Class 11

Odisha State Board CHSE Odisha Class 11 Approaches to English Book 1 Solutions Unit 3 Text A: The Telephone Textbook Activity Questions and Answers.

CHSE Odisha 11th Class Alternative English Solutions Unit 3 Text A: The Telephone

Activity-1
Text-organization

The following sentences have been removed from the passage. Decide which; paragraph each of them will fit in.
a) The telephone’s actual ring- more perhaps, than any other sound in our daily lives, evokes hope, relief, fear, anxiety, and joy according to our expectations.
b) Or perhaps- again not probably- by magnifying and extending irrational personal conflicts based on voice! contact. It has caused wars.
c) It is small and gentle- relying on how voltage and miniature parts- in times of hugeness and violence.

Answer:
(a) The telephone’s actual ringer more perhaps, than any other sound in our t daily lives, evokes hope, relief, fear, anxiety, joy, according to our expectations. (Paragraph-2)
(b) Or perhaps- again not probably- by magnifying and extending irrational personal conflicts based on voice contact. It has caused wars. (Paragraph-1)
(c) It is small and gentle- relying on how voltage and miniature parts- in times of hugeness and violence. (Paragraph-3)

Activity-2

Understanding The Relation Between The Parts Of The Text:
Complete the sentences below to show your understanding! of how the paragraphs in Text-A are related.
a) The passage as a whole deals with
b) The first paragraph decides
c) In the second the writer tells us about
d) And the concluding paragraph

Answer:
a) The passage as a whole deals with telephones and their uses.
b) The first paragraph decides the Introduction based on cricket.
c) In the second the writer tells us about telephone ringing and its role in hallucination.
d) And the concluding paragraph presents a paradoxical or opposite value.

Activity-3
Comprehension

Answer the following questions as briefly as you can:
Question (a).
Brooks mentions both positive and negative effects of the telephones on our lives. Enumerate the positive effects and the negative „ effects separately.
Answer:
The positive effects mentioned in the passage are:
1. It saves lives by getting rapid ward of illness, injury or famine from remote places.
2. By joining with the elevator, it makes possible the multi story residence or office building.
3. It has made the modem city.
4. It brings about a quantum leap in the speed and ease with information moves from place to place.
5. It accelerates the rate of scientific and technological change and growth in industry.
6. It prevents waging war.

The negative of the telephone are the following:
1. It has crippled the ancient art of letter writing.
2. It has made the waging of war chillingly more efficient.
3. It has extended the scope of human conflicts.
4. It involves personal risk.

Question (b).
How does using the telephone involves personal risk?
Answer:
Using telephone involves a tremendous personal risk because it involves exposure for some to be “hung upon” which happen to be the worst among fears. And some others also dream of a ringing phone and wake up with a heart heavily beating.

Question (c).
A paradox is a seeming contra-diction that is some how true. In what sense is the telephone “a thing of paradox”?
Answer:
Telephone involves contradictory things. It has both the brighter and darker aspect – the positive and negative effects. It is, in one sense, a metaphor for the times that helped to create and in another sense it is their polar opposite.

Extra Activity-3 (A)

Question (a).
In what way does telephone act as a source of active and rapid information?
Answer:
Telephone helps to save lives by obtaining rapid message of illness, injury or famine from distant place.

Question (b).
What does it have do with multi story building?
Answer:
Making conneetion with elevators to. make possible the multistorey buildings- residence or office.

Question (c).
How does it accelerate scientific and technological changes in the industry?
Answer:
A quantum leap in speech and ease has been achieved making information more from place to place. It has greatly accelerated the rate of scientific and technological change and growth in the industry.

Question (d).
How does telephone cripple letter writing?
Answer:
Telephone has crippled the ancient art of letter writing. It makes direct contact ‘ with both the parties at both ends. It marks the spirit of writing one’s mind through letters because it is convenient to establish contact easily and directly. The easy going people resort to the telephone and set aside writing letters.

Activity-4
Understanding The Language Functions

The activity is meant to develop your awareness of the final points of Text-A. Go back to the text after reading each question in the Activity and write your answers.

Question (i).
Brooks uses the first person plural “us” in this except instead of speaking either just about himself or about people in general. Why do you think he does this?
Answer:
The first person narrative is a powerful means of communication that includes people in general and the. another himself. He uses this technique to make communication more effective.

Question (ii).
The essay begins with a question. ’ Would a direct statement be more effective as an introduction? Write a brief justification of your as to this question.
Answer:
Beginning an essay or any other writing makes the literary piece more eye-catching and emphatic. A question sentence at the outset marks prominence and attracts the attention of the readers, Any other statement form wouldn’t have acquired the same effect as a question at the beginning of writing.

Question (iii).
How does the first sentence in paragraph 2 serve as a transition between paragraph-I and 2?
Answer:
Transition implies a change from one condition or set of circumstances to another. The first sentence ¡n paragraph-2 which begins with; “But the question remains unanswered” is quite obvious about the change of immediate context of the previous sentences.

Question (iv).
Identify the phases in the passage that explicitly point to the cause-and-effect connections.
Answer:
(a) Elevator to make possible,
(b) Personal risk because it involves exposure.

Question (v).
Brooks ends this brief introduction to the effect of the telephone with a quotation. Do you think this is an effective conclusion ? Explain.
Answer:
Brooks ends this brief introduction to the effect of the telephone with a quotation, r Using such quotations as a conclusion is likely an effective conclusion

Activity-5
(Understanding the writer’s attitude and, purpose)

Question (a).
Which of the following phrases best expresses Ibe writer’s purpose?
i) It presents objective ‘ information.
ii) to decide a situation and present others’ comment on it.
iii) to persuade the readér that the use of the telephone is an unmixed blessing.
iv) to convince the reader of the Importance of the telephone.
v) to present his personal views is the effects of the telephône.
Answer:
(i) to present objective information.

Question (b).
How would you describe the writer’s attitude towards the telephone? Choose from the following list. critical Indifferent admiring prejudiced approving disapproving condemning.
Answer:
The writer’s attitude is “critical”.

Activity-6
Indicators In Discourse/reference

What do the words ¡n’ italics refer to In the’passage?
a) “…………………….. it uns made possible for better or worse …………………… (lines – 4)
b) “……………………. by so doing, it has played a role…….(lines – 9)
c) it involves exposure ………………………” (lines 2-3)
d) ……………….. in another sense the telephone is their olar opposite.” (lines 3-2) “
Answer:
(a) It = the telephone
(b) so doing = writing letters
(c) it = personal risk
(d) their = the times.

Note
Sentences. in a text do not stand in isolation but -are opposite together with what has gone before and what comes after. One of the important ways in which a writer joins all the sentences together. to make which is one coherent text is a reference. Reference means the use of common words like “he”, “she”, 4bIt. ‘this”, ‘that”, “sà” etc. to refer to people things, or terns of information needlessly; the writer will not repeat a name where “he” or she” will do.

Obviously then to find the meaning of reference word (such as “he” or “she”), it is necessary to, look outside the sentence or the part of the sentence in which it occurs. We have her introduced reference as one of the important ways1 whiçh help tie the text together. Latçr we will look at another important device, of text cohesion, namely, the use of discourse-makers.

Activity-7
Writing A Pamphlet

Write a pamphlet, making an appeal to the people to avoid misusing the telephone. You Cad use the following outlines:
– how some people go on gossiping over the. telephone and keep the lines busy.
– the effects of such irresponsible use of the telephone.
– your suggestions regarding how the misuse of the telephone can be avoided.
– benefits of economical use of the telephone to .oneself and the others.

Answer:
It has become a common feature with some people who go on gossiping over telephones. They consider it as a play thing. They forget that telephone is a means of communication. It acts as a medium for exigencies and messages. Our gossiping over phones is just misusing it. One has just to send messages from a distance. This act of gossiping keeps the lines busy and prevents people with an urgent message from communicating. The lines are engaged and busy. Gossiping without any purpose and pursuit poses two problems.

One financial, the other dislocations. The person who gossips unnecessarily invites unwarranted financial burden to himself and the busy lines owing to his gossip brings about dislocation for others to communicate even urgent messages. An awareness among telephone users is the only powerful way of reducing the instinct for making undue gossips over telephone. Every customer has to be aware of the economical use of the telephone which will enable him to make discreet use’of the telephone. The less you talk, the less you pay. One has to decide what to talk and what to leave out.

The Telephone Summary in English

What does this unit contain?
This unit will present the following four texts, each of them related to some aspects of our living in modern times.
Text-A : The Telephone (John Brooks)
Text-B : Saturday Morning Violence (Nonnanrovisor).
Text-C : New Superstitiohi for old (Margaret Mead).
Text-D : Burnout (A Newspaper article)

The aim of this unit is to help you. practice writing pamphlets and dialogues and develop the following sub-skills of reading:
1. Understanding relations between the parts of a reading text.
2. Recognising indicators in discourse.
3. Understanding the writer’s purpose and attitude; and
4. Identifying the structure of a text in respect of cause-and-effect relationships and the development of arguments.

Section – A
Pre-reading Activity:

The telephone has come almost an indispensable part of modern life. Can you think of three positive effects and three negative effects of the telephone on our lives?

positive effects	Negative effects
1	1
2	2
3	3

Now read the following brief excerpt from the book Telephone the First Hundred Years by John Brook and check whether the writer includes your points about the effects of the telephone on our life.

Text-A
By Johe Brooks Summary
The telephone has saved the lives of people from illness or farming from distance. Joining with the elevator, has made possible the multi-story residence, or office building. It has also made the modern, city possible. It has quickened the rate of scientific and technological changes and growth in the industry. But it has handicapped the ancient art of letter writing. It has played a great role in the social, changes of a country. Waging war has become more efficient for this.

It has also extended the scope of human conflicts since it impartially disseminates the useful knowledge of scientists and the babble of bores, the affection of the affectionate and the malice of the malicious. The use of the telephone also values personal risk for it involves exposure. The telephone is our connecting link with society Marshall Meluhan has said that the telephone creates “a king of extrasensory- perception.”

Analytical outlines of the Text:

• The telephone has saved the life of people.
• It saved people from illness.
• It also saved people from famine from distance.
• They are joined by the elevator.
• It has made possible the multistorey residence or building.
• It has also made the modern city possible.
• It has quickened the rate of scientific changes.
• It has quickened the technology. changes.
• It has also quickened the growth of the industry.
• But it has handicapped the ancient art of letter writing.
• It has played a great role in the social changes of a country.
• Waging war has become more efficient for this.
• It has also extended the scope of human conflicts.
• Since it impartially propagates the useful knowledge of scientists.
• It also propagates the babble of bores.
• It also propagates the affection of the affectionate.
• It also propagates the malice of the malicious.
• The use of telephones also values personal risks.
• It values as it involves exposure.
• The telephone is our connecting link with society.
• Marshal McLuhan has said that the telephone creates, “a kind of extra-sensory perception.”

Meaning of difficult words:
remote – separated, indirect, distant, a long way away.
elevator – a lift in a big building to carry people up and down.
multi-story – many-storeyed, high building with a lot of stories.
quantum leap – a very large improvement
ease – comfort, a happy state of affairs.
accelerated – hasten, the progress of, to increase the speed of, quickened.
crippled – handicapped, disabled, unable to do things.
evanescent – disappear, vanish from sight or memory.
nerve-end – connecting link, bridging gap.
paradox – two apparently contrad¬ictory elements made a paradox of they can be reconceived to give a truth.

Odisha State Board CHSE Odisha Class 11 Math Notes Chapter 6 Complex Numbers and Quadratic Equations will enable students to study smartly.

CHSE Odisha 11th Class Math Notes Chapter 6 Complex Numbers and Quadratic Equations

Unit imaginary number ‘i’.
The unit imaginary number i = √-1
i² = -1
i³ = -i
i⁴ = 1
In general (i)⁴ⁿ = 1, (i)⁴ⁿ⁺¹ = i, (i)⁴ⁿ⁺² = -1, and (i)⁴ⁿ⁺³ = -i.
⇒ If a and b are positive real numbers then
√-a × √-b = -√ab
√a × √b = √ab

Complex Number
General form: = z = a +ib

• a = Real part of (z)  = Re (z)
• b = Imaginary part of (z) = I_m(z)
• a + i0 is purely real and 0 + ib is purely imaginary .
• a + ib = c + id iff a = c and b = d

Complex Algebra
(a) Addition of complex numbers
If z₁ = a + ib and z₂ = c + id then z₁ + z₂ = (a + c) + i(b + d)

Properties:

• Addition is commutative: z₁ + z₂ = z₂ + z₁
• Addition is associative: (z₁ + z₂) + z₃ = z₁ + (z₂ + z₃)
• 0 + i0 is the additive identity.
• -z is the additive inverse of z.

(b) Subtraction of complex numbers:
z₁ = a + ib and z₂ = c + id then z₁ – z₂ = (a – c) + i(b – d)

(c) Multiplication of complex numbers:
z₁ = a + ib and z₂ = c + id then z₁z₂ = (ac – bd) + i(bc + ad)

Properties:

• Multiplication is commutative: z₁z₂ = z₂z₁
• Multiplication is associative: z₁(z₂z₃) = z₁z₂(z₃)
• 1 = 1 + i0 is the multiplicative identity.
• If z = a + ib then the inverse of z.
  z^-1 = \(\frac{1}{a+i b}=\frac{a-i b}{(a+i b)(a-i b)}\)
  = \(\frac{a-i b}{a^2+b^2}=\frac{a}{a^2+b^2}-\frac{i b}{a^2+b^2}\)
• Multiplication is distributive over addition. z₁(z₂ + z₃) = z₁z₂ + z₁z₃

Conjugate and modulus of a complex number:
If  z = a + ib the conjugate of z is \(\bar{Z}\) = a – ib.
⇒ We get conjugate by replacing i by (-i) Modulus of z = a + ib is denoted by |z| and |z| = \(\sqrt{a^2+b^2}\)

Properties Of Conjugate:
(i) \((\overline{\bar{z}})\) = z
(ii) z + \(\bar{z}\) = 2 Re (z)
(iii) z – \(\bar{z}\) = 2i m̂ (z)
(iv) z – \(\bar{z}\) ⇔ z is purely real
(v) Conjugate of real number is itself.
(vi) z + \(\bar{z}\) = 0 ⇒ z is purely imaginary.
(vii) z. \(\bar{z}\) = [Re(z)]² + [m̂(z)]²
= a² + b²
= |z|²
(viii) \(\overline{z_1+z_2}=\overline{z_1}+\overline{z_2}\)
(ix) \(\overline{z_1-z_2}=\overline{z_1}-\overline{z_2}\)
(x) \(\overline{z_1z_2}=\overline{z_1}\overline{z_2}\)
(xi) \(\left(\overline{\frac{z_1}{z_2}}\right)=\frac{\overline{z_1}}{\overline{z_2}}\)

Properties of modulus:
(1) Order relations are not defined for complex numbers. i,e,. z₁ > z₂ or z₁ < z₂ has no meaning but |z₁| < |z₂| or |z₁| > |z₂| is meaningful because |z₁| and |z₂| are real numbers.
(2) |z|  = 0 ⇔ z = 0
(3) |z| = |\(\bar{z}\)| = |-z|
(4) |z| ≤ Re (z) ≤ |z| and -|z| ≤ m̂ (z) ≤ |z|
(5) |z₁z₂| = |z₁| |z₂|
(6) \(\left|\frac{z_1}{z_2}\right|=\frac{\left|z_1\right|}{\left|z_2\right|}\)
(7) |z₁ ± z₂|² = |z₁|² + |z₂|² ± 2 Re (z₁\(\bar{z}_2\))
(8) |z₁ + z₂|² = |z₁ – z₂|² = 2(|z₁|² + |z₂|²)
(9) |z₁ + z₂|² ≤ |z₁| + |z₂|

Square Root Of Complex Number:
Let z = a + ib
Let √z = x + iy

If b > 0 then x and y are taken as same sign.
If b < 0 then x and y are of opposite sign.

Representation of a complex number:
We represent a complex number in different forms like
(i) Geometrical form
(ii) Vector form
(iii) Polar form
(iv) Eulerian form or Exponential form

(i) Geometrical form:
Geometrically z = x + iy = (x, y) represents a point in a coordinate plane known as Argand plane or Gaussian plane.

(ii) Vector form:
In vector form a complex number z = x + iy is the vector \(\overrightarrow{\mathrm{OP}}\) where p(x, y) is the point in the cartesian plane.

(iii) Polar form:
A complex number z = x + iy  in polar form can be written as z = r(cos θ + i sin θ) where r = \(\sqrt{x^2+y^2}\) = |z| and θ is called the argument and -π < θ ≤ π. Technique to write z = x + iy in polar form.
Step – 1: Find r = |z| = \(\sqrt{x^2+y^2}\)
Step – 2: Find α = tan^-1 \(\left|\frac{y}{x}\right|\)
Step – 3:
θ = α for x > 0, y > 0
θ = π – α for x > 0, y > 0
θ = -π + α for x > 0, y > 0
θ = -α for x > 0, y > 0
Step – 4: Write z = r(cos θ + i sin θ)

(iv) Eulerian form or Exponential form z = r e^iθ, because e^iθ = cos θ + i sin θ where θ is the argument and r is the modulus if z.

Note:
(1) |z₁ z₂ z₃ ….. z_n| = |z₁||z₂| …. |z_n|
(2) arg (z₁z₂ …. Z_n) = arg (z₁) + arg (z₂) + ….. + arg (z_n)
(3) arg \(\left(\frac{z_1}{z_2}\right)\) = arg (z₁) – arg (z₂)
(4) arg \((\bar{z})\) = -arg (z)

Cube Roots Of Unity:
Cube roots of unity are 1, ω, ω² where ω = \(\frac{-1 \pm i \sqrt{3}}{2}\)

Properties of Cube roots of unity:
(i) Cube roots of unity lie on unit circle |z| = 1
(ii) 1 + ω + ω² = 0
(iii) Cube roots of -1 are -1, -ω, -ω²
(iv) 1 + ωⁿ + ω²ⁿ \(=\left\{\begin{array}{l}
0 \text { if } n \text { is not a multiple of } 3 \\
3 \text { if } n \text { is a multiple of } 3
\end{array}\right.\)
(v) z³ + 1 = (z + 1) (z + ω) (z + ω²)
(vi) -ω and -ω² are roots of z² – z + 1  = 0.

De-moivre’s theorem:
(a) (De-moivre’s theorem for integral index)
(cos θ + i sin θ)ⁿ = cos (nθ) + i sin (nθ)

(b) (De-moivre’s theorem for rational index)
cos (nθ) + i sin (nθ) is one of the values of (cos θ + i sin θ)ⁿ

(c) nth roots of unity
nth roots of unity are 1, α, α², α³ …..α^n-1. where α = e^{i\(\frac{2 \pi}{n}\)} = cos \(\frac{2 \pi}{n}\) + i sin \(\frac{2 \pi}{n}\)

Properties:

• 1 + α + α² ….. + α^n-1 = 0
• 1 + α^p + α^2p + ….. + α^(n-1)p \(= \begin{cases}0 & \text { if } p \text { is not a multiple of } n \\ n & \text { if } p \text { is a multiple of } n\end{cases}\)
• 1. α. α² ….. α^n-1 = (-1)^n-1
• zⁿ – 1 = (z – 1) (z – α) (z – α²) …..(z – α^n-1)

Quadratic Equations:
The general form: ax² + bx + c = 0  …(i)
Solutions of quadratic equation(1) are
x = \(\frac{-b \pm \sqrt{b^2-4 a c}}{2 a}\)
D = b² – 4ac is called the discrimination of a quadratic equation.
D > 0 ⇒ The equation has real and distinct roots.
D = 0 ⇒ The equation has real and equal roots.
D < 0 ⇒ The equation has complex roots.

Note:
In a quadratic equation with real coefficients, the complex roots occur in conjugate pairs.

Odisha State Board CHSE Odisha Class 11 Approaches to English Book 1 Solutions Unit 1 Text B: Make me a child again, Just for tonight Textbook Activity Questions and Answers.

CHSE Odisha 11th Class Alternative English Solutions Unit 1 Text B: Make me a child again, Just for tonight

Activity – 5

Getting the Main Idea:
Complete the following sentence in your own words to state the central idea of the above selection :
Adults would learn ________________
if they _________________________
and if they ______________________

Activity – 6

Understanding The Title:
Go back to the text and answer the following questions:
(a) Why does the author want to become a child again?
Answer:
Childhood is the best period of learning. Children learn quicker and better than adults. Adults encounter a large number of problems and difficulties in learning. Hence, the author wants to become a child again.

(b) And why “just for tonight”.
Answer:
The author talks about the learning by the adults in the evening colleges at night. Since it is difficult for adults to learn and children are prone to learn things easily, the author wants to become a child again for tonight.

(c) What, according to Stern, are the points of contrast between the learning of children and that of adults?
Answer:
Children:
(1) Children are fast learners.
(2) Children are fast learners.
(3) They have, less responsibilities in life.
(4) They get delight in the discovery of * new words, ideas etc.
(5) In an unhappy situation, children may quickly acquire the habit of lying.
(6) They provide enthusiastic curiosity to both teachers and parents.
(7) They have not too much demands in study.
Adults:
1) Adults are very slow learners.
2) They are timid or fearful to questions.
3) They have much responsibilities in life.
4) They do not have much delight in the discovery of new words, ideas etc.
5) In an unhappy situation, adults may not quickly acquire the habit of lying.
6) They don’t provide enthusiastic curiosity to both teachers and parents.
7) They have too much demands in study.

Extra Activity – 6(A)

Question 1.
How does the writer justify that childhood is such a difficult time of life?
Answer:
In fact, childhood is held as the difficult time of life, for this reason, there are child-guidance associations, child study clinics, child psychologists, child-centered homes and many other institutions for this purpose.

Question 2.
Why does the author say that adults are handicapped as students?
Answer:
The author is of the view that adults are handicapped as students because they are no longer patient listners. They have also forgotten how to make use of their teachers who can give a ready knowledge of tilings.

Question 3.
What views do the evening college administrators put forth?
Answer:
The evening college administrators are of the opinion that adult experience is a priceless commodity in the classroom for both teacher and students. But experience does not only count. Attitude is equally important.

Question 4.
What is the role of motivation in learning?
Answer:
Motivation is a powerful driving force which propel in learning things. Children easily make use of it. But the adult students do not have that motivation in them.

Activity -7

Usage:
(a) Study how the following expressions have been used in Text-B and guess their meanings. The number of the paragraph in which each of these expressions occur is given against each within square brackets.

(i)	afford to	[2]
(ii)	be wedded to	[5]
(iii)	on the other -hand	[6]
(iv)	by and large	[6]
(v)	be equipped with	[7]
(vi)	take pride in	[11]
(vii)	not only…. but…	[11]

Answer:
(i) afford to – spare or find enough time or money for, run a risk by doing something.
(ii) be wedded to – devoted to, unable to give up.
(iii) on the other hand – on the contrary.
(iv) by and large – on the whole, taking everything into consideration.
(v) be equipped with – supply (a person, oneself, a ship, etc.) (with what is need, for a purpose)
(vi) take pride in – having some pride about somebody or something.
(vii) not only but also – both… and

(b) In the conversation between Meera and Sonali below, fill in the blanks with appropriate forms of the suitable expressions from the above list.

Meera: Hi, Sonali, how are you?
Sonali: Hi, I’m fine, how are you?
Meera: Fine, Sonali, are you coming to the Bollywood concert tonight?
Sonali: I’m sorry, I can’t buy a ticket for the concert. I have not got money enough to pay my mess bills for this month.
Meera: Would you mind sharing my complimentary pass? It will admit me you too.
Sonali: Thank you very much for this, friendly gesture. Do you know who will conduct the opera?
Meera: Oh, yes, Haven’t you heard of Pandit Ravi Shankar? The whole country this great Indian musicians accomplishments.
Sonali: Oh, it would be wonderful then. Will it be a botheration for you to pick me up on your way to the concert?
Meera: Not at all it’ll be a pleasure. See you then.
Sonali: Goodbye, Meera.
Answer:
Meera: Hi, Sonali, how are you?
Sonali: Hi, I’m fine, how are you?
Meera: Fine, Sonali, are you coming to the Bollywood concert tonight?
Sonali: I’m sorry, I can’t afford to buy a ticket for the concert. I haven’t got money enough to pay my mess bills for this month.
Meera: Would you mind sharing my complimentary pass? It will admit not only me but also you too.
Sonali: Thank you very much for this friendly gesture. Do you know who will conduct the opera?
Meera: Oh, yes, Haven’t you heard of Pandit Ravi Shankar? The whole country take pride in this great Indian musician’s accomplishments.
Sonali: Oh, it would be wonderful then. Will it be a botheration for you to pick me up on your way to the concert? Meera: Not at all on the other hand it’ll a pleasure. See you then.
Sonali: Goodbye, Meera.

Extra Activity – 7(A)
(a) Question tags are short questions that are tagged with the statements in order to seek opinion from the listener. However, positive statements take negative tags and negative statements take positive tags.

It is hot now, isn’t it?
He is here, isn’t he?
They reached in time, didn’t they?
We made a plan, didn’t we?
It is not hot now, is it?
You are not my friend, aren’t you?

(b) Negative statements without ‘not’.
None of the boys can do it, can they? He saw no one there, did he?

(c) Negativisers (little, few, hardly, scarcely, rarely, seldom, etc.) Sentences with these items are negative in meaning.
Hence, they take positive tags.
Few boys were present, we aren’t they?
He has little knowledge about the matter, hasn’t he?
He has already any money, has he?
Then scarcely comes here, do they?
My friends rarely make a quarrel, do they?
He seldom speaks English, does he?

(d) Imperatives:
Open the door, will you?
Shut your mouth, will you?
Don’t go there, will you?

(e)’Let’ sentence:
Let’s go, shall we?
Let’s dig a well, shall we?

(f) Subject for nobody, somebody, everybody, we use ‘they’ in tag question.
Nobody phoned, did they?
Somebody helped her, didn’t they?
Everybody can’t do it, can they?

(g) ‘I am’ takes a question tag, aren’t I?
I am innocent, aren’t I?
I am happy, aren’t I?
I am regular, aren’t I?

Discussion of Text-B :
Read the Text-B once again this time slowly and notice the pattern Mr. Stem has followed in presenting his central idea which may be stated as follows :

“Adults would learn faster and more easily if they reacted the way children do, and if they had the same purpose and drive that motivated their own learning when they were children.”

You can use the glossary following activity 6 while reading the text slowly to understand how its parts are related. Before preceding further, you must go through Text- B. If you have finished your second reading of Text-B, you can read the following discussion of its structure.

An Overview of the Structure of Text B:
Paragraph (1-2): Two lines from a poem lead to a discussion of childhood and function as a spring bird to the central idea.

Paragraph (3): One part of the central idea is explicitly stated- “adults in evening classes would learn faster and more easily if they reached’ the way children do.”

Paragraph (4): A transition has been affected through an invitation to the reader to notice the difference between the way children learn and they way adults learn.

Paragraph (5-8): First part of the elaboration of the central theme contrast between children and adults in respect of attitudes and habits. ^

Paragraph (9): The last sentence of paragraph-8, is the form of a question and the answer to that question in paragraph 9 serves as transition to paragraph 10.

Paragraph (10-17): Second part of the. elaboration of the .central idea- “adults, if they wish to learn as effectively as the children do, should have the same purposiveness and drive that motivated learning in childhood.”

Paragraph (18-19): The central idea is restated and then to notice the pattern the author is using in presenting his or her thinking

Learning to grasp the structure of a text as you read is not easy. But with every text you practice on, you will become a little surer, a little better, you can test your ability to grasp the central idea of a text and to understand the relationship between its parts when you come to Text-C in the next section.

Section – C

Pre-reading Activity:
Text-C, which follows, is by Mera Welynski, a freelance writer who attended a small private school in New York City. In confessions of ‘Misspent Youth’ an article published in Newsweek magazine in 1976. She contrasts her progressive education with the basic education of traditional schools.

Add question tags to the following statements :
1. Come my boy.
2. He will be late.
3. He had played well.
4. You aren’t busy now.
5. She was dancing fine.
6. He only can do it.
7. Everybody had it.
8. He cut a tree.
9. Pradeep would do it.
10. I have a pen.
11. Shut the door.
12. Few people can do it.
13. He had a bad experience.
14. A few boys were present.
15. There is a cow.
16. He has a long journey to cover.
17. That is mine.
18. It rains in July.
19. Let’s have picnic.
20. Man is mortal.
21. A cow is a domestic animal.
22. He rarely comes here.
23. Come at once.
24. I am not a poet.
25. Please do it.
26. Stop going there.
27. She made all possible.
28. Read every day.
29. You needn’t do it.
30. The place was isolated.
31. He can’t lift it.
32. She is ten years old.
33. You must do it.
34. Nothing went wrong.
35. He hardly drinks tea.
36. He never drinks coffee.
37. He used to wear a coat.
38. Rana does not play cricket.
39. Sita does the work.
40. You’d like a drink.
Answer:

1. Come my boy, will you?
2. He will be late, won’t he?
3. He had played well, hadn’t he?
4. You aren’t busy now, are you?
5. She was dancing fine, wasn’t she?
6. He only can do it, can he?
7. Everybody had it, hadn’t they?
8. He cut a tree, didn’t he?
9. Pradeep would do it, wouldn’t he?
10. I have a pen, haven’t I.
11. Shut the door, will you?
12. Few people can do it, can they?
13. He had a bad experience, didn’t he?
14. A few boys were present, were they?
15. There is a cow, isn’t there?
16. He has a long journey to cover, doesn’t he?
17. That is mine, isn’t that
18. It rains in July, doesn’t it?
19. Let’s have a picnic, shall we?
20. Man is mortal, isn’t he?
21. A cow is a domestic animal, isn’t it?
22. He rarely comes here, doesn’t he?
23. Come at once, will you?
24. I am not a poet, am I?
25. Please do it, will you?
26. Stop going there, will you?
27. She made all possible, didn’t she?
28. Read every day, will you? ,
29. You needn’t do it, need you?
30. The place was isolated, wasn’t it?
31. He can’t lift it, can he?
32. She is ten years old, isn’t she?
33. You must do it, mustn’t you?
34. Nothing went wrong, did it?
35. He hardly drinks tea, does he?
36. He never drinks coffee, does he?
37. He used to wear a coat, did he?
38. Rana does not play cricket, does he?
39. Sita does the work, doesn’t he?
40. You’d like a drink, wouldn’t you?

Make me a child again, Just for tonight Summary in English

“Backward turn backward,
O Time, in your flight.
Make me a child again just for
tonight.”

The above lines appear in Elizabeth Akers Allen’s poem “Rock Me to Sleep, Mother”. Childhood is held at the high esteem as a pivotal period. We have child guidance associations, child study clinics, child psychologists and child centered homes. These can make childhood endurable for infants. Actually, it is true that children usually learn faster than adults. But on the other hand, adults are very careful learners. Of course, they are more frightful than youngsters about asking questions. Adults are also shrouded with responsibilities. This obstructs him from learning. Adults have also a lot of bad habits unconsciously. Some adults have almost a compulsion to be ‘experts’ on every subject and others accept what that ‘experts’ say. Adults are also handicapped as students by the fact that they do not prove patient listeners.

According to evening college administrators, the adults experience in a priceless commodity in the classroom for both teachers and students. But experience is not all. Attitude is as important as experience. on each subject, Mature people need the enthusiasm and self-consciousness of the child in order to capitalize fully in the classroom on their life experience.

However, motivation is another fundamental and essential thing that takes place in learning with children. Children have a natural and built in motivation. They went to communicate and to be communicated with. They want mastery and power. They also want to understand and to be understood. But purpose does not operate in such an automatic and instinctive way wi the adult students in evening classes. If an adult student does not speak English well and he wants a good job, then he should attend an evening college to improve his standard. This can be the actual way for him to fulfill his aim of life in a better and greater way.

Outlines of the passage :

• Childhood is held at the high esteem as a pivotal period.
• We have child guidance associations, child study clinics, child psychologists and child centered homes.
• These can make childhood endurable for infants.
• Actually, children usually learn faster than adults.
• But, adults are very careful learners.
• Again, they are more fearful than youngsters about asking questions.
• Adults are full of responsibilities.
• This disturbs his learning.
• Of course, adults have some bad habits.
• Some adults are compelled to be expert on each subject,
• Adults are not patient listeners.
• According to evening college administrator, the adults experience in a priced commodity.
• But experience is not all in all.
• Attitude is as important as experience.
• Mature people need the enthusiasm and self-consciousness of their child.
• Motivation is another, fundamental and essential thing in learning with children.
• Children have a natural and built-in motivation.
• They want to communicate and to be communicated.
• They want mastery and power.
• They also want to understand and to be understood.
• The purpose operates differently in case of adult students.
• An adult student who does not speak English well and wants a job should attend evening college.
• It can fulfill his aim of life.

Words meaning :

flight – noun form of fly, here passage of time.
modem standards – something used as a test or measure for qualities or for the required degree of, excellence.
inexcusably – unpardonably, that what can’t be excused.
sentimental – emotional, having to do with feeling.
inaccurate – inexact, improper, inadequate.
carefree – free from, affliction, anxiety, bondage.
solemn – grave, serious-looking,
extreme – highest degree, absolute, entire.
clinics – places were pathological tests and diagnosis of diseases are made.
psychologists – experts in psychology,
people having high proficiency in mental functioning, here of children.
endurable – lasts for a long time, lasting.
infants – children,
soap opera – a television drama serial about the happenings in the lives of a group of people.
scarcely – rarely, hardly, seldom.
Late Victorians – English people living towards the end of Queen Victoria’s reign. That is, in the last quarter of the 19th century, who had strict moral attitudes as well as sentimentality.
faster – quicker, speedier,
adults – grown-ups, mature men and women.
defense mechanism- the way of defending oneself to escape imminent problems.
contemporaries – belonging to the same time.
delight in – taking pleasure in.
curiosity – anxious to learn, inquisitiveness.
enormous – a large number of, plenty of, great deal of.
grasp – understand, comprehend
cautious – needful, careful, meticulous.
timid – coward, afraid, not boold enough
ridiculous – absurd, grotesque, unnatural, mocking.
handicapped – weak and disabled, disadvantaged.
shrugged off – dismiss something as not deserving attention as something trivial.
unconsciously – without awareness or sense.
equipped with – well furnished with, full of.
skeptical – inclined not to believe, in the habit of questioning the truth of claims, statements, etc. opposed to stoic.
oneself – consciousness- not worried about what other people think of oneself.
spontaneity – noun form of ‘spontaneous’, automatic flow.
subtle – delicate, soft, tender.
persuade – to, induce by argument, to convince.
omnipresent – the act of being present everywhere
mockery – jest, joke, ridicule, absurdity, etc.
obvious – clear, terse, succinct.
dissuade – opposed to persuade, distract.
egg-head – a person who is very intelligent and is interested only in theories and books.
cliche – an expression used so often.

Odisha State Board CHSE Odisha Class 11 Approaches to English Book 1 Solutions Unit 1 Text A: Memories of Crossgates School Textbook Activity Questions and Answers.

CHSE Odisha 11th Class Alternative English Solutions Unit 1 Text A: Memories of Crossgates School

Activity – 1

Comprehension:
Look at the passage again and put a tick mark against the correct alternative.
(a) The writer describes the pleasure of school life enjoyed during.
(i) Winter,
(ii) Summer,
(iii) Summer and Winter.
Answer:
Summer (✓)

(b) The phrase ‘taking fee walk’ in line 15 is given within question marks to indicate.
(i) that the writer does not approve of the master’s talking a walk.
(ii) that the mortar was actually keeping watch over the boys on fee pretext of taking the walk.
(iii) fee mortar had gone on a visit to the town on the pretex of taking a walk.
Answer:
That the mortar was actually keeping watch over the boys on fee pretext of taking the walks. (✓)

Activity – 2

Comprehension:
Answer the following question briefly as you can.
(a) At what time of the day did the writer bath in fee sea?
Answer:
In the afternoon of the day, the writer took bath in fee sea.

(b) Did he enjoy swimming among the chalk boulders or have a sense suffering for it? What words in the text tell you the’ answer?
Answer:
The author had a pleasant pain for his swimming among the’ chalk boulder. lt was painful because he was full of cuts when he was back at home. The words – “bathed dangerously” and “covered; with cuts” tell us this answer in the text.

(c) What was the boys’ “special treat” in mid-summer evening?
Answer:
The boy had a “special treat” in mid-summer evenings when they were not driven off to hed arid were allowed to go about the grounds in the long twilight terminating in a drive into the swimming bath at about 9 O’clock.

(d) Who were Orwell’s favourite authors in his boyhood ?”
Answer:
Ian Hay, Thackeray, Kipling and H.GWells were Orwell’s favourite authors in his boyhood.

(e) What are the different kinds of caterpillars ‘ that’ Orwell mentions in the passage?
Answer:
The silky green and purple puss moth, the ghostly green popular lewk, the private hawk are the kinds of caterpillars that Orwell mentions in the passage.

(f) What does Orwell say about his “hopeless love affair”?
Answer:
Orwell had a proud love for cricket till he was eighteen. But he was not cricket till he was eighteen. But he was not at all well at the game. Therefore it was “hopeless Ipye affair- a relentless love without any yield.

(g) What does Orwell learn from his experiences at Crossgates School?
Answer:
Orwell learnt a lot of sweet and bitter experiences at Crossgates school. The childhood days were the formative years when children enjoy moving about scot-free. This experience helps to build up the conviction that the things one wants ‘most are always unattainable.

Activity – 3

Writing:
Write in points, the six pleasures of school life that the writer mentions in the passage.
Answer:
The six pleasures of school life that the writer mentions in the passage are:
(1) Bathing dangerously among the chalk boulder in the sea in the returning afternoon.
(2) Deriving special pleasure from mid summer evenings to move about the grounds in the long twilight and taking a dive into the swimming bath at about 9 O’clock.
(3) Walking early on summer morning and having an hour’s undisturbed reading of his favourite authors like Ian Hary, Thackery, Kipling and H.G Wells.
(4) Devoting to cricket hopelessly because he was not good at this game in spite of his sincere efforts.
(5) Keeping caterpillars and dragging a beg-net along the bottom the dew ponds to collect biological specimens.
(6) Escaping long enough from the master with the illicitly purchased of tiny insects.

Activity – 4

Grammar:
Subject-verb Agreement in ’There’ constructions will you please give Text-A another look to find out how many ‘there….’ sentences are used by the author? Notice that sentence-2 in the text-Sentences on summer afternoons there were wonderful expeditions across the Downs…. is a ‘there…’ sentence. Write the sentence number as well as the ‘there…’ sentence you find in the text.
Answer:
Sentence-3: And there were still more wonderful midsummer evening when as a special treat, we were not driven off to bed as usual but allowed to wander about the grounds in the long twilight, ending up with a plunge into the swimming bath at about 9 O’clock.

Sentence-4: There was the joy of walking early on summer morning and getting in an hour’s undisturbed reading (Ian Hay, Thackery, Kipling and H.G. Wells were, the favourite authors of my boyhood) in the sunlight sleeping dormitory.

Sentence-5: There was also cricket which I was no good act but with which 1 conducted a sort of hopeless love affair upto the age of about eighteen.

Sentence-6: And there was the pleasure of keeping caterpillars-the silky iron and purple puss-math, the ghostly green popular park, the privet hawk, large as one’s third finger specimens of which could be illicitly purchased for sixpence at a shop in the town-and when one could escape long enough from the morter who was “taking the walk” share was the excitement of dredging the dew-ponds on the Downs for enormous newts with orange coloured Bellies.

You can very well notice that the verb ‘there’- sentences is either was or more. As you known the verb in an English sentence agrees with the subject in number and person.

Example:
(1) He was ready.
(2) They were ready,
(3) I am reading English grammar today.
(4) She is writing a letter to his friend now.

The subject ‘he’ in sentence-1 is singular in number and so, a singular verb ‘was’ has been used- The sentence would. be wrong if a ‘plural’ verb, such as were, was used. Also notice that verb ‘were’ in sentence two agrees with the subject they in respect of plurality.

The verbs in sentences (3) and (4), do not differ in number i.e., both of them have singular subjects, but they do differ in person. ‘I’ is a first person pronoun, while ‘she’ is a third-person pronoun. So the difference in the verb ‘am’ and ‘is’ are according to them.

In sentences 1-4, the subject comes before the verb. In these sentences, however,’ there’ comes in the subject position while the real subject (which is .also. known as the postponed subject), occurs after the verb. You can, now look back at your list of these sentences above and notice that the verb after there is ‘was’ or ‘were’ according to whether the postponed subject is singular or plural. Thus, in sentence 2 of the text-Sometimes on summer afternoon, there were wonderful expeditions…. the verb ‘were’ is plural because the postponed subject wonderful expeditions is plural. In sentence 4 of the text-there was the Joy of walking easily the singular verb was agrees with the singular postponed subject the Joy of walking early.

Now fill in each blank in the following letter with an appropriate verb from the following list:
is required
is take
do not object
are
attract
is wanted.

Dated the 11th June 2001
To
The Chairperson.
Bhimpur Municipality,
Bhimpur.
Madam,
I would like to express my concern at the growing number of so-called lottery centres in our town.
There__________ several reasons why I object to these places. Firstly, die operators under the false promise Of an easy fortune, _________quite substantial amounts of money away from the poor people who ate least able to afford it. Secondly, While I __________ to gambling in principle, I feel that this particular kind, where no skill __________ on the part of the betting person, is especially Offensive and deadening to the intellect. Thirdly, these establishments often ____________ undesirable individuals into the neighbourhood. Lastly, die physical appearance of these places is must unattractive, and there __________ almost always a lot of noise around. I hope that the Municipal Council will consider very carefully whether this mindless kind of gambling is what _______________ in this traditionally peaceful town.

Yours faithfully,
Smita Samantray.

Answer:
To
The Chairperson
Bhimpur Municipality,
Bhimpur.
Madam,
I would like to express my concern at the growing number of so-called lottery centres in our town.
There are several reasons why I object to these places. Firstly, the operators under the false promise of an easy fortune, take quite substantial amounts of money away from the poor people who are least able to afford it. Secondly, while I do not object to gambling in principle, I feel that this particular kind, where no skill is required in the part of the betting person, is especially offensive and deadening to the intellect. Thirdly, these establishments often attract undersirable individuals into the neighbourhood. Lastly, the physical appearance of these places is must unattractive, and there is almost always a lot of noise around. I hope that the Municipal Council will consider very carefully whether this mindless kind of gambling is what is wanted in this traditionally peaceful town.

Yours faithfully,
Smita Samantray.

Section- A
Pre-reading Activity:
Before reading the following passage which presents George Orwell’s recollection of his school days, write three main things which you remember about your school days.
(1)
(2)
(3)
Now read Text A below quickly to see what Orwell remembers about his school days.

Section- B
Pre-reading Activity:
The text you are going to read in this section has the title “Make Me a Child Again. Just for Tonight”. Almost every one of us has fond memories of childhood, and therefore wants to become a child again. Can you write below three possible reasons why people want to become children again?
(1)
(2)
(3)

Pre-reading Tips:
In this section, you will read a passage or learning. Your primary aim here is to grasp the central meaning of the passage. To do this rapidly and accurately, you have to adopt aggressive reading. In other words, you must get into the habit of speeding through words and you must try to focus on the broad structure, of the passage while reading.

Try to recognise the author’s pattern of thinking and to realise how the detail serve to highlight the main ideas, but don’t let these details slow you down Skim or Skip whenever you feel, you can safely do so, that is, where, you are reasonably sure that you know what the next sentence or paragraph will contain. In order to detect the author’s pattern of thinking and to understand what, in essence,the author is trying to say, you don’t have to read every single word or every single sentence. By thinking along with the author’s pattern of thinking, you can predict what is likely to come next, and whether it is important, less important or completely unimportant to the central meaning of the text. Now cruise through Text-B with conscious pressure on your speed, with an awareness that you must go first. As average learner on the Alternative English course needs 4 to 6 minutes to complete a quick reading of the passage with the aim of grasping the central idea of the piece.

Extra Activity – 4(A)

A. Make sentences of your own using the following:
horde
expeditions
bath
wander
plunge
ghostly
specimen
illicit
escape
excitement

Answer:
horde – He encountered a horde of critical situations.
expeditions – We pray God before starting our expeditions.
wander – The mad man wanders the whole Say.
bath – We should have our baths before praying God.
plunge – We plunge into a river before swimming.
ghostly – The terrorists made the atmosphere ghostly.
specimen – The bank requires specimen signature before opening an account.
illicit – It is illicit to keep gun without licence.
escape – The criminal made all possible efforts to escape punishment
excitement – One cant act well without excitement.

B. Supply appropriate prepositions in the blanks:

(a) We go by the river __________ summer evenings.
(b) He came back from the forest covered __________ cuts.
(c) We were not driven ___________ to bed as used but allowed to wander __________ the grounds _______the long twilight ending up with a plunge ___________ the swimming bath ____________ 9 O’clock.
(d) There was also cricket which I was no good ‘ but ___________ which I conducted a sort of hopeless love affair ____________ the a sort of hopeless love affair ___________ the age of about eighteen.
(e) This business __________ being out ___________ a walk coming across something _________ fascinating interest and then being dragged __________ it by a yell ___________the mortar, like a dog jerked onwards ___________ die leash, is an important feature ___________ school life.

Answer:
(a) We go by the river on summer evening.
(b) He came back from the forest covered with cuts.
(c) We were not driven off to bed as usual. but allowed to wander about the grounds In the long twilight ending up with a plunge into the swimming bath at about 9 O’clock.
(d) There was also cricket which 1 was no good at but with which I conducted a sort of hopeless love affairs upto the age of about eighteen.
(e) This business of being out for a week coming across something of fascinating interest and then being dogged away from it by a yell from the mortar like a dog jerked onwards by the leash. It is an important feature of school life.

C. Use the following :
cover
finger
treat
purchase
ground
escape
end
master
conduct
walk
Answer:
cover :
(N) The cover of this book is attractive.
(V) They covered the idol with flowers.
treat:
(N) Going on a horse’s back gives a special treat.
(V) They treated their guests with respect.
ground:
(N) Your argument has no ground.
(V) This building grounds on a solid structure.
end:
(N) The play has a tragic end.
(V) The play ends with a tragic note.
finger:
(N) Our fingers are not equal.
(V) He fingers well in typing.
purchase:
(N) Business centres round.
(V) He purchased a car yesterday.
escape:
(N) He had no escape from such a crime.
(V) At last, the criminal escaped from the prison.
Master:
(N) The master scolded his servant for his mistake.
(V) He has mastered English language well.

Memories of Crossgates School Summary in English

However, George Orwell had memories at Crossgates School. They were both sweet and bitter memories. He, very often, had beautiful expeditions to different spots. These were Downs, a vast grassy land, and Beachy Head an attractive spot for summer days. He also took baths in Beachy Head. Sometimes, he returned home with injuries. They also had special pleasure from the midsummer evenings. He also very often wander about in the twilight. He also derived great pleasure by taking a dive into the swimming bath. He had also great mirth of getting up early on summer mornings and also reading Ian Hay. Thacjceran, Kipling and H.G. Wells works. Actually, he was not very good at cricket, but this did not imply that he did not love it. On the other hand, he loved cricket upto eighteen. He also took tremendous joy in the forbidden churphase of tiny insects and ran ‘away from the mortar who was “taking the walk”. He had pleasant experiences in bringing up bag nets to collect newts with dredges. He was also bringing up other biological specimens.

Outlines of the Eassay :

• George Or well had memories at Crossgates School.
• They were both sweet and bitter memories.
• He, very often, had beautiful expeditions to different spots.
• These were Downs, a vast grassy land and Beachy Head, an attractive spot for summer days.
• He also took baths in Beachy Head.
• Sometimes, he returned home with injuries.
• He also had special pleasure from the mid-summer evening.
• He also very often wandered about in the twilight.
• He also derived great pleasure by diving into the summer bath.
• He had also great joy of getting up early on summer mornings.
• He derived great joy by reading land Hay, Thackery, Kipling and H.G Wells works.
• However, he was not very good at cricket.
• This did not imply that he did not love it.
• Actually, he loved cricket upto eighteen.
• He also took tremendous joy in the forbidden purchase of tiny insects.
• He ran away from the mortar who was ‘taking the walk’.
• He had pleasant experiences in bringing up beg nets to collect newts with dredges.
• He was also bringing up other biological specimens.

Word Meaning:
horde – a lot of, a great number, crowd, a group of.
expeditions – journeys, movement, wanderings.
the Downs – vast- grassy land.
treat (n) – joy, pleasure, pleasant experience.
wander – roam, move, travel, go about.
twilight – faint half-light after sunset or before sunrise.
walking – getting up from bed, go without sleep.
illicitly – illegally, unlawfully.
purchased – bought, had or exchanged for money.
escape – run away from free sight of
excitement – stirring up, provocation.
dredging – dragging a bag net along the bottom to collect biological specimens.
depends – small ponds.
enormous – large, huge, very big in size.
newts – small aquatic animals wife a moist skin, short legs a long tail.
yell – sound, shout or noise produces from a.distance.
unattainable – which, can not be obtained easily.

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 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 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 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 Invitation to English 4 Solutions Grammar Story Developing Textbook Activity Questions and Answers.

CHSE Odisha 11th Class English Grammar Story Developing

A. Introduction:

A story is made up of a number of events or happenings. Let us look at your own story or part of it. On Sundays, I get up at six in the morning. After a quick wash, I get into my jugging rig and go for a run. By 6.30 I am on the road. I run for half an hour. I return home and have a leisurely bathing, a luxury I cannot afford on weekdays. The bath is over, I get ready quickly. What have you done? You have described the events or your activities on a Sunday morning, in the order in which they take place. You can start with what you do first, then go on to what you do next, and so on until you come to your test activity.

B. A beginning, a middle, and an end:

Like any other piece of information, a story has a beginning, a middle, and an end – it is a complete whole. It invites you and opens the door for you; then it leads you through the plot; and finally, it shows you out at the exit, you walk out happy and satisfied and a door gently shuts behind you. The story ‘Jangled Bells’ is a case in point. It shows three stages. The transition from one stage to the next is not abrupt or sudden; it is smooth and natural.

C. The beginning of a story:

How a story begins is very important. The beginning should catch the reader’s attention and urge him to read on. It should set the scene for the action and the mood.

D. Sequence of events:

A story is the narration of action. All action occurs in time. The most natural way of narrating a story is to give the events strictly in the order in which they happened; with one event leading naturally and logically to the next. “What happened then ?” is the question the storyteller should ask himself at every stage.

E. Paragraphing a story:

The events in a story will fall into a few clusters of happenings, each cluster will have unity of time, place, and action. Each cluster can be put into a paragraph. Paragraphing a story is really a simple thing. Just remember that each paragraph tells one part of the story.

F. The Background:

The story must have a background in which the plot takes place. The background is usually set at the very beginning: It is sometimes done through descriptions of the place, the season, the time, etc. The descriptions should be rich in sensory impression — the reader should see, hear, smell, taste, and feel the atmosphere.

For example:
I was traveling across the desert with Mehmood Ah and his caravan of eighty camels and eighteen men. Ah was a dignified old man with fierce dark eyes and a white beard. His commands were the only laws that the men of the caravan knew.
How is the scene of the story set? (desert, caravan — camels, men, etc.)

G. The characters in a story:

The characters are the people in the story. The story is about ‘them — what they do. how they feel, what they say, etc. The characters must seem to be alive and doing things — not just moving through the story like puppets. They must look like people we see in real life; they must be believable. Each character should have special traits. The way he tells, the way he feels, and the way he reacts to things must be distinctly his own. Only then would he become an individual, not just one of the crowd.

H. Attitude:

She did the right thing!
That was an awful thing to do!
‘Poor woman! How she suffered at his hands?’

These are three different reactions to what someone did: approval, condemnation, and sympathy. But the action was the same; the difference is in the way the three people looked at it. It is the storyteller’s attitude that can change the story very much.

I. Dialogue:

It is possible for the storyteller to report what the characters say. But if this is done throughout he story it can become monotonous. Quoting the actual words of the characters will inject life into the story.

J. The end of a story:

A story must have a natural and definite ending. It should come to an end, not just stop suddenly. It should not leave the reader in the air – unhappily and wondering. The ending should give the feeling of completeness like the final knot on a garland of flowers.

Questions :

Question 1.
Complete a story that ends with the following paragraph :
The tail of the plane was in flames and the pilot knew he would not be able to land safely. There was another loud bang somewhere behind him. He made up his mind. He pulled the rip cord. In seconds he found himself sailing through space, the parachute billowing above him. Below him, he saw the plane crash on the field and explode like a bomb.

Answer:
Nick was bored with life. Everybody was exactly the same. He now wanted to break the monotony. It was summer vacation. He decided to go to Kashmir to enjoy its picturesque details. He arranged a plane ticket for the purpose. It was Sunday. He woke up at 6.30. The sun was shining, and the birds were singing. A gentle breeze was blowing. He got ready to catch up on his flight at 9.30. He. reached the airport one hour before the take-off time.

Nick’s mind wandered in the beautiful valley. His spirit soared. He was looking at his watch on and on. The moment he had been waiting for had come at last. The loudspeakers announced her flight. With a heart of excitement, Nick boarded the plane and sat near the window. The plane took off. Nick looked out of the window. What a beautiful sight! All of a sudden, ominous whispering among the passengers caught his attention. Then they gave loud shrieks. Nick felt terribly confused.

To his stunned disbelief, Nick noticed the plane swing violently. The tail of the plane was in flames and the pilot knew he would not be able to land safely. There was another loud bang somewhere behind him. He made up his mind. He pulled the rip cord. In seconds he found himself sailing through space, the parachute billowing above him. Below him, he saw the plane crash on the field and explore like a bomb.

Question 2.
The day was fine and the clock struck nine. I had an interview at ten. What if I got late? I was in the middle of a busy street, (continue …………)

Answer:
The day was fine and the clock struck nine. I had my interview at ten. What if I got late? I was in the middle of a busy street. The vehicles moved at a snail’s pace. I was getting more and more tensed. If I couldn’t make it, my career would be at stake. I had no other option but to wait. I checked out my watch. “God ! it was already 9.30.” My heart started hammering within my ribs. I had to do something. Just then I was reminded of a narrow lane some yards away which was a shortcut.

With much difficulty, I parked my car. But that was not the end to my problems. The traffic police stopped me for the wrong parking. When the senior officer came, I explained the situation to him. Thankfully, he understood and even helped me get there. When I reached there it was just two minutes to ten. I heared a sigh of sweet relief. The moment of truth came at last. 1 was called for the interview. It was a hectic one. I responded to the questions of the members with a great deal of confidence.

They shook hands with me. I noticed smiles on their faces at the time of my departure. A thrilling experience indeed! A month passed by. I had been waiting for the moment when my appointment would come. Sincerity never goes unrewarded. The moment I had been waiting for came at last. I got my appointment letter. Excitement was in the air. I was really on the moon. That day’s experience still lingers in my memory.

Question 3.
Provide a suitable ending to the following story.
As Sandhya was sitting on the steps at the temple two terrorists appeared. They were armed with AK-56 rifles and hand grenades. She was driven into panic at their sight. To her horror, she found them entering the temple. They fired their shots indiscriminately. The silent prayers of the devotees turned into wailing in a flash. Thousands of them ran hither and thither to save themselves from the brutal attacks of the two dreaded terrorists.

The pitiable cries of women and children moved Sandhya to tears. She was fortunate to leave that spot in a flash. I stood at a distance and was a silent spectator to the ghastly scene. They killed 29 innocent devotees and 3 children and injured 74 others. In the night-long operation to flush out the terrorists, the national security guards lost two of its commandos and the State Reserve Police two of its personnel, the terrorists were killed early the next morning.

Answer:
Whenever we meet by chance, Sandhya tells me she cannot forget the harrowing moment of that day. She still remembers when she was having a close look at the snow-white Akshardham temple, sitting on its marble steps. It is the pride of the Swaminarayan Sect. The bloodshed in the place of worship, which stands for universal peace and brotherhood has left many questions unanswered. There should be conceited efforts to eliminate terrorism for all time to come.

Question 4.
Provide a suitable beginning to the story.
At last, a foolish Brahmin passed by that way. Seeing him the tiger begged him to let him come out from the cage. He took pity on the tiger and opened the door of the cage. As soon as the door was opened the tiger came out and wanted to eat the Brahmin. The Brahmin now realized that he had acted foolishly. However, he told the tiger that he had done a good service to him, so he should not eat him. But the ungrateful tiger would not listen to his argument.

He said that he was very hungry. So he must eat him. The Brahmin was quite helpless. At this time a fox came there. He heard from the Brahmin what had happened. He wanted to decide the matter. But first of all, he must see how the tiger got into the cage and how the Brahmin helped him to get out. They agreed. The tiger then got into the cage through the open door. The fox then shut the door.

The tiger was thus again trapped in the cage. The fox now asked the Brahmin to throw the cage into the river. He called other men to help him. They heard the story and dragged the cage to the bank of the nearby river. Then they threw it into the water. The ungrateful tiger was drowned. Thus, the clever fox saved the foolish Brahmin’s life.

Answer:
Once a tiger was caught in a cage. He tried much to get out. The door of the cage was shut and the iron bars of the cage were very strong. So he could not come out. He asked the passers-by to let him out. But none dared to do so. They feared that the tiger would eat them if he could come out once.