Class 12

Odisha State Board CHSE Odisha Class 12 Approaches to English Book 1 Solutions Unit 3 Text A: How to Write a Winning ‘Resume’ Textbook Activity Questions and Answers.

CHSE Odisha 12th Class Alternative English Solutions Unit 3 Text A: How to Write a Winning ‘Resume’

Activity-1
Comprehension
Answer the following questions as briefly as you can:

Question (a)
The writer talks of two kinds of resume in the first paragraph, which of them does he think more useful and why?
Answer:
The two resumes mentioned in the first paragraph are ‘tombstone’ and ‘functional’. The‘functional’happens to be more preferable to the tombstone.

Question (b)
Why does the writer advise the job seekers never to be apologetic in a resume?
Answer:
The writer advises the job seekers never to apologize because it is the hardest job of all. It creates a negative impression.

Question (c)
Who are the intended audience for this article? How do you know this?
Answer:
The intended audience for this article are job seekers of any shade and variety. The personal pronoun ‘you’ makes it explicit.

Question (d)
What does the writer want to done between the preparation of the first draft of the } resume and its despatch to the prospective employer?
Answer:
Clustering accomplishments like leadership skills, budget management skills, child development skills, sending it a printer because a printed resume is superior to photocopies are to be done between the first draft of the resume and its despatch to the prospective employer.

Activity-2
(Guessing The Meaning Of)

Choose the best answer:

Question (a)
One type of resume is called a ‘tombstone’ (Para -1) because:
(i) It lists what you have done in the post rather than what you can do in the future.
(ii) It lists your achievements in chronological order.
(iii) It leads you to failure in the job market.
Answer:
(iii) It leads you to failure in the job market.

Question (b)
‘Quick’in para 2 means:
(i) fast
(ii) efficient
(iii) alive
Answer:
(iii) alive

Question (c)
Took your own horn! (para – 6) means:
(i) don’t be modest
(ii) boast about your achievements
(iii) tell the employer what you have done in the past.
Answer:
(ii) boast about your achievements

Question (d)
When people clutch when asked to…. ? (para – 6) means:
(i) many people underestimate themselves.
(ii) many people panic.
(iii) many people hold into their old jobs when asked to leave.
Answer:
(i) many people underestimate themselves.

Question (e)
‘How to psych yourself up ’ (para – 9) means:
(i) How to make a list of your abilities before you write your resume.
(ii) How to write your resume most enthusiastically.
(iii) How to prepare yourself mentally before writing your resume.
Answer:
(i) How to make a list of your abilities before you write your resume.

Activity-3
Cohesive Devices

Say what the italicized words refer to in the passage.
(a) “Everyone does” __________ (para 6)
(b) ‘Oneofyours’ ____________ (para 6)
(c) ____________ about what it all means (para 9)
(d) ‘It shows an employer’ ____________ (para 9) ‘
Answer:
(i) does – clutch
(ii) yours – abilities
(iii) it – accomplishment
(iv) it- listing of accomplishments in the final resume.

Activity-4

Look at the following note, made by someone to repairing a resume. Decide which points should be included in the final vision.

• A job as Manager (Research and Development)
• Joined M. A. (Economics) in Delhi University
• Left after the first year because of father’s death.
• Studied Business Adm. at Indira Gandhi National Open University.
• Specialised in business application of computers.
• Not sure that I’d be good at doing!
• Have been without a job for three months.
• Can play violin.
• Chairperson of the parents Association of my son’s school.
• Helped friends to learn computers.
• Worked for three years as a sales representative for a computer film.
• Hated the hardware selling job.
• Get boarded quickly.
• Look after the local club’s budget and save 20% of its yearly income.
• Like working in a team.
• Can inspire young people to complete a task on time.
• Teach Economics to a group of 20 poor students of the local colleges without fees.
• 36 years old and only 5 years of salaried employment.

Now group the points you have chosen under the following heading:

• Vocational objectives
• Man management skills.
• Competition and team skills.
• Management skills
• Summary of background and the others.

Answer:
Vocational objectives:
(a) Specialised in the business application of computers.
(b) Helped a friend to learn computers.
(c) A job is manager (Research and Development)

VLAN management skills:
(a) Chairperson of the Parents Association of my sons’ school
(b) Look after the local club’s budget and save 20% of its yearly income.
(c) Can inspire young people to complete a task in time.

Competition and team skills:
(i) Like working in a team

Management skills:
(a) Worked for 3 years as a sales representative for a computer firm

Summary of backgrounds and the others:
(a) Joined M.A (Economics) in Delhi University.
(b) Left after the first year because of father’s death.
(c) Not sure what I’d be good at doing!
(d) Have been without a job for three months.
(e) Can pay for violin.
(f) Hated the hardware selling job.
(g) Get bored quickly.
(h) 36 years old and only 5 years of salaried employment.

Extra Activity – 4(A)

Give the derivatives of the following words in the Text – A

purpose	characterize
convince	accomplishments
employer	Skill
interview	world
familiar	people
quick	specific
writing	brief
functional	new
resume	education
important	necessarily
translating

Answer:

Words	Derivatives
purpose	Purposeful, purposive, purposefully
convince	conviction, convincing, convincingly
employer	employee, employment, employed
interview	interviewer, interviewee
familiar	familiarity, familiarise, familiarly
quick	quicken, quickly
writing	write, written
functional	functionalised, function, defunct, malfunctioned
resume	resumption, resuniptive
important	importance, importantly
translating	translate, translation
characterize	character, characteristic
accomplishments	accomplish, accomplished
Skill	skillful, skilled, unskilled, skillfulness
world	worldly, world-wide
people	popular, popularise, population, populate
specific	specifically, specify, specification
brief	briefly, briefness, brevity
new	newly, a new, newness
education	educational, educated, educationally, educationist, educate
necessarily	necessary, necessity, necessitate
identity	Identification, identity

Extra Activity – 4(B)

(i) Use the following phrases in sentences of your own:
drop in,
as well as,
of all shades and variety,
in particular,
have a good look at
Answer:
drop in – My friend occasionally drops in at my residence,
as well as – I as well as any friend can attend the meeting.
of all shades and variety – Sachin Tendulkar tormented the bowlers of all shades and variety.
in particular – I relish tea, in particular Tata Tea.
have a good look at – A good batsman should have a good at the pitch, before batting.

(ii)Derive adjectives from the following nouns:

Legend	comfort
pride	modesty
explosiveness	temperament
critic	nightmare

Answer:

Legend	:	legendary
pride	:	proud
explosiveness	:	explosive
critic	:	critical
comfort	:	comfortable
modesty	:	modest
temperament	:	temperamental
nightmare	:	nightmarish

(iii)Derive nouns from the following verbs:

reverse	encourage
achieve	admit
behave	react
endorse	expect
reflect	succeed

Answer:

reverse	:	reverence
achieve	:	achievement
behave	:	behavior
endorse	:	endorsement
reflect	:	admission
react	:	reaction
expect	:	expectation
succeed	:	success

(iv) Derive adjectives form the following nouns:

quintessence	passion
regret	culture
example	benefit
privacy	importance
pride	modesty

Answer:

Noun		Adjectives
quintessence	:	quiertessential
regret	:	regretful
example	:	exemplary
privacy	:	private
pride	:	proud
passion	:	passionate
culture	:	cultural
benefit	:	beneficial
importance	:	important
modesty	:	modest

(v) Derive verbs from the following nouns:
proposal
operator
introduction
requirement
provision
emission
advice
suspension

Answer:

Nouns	Verb forms
proposal	propose
operator	operate
introduction	introduce
requirement	require
provision	provide
emission	emit
advice	advise
suspension	suspend

(E)(i) Give the antonyms of the following words:

opening	reject
earlier	significant
indoor	success
cause	present
foul	before

Answer:

Words	Antonyms
opening	closing
earlier	later
indoor	outdoor
cause	effect
foul	fair
significant	insignificant
reject	accept
success	Mure
present	absent
before	after

(ii) Substitute the following expressions with one word each.
1. the system by which something can be measured.
2 Any substance that causes pollution.
3. Make people angiy.
4. Public warning to make people careful.
5. A group of people joining together to influence someone in power:

Answer:
1. index
2. pollutant
3. gall
4. alert
5. lobby

(iii) Insert articles, wherever, necessary, in the following sentences:
1. After college career, I shall join university.
2. Mr. Ahuja is Indian, his wife is European.
3. Business of thinking new thoughts is sign of civilization.
4. Japanese are industrious nation.
5. Democracy consists in giving people, things they want.
6. Freedom is spiritual quality.
7. Going abroad is unique opportunity.
8. Ganges is holiest among Indian rivers.
9. We had better have new look at whole problem.
10. He has read Upanishads.

Answer
1. ___________ a university
2. _________ an Indian __________ a European.
3. The business ___________ a sing __________.
4. The Japanese are an industrious nation.
5. __________the things __________.
6. ___________ a spiritual quality.
7. _________ a unique opportunity
8. Ganges is the holiest _________.
9. _________ a new look at the __________.
10. ___________ the Upanishads.

(iv) Derive nouns from the following verbs:

die	solve
enforce	emit
regulate	appear
exist	reduce
maintain	pollute

Answer:

Verbs	Noun Forms
die	death
enforce	enforcement
regulate	regulation
exist	existence
maintain	maintenance
solve	solution
enforce	emission
appear	appearance
reduce	reduction
pollute	pollution

(v)Match the words in column ‘A’ with their antonyms in column ‘B’

‘A’	‘B’
rising	easy
increase	rural
warm	harmless
urban	falling
offenders	cool
difficult	decrease
reduce	irregular
major	enhance
regular	defenders
harmful	minor

Answer:

‘A’	‘B’
rising	falling
increase	decrease
warm	cool
urban	rural
offenders	defenders
difficult	easy
reduce	enhance
major	minor
regular	irregular
harmful	harmless

What Does This Unit Contain?
In this unit you will have further practice in skimming and scanning as reading subskills. You will also have practice in referring and is assessing the communicative value of a Text.
The unit comprises the following sections:
(A) How to write a wining resume-Dick Irish
(B) Advertisement samples (From newspapers and magazines)
(C) On the education ofa man of business – Sir Arthur helps

Section-(A)

Imagine that you are the owner of a small factory producing garments. You have to select and engage five employees in your factory. What are the most important qualities that you would like your employees to possess. Next, suppose you have received more than twenty applications in response to your advertisement for five jobs and you can interview .only ten candidates, how are you going to short-list the ten candidates on the basis of their job applications (also called resumes). While reading Text – A, your focus should be on getting tips on how to select these ten interviewees.

How to Write a Winning ‘Resume’ Summary in English

The main purpose of a resume is to convince an employer to grant you an interview. There are two kinds. One is the familiar ‘tombstone’ that lists where you went to school and where you have worked in the chronological order. The other is what is called the ‘functional’ resume – descriptive, fon to read, unique to you and much more likely to land you and interview. It is handy to have a ‘tombstone ’ for certain occasions. But prospective employers throw away most of those unrequested ‘tombstone’ lists, preferring to interview the quick rather than the dead.

Put Yourself First:
While writing a resume, you have to feel yourself important. ‘Sell what you can do, not who you are’ practice translating your personality traits, character accomplishment and achievements into still areas in the world of work.

Took Your Own Horn:
Many people clutch when asked to think about their abilities. Some think they have none at all

Be specific, be concrete and be brief:
Turn bad news into good:
If you to mention your disappointment in work look for the positive side.

Never Apologize:
If you are returning to the work force after fifteen years as a parents, simply write a short paragraph in place of a chronology of experience. Don’t apologize for working at a being a mother; it’s the hardest job of all. If you have not special training or higher education, just don’t mention education.

How to Psych yourself up:
The secret is to think about the self before you start writing about yourself. Take four or five hours off not necessarily consecutive and simply write down every accomplishment in your life, on or off the job, that made you feel effective. Study the list and try to spot patterns. While studying your list, you will come closer to the meaning: identifying your marketable skills. Once you discover patterns, give names to your cluster of accomplishments (leadership skills, budget management skills, child development skill etc.). Try to list atleast three accomplishments under the same skills heading. You may take your drafts or more and several weeks, before you are already to show it to a stranger for a reaction. When you are satisfied, send it to a printer, a printed resume is far superior to photocopies.

Analytical Outlines

• The main purpose of a resume is to convince an employer.
• He is convinced to grant you an interview.
• There are two kinds.
• One is the familiar‘tombstone’.
• It can list where you went to school.
• It can also list where you have worked in the chronological order.
• The other is ‘functional’ resume.
• It is descriptive.
• It is‘fun to read’.
• It is also unique to you.
• It is much more likely to land you in an interview.
• It is handy to have a ‘tombstone’
• This we find in certain occasions.
• But prospective employers throw away these.
• They throw away most of those unrequired ‘tombstone’ list.
• They prefer to do the interview quick.
• For this reason, they throw away these lists.
• You have to feel yourself important while writing a resume.
• Sell what you can do, not who you are.
• you should practise translating your personality traits.
• You should practise character accomplishments.
• You should practise achievements into skill areas in the world of work.
• Many people clutch thinking about their abilities.
• Some think they have none at all.
• In writing a resume, one should be specific.
• He also should be concrete.
• He should be brief too.
• One should turn bad news into good one.
• You should always look for the positive side.
• You should never apologize.
• You are returning to the work force after fifteen years.
• Simply write a short paragraph.
• You can write it in place of chronology of experience.
• Don’t apologize for working at being a mother.
• It is the hardest job of all.
• You have no special training.
• You have no higher education.
• Then, just don’t mention education.
• You are going to write about yourself.
• So, first you have to think about the self.
• Take four or five hours off
• It should not be necessarily consecutive.
• Simply write down every accomplishment of your life.
• It is about on or off the job.
• The job that make you feel affective.
• Then study the list.
• Try to sport patterns.
• You are studying the list now.
• You will come closer to the meaning.
• Identify your marketable skills.
• Give name to you cluster of accomplishments.
• It may be just like leadership skills.
• It may be budget management skills.
• It may be child development skills.
• Try to list of least three accomplishments.
• These three must be written under the same skill heading.
• You may take your draft.
• You may take it to several weeks.
• Then you can show this to a stranger.
• You have to do this to know others’ reactions for it.
• Then send it to a printer.
• A printed resume is far superior to photocopies.

Meaning Of difficult Words
Resume – summary
convince – to persuade fully, to cause to believe
grant – to permit, to allow
familiar – well – known, general, well-acquainted
chronology – the science of computing time, a scheme of the table of time, order of time.
functional – active, professional
descriptive – narrative, explaining in detail
unique – stole, special
handy – convenient, dexterous
prospective – expecting for future
accomplishments – completeness, happenings, fulfillment
achievement – performance, gain.
clutch – to grasp, tightly, to hatch, to snatch
concrete – material, not abstract, real
positive side – bright side, good side
mention – to refer to, to speak about
disappoint, – to be hopeless
apologize – make frank acknowledgment
consecutive – following in regular order, or one after another
pattern – type, order
effective – fruitful, necessary
identify – to find out, recognize
cluster – a bunch, swarm, crowd

Odisha State Board CHSE Odisha Class 12 Approaches to English Book 1 Solutions Unit 1 Text B: Typing your own Blood Textbook Activity Questions and Answers.

CHSE Odisha 12th Class Alternative English Solutions Unit 1 Text B: Typing your own Blood

Activity – 8

Comprehension:
Question 1.
What does typing someone’s blood mean?
Answer:
Typing someone’s blood means determining the exact type of blood a person usually has. It was make one know one’s blood – group whether ‘A’ or ‘B’ or ‘O’.

Question 2.
What materials are necessary to type one’s blood? Which paragraphs tell you about these materials?
Answer:
Alcohol – soaked cotton balls, sterile lancet, a small test tube containing 1ml. of saline solution, anti-A anti- B and anti- Rhserum with individual eye droppers, two microscope slides, a grease pencil, a posture pipette, three applicator sticks and a warm fluorescent light or other low-heat sources are used on typing one’s blood.

Question 3.
What are the three stages of experimental process described in this text? Name them.
Answer:
First label one slide Rh with a grease pencil and place it under the low-heat source. Divide the cool slide into two equal portions labeling one side A and B and a drop of anti- Rh to warm the Rh slide. In the second stage, use an alcohol-soaked cotton ball to swab your middle or ring finger opening the sterile lancet prick the sterile finger once. Collect several drop of blood in the tube containing saline solution. In the third stage, using the porture pipette, transfer one drop of saline solution containing blood to each of the anti- A, anti- B and anti-Rh serums using a separate applicator stick. Two or three minutes after clumping should have appeared in one or three of the areas. This clumping determines what kind of blood a person has. The stages can be named as preparatory stage, experimental stage and conclusive stage.

Activity – 9

Remedial Grammar:
Like your Rh- slide experiment, you have only two tense forms of most of the English verbs, e.g. “go” and “went”. “Gone” is not a tense form. In association with the other auxiliary verbs, it gives a sense of completion of an activity“has gone”ora passive sense “is done”. Hence like Rh+ or Rh-. English verbs can be either in past tense or non-past tense.

Similarly like your blood grouping. A, B, AB or O, we can have the aspects of perfect (have + V + en), progressive (be + V + ing), perfect progressive (both combined or simple neither, perfect, nor progressive). These four aspects of either past or non-past give us the 8 types of verb groups. In addition to these two tenses and four aspects we can find do operations or model auxiliaries as elements ofa very group.

In the first sentences of the text, the verb is……used. You can see that it is be + v + en structure in simple non-past tense form. Hence, is a simple non-past passive structure. Similarly, find out the aspect, tense and voice of the following verb groups: Illustrates has finished is doing had been completed was being conducted.

Tense	Aspects	Voice
(i) Past	(a) simple	(i) Active
	(b)perfect
(ii) Non-past	(c) progressive	(ii) Passive
	(d)Perfect Progressive

Answer:

Verb groups	Tense	Aspects	Voice
Illustrate	Non-past	Simple	Active
has finished	Non-past	Perfective	Active
is doing	Non-past	Progressive	Active
had been completed	Past	Perfective	Passive
was being conducted	Past	Progressive	Passive

Activity -10

Composition:
In the passage you have step-by-step instructions on how to test and categorize your blood. Write instructions to carry out one of the following tasks.
(a) Teaching your friend how to make tea/cake/an omelette.
(b) Instructing a new friend how to reach your home.
(c) How to fix a fuse wire on your main switch.
Answer:
(a) How to make tea:
Ingredients: water, sugar, tea dust, boiled milk.
Instruments: stove, fry pan, a flat metal piece, spoon, a seive.

Preparation:
(i) Fire the stove.
(ii) Pour required cups of water.
(iii) Mix spoons of sugar as required.
(iv) Add one/two spoons of tea or as required.
(v) Serve the hot solution.
(vi) Add boiled milk to it.
(vii) Serve it in cups.

Extra Activity – 10(A)

B.(i) Derive adjectives from the following words in the text:
words – adjectives
thank – thankful
prepare – preparatory
talk – talkative
servility- servile
compel – compulsory
wisdom – wise
pleasure – pleasant
value- valuable
importance- important
success- successful
luck- lucky
proportion- proportional
enthusiasm- enthusiastic
completion- complete
reproach- reproachful
satisfy- satisfactory
reluctantly- reluctant
pleasure- pleasant
hastiness- haste
trouble- troublesome
persuade- persuasive
purpose- perposefiil
anger- angry
thought- thoughtful
child- children
despise- despicable
triviality- trivial
poverty- poor
necessity- necessary
deceive- deceptive/deceitful
sympathy- sympathetic
passion- passionate
clarity- clear
day- diurnal
night- nocturnal
truth- true
regularity- regular
respect- respectful
forget- forgetful
exhaust- exhaustive
fool- foolish
contempt- contemptuous
falsity- false
money- monetary
anxiety- anxious
continually- continual
pretend – pretentious
superiority- superior
misery- miserable

(ii) Derive adverbs from the following:

Words – Adverbs
thoughtful- thoughtfully
pleasant- pleasantly
reproachful- reproachfully
complete- completely
gradual- gradually
real- realty
excellent- excellently
passionate- passionately
filth- filthily
deep- deeply
full- folly
attract- attractively
possible- possibly
hunger- hungrily
exhaust- exhaustively
hesitate- hesitatingly
watch- watchfully
sharp- sharply
transitory- transitorily
rich- richly
strange- strangely
ordinary- ordinarily
desire- desirety
force- forcefully
strength- strongly
empty- emptity
foolish- foolishly
continual- continually
eternal- eternally
wonder- wonderfully
compel- compulsorily
respect- respectfully
necessary- necessarily
despicable- despicably
regular- regularly
contempt- contemptuously
anxiety- anxiously
misery- miserable

(iii) Say which words of the following in the text are nouns and which are adjectives:

happiness- Noun
good- Adjective
long- Adjective
flight- Noun
excess- Adjective
horrible- Adjective
ugly- Adjective
praise- Noun
bitter- Adjective
sleep- Noun
transitory- Adjective
happy- Adjective
mild- Noun
appearance- Noun
water- Noun
river- Noun
empty- Adjective
foolish- Adjective
years- Noun
folly- Noun
knowledge- Noun
Mortification- Noun
arrogance- Noun
intellectual- Adjective
penitence- Noun
voice- Noun
inward- Adjective
salvation- Adjective
power- Adjective
priest- Noun
madness- Noun
futile- Adjective
special- Adjective
crystal- Noun
depth- Noun
grateful- Adjective
new- Adjective
guest- Noun
hut- Noun
clothes – Noun
current- Noun
affection- Noun
secure- Adjective
bread- Noun
enjoyment- Noun
origin- Noun
despair- Noun
night- Noun
studies- Noun

(iv) Write antonyms of the following:
greatest- smallest
long- short
successful- unsuccessful
everywhere- nowhere
mature- immature
old- new
reality- appearance
real- unreal
presence- absence
true- false
agree- disagree
begin- end
wise- foolish
quickly- slowly
reach- depart
join- separate
remember- forget
small- big /great
back- front
conscious- unconscious
pallid- bright
obtain- lose
compare- contrast
fresh- stale
straighten- bend
hope- hopelessness/despair
win-lose /defeat
injustice- justice
take- give
sorrow- pleasure
much- less
stronger- weaker
completely- incompletely
remember- forget
compared- contrasted
difference- similarity
disappeared- appeared
secure- insecure
knowledge- ignorance
inward- outward
new-old
despair- hope

Section – C
Between men and women, who are the stronger? Who are more intelligent? Who are biologically superior? Why do you think so?
Discuss these questions in small groups and write down your most important arguments. Now read the following title on the differences between men and women.

Typing your own Blood Summary in English

Even if you do not wish to learn your blood type, the exercise is useful, because it familiarises you with some simple laboratory techniques, illustrates the use of basic equipment and prepare you to follow the stages of an orderly scientific procedure. In order to type your own blood, you need alcohol-soaked, soaked cotton balls, a sterile lancet, a small test tube containing 1 ml. of saline solution; anti – A, anti – B and anti – Rh serums with individual eye droppers, two microscope slides; a greased pencil; a pasture pipette, three applicator sticks and a warm fluorescent light or other low heat sources. One can make a typing of one’s own blood with these terms.

Analytical Outlines

• Of course, we don’t wish to learn about our blood type.
• Exercise is useful to us.
• It familiarises us with some sample laboratory techniques.
• It illustrates the use of basic equipment.
• It prepares us to follow the stages of an orderly scientific procedure.
• We can type our own blood.
• We need some equipment to do this.
• We require alcohol-soaked cotton balls.
• We require sterile lancet.
• We need a small test tube containing 1 ml of saline solution.
• We require anti-A
• We need anti-B
• We require anti-Rh serums with individual eye droppers.
• We require two microscope slides.
• We need a grease pencil.
• We require a Pasteur pipette.
• We need three applicator sticks.
• We also require a warm fluorescent light.
• We also need other low-heat sources.
• We require label one slide Rh with grease pencil.
• We have to place this slide under the low heat source.
• We have to divide the cool slide into two equal portion.
• We have to label one side as A and the other side B.
• We have to apply one drop of anti-A – A serum to slide – A.
• We have to apply one drop anti-B serum to slide – B.
• We have to apply one drop of anti – Rh serum to work Rh slide.
• We have to use an alcohol-soaked cotton ball to swab the middle or ring finger.
• We have to allow the excess alcohol to evaporate.
• After opening the sterile lancet, prick the sterile finger once.
• Now we have to collect several drops of blood in the test tube.
• The test tube also contains the saline solution.
• Now, we have to mix the solution.
• Again, we have to hold another sterile cotton ball over the cut.
• We have to allow the blood to clot.
• We have to transfer a drop of saline solution.
• It contains blood.
• It is transferred to anti-A.
• One drop is transferred to anti-B.
• Another drop is transferred to anti – Rh
• It is mixed using a separate applicator stick.
• It is allowed two or three minutes.
• Now, clumping should have appeared in A and B.
• Clumping denotes O blood.
• Rh – clumping means the blood is Rh- positive.
• The absence of Rh- clumping indicates it is Rh – negative

Meaning Of Difficult Words:

blood type – blood group
familiarise – intimate, make well-known
techniques – principles processes
illustrate – to explain, exemplify, show
I basic – fundamental, main, original
equipment – necessary instruments
procedure – principles, techniques
alcohol – pure spirit
soaked – absorbed
sterile – completely free from the seeds of disease
contain – comprise
saline – pertaining to salt
serum- liquid from of blood.
microscope – a magnifying instrument

Odisha State Board CHSE Odisha Class 12 Approaches to English Book 1 Solutions Unit 1 Text A: ‘Cures’ for the Common Cold Textbook Activity Questions and Answers.

CHSE Odisha 12th Class Alternative English Solutions Unit 1 Text A: ‘Cures’ for the Common Cold

Activity -1

Relation Between Parts of a Text:
If you are asked to divide the lesson into 5 sections in order to make notes, where possibly could you draw the lines separating the sections?
Write the paragraph number and the last word of the paragraph after which you will start a new section. Suggest a title for each section.
Answer:
Section- 1: Paragraph-1 …………… harmful
Title: Old Fashioned Remedies for Cold
Section- 2: Paragraphs – 2-4 …………avoided
Title: Morphine, Codeine and Papaverine as Remedies.
Section- 3: Paragraphs 5-6 ………… cold
Title: What The Scientists Studied.
Section- 4: Paragraphs 7-8 ………………..complications
Title: Opium Derivatives and Bed Rest.
Section- 5: Paragraph-9 ……………………..before
Title: Hot Baths and Cold Measurement

Activity – 2

Summary skill:
Of the following six statements only three are main points of the passage. Identify them:
(i) Many widely advertised cures and home remedies for cold are worthless or harmful
(ii) Students treated with sugar tablets showed little improvement.
(iii) Neither vaccines, nor vitamins and any other dietary measures prevent cold.
(iv) Nasal drops and sprays are found to be dangerous.
(v) Staying in bed for the duration of the cold was the only remedy that showed any result.

Activity-3

Comprehension:
Question 1.
The second paragraph possesses a question, what is it?
Answer:
The question is: Is there any remedy, then, of value in the treatment of colds?

Question 2.
What answer does the writer suggest?
Answer:
The writer says that there is scarcely any viable alternative for the treatment of common cold. However, there are a good many kinds of medicines which can be administered against cold.

Question 3.
How does the writer establish his answer?
Answer:
The writer picks up names like Morphine, Codeine, Papaverine combination, quinine hot water, air and stream baths were used as common therapies for cold but not as permanent cures.

Question 4.
What further recommendations did the writer make?
Answer:
The writer also brought out the names of different experts such as Dr. Russell Cecil, Dr. FitzHutter, De Quineeywhose findings were the best recommendations in the treatment of cold.

Activity – 4

Sequence In an Experiment:
What are the steps of the experiments mentioned in paragraph 4 and 5? Rearrange the steps given below in proper order:
(i) Record the health conditions of the patients at regular intervals.
(ii) Record initial health conditions of all the patients.
(iii) Compare the health conditions of the experiments group with that of the control group.
(iv) Prepare dummy to mixture.
(v) Draw inference after analysis of findings.
(vi) Divide the patients into experimental and control groups.
(vii) Select sample patients.
(viii) Prepare Codeine- Papaverine mixture.

Activity -5

Composition:
In this part of the country Tulsi leaves with honey are considered remedies for common cold. If you have to conduct an investigation to ascertain the truth of this belief, how will you organise the experiment? You can take clues from the reading passage and write down the steps of your proposed experiment.

Activity – 6

Remedial Grammar:
Morphine (which is) a derivative of opium, showed excellent results. (Paragraph – 3)
This preparation (which is) common called copavin, is not advertised to the public (Paragraph- 5)
In these sentences you have seen examples of non-defining relative clauses. Such clauses are separated from the main clauses with the help of commas. Secondly, the relative pronoun (like ‘which) and the ‘be’ verb can be omitted. The relative clauses without the relative pronoun and the ‘be’ verb are called the reduced relative clause. Similar reduction is possible in defining relative clauses also. Now reduce the relative clauses in the following sentences:
(a) They stood on the bridge which was connecting Cuttack with Jagatpur.
(b) The girl who is standing at the bus stop over there is my sister.
(c) The weapon that was used in the murder has been found.
(d) The boys who are being chosen for the college team are all under 18.
(e) The wooden beams which were holding up the roof have been damaged.

Activity – 7

Remedial Grammar:
1. Nasal congestion and stillness are reduced.
2. It was found that powered opium and Dover’s powder were beneficial.
3. The progress of the cold seemed to be arrested.
4. Commercial remedies are still sold.
In scientific tests were offer to see the examples of passive sentences. Whatever reduced nasal congestion, whoever found it out are unimportant in the first two sentences above. Similarly, we get examples of get-passive and have-passive scientific texts e.g.

When the boy gets chilled ___________.
I had my eyes tested.
Now rewrite the following sentences using passive structures like have/get + v + past participle.

The first one has been done for you.
1 . Our houses looked ugly. Its paint was pelling off.
So we got /had it painted.
2. Raman’s watch book. He could not afford to buy a new one.
So _____________
3. Lili split coffee on her favourite dress. She could not wash it by hand.
So _____________
4. In the super cyclone the roof was flown flourished and a wall fell down.
So _____________
5. Sharukh’s car was not starting well and seemed to be using too much petrol. But he did not want to sell his lucky car.
So _____________

Answer:
2. Raman’s watch broke. He could not afford to buy a new one.
So he had it repaired.
3. Lili split coffee on her favourite dress. She could not wash it by hand.
So she got it washed.
4. In the super cyclone the roof was flown of four shed and a wall fell down.
So we had it rebuilt.
5. Sharukh’s car was not starting well and seemed to be using too much petrol But he did not want to sell his lucky car.
So he got it repaired.

Section – A
New look at the little of the first passage. “Cures for The Common Cold.” What possible cures can you think of? Do you know that science has not yet brought us a cure for this disease? However, the quest continues to find a possible remedy, can you guess any home remedy that may cure common cold?
Now go through the text quickly and see if you guess right. You have only two minutes to do so. Read the text again and identify the cures that have been short-listed.

Section – B
In section A we read about a sequence of experiments to find a cure for the common cold. In Section B we shall read about a different kind of experiment whose purpose is to find out the types of human blood. What’s more interesting, you can learn how to determine your blood type as well as that of others.

‘Cures’ for the Common Cold Summary in English

Cures for the common cold comprise general skepticism. Millions of dollars is being spent for this every year. Obsolete cures like asafetida and camphor are not longer in vogue and popular remedies like vitamins, vaccines, nasal medications and other drugs have substituted them. Advertised remedies now available in the market sometimes prove worthless and harmful. There is absolutely to effective prevention of the common cold. Morphine which is a derivative of opium showed excellent results, but was rejected on account of its danger. But some other derivatives of opium which are less toxic and carry no practical danger of habituation proved to be definitely valuable. Codline and papaverine both proved valuable in the treatment of acute colds.

The codlin-papaverine combination proved to be, after Morphine, the most valuable of all cold medications. A preparation, consisting of one quarter grain of codeine and one quarter of grain of papaverine was finally selected as the most effective dosage. The main efficacy was a marked decrease or complete disappearance of nasal congestion and discharge. Most of the students were up and doing while taking this medication. Had they remained in bed while using it is probable that even better result have been obtained. This preparation commonly called copavin, is not advertised to the public. But it is available through physicians who should decide when and in what dosage it should be used. Dr. Russel CecilofNew York and Dr. Fritz Hutter of Vienna, both found that the codeine, paparine mixture was particularly beneficial if used by their patients at the very beginning of the affection.

Dr. Quincy, in his “Confessions” wrote that during the years in which he had taken opium he “never once caught cold, one the phrase in nor even the slightest cough. But after discontinuing the use of opium, a violent cold attacked me and a cough soon after.” Less effective, but still of moderate value were several other opium derivatives. In addition to codeine and papaverine it was found that powered opium and the old fashioned Dover’s powder were beneficial. Quinine also came to be included in this group of moderately valuable medications. In the end, certain general hygienic measures are helpful in the treatment of colds. Going to bed and remaining there until recovery is good advice.

The value of bed rest lies in protecting others from exposure, in necessary general resistance and in keeping the body warm. Hot baths for the treatment of colds may consist of hot water, hot air stream. The effect of these baths is to dialate the blood vessels of the skin and to increase blood flow through them. As a result, nasal congestion and stiffness are reduced. Other effects may be obtained with message of or other forms of physiotherapy, with hot or cold compresses, mustard plasters and certain, medicated ointments. If such treatments are followed by rest in bed with sufficient covers to prevent cooling, the effect is prolonged and the possibility of their being more than temporary benefit is increased.

Analytical Outlines:

• Cures for the common cold comprise of general skepticism.
• Millions of dollars is being spent for this every year.
• Asafetida and camphor are considered as obsole cures.
• These are no longer in vogue.
• These have been so far substituted.
• The substitutions are popular remedies.
• These are vitamins, vaccines, nasal medications etc.
• Now advertised remedies are available in the market.
• These are proved worthless and harmful.
• There is absolutely no effective prevention of common cold.
• Morphine is a derivative of opium.
• Morphine should excellent results.
• But it was rejected on account of its danger.
• However, some other derivatives of opium are taken.
• These are less toxic.
• They also carry no practical danger of habituation.
• Hence, it proved to be definitely valuable.
• Codeine and papaverine both proved valuable in the treatment of acute cold.
• The codeine-papaverine combination proved to be the most valuable of all cold medication after morphine.
• The preparation is made.
• One-quarter grain of codeine and one-quarter grain of papaverine are prepared together.
• It is finally selected as the most effective dosage.
• The main result was the marked decrease or complete disappearance of nasal congestion and discharge.
• Most of the students were up and doing while taking this medication.
• They had to remain in bed.
• So that they would have obtained better results.
• This preparation is commonly called copavin.
• It is not advertised to the public.
• But it is available through physicians.
• He is to decide about the dosage.
• Dr. Russell of New York and Dr. Fritz Hutter of Vienna found something about it.
• They found something beneficial about the mixture of codeine and papaverine.
• It is particularly beneficial for the patients at the very beginning of affection.
• Dr. Quincy in his “Confessions” wrote something.
• He wrote that the had taken opium for something.
• He marked that the had never caught by cold once.
• There was not even the slighest cough.
• But he discontinued the use of opium.
• He was attacked by a variant cold then.
• It was followed by a cough soon after.
• Several other opium derivative were less effective.
• They were still having with moderate value.
• It was found that powered opium and the old fashioned Dover’s powder were beneficial.
• Quinine belongs to this group.
• It is also accepted as the moderately valuable medication
• Certain general hygienic measures are considered.
• They are found helpful in the treatment of cold.
• Complete bed rest up to full recovery is a good advice.
• It lies in protecting others from exposure.
• It is necessary for general resistance.
• Again it keeps the body warm.
• We can accept hot baths for the treatment of cold.
• It may consist of hot water, hot air or stream
• Its effect is very important.
• It can dialate the blood vessels of the skin.
• Again, it can increase blood flow through them
• As a result of this, nasal congestion and stiffness are cured.
• Other effects may be obtained with message.
• We can also adopt other forms of physiotherapy.
• This can be done with hot or cold compresses.
• This can be done with hot or cold compresses.
• It can also be done by other medicated ointments.
• Such treatment should be followed with complete bed rest.
• It should be with sufficient covers to prevent cooling.
• Its effect to some extent prolonged.
• Again, the possibility of temporary benefit is increased.

Meaning Of Difficult Words:

remedies – panaceas: ways and means of cure.
in vogue – in prevalence, in operation
investigation – searching or examining carefully, enquiry into a matter.
uniformly – identically, almost the same, equally
distinct – distinguished, different, separate
discarded – cast off rejected, thrown away, not accepted
one- quarter- one-fourth something
De Quincey – Thomas De Quincey (1 785 – 59), English essayist and critic famous for “Confession of an English Opium- eater”, fascinating memories of distinguished by great imaginative power and splendid prose.
Octean – Jean Octean (1 889 – 1963) French poet, novelist, dramatist, film writer and director who was in the vanguard of almost every experimental artistic movement of the 20th century.

Odisha State Board Elements of Mathematics Class 12 Solutions CHSE Odisha Chapter 11 Differential Equations Ex 11(c) Textbook Exercise Questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Exercise 11(c)

Find the solutions of the following differential equations:
Question 1.
(x + y) dy + (x – y) dx = 0
Solution:
Given equation can be written as

Question 2.
\(\frac{d y}{d x}\) = \(\frac{1}{2}\left(\frac{y}{x}+\frac{y^2}{x^2}\right)\)
Solution:

Question 3.
(x² – y²) dx + 2xy dy = 0
Solution:
Given equation can be written as

Question 4.
x\(\frac{d y}{d x}\) + \(\sqrt{x^2+y^2}\) = y
Solution:
Given equation can be written as

Question 5.
x (x + y) dy = (x² + y²) dx
Solution:


This is the required solution.

Question 6.
y² + x² \(\frac{d y}{d x}\) = xy \(\frac{d y}{d x}\)
Solution:
Given equation can be written as

This is the required solution.

Question 7.
x sin\(\frac{y}{x}\) dy = \(\left(y \sin \frac{y}{x}-x\right)\)dx
Solution:
Given equation can be written as

Question 8.
x dy – y dx= \(\sqrt{x^2+y^2}\) dx
Solution:

This is the required solution.

Question 9.
\(\frac{d y}{d x}\) = \(\frac{y-x+1}{y+x+5}\)
Solution:
Given equation is


This is the required solution.

Question 10.
(x – y) dy = (x + y + 1) dx
Solution:
Given equation can be written as


This is the required solution.

Question 11.
(x – y – 2) dx + (x – 2y – 3) dy = 0
Solution:
Given equation can be written as


This is the required solution.

Question 12.
\(\frac{d y}{d x}\) = \(\frac{3 x-7 y+7}{3 y-7 x-3}\)
Solution:
Given equation can be written as


This is the required solution.

Question 13.
(2x + y + 1) dx + (4x + 2y – 1) dy = 0
Solution:
Given equation can be written as

⇒ 2z + ln (z – 1) = 3x + C
⇒ 2 (2x + y) + ln (2x + y – 1 ) = 3x + C
⇒ (x + 2y) + ln (2x + y – 1 ) = C
This is the required solution.

Question 14.
(2x + 3y – 5)\(\frac{d y}{d x}\) + 3x + 2y – 5 = 0
Solution:
Given equation can be written as

Question 15.
(4x + 6y + 5) dx – (2x + 3y + 4) dy = 0
Solution:
Given equation can be written as


This is the required solution.

Odisha State Board Elements of Mathematics Class 12 Solutions CHSE Odisha Chapter 11 Differential Equations Ex 11(b) Textbook Exercise Questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Exercise 11(b)

Solve the following differential equations.
Question 1.
\(\frac{d y}{d x}\) + y = e^-x
Solution:
Given equation is \(\frac{d y}{d x}\) + y = e^-x … (1)
This is a linear differential equation.
Here P = 1, Q = e^-x
So the integrating factor
I.F. = e^{∫P dx} = e^∫dx = e^x
The solution of (1) is given by
ye^x = ∫e^-x . e^x dx = ∫dx = x + C
⇒ y – xe^-x + Ce^-x

Question 2.
(x² – 1)\(\frac{d y}{d x}\) + 2xy = 1
Solution:
Given equation is (x² – 1)\(\frac{d y}{d x}\) + 2xy = 1

Question 3.
(1 – x²)\(\frac{d y}{d x}\) + 2xy = x \(\sqrt{1-x^2}\)
Solution:
Given equation is

Question 4.
x log x \(\frac{d y}{d x}\) + y = 2 log x
Solution:
Given equation is

Question 5.
(1 + x²)\(\frac{d y}{d x}\) + 2xy = cos x
Solution:

Question 6.
\(\frac{d y}{d x}\) + y sec x = tan x
Solution:
Given equation is
\(\frac{d y}{d x}\) + y sec x = tan x
This is a linear equation where
P = sec x, Q = tan x
I.F. = e^{∫sec dx}
= e^{(sec x + tan x)} = sec x + tan x
The solution is y . (sec x + tan x)
= ∫(sec x + tan x) tan x dx
= ∫(sec x tan x + tan² x) dx
= ∫(sec x . tan x + sec² x – 1) dx
= ∫(sec x + tan x) – x + C
⇒ (y – 1) (sec x + tan x) + x = C

Question 7.
(x + tan y) dy = sin 2y dx
Given equation can be written as
Solution:

Question 8.
(x + 2y³)\(\frac{d y}{d x}\) = y
Solution:
Given equation can be written as

Question 9.
sin x\(\frac{d y}{d x}\)+ 3y = cos x
Solution:

Question 10.
(x + y + 1)\(\frac{d y}{d x}\) = 1
Solution:

Question 11.
(1 + y²) dx + (x – \(e^{-\tan ^{-1} y}\)) dy = 0
Solution:
Given equation can be written as

Question 12.
x\(\frac{d y}{d x}\) + y = xy²
Solution:
Given equation can be written as

⇒ z = -x ln x + Cx
⇒ \(\frac{1}{y}\) = -x ln x + Cx
⇒ 1 = -xy ln x + Cxy
∴ The solution is (C – ln x) xy = 1

Question 13.
\(\frac{d y}{d x}\) + y = y² log x
Solution:

Question 14.
(1 + x²)\(\frac{d y}{d x}\) = xy – y²
Solution:
The given equation can be written as

Question 15.
\(\frac{d y}{d x}\) + \(\frac{y}{x-1}\) = \(x y^{\frac{1}{2}}\)
Solution:
The given equation can be written as

Question 16.
\(\frac{d y}{d x}\) + \(\frac{y}{x}\) = x², y(1) = 1
Solution:
The given equation can be written as
\(\frac{d y}{d x}\) + \(\frac{y}{x}\) = x², y(1) = 1 … (1)
This is a linear equation.

Question 17.
\(\frac{d y}{d x}\) + 2y tan x = sin x, y\(\left(\frac{\pi}{3}\right)\) = 0.
Solution:

Odisha State Board Elements of Mathematics Class 12 Solutions CHSE Odisha Chapter 11 Differential Equations Ex 11(a) Textbook Exercise Questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Exercise 11(a)

Question 1.
Determine the order and degree of each of the following differential equations.
(i) y sec² x dx + tan x dy = 0
Solution:
Order: 1, Degree: 1

(ii) \(\left(\frac{d y}{d x}\right)^4\) + y⁵ = \(\frac{d^3 y}{d x^3}\)
Solution:
Order: 3, Degree: 1

(iii) a\(\frac{d^2 y}{d x^2}\) = \(\left\{1+\left(\frac{d y}{d x}\right)^2\right\}^{\frac{3}{2}}\)
Solution:
Order: 2, Degree: 2

(iv) tan^-1\(\sqrt{\frac{d y}{d x}}\) = x
Solution:
Order: 1, Degree: 1

(v) ln\(\left(\frac{d^2 y}{d x^2}\right)\) = y
Solution:
Order: 2, Degree: 1

(vi) \(\frac{\frac{d y}{d t}}{y+\frac{d y}{d t}}\) = \(\frac{y t}{d y}\)
Solution:
Order: 1, Degree: 2

(vii) \(\frac{d^2 y}{d u^2}\) = \(\frac{3 y+\frac{d y}{d u}}{\sqrt{\frac{d^2 y}{d u^2}}}\)
Solution:
Order: 2, Degree: 3

(viii) \(e^{\frac{d z}{d x}}\) = x²
Solution:
Order: 1, Degree: 1

Question 2.
Form the differential equation by eliminating the arbitrary constants in each of the following cases.
(i) y = A sec x
Solution:
y = A sec x
Then \(\frac{d y}{d x}\) = A sec x tan x = y tan x

(ii) y = C tan^-1 x
Solution:

(iii) y = Ae^t + Be^2t
Solution:

(iv) y = Ax² + Bx
Solution:

(v) y = -acos x + b sin x
Solution:

(vi) y = a sin^-1 x + b cos^-1 x
Solution:

(vii) y = at + be^t
Solution:

(viii) y = a sin t + be^t
Solution:

(ix) ax² + by = 1
Solution:

Question 3.
Find the general solution ofthe following differential equations.
(i) \(\frac{d y}{d x}\) = \(\frac{e^{2 x}+1}{e^x}\)
Solution:
\(\frac{d y}{d x}\) = \(\frac{e^{2 x}+1}{e^x}\)
⇒ y = ∫(e^x + e^-x) dx = e^x – e^-x + C

(ii) \(\frac{d y}{d x}\) = x cos x
Solution:
\(\frac{d y}{d x}\) = x cos x
⇒ y = ∫x cos x dx
= x . sin x – ∫sin x dx – x sin x + cos x + C

(iii) \(\frac{d y}{d x}\) = t⁵ log t
Solution:

(iv) \(\frac{d y}{d x}\) = 3t² + 4t + sec² t
Solution:
\(\frac{d y}{d x}\) = 3t² + 4t + sec² t
⇒ y = t³ + 2t² + tan t + C

(v) \(\frac{d y}{d x}\) = \(\frac{1}{x^2-7 x+12}\)
Solution:

(vi) \(\frac{d y}{d u}\) = \(\frac{u+1}{\sqrt{3 u^2+6 u+5}}\)
Solution:

(vii) (x² + 3x + 2) dy – dx = 0
Solution:

(viii) \(\frac{d y}{d t}\) = \(\frac{\sin ^{-1} t e^{\sin ^{-1} t}}{\sqrt{1-t^2}}\)
Solution:

Question 4.
Solve the following differential equations.
(i) \(\frac{d y}{d x}\) = y + 2
Solution:

(ii) \(\frac{d y}{d t}\) = \(\sqrt{1-y^2}\)
Solution:
\(\frac{d y}{d t}\) = \(\sqrt{1-y^2}\)
⇒ \(\frac{d y}{\sqrt{1-y^2}}\) = dt
⇒ sin^-1 y = t + C

(iii) \(\frac{d y}{d z}\) = sec y
Solution:
\(\frac{d y}{d z}\) = sec y
⇒ cos y dy = dz
⇒ sin y = z + C

(iv) \(\frac{d y}{d x}\) = e^y
Solution:

(v) \(\frac{d y}{d x}\) = y² + 2y
Solution:

(vi) dy + (y² + 1) dx = 0
Solution:

(vii) \(\frac{d y}{d x}\) + \(\frac{e^y}{y}\) = 0
Solution:

(viii) dx + cot x dt = 0
Solution:
dx + cot x dt = 0
⇒ tan x dx + dt = 0
⇒ ∫tan x dx + ∫dt = C₁
⇒ In sec x + t = C₁
⇒ In sec x = C₁ – t
⇒ sec x = \(e^{C_1}\) . e^-t
⇒ cos x = \(e^{-C_1}\) . e^t
⇒ cos x = Ce^t where C = \(e^{-C_1}\)

Question 5.
Obtain the general solution of the following differential equations.
(i) \(\frac{d y}{d x}\) = (x² + 1) (y² + 1)
Solution:

(ii) \(\frac{d y}{d t}\) = e^2t+3y
Solution:

⇒ 2e^-3y + 3e^2t + 6C₁ = 0
⇒ 2e^-3y + 3e^2t = C
where C = -6C₁

(iii) \(\frac{d y}{d z}\) = \(\frac{\sqrt{1-y^2}}{\sqrt{1-z^2}}\)
Solution:

(iv) \(\frac{d y}{d z}\) = \(\frac{x \log x}{3 y^2+4 y}\)
Solution:

(v) x²\(\sqrt{y^2+3}\) dx + y\(\sqrt{x^3+1}\) dy = 0
Solution:

(vi) tan y dx + cot x dy = 0
Solution:
tan y dx + cot x dy = 0
⇒ tan x . dx + cot y dy = 0
⇒ ∫tan x dx + ∫cot y dy = 0
⇒ -ln cos x + ln siny = ln C
⇒ ln\(\frac{\sin y}{\cos x}\) = ln C
⇒ \(\frac{\sin y}{\cos x}\) = C
⇒ sin y = C cos x

(vii) (x² + 7x + 12) dy + (y² – 6y + 5) dx = 0
Solution:

(viii) y dy + e^-y x sin x dx = 0
Solution:
y dy + e^-y x sin x dx = 0
⇒ ye^y dy + x sin x dx = 0
⇒ ∫ye^y dy + ∫x sin dx = C
[Integrating by parts.
⇒ ye^y – ∫e^y dy + x(-cos x) – ∫(-cos x) dx = C
⇒ ye^y – e^y – x cos x + sin x = C
⇒ (y – 1) e^y – x cos x + sin x = C

Question 6.
Solve the following second order equations.
(i) \(\frac{d^2 y}{d x^2}\) = 12x² + 2x
Solution:

(ii) \(\frac{d^2 y}{d t^2}\) =e^2t +e^-t
Solution:

(iii) \(\frac{d^2 y}{d \vartheta^2}\) = -sin υ + cos υ + sec² υ
Solution:
\(\frac{d^2 y}{d \vartheta^2}\) = -sin υ + cos υ + sec² υ
Integrating we get
\(\frac{d y}{d υ}\) = ∫sin υ dυ + ∫cos υ dυ + ∫sec² υ dυ
= cos υ + sin υ + tan υ + A
Again integratingwe get
y = ∫(cos υ + sin υ + tan υ + A)dυ + B
where A, B are arbritrary constants.
⇒ y = sin υ – cos υ + ln |sec υ| + A.υ. + B

(iv) cosec x \(\frac{d^2 y}{d x^2}\) = x
Solution:
cosec x \(\frac{d^2 y}{d x^2}\) = x
\(\frac{d^2 y}{d x^2}\) = x sin x
Integrating we get
\(\frac{d y}{d x}\) = ∫x sin x dx + A
= x . (-cos x) – ∫(-cos x) dx + A
= -x cos x + ∫cos x dx + A
= -x cos x + sin x + A
Again integrating we get
y = -∫x cos x dx + ∫sin x + ∫A dx + B
= -{x sin x -∫1 . sin x dx} – cos x + Ax + B
= -x sin x – 2cos x + Ax + B

(v) x²\(\frac{d^2 y}{d x^2}\) + 2 = 0
Solution:

(vi) sec x \(\frac{d^2 y}{d x^2}\) = sec 3x
Solution:

(vii) \(\frac{d^2 y}{d x^2}\) = sec² x + cos² x
Solution:

(viii) e^-x\(\frac{d^2 y}{d x^2}\) = x
Solution:
e^–x\(\frac{d^2 y}{d x^2}\) = x
⇒ \(\frac{d^2 y}{d x^2}\) = xe^x
Integrating we get
\(\frac{d y}{d x}\) = ∫xe^x dx = ∫e^x dx + Ax + B
= xe^x – e^x – e^x + Ax + B
= (x – 2)e^x + Ax + B

Question 7.
Find the particular solutions of the following equations subject to the given conditions.
(i) \(\frac{d y}{d x}\) = cos x, given that y = 2 when x = 0.
Solution:
\(\frac{d y}{d x}\) = cos x
Integrating we get
y = ∫cos x dx = sin x + C
Given that when x = 0, y = 2
So 2 = C
∴ The particular solution is y = sin x + 2

(ii) \(\frac{d y}{d t}\) = cos² y subject to y = \(\frac{\pi}{4}\) when t = 0.
Solution:
\(\frac{d y}{d t}\) = cos² y
⇒ sec² y dy = dt
∫sec² dy = ∫dt
⇒ tan y = t + C
When t = 0, y = \(\frac{\pi}{4}\)
So tan \(\frac{\pi}{4}\) = C ⇒ C = 1
∴ The particular solution is tan y = t + 1

(iii) \(\frac{d y}{d x}\) = \(\frac{1+y^2}{1+x^2}\) given that y = √3 when x = 1.
Solution:

(iv) \(\frac{d^2 y}{d x^2}\) = 6x given that y = 1 and \(\frac{d y}{d x}\) = 2 when x = 0.
Solution:
\(\frac{d^2 y}{d x^2}\) = 6x ⇒ \(\frac{d y}{d x}\) = 3x² + 2
When x = 0, \(\frac{d y}{d x}\) = 2
So 2 = A
∴ \(\frac{d y}{d x}\) = 3x² + 2
Again integrating we get
y = x³ + 2x + B
When x = 0, y = 1
So B = 1.
∴ The particular solution is y = x³ + 2x + 1

Question 8.
(i) Solve : \(\frac{d y}{d x}\) = sec (x + y)
Solution:

(ii) Solve : \(\frac{d y}{d x}\) = sin(x + y) + cos(x + y)
Solution:

(iii) Solve : \(\frac{d y}{d x}\) = cos (x + y)
Solution:

(iv) Solve : \(\frac{d y}{d x}\) + 1 = e^x+y
Solution:

Odisha State Board Elements of Mathematics Class 12 CHSE Odisha Solutions Chapter 10 Area Under Plane Curves Ex 10 Textbook Exercise Questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 10 Area Under Plane Curves Exercise 10

Question 1.
Find the area bounded by
(i) y = e^x, y = 0, x = 4, x = 2
Solution:
Area = \(\int_2^4\)e^x dx
= \(\left[e^x\right]_2^4\)
= e⁴ – e²

(ii) y = x², y = 0, x = 1
Solution:
Area = \(\int_0^1\)x² dx
= \(\left[\frac{x^3}{3}\right]_0^1\)
= \(\frac{1}{3}\)

(iii) xy = a², y = 0, x = α, x = β (β > α > 0)
Solution:
Area = \(\int_\alpha^\beta y\)y dx
= \(\int_\alpha^\beta \frac{a^2}{x}\) dx
= a²\([\ln x]_\alpha^\beta\)
= a² ln (β/α)

(iv) y = sin x, y = 0, x = \(\frac{\pi}{2}\)
Solution:
Area = \(\int_0^{\frac{\pi}{2}}\)sin x dx
= \([-\cos x]_0^{\frac{\pi}{2}}\)
= -cos\(\frac{\pi}{2}\) + cos θ = 1

Question 2.
Find the area enclosed by
(i) y = e^x, x = 0, y = 2, y = 3
Solution:

(ii) y² = x, x = 0, y = 1
Solution:
Area = \(\int_0^1\)x dy
= \(\int_0^1\)y² dy
= \(\left[\frac{y^3}{3}\right]_0^1\)
= \(\frac{1}{3}\)

(iii) xy = a², x = 0, y = α, y = β (β > α > 0)
Solution:
Area = \(\int_\alpha^\beta\)x dy
= \(\int_\alpha^\beta \frac{a^2}{y}\)dy
= a²\([\ln y]_\alpha^\beta\)
= a² ln (β/α)

(iv) y² = x³, x = 0, y = 1
Solution:
Given curve is y² = x³
⇒ x = y^2/3
It passes through the origin. So the required area
= \(\int_0^1\)x dy
= \(\int_0^1 y^{\frac{2}{3}}\) dy
= \(\left[\frac{y^{\frac{5}{3}}}{5 / 3}\right]_0^1\)
= \(\frac{3}{5}\)

Question 3.
(i) Determineellipse the area with in the ellipse
\(\frac{x^2}{a^2}\) + \(\frac{y^2}{b^2}\) = 1
Solution:

The ellipse is symmetrical about x-axis and y-axis.It is divided into 4 equal parts by the coordinate axes. So required area

(ii) Find the area of the circle x² + y² = 2ax.
Solution:
Given circle is x² + y² = 2ax
⇒ x² – 2ax + a² + y² = a²
⇒ (x – a)² + y² = a² … (1)
The centre of the circle is (a, 0) and radius is a.

(iii) Find the area of the portion of the parabola y² = 4x bounded by the double ordinate through (3, 0).
Solution:
Given parabola is y² = 4x

(iv) Determine the area of the region bounded by y² = x³ and the double ordinate through (2, 0)
Solution:
Given curve is y² = x³
⇒ y = ±x^3/2 … (1)
The curve passes through the origin and symmetrical about x-axis because the power of y is even.

Question 4.
(i) Find the area of the regions into which the circle x² + y² = 4 is divided by the line x + √3y = 2.
Solution:
Given circle and the straight line are x²+ y² = 4 and x+ √3y = 2
The circle has the centre at (0, 0) and radius ‘2’.
The eqn. (2) can be written as
y = –\(\frac{1}{\sqrt{3}}\)x + \(\frac{2}{\sqrt{3}}\)
Slope of the strainght line = –\(\frac{1}{\sqrt{3}}\)
The line makes and angle of 150° with x-axis making intercept \(\frac{2}{\sqrt{3}}\) from y-axis.
It intersects x-axis at (2, 0).

Solveing (1) and (2),
ge wet (2 – 3√y)² + y² = 4
4 + 3√y² – 4√3y + y² = 4
4y² – 4√3y = 0
y(y – √3) = 0
y = 0 or y = √3
When y = 0, x = 2
When y = √3, x = -1
Thus the straight line intersects the circle at (2, 0) and (-1, √3).
Area of the portion ACBA.

(ii) Determine the area of the region between the curves y = cos x and y = sin x, bounded by x = 0.
Solution:

The curves y = cos x and y = sin x are shown in the above figure. The region included between these two curves in [0, \(\frac{\pi}{4}\)] is OABO.

(iii) Find the area enclosed by the two parabolas y² = 4 ax and x² = 4ay.
Solution:
The given parabolas are y² = 4ax and x² = 4ay.
The graphs of the two parabolas are shown in the figure.

⇒ x⁴ = 64 a⁴
⇒ x⁴ – 64 a³x = 0
⇒ x (x³ – (4a)³) = 0
⇒ x (x – 4a) (x² + 4ax + 16a²) = 0
⇒ x = 0, 4a
When x = 0, y = 0 and
when x = 4a, y = 4a
Thus the two parabolas intersect at (0, 0) and (4a, 4a).
Area between two parabolas

(iv) Determine the area common to the parabola y² = x and the circle x²+ y² = 2x.
Solution:
Gien parabola is y² = x
Given circle is
x² + y² = 2x ⇒ (x – 1)² + y² = 1
The centre is at (1, 0) and radius is 1.

Solving (1) and (2) we get
x² + x = 2x ⇒ x² – x = 0 ⇒ x(x – 1) = 0
⇒ x = 0, x = 1
When x = 0, y = 0 and when x = 1, y = 1.
Thus both the parabola and circle intersect at (0, 0) and (1, 1).
Required Area

Odisha State Board Elements of Mathematics Class 12 Solutions CHSE Odisha Chapter 9 Integration Additional Exercise Textbook Exercise questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Additional Exercise

Question 1.
∫\(\sqrt{1-\sin 2 x}\) dx
Solution:
I = ∫\(\sqrt{1-\sin 2 x}\) dx
= ∫\(\sqrt{(\cos x-\sin x)^2}\) dx
= ∫(cos x – sin x) dx
= sin x + cos x + c

Question 2.
∫\(\frac{d x}{1+\sin x}\)
Solution:
I = ∫\(\frac{d x}{1+\sin x}\)
= ∫\(\frac{1-\sin x}{\cos ^2 x}\)
= ∫sec² x – sec x tan x dx
= tan x – sec x + c

Question 3.
∫\(\frac{\sin x}{1+\sin x}\) dx
Solution:

Question 4.
∫\(\frac{\sec x}{\sec x+\tan x}\) dx
Solution:
I = ∫\(\frac{\sec x}{\sec x+\tan x}\) dx
= ∫\(\frac{\sec x(\sec x-\tan x)}{\sec ^2 x-\tan ^2 x}\) dx
= ∫sec² x – sec x tan x dx
= tan x – sec x + c

Question 5.
∫\(\frac{1+\sin x}{1-\sin x}\) dx
Solution:
I = ∫\(\frac{1+\sin x}{1-\sin x}\) dx
= ∫\(\frac{(1+\sin x)^2}{\cos ^2 x}\) dx
= ∫[sec² x+ tan² x+ 2sec x tan x) dx
= ∫[2sec² x – 1 + 2sec x tan x) dx
= 2tan x – x + 2sec x + c

Question 6.
∫tan^-1 (sec x + tan x) dx
Solution:

Question 7.
∫\(\frac{\cos 2 x-\cos 2 \alpha}{\cos x-\cos \alpha}\) dx
Solution:
I = ∫\(\frac{\cos 2 x-\cos 2 \alpha}{\cos x-\cos \alpha}\) dx
= ∫\(\frac{\left(2 \cos ^2 x-1\right)-\left(2 \cos ^2 \alpha-1\right)}{\cos x-\cos \alpha}\) dx
= 2 ∫(cos x + cos α) dx
= 2 sin x + 2x cos α + c

Question 8.
∫tan-1\(\sqrt{\frac{1-\cos 2 x}{1+\cos 2 x}}\) dx
Solution:

Question 9.
∫\(\frac{d x}{\sqrt{x+1+} \sqrt{x+2}}\)
Solution:

Question 10.
∫\(\frac{2+3 x}{3-2 x}\) dx
Solution:

Question 11.
∫\(\frac{d x}{\sqrt{x}+x}\)
Solution:

Question 12.
∫\(\frac{d x}{1+\tan x}\)
Solution:

Question 13.
∫\(\frac{x+\sqrt{x+1}}{x+2}\) dx (Hints put : \(\sqrt{x+1}\) = t)
Solution:

Question 14.
∫sin^-1\(\sqrt{\frac{x}{a+x}}\) dx (Hints put : x = a tan² t)
Solution:

Question 15.
∫e^x\(\left(\frac{2+\sin 2 x}{1+\cos 2 x}\right)\) dx
Solution:

Question 16.
∫\(\frac{\left(x^2+1\right) e^x}{(x+1)^2}\) dx
Solution:

Question 17.
∫\(\frac{x^2-1}{x^4+x^2+1}\) dx
Solution:

Question 18.
∫\(\frac{x^2 d x}{x^4+x^2+1}\)
Solution:

Question 19.
∫\(\sqrt{\cot x}\) dx
Solution:

Question 20.
∫\((\sqrt{\tan x}+\sqrt{\cot x})\) dx
Solution:

Question 21.
∫\(\frac{\mathrm{dx}}{x\left(x^4+1\right)}\)
Solution:

Question 22.
∫\(\frac{\mathrm{dx}}{e^x-1}\)
Solution:

Question 23.
∫\(\frac{(x-1)(x-2)(x-3)}{(x+4)(x-5)(x-6)}\) dx
Solution:

Question 24.
∫\(\frac{d x}{\left(e^x-1\right)^2}\)
Solution:

Question 25.
∫\(\frac{d x}{\sin x \cos ^2 x}\)
Solution:

Question 26.
\(\int_2^4 \frac{\left(x^2+x\right) d x}{\sqrt{2 x+1}}\)
Solution:

Question 27.
\(\int_{-a}^a \sqrt{\frac{a-x}{a+x}}\) dx
Solution:

Let a² – x² = t²
-2x dx = 2t dt
x = -a ⇒ 0 t = 0
x = a ⇒ t = 0
= 0
I = aI₁ – I₂ = aπ

Question 28.
\(\int_0^{\pi / 2}(\sqrt{\tan x}+\sqrt{\cot x})\) dx
Solution:

Question 29.
\(\int_0^{\pi / 2} \frac{\cos x d x}{1+\cos x+\sin x}\)
Solution:

Question 30.
\(\int_0^1\)x (1 – x)ⁿ dx
Solution:

Question 31.
\(\int_0^{\pi / 2}\)sin 2x log (tan x) dx
Solution:

Question 32.
\(\int_0^{\pi / 2} \frac{\sin ^2 x d x}{\sin x+\cos x}\)
Solution:

Question 33.
\(\int_0^{\pi / 2} \frac{\sin ^2 x d x}{1+\sin x \cos x}\)
Solution:

Question 34.
\(\int_0^{\pi / 2} \frac{x d x}{\sin x+\cos x}\)
Solution:

Question 35.
Prove that \(\int_0^\pi\) x sin³ x dx = \(\frac{2 \pi}{3}\)
Solution:

Question 36.
\(\int_{\pi / 5}^{3 \pi / 10} \frac{\sin x d x}{\sin x+\cos x}\)
Solution:

Question 37.
\(\int_0^\pi\)|cos x| dx
Solution:

Question 38.
\(\int_1^4\)(|x – 1| + |x – 2| + |x – 3|) dx
Solution:

Question 39.
\(\int_{-\pi / 2}^{\pi / 2}\)(sin |x| + cos |x|) dx
Solution:

Question 40.
\(\int_0^\pi\)log (1 + cos x) dx
Solution:

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

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

Question 1.
\(\int_0^{\frac{\pi}{2}}\)sin¹⁰ θ dθ
Solution:
\(\int_0^{\frac{\pi}{2}}\)sin¹⁰ θ dθ = \(\frac{9}{10} \cdot \frac{7}{8} \cdot \frac{5}{6} \cdot \frac{3}{4} \cdot \frac{1}{2} \cdot \frac{\pi}{2}\) = \(\frac{405 \pi}{7680}\)

Question 2.
\(\int_0^{\frac{\pi}{2}}\)cos¹² θ dθ
Solution:
\(\int_0^{\frac{\pi}{2}}\)cos¹² θ dθ = \(\frac{11}{12} \cdot \frac{9}{10} \cdot \frac{7}{8} \cdot \frac{5}{6} \cdot \frac{3}{4} \cdot \frac{1}{2} \cdot \frac{\pi}{2}\) = \(\frac{4455 \pi}{92160}\)

Question 3.
\(\int_0^{\frac{\pi}{2}}\)sin¹¹ θ dθ
Solution:
\(\int_0^{\frac{\pi}{2}}\)sin¹¹ θ dθ = \(\frac{10}{11} \cdot \frac{8}{9} \cdot \frac{6}{7} \cdot \frac{4}{5} \cdot \frac{2}{3}\) = \(\frac{3840}{4455}\)

Question 4.
\(\int_0^{\frac{\pi}{2}}\)cos⁹ θ dθ
Solution:
\(\int_0^{\frac{\pi}{2}}\)cos⁹ θ dθ = \(\frac{8}{9} \cdot \frac{6}{7} \cdot \frac{4}{5} \cdot \frac{2}{3}\) = \(\frac{384}{405}\)

Question 5.
\(\int_0^1 \frac{x^7}{\sqrt{1-x^2}}\) dx
Solution:

Question 6.
\(\int_0^1 \frac{x^5\left(4-x^2\right)}{\sqrt{1-x^2}}\) dx
Solution:

Question 7.
\(\int_0^a x^3\left(a^2-x^2\right)^{\frac{5}{2}}\) dx
Solution:

Question 8.
\(\int_0^1 x^5 \sqrt{\frac{1+x^2}{1-x^2}}\) dx
Solution:

Question 9.
\(\int_0^{\infty} \frac{x^2}{\left(1+x^6\right)^n}\) dx
Solution:

Question 10.
\(\int_0^\pi\)sin⁸ θ dθ
Solution:

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

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

Evaluate the following Integrals:
Question 1.
(i) \(\int_0^{\frac{\pi}{2}} \frac{d x}{1+\tan x}\)dx
Solution:

(ii) \(\int_0^{\frac{\pi}{2}} \frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}\)dx
Solution:

(iii) \(\int_0^1 \frac{\ln (1+x)}{2+x^2}\)dx (x = tan θ)
Solution:

(iv) \(\int_0^\pi \frac{x d x}{1+\sin x}\)
Solution:

Question 2.
(i) \(\int_{-a}^a\)x⁴ dx
Solution:

(ii) \(\int_{-a}^a\)(x⁵ + 2x² + x) dx
Solution:

(iii) \(\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}}\)cos² x dx
Solution:

(iv) \(\int_{-\frac{\pi}{6}}^{\frac{\pi}{6}}\)sin⁵ x dx
Solution:
Let f(x) = sin⁵ x
Then f(-x) = sin⁵ (-x)
= -sin⁵ x = -f(x)
So f(x) is an odd function.
Thus \(\int_{-a}^a\)f(x) dx = 0
\(\int_{-\frac{\pi}{6}}^{\frac{\pi}{6}}\)sin⁵ x dx = 0

Question 3.
(i) \(\int_0^\pi\)cos³ x dx
Solution:

(ii) \(\int_0^\pi\)cos² x dx
Solution:

(iii) \(\int_0^\pi\)sin³ x cos x dx
Solution:
\(\int_0^\pi\)sin³ x cos x dx
[Put sin x = t, then cos x dx = dt
When x = 0, t = 0, when x = π, t = 0
\(\int_0^\pi\)t³ dt = 0

(iv) \(\int_0^\pi\)sin x cos² x dx
Solution:

Question 4.
Show that
(i) \(\int_0^1 \frac{\ln x}{\sqrt{1-x^2}}\) dx = \(\frac{\pi}{2} \ln \frac{1}{2}\)
Solution:

(ii) \(\int_0^{\frac{\pi}{2}} \frac{\cos x-\sin x}{1+\sin x \cos x}\) dx = 0
Solution:

(iii) \(\int_0^\pi\)x ln sin x dx = \(\frac{\pi^2}{2} \ln \frac{1}{2}\)
Solution:

Question 5.
(i) \(\int_0^{\pi / 2}\)ln (tan x + cot x) dx
Solution:

(ii) \(\int_0^\pi \frac{x \tan x-\sin x}{1+\sin x \cos x}\) dx
Solution:

(iii) \(\int_1^3 \frac{\sqrt{x} d x}{\sqrt{4-x}+\sqrt{x}}\)
Solution:

(iv) \(\int_0^\pi \frac{x \sin x d x}{1+\cos ^2 x}\)
Solution:

(v) \(\int_0^1\)x (1 – x)¹⁰⁰ dx
Solution:

(vi) \(\int_{\pi / 6}^{\pi / 3} \frac{d x}{1+\sqrt{\cot x}}\)
Solution:

(vii) \(\int_0^{50}\)e^x-[x] dx
Solution:

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

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

Evaluate the following Integrals.
Question 1.
(i) ∫\(\frac{d x}{4+5 \cos x}\)
Solution:

(ii) ∫\(\frac{d x}{3+\cos x}\)
Solution:

(iii) ∫\(\frac{d x}{3+\sin x}\)
Solution:

(iv) ∫\(\frac{d x}{1+2 \sin x}\)
Solution:

(v) ∫\(\frac{d x}{2 \sin x+3 \cos x}\)
Solution:

(vi) ∫\(\frac{d x}{1+\cos x+\sin x}\)
Solution:

Question 2.
(i) ∫\(\frac{3 \sin x+28 \cos x}{5 \sin x+6 \cos x}\) dx
Solution:

(ii) ∫\(\frac{12 \sin x-2 \cos x+3}{\sin x+\cos x}\) dx
Solution:
Let 12 sin x – 2 cos x = A (sin x + cos x) + B ( cos x – sin x)
[Note that cos x – sin x is the derivative of sin x + cos x]
Then A – B = 12, A + B = -2
⇒ 2A = 10
⇒ A = 5, B = -7
Thus 12 sin x – 2 cos x = 5 (sin x + cos x) – 7 (cos x – sin x)

(iii) ∫\(\frac{5 \sin x}{3-2 \sin x}\) dx
Solution:

(iv) ∫\(\frac{2 \cos x+7}{4-\sin x}\) dx
Solution:

Question 3.
(i) ∫\(\frac{d x}{2 \cos ^2 x+3 \cos x}\)
Solution:

(ii) ∫\(\frac{d x}{4 \sin ^2 x-\sin x}\)
Solution:

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

(iv) ∫\(\frac{d x}{\cos x-\cos 3 x}\)
Solution:

Question 4.
(i) ∫\(\frac{d \theta}{4+3 \sin ^2 \theta}\)
Solution:

(ii) ∫\(\frac{d \theta}{2-3 \cos ^2 \theta}\)
Solution:

(iii) ∫\(\frac{d \theta}{4 \cos ^2 \theta+9 \sin ^2 \theta}\)
Solution:

(iv) ∫\(\frac{d \theta}{2+3 \cos ^2 \theta-4 \sin ^2 \theta}\)
Solution:

Question 5.
(i) ∫\(\frac{\sin 3 x}{\cos 7 x \cos 4 x}\) dx
Solution:

(ii) ∫\(\frac{\cos 2 x}{\sin 7 x \cos 5 x}\) dx
Solution:

Question 6.
(i) ∫\(\frac{d x}{\cos x(5+3 \cos x)}\)
Solution:

(ii) ∫\(\frac{d x}{\cos x(1+2 \sin x)}\)
Solution: