Class 12

Odisha State Board CHSE Odisha Class 12 Invitation to English 4 Solutions Grammar Fundamental Rules for Correcting Grammatical Errors Textbook Activity Questions and Answers.

CHSE Odisha 12th Class English Grammar Fundamental Rules for Correcting Grammatical Errors

Question 1.
In English Grammar verbs always agree with their subject in persons and numbers.
Example:
(a) I am a teacher.
subject – I
verb – am
(b) He is a student.
subject – He
verb – is
(c) They are farmers.
subject – They
verb – are
So while correcting the errors first underline the subject and verb of each sentence and see whether the verb agrees with its subject.

Question 2.
Prepositions always follow either a noun or an adjective or a verb. So find out preposition in each sentence and see their appropriate use.
Example:
(a) He is jealous of my success.
Adjective – jealous
Preposition – of
(b) The boy always depends on his father’s help.
Adjective – depends
Preposition – on
(c) He has confidence in me.
Noun- Confidence
Preposition – in

Question 3.
To find out errors in sentences articles play important role. In some sentences: there is wrong use of articles i.e., wrong use of ‘a’, ‘an’, ‘the’. So go through the use of articles meticulously and get it corrected.
Example:
A hermit live in a small cottage in a remote village. There is an orchard behind the cottage. The orchard has many rare trees.

Question 4.
Sometimes the use of quantifiers like ‘much’, ‘many’, ‘a lot of, ‘little’, ‘a little’, ‘few’ and ‘a few’ is wrongly used in sentences. It should be corrected carefully.
Example:
(a) They don’t find much time to work in the garden.
Uncountable – time
(b) Is there much water in the pond?
Uncountable – water

From the above examples it is found that ‘much’ is used before uncountable nouns in negative and interrogative sentences.
Example:
(c) He doesn’t have many shirts.
PI. Count. – shirts
(d) Does he have many books in his library.
PI. Count. – books

From the above examples it is found that ‘many’ is used before plural countable noun in negative and interrogative sentences.
Example:
(e) The mechanic repairs a lot of bikes everyday.
PI. Count. – bikes
(f) He has got a lot of money.
Uncountable – money

From the above examples it is found that ‘a lot of can be used with plural countable nouns as well as uncountable nouns in affirmative sentences.

Question 5.
Some verbal expressions like would rather and had better always go with the Bare Infinitive form of a non-finite verbs.
Example:
(a) You would rather go than stay here.
(going, go, gone, to go)
(Choose the correct alternative)
(b) They had better do their duty in time.
(doing, done, do, to do)
(Choose the correct alternative)

Question 6.
An imperative sentence beginning with “Let” can be followed by objective case of a pronoun.
Example:
(a) (i) Let he do whatever he wants to do. (Incorrect)
(ii) Let him do whatever he wants to do. (Correct)
(b) (i) Let you and I solve the problem. (Incorrect)
(ii) Let you and me solve the problem. (Correct).
(c) (i) Let there be no secret between you and we. (Incorrect)
(ii) Let there be no secret between you and us. (Correct)

Pronouns
Subjective case	Objective Case
I	Me
We	Us
You	You
He	Him
She	Her
It	It

Question 7.
The expression “I wish” always goes with “I were” i.e., past simple or “had + past participle” i.e., past perfect.
Example:
(a) (i) I wish I am the king. (Incorrect).
(ii) I wish I were the king. (Correct)
(b) (i) I wish I know your name. (Incorrect)
(ii) I wish I knew /had known your name. (Correct)

Question 8.
When the singular pronouns of first, second and third person go together their order should be “2nd person, third person and first person” i.e., “you + he and I”.
Example: (a) (i) I, you and he are friends. (Incorrect)
(ii) You, he and I are friends. (Correct)
(b) (i) You, they and we must work together. (Incorrect)
(ii) We, you and they must work together. (Correct)

Question 9.
Verbs like absent, apply, acquit, enjoy, resign pride, avail always go with reflexive pronouns (self/selves forms).
I – myself
he – himself
she – herself
you – yourself (singular) yourselves (plural)
we – ourselves
they – themselves
child/animal/ object (singular) – itself.

Example:
(a) (i) He prides on his money. (Incorrect)
(ii) He prides himself on his money. (Correct)

(b) (i) He availed of the chance. (Incorrect)
(ii) He availed himself of the chance. (Correct)

(c) (i) The children are expected to behave in the class. (Incorrect)
(ii) The children are expected to behave themselves in the class. (Correct)

(d) (i) The man resigned to the will of God. (Incorrect)
(ii) The man resigned himself to the will of God. (Correct)

(e) (i) My sister absented from her chemistry class yesterday. (Incorrect)
(ii) My sister absented herself from her chemistry class yesterday. (Correct)

Question 10.
The idiomatic expressions like “with a view to”, “look forward to”, “used to”, “habituated to”, and “objected to” always go with the “V+ing” form of a non-finite verb.
Example:
(a) (i) Sulipta is working hard with a view to win the match. (Incorrect)
(ii) Sulipta is working hard with a view to i. (Correct)
(b) (i) I look forward to see the doctor next month. (Incorrect).
(ii) I look forward to seeing the doctor next month. (Correct)
(c) (i) He is used to get up late. (Incorrect)
(ii) He is used to getting up late. (Correct)

Question 11.
(a) Some nouns look singular but they are originally plural. These nouns are people, police, public, cattle, swine, geese, teeth, oxen, mice and they can take plural verbs after them.
(b) The + Adjective and ‘The + Nationality word’ used as the subjects can take plural verbs.
Example:
(i) The police are on duty.
(ii) The people are shouting.
(iii) The cattle are grazing.
(iv) The poor are helpful.
(v) The wounded have been admitted in the hospital.
(vi) The English are a nation of traders.
(vii) The Japanese are industrious.
(viii) The geese look hungry.
(ix) The swine are the lovers of filth.
(x) When the cat is away, mice dance in joy.
(xi) The blind are the most unfortunate.

Question 12.
Grammatical expressions like “It is no use”, “There is no use” and “There is no point” always go with “-ing” form of non-finite verbs.
Example:
(i) It is no use attending classes today.
(ii) There is no point talking with her.

Question 13.
Some verbs tell us about our feelings, emotions, opinions, relations or about a permanent state. Such verbs are called stative verbs. These verbs are usually used in the simple present form.
(a) Verbs of possession: have, own, possess, belong to, contain, consist.
(b) Verbs of liking/disliking: like, dislike, love, hate, prefer, admire and want.
(c) Verbs of perception: see, hear, smell, taste, feel.
(d) Verbs of thinking: think, believe, understand, know.
(e) Verbs of mental activity: hope, forget, remember.
(f) Verbs of appearance: appear, seem, look (like) resemble.
(g) Some other verbs: depend, weight, cost, measure, sound.

Example:
(i) He belongs to Cuttack.
(ii) A year consists of six seasons.
(iii) Lipsa hates telling a lie.
(iv) Honey tastes sweet.
(v) Rose smells sweet.
(vi) Batu loves this girl.
(vii) Mama resembles her mother.
(viii) He knows me.
(ix) Empty vessel sounds much.
(x) I never believe in ghosts.

Question 14.
Adverb like No sooner, hardly/seldom, little, under no circumstances, nowhere and conjunctions like neither, not only etc. are used with inversion if they are used at the beginning of the sentences, i.e., Adv. + Aux. Verb + Sub. + Main verb.
Example:
(a) (i) No sooner the bell rang than the students came out of the class. (Incorrect)
(ii) No sooner did the bell rang than the students came out of the class. (Correct)

(b) (i) Hardly father had come out of the house when it began to rain. (Incorrect)
(ii) Hardly had father come out of the house when it began to rain. (Correct)

(c) (i) Little he understands my problem. (Incorrect)
(ii) Little does he understand my problem. (Correct)

(d) (i) Under no circumstances I shall compel you to do this work. (Incorrect)
(ii) Under no circumstances shall I compel you to do this work. (Correct)

(e) (i) Not only the robbers captured the city but also destroyed it. (Incorrect)
(ii) Not only did the robbers captured the city but also destroyed it. (Correct)

(f) (i) Neither the girl can dance nor sing. (Incorrect)
(ii) Neither can the girl dance nor sing. (Correct)

Question 15.
Some nouns look like plural but originally they are singular and take singular verbs. News, physics, politics, economics, measles, mathematics, diabetes, gymnastics, mumps, rabies, itches, scabies, electronics, athletics, sports, billiards, hurdles, cards.
Example:
(i) Mathematics is my favourite subjects.
(ii) Diabetes is a disease.
(iii) Gymnastics is good for health.
(iv) Billiards is my favourite indoor game.
(v) Physics is a tough subject.
(vi) The new has been greeted with cheers.
(viii)Politics is dirty game.

Question 16.
When the pronoun is the object of a verb or a preposition, it should be in the objective case.
Example:
(a) (i) These books are for you and I. (Incorrect)
(ii) These books are for you and me. (Correct)
(b) (i) Between he and I there is an understanding. (Incorrect)
(ii) Between he and me, there is an understanding. (Correct)

Question 17.
Subject form of the pronouns like (I/he/she/ you/they/we, etc.) is used with the sentences taking than.
Example:
(a) (i) He is taller than me. (Incorrect)
(ii) He is taller than I. (Correct)
(b) (i) I love you more than him. (Incorrect)
(ii) I love you more than he (Correct)

Question 18.
To avoid repetition of a noun, we use that for singular noun and those for the plural noun.
Example:
(a) (i) The climate of Simla is better than Darjeeling. (Incorrect)
(ii) The climate of Simla is better than that of Darjeeling. (Correct)
(b) (i) The roads of Bhubaneswar are wider than Cuttack. (Incorrect)
(ii) The roads of Bhubaneswar are wider than those of Cuttack. (Correct)

Question 19.
‘Each other’, the reciprocal pronoun is used in speaking of two persons/things, ‘one another’ for more than two.
Example:
(a) (i) Jack and Jill love one another. (Incorrect)
(ii) Jack and Jill love each other. (Correct)
(b) (i) The street dogs barked at each other. (Incorrect)
(ii) The street dogs barked at one another. (Correct)

Question 20.
The expressions like each (of), everyone (of), neither of (of the two), either of (of the two), the number of, one of etc. always take singular verb after them.
Example:
(a) (i) Each boy were given a prize. (Incorrect)
(ii) Each boy was given a prize. (Correct)

(b) (i) Each man and each woman have been sent to the workfield. (Incorrect)
(ii) Each man and each woman has been sent to the workfield. (Correct)

(c) (i) Everyone of the cars look attractive. (Incorrect)
(ii) Everyone of the cars looks attractive. (Correct)

(d) (i) Either of the books have a lot of pictures. (Incorrect)
(ii) Either of the books has a lot of pictures. (Correct)

(e) (i) Neither of the girls were called to appear the test. (Incorrect)
(ii) Neither of the girls was called to appear the test. (Correct)

(f) (i) The number of books stolen are forty. (Incorrect)
(ii) The number of books stolen is forty. (Correct)

(g) (i) One of the chairs have a broken leg. (Incorrect)
(ii) One of the chairs has a broken leg. (Correct)

(h) (i) One of my brothers are in the Indian army. (Incorrect)
(ii) One of my brothers is in the Indian army. (Correct)

Question 21.
The name of the shops (books/ opticals) always takes singular verb after them.
Example:
(a) (i) The Books and Books stand on the main road, Bhubaneswar. (Incorrect)
(ii) The Books and Books sstands on the main road, Bhubaneswar. (Correct)
(b) (i) The Giri Opticals have a good name in the whole town. (Incorrect)
(ii) The Giri Opticals has a good name in the whole town. (Correct)
(c) (i) ‘Gulliver’s Travels’ were written by Jonathan Swift. (Incorrect)
(ii) ‘Gulliver’s Travels’ was written by Jonathan Swift. (Correct)

Question 22.
After same or such, relative pronoun as or that is used.
Example:
(a) (i) This isn’t such a good book with I expected. (Incorrect)
(ii) This isn’t such a good book as I expected. (Correct)
(b) (i) This is the same beggar who came to our house last week. (Incorrect)
(ii) this is the same beggar that came to our house yesterday. (Correct)
(c) (i) My problem is the same which yours. (Incorrect)
(ii) My problem is the same as yours. (Correct)

Question 23.
We use relative pronoun that in the noun phrase on antecedent have the + superlative degree, ordinal (like first/second/ tenth /last, etc.) something/ anything/ all/nothing/ somebody etc. (Indefinite pronouns)
Example:
(a) (i) This is the best film which I have ever seen. (Incorrect)
(ii) This is the best film that I have ever seen. (Correct)
(b) (i) Love is something which money can’t buy. (Incorrect)
(ii) Love is something that money can’t buy. (Correct)
(c) (i) She is the most beautiful girl which has ever lived. (Incorrect)
(ii) She is the most beautiful girl that has ever lived. (Correct)
(d) (i) This is all which is yours. (Incorrect)
(ii) This is all that is yours. (Correct)
(e) (i) The second train which left for Puri just now, has faced an accident. (Incorrect)
(ii) The second train that left for Puri just now, has faced an accident. (Correct)

Question 24.
Comparative degree is used for two persons things/ animals and superlative degree for more than two.
Example:
(a) (i) It is the best of the two books. (Incorrect)
(ii) It is better of the two books. (Correct)
(b) (i) He is the better of the three boys. (Incorrect)
(ii) He is the best of the three boys. (Correct)
(c) (i) Which is the best: bread or butter? (Incorrect)
(ii) Which is better: bread or butter? (Correct)
(d) (i) Out of these two watches, which is the best? (Incorrect)
(ii) Out of these two watches, which is better? (Correct)

Question 25.
We use Fewer/Few/ A Few to denote number and less for quantity.
Example:
(a) (i) There are no less than twenty boys in this class. (Incorrect)
(ii) There are no fewer than twenty boys in this class. (Correct)
(b) (i) He takes no fewer than one litre of milk. (Incorrect)
(ii) He takes no jess than one litre of milk, (milk – u.n.) (Correct)

Question 26.
When comparative degree is used in the superlative sense, it is followed by ‘any other’.
Example:
(a) (i) Akram is better than any bowler. (Incorrect)
(ii) Akram is better than any other bowler. (Correct)
(b) (i) Raman is better than any student in the class. (Incorrect)
(ii) Raman is better than any other student in the class. (Correct)

Question 27.
Comparative Adjectives like senior, junior, superior, inferior take to after them instead of than’.
Example:
(a) (i) My father is senior than you in service. (Incorrect)
(ii) My father is senior to you in service. (Correct)
(b) (i) I am junior than her. (Incorrect)
(ii) I am junior to her. (Correct)
(c) (i) This book is inferior than that book. (Incorrect)
(ii) This book is inferior to that book. (Correct)

Question 28.
When expression of measurement, amount, and quantity are used as adjectives, the nouns occuring after the hyphen (-) are always singular.
Example:
(a) (i) The Prime Minister went on a three day state visit to America. (Incorrect)
(ii) The Prime Minister went on a three day state visit to America. (Correct)
(b) (i) A seven-members jury decided the case. (Incorrect)
(ii) A seven-members jury decided the case. (correct)
(c) (i) I came upon a hundred-rupees note. (Incorrect)
(ii) I came upon a hundred-rupee note. (Correct)

Question 29.
We usually use ‘but’ (not ‘than’) after ‘else’.
Example:
(a) (i) It is nothing else than pride. (Incorrect)
(ii) It is nothing else but pride. (Correct)
(b) (i) Call me anything else than a thief. (Incorrect)
(ii) Call me anything else but a thief. (Correct)

Question 30.
‘Very’ is used with adjectives and adverbs in the positive degree with precedent and much is used with Adjectives and adverbs in the comparative degree with past participle.
Example:
(a) (i) She is very slower than Reena. (Incorrect)
(ii) She is much slower (c.d) than Reena. (Correct)
(b) (i) You are very older than me. (Incorrect)
(ii) You are much older (c.d.) than me. (Correct)
(c) (i) The policeman was walking much slowly. (Incorrect)
(ii) The policeman, was walking very slowly. (Correct)

Question 31.
The verb ‘know’ is followed by ‘how to + infinitive’.
Example:
(a) (i) The professor knows to teach the topic. (Incorrect)
(ii) The professor knows how to teach the topic. (Correct)
(b) (i) An expert pilot knows to land the flight in hostile weather. (Incorrect)
(ii) An expert pilot knows how to land the flight in hostile weather. (Correct)

Question 32.
When a noun or pronoun is placed before a gerund (verb + ing working like a noun is called ‘gerund’), it (that noun/pronoun) should be put in the ‘possessive case’, (my/his/her/our/their/teacher’s/one’s etc)
Example:
(a) (i) Please excuse me being late. (Incorrect)
(ii) Please excuse my being late. (Correct)
(b) (i) I would like you stopping smoking. (Incorrect)
(ii) I would like your stopping smoking. (Correct)
(c) (i) The professor disliked us coming late. (Incorrect)
(ii) The professor disliked our coming late. (Correct)

Question 33.
Certain verbs looking alike have different meanings. They are as follows :
Example:
(a) (i) He hanged the lamp on the wall. (Incorrect)
(ii) He hung the lamp on the wall. (Correct)
(b) (i) The criminal was hung for murder. (Incorrect)
(ii) The criminal was hanged for murder. (Correct)
(c) (i) The hens have laid no eggs today. (Incorrect)
(ii) The hens have lain no eggs today. (Correct)
(d) (i) Let me lay on the bed. (Incorrect)
(ii) Let me lay on the bed. (Correct)
Look at the list of following verbs with their past and perfect or past participle forms.

Present	Past	P.P. / Perfect
lie (rest/lie down)	lay	lain
lay (place, arrange/deposit)	laid	laid
lie (to tell a lie)	lied	lied
hang (put up)	hung	hung
hang (execute the order of death sentence)	hanged	hanged
flow (water)	flowed	flowed
fly (bird)	flew	flown
flee (run away).	fled	fled
bear (put up with/tolerate)	bore	borne
bore (make a hole/make tied)	bored	bored
find (to discover)	found	found
found (establish)	founded	founded
fall (drop in the ground)	fell	fallen
fell (cut down a tree)	felled	felled
feel (sensitize)	felt	felt
fill (to pour till the end)	filled	filled
awake (intransitive verb)	awoke	awoke
awake (transitive verb)	awaked	awaked

Question 34.
Indirect questions have the usual Wh-word + subject + verb order.
Example:
(a) (i) Tell me where are you going. (Incorrect)
(ii) Tell me where you are going. (Correct)
(b) (i) Father asked the servant where had he gone. (Incorrect)
(ii) Father asked the servant where he had gone. (Correct)
(c) (i) Can you tell me why does the girl cry bitterly. (Incorrect)
(ii) Can you tell me why the girl cries bitterly. (Correct)
(d) (i) The gentleman asked the station master when was the next train. (Incorrect)
(ii) The gentleman asked the station master when next train was. (Correct)

Question 35.
The past tense in the Principal Clause or Main Clause is used with the same past tense in the Subordinating or Dependent Clause.
Example:
(a) (i) She knew that I am coming. (Incorrect)
(ii) She knew that I was coming. (Correct)
(b) (i) Father said that he won’t go to office that day. (Incorrect)
(ii) Father said that he wouldn’t go to office that day. (Correct)
(c) (i) The doctor asked the patient if he (the patient) has taken medicine at regular
intervals. (Incorrect)
(ii) The doctor asked the patient if he had taken medicine at regular intervals.(Correct)

Question 36.
It is time + subject + past tense verb.
Example:
(a) (i) It is time you go to bed. (Incorrect)
(ii) It is time you went to bed. (Correct)
(b) (i) It is time the doctor operates the patient. (Incorrect)
(ii) It is time the doctor operated the patient. (Correct)

Question 37.
After imperatives (order, advice, request) we use won’t you? (to invite people to do things) and will you/would you/could you? (to tell people to do things).
Example:
(a) (i) Do sit down, will you? (Incorrect)
(ii) Do sit down, won’t you? (Correct)
(b) (i) Give me sufficient time, won’t you? (Incorrect)
(ii) Give me sufficient time, will you? (Correct)
(c) (i) Shut up, can you? (Incorrect)
(ii) Shut up, can’t you? (Correct)

Question 38.
Certain verbs in English are not used in their progressive forms. They are :
Example:
(a) (i) I amn’t seeing you these days. (Incorrect)
(ii) I don’t see you these days. (Correct)

(b) (i) This bag is containing a lot of story books. (Incorrect)
(ii) This bag is contains a lot of story books. (Correct)

(c) (i) Are you appearing disappointed? (Incorrect)
(ii) Do you appear disappointed? (Correct)

(d) (i) Why is the girl hating me so much? (Incorrect)
(ii) Why does the girl hate me so much? (Correct)

(e) (i) Honey is tasting sweet. (Incorrect)
(ii) Honey tastes sweet. (Correct)

(f) (i) Rose is smelling sweet. (Incorrect)
(ii) Rose smells sweet. (Correct)

(g) (i) We are not believing in ghost. (Incorrect)
(ii) We don’t believe in ghost. (Correct)

(h) (i) I am loving this girl. (Incorrect)
(ii) I love this girl. (Correct)

Question 39.
Would rather + subject takes past simple tense of the verb.
Example:
(a) (i) I would rather you resign the job. (Incorrect)
(ii) I would rather you resigned the job. (Correct)
(b) (i) I would rather he leaves this place. (Incorrect)
(ii) I would rather he left this place. (Correct)

Some incorrect words/expressions and their correct use.

Incorrect – Correct
arm (weapon) – arms
blotting – blotting paper
boarding – boarding school
bowel – bowels
foundation – foundations
arrear – arrears
furnitures – furniture
luggages – luggage
earning – earnings
breads – pieces/slices/loaves of bread
equipments – equipment
informations – information
gentries – gentry
machineries – machinery/machines
poetries – poetry
sceneries – scenery
traffics – traffic
scissor – scissors
trouser – trousers
a coward man – a coward/ a cowardly man
a miser man – a miser/ a miserly man
a man of letter – a man of letters(literate)
arrear bill – arrears bill
birth date – date of birth
cousin brother/sister – cousin
custom duty – customs duty
famous criminal – notorious criminal
in the campus – on the campus
in the committee – on the committee
in leave – on leave
in holiday – on holiday
tennis field – tennis court
Cheque of Rs. 200/- – Cheque for Rs. 200/-
no place (bus/train etc.) – no room (bus/train etc.)
today morning/ – this morning/afternoon/
afternoon/evening – evening
this night – tonight
two dozens pens – two dozen pens
saving bank – savings bank
make noise – make a noise
tell lie – tell a lie/tell lies
white hair – grey hair
miles after miles – mile after mile
in hurry – in a hurry
bad in studies – bad at studies
strong/weak of/ good in – strong/weak in/ good at
what to speak of – not to speak of
with bag and baggage – bag and baggage
with heart and soul – heart and soul
with tooth and nail (Completely) – tooth and nail
with black and blue (mercilessly/beat) – black and blue
clever in figure works – clever at figure works
build a home – build a house
cut jokes – crack jokes
cut the pencil – mend the pencil
cook bread – bake bread
describe about – describe
discuss about – discuss
attack on – attack
give a speech – deliver a speech
goodbye – bid goodbye
eat the poor – feed the poor
give order – give orders
make a goal – score a goal
rise the lid – raise the lid
see the pulse – feel the pulse
to have headache – to have a headache
to have temperature – to have a temperature
speak a lie – tell a lie
make prayers – say prayers
ladies bicycle – ladies’ bicycle
women’s college – womens’ college
family members – members of the family
passing marks – pass marks
decrease fear – allay fear
in class tenth – in class ten/in the tenth class
lecture (person) – lecturer
out of orders – out of order (technically defect)
deny request/invitation – refuse request/invitation
refuse stealing/lying – deny stealing/lying
refuse – reject/not to accept
deny – not acknowledge/tell that something is untrue.
a furniture – a piece of furniture
a luggage – a piece of luggage
an information – a piece of information.
a wood – a piece of wood
a grass – a blade of grass
a chocolate – a bar of chocolate
a paper – a sheet of paper
a news – a piece of news/some news
a work – a piece of work / some work
a bread – a slice of bread
a meat – a piece of meat
a toothbrush – a stick of toothbrush
a toothpaste – a tube of toothpaste

Rewrite the following passage, correcting all the grammatical errors in it:

Question 1.
The streets crowd by traffic even before the daybreak. Taxis have been bringing tired people from the airport and the railway station to hotels. They hope sleeping for a few hours before their busy day in the big city is beginning. Trucks are bringing fresh fruits and vegetables in the city. Ships loaded with food and fuel tie up in the dock. Towards morning the streets are quieter, and they are never deserted in the big city.
Answer:
The streets are crowded by traffic even before the daybreak. Taxis bring tired people from the airport and the railway station to the hotels. They hope to sleep for a few hours before their busy day in the big city begins. Trucks bring fresh fruits and vegetables to the city. Ships loaded with food and fuel tied up at the dock. Towards the morning the streets are quieter, and they are never deserted in the big city.

Question 2.
Every country in the world want rapid economic development today. Some economists tells us that it is possible to remove poverty and make everyone prosperous, provide we adopt the right economic policies. The key to prosperity, we are also told, lies in rapid and large-scale industrialization: setting up more factories which will churn out an endless stream of consumer good – products designed to make life more pleasant; motor cars to carried us in comfort and at high speed along smooth super highways; air conditioners to keeps us cool in summer, television sets which will keep us informed as well as entertain and so on. It is believed that as more and more consumers buy the goods that these factories will produce, more and more workers find employment in them; and as their levels of income rise, they will, in their turn create a farther demand for yet more goods. In this way, everyone becomes rich. There are no limits to economical growth and prosperity.
Answer:
Every country in the world wants rapid economic development today. Some economists tell us that it is possible to remove poverty and make everyone prosperous, provided we adopt the right economic policies. The key to prosperity, we are also told, lies in rapid and large-scale industrialization: setting up more factories which will churn out an endless stream of consumer goods – products designed to make life more pleasant; motor cars to carry us in comfort and at high speed along smooth super highways.

air conditioners to keep us cool in summer, television sets which will keep us informed as well as entertained and so on. It is believed that as more and more consumers buy the goods that these factories will produce, more and more workers find employment in them; and as their level of income rises, they will, in their turn create a further demand for yet more goods. In this way, everyone becomes rich. There are no limits to economical growth and prosperity.

Question 3.
Thrift means regulating expenses by such a way that there might be some saving in income. There could be no hard and fast standards for what shall be one’s expenditure and saving. It varied according to one’s circumstances. The rich man may neglect the duty of saving in special occasions because he has power to make up for this neglect. And people with limited income need thrift the most. It gives them strength by relieving them of anxieties for the future. Good housewives in poor families are found to lay off something from daily expenses.
Answer:
Thrift means regulating expenses in such a way that there might be some saving in income. There should be no hard and fast standard for what could be one’s expenditure and saving. It varies according to one’s circumstances. The rich man may neglect the duty of saving on special occasions because he has the power to make up for this neglect. And people with limited income need thrift the most. It gives them strength by relieving them from anxieties for future. Good housewives in poor families are found laying off something from daily expenses.

Question 4.
Preschools have offered good basic education as well as help the child in becoming much independent but confident. Parents may rely in preschools for all-round development of the children. The pre-primary education of the child generally began in home by parents and grandparents. But the picture changes rapidly.
Answer:
Preschools offer good basic education as well as help the child in becoming much independent and confident. Parents rely on preschools for an all-round development of their children. The pre-primary education of the child begins at home by parents and grandparents. But the picture has changed rapidly.

Question 5.
Once a lion was enjoy a nap in his den. A mouse came out in its hole in the den. It start frisking about. In so doing it leap upon the lion’s face. The lion’s sleep disturbs. He wake up furious. He caught the mouse and had been killed it, but the mouse entreated “Your Majesty, I humbly beg your pardon, I’m a poor and little subject of you. But a tiny creature as 1 am, I shall be of some help to you in time. So, please let me go.” The lion laughed aloud for this, but he released the mouse all the same.
Answer:
Once a lion was enjoying a nap in his den. A mouse came out of its hole in the den. It started frisking about. In so doing it leapt upon the lion’s face. The lion’s sleep was disturbed. He woke up furiously. He caught the mouse and was about to kill it. But the mouse entreated “Your Majesty, I humbly beg your pardon, I’m a poor and little subject of yours. But a tiny creature as I am, I shall be of some help to you in time. So, please let me go.” The lion laughed aloud for this and he released the mouse all the same.

Question 6.
Man is at first like an animal. His power then rested only in physical strength. But in this respect he was no match to many beasts. So he has to live in constant fear of them. In course of time came knowledge and it gave him the power to get mastery the entire animal kingdom. He invented weapons with which he not only scared off beasts but even killed them. He learnt how to hunt beasts like a deer which may run more faster than he did Now man could achieve things which were considered impossible then.
Answer:
Man was at first like an animal. His power then rested only on physical strength. But in this respect he was no match for many beasts. So he had to live in constant fear of them. In course of time came knowledge and it gave him a power to get mastery upon the entire animal kingdom. He invented weapons with which he not only scared off beasts but also killed them. He learnt how to hunt beasts like a deer who could run faster than he. Now man has achieved things which were considered impossible then.

Question 7.
The foodbazar took the entire responsibility in sending a farmer’s produce to the customer, without depending on a chain of middlemen. The private company with its vast resources could set in cold storages, acquire refrigerated trucks to transport the produce to cities. In the end of the food chain there are airconditioned supermarkets where consumers can buy the produce at good condition. A kilo of tomato which a customer will buy for 10 rupees may be available for only 7 rupees in a supermarket. Don’t imagine that any private company will do all this out from charity or love towards the farmers. In time food chains might come on away from the cities and closer to the farms.
Answer:
The foodbazar is taking the entire responsibility of sending a farmer’s produce to the customer, without depending on a chain of middlemen. The private company with its vast resources can set up cold storages and acquire refrigerated trucks to transport the produce to cities. At the end of the food chain there are airconditioned supermarkets where consumers can buy the produce in good condition. A kilo of tomato which a customer will buy for 10 rupees can be available for only 7 rupees in a supermarket. We can’t imagine that any private company will do all this out of charity or love for the farmers. With time food chains may come away from the cities and closer to the farms.

Question 8.
A big fat hen lived of a farmyard in a village. A dove also lived in a large tree besides the same farmyard. Soon both of them became good friend. They use to meet every evening to share their thoughts. Oneday, the hen see by a fox. He enters the farmyard secretly and was able to catch the hen. The clever fox put the hen in a sack, carried at his back and the hen started crying aloud in protest. The dove heard the crying of the hen from the top of the tree. She at once realised that the hen fell in danger. The dove thought off an idea to save her friend. She went ahead and lay on the path motionless as of she was dead. The fox put his sack down at the wayside and went near the dove. In the meantime, the hen managed to escape, and hide behind a bush. When the fox was about to catch the ‘dead’ dove, he surprised to see that the dove quickly flew away. In the evening the dove met the hen at the farmyard and the two began to laugh for the foolishness of the fox.
Answer:
A big fat hen lived in a farmyard in a village. A dove also lived in a large tree beside the same farmyard. Soon both of them became good friends. They used to meet every evening to share their thoughts. Oneday, the hen was seen by a fox. He entered the farmyard secretly and was able to catch the hen. The clever fox put the hen in a sack, carried on his back and the hen started crying aloud in protest. The dove heard the cry of the hen from the top of the tree.

She at once realised that the hen had fallen into danger. The dove thought of an idea to save her friend. She went ahead and lay on the path motionless as if she was dead. The fox put his sack down on the wayside and went near the dove. In the meantime, the hen managed to escape, and hid behind a bush. When the fox was about to catch the ‘dead’ dove, he was surprised to see that the dove quickly flew away. In the evening the dove met the hen at the farmyard and the two began to laugh for the foolishness of the fox.

Question 9.
The food bazaar is taking entire responsibility in sending the farmer’s produce to the consumer. The private company with its vast resource, may set up cold storages, acquire fleets of refrigerated trucks to transport the produce into cities and even construct roads for speedy transportation. In the end of the food chain, there have been air-conditioned supermarkets where consumers could buy produce of high quality, in good condition, at comparatively reasonable prices, in clean and hygienic surrounding. A kilo of tomatoes which a customer could buy from a vegetable-vendor for ten rupees must be available, weighed but neatly packed for only t 7.50 in a super-market. Out of that amount, the farmer is likely to have got at least ? 3.50 a much higher price than he would get if he would sell his produce to a middleman.
Answer:
The food bazaar is taking an responsibility in sending a farmer’s produce to the consumer. The private company with its vast resources, can set up cold storages, acquire a fleet of refrigerated trucks to transport the produce to the cities and even construct roads for speedy transportation. At the end of the food chain, there are air-conditioned supermarkets where consumers can buy produce of high quality, in good condition, in a comparatively reasonable prices, in a clean and hygienic surrounding.

A kilo of tomato which a customer can buy from a vegetable-vendor for ten rupees may be available, weighed but neatly packed for only t 7.50 in a super-market. Out of that amount, the farmer is likely get at least ? 3.50 a much higher price than he would get if he would sell his produce to a middleman.

Question 10.
Few years ago I was spending a week at Port Blair. The week had been over but I was in the airport ready for leaving when I discovered, in my dismay, that I forgot one of my suitcase in my hotel. Quickly, I jumped into a taxi and had explained my situation to the taxi driver. We sped away in the direction of my hotel. Suddenly the taxi driver slowed down so that he would talk with the driver of a truck moving through a road next to us. The truck contained live chickens.
Answer:
A few years ago I was spending a week in Port Blair. As the week was over but 1 was in the airport ready to leave when I discovered, to my dismay, that I had forgotten one of my suitcases in my hotel. Quickly, I jumped into a taxi and explained my situation to the taxi driver. We sped away in the direction of my hotel. Suddenly the taxi driver slowed down so that he might talk to the driver of a truck moving through the road next to us. The truck carried live chichens.

Question 11.
Rahul and Ramesh were best friend studying in school. Rahul was good in sports but poor in studies. Ramesh is good at both. At any competition, Ramesh always managed to win. As a result, Rahul became jealous with Ramesh. The sports day was near. Rahul and Ramesh were practising for it. Rahul was not sure if he score a win over Ramesh. Slowly they talked less among themselves. Ramesh asked him the reason many time but Rahul always put him of with a excuse or other. One day before the sports event, Rahul hit at a plan to defeat Ramesh. He went to the ground before anyone has arrived there. He dug a small pit on the path where Ramesh was supposed to run. Then he covered it with leaves and went back to his classroom. When the race started. Ramesh was ahead of Rahul but after sometimes he stepped on the pit and fell down. Rahul took over him. He was very happy as his plan worked and he had own the first prize. Soonafter, Rahul realized his fault and begged forgiveness.
Answer:
Rahul and Ramesh were best friends when studying in school. Rahul was good at sports but poor in studies. Ramesh was good at both. In any competition, Ramesh always managed to win. As a result, Rahul became jealous of Ramesh. The sports day was near. So Rahul and Ramesh were practising for it. Rahul was not sure if he would score a win over Ramesh. Gradually they talked less between themselves. Ramesh asked him the reason many a time but Rahul always put him off with an excuse or other.

One day before the sports event, Rahul hit upon a plan to defeat Ramesh. He went to the ground before anyone had arrived there. He dug a small pit on the path where Ramesh was supposed to run. Then he covered it with leaves and went back to his classroom. When the race started. Ramesh was ahead of Rahul but after sometime he stepped on the pit and fell down. Rahul took over him. He was very happy as his plan worked and he had won the first prize. Soonafter, Rahul realized his fault and begged forgiveness.

Question 12.
Generally the word ‘superstition’ was used as the term of disgrace or reproach to senseless belief based in ignorant fear. We look down to the uncivilized people because they worshipped the different aspect of nature and are filled in awe by things that are simple enough for us. Many Hindus considered a sneeze, the cry of a lizard as ominous. For the Westerners, number 13 is so unlucky that it can be avoided by all means. If the cat crosses their path they call up their tour. Indeed, every community is subject to some superstition and we could not conceive for any community being completely free of it.
Answer:
Generally the word ‘superstition’ is used as a term of disgrace or reproach to senseless belief based on ignorant fear. We look down upon the uncivilized people because they worshipped different aspects of nature and were fdled with awe by the things that are simple enough for us.

Many Hindus consider a sneeze, the cry of a lizard as ominous. For the Westerners, number 13 is so unlucky that it should be avoided by all means. If the cat crosses their path they call off their tour. Indeed, every community is subject to some superstition and we can not conceive of any community completely free from it.

Odisha State Board Elements of Mathematics Class 12 Solutions CHSE Odisha Chapter 9 Integration Ex 9(a) Textbook Exercise Questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Exercise 9(a)

Question 1.
(i) ∫2 dx
Solution:
∫2 dx = 2x + C

(ii) ∫3x² dx
Solution:
∫3x² dx = 3 \(\frac{x^3}{3}\) + C = x³ + C

(iii) ∫4x³ dx
Solution:
∫4x³ dx = 4 \(\frac{x^4}{4}\) + C = x⁴ + C

(iv) ∫x⁵ dx
Solution:
∫x⁵ dx = \(\frac{x^6}{6}\) + C

(v) ∫x³¹ dx
Solution:
∫x³¹ dx = \(\frac{x^{32}}{32}\) + C

(vi) ∫\(\left(2 \sqrt{x}+\frac{3}{\sqrt{x}}\right)\) dx
Solution:

(vii) ∫\(\frac{1}{x \sqrt{x}}\) dx
Solution:

(viii) ∫\(\left(x^{\frac{4}{7}}+\frac{1}{x^{\frac{1}{3}}}\right)\) dx
Solution:

(ix) ∫\(\frac{4}{x}\) dx
Solution:
∫\(\frac{4}{x}\) dx = 4 ln |x| + C

(x) ∫\(\left(\frac{3}{x^2}+\frac{1}{x^{12}}\right)\) dx
Solution:

(xi) ∫(x² + √x)² dx
Solution:

(xii) ∫(x + 3) (2 – x) dx
Solution:
∫(x + 3) (2 – x) dx
= ∫(2x + 6 – x² – 3x) dx
= ∫(-x² – x + 6) dx
= –\(\frac{1}{3}\)x³ – \(\frac{1}{2}\)x² + 6x + C

(xiii) ∫\(\frac{(\sqrt{x}+2)^2}{x^4}\) dx
Solution:

(xiv) ∫\(\frac{(x+\sqrt{x})(2 x+1)}{x^2}\) dx
Solution:

Question 2.
(i) ∫cos dx
Solution:
∫cos dx = sin x + C

(ii) ∫\(\frac{d x}{\cos ^2 x}\)
Solution:
∫\(\frac{d x}{\cos ^2 x}\) = ∫sec² x dx = tan x + C

(iii) ∫\(\frac{d x}{1-\cos ^2 x}\)
Solution:
∫\(\frac{d x}{1-\cos ^2 x}\) = ∫\(\frac{d x}{\sin ^2 x}\)
= ∫cosec² x dx
= -cot x + C

(iv) ∫\(\frac{\sin x}{\cos ^2 x}\) dx
Solution:
∫\(\frac{\sin x}{\cos ^2 x}\) dx = ∫\(\frac{\sin x}{\cos x} \cdot \frac{1}{\cos x}\) dx
= ∫sec x . tan x dx = sec x + C

(v) ∫\(\frac{2 \cos x}{1-\cos ^2 x}\) dx
Solution:

(vi) ∫\(\frac{1-\sin ^3 x}{\sin ^2 x}\) dx
Solution:
∫\(\frac{1-\sin ^3 x}{\sin ^2 x}\) dx = ∫(cosec² x – sin x) dx
= -cot x + cos x + C

(vii) ∫\(\frac{\sin ^2 x}{1+\cos x}\) dx
Solution:

(viii) ∫\(\frac{\cos 2 x}{\cos x+\sin x}\) dx
Solution:

(ix) ∫\(\frac{\cos ^4 x-\sin ^4 x}{\cos x-\sin x}\) dx
Solution:

(x) ∫\(\frac{\cos 2 x}{\sin ^2 x \cdot \cos ^2 x}\) dx
Solution:

(xi) ∫\(\frac{a^2 \sin ^2 x+b^2 \cos ^2 x}{\sin ^2 2 x}\) dx
Solution:

(xii) ∫(tan x + cot x)² dx
Solution:

(xiii) ∫\(\frac{1-\cos 2 x}{1+\cos 2 x}\) dx
Solution:

(xiv) ∫sec² x . cosec² x dx
Solution:

(xv) ∫\(\frac{\sin ^3 x+b \cos ^3 x}{\sin ^2 x \cos ^2 x}\) dx
Solution:

(xvi) ∫\(\sqrt{1+\sin 2 x}\) dx
Solution:

(xvii) ∫\(\sqrt{1-\cos 2 x}\) dx
Solution:

(xviii) ∫\(\sqrt{1+\cos 2 x}\) dx
Solution:
∫\(\sqrt{1+\cos 2 x}\) dx = ∫√2 cos x dx
= √2 sin x + C

(xix) ∫\(\frac{\cos 3 x \cos 2 x+\sin 3 x \sin 2 x}{1-\cos ^2 x}\) dx
Solution:

(xx) ∫(a cot x + b tan x)² dx
Solution:
∫(a cot x + b tan x)² dx
= ∫{a² cot² x + b² tan² x + 2ab cot x – tanx} xdx
= ∫(a² cot² x + b² tan² x + 2ab) dx
= a² ∫cot² x dx + b² ∫tan² x dx + 2ab ∫dx
= a² ∫(cosec² x – 1) dx + b² ∫(sec² x – 1) dx + 2ab x + C
= a² (-cot x – x) + b² (tan x – x) + 2ab x + C
= b² tan x – a² cot x – x (a² + b² – 2ab) + C
= b² tan x – a² cot x – (a – b)² x + C
= b² tan x – a² cot x + C

Question 3.
(i) ∫(e^x + 2) dx
Solution:
∫(e^x +2) dx = ∫e^x dx + 2 ∫dx
= e^x + 2x + C

(ii) ∫3^x dx
Solution:
∫3^x dx = ∫\(\frac{3^x}{\ln 3}\) + C

(iii) ∫a^x+2 dx
Solution:
∫a^x+2 dx = ∫a^x a² dx
= a² . \(\frac{a^x}{\ln a}\) + C
= \(\frac{a^{x+2}}{\ln a}\) + C

(iv) ∫a^3x dx
Solution:
∫a^3x dx = \(\frac{1}{3} \frac{a^{3 x}}{\ln a}\) + C

(v) ∫\(\frac{e^{2 x}+1}{e^x}\) dx
Solution:
∫\(\frac{e^{2 x}+1}{e^x}\) dx = ∫(e^x + e^-x) dx
= e^x – e^-x + C

Question 4.
(i) ∫\(\left(\frac{5}{\sqrt{1-x^2}}+\frac{7}{1+x^2}\right)\) dx
Solution:

(ii) ∫\(\frac{3 x^2}{x^2+1}\) dx
Solution:

(iii) ∫\(\frac{x^6}{x^2+1}\) dx
Solution:

(iv) ∫\(\frac{x^4+x^3+x^2+x+2}{x^2+1}\) dx
Solution:

(v) ∫\(\left(\sqrt{1-x^2}+\frac{x^2}{\sqrt{1-x^2}}\right)\) dx
Solution:

(vi) ∫\(\frac{x^2+\sqrt{x^2+1}}{x^3 \sqrt{x^2-1}}\) dx
Solution:

Question 5.
Find the unique antiderivative F(x) of f(x) = 2x² + 1, whese F(o) = -2.
Solution:
F(x) = ∫f(x) dx = ∫(2x² + 1) dx
= \(\frac{2}{3}\)x³ + x + C
But F(0) = -2
∴ -2 = C
Thus F(x) = \(\frac{2}{3}\)x³ + x – 2

Odisha State Board CHSE Odisha Class 12 Math Solutions Chapter 8 Application of Derivatives Additional Exercise Textbook Exercise Questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 8 Application of Derivatives Additional Exercise

(A) Multiple Choice Questions (Mcqs) With Answers

Question 1.
Write the maximum value of the function y = x⁵ in the interval [1, 5].
(a) 125
(b) 3125
(c) 625
(d) 225
Solution:
(b) 3125

Question 2.
Differentiate sin^-1 (cos x) with respect to x.
(a) -1
(b) 0
(c) 1
(d) None of the above
Solution:
(a) -1

Question 3.
Write the equation of the tangent to the curve y = |x| at the point (-2, 2).
(a) 2x + 2y = 0
(b) 2x + y = 0
(c) x + y = 1
(d) x + y = 0
Solution:
(d) x + y = 0

Question 4.
Write the value of \(\frac{d}{d x}\)(sin x)
(a) tan x
(b) sin x
(c) cos x
(d) cot x
Solution:
(c) cos x

Question 5.
Write the value of \(\frac{d}{d x}\)(cos x)
(a) cos x
(b) -sin x
(c) sin x
(d) cot x
Solution:
(b) -sin x

Question 6.
Write the value of \(\frac{d}{d x}\)(tan x)
(a) tan x
(b) sec² x
(c) sin x
(d) cot x
Solution:
(c) sin x

Question 7.
Write the value of \(\frac{d}{d x}\)(sec x)
(a) sin x cos x
(b) sin x tan x
(c) cot x tan x
(d) sec x tan x
Solution:
(d) sec x tan x

Question 8.
Write the value of \(\frac{d}{d x}\)(cot x) = -cosec²x
(a) -sin² x
(b) -cos² x
(c) -cosec² x
(d) -tan² x
Solution:
(c) -cosec² x

Question 9.
Write the value of \(\frac{d}{d x}\)(cosec x)
(a) -cosec x cot x
(b) -sin x cot x
(c) -tan x cot x
(d) -cos x cot x
Solution:
(a) -cosec x cot x

Question 10.
Write the value of \(\frac{d}{d x}\)(e^x) = ex
(a) e
(b) x
(c) e^x
(d) e^2x
Solution:
(c) e^x

Question 11.
Write the value of \(\frac{d}{d x}\)(a^x)
(a) a^2x log a
(b) a^x log a
(c) log a
(d) log a^x
Solution:
(b) a^x log a

Question 12.
Write the value of \(\frac{d}{d x}\)(log x)
(a) x
(b) \(\frac{1}{2 x}\)
(c) \(\frac{1}{x^2}\)
(d) \(\frac{1}{x}\)
Solution:
(d) \(\frac{1}{x}\)

Question 13.
Write the value of \(\frac{d}{d x}\)(log_a x)
(a) \(\frac{x}{\log a}\)
(b) \(\frac{1}{x \log a}\)
(c) \(\frac{1}{x^2 \log a}\)
(d) \(\frac{1}{2 x^2 \log a}\)
Solution:
(b) \(\frac{1}{x \log a}\)

Question 14.
Write the value of \(\frac{d}{d x}\)(cot^-1 x) = \(\frac{-1}{1+x^2}\)
(a) \(\frac{1}{1-x^2}\)
(b) \(\frac{1}{1+x^2}\)
(c) \(\frac{-1}{1-x^2}\)
(d) \(\frac{-1}{1+x^2}\)
Solution:
(d) \(\frac{-1}{1+x^2}\)

Question 15.
Find \(\frac{\mathrm{dy}}{\mathrm{dx}}\) for √x + √y = √c.
(a) \(\frac{\sqrt{y}}{\sqrt{x}}\)
(b) \(\frac{\sqrt{x}}{\sqrt{y}}\)
(c) –\(\frac{\sqrt{y}}{\sqrt{x}}\)
(d) –\(\frac{\sqrt{x}}{\sqrt{y}}\)
Solution:
(c) –\(\frac{\sqrt{y}}{\sqrt{x}}\)

Question 16.
If f(x) = \(\log _{x^2}\) (log x), then find f'(e).
(a) \(\frac{1}{e}\)
(b) \(\frac{1}{\mathrm{e}^2}\)
(c) \(\frac{1}{2 \mathrm{e}}\)
(d) \(\frac{2}{\mathrm{e}}\)
Solution:
(c) \(\frac{1}{2 \mathrm{e}}\)

Question 17.
If y = \(\sqrt{\sin x+y}\), then what is \(\frac{\mathrm{dy}}{\mathrm{dx}}\)?
(a) \(\frac{\cos x}{2 y+1}\)
(b) \(\frac{\cos x}{2 y-1}\)
(c) \(\frac{\sin x}{2 y-1}\)
(d) \(\frac{\sin x}{2 y+1}\)
Solution:
(b) \(\frac{\cos x}{2 y-1}\)

Question 18.
If y = 10^x and z = 100^x/2 then find \(\frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{\mathrm{y}^2}{\mathrm{z}}\right)\)
(a) 10^x log 10
(b) 10^2x log 10
(c) x log 10
(d) x² log 10
Solution:
(a) 10^x log 10

Question 19.
Find the differential of y = sin² x.
(a) 2sinx cot x dx
(b) 2sinx tan x dx
(c) 2sinx cosx dx
(d) 2cosx sinx dx
Solution:
(c) 2sinx cosx dx

Question 20.
Differentiate \(\sqrt{\sin \sqrt{x}}\)
(a) \(\frac{\cos \sqrt{x}}{4 \sqrt{x \sin \sqrt{x}}}\)
(b) \(\frac{\sin x}{4 \sqrt{x \cos \sqrt{x}}}\)
(c) \(\frac{\sin \sqrt{x}}{4 \sqrt{x \cos \sqrt{x}}}\)
(d) \(\frac{\sin x}{4 \sqrt{x \cos x}}\)
Solution:
(a) \(\frac{\cos \sqrt{x}}{4 \sqrt{x \sin \sqrt{x}}}\)

Question 21.
If f(x) = [tan² x], what is f'(0)?
(a) 0
(b) 1
(c) -1
(d) None of the above
Solution:
(a) 0

Question 22.
If xy = e^x-y then find \(\frac{\mathrm{dy}}{\mathrm{dx}}\).
(a) \(\frac{\log x}{(1-\log x)^2}\)
(b) \(\frac{\log x}{(1+\log x)^2}\)
(c) \(\frac{\log x}{(1+\log x)}\)
(d) \(\frac{\log x}{(1-\log x)}\)
Solution:
(b) \(\frac{\log x}{(1+\log x)^2}\)

Question 23.
Find \(\frac{\mathrm{dy}}{\mathrm{dx}}\) if y = cot^-1\(\left(\frac{1+\cos x}{1-\cos x}\right)^{\frac{1}{2}}\)
(a) 0
(b) \(\frac{1}{2}\)
(c) 1
(d) \(\frac{1}{4}\)
Solution:
(b) \(\frac{1}{2}\)

Question 24.
Find the derivative of sin x° w.r.t. x.
(a) \(\frac{\pi}{90}\) cos x°
(b) π cos x°
(c) \(\frac{\pi}{180}\) cos x°
(d) π sin x°
Solution:
(c) \(\frac{\pi}{180}\) cos x°

Question 25.
The least value of a such that f(x) = x² + ax + 1 is strictly increasing on (1, 2) is
(a) -2
(b) -4
(c) 2
(d) 4
Solution:
(a) -2

Question 26.
If y = sec^-1\(\left(\frac{x+1}{x-1}\right)\) + sin^-1\(\left(\frac{x-1}{x+1}\right)\) then find \(\frac{\mathrm{dy}}{\mathrm{dx}}\).
(a) 0
(b) -1
(c) 1
(d) None of the above
Solution:
(a) 0

Question 27.
Let f(x) = e^x g(x), g(0) = 2 and g'(0) = 1, then find f'(0).
(a) 1
(b) 2
(c) 3
(d) 0
Solution:
(c) 3

Question 28.
If x ∈ (0, π/2) then find \(\frac{\mathrm{d}}{\mathrm{dx}}\)[sin x]
(a) 0
(b) 1
(c) 7C
(d) \(\frac{\pi}{2}\)
Solution:
(a) 0

Question 29.
If f(x) = [x²] then f’\(\left(\frac{3}{2}\right)\) = ________.
(a) 1
(b) \(\frac{3}{2}\)
(c) 0
(d) –\(\frac{3}{2}\)
Solution:
(c) 0

Question 30.
A differentiable function f defined for all x > 0 satisfies f(x²) = x³ for all x > 0. What is f'(b)?
(a) 3
(b) 4
(c) 6
(d) 12
Solution:
(c) 6

Question 31.
What is the derivative of f'(In x) w.r.t x where f(x) = ln x.
(a) \(\frac{1}{x^2 \ln x}\)
(b) \(\frac{1}{2 x \ln x}\)
(c) \(\frac{1}{x \ln x^2}\)
(d) \(\frac{1}{x \ln x}\)
Solution:
(d) \(\frac{1}{x \ln x}\)

Question 32.
For what value of a, f(x) = log_a (x) is increasing on R?
(a) a > 1
(c) a < 1
(b) a = 1
(d) a > 1
Solution:
(a) a ≥ 1

Question 33.
Write the points at which tangent to the curve y = x² – 3x is parallel to x – axis.
(a) \(\left(\frac{3}{2}, \frac{9}{4}\right)\)
(b) \(\left(\frac{3}{2}, \frac{-9}{4}\right)\)
(c) \(\left(\frac{-3}{2}, \frac{9}{4}\right)\)
(d) \(\left(\frac{-3}{2}, \frac{-9}{4}\right)\)
Solution:
(b) \(\left(\frac{3}{2}, \frac{-9}{4}\right)\)

Question 34.
If y = e^x + e^-x + 2 has a tangent parallel to x – axis at (α, β) then find the value of α.
(a) 0
(b) 2
(c) 1
(d) -1
Solution:
(a) 0

Question 35.
Write slope of the tangent to the curve y = √3 sin x + cos x at (\(\frac{\pi}{3}\), 2)
(a) 1
(b) -1
(c) 0
(d) None of the above
Solution:
(c) 0

Question 36.
Find the value of x for which f(x) is either a local maximum or a local minimum when
f(x) = x³ – 3x² – 9x + 6.
(a) (3, -1)
(b) (3, 1)
(c) (-1, 3)
(d) (1, -3)
Solution:
(a) (3, -1)

Question 37.
Find the x – coordinates of the extreme points of the function.
y = cos x + sin x , x ∈ [0, \(\frac{\pi}{2}\)]
(a) \(\frac{\pi}{4}\)
(b) \(\frac{2 \pi}{3}\)
(c) \(\frac{\pi}{2}\)
(d) π
Solution:
(a) \(\frac{\pi}{4}\)

Question 38.
Find the equation of tangent to the curve x = y² – 2 at the points where slope of the normal equal to (-2).
(a) x + y – 1 = 0
(b) 2x + y – 1 = 0
(c) 2x – y + 1 = 0
(d) x – y + 1 = 0
Solution:
(b) 2x + y – 1 = 0

Question 39.
f(x) = x⁴ – 62x² + ax + 9 attains its maximum value at x = 1 in the interval [0, 2]. Find the value of ‘a’.
(a) 12
(b) 120
(c) 1
(d) 1200
Solution:
(b) 120

Question 40.
If sin (x + y) = log (x + y) then \(\frac{d y}{d x}\) is:
(a) 2
(b) -2
(c) 1
(d) -1
Solution:
(d) -1

Question 41.
If f(x) = 2x and g(x) = \(\frac{x^2+1}{2}\) then which of the following can be a discontinuous function?
(a) F(x) + g(x)
(b) f(x) – g(x)
(c) f(x) . g(x)
(d) \(\frac{f(x)}{g(x)}\)
Solution:
(d) \(\frac{f(x)}{g(x)}\)

Question 42.
If f(x) = x² sin (‘) where x ≠ 0 then the value of the function f at x = 0, so that the function is continuous at x = 0 is
(a) 0
(b) -1
(c) 1
(d) none of these
Solution:
(a) 0

Question 43.
The derivative of cos^-1 (2x² – 1) with respect to cos^-1 x
(a) 2
(b) \(\frac{-1}{2 \sqrt{1-x^2}}\)
(c) \(\frac{2}{x}\)
(d) 1 – x²
Solution:
(a) 2

Question 44.
If y = \(\sqrt{\sin x+y}\) then \(\frac{d y}{d x}\) is equal to:
(a) \(\frac{\cos x}{2 y-1}\)
(b) \(\frac{\cos x}{1-2 y}\)
(c) \(\frac{\sin x}{1-2 y}\)
(d) \(\frac{\sin x}{2 y-1}\)
Solution:
(a) \(\frac{\cos x}{2 y-1}\)

Question 45.
Find the value of \(\frac{d y}{d x}\) at θ = \(\frac{\pi}{3}\) if x = asec³ θ and x = atan³ θ is:
(a) \(\frac{\sqrt{3}}{2}\)
(b) –\(\frac{\sqrt{3}}{2}\)
(c) \(\frac{1}{2}\)
(d) 1
Solution:
(a) \(\frac{\sqrt{3}}{2}\)

Question 46.
If x²y = e^x-y then \(\frac{d y}{d x}\) is:
(a) \(\frac{1+x}{1+\log x}\)
(b) \(\frac{1-x}{1+\log x}\)
(c) \(\frac{x}{1+\log x}\)
(d) \(\frac{x}{(1+\log x)^2}\)
Solution:
(d) \(\frac{x}{(1+\log x)^2}\)

Question 47.
Differential coefficient of sec (tan^-1 x) is:
(a) \(\frac{x}{1+x^2}\)
(b) \(x \sqrt{1+x^2}\)
(c) \(\frac{x}{\sqrt{1+x^2}}\)
(d) \(\frac{1}{\sqrt{1+x^2}}\)
Solution:
(c) \(\frac{x}{\sqrt{1+x^2}}\)

Question 48.
The function f(x) = x^x is decreasing in the interval:
(a) (0, e)
(b) (0, \(\frac{1}{e}\))
(c) (0, 1)
(d) none of these
Solution:
(b) (0, \(\frac{1}{e}\))

(B) Very Short Type Questions With Answers

Question 1.
If f'(2⁺) = 0 and f'(2^–) = 0, then is f(x) continuous at x = 2?
Solution:
f'(2⁺) = 0 and f'(2^–) = 0
⇒ f is differentiable at x = 2
⇒ f is continuous at x = 2.

Question 2.
If Φ(x) = f(x) + f(1 – x), f'(x) = 0 for 0 < x < 1, then is x = a point of maxima or minima of Φ(x)?
Solution:
Let Φ(x) = f(x) + f(1 – x)
⇒ Φ’ = f'(x) – f'(1 – x)
Φ”(x) = f”(x) + f”(1 – x)
Now Φ'(\(\frac{1}{2}\)) = f'(\(\frac{1}{2}\)) – f'(\(\frac{1}{2}\)) = 0
⇒ \(\frac{1}{2}\) is a critical point.
Φ”(\(\frac{1}{2}\)) = f”(\(\frac{1}{2}\)) – f”(\(\frac{1}{2}\)) = 0
⇒ x = \(\frac{1}{2}\) is neither a point of local maxima nor a point of local minima.

Question 3.
Write the interval in which the function f(x) = sin^-1 (2 – x) is differentiable.
Solution:
f(x) = sin^-1 (2 – x) is differentiable for
1 – (2 – x)² > 0.
⇒ (2 – x)² < 1
⇒ -1 < 2 – x < 1
⇒ -3 < -x < -1
⇒ 3 > x < 1
⇒ 1 < x < 3
⇒ x ∈ (1, 3)
∴ f is differentiable on (1, 3)

Question 4.
Write the set of values of x for which the function f(x) = sin x – x is increasing.
Solution:
f(x) = sin x – x
⇒ f'(x) = cos x – 1 ≤ 0 for all x ∈ R
∴ f(x) is increasing for all x ∈ R

Question 5.
Write the differential coefficient of e^{[x] ln (x + 1)} where 3 ≤ x < 4 with respect to x.
Solution:
Let y = e
= e^3ln(x+1) (∵ 3 ≤ x < 4)
= \(e^{\ln (x+1)^3}\) = (x + 1)³
\(\frac{d y}{d x}\) = 3(x + 1)²

Question 6.
Write a logarithmic function which is differentiable only in the open interval (-1, 1).
Solution:
f(x) = \(\left\{\begin{array}{l}
\ln \left|\sin ^{-1} x\right|, x \neq 0 \\
1, x=0
\end{array}\right.\) is differentiable only on (-1, 1).

Question 7.
Write a function which has both relative and absolute maximum at the point (1, 2).
Solution:
f(x) = -x² + 2x + 1 on [0, 1] has relative and absolute maximum at (1, 2).

Question 8.
Write the derivation of e^3logx with respect to x².
Solution:
Let u = e^3lnx = e^lnx = \(e^{\ln x^3}\)= x³ and v = x²
∴ \(\frac{d u}{d x}\) = 3x² and \(\frac{d v}{d x}\) = 2x
∴ Derivative of u w.r.t. v is
\(\frac{d u}{d v}\) = \(\frac{3 x^2}{2 x}\) = \(\frac{3 x}{2}\)

Question 9.
Write the maximum value of the function y = x⁵ in the interval [1, 5]:
Solution:
y = x⁵
⇒ \(\frac{d y}{d x}\) = 5x⁴ > 0 ∀ x ∈ [1, 5]
i.e y is strictly increasing in the given interval.
Thus the maximum value of
y = x⁵ = 5⁵ = 3125

Question 10.
Differentiate a^lnx with respect to x
Solution:
\(\frac{d}{d x}\)(a^{ln x}) = a^{ln x} (ln a)\(\frac{d}{d x}\)ln x
= \(\frac{a^{\ln x} \ln a}{x}\)

Question 11.
Mention the values of x for which x the function f(x) = x³ – 12x is increasing.
Solution:
Given f(x) = x³ – 12x
f is increasing if f'(x) > 0
⇒ 3x² – 12 > 0
⇒ x² > 4
⇒ -2 > x > 2
⇒ x ∈ (-∞, -2) ∪ (2, ∞)

Question 12.
Differentiate sin^-1 (cos x) with respect to x.
Solution:
\(\frac{d}{d x}\)sin^-1 (cos x) = \(\frac{d}{d x}\)sin^-1 sin (\(\frac{\pi}{2}\) – x)
= \(\frac{d}{d x}\) (\(\frac{\pi}{2}\) – x) = 1

Question 13.
Write the equation of the tangent to the curve y = |x| at the point (-2, 2).
Solution:
y = |x| = \(\left\{\begin{aligned}
x, & x \geq 0 \\
-x, & x<0
\end{aligned}\right.\)
When x < 0, \(\frac{d y}{d x}\) = -1
which is the slope of the tangent at (-2, 2).
Equation of the required tangent is
y – 2 = (-1) (x + 2)
⇒ y = -x
⇒ x + y = 0.

Question 14.
What is the derivative of sec^-1 x with respect to x if x < -1?
Solution:
Let y = sec^-1 x
⇒ \(\frac{d y}{d x}\) = \(\frac{1}{|x| \sqrt{x^2-1}}\)
when x < (-1) we have |x| = -x
∴ \(\left.\frac{\mathrm{dy}}{\mathrm{dx}}\right]_{\mathrm{x}<(-1)}\) = \(\frac{(-1)}{x \sqrt{x^2-1}}\)

Question 15.
Write the set of points where the function f(x) = x³ has relative (local) extrema.
Solution:
f(x) = x³
⇒ f'(x) = 3x²
f'(x) = 0
⇒ x = 0
But f”(x) = 6x = 0 when x = 0
∴ x = 0 is not a point of local extrema.
∴ Hence the set of points where f has relative extrema = Φ.

Question 16.
In which interval of x the function \(\frac{\ln \mathbf{x}}{x}\) is decreasing?
Solution:
Let f(x) = \(\frac{\ln x}{x}\)
⇒ f'(x) = \(\frac{x \cdot \frac{1}{x}-\ln x}{x^2}\) = \(\frac{1-\ln x}{x^2}\)
f is decreasing for
f'(x) < 0 ⇒ \(\frac{1-\ln x}{x^2}\) < 0
⇒ 1 – ln x < 0 (∵ x² > 0)
⇒ 1 < ln x
⇒ ln x > 1
⇒ x e (e, ∞).

(C) Short Type Questions With Answers

Question 1.
If y = \(e^{x^{e^{e^{e^{x^x…..}}}}}\), then find \(\frac{d \mathbf{y}}{\mathbf{d x}}\).
Solution:

Question 2.
If \(\frac{\mathbf{d}^2 \mathbf{y}}{\mathbf{d x}^2}\), if x = a cosΦ and y = b sinΦ.
Solution:
x = a cosΦ, y = b sinΦ

Question 3.
Find the point on the curve x² + y² – 4xy + 2 = 0, where the normal to the curve is parallel to the x – axis.
Solution:
Given equation of the curve is
x² + y² – 4xy + 2 = 0 … (1)
Differentiang we get

Question 4.
Find the intervals in which the function y = \(\frac{\ln x}{x}\) is increasing and decreasing.
Solution:

Question 5.
Differentiate y = tan^-1\(\frac{\sqrt{1+x^2}+\sqrt{1-x^2}}{\sqrt{1+x^2}-\sqrt{1-x^2}}\)
Solution:

Question 6.
Differentiate y = (sin y)^{sin 2x}
Solution:
y = (sin y)^{sin 2x}
⇒ In y = sin 2x ln (sin y)

Question 7.
Test the differentiability and continuity of the following function at x = 0:
f(x) = \(\left\{\begin{array}{cc}
\frac{1-e^{-x}}{x} & x \neq 0 \\
1 & x=0
\end{array}\right.\)
Solution:
Given function is


As LHD = RHD, we have the given function is differentiable at x = 0 and f'(0) = –\(\frac{1}{2}\).

Question 8.
Show that the sum of intercepts on the co-ordinate axes of any tangent to the curve
√x + √y = √a is constant.
Solution:
Given curve √x + √y = √a … (1)

Question 9.
Find the equation of the normal to the curve y = (log x)² at x = \(\frac{1}{e}\).
Solution:
Given equation of the curve is
y = (log x)²

Question 10.
If y = x + \(\frac{1}{x+\frac{1}{x+\cdots \cdots \infty}}\), find \(\frac{\mathbf{d y}}{\mathbf{d x}}\), the rhs being a valid expression.
Solution:

Question 11.
Differentiate sec^-1 (\(\frac{1}{2 x^2-1}\)) with respect to \(\sqrt{1-x^2}\).
Solution:

Question 12.
Find \(\frac{d \mathbf{y}}{\mathbf{d t}}\) , when y = sin^-1 \(\left(\frac{2 \sqrt{t}-1}{t^2}\right)\)
Solution:

Question 13.
Find \(\frac{\mathbf{d y}}{\mathbf{d x}}\) , if x^myⁿ = \(\left(\frac{\mathbf{x}}{\mathbf{y}}\right)^{\mathrm{m}+\mathrm{n}}\)
Solution:

Question 14.
If x = a sec θ, y = b tan θ, prove that:
\(\frac{d^2 y}{d x^2}\) = –\(\frac{b^4}{a^2 y^3}\)
Solution:

Question 15.
Find the interval where the following function is increasing:
y = sin x + cos x, x ∈ [0, 2π].
Solution:
y = sin x + cos x, x ∈ [0, 2π].

Question 16.
Find \(\frac{d y}{d x}\), when y^x = x^{sin y}.
Solution:
y^x = x^{sin y}
Taking log of both sides we get,
x ln y = (sin y) ln x
Differentiating with respect to x we get,

Question 17.
Show that \(\frac{d y}{d x}\) is independent of t,
if cos x = \(\sqrt{\frac{1}{1+t^2}}\), sin y = \(\frac{2 t}{1+t^2}\)
Solution:
cos x = \(\sqrt{\frac{1}{1+t^2}}\), sin y = \(\frac{2 t}{1+t^2}\)
Let t = tan θ
∴ cos x = cos θ, sin y = sin 2θ
⇒ y = 2θ, x = θ
⇒ y = 2x
⇒ \(\frac{d y}{d x}\) = 2 which is independent of t.

Question 18.
Show that 2sin x + tan x > 3x for all x ∈ (0, \(\frac{\pi}{2}\))
Solution:
Let f(x) = 2sin x + tan x – 3x
⇒ f'(x) = 2cos x + sec² x – 3
Let g(x) = f'(x) = 2cos x + sec² x – 3
⇒ g'(x) = -2sin x + 2sec² x tan x
= 2sin x (sec³ x – 1) ≥ 0
for all x ∈ (0, \(\frac{\pi}{2}\))
∴ g is an increasing function on (0, \(\frac{\pi}{2}\))
But g(0) = 0
∴ g(x) ≥ 0 for all x ∈ (0, \(\frac{\pi}{2}\))
⇒ f(x) ≥ 0 for all x ∈ (0, \(\frac{\pi}{2}\))
⇒ f is an increasing function on (0, \(\frac{\pi}{2}\))
But f(0) = 0
∴ f(x) ≥ 0 for all x ∈ (0, \(\frac{\pi}{2}\))
⇒ 2sin x + tan x ≥ 3x for x ∈ (0, \(\frac{\pi}{2}\))

Question 19.
Show that no two normals to a parabola are parallel.
Solution:
Let us consider the parabola y² = 4ax – (1) and A(at₁², 2at₁) and B (at₂², 2at₂) are any two points on its from (1)

As t₁ ≠ t₂ we have no two normals of a parabola are parallel.

Question 20.
Examine the differentiability of In x² for all real values of x.
Solution:
Let f(x) = In (x²) = 2 ln |x|
Clearly Dom f = R – {0}
Let any a ∈ Dom f.

Question 21.
Find the derivative of x^{sin x} with respect to x.
Solution:
Let y = x^{sin x} = e^{(sin x)/nx}

Question 22.
Differentiate sin^-1 \(\left(\frac{2 x}{1+x^2}\right)\) with respect to cos^-1 \(\left(\frac{1-x^2}{1+x^2}\right)\).
Solution:

Question 23.
Find the slope of the tangent to the curve x = 2(t – sin t), y = 2(1 – cos t) at t = \(\frac{\pi}{4}\).
Solution:

Question 24.
If cos y = x cos(a + y), then show that \(\frac{d y}{d x}\) = \(\frac{\cos ^2(a+y)}{\sin a}\).
Solution:

Question 25.
Find the extreme values of the function y = x + \(\frac{y}{x}\).
Solution:

Question 26.
Find the intervals where the following function is (i) increasing, (ii) decreasing:
f(x) = \(\begin{cases}\mathbf{x}^2+1, & x \leq-3 \\ x^3-8 x+13, & x>-3\end{cases}\)
Solution:
Given f(x) = \(\begin{cases}\mathbf{x}^2+1, & x \leq-3 \\ x^3-8 x+13, & x>-3\end{cases}\)
Clearly f is not differentiable at x = -3.
and f ‘(x) = \(\begin{cases}2 \mathrm{x}, & \mathrm{x}>-3 \\ 3 \mathrm{x}^2-8 & \mathrm{x}<-3 \\ \text { does not exist } & \text { for } \mathrm{x}=-3\end{cases}\)
Case – 1: x > -3
Clearly f'(x) > 0 for x > 0 and f'(x) < 0 for -3 < x < 0
∴ f is increasing in (0, ∞) and decreasing in (-3, 0).
Case – 2: x < -3
Clearly for x < -3, f(x) = 3x² – 8 > 0
i.e., f is increasing in (-∞, -3)
Thus f is increasing in (-∞, -3) ∪ (0, ∞) and decreasing in (-3, 0).

Question 27.
Prove that:
y = In tan \(\left(\frac{\pi}{4}+\frac{x}{2}\right)\) ⇒ \(\frac{d y}{d x}\) = sec x
Solution:

Question 28.
Differentiate with respect to x:
y = \(2^{x^2}\) + tan^-1 \(\left(\frac{\cos x-\sin x}{\cos x+\sin x}\right)\)
Solution:

Question 29.
Find the equation of the tangent to the curve x = y² – 1 at the point where the slope of the normal to the curve is (-2).
Solution:
Given equation of the curve is x = y² – 1.

Question 30.
Find \(\frac{d y}{d x}\) if y = \(\log _{\left(x^2\right)}\)3.
Solution:

Question 31.
Write why the function sin^-1 \(\frac{1}{\sqrt{1-x^2}}\) cannot be differentiated anywhere.
Solution:

⇒ 1 – x² ≥ 1
⇒ x = 0
∴ DOM of {0 3}
But f is not defined in a deleted interval of x = 0.
Hence f is not differentiable at x = 0 and hence not differentiable anywhere.

Odisha State Board Elements of Mathematics Class 12 CHSE Odisha Solutions Chapter 7 Continuity and Differentiability Ex 7(c) Textbook Exercise Questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 7 Continuity and Differentiability Exercise 7(c)

Find derivatives of the following functions.
Question 1.
(x² +5)⁸
Solution:
y = (x² +5)⁸
\(\frac{d y}{d x}\) = \(\frac{d}{d x}\)(x² +5)⁸
= 8(x² +5)⁷ × \(\frac{d}{d x}\)(x² +5) by chain rule
= 8(x² +5)⁷ . 2x
= 16x (x² +5)⁷

Question 2.
\(\frac{1}{\left(x^3+\sin x\right)^2}\)
Solution:

Question 3.
In (√x+1)
Solution:

Question 4.
sin 5x + cos 7x
Solution:
sin 5x + cos 7x
\(\frac{d y}{d x}\) = \(\frac{d}{d x}\)(sin 5x) + \(\frac{d}{d x}\)(cos 7x)
= cos 5x . \(\frac{d}{d x}\)(5x) – sin 7x . \(\frac{d}{d x}\)(7x)
= 5 cos 5x – 7 sin 7x

Question 5.
e^{sin t}
Solution:
y = e^{sin t}
\(\frac{d y}{d x}\) = \(\frac{d}{d t}\)(e^{sin t}) = e^{sin t} . \(\frac{d}{d t}\)(sin t)
= e^{sin t} . cos t

Question 6.
\(\sqrt{a x^2+b x+c}\)
Solution:
y = \(\sqrt{a x^2+b x+c}\)

Question 7.
\(\left(\frac{x+1}{x^2+3}\right)^{-3}\)
Solution:

Question 8.
sec (tan θ)
Solution:
y = sec (tan θ)
\(\frac{d y}{d θ}\) = \(\frac{d}{d θ}\) {sec (tan θ)}
= sec (tan θ) . tan (tan θ) . \(\frac{d}{d θ}\)(tan θ)
= sec (tan θ) . tan (tan θ) . sec² θ

Question 9.
sin \(\left(\frac{1-x^2}{1+x^2}\right)\)
Solution:

Question 10.
\(\sqrt{\tan (3 z)}\)
Solution:

Question 11.
tan³ x
Solution:
y = tan³x = (tan x)³
\(\frac{d y}{d x}\) = 3(tan x)². \(\frac{d}{d x}\)(tan x)
= 3tan² x . sec² x

Question 12.
sin⁴ x
Solution:
y = sin⁴x
\(\frac{d y}{d x}\) = \(\frac{d}{d x}\)(sin⁴ x)
= 4 (sin x)³ . \(\frac{d}{d x}\)(sin x)
= 4 sin³ x . cos x

Question 13.
sin² x cos² x
Solution:
y = sin² x cos² x = \(\frac{1}{4}\)sin² 2x
\(\frac{d y}{d x}\) = \(\frac{1}{4}\) \(\frac{d}{d x}\)(sin 2x)²
= \(\frac{1}{4}\). 2sin 2x . \(\frac{d}{d x}\)(sin 2x)
= \(\frac{1}{2}\) sin 2x . cos 2x . \(\frac{d}{d x}\)(2x)
= \(\frac{1}{2}\) sin 2x cos 2x . 2 = sin 2x . cos 2x

Question 14.
sin 5x cos 7x
Solution:
y = sin 5x cos 7x
\(\frac{d y}{d x}\) = \(\frac{d}{d x}\)(sin 5x) . cos 7x + \(\frac{d}{d x}\)sin 5x . (cos 7x)
= cos 5x . \(\frac{d}{d x}\)(5x) . cos 7x + sin 5x . (-sin 7x) . \(\frac{d}{d x}\)(7x)
= 5 cos 5x . cos 7x – 7 sin 5x . sin 7x

Question 15.
tan x cot 2x
Solution:
y = tan x. cot 2x
\(\frac{d y}{d x}\) = \(\frac{d}{d x}\)(tan x) . cot 2x + tan x . \(\frac{d}{d x}\)(cot 2x)
= sec² x / cot 2x + tan x . (-cosec² x) \(\frac{d}{d x}\)(2x)
= sec² x . cot 2x – 2 tan x . cosec² x

Question 16.
\(\sqrt{\sin \sqrt{x}}\)
Solution:

Question 17.
\(\sqrt{\sec (2 x+1)}\)
Solution:

Question 18.
cosec (ax + b)²
Solution:
y = cosec (ax + b)²
\(\frac{d y}{d x}\) = – cosec (ax + b)² cot (ax + b)² . \(\frac{d}{d x}\)(ax + b)²
[ ∵ \(\frac{d}{d x}\)(cosec u) = -cosec u . cot u . \(\frac{d u}{d x}\)
= – cosec (ax + b)² . cot (ax + b)² . 2(ax + b) . \(\frac{d}{d x}\)(ax + b)
[ ∵ \(\frac{d}{d x}\)(u²) = 2u \(\frac{d u}{d x}\)
= – cosec (ax + b)² . cot (ax + b)² . 2(ax + b) . a
= – 2a (ax + b) cosec (ax + b)² cot (ax + b)²

Question 19.
a^{In x}
Solution:
y = a^{In x} . In a . \(\frac{d}{d x}\)(In x)
[ ∵ \(\frac{d}{d x}\)(a^u) = a^u . In a . \(\frac{d u}{d x}\)
= a^{In x} . In a . \(\frac{1}{x}\) = \(\frac{a^{\ln x} \ln a}{x}\)

Question 20.
\(a^{x^2} b^{x^3}\)
Solution:

Question 21.
In tan x
Solution:

Question 22.
\(5^{\sin x^2}\)
Solution:

Question 23.
In tan\(\left(\frac{\pi}{4}+\frac{x}{2}\right)\)
Solution:

Question 24.
\(\sqrt{\left(a^{\sqrt{x}}\right)}\)
Solution:

Question 25.
In (e^nx + e^-nx)
Solution:
y = In (e^nx + e^-nx)
\(\frac{d y}{d x}\) = \(\frac{1}{e^{n x}+e^{-n x}}\) . \(\frac{d}{d x}\) (e^nx + e^-nx)
= \(\frac{n\left(e^{n x}-e^{-n x}\right)}{e^{n x}+e^{-n x}}\)

Question 26.
\(e^{\sqrt{a x}}\)
Solution:

Question 27.
\(\sqrt{\log x}\)
Solution:
y = \(\sqrt{\log x}\)
\(\frac{d y}{d x}\) = \(\frac{1}{2 \sqrt{\log x}}\) . \(\frac{d}{d x}\)(log x)
= \(\frac{1}{2 \sqrt{\log x}}\) . \(\frac{1}{x}\)

Question 28.
e^{sin x} – a^{cos x}
Solution:
y = e^{sin x} – a^{cos x}
\(\frac{d y}{d x}\) = \(\frac{d}{d x}\)(e^{sin x}) – \(\frac{d}{d x}\)(a^{cos x})
= e^{sin x} . \(\frac{d}{d x}\)(sin x) – a^{cos x} . In a . \(\frac{d}{d x}\)(cos x)
= e^{sin x} . cos x + a^{cos x} . In a . sin x

Question 29.
\(\frac{e^{3 x^2}}{\ln \sin x}\)
Solution:

Question 30.
Prove that
\(\frac{d}{d x}\left[\frac{1-\tan x}{1+\tan x}\right]^{\frac{1}{2}}\) = 1 / \(\sqrt{\cos 2 x}\) (cos x + sin x)
Solution:

Odisha State Board CHSE Odisha Class 12 Math Solutions Chapter 8 Application of Derivatives Ex 8(c) Textbook Exercise Questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 8 Application of Derivatives Exercise 8(c)

Question 1.
Find the intervals where the following functions are (a) increasing and (b) decreasing.
(i) y = sin x, x ∈ [0, 2π]
Solution:

(ii) y = In x, x ∈ R₊
Solution:
⇒ \(\frac{d y}{d x}\) = \(\frac{1}{x}\) > 0
for all x ∈ R₊.
Thus y = In x is increasing in (0, ∞) and decreasing nowhere.

(iii) y = a^x, a > 0, x ∈ R
Solution:
⇒ \(\frac{d y}{d x}\) = a^x In a > 0 for all x ∈ R provided a > 1.
The function is increasing in (-∞, ∞) is a > 1.
Again \(\frac{d y}{d x}\) = a^x In a < 0 if 0 < a < 1.
Thus the function is decreasing in (-∞, ∞) if (0 < a < 1)

(iv) y = sin x + cos x, x ∈ [0, 2π]
Solution:

(v) y = 2x³ + 3x² – 36x – 7
Solution:
\(\frac{d y}{d x}\) = 6x² – 6x – 36
y is increasing if \(\frac{d y}{d x}\) > 0
⇒ x² + x – 6>0
⇒ x² + 3x – 2x – 6>0
⇒ x (x + 3) -2 (x + 3) > 0
⇒ (x + 3) (x – 2) > 0
for x > 2 or x < -3 and (x + 3 ) (x – 2) < 0 for -3 < x < 2
∴ The function is increasing in (-∞, -3) ∪ (2, ∞) and is decreasing in (-3, 2).

(vi) y = \(\frac{1}{x-1^{\prime}}\) x ≠ 1
Solution:
\(\frac{d y}{d x}\) = \(\frac{-1}{(x-1)^2}\)
The function is increasing if \(\frac{d y}{d x}\) > 0
⇒ \(\frac{-1}{(x-1)^2}\) > 0 which is impossible because \(\frac{-1}{(x-1)^2}\) is always -ve for all x ≠ 1. So the function is increasing nowhere. It is decreasing in R – {1}

(vii) y = \(\left\{\begin{array}{cc}
x^2+1, & x \leq-3 \\
x^3-8 x+13, & x>-3
\end{array}\right.\)
Solution:

(viii) y = 4x² + \(\frac{1}{x}\)
Solution:

(ix) y = (x – 1)² (x + 2)
Solution:
\(\frac{d y}{d x}\) = 2 (x – 1) (x + 2) + (x – 1)²
= (x – 1) (2x + 4 + x – 1)
= (x – 1 ) (3x + 3)
= 3 (x – 1) (x + 1 )
The function is increasing if \(\frac{d y}{d x}\) > 0.
⇒ (x + 1) (x – 1) > 0
⇒ x < -1 or x > 1
∴ The function is increasing in (-∞, -1) ∪ (1, ∞)
It is decreasing in (-1, 1).

(x) y = \(\frac{\ln x}{x}\), x > 0
Solution:

(xi) y = tan x – 4 (x – 2), x ∈ (-\(\frac{\pi}{2}\), \(\frac{\pi}{2}\))
Solution:

(xii) y = sin 2x – cos 2x, x ∈ [0, 2π]
Solution:

Question 2.
Give a rough sketch of the functions given in question 1.
Solution:
Do yourself.

Question 3.
Show that the function \(\frac{e^x}{x^p}\) is strictly increasing for x > p > 0.
Solution:

Question 4.
Show that 2 sin x + tan x ≥ 3x for all x ∈ (0, \(\frac{\pi}{2}\)).
Solution:
Let f(x) = 2 sin x + tan x – 3x
Then f(x) = 2 cos x + sec² x – 3

Odisha State Board CHSE Odisha Class 12 Math Solutions Chapter 6 Probability Ex 6(d) Textbook Exercise Questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 6 Probability Exercise 6(d)

Question 1.
State which of the following is the probability distribution of a random variable X with reasons to your answer:
(a)

X = x	0	1	2	3	4
p(x)	0.1	0.2	0.3	0.4	0.1

(b)

X = x	0	1	2	3
p(x)	0.15	0.35	0.25	0.2

(c)

X = x	0	1	2	3	4	5
p(x)	0.4	R	0.6	R2	0.7	0.3

Solution:
(a) As ∑p_i = 1.1 > 1, the given distribution is not a probability distribution.
(b) As ∑p_i = 0.95 < 1, the given distribution is not a probability distribution.
(c) As the value of R is not known, the given distribution is not a probability distribution.

Question 2.
Find the probability distribution of number of doublets in four throws of a pair of dice. Find also the mean and the variance of the number of doublets.
Solution:
Let the random variable X represents the number of doublets in 4 throws of a pair of dice.
X can take values 0, 1, 2, 3, 4
In a single throw of two dice
X can take values 0, 1, 2, 3, 4
In a single throw of two dice.
P(doublet) = \(\frac{6}{36}\)
P(non-doublet) = 1 – P (doublet) = \(\frac{5}{36}\)
Clearly the given experiment is a binomial experiment with n = 4,
p = \(\frac{1}{6}\), q = \(\frac{5}{6}\)

N.B. We can use the definition to find mean and variance.

Question 3.
Four cards are drawn successively with replacement from a well-shuffled pack of 52 cards. Find the probability distribution of the number of aces. Calculate the mean and variance of the number of aces.
Solution:
Let the random variable X denotes the number of aces.
Thus X can take values 0, 1, 2, 3, 4.
Clearly the given experiment is a binomial experiment with n = 4

N.B. We can use the definition to find mean and variance.

Question 4.
Find the probability distribution of
(a) number of heads in three tosses of a coin.
Solution:
Let the random variable X denotes the number of heads in three tosses of a coin.
X can take values 0, 1, 2, 3
In one toss p(H) = \(\frac{1}{2}\), p(T) = \(\frac{1}{2}\)
This experiment is a binomial experiment with n = 3, p = \(\frac{1}{2}\) and q = \(\frac{1}{2}\)

(b) number of heads in simultaneous tosses of four coins.
Solution:
When 4 coins tossed simultaneously, let the random variable X denotes the number of heads.
X can take values 0, 1. 2, 3, 4
This experiment is a binomial experiment with n = 4, p = \(\frac{1}{2}\) and q = \(\frac{1}{2}\)

Question 5.
A biased coin where the head is twice as likely to occur as the tail is, tossed thrice. Find the probability distribution of number of heads.
Solution:
Let in a toss P(T) = x
According to the question P(H) = 2x
Now 2x + x = 1
⇒ x = \(\frac{1}{3}\)
Thus P(T) = \(\frac{1}{3}\), P(H) = \(\frac{3}{3}\)
Clearly the given experiment is a binomial experiment with n = 3, p = P(H) = \(\frac{2}{3}\), q = P(T) = \(\frac{1}{3}\)
Let the random variable X denotes the number of heads in three throws of that coin.
X can take values 0, 1, 2, 3

Question 6.
Find the probability distribution of the number of aces in question no. 3 if the cards are drawn successively without replacement.
Solution:
Total number of cards = 52
Number of aces = 4
Number of cards drawn = 4 (one by one without replacement)
Let X = the random variable of number of aces drawn.
X can take values 0, 1, 2, 3 or 4

Question 7.
From a box containing 32 bulbs out of which 8 are defective 4 bulbs are drawn at random successively one after another with replacement. Find the probability distribution of the number of defective bulbs.
Solution:
Total number of bulbs = 32
Number of defective bulbs = 8
∴ Number of nondefective bulbs = 24
Number of bulbs drawn = 4 (one after another with replacement)
Clearly the experiment is a binomial experiment with n = 4
p = p ((drawing a defective bulb) = \(\frac{8}{32}\) = \(\frac{1}{4}\)
q = p (drawing a non defective bulb) = \(\frac{3}{4}\)
Let the random variable X denotes the number of defective bulbs.
∴ X can take values 0, 1, 2, 3, 4.

Question 8.
A random variable X has the following probability distribution.

X = x	0	1	2	3	4	5
p(x)	0	R	2R	3R	3R	R

Determine
(a) R
(b) P(X < 4)
(d) P(2 ≤ X ≤ 5)
(c) P(X ≥ 2)
Solution:
(a) Clearly ∑P_i = 1
⇒ R + 2R + 3R + 3R + R = 1

Question 9.
Find the mean and the variance of the number obtained on a throw of an unbiased coin.
Solution:
When an unbiased coin is tossed we donot get any number.
p(getting a number) = 0
Thus mean = variance = 0
If instead of coin it would be an unbiased die
Then let X = The number obtained in the throw.
X can take values 1, 2, 3, 4, 5 or 6.
P(X = 1) = P(X = 2) = P(X = 3) = P(X = 4)

Question 10.
A pair of coins is tossed 7 times. Find the probability of getting
(i) exactly five tails
(ii) at least five tails
(iii) at most five tails
Solution:
Clearly the given experiment is a binomial experiment with

Question 11.
If a pair of dice is thrown 5 times then find the probability of getting three doublets.
Solution:
In a single through of a pair of dice
p (a doublet) = \(\frac{6}{36}\) = \(\frac{1}{6}\)
p (a non doublet) = 1 – p (a doublet) = 1 – \(\frac{1}{6}\) = \(\frac{5}{6}\)
Clearly the given experiment is a binomial experiment with n = 5
p = \(\frac{1}{6}\) and q = \(\frac{5}{6}\)
p(3 doublets in 5 throw) = ⁵C₃ p³q²
= 10. \(\left(\frac{1}{6}\right)^3\) \(\left(\frac{5}{6}\right)^2\) = \(\frac{250}{6^5}\) = \(\frac{125}{3888}\)

Question 12.
Four cards are drawn successively with replacement from a well-shuffled pack of 52 cards. What is the probability that:
(i) all the four cards are diamonds
(ii) only two cards are diamonds
(iii) none of the cards is a diamond.
Solution:
Out of 52 cards there are 13 diamonds. 4 cards are drawn one by one with replacement.
When we draw a card

Question 13.
In an examination, there are twenty multiple-choice questions each of which is supplied with four possible answers. What is the probability that a candidate would score 80% or more in the answers to these questions?
Solution:
Total number of questions = 20 for each question
p (correct answer) = \(\frac{1}{4}\)
p (wrong answer) = \(\frac{3}{4}\)
p (the score is > 80%)
= p (no. of correct answer > 16)
= p (16 correct answers)
+ p(17 correct answers)
+ p (18 correct answers)
+ p (19 correct answers)
+ p (20 correct answers)

Question 14.
A bag contains 7 balls of different colours. If five balls are drawn successively with replacement then what is the probability that none of the balls drawn is white?
Solution:
Total number of balls = 7 (The colours are different)
Number of balls drawn = 5 (one by one with replacement)
Case – 1
(If a white coloured ball is not present)
p(non is white) = 1
Case – 2
(If one ball is white).
In one draw p(white ball) = \(\frac{1}{7}\)
p (non white ball) = \(\frac{6}{7}\)
When 5 balls are drawn
p (none of the balls drawn is white) = ⁵C₀ \(\left(\frac{1}{7}\right)^0\) \(\left(\frac{6}{5}\right)^5\) = \(\left(\frac{6}{7}\right)^5\)

Question 15.
Find the probability ofthrowing at least 3 sixes in 5 throws of a die.
Solution:
In one throw of a die

Question 16.
The probability that a student securing first division in an examination is \(\). What is the probability that out of 100 students twenty pass in first division?
Solution:
Clearly the given experiment is a binomial experiment with

Question 17.
Sita and Gita throw a die alternatively till one of them gets a 6 to win the game. Find their respective probability of winning if Sita starts first.
Solution:
In one throw of a die
p (getting a 6) = \(\frac{1}{6}\)
p (not getting a 6) = \(\frac{5}{6}\)
If Sita begins the game she may win in first round or second round or third round, etc.
Thus p(Sita wins)

Question 18.
If a random variable X has a binomial distribution B(8, \(\frac{1}{2}\)) then find X for which the outcome is the most likely. [Hint: Find X=x for which P(X = x) is the maximum, x = 0, 1, 2, 3,…….. 8.]
Solution:
Given binomial distribution is B(8, \(\frac{1}{2}\))
Thus n = 8, p = \(\frac{1}{2}\), q = \(\frac{1}{2}\)
We shall find X which is most likely i.e. for which p(x = r) is maximum where r = 0, 1, 2, ….., 8
As p = q, the probability is maximum when ⁸Cr is maximum
But ⁸Cr, r = 0, 1, 2, ….. ,8 is maximum when r = 4.
Thus the most likely outcome is x = 4.

Odisha State Board Elements of Mathematics Class 12 CHSE Odisha Solutions Chapter 7 Continuity and Differentiability Ex 7(j) Textbook Exercise Questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 7 Continuity and Differentiability Exercise 7(j)

Test differentiability and continuity of the following functions.
Question 1.
\(\left|1-\frac{1}{x}\right|\) at x = 1
Solution:

Question 2.
x² |x| at x = 0.
Solution:
Let f(x) = x² |x|
Then f(0) = 0

As L.H.D.=R.H.D., f(x) is differentiable at x = 0. We know that every differentiable function is continuous. So f(x) is also continuous at x = 0.

Question 3.
f(x) = tan x at x = \(\frac{\pi}{2}\)
Solution:
f(x) = tan x
f(\(\frac{\pi}{2}\)) = tan \(\frac{\pi}{2}\) which does not exist.
So f(x) is neither continuous not differentiable.

Question 4.
f(x) = cot x at x = \(\frac{\pi}{2}\).
Solution:

Question 5.
f(x) = |sin x| at x = π.
Solution:
Differentiability:
f(x) = |sin x|, x = π
f(π) |sin π| = 0

Question 6.
f(x) = latex]\frac{x}{1+|x|}[/latex] at x = 0
Solution:

As L.H.D. = R.H.D., f(x) is differentiable at x = 0.
Every differentiable function is continuous.
So f(x) is also continuous at x = 0.

Question 7.
f(x) = \(\begin{cases}x \sin \frac{1}{x}, & x \neq 0 \\ 0, & x=0\end{cases}\) at x = 0
Solution:

Question 8.
f(x) = \(\begin{cases}\frac{1-e^{-x}}{x}, & x \neq 0 \\ 1 & x=0\end{cases}\) at x = 0
Solution:
f(0) = 1

As L.H.D. = R.H.D., f(x) is differentiable at the origin. Again every differentiable function is continuous. So f(x) is continuous at the origin.

Odisha State Board CHSE Odisha Class 12 Math Solutions Chapter 8 Application of Derivatives Ex 8(b) Textbook Exercise Questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 8 Application of Derivatives Exercise 8(b)

Question 1.
Find the equations to the tangents and normals to the following curves at the indicated points.
(i) y = 2x² + 3 at x = -1
Solution:
y = 2x² + 3 at x = -1
⇒ \(\frac{d y}{d x}\) = 4        ∴\(\left.\frac{d y}{d x}\right]_{-1}\) = -4
Thus slope of the tangent at x = -1 is -4.
Angin for x = -1 , y = 2 + 3 = 5
The point of contact is (-1, 5).
Equation of the tangent is y – 5 = -4 (x + 1)
⇒ 4x + y – 1 = 0
Slope of the normal = \(\frac{1}{4}\)
Eqn. of the normal is y – 5 = \(\frac{1}{4}\) (x + 1)
⇒ 4y – 20 = x + 1
⇒ x – 4y + 21 = 0

(ii) y = x³ – x at x = 2
Solution:
y = x³ – x at x = 2
When x = 2, y = 2³ – 2 = 6
The point is (2, 6).
Again \(\frac{d y}{d x}\) = 3x² – 1
\(\left.\frac{d y}{d x}\right]_{x=2}\) = 12 – 1 = 11
Slope of the tangent = 11.
Slope of the normal = –\(\frac{1}{11}\)
Equation of the tangent is
y – 6 = 11 (x – 2)
⇒ 11x – y – 16 = 0
Equation of the normal is
y – 6 = –\(\frac{1}{11}\)
⇒ 11y – 66 = – x + 2
⇒ x + 11y – 68 = 0

(iii) y = √x + 2x + 6 at x = 4
Solution:
y = √x + 2x + 6 at x = 4
For x = 4, y = 2 + 8 + 6 = 16
The point is (4, 16)
Again \(\frac{d y}{d x}\) = \(\frac{1}{2 \sqrt{x}}\) + 2
\(\left.\frac{d y}{d x}\right]_{x=4}\) = \(\frac{1}{4}\) + 2 = \(\frac{9}{4}\)
Slope of the tangent = \(\frac{9}{4}\)
Slope of the normal = –\(\frac{4}{9}\)
Equation of the tangent is
y – 16 = \(\frac{9}{4}\)(x – 4)
⇒ 9x – 4y + 28 = 0
Equation of the normal is
y – 16 = – \(\frac{4}{9}\)(x – 4)
⇒ 9y – 144 = -4x + 16
⇒ 4x + 9y – 160 = 0

(iv) y = √3 sin x + cos x at x = \(\frac{\pi}{3}\)
Solution:

(v) y = (log x)² at x = \(\frac{1}{e}\)
Solution:

(vi) y = \(\frac{1}{\log x}\) at x = 2
Solution:

(vii) y = xe^-x at x = 0
Solution:
y = xe^-x at x = 0
For x = 0, y = 0
The point is (0, 0).
\(\frac{d y}{d x}\) = e^-x + x . (-e^-x)
\(\left.\frac{d y}{d x}\right]_{x=0}\) = 1
Slope of the tangent = 1
Slope of the normal = -1
Equation of the tangent is y = x and equation of the normal is y = -x

(viii) y = a (θ – sin θ), y = a (1 – cos θ) at θ = \(\frac{\pi}{2}\)
Solution:
y = a (θ – sin θ), y = a (1 – cos θ) at θ = \(\frac{\pi}{2}\)

(ix) \(\left(\frac{x}{a}\right)^{2 / 3}\) + \(\left(\frac{y}{b}\right)^{2 / 3}\) = 1 at (a cos³ θ, b sin³ θ)
Solution:
\(\left(\frac{x}{a}\right)^{2 / 3}\) + \(\left(\frac{y}{b}\right)^{2 / 3}\) = 1 at (a cos³ θ, b sin³ θ)

Question 2.
Find the point on the curve y² – x² + 2x – 1 = 0 where the tangent is parallel to the x – axis.
Solution:
Given curve is y² – x² + 2x – 1 = 0 … (1)

Putting x = 1 in (1) we get
y² – 1 + 2 – 1 = 0 ⇒ y = 0
∴ The point is (1, 0).

Question 3.
Find the point (s) on the curve x = \(\frac{3 a t}{1+t^2}\), y = \(\frac{3 a t^2}{1+t^2}\) where the tangent is perpendicular to the line 4x + 3y + 5 = 0.
Solution:

Question 4.
Find the point on the curve x² + y² – 4xy + 2 = 0 where the normal is parallel to the x axis.
Solution:
x² + y² – 4xy + 2 = 0

Question 5.
Show that the line y = mx + c touches the parabola y² = 4ax if c = \(\frac{a}{m}\).
Solution:
Given line is y = mx + c … (1)
Given parabola is y² = 4ax

Question 6.
Show that the line y = mx + c touches the ellipse
\(\frac{x^2}{a^2}\) + \(\frac{y^2}{b^2}\) = 1 if c² = a²m² + b²
[Hints: Find equation to tangent at a point (x’, y’) of the curve and compare it with
y = mx + c].
Solution:

Question 7.
Show that the sum of the intercepts on the coordinate axes of any tangent to the curve
√x + √y = √a is constant.
Solution:

Question 8.
Show that the curves y = 2^x and y = 5^x intersect at an angle
tan^-1 \(\left|\frac{\ln \left(\frac{5}{2}\right)}{1+\ln 2 \ln 5}\right|\)
(Note: Angle between two curves is the angle between their tangents at the point of intersection)
Solution:
Given curves are
y = 2^x … (1)
and y = 5^x … (2)
Differentiating (1) we get \(\frac{d y}{d x}\) = 2^x . In 2
Differentiating (2) we get \(\frac{d y}{d x}\) = 5^x . In 2
Slope of the tangent to the first curve at (x, y) = 2^x . In 2
Slope of the tangent to the second curve at (x, y) = 5^x . In 2
Solving (1) and (2) we get 2^x = 5^x ⇒ x = 0
For x = 0, y = 1
∴ The point of intersection is (0, 1).
At (0, 1) slope of the 1st tangent = In 2
Slope of the second tangent = In 5
If θ is the angle between two tangents
then \(\frac{\ln 5-\ln 2}{1+\ln 5 \cdot \ln 2}\) = \(\frac{\ln \frac{5}{2}}{1+\ln 5 \cdot \ln 2}\)
= tan^-1 \(\left(\frac{\ln \frac{5}{2}}{1+\ln 5 \cdot \ln 2}\right)\)
We know that angle between two curves is the angle between their tangents at the point of intersection.
Hence the two curves intersect at an angle.
\(\left(\tan ^{-1} \cdot \frac{\ln \frac{5}{2}}{1+\ln 5 \cdot \ln 2}\right)\)

Question 9.
Show that the curves ax² + by² =1 and a’x² + b’y² = 1. Intersect at right angles if
\(\frac{1}{a}\) – \(\frac{1}{a}\) = \(\frac{1}{a^{\prime}}\) – \(\frac{1}{b^{\prime}}\).
Solution:
Given curves are
ax²+ by²= 1 … (1)
and a’x² + b’y² = 1 … (2)
Differentiating (1) we get

Question 10.
Find the equation of the tangents drawn from the point (1, 2) to the curve.
y² – 2x³ – 4y + 8 = 0
Solution:
Given curve is
y² – 2x³ – 4y + 8 = 0 … (1)

⇒ -(k – 2)² = 3h² – 3h³
⇒ (k – 2)² = 3h³ – 3h²
Putting it in (2) we get
k² – 2h³ – 4k + 8 = 0
⇒ k² – 4k + 4 – 2h³ + 4 = 0
⇒ (k – 2)² – 2h³ + 4 = 0
⇒ 3h³ – 3h² – 2h³ + 4 = 0
⇒ h³ – 3h² + 4 = 0
⇒ h³ + h² – 4h² + 4 = 0
⇒ h² (h + 1) – 4 (h² – 1) = 0
⇒ h² (h + 1) – 4 (h + 1) (h – 1) = 0
⇒ (h + 1) (h² – 4h + 4) = 0
⇒ (h + 1) (h – 2)² = 0
⇒ h = -1, 2.
For h = -1, k is imaginary.
For h = 2, k = 2 + 2√3
The point at which the tangent is drawn is (2, 2 + 2√3).
Slope of the tangents
= \(\frac{3 h^2}{k-2}\) = \(\frac{12}{\pm 2 \sqrt{3}}\) = ± 2√3
Equations of the tangental
y – (2 ± 2√3) = ± 2√3(x – 2)

Question 11.
Show that the equation to the normal to x^2/3 + y^2/3 = a^2/3 is y cos θ – x sin θ = a cos 2θ where θ is the inclination of the normal to x – axis.
Solution:

The point is (a² sin³ θ, a² cos³ θ).
Equation of the normal is
y – a² cos³ θ = tan θ (x – a² sin³ θ)
⇒ y cos θ – a² cos⁴ θ = x sin θ – a² sin⁴ θ
⇒ y cos θ – x sin θ = a² (cos⁴ θ – sin⁴ θ)
= a² (cos² θ + sin² θ) (cos² θ – sin² θ)
= a² cos 2θ.
∴ Equation of the normal is
y cos θ – x sin θ = a² cos 2θ.(Proved)

Question 12.
Show that the length of the portion of the tangent to x^2/3 + y^2/3 = a^2/3 intercepted between the axes is constant.
Solution:

Question 13.
Find the tangent to the curve y = cos (x + y), 0 < x ≤ 2π which is parallel to the line
x + 2y = 0.
Solution:
y = cos (x + y) … (1)

Question 14.
If tangents are drawn from the origin to the curve y = sin x then show that the locus of the points of contact is x²y² = x² – y²
Solution:
Given curve is
y = sin x … (1)
\(\frac{d y}{d x}\) = cos x
Slope of the tangent to the cure (1) at any point (x, y) is cos x.
Let (h, k) be the point of contact.
Then slope of the tangent at (h, k) = cos h.
Equation of the tangent to the curve (1) at (h, k) is
y – k = cos h (x – h) … (2)
If the tangent is drawn from the origin then -k = -h cos h
⇒ k = h cos h … (3)
As (h, k) is the poitn of contact then we have
k = sin h … (4)
From (3) we get,
k = h\(\sqrt{1-\sin ^2 h}\) = h\(\sqrt{1-k^2}\) by (4).
Squaring both sides we get
k² = h² (1 – k²)
⇒ k² = h² – h²k²
⇒ h²k² = h² – k²
The locus of (h, k) is x².y² = x² – y² (Proved)

Question 15.
Find the equation of the normal to the curve given by
x = 3 cos θ – cos³ θ
y = 3 sin θ – sin³ θ at θ = \(\frac{\pi}{4}\)
Solution:
x = 3 cos θ – cos³ θ
y = 3 sin θ – sin³ θ
Differentiating we get
\(\frac{d x}{d \theta}\) = -3 sin θ + 3 cos² θ . sin θ
\(\frac{d x}{d \theta}\) = 3 cos θ – 3 sin² θ . cos θ

Question 16.
If x cos α + y sin α = p is a tangent to the curve \(\left(\frac{x}{a}\right)^{\frac{n}{n-1}}\) + \(\left(\frac{y}{b}\right)^{\frac{n}{n-1}}\) = 1
then show that (a cos α)ⁿ + (b sin α)ⁿ = pⁿ
Solution:
Given straight line is x cos α + y sin α = p … (1)
Given curve is

Question 17.
Show that the tangent to the curve
x = a (t – sin t), y = at (1 + cos t) at t = \(\frac{\pi}{2}\) has slope. (1 – \(\frac{\pi}{2}\))
Solution:
Given curve is x = a (t – sin t)
y = at (1 + cos t)

Odisha State Board CHSE Odisha Class 12 Invitation to English 4 Solutions Grammar Tense Patterns Unit 6 The Past Simple and the Past Perfect Textbook Activity Questions and Answers.

CHSE Odisha 12th Class English Grammar Tense Patterns Unit 6 The Past Simple and the Past Perfect

SECTION – 1

Look at the sentences below.
(a) I reached the hostel in the morning and found that somebody had broken into my room during the night.
(b) She said that her friend had published a book.
(c) He had lived in this town for ten years; then he migrated to Japan.
Can you find the Past Perfect Tense in each sentence? Note that the sentence in which it occurs refers to two actions — the action expressed by the Past Perfect and another action expressed by the Past Simple. Of the two actions, which takes place earlier and which takes place later? List them below.

(a) (1) I reached the hotel in the morning and found. (later).
(2) that somebody had broken (earlier action) into my room during the night.
(b) (1) She said _________(later action).
(2) that her friend had published a book _________(earlier action).
(c) (1) _________then he migrated to Japan _________(later action).
(2) He had lived in this town for ten years, (earlier action).
Can you answer now? Which action does the Past Perfect refer to — the earlier one or the later one? Which action does the Past Simple refer to?
Answer:
The earlier action refers to Past Perfect and the later one refers to Past Simple Tense.

Activity – 23
Combine each pair of sentences below into a single sentence, using the Past Perfect to show which action took place earlier. (You may have to use words like after; when etc.
(a) (i) I finished my homework.
(ii) Then I went to buy a pen.
_____________________________________.
(b) (i) Then the doctor gave some medicine to the patient,
(ii) Then the patient regained his senses.
_____________________________________.
(c) (i) I read a few pages from the book.
(ii) After that I returned it to the librarian.
_____________________________________.
(d) (i) I worked in the garden for some time.
(ii) After that I had my breakfast.
____________________________
(e) (i) He left the place in a hurry.
(ii) After that his friend arrived.
____________________________
(f) (i) The young girl finished shopping.
(ii) Then she met with an accident.
____________________________
(g) (i) The thief ran away with the gold.
(ii) After that the police arrived.
____________________________
Answer:
(a) After I had finished my homework, I went to buy a pen.
(b) After the doctor had given some medicine to the patient, the patient regained his senses.
(c) After I had read a few pages from the book, I returned to the librarian.
(d) I had worked in the garden for some time before I had my breakfast.
(e) When he had left the place in a hurry. his friend arrived.
(f) After the young girl had finished shopping. she met with an accident.
(g) The thief had run away with the gold before the police arrived.

Activity — 24
Jatin arrived late at different places yesterday. What did he find when he arrived
at each place?
Example — When he arrived at the cricket stadium, the game had ended.
(a) the hank it / already I close.
____________________________
(b) his uncle’s house his uncle I go the sleep.
____________________________
(c) the bus stops the bus I already / leave.
____________________________
(d) book shop the book he wanted/sold out already.
_________________________________
(e) the club his friends/leave.
_________________________________
(f) the hostel everyone / go to bed.
_________________________________
Answer:
(a) When he arrived at the bank, it had already closed.
(b) When he got to his uncle’s house, his uncle had gone to sleep.
(c) When he reached the bus stop, the bus had already left.
(d) When he came to the bookshop, the book he wanted had been sold out already.
(e) When he arrived at the club, his friends had left.
(f) When he came to the hostel, everyone had gone to bed.

Activity – 25
Use the verb supplied in brackets in the appropriate form.
(a) We went to Anil’s house and _____________(knock) on the door but there _____________ (be) no answer. Either he _____________ (go) out or he _____________ (not want) to see anyone.
(b) Sadhan _____________(go) for a walk yesterday because the doctor _____________(tell) him last week that he _____________(need) exercise.
(c) A : _____________(Seema / arrive) at the party in time last night ?
B: No, she was late. By the time, we got there, everyone _____________(leave).
Answer:
(a) We went to Anil’s house and knocked on the door but there was no answer. Either he had gone out or he did not want to see anyone.
(b) Sadhan went for a walk yesterday because the doctor told him last week that he needed exercise.
(c) A: Did Seema arrive at the party in time last night?
B: No, she was late. By the time, we got there, everyone had left.

Odisha State Board CHSE Odisha Class 12 Invitation to English 3 Solutions Writing Reporting Events and Business Matters Textbook Activity Questions and Answers.

CHSE Odisha 12th Class English Writing Reporting Events and Business Matters

8.1 Reporting Events

Generally, we come across events in daily newspapers. Reporting events requires a special skill in presenting the event accurately, concisely, and authentically. Various types of events take different forms of expression.

Now read the following news report from a staff reporter of a newspaper.

Answer these questions on the news item :

Question 1.
Who has been sentenced?
Answer:
A doctor belonging to Ranchi has been sentenced.

Question 2.
How long is the sentence?
Answer:
The sentence is for five years’ rigorous imprisonment.

Question 3.
Why has he been sentenced?
Answer:
He has been sentenced because he was caught in the act of removing a kidney of one Nasir Ali when he was unconscious.

Question 4.
When was the case detected?
Answer:
The case was detected by the police on May 8, 2008.

Question 5.
What was the doctor doing?
Answer:
The doctor was trying to remove the kidney of the unconscious Nasir Ali.

Activity – 1

A report about smuggling of fake currency notes appeared in The Times of India on 2 March 2009. The bare facts are given below. Write the report, using these facts.
Three Pakistani nationals have been arrested.
They were arrested on Monday.
Place of arrest, Amritsar.
They were attempting to smuggle fake currency notes.
The value of the currency notes was Rs. 20 lakh.
They were travelling by the Samjhauta Express.
The customs officials seized the fake currency notes.
This was the fifth major seizure.

Answer:
Fake Currency Notes Siezed The Times of India News Service:
Amritsar March 2 : Custom officials seized currency notes worth Rs. 20 lakhs from three Pakistani nationals who were travelling by Samjhauta Express. They were arrested on Monday at Amritsar. This was the custom officials fifth major seizure.

Activity – 2

A newspaper reporter sent a report of an incident on March 12, 2009 over telephone. A sub-editor in the newspaper office, who received the telephone message, took the following notes. Use them to write up a report for the newspaper.
Five pick-pockets caught red-handed
In north Calcutta
Wednesday
Robbing passengers on a private bus
45 wallets and jewellery recovered
police arrested them.

Answer:
Pick-pockets caught Red-handed
Calcutta March 12 : Police caught five pick-pockets in north Calcutta this Wednesday while they were trying to flee after having robbed passengers on a private bus. 45 wallets and jewellery were recovered from them. Police arrested them.

Activity – 3

There was a train accident in the area where you work as a news reporter. You went to the spot and talked to different people including some of the passengers. You also met the railway officials. The following are the points you noted down.
Train accident at Retang at 7.30 p.m. on 11 April.
Three bogies of the East Coast Express derailed.
7 bodies recovered so far
25 injured sent to hospital in Cuttack; 5 serious cases.
Rescue operations still on.
More bodies suspected trapped inside wreakage.

Answer:
Train Derailed : Many Feared Dead, 25 Injured.
Retang, April 1 1 : Three bogies of the East Coast Express derailed here at 7 30 p.m. this morning. 7 bodies have been recovered so far but many are feared to be still trapped under the wreckage. The injured have been sent to the SC B Hospital Cuttack Oi 25 injured, five are in an extremely serious condition. Rescue operations are still on.

Activity – 4

Below you can see a picture of an incident that happened in front of Badanibadi Bus Stand, Cuttack. Report the incident, using the hints given below the picture.

Hints :
Badanibadi, Cuttack  12 December, 2008.  Morning 9
bulls ready to fight traffic held up
Answer:
Bulls Hold up Traffic
Cuttack 12 : Traffic was held up for almost an hour from 9.00 am this morning at Badanibadi as two bulls were poised to fight in the middle of the road. The bull had locked horns while bystanders cheered them.

Activity – 5

Very often, as a students ’ representative, you may have to read out a report in a function, ceremony, meeting etc. At that time, you don’t have to make the mention of the place, date, etc. as is done in writing a news report. It is normally written like an essay.

A. The Minister of Education was the Chief Guest in the Annual Day Celebration of your college. Write a news report to be sent to a newspaper.
Answer:

To,
The Chief Editor,
The Times of Odisha, Bhubaneswar

Report on Annual Day Celebration

Cuttack: 12 March
The Annual Day celebration of A.K.B. College, Govindpur came off smoothly. The Hon’ble Education Minister Sri N. K. Patra was the Chief Guest. The celebration started with the national anthem and devotional song. The local Collector and Dist. Magistrate presided over the meeting. After a short speech made by the Hon’ble Chief Guest on the all-round development of the college, gave away the prizes for curricular and extracurricular activities of the students. The celebration was a grand success.

B. During the Annual Day celebration, you were asked to present a report on the students’ activities during the year. Draft your report.
Answer:

A Report on Students’ Activities

The A.K.B. College, Govindpur has already been recognized as a unique educational institution imparting teaching on all branches/streams in the State. Elections to the students’ union, the dramatic society, the day-scholars’ Association, the +2 Cultural Association, etc. were held without any untoward incident. The performance of the students in curricular activities was excellent. Most of the students were awarded gold medals for their outstanding results in different streams. The mention of the co-operation and mutual understanding among the students in the college and outside would not be out of place here,

8.2 Business Reports

Business reports are mainly based on market condition, market surveys or market- analyses. We usually take into account the background information, method of investigation, findings, and recommendations while preparing business reports. Besides, we follow direct and factual language based on the market-surveys and market analyses. There is no scope of rendering personal feelings or attributing subjective interpretations.

Activity – 6

Here is a report about the introduction of a new mosquito-repellant. Read the report, paying attention to various parts.

Quality Marketing Agency
27 Janpath, Bhubaneswar

4 March, 2013

To
Mr. M. Pradhan
Managing Director
Home Products India Ltd.
Industrial Estate
Mancheswar, Bhubaneswar.
Dear Mr. Pradhan,
As requested by you, vide your letter No. MD/NS/2233 dated 2.2.2013, we have carried out a market survey to test the public acceptance of the mosquito repellant which your company plans to manufacture. We conducted an opinion poll covering 1000 families in the coastal districts of Odisha. Forty percent of these families use mosquito repellants, but most of them are unhappy with the existing products in the market. They find the electronic repellants too expensive while the coil-based ones emit too much smoke.

The preference is for a less expensive product, preferably one that produces no smoke. Our study suggests there may be a good market for a new repellant, provided these requirements are kept in mind. We recommend that your company should concentrate on manufacturing an improved kind of smoke-free mosquito coil, preferably one that produces a pleasant fragrance.
Yours sincerely,
S.K. Patnaik
Director of Research
Quality Marketing Agency

You must have observed that the report has been written in the format of a business letter. However, it could be written in a different format as given below.

Date : 4 March 20
To : Mr M Pradhan, Managing Director, Home Products India Ltd. Industrial Estate, Mancheswar, Bhubaneswar
From : S.K. Patnaik, Director of Research, Quality Marketing Agency 27 Janpath, Bhubaneswar
Subject : Survey report on the Introduction of a new mosquito repellant.

A Report On The Introduction Of A New Mosquito Repellant

A market survey to test the public’s acceptance of a new mosquito repellant was conducted in the coastal districts of Odisha on 20 February, 20 by Quality Marketing
Agency, Bhubaneswar.
An Opinion poll covering 100 families ……………………. etc.
______________________________________
______________________________________
______________________________________
An improved kind of smoke-free mosquito coil, preferably one that produces a pleasant fragrance is likely to be widely accepted by the public.
Sd/-
(Director of Research)

Activity – 7

Imagine that you are the President of the Literary Society of your College. Your Society plans to publish a journal. You have asked the Secretary of the Society to contact all the printing firms in town and to select one of them to print your journal.
Answer:

Literary Society, Kulailey College, Kailey

5 February 2012

To,
Prof. B. Pujari
President, Literary Society
Sir,
As desired by you a team consisting of the Secretary and Assistant Secretary of the society contacted all five printing firms in town and obtained quotations from them for printing of the proposed journal. All the firms quoted the same price, that is Rs. 5000/- for 1000 copies. Rasmita Printers, however, offered a discount of ten per cent, provided we allowed them an extra period of fifteen days for printing. Since we do not need the copies of the journal till a month later, we could consider the offer of Rasmita Printers as it will cost us Rs. 500/- less than the offers quoted by other printing firms.
Yours faithfully,
S. Pujari
Secretary

Activity – 8

A customer approached a bank for a house building loan. Before sanctioning the loan, the Branch Manager asked the Field Officer to examine the application and suggest whether the loan should be sanctioned.

The following is the report that the Field Officer wrote. Some parts of the report are missing. Re-write the missing parts, using the hints supplied. (See the letter at textbook page 75)
Answer:

SBI PD BRANCH
CRP Square
Bhubaneswar

3 March, 2012

To,
Prof. M. Mishra
Branch Manager
SBI PD Branch
CRP Square
Bhubaneswar
Sir,

1. As desired by you in your letter No. 254 dt. 24 February 2010, I examined the application of Mr. J.K. Panda for a house-building loan. I also personally inspected the site, interviewed Mr Panda and examined the documents relating to the plot.
2. My examination of the application and the relevant documents reveal that the site is an undisputed one. Till date all land cess has been paid and the plot is litigation free. Mr J.K. Panda is the owner of the plot and he has clear papers certifying its ownership. The plot is 112 decimals in size and its market value is around Rs. 8.00 lakhs. Mr Panda also has a regular income of Rs. 15,000 and has no outstanding loan to his account.
3. As Mr Panda is a deserving party, the sanction of the loan is recommended
Yours faithfully,
K.C. Panigrahy
Field Officer

Activity – 9

Imagine that you are the foreman in a factory. There has been a fire in the Factory and one of the workers has been badly burnt and is in hospital. Your General Manager has asked you to send him a report on the fire. Write a report.
Answer:

JAYASHREE CHEMICALS
Corporate Office, Metro Towers
Bhubaneswar – 751007

28.3. 2012

To,
Mr J.B. Sahu
General Manager
Jayashree Chemicals
Corporate Office, Bhubaneswar
Sir,
1. As desired by you vide your letter No. JC 289 dt. 20.3.2012, I conducted an enquiry into the fire that ravaged the factory at the Mancheswar Industrial Estate.
2. On inspection of the fire ravaged site and after interviewing the workers present during the incident, I discovered that the fire was caused by the Careless throwing of a lighted cigarette into the boiler room where it ignited a dry rope. The rope was lying in contact with a jar of petrol kept by an employee for his personal use. The jar of petrol caught fire immediately and broke into high flames that ravaged the electrical circuit. Fortunately, there was no one in the boiler room as it was lunch time except Mr Lingaraj Patra who was cleaning some machine part.
3. He has received 40 per cent bums in his body and has been admitted at the Kalinga hospital. He is presently out of danger and the personnel department has already sanctioned Rs. 50,000 for his treatment.
4. Damage to the boiler section of the factory was limited to the minimum by the quick action of our fire personnel. A few chairs and benches must be replaced and the Electrical circuit done up again.
This is for your kind information.
Yours faithfully
D.P. Dehury
Chief Security Officer
Jayashree Chemicals

Activity – 10

Your club wants to buy a CTV (Colour TV) set. You have been asked to contact various firms marketing such sets and make your recommendation on the brand to be bought. Make a comparative study of the price, quality and durability of different brands of CTV and make a report.
Answer:

YOUNG FOLKS CLUB
213 Kharvela Nagar
Bhubaneswar

21 April 2012

Mr P. Khuntia
The Secretary
Young Folks Club
213 Kharavela Nagar
Bhubaneswar
Dear Mr Khuntia,

As requested by you I contacted 10 dealers of colour television in the town and personally inspected several makes of colour televisions in their showrooms. LG, Samsung, Philips and Onida companies all offer 25” CTVs at competitive prices. Of course L.G. is the costliest at Rs. 58,000 but it employs the golden eye technology which is truly soothing to the eyes. Samsung and Onida brands are also good. The unique thing about the Samsung T V. is that it offers two games in its set but we cannot afford members fiddling with the game program of the T.V. while others want to watch programs.

Except for its woofer impacted quality the Videocon CTV’s picture quality is no match for those of other brands. The Onida brand is good. It commands a price tag of Rs. 50,000. However its after sales-service is doubtful as the dealer here does not guarantee qualified personnel from the company to give service. However with a price tag of Rs. 48,000/- and a 5 year warranty contract, quality picture and sound, the Philips flatron CTV is perhaps the best that our town has to offer.

The concerned dealer is offering a free Antennae along with booster as well as 10 VCD’s with the T.V. He has also agreed to transport the CTV and to install it in our conference Hall free of cost. It is therefore recommended that the Shanti Electronics, 105, Bapuji Nagar, Bhubaneswar be contacted and the Philips Flatron CTV be bought from them at the price as well as the terms and conditions offered by them.
Yours faithfully
Girish Mishra
Joint Secretary
Young Folks Club

Odisha State Board CHSE Odisha Class 12 Invitation to English 4 Solutions Grammar Tense Patterns Unit 5 Past Simple and Past Progressive Textbook Activity Questions and Answers.

CHSE Odisha 12th Class English Grammar Tense Patterns Unit 5 Past Simple and Past Progressive

SECTION – 1

Study the sentences below :
(a) It started to rain while we were walking home.
(b) My sister was tidying my room when I saw your letter.
(c) Anita was walking along the road when suddenly she heard footsteps behind her. Someone was following her. She was frightened and started to run.
What do you think the use of the Past Simple and the Past Progressive indicates in these sentences?
(Hint: Think of a point of time and a duration of time in the past and relate them to the action.)

Note:
The Past Simple tells us that the work/action started and finished in the past. The speaker has a definite time in mind. But in Past Progressive the time of beginning or completion of the activity is not mentioned. The activity was in progress for that hour.

Activity – 21
Put the verbs into the correct form, Past Progressive or Past Simple.
(a) My friend ______________(meet) Anima and Amiya at the bus stop four days ago. They ______________(go) to Paradeep and my friend ______________ (go) to Bolangir. They ______________(have) a chat while they ______________(wait) for their buses.
(b) My brother ______________ (cycle) to school last Monday when suddenly an old woman ______________(step) out into the road in front of him. He ______________(go) quite fast but luckily he ______________ (manage) to stop in time and ______________(not / hit) her.
Answer:
(a) My friend met Anima and Amiya at the bus stop four days ago. They were going to Paradeep and my friend was going to Bolangir. They had a chat while they were waiting for their buses.
(b) My brother was cycling to school last Monday when suddenly an old woman stepped out into the road in front of him. He was going quite fast but luckily he managed to stop in time and did not hit her.

Activity – 22
Here is a true story.
An old couple …………………………living in a flat in Bhubaneswar. ………………………… locked up in one room ………………………….. Some unknown people took away everything ………………………… police arrived ………………………… climbed ………………………… rescued ………………………… broke open a door ………………………… one dacoit was killed ………………………… detective was called …………………………interviewed a witness.
Imagine that you are being questioned by the police as if you were a witness to the crime. A police Inspector is recording your statements in a notebook. Think about the situation and write the appropriate answers.
Inspector: Where were you standing at that time?
Answer: _____________________________.
Inspector: Why did you come here?
Answer: _____________________________.
Inspector: What was the old man doing at the time?
Answer: _____________________________.
Inspector: How did you see that?
Answer: _____________________________.
Inspector: How long were you standing there?
Answer: _____________________________.

Answer:
An old couple _____ living in a flat in Bhubaneswar, _____ Orissa locked up in one room _____. Some unknown people took away everything. _____ police arrived _____, limbed _____ rescued _____ , broke open a door _____one dacoit killed, _____detective was called _____ interviewed a witness.
Inspector: Where were you standing at that time?
Answer: I was standing near the flat.
Inspector: Why did you come here?
Answer: I came here to play.
Inspector: What was the old man doing at that time?
Answer: The old man was shouting and trembling out of fear.
Inspector: How did you see that?
Answer: I heard him shouting and saw him trembling.
Inspector: How long were you standing there?
Answer: I stood there till your arrival.