Tamilnadu Board

Tamilnadu State Board Class 9 Maths Solutions Chapter 2 Real Numbers Ex 2.2

Question 1.
Express the following rational numbers into decimal and state the kind of decimal expansion.
(i) \(\frac { 2 }{ 7 }\)
(ii) \(-5 \frac{3}{11}\)
(iii) \(\frac { 22 }{ 3 }\)
(iv) \(\frac { 327 }{ 200 }\)
Solution:
(i) \(\frac { 2 }{ 7 }\)

\(\frac{2}{7}=0 . \overline{285714}\)
Nen-terminating and recurring

(ii) \(-5 \frac{3}{11}\)

\(-5 \frac{3}{11}=-5 . \overline{27}\)
Nen-terminating and recurring

(iii) \(\frac { 22 }{ 3 }\)

\(\frac{22}{3}=7 . \overline{3}\)
Nen-terminating and recurring

(iv) \(\frac { 327 }{ 200 }\)

\(\frac { 327 }{ 200 }\) = 1.635, Terminating.

Question 2.
Express \(\frac { 1 }{ 13 }\) in decimal form. Find the length of the period of decimals.
Solution:

\(\frac{1}{13}=0 . \overline{076923}\) has the length of the period of decimals = 6.

Question 3.
Express the rational number \(\frac { 1 }{ 13 }\) in recurring decimal form by using the recurring decimal expansion of \(\frac { 1 }{ 11 }\) . Hence write \(\frac { 71 }{ 33 }\) in recurring decimal form.
Solution:
The recurring decimal expansion of \(\frac { 1 }{ 11 }\) = 0.09090909…. = \(0.\overline { 09 }\)

Question 4.
Express the following decimal expression into rational numbers.
(i) \(0.\overline { 24 }\)
(ii) \(2.\overline { 327 }\)
(iii) -5.132
(iv) \(3.1\overline { 7 }\)
(v) \(17.\overline { 215 }\)
(vi) \(-21.213\overline { 7 }\)
Solution:
(i) \(0.\overline { 24 }\)
Let x = \(0.\overline { 24 }\) = 0.24242424……… ….(1)
(Here period of decimal is 2, multiply equation (1) by 100)
100x = 24.242424 ………. ….(2)
(2) – (1)
100x – x = 24.242424…. – 0.242424….
99x = 24
x = \(\frac { 24 }{ 99 }\)

ii. \(2.\overline { 327 }\)
Let x = 2.327327327…… ….(1)
(Here period of decimal is 3, multiply equation (1) by 1000)
1000x = 2327.327…. ….(2)
(2) – (1)
1000x – x = 2327.327327… – 2.327327….
999x = 2325
x = \(\frac { 2325 }{ 999 }\)

(iii) -5.132
\(x=-5.132=\frac{-5132}{1000}=\frac{-1283}{250}\)

(iv) \(3.1\overline { 7 }\)
Let x = 3.1777 ……. ..(1)
(Here the repeating decimal digit is 7, which is the second digit after the decimal point, multiply equation (1) by 10)
10x = 31.7777 …….. ..(2)
(Now period of decimal is 1, multiply equation (2) by 10)
100x = 317.7777…….. …(3)
(3) – (2)
100x – 10x = 317.777…. – 31.777….
90x = 286
\(x=\frac{286}{90}=\frac{143}{45}\)

(v) \(17.\overline { 215 }\)
Let x = 17.215215 ……. ….(1)
1000x = 17215.215215…… ….(2)
(2) – (1)
1000x – x = 17215.215215… – 17.215…
999x = 17198
x = \(\frac { 17198 }{ 999 }\)

(vi) \(-21.213\overline { 7 }\)
Let x = -21.2137777… …(1)
10x = -212.137777…… …(2)
100x = -2121.37777…… …(3)
1000x = -21213.77777…. …(4)
10000x = 212137.77777….. ..(5)
(Now period of decimal is 1, multiply equation (4) it by 10)
(5) – (4)
10000x – 1000x = (-212137.7777…) – (-21213.7777…)
9000x = -190924
x = –\(\frac { 190924 }{ 9000 }\)

Question 5.
Without actual division, find which of the following rational numbers have terminating decimal expansion.
(i) \(\frac { 7 }{ 128 }\)
(ii) \(\frac { 21 }{ 15 }\)
(iii) 4\(\frac { 9 }{ 35 }\)
(iv) \(\frac { 219 }{ 2200 }\)
Solution:
(i) \(\frac { 7 }{ 128 }\)

So \(\frac{7}{128}=\frac{7}{2^{7} 5^{0}}\)
This of the form 4m, n ∈ W
So \(\frac { 7 }{ 128 }\) has a terminating decimal expansion.

(ii) \(\frac { 21 }{ 15 }\)

So \(\frac { 21 }{ 15 }\) has a terminating decimal expansion.

(iii) 4\(\frac { 9 }{ 35 }\) = \(\frac { 149 }{ 35 }\)

\(\frac{49}{35}=\frac{149}{5^{1} 7^{1}}\)
∴ This is not of the form \(\frac{p}{5^{1} 7^{1}}\)
So 4\(\frac { 9 }{ 35 }\) has a non-terminating recurring decimal expansion.

(iv) \(\frac { 219 }{ 2200 }\)

\(\frac{219}{2200}=\frac{219}{2^{3} 5^{2} 11^{1}}\)
∴ This is not of the form \(\frac{p}{2^{m} 5^{n}}\)
So \(\frac { 219 }{ 2200 }\) has a non-terminating recurring decimal expansion.

Samacheer Kalvi 9th Maths Book Answers

Tamilnadu State Board Class 10 English Grammar Pronouns

♦ Pronouns (Text Book Page No.: 142-144)

A pronoun is a word or phrase that substitutes a noun or a noun phrase. There are ten types of pronouns generally used.

A pronoun is a word that replaces a noun or noun phrase. Using pronouns correctly eliminates unnecessary noun repetition in your writing.

Unnecessary repetition:
e.g. Mary is in Mary’s office, but Mary asked not to be interrupted.

Usage of Pronoun eliminates repetition:
e.g. Mary is in her office, but she asked not to be interrupted.

Personal Pronouns:
Personal pronouns refer to people and things; they can be singular or plural, and their form often changes according to their grammatical function in a sentence, as seen in the table below:

Pronoun as Subject:

• we
• you
• he
• she
• it
• they

Pronoun as Subject:

• Me
• us
• you
• him
• her
• it
• them

Possessive:

• my/mine
• our / ours
• your/yours
• his
• her/hers
• its
• our/ours
• your/yours
• their /theirs

Demonstrative Pronouns (this, that, these, those)
Demonstrative pronouns identify or point to a noun.
e.g., This lab report is due tomorrow.
e.g., That looks like the computer I used to have.
e.g., Put these pencils on your desk.
e.g., Those were the best days of my life.

Relative Pronouns (who, whom, whose, which, that)
Relative pronouns begin adjective clauses and usually refer to the noun that comes right before them.
e.g., Senthil Kumar is a manager whom everyone respects.
e.g., I don’t know who is responsible for setting up the lab equipment.
e.g., You need to talk to the students whose laptops were stolen.
e.g., My textbook, which costs Rs. 125, is missing.
e.g., The new software lacks many of the benefits that the company promised.

Exercises: Type -1

Fill in with relative pronouns.

Question 1.
I met a woman _______ can speak six languages.
Answer:
who

Question 2.
What’s the name of the man _______ lives next door?
Answer:
who

Question 3.
What’s the name of the river _______ goes through the city?
Answer:
which

Question 4.
Everybody _______ I met at the party enjoyed it very much.
Answer:
whom

Question 5.
Do you know anybody _______ wants to buy a car?
Answer:
who

Question 6.
Where is the picture _______ was on the wall?
Answer:
which

Question 7.
She always asks me questions _______ are difficult to answer.
Answer:
that

Question 8.
I have a friend _______ is very good at repairing cars.
Answer:
who

Question 9.
A coffee-maker is a machine _______ makes coffee.
Answer:
that

Question 10.
Why does he always wear clothes _______ are too small for him?
Answer:
which

Exercises: Type – 2

Combine the following sentences using a relative pronoun.

Question 1.
A girl is now in hospital. She was injured in the accident.
Answer:
A girl who was injured in the accident is now in hospital.

Question 2.
A man told me you were away. He answered the phone.
Answer:
A man who answered the phone told me you were away.

Question 3.
A waitress Was very impolite and impatient. She served us.
Answer:
A waitress who served us was very impolite and impatient.

Question 4.
A building was destroyed in the fire. It has now been rebuilt.
Answer:
A building which has now been rebuilt was destroyed in the fire.

Question 5.
A bus goes to the airport. It runs every half hour.
Answer:
A bus which goes to the airport rims every half hour.

Question 6.
I saw the man. He closed the door.
Answer:
I saw the man who closed the door.

Question 7.
The girl is happy. She won the race.
Answer:
The girl who won the race is happy.

Question 8.
The student is from Chennai. He sits next to me.
Answer:
The student who sits next to me is from Chennai.

Question 9.
We are studying sentences. They contain adjective clauses.
Answer:
We are studying sentences that contain adjective clauses.

Question 10.
The taxi driver was friendly. He took me to the airport.
Answer:
The taxi driver who took me to the airport was friendly.

Tamilnadu Board Class 10 English Solutions

Tamilnadu State Board Class 10 English Grammar Finite, Non-finite Verbs

Finite, Non-Finite Verbs (Text Book Page No.: 176-177)

1. Finite verbs :
a. My brother goes to temple daily.
b. We have already finished the project.

The words printed in bold letters are finite verbs.
1. Finite verbs indicate the tense and time of actions.
2. Finite verbs undergo a change as and when the Subject (number or person) changes.

2. Non-Finite Verbs :
1. Non-finite verbs do not indicate the tense and time of actions.
2. Non-finite verbs do not change even when the Subject (number or person)changes.

There are three kinds of non-finite verbs.
1. An infinitive (to + verb)
2. A gerund (verb + ing)
3. A participle

Example:
a. My son likes to watch cricket matches. (Infinitive)
b. Playing chess is my hobby. (Gerund)
c. Driven out of the kingdom, the king hid himself in a forest. (Participle)

Exercises:
In the following sentences, state whether the underlined verb is finite or non-finite.

Question 1.
My little brother wants to be an actor.
Answer:
wants – finite; to be – non-finite

Question 2.
She worked hard to pass the test.
Answer:
worked – finite; to pass – non-finite

Question 3.
I couldn’t solve the problem.
Answer:
couldn’t solve – finite

Question 4.
To err is human.
Answer:
to err – non-finite; is – finite

Question 5.
Your duty is to cross the river without getting noticed.
Answer:
is – finite; to cross – non-finite; getting – non-finite

Question 6.
The doctor is attending to the injured people.
Answer:
is attending – finite

Question 7.
She opened the door.
Answer:
opened – finite

Question 8.
The students were asked to submit their assignments by Friday.
Answer:
were asked – finite; to submit – non-finite

Question 9.
The teacher encouraged the students to work hard.
Answer:
encouraged – finite; to work – non-finite

Question 10.
The dog wagged its tail to show its happiness.
Answer:
wagged – finite; to show – non-finite

Tamilnadu Board Class 10 English Solutions

Tamilnadu State Board Class 10 English Grammar Nominalisation

♦ Nominalisation (Text Book Page No.: 105-107)

Formal written English uses nouns more than verbs. For example, “judgement” rather than “judge”, “development” rather than “develop”, “admiration” rather than “admire”.

Changing a verb or other word into a noun is called nominalisation.

Instead of:
This information enables us to formulate precise questions.

We would write:
This information enables the formulation of precise questions.

Compare these sentences:
1. Organisms reproduce. This is a major characteristic of life.
2. Reproduction is a major characteristic of life.

In general, they mean the same, but sentence 2 is expressed more concisely. It uses the word “reproduction”, whereas sentence 1 uses the word ’’reproduce”. Here the word “reproduce” is a verb. It would change to “reproduces” if “organisms” changed to “an organism”. “Reproduction” is a noun made from the verb “reproduce”; we call this process nominalisation.

Word	Nomination
vary	variation
duplicate	duplication
divide	Division
mutate	mutation
combine	combination
adapt	adaptation
contract	contraction
expand	expansion
react	reaction
rotate	rotation
appropriate	appropriateness
active	activism
complex	complexity
desirable	desirability
sceptical	scepticism
willing	willingness

Exercises: Type -1

Choose the correct nominalisation forms.

Question 1.
resolve
(a) Solve
(b) Solution
(c) Resolves
(d) Resolution
Answer:
(d) Resolution

Question 2.
imperil
(a) Peril
(b) Pearl
(c) Perish
(d) Impediment
Answer:
(d) Impediment

Question 3.
accustom
(a) Customization
(b) Customary
(c) Accustomed
(d) Custom
Answer:
(d) Custom

Question 4.
holy
(a) Hole
(b) Holiness
(c) Notification
(d) Whole
Answer:
(b) Holiness

Question 5.
absurd
(a) Absurdity
(b) Absurdness
(c) Sturdy
(d) Sturdiness
Answer:
(a) Absurdity

Question 6.
furious
(a) Furiously
(b) Furiousness
(c) Fury
(d) Furry
Answer:
(c) Fury

Question 7.
When you overlook his mistake it gives him (‘encourage’) to repeat it.
(a) Courage
(b) Courageous
(c) Courageously
(d) Encouragement
Answer:
(d) Encouragement

Question 8.
Do you have the (prove) of his crime?
(a) Proved
(b) Proving
(c) Proofly
(d) Proof
Answer:
(d) Proof

Question 9.
There is a __________ (sell) of cheap plasticware
(a) Seal
(b) Sale
(c) Sold
(d) Sells
Answer:
(b) Sale

Question 10.
Can you arrange (lend) for my mother’s operation?
(a) Loan
(b) Lent
(c) Lens
(d) Lean
Answer:
(a) Loan

Exercises: Type – 2

Combine the following pairs of sentences nominalising the first verb.

Question 1.
The poor live unhygienically. They suffer from diseases.
(a) The poor’s suffering from diseases because of their life’s unhygienic.
(b) The poor suffer from diseases because of their unhygienic life.
(c) Because of their unhygienic living, the poor suffer from diseases.
(d) The poor lived suffering from diseases because of their unhygienic.
Answer:
(b) The poor suffer from diseases because of their unhygienic life.

Question 2.
The daughter resembles her mother. The father often gets confused.
(a) Resemblance between the daughter and mother confuses the father.
(b) Resembled the daughter with the mother confuses the father.
(c) Resembling between the daughter and mother confuses the father.
(d) The daughter and mother confuse the father because they are resembling.
Answer:
(a) Resemblance between the daughter and mother confuses the father.

Question 3.
Ravi has been pursuing constantly. He wants a job in Infosys.
(a) Ravi has been pursuing constantly to get a job in Infosys.
(b) Ravi’s constant pursuance is to get a job in Infosys.
(c) Ravi’s constant pursuit is to get a job in Infosys.
(d) Constant pursuance of Ravi is to get a job in Infosys.
Answer:
(c) Ravi’s constant pursuit is to get a job in Infosys.

Question 4.
Population grows fast around industrial centres. The industrial centres provide better employment opportunities.
(a) Growing population around industrial centres is fast due to its better employment opportunities
(b) The population growthness around industrial centres is fast due to its better employment opportunities
(c) The population growness around industrial centres is fast due to its better employment opportunities
(d) The population growth around industrial centres is fast due to its better employment opportunities.
Answer:
(d) The population growth around industrial centres is fast due to its better employment opportunities.

Question 5.
He worked hard and got the job.
(a) His working hard got the job.
(b) His work hardly to get the job.
(c) His hard work got the job.
(d) He works hardly and got the job.
Answer:
(c) His hard work got the job

Tamilnadu Board Class 10 English Solutions

Tamilnadu State Board Class 10 English Grammar Conjunctions

Conjunctions (Text Book Page No.: 102-105)

Conjunctionsoins words, phrases and clauses together.
1. Coordinating Conjunctions

• but
• so
• of
• and

Coordinating conjunctionsoin words, phrases, or independent clauses of a sentence together. They often link similar grammatical parts of a sentence together (parts of speech + parts of speech/phrase + phrase/clause + clause ).In a sentence, Coordinating conjunctions come in between the individual words, phrases, and independent clauses they are joining.

Here are some example sentences:
We went to the park, but we did not have time for the museum.
She has to work late tonight, so she cannot get up early.
Let’s meet at the beach or in front of the hotel.
On Friday night we watched a drama and a movie.

2. Subordinating Conjunctions

• after/before
• although/even though
• because/as
• if
• as long as/provided that
• till/until unless
• when/once/as soon as
• while/whereas
• in spite of/despite

Subordinating conjunctionsoin an independent clause (which contains both a subject and a verb and can act as a complete sentence) and a dependent clause (which also contains a subject and a verb, but is not a complete sentence),

e.g They went running, although I even though it was very hot.

Coordinating conjunctionsoin parts of sentence that are similar whereas subordinating conjunctions often show a contrasting or unequal relationship.

Exercises: Type – 1
Fill in the blanks with subordinating conjunctions.

Question 1.
We played the match __________ the drizzle.
Answer:
in spite of/despite

Question 2.
We went to the beach __________ the fine weather.
Answer:
due to I on account of

Question 3.
We decided to go to the cinema __________ it was too hot outside.
Answer:
because I as

Question 4.
Monisha lives in Chennai, _________ her brother lives in Coimbatore.
Answer:
while/whereas

Question 5.
He went to work _________ he had finished his breakfast.
Answer:
after

Question 6.
He had finished breakfast _________ he went to work.
Answer:
before

Question 7.
I will go there _________ as you go too.
Answer:
if/provided that/as long

Question 8.
I wont o there _________ you go.
Answer:
unless

Question 9.
We can leave __________ you are ready.
Answer:
when/once

Question 10.
You can’t drive _________ you get your licence.
Answer:
until I till

Exercises: Type – 2

Choose the correct option.

Question 1.
We ate a pizza a kebab.
(a) But
(b) and
(c) so
Answer:
(b) and

Question 2.
We had some cake ________ we didn’t have any coffee.
(a) unless
(b) until
(c) but
Answer:
(c) but

Question 3.
I had a headache ________ I didn’t go to the party.
(a) when
(b) so
(c) whereas
Answer:
(b) so

Question 4.
You can have a coffee ________ a tea but not both.
(a) or
(b) till
(c) but
Answer:
(a) or

Question 5.
I can’t come to school ________ I have an important appointment.
(a) so
(b) because
(c) unless
Answer:
(b) because

Question 6.
I will call you ________ I get home.
(a) as
(b) and
(c) when
Answer:
(c) when

Question 7.
you do your homework, you will pass the course.
(a) unless
(b) until
(c) as long as
Answer:
(c) as long as

Question 8.
I wanted to eat fried rice _________ my sister wanted to eat noodles.
(a) so
(b) when
(c) whereas
Answer:
(c) whereas

Question 9.
You cannot vote ________ you are 18.
(a) when
(b) until
(c) as
Answer:
(b) until

Question 10.
She still went to work ________ she was sick.
(a) even though
(b) until
(c) if
Answer:
(a) even though

Question 11.
________ the bad weather, they decided to have a picnic.
(a) because
(b) despite
(c) whereas
Answer:
(b) despite

Question 12.
Wash your hands ________ you eat your dinner.
(a) till
(b) when
(c) before
Answer:
(c) before

Question 13.
I like milk, butter, cream and yogurt., I don’t like cheese.
(a) so
(b) however
(c) and
Answer:
(b) however

Question 14.
You can have an ice-cream ________ you have finished you homework.
(a) so
(b) provided that
(c) or
Answer:
(b) provided that

Question 15.
I will take a shower _________ i reach home.
(a) but
(b) as soon as
(c) until
Answer:
(b) as soon as

Tamilnadu Board Class 10 English Solutions

Tamilnadu State Board Class 10 English Grammar Tenses

Tenses (Text Book Page No.: 75-83)
Tenses denote the time of action. They show when the work is done.

They are:

1. Present Tense
2. Past Tense
3. Future Tense

Present Tense

The Present Tense form is further divided into:

(1) Simple Present:
It is used to denote scientific facts, universal truths and work done on a daily basis.

Example:
She writes a letter. (Assertive) She does not write a letter. (Negative)
Does she write a letter( Interrogative) Does she not write a letter? (Interrogative -ve)

(2) Present Continuous:
It is used to express an action taking place at the time of speaking.

Example:
She is writing a letter. She is not writing a letter.
Is she writing a letter? Is she not writing a letter?

(3) Present Perfect:
It is used to show an action that started in the past and has just finished.

Example:
She has( recently) written a letter. She has not written a letter (recently).
Has she written a letter? Has she not written a letter?

(4) Present Perfect Continuous: This tense shows the action which started in the past and is still continuing.
Example:
She has been writing a letter since morning. She has not been writing a letter.
Has she been writing a letter for an hour? Has she not been writing a letter?

Past tense:
The past tense form is further divided into Simple Past, Past Continuous Tense and Past Perfect Tense.

Simple Past:
Used to indicate an action completed in the past. It often occurs with an adverb of time. Sometimes it is used without an adverb of time.

Used for past habits.

Example:
She wrote a letter. – She didn’t write a letter.
Did she write a letter? – Did she not/Didn’t she write a letter?

Past Continuous Tense:
Used to denote an action going on at some time in the past.

Example:
She was writing a letter. – She was not writing a letter.
Was she writing a letter? – Was she not writing a letter?

Past Perfect Tense:
Used to describe an action completed before a certain moment in the past, usually a long time ago. If two actions happened in the past, past perfect is used to show the action that took place earlier.

Example:
She had already written a letter. – She had not written a letter.
Had she written a letter? – Had she not written a letter?

Past Perfect Continuous Tense:
Used to denote an action that began before a certain point in the past and continued up to some time in past.

Example:
She had been writing a letter yesterday for an hour. She had not been writing a letter.
Had she been writing a letter? Had she not been writing a letter?

Future Tense:
This tense tells us about an action which has not occurred yet and will occur after saying or in future Tense helping verb ‘Shall’ is used with ‘I’ and ‘We’. Helping verb ‘Will’ is used with all others.

When you are to make a commitment or warn someone or emphasize something, use of ‘will/shall’ is reversed. ‘Will’ is used with ‘I’ & ‘We’ and ‘shall’ is used with others.

In general speaking, there is hardly any difference between ‘shall & will’.

The future form is further divided into Simple Future, Future Continuous Tense and Future Perfect Tense. Simple Future
Example;
She will write a letter. – She will not write a letter.
Will she write a letter? – Will, she not write a letter?

Future Continuous Tense:
It is used to express an ongoing or continued action in future.

Example:
She will be writing a letter tomorrow by this time. – She will not be writing a letter.
Will she be writing a letter? – Will she not be writing a letter?

Future Perfect Tense:
It is used to express an action which will happen/occur in future and will be completed by a certain time in future.

We use the future perfect to say that something will be finished by a particular time in the future.

Example:
She will have written the letter by Sunday morning.
Tense symbolizes the ever-moving, non-stop wheel of time which is forever busy gathering moments of future and throwing them into the dustbin of the past.

Exercises: Type – 1

Fill in with the simple present or present continuous form of the verb in brackets.

Question 1.
Please be quiet. I _________ (try) to read my book.
Answer:
am trying

Question 2.
I __________ (not/use) the computer at the moment so you can use it.
Answer:
am not using

Question 3.
Mary is ill so She _________ (go) to the hospital now.
Answer:
is going

Question 4.
Excuse me, I _________ (look) for a hotel.
Answer:
am looking

Question 5.
_________ (you/wait) for someone?
Answer:
are you waiting

Question 6.
Ricky, you are very careless. You _________ (always I forget) to do your homework!
Answer:
are always forgetting

Question 7.
The cost of living (rise) very fast. Every year things are more expensive.
Answer:
is rising

Question 8.
What _________ (you/do)? I _________ (clean) my shoes.
Answer:
are you doing; am cleaning

Question 9.
The post man __________ (come) everyday at 11 a.m.
Answer:
comes

Question 10.
The boys usually (play) cricket in this ground.
Answer:
play

Exercise: Type – 2

Fill in with the simple past or past continuous form of the verb in brackets

Question 1.
I __________ (sleep) when she __________ (come).
Answer:
I was sleeping when she came.

Question 2.
They __________ (wait) for me when I __________ (arrive).
Answer:
They were waiting for me when I arrived.

Question 3.
The phone __________ (ring) while I _________ (have) a shower.
Answer:
The phone rang while I was having a shower.

Question 4.
We __________ (not/go out) last Sunday because it __________ (rain).
Answer:
We did not go out last Sunday because it was raining.

Question 5.
I __________ (see) Adhira at the party She __________ (wear) a new dress.
Answer:
I saw (see) Adhira at the party. She was wearing a new dress.

Question 6.
I __________ (break) a bowl this morning. When I __________ (wash) the dishes itust (slip) out of my hand onto the floor!
Answer:
I broke a bowl this morning. When I was washing the dishes it just slipped out of my hand onto the floor!

Question 7.
Why he (carry) the table, he __________ (feel) a sharp pain in his back.
Answer:
When he was carrying the table, he felt a sharp pain in his back.

Question 8.
We ______ (watch) TV when someone _______ (come) to the door and ______ (knock) very loudly.
Answer:
We were watching TV when someone came to the door and was knocking very loudly.

Exercises: Type – 3

Complete the sentences using the present perfect continuous and add since or for.

Question 1.
Maria __________ (learn) English ___________ two years.
Answer:
Maria has been learning English for two years.

Question 2.
I ___________ (write) letters ___________ 8:00.
Answer:
I have been writing letters since 8:00.

Question 3.
anu and Ram _________ (travel) around the city _________ five weeks.
Answer:
Janu and Ram have been travelling around the city for five weeks.

Question 4.
We _________ (go) to Ooty for our holidays _________ 2010.
Answer:
We have been going to Ooty for our holidays since 2010.

Question 5.
It _________ (rain) continuously morning.
Answer:
It has been raining continuously since morning.

Exercise: Type – 4

Complete the sentences using the present perfect.

Question 1.
They __________ (build) all the main walls recently.
Answer:
have built.

Question 2.
Viman is writing a novel. He __________ (complete) the first three chapters.
Answer:
has completed

Question 3.
The painter (paint) nearly half the town hall.
Answer:
has painted

Question 4.
I am watching cartoons on TV. I _________ (watch) 4 cartoons already.
Answer:
have watched

Question 5.
I __________ (be) to the Yelagiri mountains recently.
Answer:
have been

Exercises: Type – 5

Read the situations and write sentences in the past perfect using the words In brackets.

Question 1.
I arrived at the party at 9 p.m, but Nimmy was already there. (she/arrive I before/me)
Answer:
She had arrived before me.

Question 2.
I invited Afsarto come for lunch but he wasn’t hungry. (he/already/eat I lunch)
Answer:
He had already eaten lunch.

Question 3.
Gown was late for her exam. All the other students were already there. (they I start/the exam)
Answer:
They had started the exam.

Question 4.
Kavin got home after midnight. The house was quiet. (everybody I go/to bed)
Answer:
Everybody had gone to bed.

Question 5.
I rang Vijay but he wasn’t at home. (he/already I leave I for work)
Answer:
He had already left for work.

Exercise: Type-6

Complete the sentences using the past perfect or the past perfect continuous.

Question 1.
By the time I got home they (eat) all the cake.
Answer:
had eaten

Question 2.
Mangai retired at fifty-eight, but she __________ (work) hard all her life.
Answer:
had been working

Question 3.
Elambrathi was very irritable. He __________ (look) for his contact lens for an hour and he still __________ (not I find) it!
Answer:
had been looking; had not found

Question 4.
I was furious with Umar when he arrived. I __________ (wait) for him for hours.
Answer:
had been waiting

Question 5.
Han was too sad to sell his car. He (have) it for a long time. Answer: had had I

Exercise: Type – 7

Choose the correct form of the verb.

Question 1.
What time _________ leave tomorrow? (does your train/will your train)
Answer:
will your train

Question 2.
‘I’m very thirsty.’ you something to drink. Coke or tea?’ (I’ll get/I get)
Answer:
‘I’ll get

Question 3.
I think __________ later today. (it rains/will rain)
Answer:
will rain

Question 4.
‘Why have you put the TV on?’ __________ the news.’ (I’m going to watch/I will watch)
Answer:
I’m going to watch

Question 5.
__________ I carry that for you? (Shall/Will)
Answer:
Shall

Tamilnadu Board Class 10 English Solutions

Tamilnadu State Board Class 9 Maths Solutions Chapter 2 Real Numbers Ex 2.1

Question 1.
Which arrow best shows the position of \(\frac { 11 }{ 3 }\) on the number line?

Solution:
\(\frac { 11 }{ 3 }\) = 3.666….. = 3.7 (nearly)
∴ D arrow best shows the position of \(\frac { 11 }{ 3 }\) on the number line.

Question 2.
Find any three rational numbers between \(\frac { -7 }{ 11 }\) and \(\frac { 2 }{ 11 }\).
Solution:
Three rational numbers between \(\frac { -7 }{ 11 }\) and \(\frac { 2 }{ 11 }\) are \(\frac { -6 }{ 11 }\), \(\frac { -5 }{ 11 }\), \(\frac { -4 }{ 11 }\), ……, \(\frac { -1 }{ 11 }\)

Question 3.
Find any five rational numbers between
(i) \(\frac { 1 }{ 4 }\) and \(\frac { 1 }{ 5 }\)
(ii) 0.1 and 0.11
(iii) -1 and -2
Solution:
(i) a = \(\frac { 1 }{ 4 }\), b = \(\frac { 1 }{ 5 }\)
Let q₁, q₂, q₃, q₄ and q₅ be five rational numbers, q₁ = \(\frac { 1 }{ 2 }\)(a + b)

(ii) The rational numbers between 0.1 and 0.11 are 0.101, 0.102, 0.103, …..0.109.
(iii) -1 and -2
Let q₁, q₂, q₃, q₄ and q₅ be five rational numbers.

Samacheer Kalvi 9th Maths Book Answers

Tamilnadu State Board Class 10 Maths Solutions Chapter 2 Numbers and Sequences Ex 2.6

Question 1.
Find the sum of the following
(i) 3,7,11,… up to 40 terms.
(ii) 102, 97, 92,… up to 27 terms.
(iii) 6 + 13 + 20 + … + 97
Solution:
(i) 3, 7, 11,. . . upto 40 terms.
a = 3, d = t₂ – t₁ = 7 – 3 = 4
n = 40

(ii) 102, 97, 952,… up to 27 terms
a = 102,
d = t₂ – t₁
= 97 – 102 = -5
n = 27

(iii) 6 + 13 + 20 + … + 97
a = 6,d = 7,l = 97

Question 2.
How many consecutive odd integers beginning with 5 will sum to 480?
Answer:
5,7,9, 11, 13,…
s_n = 480
a = 5, d = 2, s_n = 480

2 n² + 8 n – 960 = 0
⇒ n² + An – 480 = 0
⇒ n² + 24n – 20n – 480 = 0
⇒ n(n + 24) – 20(n + 24) = 0
⇒ (n – 20)(n + 24) = 0
⇒ n = 20,-24
No. of terms cannot be -ve.
∴ No. of consecutive odd integers beginning with 5 will sum to 480 is 20.

Question 3.
Find the sum of first 28 terms of an A.P. whose n^th term is 4n – 3.
Solution:
n = 28
t_n = 4n – 3
t₁ = 4 × 1 – 3 = 1
t₂ = 4 × 2 – 3 = 5
t₂₈ = 4 × 28 – 3
= 112 – 3 = 109
∴ a = 1, d = t₂ – t₁ = 5 – 1 = 4
l = 109.
S_n = \(\frac { n }{ 2 } \) (2a+(n-1)d)
S₂₈ = \(\frac { 28 }{ 2 } \) (2 × 1 + 27 × 4)
= 14(2 + 108)
= 14 × 110
= 1540

Question 4.
The sum of first n terms of a certain series is given as 2n2 – 3n. Show that the series is an A.P.
Solution:
Given S_n = 2n² – 3n
S₁ = 2(1)² – 3(1) = 2 – 3 = – 1
⇒ t₁ = a = – 1
S₂ = 2(2²) – 3(2) = 8 – 6 = 2
t₂ = S₂ – S₁ = 2 – (-1) = 3
∴ d = t₂ – t₁ = 3 – (-1) = 4
Consider a, a + d, a + 2d,….
-1, -1 + 4, -1 + 2(4),…
-1, 3, 7,….
Clearly this is an A.P with a = – 1, and d = 4.

Question 5.
The 104^th term and 4th term of an A.P are 125 and 0. Find the sum of first 35 terms.
Solution:
t₁₀₄ = 125
t₄ = 0
a + (n – 1)d = t_n

Question 6.
Find the sum of all odd positive integers less than 450.
Solution:
Sum of all odd positive integers less than 450 is given by
1 + 3 + 5 + … + 449
a = 1
d = 2
l = 449


= 50625
Another method:
Sum of all +ve odd integers = n².
We can use the formula n² = 225²
= 50625

Question 7.
Find the sum of all natural numbers between 602 and 902 which are not divisible by 4.
Answer:
Natural numbers between 602 and 902
603,604, …, 901
a = 603, l = 901, d = 1,

Sum of all natural numbers between 602 and 902 which are not divisible by 4.
= Sum of all natural numbers between 602 and 902
= Sum of all natural numbers between 602 and 902 which are divisible by 4.
l = 902 – 2 = 900
To make 602 divisible by 4 we have to add 2 to 602.
∴ 602 + 2 = 604 which is divisible by 4.
To make 902 divisible by 4, subtract 2 from 902.
∴ 900 is the last number divisible by 4.

Sum of all natural numbers between 602 and 902 which are not divisible 4.
= 224848 – 56400
= 168448

Question 8.
Raghu wish to buy a laptop. He can buy it by paying ₹40,000 cash or by giving it in 10 installments as ₹4800 in the first month, ₹4750 in the second month, ₹4700 in the third month and so on. If he pays the money in this fashion, find
(i) total amount paid in 10 installments.
(ii) how much extra amount that he has to pay than the cost?
Solution:
4800 + 4750 + 4700 + … 10 terms
Here a = 4800
(i) d = t₂ – t₁ = 4750 – 4800 = -50
n = 10
S_n = \(\frac { n }{ 2 } \) (2a + (n – 1)d)
s₁₀ = \(\frac { 10 }{ 2 } \) (2 × 4800 + 9 × -50)
= 5 (9600 – 450)
= 5 × 9150 = 45750
Total amount paid in 10 installments = ₹ 45750.
(ii) The extra amount he pays in installments
= ₹ 45750 – ₹ 40,000
= ₹ 5750

Question 9.
A man repays a loan of ₹65,000 by paying ₹400 in the first month and then increasing the payment by ₹300 every month. How long will it take for him to clear the loan?
Solution:
Loan Amount = ₹ 65,000
Repayment through installments
400 + 700 + 1000 + 1300 + …
a = 400
d = 300
S_n = 65000
S_n = \(\frac { n }{ 2 } \) (2a+(n – 1)d)
= 65000
\(\frac { n }{ 2 } \) (2 × 400 + (n – 1)300) = 65000
n(800 + 300n – 300) = 130000
n(500 + 300n) = 130000
500n + 300n² = 130000

Number of terms should be (+ve) and cannot be (-ve) or fractional number.
∴ He will take 20 months to clear the loans.

Question 10.
A brick staircase has a total of 30 steps. The bottom step requires 100 bricks. Each successive step requires two bricks less than the previous step.
(i) How many bricks are required for the top most step?
(ii) How many bricks are required to build the stair case?
Solution:
100 + 98 + 96 + 94 + … 30 steps.
Here
a = 100
d = -2
n = 30
∴ S_n = \(\frac { n }{ 2 } \) (2a + (n – 1)d)
S₃₀ = \(\frac { 30 }{ 2 } \) (2 × 100 + 29 × -2)
= 15(200 – 58)
= 15 × 142
= 2130
t₃₀ = a + (n – 1)d
= 100 + 29 × -2
= 100 – 58
= 42
(i) No. of bricks required for the top step are 42.
(ii) No. of bricks required to build the stair case are 2130.

Question 11.
If S₁,S₂,S₃ , …. , S_m are the sums of n terms of m A.P.’s whose first terms are 1,2,3,…,m and whose common differences are 1, 3, 5,…, (2m – 1) respectively, then show that
S₁ + S₂ + S₃ + …. + Sm = \(\frac { 1 }{ 2 } \) mn(mn + 1).
Solution:

Question 12.
Find the sum

Solution:

Samacheer Kalvi 10th Maths Book Solutions

Tamilnadu State Board Class 9 Maths Solutions Chapter 1 Set Language Additional Questions

Text Book Activities

Question 1.
Discuss and give as many examples of collections from your daily life situations, which are sets and which are not sets.
Solution:
Which are sets

1. Collection of pen
2. Collection of dolls
3. Collection of books
4. Collection of red flower etc.

Which are not sets

1. Collection of good students in a class.
2. Collection of beautiful flowers in a garden etc.

Question 2.
Write the following sets in respective forms.
Solution:

Question 3.
Fill in the blanks with appropriate cardinal numbers.
Solution:

Additional Questions and Answers

Exercise 1.1

Question 1.
Let A = {0, 1, 2, 3, 4, 5}. Insert the appropriate symbol G or g in the blank spaces,
(i) 0 ___ A
(ii) 6 ___ A
(iii) 3 ___ A
(iv) 4 ____ A
(v) 7 ____ A
Solution:
(i) 0 ∈ A
(ii) 6 ∉ A
(iii) 3 ∈ A
(iv) 4 ∈ A
(v) 7 ∉ A

Question 2.
Write the following in Set-Builder form.
(i) The set of all positive even numbers.
(ii) The set of all whole numbers less than 20.
(iii) The set of all positive integers which are multiple of 3.
(iv) The set of all odd natural numbers less than 15.
(v) The set of all letters in the word ‘computer’.
Solution:
(i) A = {x : x is a positive even number}
(ii) B = {x : x is a whole number and x < 20}
(iii) C = {x : x is a positive integer and multiple of 3}
(iv) D = {x : x is an odd natural number and x < 15}
(v) E = {x : x is a letter in the word “Computer”}

Question 3.
Write the following sets in Roster form.
(i) A = {x : x ∈ N, 2< x < 10 }
(ii) B = {x : x ∈ Z, –\(\frac { 1 }{ 2 }\) < x < \(\frac { 11 }{ 2 }\) }
(iii) C = {x : x is a prime number and a divisor of 6 }
(iv) x = {x : x = 2ⁿ, it n ∈ N and n ≤ 5}
(v) M = {x : x = 2y – 1, y ≤ 5, j ∈ W}
Solution:
(i) A = {3,4, 5, 6, 7, 8, 9}
(ii) B = {0, 1, 2, 3, 4, 5}
(iii) C = {2, 3}
(iv) Given, x = 2ⁿ, n ∈ N and n ≤ 5.
Here n = 1, 2, 3, 4, 5
n = 1 ⇒ 21 = 2
n = 2 ⇒ 22 = 4
n = 3 ⇒ 23 = 8
n = 4 ⇒ 24 = 16
n = 5 ⇒ 25 = 32
X = {2, 4, 8, 16, 32}

(v) Given, x = 2y – 1, y ≤ 5 and y ∈ W Here j = 0, 1,2, 3, 4, 5
y = 0 ⇒ x = 2 (0) – 1 = -1
y = 1 ⇒ x = 2 (1) – 1 = 2 – 1 = 1
y = 2 ⇒ x = 2 (2) – 1 = 4 – 1 = 3
y = 3 ⇒ x = 2 (3) – 1 = 6 – 1 = 5
y = 4 ⇒ x = 2 (4) – 1 = 8 – 1 = 7
y = 5 ⇒ x = 2 (5) – 1 = 10 – 1= 9
M = {-1, 1, 3, 5, 7, 9}

Exercise 1.2

Question 1.
Find the number of subsets and number of proper subsets of a set X = {a, b, c, x, y, z}.
Solution:
Given X = {a, b, c, x, y, z}.
Then, n(X) = 6
The number of subsets = n[P(X)] = 2⁶ = 64
The number of proper subsets = n[P(X)] – 1 = 2⁶ – 1 = 64 – 1 = 63

Question 2.
Find the cardinal number of the following sets.
(i) A = {x : x is a prime factor of 12}.
(ii) B = {x : x ∈ W, x ≤ 5}.
(iii) X = {x : x is an even prime number}
Solution:
(i) Factors of 12 are 1, 2, 3, 4, 6, 12. So, the prime factors of 12 are 2,3.
We write the set A in roster form as A = {2, 3} and hence n(A) = 2.
(ii) In Tabular form B = {0, 1, 2, 3, 4, 5}
The set B has six elements and hence n(B) = 6
(iii) X = {2} [2 is the only even prime number]
∴ n (X) = 1

Question 3.
State whether the following sets are finite or infinite.
(i) A = {x : x is a multiple of 5, x ∈ N}.
(ii) B = {0,1, 2, 3, 4, 75}.
(iii) The set of all positive integers greater than 50.
Solution:
(i) A = {5, 10, 15, 20, …… } ∴A is an infinite set
(ii) Finite
(iii) Let X be the set of all positive integers greater than 50
Then X= (51, 52, 53, ….. }
∴ X is an infinite set.

Question 4.
Which of the following sets are equal?
(i) A = (1, 2, 3, 4}, B = {4,3, 2,1}
(ii) A = (4, 8,12,16}, B = (8, 4,16,18}
(iii) X ={2,4, 6, 8}
Y = {x : x is a positive even integer and 0 < x < 10}
Solution:
(i) Since A and B contain exactly the same elements, A and B are equal sets.
(ii) A and B has different elements.
∴ A and B are not equal sets.
(iii) X = {2, 4, 6, 8}, Y = {2,4, 6, 8}
∴ X and Y are equal sets.

Question 5.
Write ⊆ or ⊈ in each blank to make a true statement.
(i) {4, 5, 6, 7} ____ {4, 5, 6, 7, 8}
(ii) {a, b, c} ____ {b, e, f, g}
Solution:
(i) {4, 5, 6, 7} ⊈ {4, 5, 6, 7, 8}
(ii) {a, b, c} ⊈ {b, e, f, g}

Question 6.
Write down the power set of A= {3, {4, 5}}.
Solution:
The subsets of A are {φ, {3}, {4, 5}, {3,{4, 5}}
P(A) = {, φ, {3}, {4,5}, {3{4,5}}

Exercise 1.3

Question 1.
Find the union of the following sets.
(i) A = {1, 2, 3, 5, 6} and B = {4, 5, 6, 7, 8}
(ii) X = {3, 4, 5} and Y = φ
Solution:
(i) A ∪ B = {1, 2, 3, 4, 5, 6, 7, 8}
(ii) X ∪ Y = {3, 4, 5}

Question 2.
Find A ∩ B if (i) A = {10, 11, 12, 13}, B = {12, 13, 14, 15}, (ii) A = {5, 9, 11}, B = φ.
Solution:
(i) A ∩ B = {12, 13}
(ii) A ∩ B = φ

Question 3.
Given the sets A = {4, 5, 6, 7} and B = {1, 3, 8, 9}, find A ∩ B.
Solution:
A ∩ B = φ

Question 4.
If A= {-2, -1, 0, 3, 4}, B = {-1, 3, 5}, find (i) A – B, (ii) B – A.
Solution:
(i) A – B = {-2, 0, 4}
(ii) B – A = {5}

Question 5.
If A = {2, 3, 5, 7,11} and B = {5, 7, 9, 11, 13}, find A ∆ B.
Solution:
A ∆ B= {2, 3, 9, 13}

Question 6.
Draw a venn diagram similar to one at the side and shade the regions representing the following sets (i) A’, (ii) B’, (iii) A’ ∪ B’, (iv) (A ∪ B)’, (v) A’ ∩ B’
Solution:
(i) A’

(ii) B’

(iii) A’ ∪ B’

(iv) (A ∪ B)’

(v) A’ ∩ B’

Question 7.
State which of the following sets are disjoint.
(i) A = {2, 4, 6, 8}, B = {x : x is an even number < 10, x ∈ N}
(ii) X = {1, 3, 5, 7,9}, Y = {0, 2, 4, 6, 8, 10}
(iii) R = {a, b, c, d, e}, S = {d, e, b, c, a}
Solution:
(i) A = {2, 4, 6, 8}, B = {2, 4, 6, 8}
A ∩ B={2, 4, 6, 8} ≠ Φ
∴ A and B are not disjoint sets.
(ii) X ∩ Y = { } = Φ, X and Y are disjoint sets.
(iii) R ∩ S = {a, b, c, d, e} ≠ Φ
∴ R and S are not disjoint sets.

Question 8.
If A = {a, b, c, d, e}and B = {a, e, i, o, u} find AB.
Solution:
A ∩ B = {a, b, c, d, e} ∩ {a, e, i, o, u} = {a, e}

Exercise 1.4

Question 1.
If A and B are two sets containing 13 and 16 elements respectively, then find the minimum and maximum number of elements in A ∪B?
Solution:

n(A) = 13 ; n(B) = 16
Minimum n(A ∪ B) = 16
Maximum n(A ∪ B) = 13 + 16 = 29

Question 2.
If n(U) = 38, n(A) = 16, n(A ∩ B) = 12, n(B’) = 20, find n(A ∪ B).
Solution:

n(U) = 38
n(A) = 16
N(A ∩ B) = 12
n(B’) = 20
n(A ∪ B) = ?
n(B) = n(U) – n(B)’
n(B) = 38 – 20
n(B) = 18
n(A ∪ B) = n(A) + n(B) – n (A ∩ B)
n(A ∪ B) = 16 – 12 + 18
n(A ∪ B) = 4 + 18 = 22.

Question 3.
Let A = {b, d, e, g, h} and B = {a, e, c, h} verify that n(A – B) = n(A) – n(A ∩ B)
Solution:
A = {b, d, e, g, h),
B = {a, e, c, h}
A ∩ B = {b, d, g}
n(A ∩ B) = 3 …(1)
A ∩ B = {e, h}
n(A ∩ B) = 2, n(A) = 5
n(A) – n(A ∩ B) = 5 – 2 = 3 …(2)
Form (1) and (2) we get
n(A – B) = n(A) – n(A ∩ B)

Question 4.
If A = {2, 5, 6, 7} and B = {3, 5, 7, 8}, then verify the commutative property of
(i) union of sets
(ii) intersection of sets
Solution:
Given, A = {2, 5, 6, 7} and B = {3, 5, 7, 8}
(i) A ∪ B = {2, 3, 5, 6, 7, 8} … (1)
B ∪ A = {2, 3, 5, 6, 7, 8} … (2)
From (1) and (2) we have A ∪ B = B ∪ A
It is verified that union of sets is commutative.

(ii) A n B = {5, 7} …(3)
B n A = {5, 7} …(4)
From (3) and (4) we get, A ∩ B = B ∩ A
It is verified that intersection of sets is commutative.

Question 5.
If A = {b, c, d, e} and B = {b, c, e, g} and C = {a, c, e}, then verify A ∪ (B ∪ C) = (A ∪ B) ∪ C.
Solution:
Given, A = {b, c, d, e} and B = {b, c, e, g} and C = {a, c, e}
Now B ∪ C = {a, b, c, e, g}
Au(B ∪C) = {a, b, c, d, e, g} …(1)
Then, A ∪ B = {b, c, d, e, g}
(A ∪ B) ∪ C = {a, b, c, d, e, g} … (2)
From (1) and (2) it is verified that
A ∪ (B ∪ C) = (A ∪ B) ∪ C

Exercise 1.5

Question 1.
If A = {1, 3, 5, 7, 9}, B = {x : x is a composite number and x < 12} and C = {x : x ∈ N and 6 < x < 10} then verify A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C).
Solution:
Given, A = {1, 3, 5, 7, 9} and B = {4, 6, 8, 9, 10}and C = {6, 7, 8, 9}
B ∩ C = {4, 6, 8, 9, 10} n {6, 7, 8, 9} = {6, 8, 9}
A ∪ (B ∩ C) = {1,3, 5, 6, 7, 8,9} …(1)
Then (A ∪ B) = {1, 3, 5, 7, 9} ∪ {4, 6, 8, 9, 10}
= {1, 3, 4, 5, 6, 7, 8, 9, 10}
(A ∪ C) = {1, 3, 5, 7, 9} ∪ {6, 7, 8, 9}
= {1,3, 5, 6, 7, 8, 9}
(A ∪ B) ∩ (A ∪ C) = {1, 3,4, 5, 6, 7, 8, 9, 10} ∩ {1, 3, 5, 6, 7, 8, 9}
= {1,3, 5, 6, 7, 8,9} …(2)
From (1) and (2), it is verified that
A ∪ (B ∩C) = (A ∪ B) ∩ (A ∪ C)

Question 2.
If A, B and C are overlapping sets, draw venn diagram for : A ∩ B
Solution:

Question 3.
Draw Venn diagram for A ∩ B ∩ C
Solution:

Question 4.
If P = {x : x ∈ N and 1 < x < 11}, Q = {x : x = 2n, n ∈ N and it < 6} and R = {4, 6, 8, 9, 10, 12}, then verify P – (Q ∩ R) = (P – Q) ∪ (P – R).
Solution:
The roster form of sets P, Q and R are P = {2, 3, 4, 5, 6, 7, 8, 9, 10}, Q = {2, 4, 6, 8, 10} and R = {4, 6, 8, 9, 10, 12}
First, we find Q ∩ R = {4, 6, 8, 10}
Then, P – (Q ∩ R) = {2, 3, 5, 7, 9} …..(1)
Next, P – Q = {3, 5, 7, 9}
and P – R = {2, 3, 5, 7}
and so, (P – Q) ∪ (P – Q) = {2, 3, 5, 7, 9} … (2)
Hence from (1) and (2), it verified that P – (Q ∩R) = (P – Q) ∪ (P – R)
Finding the elements of set Q
Given, x = 2 n
n = 1 → x = 2 (1) = 2
n = 2 → x = 2(2) = 4
n = 3 → x = 2 (3) = 6
n = 4 → x = 2(4) = 8
n = 5 → x = 2(5) = 10
Therefore, x takes values such as 2, 4, 6, 8, 10

Question 5.
If U = {x : x ∈ Z, -3 < x ≤ 9}, A = {x : x = 2P + 1, P ∈ Z , -2 ≤ P ≤ 3}, B = {x : x = q + l, q ∈ Z, 0 ≤ q ≤ 3}, verify De Morgan’s laws for complementation.
Solution:
Given, U = {-3,-2,-1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
A = {-3, -1, 1, 3, 5,7} and B = {1,2, 3, 4}
Law (i) (A ∪ B)’ = A’ ∩ B’
Now, A ∪ B = {-3,-1, 1, 2, 3, 4, 5, 7}
(A ∪ B)’ = {-2, 0,6, 8, 9} …..(1)
Then, A’ = {-2, 0, 2, 4, 6, 8, 9) and
B’ = {-3, -2, -1, 0, 5, 6, 7, 8, 9}
A’ ∩ B’ = {-2, 0, 6, 8, 9} ……(2)
From (1) and (2) it is verified that
(A ∪ B)’ = A’ ∩ B’
Law (ii) (A ∩ B)’ = A’ ∪ B’
Now, A ∩ B = {1, 3}
(A ∩ B)’ = {-3,-2,-1,0, 2, 4, 5, 6, 7, 8, 9, ……..}
Then, A’ ∪ B’ = {-3, -2, -1, 0, 2, 4, 5, 6, 7, 8, 9, ……..}
From (3) and (4) it is verified that
(A ∩ B)’ = A’ ∪ B’

Exercise 1.6

Question 1.
From the given venn diagram. Find (i) A, (ii) B, (iii) A ∪ B (iv) A ∩ B also verify that n(A ∪B) = n(A) + n(B) – n(A ∩ B).

Solution:
A = {a, b, d, e, g, h}
B = {b, c, e, f, h, i, j}
A ∪ B = {a, b, c, d, e, f, g, h, i, j}
A ∩ B = {b, e, h}
So, n(A) = 6, n(B) = 7, n(A ∪B) = 10, n(A ∩ B) = 3
Now, n(A) + n(B) – n(A ∩ B) = 6 + 7 – 3 = 10
Hence, n(A) + n(B) – n(A ∩ B) = n(A ∪ B)

Question 2.
If n(A) = 12, n(B) = 17 and n(A ∪ B) = 21, find n(A ∩B).
Solution:
Given that n(A) = 12, n(B) = 17 and n(A ∪ B) =21
By using the formula n(A ∪ B) = n(A) + n(B) – n(A ∩ B)
n(A ∩ B) = 12 + 17 – 21 = 8

Question 3.
In a school, 80 students like Maths, 90 students like Science, 82 students like History, 21 like both Maths and Science, 19 like both Science and History 20 like both Maths and History and 8 liked all the three subjects. If each student like atleast one subject, then find (i) the number of students in the school (ii) the number of students who like only one subject.
Solution:
Let M, S and H represent sets of students who like Maths, Science and History respectively.
Then, n(M) = 80, n(S) = 90, n(H) = 82, n(M ∩ S) = 21, H(S ∩ H) = 19, n(M ∩ H) = 20, n(M ∩ S ∩ H) = 8
Let us represents the given data in a venn diagram.

(i) The number of student in the school = 52 + 59 + 55 + 12 + 11 + 8 + 8 = 205
(ii) The number of students who like only one subject = 52 + 59 + 55 = 166

Question 4.
State the formula to find n(A ∪ B ∪ C).
Solution:
n( A ∪ B ∪ C) = n(A) + n(B) +n(C) – n(A ∩ B) – (B ∩ C) – n(A ∩ C) + (A ∩ B ∩ B)

Question 5.
Verify n (A ∪ B ∪ C) = n (A) + n(B) + n (C) – n(A ∩ B) – (B ∩ C) – n(A ∩ C) + (A ∩ B ∩ C) for the following sets A = {1, 3, 5, 6, 8}, B = {3, 4, 5, 6} and C = {1, 2, 3, 6}
Solution:
(A ∪ B ∪ C) = {1,2, 3, 4, 5, 6, 8}
n (A ∪ B ∪ C) = 7
Also, n (A) = 5, n (B) = 4, n (C) = 4,
Further, A ∩ B = {3, 5, 6} ⇒ n(A ∩ B) = 3 = {3,6}
B ∩ C ⇒ n(B ∩ C) = 2
A ∩ C = {3, 5, 6} ⇒ n(A ∩ C) = 3 = {3, 6} ⇒ n(A ∩ B ∩ C) = 2
Now n (A) + n (B) + n (C) – n(A ∩ B) – n( B ∩ C) – n (A ∩ C) + n(A ∩ B ∩ C)
= 5 + 4 + 4 – 3 – 2 – 3 + 2 = 13 – 8 + 2
= 5 + 2
= 7
Thus verified

Exercise 1.7

Multiple Choice Questions

Question 1.
If A = {5, {5, 6}, 7} which of the following is correct?
(1) {5, 6} ∈ A
(2) {5} ∈ A
(3) {7} ∈ A
(4) {6} ∈ A
Answer:
(1) {5, 6} ∈ A
Hint: {5, 6} is an element of A.

Question 2.
If x = {a, {b, c}, d}, which of the following is a subset of X?
(1) {a, b}
(2) {b, c}
(3) {c, d}
(4) {a, d}
Answer:
(4) {a, d}
Hint: b is not an element of X. Similarly c.

Question 3.
If a finite set A has m elements, then the number of non-empty proper subset of A is
(1) 2^m
(2) 2^m – 1
(3) 2^m-1
(4) 2(2^m-1 – 1)
Answer:
(4) 2(2^m-1 – 1)
Hint: P(A) = 2m Proper non empty subset = 2m – 2 = 2 (2^m-1 – 1)

Question 4.
For any three A, B and C, A – (B ∪ C) is
(1) (A – B) ∪ (A – C)
(2) (A – B) ∩ (A ∪ C)
(3) (A – B) ∪C
(4) A ∪ (B – C)
Answer:
(2) (A – B) ∩ (A ∪ C)

Question 5.
Which of the following is true?
(1) (A ∪ B) = B ∪ A
(2) (A ∪ B)’ = A’ – B’
(3) (A ∩ B)’ = A’ ∩ B’
(4) A – (B ∩ C) = (A – B) ∩ (A – C)
Answer:
(1) (A ∪ B) = B ∪ A

Question 6.
The shaded region in the venn diagram is
(1) A ∪ B
(2) A ∩ B
(3) (A ∩ B)’
(4) (A – B) ∪ (B – A)
Answer:
(4) (A – B) ∪ (B – A)
Hint:

Samacheer Kalvi 9th Maths Book Solutions

Tamilnadu State Board Class 10 English Grammar Articles and Determiners

Articles And Determiners (Text Book Page No.: 40, 41)

Articles: (an, a, the)

An article is a word that modifies or describes the Noun.

It is used before the noun to show whether it refers to something specific or not.

So, in a way, articles can also be described as a type of adjectives as they also tell us something about the nouns, like adjectives.

These are of two kinds, namely, Indefinite Articles and Definite Articles.

(i) Indefinite Articles :
Indefinite means something which is not clear, obvious or exact. They are called indefinite because the identity of the thing or person being spoken about is left unclear or indefinite.

The indefinite articles in English are ‘a’ and W. They are Invariable.

We use ‘a’ when the next word starts with a consonant sound.

(Consonant letters in the English alphabet are B, C, D, F, G, H, J, K, L, M, N, P, Q, R, S, T, V, W, X, Y, Z.) or before words that start with u and eu as they sound you’.

Some examples are given below:

a boy, a car, a helicopter, a big elephant, a European, a unit etc.

We use ‘An’ when the next word starts with a Vowel sound (Vowel letters in the English alphabet are A, E, I, O, U.) or with a mute h or with a consonant giving the sound of a vowel.

Some examples are given below:
An apple, an egg, an inkpot, an orange, an umbrella, an hour, an M.L.A, an M.P. an M.A. etc.

Note: We use ‘a’ and a before a singular noun. We can’t use ‘a’ and ‘an’ before a plural noun because ‘a’ or ‘an’ is the weakened form of one.

(ii) Definite Articles :
Definite means to be clear, exact or obvious about something. It is called definite because it is used in relation to a particular thing or person. “The” is the definite article in English, which is used to refer to particular/specific nouns. The speaker talks about a particular thing.

For Example:
The boy sat on the sofa. The lion pounced on him and ran away.

Exercises
Fill in the blanks with an appropriate article, a, an, or the, or leave the space blank if no article is needed.

Question 1.
I want apple from that basket.
(a) a
(b) an
(c) the
(d) no article
Answer:
(b) an

Question 2.
church which we go is newly constructed.
(a) An
(b) A
(c) No article
(d) The
Answer:
(d) The

Question 3.
Miss Anne speaks _______ Chinese.
(a) the
(b) no article
(c) a
(d) an
Answer:
(b) no article

Question 4.
I borrowed _______ pencil from your pile of pencils and pens.
(a) no article
(b) an
(c) the
(d) a
Answer:
(d) a

Question 5.
One of the students shouted,” Principal is coming up.”
(a) the
(b) an
(c) no article
(d) a
Answer:
(a) the

Question 6.
Yusuf likes to play _______ volleyball.
(a) a
(b) no article
(c) an
(d) the
Answer:
(b) no article

Question 7.
I bought _______ umbrella to go out in the rain.
(a) an
(b) a
(c) the
(d) no article
Answer:
(a) an

Question 8.
My daughter is learning to play ________ violin at her school.
(a) a
(b) an
(c) the
(d) no article
Answer:
(c) the

Question 9.
Please give me ________ cake that is on display.
(a) an
(b) no article
(c) a
(d) the
Answer:
(d) the

Question 10.
I lived on ________ first Street when I first came to town.
(a) the
(b) a
(c) an
(d) no article
Answer:
(a) the

Tamilnadu Board Class 10 English Solutions