the Himalayas.
(b) India celebrated the fiftieth anniversary of its independence in the year 1997.
(c) The United Nations is an international organization.
(d) The United States of America is a republic.
(e) We visited the Nehru Park when we were in Hyderabad.
(f) Is the Pacific Ocean larger than Indian Ocean?
(g) We read the Statesman every day and the India Today every week.
(h) The University of Utkal is older than Berhampur University.
(i) The Grand Trunk Road is the oldest road in India.
(j) The Oberoi is the only 5-star hotel in Bhubaneswar.
(k) The Thar is in Rajasthan.
(l) The President visits Pun tomorrow.

SECTION – 11

Compare the sentences in each of the following pairs.
1. (a) My sister goes to school every morning.
(b) Mina’s father went to the school to meet the headmaster.
2. (a) He has passed the school final examination and will go to college in July.
(b) The new cinema is very near the college.
3. (a) The injured persons have been sent to the hospital.
(b) The hospital was badly damaged by the cyclone.

In the above examples i.e. 1. (a), 2. (a), and 3. (a) ‘the’ is not used before the singular countable nouns like school, college, or hospital. Here the purpose is particular/ primary.
But in sentences 1. (b), 2. (b), and 3. (b) the singular countable nouns school, college, and hospital, do go before them with ‘the’ because the purpose is not primary. Only a few nouns can be used in this way without ‘the’, such as school, college, university, hospital, church, market, and bed.
Do mark deviation: The singular countable noun ‘office’ usually goes with the.
Now mark the following nouns.
work, home.

• He goes to work at 10.
• He will be at home this evening.

In the above examples work and home have some special meanings somewhat similar to the meanings of school, and college. That is why they are used without ‘the’. When nouns like a car, bus, train are used as means of travel (communication), they do not go with ‘the’. We do not normally use articles with the names of different meals i.e. breakfast, lunch, brunch, tea (afternoon meal), supper, dinner, etc.

Activity – 21

Put the into the blank, where necessary.
1. They always go to ______ church on Sunday.
2. When do you plan to go to ______ college?
3. My friend has been taken to ______ hospital.
4. The prisoner was sent to ______ jail.
5. My daughter will go to ______ school next year.
6. When do you usually have ______ lunch?
7. My father goes to ______ work at 9 a.m.
8. I’m taking these books back to ______ library.
9. The weather is too bad to go out. I’m staying at ______ home today.
10. Malati has just had a baby. We are going to ______ hospital to visit her.
11. The prisoner’s wife drove to ______ prison to meet her husband.
12. They like lying on the beach. They always spend their holidays at ______ seaside.
13. ______ bed is very expensive.
14. He went to Delhi by ______ train.
Answer:
1. They always go to church on Sunday.
2. When do you plan to go to college?
3. My friend has been taken to hospital.
4. The prisoner was sent to jail.
5. My daughter will go to school .next year.
6. When do you usually have lunch?
7. My father goes to work at 9 a.m.
8. I’m taking these books back to the library.
9. The weather is too bad to go out. I’m staying at home today.
10. Malati has just had a baby. We are going to the hospital to visit her.
11. The prisoner’s wife drove to the prison to meet her husband.
12. They like lying on the beach. They always spend their holidays at the seaside.
13. The bed is very expensive.
14. He went to Delhi by train.

Activity – 22

Put in ‘the’ where necessary.
Today Alan Broome is a world-famous actor. Forty years ago he was an unhappy child. He didn’t do well at ______ school and he never went to ______ university. His greatest enjoyment was going to ______ movies. The family lived in an unattractive industrial town in England. Their home was next to ______ railway station. Alan’s father was a sailor, and he spent months at ______ sea. He was seldom at ______ home. When he did come home, he did not do much. Sometimes he would lie all day in ______ bed. His wife had to get up at 5 o’clock to go to ______ work. Mr. Broome lost his job in ______ navy and then went to ______ prison for stealing.
Answer:
Today Alan Broome is a world-famous actor. Forty years ago he was an unhappy child. He didn’t do well at school and he never went to university. His greatest enjoyment was going to movies. The family lived in an unattractive industrial town in England. Their home was next to the railway station. Alan’s father was a sailor, and he spent months at sea. He was seldom at home. When he did come home, he did not do much. Sometimes he would lie all day in bed. His wife had to get up at 5 o’clock to go to work. Mr. Broome lost his job in the navy and then went to prison for stealing.

Activity – 23

Insert a / an and the where necessary, in the passage below.
Push metal rod through cork and then put two pins into cork. Take two more corks and push nails into them. Put pins on two glasses and move cork to right place so that it balances properly. Then you need candle and some matches. Make candle stand on saucer under one side of rod and light it. Heat that comes from candle will make metal expand. Extra length will make rod fall. Experiment shows that beat makes metals expand.
Answer:
Push metal rod through a cork and then put two pins into the cork. Take two more corks and push nails into them. Put pins on two glasses and move the cork to right place so that it balances properly. Then you a need candle and some matches. Make a candle stand on a saucer under one side of the rod and light it. Heat that comes from the candle will make the metal expand. An extra length will make the rod fall. The experiment shows that the heat makes the metals expand.