Tamilnadu Board

Tamilnadu State Board Class 10 Maths Solutions Chapter 2 Numbers and Sequences Ex 2.5

Question 1.
Check whether the following sequences are in A.P.
(i) a – 3, a – 5, a – 7,…
(ii) \(\frac { 1 }{ 2 } \),\(\frac { 1 }{ 3 } \),\(\frac { 1 }{ 4 } \),\(\frac { 1 }{ 4 } \),………..
(iii) 9, 13, 17, 21, 25,…
(iv) \(\frac { -1 }{ 3 } \),0,\(\frac { 1 }{ 3 } \),\(\frac { 2 }{ 3 } \),………..
(v) 1,-1, 1,-1, 1,-1,…
Solution:
To prove it is an A.P, we have to show d = t₂ – t₁ = t₃ – t₂.
(i) a – 3, a – 5, a – 7………
t₁,t₂,t₃



(v) 1,-1, 1,-1, 1,-1,…
d = t₂ – t₁ = -1 -1 = -2
d = t₃ – t₂ = 1 – (-1) = 2
-2 ≠ 2 ∴ It is not an A.P.

Question 2.
First term a and common difference d are given below. Find the corresponding A.P.
(i) a = 5,d = 6
(ii) a = 7,d = 5
(iii) a = \(\frac { 3 }{ 4 } \), d = \(\frac { 1 }{ 2 } \)
Solution:
(i) a = 5,d = 6
A.P a, a + d, a + 2d,….
= 5,5 + 6,5 + 2 × 6,…
= 5,11,17,…
(ii) a = 7,d = -5
A.P. = a,a + d,a + 2d,…
= 7,7 + (-5), 7 + 2(-5),…
= 7,2,-3,…
(iii) a = \(\frac { 3 }{ 4 } \), d = \(\frac { 1 }{ 2 } \)

Question 3.
Find the first term and common difference of the Arithmetic Progressions whose n^th terms are given below
(i) t_n = -3 + 2n
(ii) t_n = 4 – 7n
Solution:
(i) a = t₁ = -3 + 2(1) = -3 + 2 = -1
d = t₂ – t₁
Here t₂ = -3 + 2(2) = -3 + 4 = 1
∴ d = t₂ – t₁ = 1 – (-1) = 2
(ii) a = t₁ = 4 – 7(1) = 4 – 7 = -3
d = t₂ – t₁
Here t₂ = 4 – 7(2) = 4 – 14 – 10
∴ d = t₂ – t₁ = 10 – (-3) = -7

Question 4.
Find the 19th term of an A.P. -11, -15, -19,…
Answer:
A.P = -11,-15,-19,…
a = -11
d = t₂ – t₁ =-15-(-11)
= -15 + 11
= -4
n = 19
∴ t_n = a + (n – 1)d
t₁₉ = -11 + (19 – 1)(-4)
= -11 + 18 × -4
= -11 – 72
= -83

Question 5.
Which term of an A.P. 16,11, 6,1,… is -54 ?
A.P = 16, 11,6, 1,…
It is given that
t_n = -54
a = 16, d = t₂ – t₁ = 11 – 16 = -5
∴ t_n = a + (n – 1)d
-54 = 16 + (n – 1) (-5)
-54 = 16 – 5n + 5
21 – 5n = -54
-5n = -54 -21
-5n = -75
n = \(\frac { 75 }{ 5 } \) =15
∴ 15th term is -54.

Question 6.
Find the middle term(s) of an A.P. 9, 15, 21, 27,…,183.
Solution:
A.P = 9, 15,21,27,…, 183
No. of terms in an A.P. is
n = \(\frac { l-a }{ d } \) + 1
a = 9, l = 183, d = 15 – 9 = 6
∴ n = \(\frac { 183-9 }{ 6 } \) + 1
= \(\frac { 174 }{ 6 } \) + 1
= 29 + 1 = 30
∴ No. of terms = 30. The middle must be 15th term and 16th term.
∴ t₁₅ = a + (n – 1)d
= 9 + 14 × 6
=9 + 84
= 93
t₁₆ = a + 15 d
= 9 + 15 × 6
= 9 + 90 = 99
∴ The middle terms are 93, 99.

Question 7.
If nine times ninth term is equal to the fifteen times fifteenth term, show that six times twenty fourth term is zero.
Solution:
Nine times ninth term = Fifteen times fifteenth term
9t₉ = 15t₁₅
9(a + 8d) = 5(a + 14d)
9a + 72d = 15a + 210
15a + 210d – 9a – 72d = 0
⇒ 6a + 138 d = 0
⇒ 6(a + 23 d) = 0
⇒ 6(a + (24 – 1)d) = 0
⇒ 6t₂₄ = 0. Hence it is proved.

Question 8.
If 3 + k, 18 – k, 5k + 1 are in A.P. then find k.
Solution:
3 + k, 18 – k, 5k + 1 are in A.P
⇒ 2b = a + c if a, b, c are in A.P

Question 9.
Find x, y and z, given that the numbers x, 10, y, 24, z are in A.P.
Solution:
A.P = x, 10, y, 24, z,…
d = t₂ – t₁ = 10 – x …(1)
= t₃ – t₂ = y – 10 …(2)
= t₄ – t₃ = 24 – y …(3)
= t₅ – t₄ = z – 24 ….(4)
(2) and (3)
⇒ y – 10 = 24 – y
2y = 24 + 10 = 34
y = \(\frac { 34 }{ 2 } \) = 17

From (3) and (4)
24 – y = z – 24
24 – 17 = z – 24
7 = z – 24
∴ z = 7 + 24 = 31
∴ Solutions x = 3
y = 17
z = 31

Question 10.
In a theatre, there are 20 seats in the front row and 30 rows were allotted. Each successive row contains two additional seats than its front row. How many seats are there in the last row?
Solution:
t₁ = a = 20
t₂ = a + 2 = 22
t₃ = a + 2 + 2 = 24 ⇒ d = 2
∴ There are 30 rows.
t₃₀ = a + 29d
= 20 + 29 × 2
= 20 + 58
= 78
∴ There will be 78 seats in the last row.

Question 11.
The sum of three consecutive terms that are in A.P. is 27 and their product is 288. Find the three terms.
Solution:
Let the three consecutive terms be a – d, a, a + d

Question 12.
The ratio of 6th and 8^th term of an A.P is 7:9. Find the ratio of 9^th term to 13^th term.
Solution:
\(\frac{t_{6}}{t_{8}}=\frac{7}{9}\)
\(\frac { a+5d }{ a+7d } \) = \(\frac { 7 }{ 9 } \)
9a + 45d = 7a + 49d
9 a + 45 – 7d = 7a + 49 d
9a + 45d – 7a – 49d = 0
2a – 4d = 0 ⇒ 2a = 4d
a = 2d
Substitue a = 2d in

Question 13.
In a winter season let us take the temperature of Ooty from Monday to Friday to be in A.P. The sum of temperatures from Monday to Wednesday is 0° C and the sum of the temperatures from Wednesday to Friday is 18° C. Find the temperature on each of the five days.
Solution:
Let the five days temperature be (a – d), a, a + d, a + 2d, a + 3d.
The three days sum = a – a + a + a + a = 0
⇒ 3a = 0 ⇒ a = 0. (given)
a + d + a + 2d + a + 3d = 18
3a + 6d = 18
3(0) + 6 d = 18
6d = 18
d = \(\frac { 18 }{ 6 } \) = 3
∴ The temperature of each five days is a – d, a, a + d, a + 2d, a + 3d
0 – 3, 0, 0 + 3, 0 + 2(3), 0 + 3(3)
= -3°C, 0°C, 3°C, 6°C, 9°C

Question 14.
Priya earned ₹ 15,000 in the first month. Thereafter her salary increased by ₹1500 per year. Her expenses are ₹13,000 during the first year and the expenses increases by ₹900 per year. How long will it take for her to save ₹20,000 per month.
Solution:

We find that the yearly savings is in A.P with a₁ = 2000 and d = 600.
We are required to find how many years are required to save 20,000 a year …………..
a_n = 20,000
a_n = a + (n – 1)d
20000 = 2000 + (n – 1)600
(n – 1)600 = 18000
n – 1 = \(\frac { 18000 }{ 600 } \) = 30
n = 31 years

Samacheer Kalvi 10th Maths Book Solutions

Tamilnadu State Board Class 9 Maths Solutions Chapter 1 Set Language Ex 1.7

Tamilnadu Board Class 9 Maths Chapter 1 Set Language Ex 1.7 Multiple Choice Questions

Question 1.
Which of the following is correct?
(1) {7} ∈ {1,2,3,4,5,6,7,8,9,10}
(2) 7 ∈ {1,2,3,4,5,6,7,8,9,10}
(3) 7 ∉ {1,2,3,4,5,6,7,8,9,10}
(4) {7} ⊈ {1,2,3,4,5,6,7,8,9,10}
Answer:
(2) 7 ∈ {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

Question 2.
The set P = {x | x ∈ Z, -1 < X < 1} is a
(1) Singleton set
(2) Power set
(3) Null set
(4) Subset
Answer:
(I) Singleton set
Hint: P = {0}

Question 3.
If U = {x | x ∈ N, x < 10} and A = {x | x ∈ N, 2 < x < 6} then (A’)’ is
(1) {1, 6, 7, 8, 9}
(2) {1, 2, 3, 4}
(3) {2, 3, 4, 5}
(4) { }
Answer:
(3) {2, 3, 4, 5}
Hint: (A’) = A= {2, 3, 4, 5}

Question 4.
If B ⊆ A then n(A ∩ B) is
(1) n(A – B)
(2) n(B)
(3) n(B – A)
(4) n(A)
Answer:
(2) n(B)
Hint: B ⊆ A ⇒ A ∩ B = B

Question 5.
If A= {x, y, z} then the number of non-empty subsets of A is
(1) 8
(2) 5
(3) 6
(4) 7
Answer:
(4) 7
Hint: Number of non-empty subsets = 2 – 1 = 8 – 1 = 7

Question 6.
Which of the following is correct ?
(1) Φ ⊆ {a,b}
(2) Φ ∈ {a, b}
(3) {a} ∈ {a, b}
(4) a ⊆ {a, b}
Answer:
(1) Φ ⊆ {a,b}
Hint: Empty set is an improper subset

Question 7.
If A ∪ B = A ∩ B, then
(1) A ≠ B
(2) A = B
(3) A ⊂ B
(4) B ⊂ A
Answer:
(2) A = B

Question 8.
If B – A is B, then A ∩ B is
(1) A
(2) B
(3) U
(4) Φ
Answer:
(4) Φ
Hint: B – A = B ⇒ A and B are disjoint sets.

Question 9.
From the adjacent diagram n[P(A ∆ B) is
(1) 8
(2) 16
(3) 32
(4) 64

Answer:
(3) 32
Hint: A ∆ B = { 60, 85, 75, 90, 70}
⇒ n(A ∆ B) = 5
⇒ n(P(A ∆ B)) = 2⁵ = 32 [Ans. (3) 32]

Question 10.
If n(A) = 10 and n(B) = 15 then the minimum and maximum number of elements in A ∩ B is
(1) (10, 15)
(2) (15, 10)
(3) (10, 0)
(4) (0, 10)
Answer:
(4) (0, 10)

Question 11.
Let A = {Φ} and B = P(A) then A ∩ B is
(1) {Φ, {Φ} }
(2) {Φ}
(3) Φ
(4) {0}
Answer:
(2) {Φ}
Hint: P(A) = {Φ {Φ}}

Question 12.
In a class of 50 boys, 35 boys play carom and 20 boys play chess then the number of boys play both games is
(1) 5
(2) 30
(3) 15
(4) 10
Answer:
(1) 5
Hint: n(A ∪ B) = n(A) + n(B) – n(A n B) ⇒ 50 = 35 + 20 – n(A ∩ B) ⇒ n(A ∩ B) = 5

Question 13.
If U = {x : x G N and x < 10}, A ={1, 2, 3, 5, 8} and B = {2, 5, 6, 7, 9}, then n[(A ∪ B)’] is
(1) 1
(2) 2
(3) 4
(4) 8
Answer:
(1) 1
Hint: U = {1, 2, 3, 4, 5, 6, 7, 8, 9}
A = {1, 2, 3, 5, 8}
B = {2, 5, 6, 7, 9}
A ∪ B = {1,2, 3, 5, 6, 7, 8,9}
(A ∪ B)’ = {4},
n(A ∪ B)’ = 1

Question 14.
For any three sets P, Q and R, P – (Q ∩ R) is
(1), P – (Q ∪ R)
(2) (P ∩ Q) – R
(3) (P – Q) ∪ (P – R)
(4) (P – Q) ∩ (P – R)
Answer:
(3) (P – Q) ∪ (P – R)
Hint: P – (Q ∩ R) = (P – Q) ∪ (P – R)

Question 15.
Which of the following is true?
(1) A – B = A ∩ B
(2) A – B = B-A X
(3) (A ∪ B)’ = A’ ∪ B’
(4) (A ∩ B)’ = A’ ∪ B’
Answer:
(4) (A ∩ B)’ = A’ ∪ B’

Hint: (1) (A – B) = A ∩ B ✘
(2) A – B = B – A ✘
(3) (A ∪ B) = A’ ∪ B’ ✘
(4) (A ∩ B)’ = A’ ∪ B’ ✓

Question 16.
If n(A ∪ B ∪ C) = 100, n(A) = 4x, n(B) = 6x, n(C) = 5x, n(A ∩ B) = 20, n(B ∩ C) = 15, n(A ∩ C) = 25 and n(A ∩ B ∩ C)= 10 , then the value of x is
(A) 10
(B) 15
(C) 25
(D) 30
Answer:
(A) 10
Hint:
n(A ∪ B ∪ C) = n(A) + n(B) + n(C) – n(A ∩ B) – n(B ∩ C) – n(C ∩ A) + n(A ∩ B ∩ C)
100 = 4x + 6x + 5x – 20 – 15 – 25 + 10
100 = 15x -60 + 10
100 = 15x – 50
∴ 15x = 100 + 50 = 150
x = 10

Question 17.
For any three sets A, B and C, (A – B) ∩ (B – C) is equal to
(1) A only
(2) B only
(3) C only
(4) Φ
Answer:
(4) Φ
Hint: (A – B) ∩ (B – C) is equal to Φ

Question 18.
If J = Set of three sided shapes, K = Set of shapes with two equal sides and L = Set of shapes with right angle, then J ∩ K ∩ L is
(1) Set of isoceles triangles
(2) Set of equilateral triangles
(3) Set of isoceles right triangles
(4) Set of right angled triangles
Answer:
(3) Set of isoceles right triangles
Hint:

Question 19.
The shaded region in the Venn diagram is

(1) Z – (X ∪ Y)
(2) (X ∪ Y) ∩ Z
(3) Z – (X ∩ Y)
(4) Z ∪ (X ∩ Y)
Answer:
(3) Z – (X ∩ Y)
Hint: Z – (X ∩ Y)

Question 20.
In a city, 40% people like only one fruit, 35% people like only two fruits, 20% people like all the three fruits. How many percentage of people do not like any one of the above three fruits?
(1) 5
(2) 8
(3) 10
(4) 15
Answer:
(1) 5
Hint:

40 + 35 + 20 + x = 100%
95% + x = 100%
x = 5%

Samacheer Kalvi 9th Maths Book Solutions

Tamilnadu State Board Class 10 English Grammar Modals

♦ Modals (Text Book Page No.: 12–14)

A modal verb is used to indicate modality (that expresses a speaker s general intention) i.e. likelihood, ability, permission, request, capacity, suggestions, order, obligation, advice etc.

We use modals to show if we believe something is certain, probable or possible. Modals are, can, could, may, might, will, would, shall, should, must, ought to, need, used to & dare to.

Auxiliary verbs are divided into 2 categories
(a) Primary auxiliaries
(b) Modal auxiliaries,

Primary auxiliaries are the forms of the verbs to be, to have, to do Modal Auxiliaries are shall, should, will, would, can, could, may, might, dare, need, ought to, used to.

Modals are followed by principal or main verbs They express the mode or manner of happening of an action or event. They are used to give expression to ideas of ability, probability, possibility, permission, obligation, duty, determination, etc.

• Modals do not change according to the number or person of the subject. The modal does not go alone. It always goes with a principal verb.
• Can/could, may/might, will/would, shall/should and must are followed by infinitive without ‘to’.
  Geetha must go to hospital.
  Santhosh will help you.
  Devi should keep up her word.
  Rohit might leave India shortly.
• Ought and used are followed by the infinitive with ‘to’.
  You ought to keep up your promise.
  He used to teach French.
• The negative is formed by putting ‘not’ after the auxiliary.
  Students must not wear jeans.
  She couldn’t walk as she fell down recently.

Can is used to express
(a) Ability – I can speak English fluently.
(b) Permission – Can I go out? You can go home now.
(c) Possibility – Malaria can be dangerous.

Could is the past tense of can. It is used to express:
(a) Ability in the past – I could easily run fast those days.
(b) A polite request in the present; in the form of a question – Could I see the manager for a minute?

May is used for all persons in the present and future. It is followed by the infinitive without ‘to’. May used to
(a) Seek or grant permission – May I go home in the afternoon?
(b) Express possibility – It may rain in the afternoon.
(c) Express a wish or prayer – May God bless you.
(d) Express a purpose – He goes to gym so that he may become lean.

Might is the past tense of May. It is used to express
(a) A doubtful possibility – He might go to a movie.
(b) Permission in the present tense – Might I ask a doubt?
(c) To show permission – He told me that I might use his car.
(d) Purpose – He left office early so that he might go for a function.
(e) Suggestion – you might visit the big temple which is on your way to college.

Notice the difference between May and Might.
You may put the lights on. (Giving permission)
You might put the lights on. (Suggesting)
Might is used as the past tense of May.

My mother said, “You may go to movie at three.” My mother told The* thkt I might go to the movie at three–o–clock.

Shall is used to express the future when the subject is I or We.
I shall be sixteen next year.
Shall is used to express:

(a) Certainty – If you sing well, you shall be the winner.
(b) Assurance – Take the medicine regularly and you shall be well.
(c) Promise – If you win the match, you shall have a holiday.
(d) Threat – If you do not pass in the exam, you shall be retained in the same class.
(e) Command – All students shall reach the prayer hall by 4.00 p.m.

When shall is used in questions, it asks the will or the desire of the person addressed.
Shall I go with you for the movie? Shall we go to the market?

Should is the past tense of shall. Should is used to express:
(a) Duty or moral obligation.
We should take care of our parents during old age.
(b) Advice or instruction or opinion.
You should lock the door when you leave home.
(c) High degree probability.
The team should win the trophy for all the efforts taken.
Should is used as the past tense of shall.
You should have worn a grand dress for the party.

Must refers to the present or the future tense. It is used to express:
(a) Obligation or necessity – We must obey our parents.
(b) Compulsion – You must finish your assignment by tomorrow.
(c) Possibility – She should have reached home by now.

Ought is used nearly in the same sense as should. It is followed by the infinitive ‘to’. It is
(a) Used to express duty or moral binding – You ought to send money to your parents.
(b) An inference – The train ought to be arriving late, hence the platform is crowded.
(c) Possibility – He has done well in the interview. He ought to get the job.

Exercises: Type –1

Complete the following sentence choosing the most appropriate Modal Verb.

Question 1.
We _______ obey our teachers
(a) can
(b) may
(c) must
(d)should
Answer:
(c) must

Question 2.
She _______ pass this exam
(a) ought to
(b) used to
(c) dare to
(d) need
Answer:
(a) ought to

Question 3.
He _______ not buy a car.
(a) need
(b) dare
(c) ought
(d) used
Answer:
(a) need

Question 4.
He works hard lest he _______ fail.
(a) Can
(b) shall
(c) should
(d) may
Answer:
(c) should

Question 5.
_______ you cook your own meal?
(a) Shall
(b) May
(c) Can
(d) Could
Answer:
(c) Can

Question 6.
The villagers _______ buy kerosene lamps a few years ago.
(a) can
(b) will
(c) could
(d) would
Answer:
(c) could

Question 7.
The old lady _______ take a bath every day before taking meal during fasting days.
(a) used to
(b) dare to
(c) need to
(d) need
Answer:
(a) used to

Question 8.
She _______ finish this work before I go.
(a) can
(b) will
(c) must
(d) shall
Answer:
(c) must

Question 9.
Ramesh said that they _______ report for duty on Monday.
(a) may
(b) should
(c) might
Answer:
(b) should

Question 10.
We _______ prepare our lessons well before examination.
(a) can
(b) ought to
(c) dare to
Answer:
(b) ought to

Exercises: Type – 2

Fill in the blanks with appropriate modals:

Question 1.
We _______ pay attention to our studies, (ought to, should)
Answer:
should

Question 2.
You _______ not litter the classroom, (should, could)
Answer:
should

Question 3.
I _______ talk to you immediately, (need to, ought to)
Answer:
nedd to

Question 4.
They _______ not enter the exam hall without hall ticket (May, will, must)
Answer:
must

Question 5.
He _______ take those medicines when he had an operation, (used to, dare to , ought to)
Answer:
ought to

Question 6.
You _______ consult a physician, (should, ought to)
Answer:
should

Question 7.
You _______ exercise daily, (dare to, need to)
Answer:
need to

Question 8.
They _______ not send the letter now. (need, would)
Answer:
would

Question 9.
Sahil ran fast so that he _______ catch the train, (should, might)
Answer:
should

Question 10.
You _______ not use the office phone for private calls, (must, could)
Answer:
must

Exercises: Type – 3

Fill in with modals.

Question 1.
You _______ arrive any time, but you _______ want to be early to get a good seat.
Answer:
can; may

Question 2.
She _______ recite the entire poem from memory.
Answer:
could

Question 3.
If she receives a scholarship, she _______ continue with her studies.
Answer:
will

Question 4.
_______ I open it for you?
Answer:
Shall

Question 5.
I _______ be surprised if he is not elected.
Answer:
would

Question 6.
_______ you leave early tomorrow?
Answer:
Can

Question 7.
_______ you give me a ride?
Answer:
Could

Question 8.
You _______ take a day off tomorrow if you can’t find a home for rent.
Answer:
can

Question 9.
If necessary, you _______ ask for an extension.
Answer:
could

Question 10.
If you work hard, you _______ relax for the rest of the semester.
Answer:
can

Question 11.
If you continue to improve your grades, you _______ graduate with honors.
Answer:
could

Question 12.
We _______ overcome.
Answer:
shall

Question 13.
You _______ receive a new credit card in the mail.
Answer:
should

Question 14.
_______ you join me for lunch?
Answer:
Will

Question 15.
I _______ appreciate your assistance.
Answer:
would

Tamilnadu Board Class 10 English Solutions

Tamilnadu State Board Class 9 Maths Solutions Chapter 1 Set Language Ex 1.6

Question 1.
(i) If n(A) = 25, n(B) = 40, n(A ∪ B) = 50 and n(B’) = 25 , find n(A ∩ B) and ii(U).
(ii) If n(A) = 300, n(A ∪ B) = 500, n(A ∩ B) = 50 and n(B’) = 350, find n(B) and n(U).
Solution:
(i) n(A ∩ B) = n(A) + n(B) – n(A ∪ B)
n(A ∩ B) = 25 + 40 – 50 = 65 – 50= 15
n(U) = n(B) + n(B’) = 40 + 25 = 65

(ii) n(U) = n(B) + n(B’)
n(A ∩ B) = n(A) + n(B) – n(A B)
n(B) = n(A ∪ B) + n(A ∩ B) – n(A) = 500 + 50 – 300 = 250
n(U) = 250 + 350 = 600

Question 2.
If U = {x : x ∈ N, x ≤ 10},A = {2, 3,4, 8, 10) and b = {1, 2, 5, 8, 10}, then verify that n(A ∪ B)= n(A) + n(B) – n(A ∩ B)
Solution:
n(A) = 5, n(B) = 5
A ∪ B = {1, 2, 3, 4, 5, 8, 10}, A ∩ B= {2,8, 10}
n(A ∪ B) = 7, n(A ∩ B) = 3
L.H.S n(A ∪ B) = 7
R.H.S = n(A) + n(B) – n(A ∩ B) = 5 + 5 – 3 = 7
∴ L.H.S = R.H.S proved.

Question 3.
Verify n(A ∪ B ∪ C) = n(A) + n(B) + n(C) – n(A ∩ B) – n(B ∩ C) – n(A ∩ C) + n(A ∩ B ∩ C) for the following sets.
(i) A = {a, c, e, f, h}, B = {c, d, e,f} and C = {a, b, c,f}
(ii) A = {1, 3, 5} B = {2, 3, 5, 6} and C = {1, 5, 6, 7}.
Solution:
n(A ∪ B ∪ C) = n(A) + n(B) + n(C) – n(A ∩ B) – n(B ∩ C) – n(A ∩ C) + n(A ∩ B ∩ C)

(i) A = {a, c, e,f, h}, B = {c, d, e, f}, C = {a, b, c, f}
n (A) = 5, n (B) = 4, n (C) = 4
n( A ∩ B) =3
n(B ∩ C) = 2
n( A ∩ C) =3
n( A ∩ B ∩ C) = 2
A ∩ B = {c,e,J}
B ∩ C = {c,f}
A ∩ C = {a, c,f}
A ∩ B ∩ C = {c,f}
A ∪ B ∪ C = {a, c,d, e,f, b, h}
∴ n(A ∪ B ∪ C) = 7 … (1)
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 = 15 – 8
∴ (1) =(2)
⇒ n(A ∪ B ∪ C) = n(A) + n(B) + n(C) – n(A ∩ B) – n(B ∩ C) – n(A ∩ C) + n(A ∩ B ∩ C)
Hence it is verified.

(ii) A = {1, 3, 5}, B = {2, 3, 5, 6 }, C = {1, 5, 6,7} = 3, n (B) = 4, n (C) = 4
n(A ∩ B) = 2
n(B ∩ C) = 2
n(C ∩ A) = 2
n(A ∩ B ∩ C) = 1
n(A ∪ B ∪ C) = 6
n(A ∪ B ∪ C) = n(A) + n(B) + n(C) – n(A ∩ B) – n(B ∩ C) – n(A ∩ C) + n(A ∩ B ∩ C)
6 = 3 + 4 + 4 – 2 – 2 – 2 + 1 = 12 – 6 = 6
Hence it is verified.

Question 4.
In a class, all students take part in either music or drama or both. 25 students take part in music, 30 students take part in drama and 8 students take part in both music and drama. Find
(i) The number of students who take part in only music.
(ii) The number of students who take part in only drama.
(iii) The total number of students in the class.
Solution:
Let the number of students take part in music is M.
Let the number of students take part in drama is D.

(i) The number of students take part in only music is 17.
(ii) The number of students take part in only drama is 22.
(iii) The total number of students in the class is 17 + 8 + 22 = 47.

Question 5.
In a party of 45 people, each one likes tea or coffee or both. 35 people like tea and 20 people like coffee. Find the number of people who
(i) like both tea and coffee.
(ii) do not like tea.
(iii) do not like coffee.
Solution:
Let the people who like tea be T.
Let the people who like coffee be C
By using formula
n( A ∪ B) = n(A) + n(B) – n(A ∩ B)

(i) n(T ∩ C) = n(T) + n(C) – n(T ∪ C) = 35 + 20 – 45 = 55 – 45 = 10
The number of people who like both coffee and tea = 10.

(ii) The number of people who do not like Tea
n(T) = n(U) – n(T) = 45 – 35 = 10

(iii) The number of people who do not like coffee
n(C’) = n(U) – n(C) = 45 – 20 = 25.

Question 6.
In an examination 50% of the students passed in Mathematics and 70% of students passed in Science while 10% students failed in both subjects. 300 students passed in at least one subjects. Find the total number of students who appeared in the examination, if they took examination in only two subjects.
Solution:
Let the students who appeared in the examination be 100%.
Let the percentage of students who failed in mathematics be M.
Let the percentage of students who failed in science be S.
Failed in Maths = 100 % – Pass% = 100% – 50% = 50%
Failed in Science% 100% – 70% = 30%
Failed in both% = 10%
n(M ∪ S) = n(M) + n(S) – n(M ∩ S)
= 50% + 30% – 10% = 70%
% of students failed in atleast one subject = 70%
∴ The % of students who have passed in atleast one subject = 100% – 70% = 30%
30% = 300
∴ \(100 \%=\frac{100 \times 300^{10}}{30}=1000\)
∴ The total number of students who appeared in the examination = 1000 students.

Question 7.
A and B are two sets such that n(A – B) = 32 + x, n(B – A) = 5x and n(A ∩ B) = x. Illustrate the information by means of a venn diagram. Given that n(A) = n(B), calculate the value of x.
Solution:

n (A – B) = 32 +x
n(B – A) = 5x
n(A ∩ B) = x
n( A) = n(B)
32 + x + x = 5x + x
32 + 2x = 6x
4x = 32
x = 8

Question 8.
Out of 500 car owners investigated, 400 owned car A and 200 owned car B, 50 owned both A and B cars. Is this data correct?
Solution:
n( A ∪ B) = n(A) + n(B) – n( A ∩ B)
n(A ∪ B) = 500 (given) … (1).
n( A) = 400
n(B) = 200
n( A ∩ B) =50
∴ n(A ∩ B) = 400 + 200 – 50 = 550 …(2)
1 ≠ 2
∴ This data is incorrect.

Question 9.
In a colony, 275 families buy Tamil newspaper, 150 families buy English newspaper, 45 families buy Hindi newspaper, 125 families buy Tamil and English newspapers, 17 families buy English and Hindi newspapers, 5 families buy Tamil and Hindi newspapers and 3 families buy all the three newspapers. If each family buy atleast one of these newspapers then find
(i) Number of families buy only one newspaper
(ii) Number of families buy atleast two newspapers
(iii) Total number of families in the colony.
Solution:
(i) Tamil Newspaper buyers n(A) = 275
English Newspaper buyers n(B) = 150
Hindi Newspaper buyers n(C) = 45
Tamil and English Newspaper buyers n(A ∩ B) = 125
English and Hindi Newspaper buyers n(B ∩ C) = 17
Hindi and Tamil Newspaper buyers n(C ∩ A) = 5
All the three Newspaper buyers n(A ∩ B ∩ C) = 3

(i) Number of families buy only one newspaper = 148 + 11 + 26 = 185
(ii) Number of families buy atleast two news papers = 122 + 14 + 2 + 3 = 141
(iii) Total number of families in the colony = 148 + 11 + 26 + 122 + 14 + 2 + 3 = 326

Question 10.
A survey of 1000 farmers found that 600 grew paddy, 350 grew ragi, 280 grew corn, 120 grew paddy and ragi, 100 grew ragi and corn, 80 grew paddy and corn. If each farmer grew atleast any one of the above three, then find the number of farmers who grew all the three.
Solution:

a = 600 – (120 – x + x + 80 – x)
= 600 – (200 – x)
= 600 – 200 + x
= 400 + x
b = 350 – (120 – x + x + 100 – x)
= 350 – (220 – x)
= 350 – 230 + x
= 130 + x
c = 280 – (80 – x + x + 100 – x)
= 2800 – (180 – x) = 280 – 180 + x = 100 + x
Each farmer grew atleast one of the above three, the number of farmers who grew all the three is x.
= a + b + c + 120 – x + 100 – x + 80 – x + x = 1000
400 + x + 130 + x + 100 + x + 120 – x + 100 – x + 80 – x + x= 1000
∴ 930 + x = 1000
x = 1000 – 930 = 70
∴ 70 farmers grew all the three crops

Question 11.
In the adjacent diagram, if n(U) = 125,y is two times of x and z is 10 more than x, then find the value of x, y and z.

Solution:
n(U) = 125
y = 2x
z = x + 10
∴ x + y + z + 4 + 17 + 6 + 3 + 5 = 125
x + 2x + x + 10 + 35 = 125
4x + 45 = 125
4x = 125 – 45
4x = 80
x = 20
∴ y = 2x = 2 x 20 = 40
z = x + 10 = 20 + 10 = 30
Hence x = 20 ; y = 40; z = 30

Question 12.
Each student in a class of 35 plays atleast one game among chess, carrom and table tennis. 22 play chess, 21 play carrom, 15 play table tennis, 10 play chess and table tennis, 8 play carrom and table tennis and 6 play all the three games. Find the number of students who play (i) chess and carrom but not table tennis (ii) only chess (iii) only carrom (Hint: Use Venn diagram)
Solution:

A – Chess
B – Carrom
C – Table Tennis
n(A) = 22
n(B) = 21
n(C) = 15
n(A ∩ C) = 10
n(B ∩ C) = 8
n(A ∩ B ∩ C) = 6
(i) y = 22 – (x + 6 + 4) = 22 – (x + 10)
= 22 – x – 10
= 12 – x
z = 21 – (x + 6 + 2)21 – (8 + x)
21 – 8 – x 13 – x
y + z + 3 + x + 2 + 4 + 6 = 35
12 – x + 13 – 1 + 15 + x = 35
40 – x = 35
x = 40 – 35 = 5
(i) Number of students who pay only chess and Carrom but not table tennis = 5
(ii) Number of students who play only chess = 12 – x = 12 – 5 = 7
(iii) Number of students who play only carrom = 13 – x = 13 – 5 = 8

Question 13.
In a class of 50 students, each one come to school by bus or by bicycle or on foot. 25 by bus, 20 by bicycle, 30 on foot and 10 students by all the three. Now how many students come to school exactly by two modes of transport?
Solution:

A – by bus
B – by bicycle
C – on foot
n(A) = 25
n(B) = 20
n(C) = 30
n(A ∩ B ∩ C ) = 10
n(A ∪ B ∪ C ) = n(A) + n(B) + n(C) – n(A ∩ B) – n(B ∩ C) – n(C ∩ A) + n(A ∩ B ∩ C)
50 = 25 + 20 + 30 – (10 + x) – (10 +y) – (10 + z) + 10
50 = 75 – 10 – x – 10 – y – 10 – z + 10
= 75 – 20 – (x + y + z)
= 55 – (x + y + z)
x + y + z = 55 – 50 = 5
∴ The number of students who come to school exactly by two modes of transport = 5.

Samacheer Kalvi 9th Maths Book Solutions

Tamilnadu State Board Class 10 English Solutions Poem Chapter 6 No Men Are Foreign

No Men Are Foreign Cricket Textual Questions

Based on the understanding of the poem, read the following lines and answer the questions given below.

1. Beneath all uniforms, a single body breathes
Like ours: the land our brothers walk upon
Is earth like this, in which we all shall lie.
(a) What is found beneath all uniforms?
(b) What is same for every one of us?
(c) Where are we all going to lie finally?
Answer:
(a) The same human body that breathes is found beneath all uniforms.
(b) The earth we walk upon is the same for all of us.
(c) We are all going to lie beneath the earth.

2. They, too, aware of sun and air and water,
Are fed by peaceful harvests, by war’s long winter starv’d.
(a) What is common for all of us?
(b) How are we fed?
(c) Mention the season referred here?
Answer:
(a) The sun, air and water are common for all of us.
(b) When a country is peaceful and harvest can be reaped we are fed.
(c) Winter is the season referred here.

3. Their hands are ours, and in their lines we read
A labour not different from our own.
(a) Who does ‘their’ refer to?
(b) What does the poet mean by Tines we read’?
(c) What does not differ?
Answer:
(a) All the people from other countries of the world.
(b) The poet means lines that show hard work done by the hands.
(c) Labour does not differ. Hard work is the same for all human beings.

4. Let us remember, whenever we are told
To hate our brothers, it is ourselves
That we shall dispossess, betray, condemn.
(a) Who tells us to hate our brothers?
(b) What happens when we hate our brothers?
(c) What do we do to ourselves?
Answer:
(a) War tells us to hate our brothers.
(b) When we hate our brothers We hate ourselves.
(c) We disown ourselves, we cheat ourselves and hate ourselves.

5. Our hells of fire and dust outrage the innocence
Of air that is everywhere our own,
Remember, no men are foreign, and no countries strange.
(a) What outrages the innocence?
(b) Who are not foreign?
(c) What is not strange?
Answer:
(a) The fire and dust caused during a war when innocent people as well the earth suffer the purity is lost.
(b) The people who belong to earth are not foreign.
(c) All the countries on earth are not strange.

Poetic Comprehension – Additional

1. Remember they have eyes like ours that wake
Or sleep, and strength that can be won By love.
(a) Who do the words ‘they’ refer to?
(b) How can we win other people?
(c) Explain’They have eyes like ours that wake or sleep”.
Answer:
(a) They’ refer to people of different countries whom we consider to be strangers.
(b) The strength of other people can be won by love and kindness.
(c) The strangers or foreigners are similar to us in every way. They sleep and wake up each new day just like us.

2. In every land is common life
That all can recognise and understand.
(a) What do you understand by “common life”?
(b) Who experiences the common life?
(c) What does the poet is trying to convey in these lines?
Answer:
(a) Common life means life anywhere in the world that has similar patterns and features like birth, death, joys, sorrows, youth and old age.
(b) ‘The common life’ is experienced by all the people of the world regardless of the country in which they live.
(c) He is trying to convey that even our strangers are similar to us in every way.

3. Remember, we who take arms against each other It is the human earth that we defile.
(a) Why do we take arms against each other?
(b) What is ‘human earth’?
(c) How do we defile ‘human earth’?
Answer:
(a) We take arms against each other when we decide to harm each other.
(b) ‘Human earth’ is the human world that is comprised of all countries, races, cultures and creeds.
(c) We defile or pollute the human earth by using arms and ammunition to cause widespread death and destruction.

Poetic Devices – Additional

1. Beneath all uniforms, a single body breathes
Like ours: the land our brothers walk upon
(a) What is the figure of speech employed ?
(b) Pick out the alliterated words.
Answer:
(a) ‘Like ours’ – similie
(b) Beneath – body – breathes are alliterated words.

2. Our hells of fire and dust outrage the innocence
(a) What is the figure of speech used in this line?
Answer:
Metaphor is used here in the words ‘Hells of fire’. Hells are compared to fire.

3. They, too, aware of sun and air and water,
Are fed by peaceful harvests, by war’s long winter starv’d.
(a) What is the poetic device used here?
Answer:
winter – starv’d is the “transferred epithet” used here. Winter is not starved here, but the people . are starved.

4. Remember, no men are strange, no countries foreign
Remember, no men are foreign, and no countries strange.
(a) What is the figure of speech used in these lines?
Answer:
Repetition. It emphasizes the core message of the oneness of mankind.

Based on your understanding of the poem complete the following by choosing the appropriate words/phrases given in brackets:

This poem is about the (i) ______ of all men. The subject of the poem is the (ii) ______race, despite of the difference in colour, caste, creed, religion, country etc. All human beings are same. We walk on the (iii) ______and we will be buried under it. Each and everyone of us are related to the other. We all are born same and die in the same way. We may wear different uniforms like (iv) ‘ ______’ during wars the opposing side will also have the same (v) ______ like ours. We as human do they same labour with (vi) ______and look at the world with the (vii)______. Waging war against others as they belong to a different country is like attacking our own selves. It is the (viii) ______ we impair. We all share the same (ix) ______. We are similar to each other. So the poet concludes that we shouldn’t have wars as it is(x) ________ to fight against us.

Answer:
(i) brotherhood
(ii) unity of human
(iii) same land
(iv) language
(v) dreams and aspirations
(vi) our hand
(vii) same eye
(viii) breathing body
(ix) human earth
(x) unnatural

Based on your understanding of the poem answer the following questions in a paragraph of about 100-150 words.

Question 1.
What is the central theme of the poem ‘No men are foreign’?
Answer:
The central theme of the poem is about the oneness of mankind. Though there are many countries, races, colour, languages, castes and creeds all are human beings. We occupy the same land and die in the same land. Hands work hard, eyes witness the same life, we are all fed by peaceful harvests. Attacking others who are neighbours in the same planet is nothing but harming one self. Hence, no land is foreign, no man is a stranger.

Question 2.
The poem ‘No men are foreign’ has a greater relevance in todays world. Elucidate.
Answer:
The poem “No men are foreign’ has a greater relevance in today’s world that witnesses major disaster due to internal conflicts and external factors that destroy the peace of nations of the world. Factors like racism, terrorism etc., alienate people to the extent of ruining places and civilizations that had taken centuries of hard work to be built. The very fact that once we consider the entire world as a single living place for all who exist on it. It becomes a commonplace and people will stay united understanding the truth that all are human with the same dreams and aspirations.

Paragraph Question & Answer – Additional

Question 1.
How does the title sum up the theme of the poem ‘No men are Foreign’?
Answer:
‘No men are Foreign’ is an apt title for the poem, because it deals with the theme of universal brotherhood. Right through the poem, the poet talks about the concept of universal brotherhood ‘ and peaceful co-existence. He insists that we should not give any place for any kind of prejudice. He emphasises the fact that all human beings are naturally the same. The divisions based on nation, caste, colour, creed or religion are baseless. Every human will respond positively to love. The poet James Kirkup, has beautifully conveyed these ideas through the title of the poem “No Men are Foreign”. He gives a strong message to stop all wars and look upon humanity as a unified entity.

No Men Are Foreign By James Falconer Kirkup

James Falconer Kirkup (1918 – 2009), born James Harold Kirkup, is an English poet, translator and travel writer. He wrote over 30 books, including autobiographies, novels and plays. He wrote his first book of poetry, “The Drowned Sailor” which was published in 1947. He left England in a 1956 to live and work in Europe, America and the Far East. He settled in Japan for 30 years, lecturing in English literature at several universities. He has won many awards such as Atlantic-Rockefeller award, (1950), Japan P.E.N. Club International literary prize, (1965), Batchelder award, for translation, (1968), Keats prize, (1974), Scott-Moncrieff prize for translation, (1993). He became a Fellow of the Royal Society of Literature in 1962. “No men are Foreign” was published in the year 1966. ‘

No Men Are Foreign Key Points

• This poem tells us why we must not hate our brothers and sisters who live in different countries of the world.
• No man or country is strange.
• Beneath die various colours of uniform, it is the same human body.
• The earth that every human walks on is the same.
• The otherS hround the world depend on the same air, water and sun.
• They eat out of peaceful harvests and are starved during war.
• Their hands have the same lines of hard work.
• Their eyes see, sleep and wake like ours.
• Like everyone else the strength can be overcome by loving each other.
• The common life is understood by everyone.
• We must remember if we try to hate others it means we hate ourselves.
• When we try to use weapons against others we are ruining human beings and earth.
• Fire and dust destroy the purity of air that belongs to all.
• So we must remember that no one is our enemy or foreigner and no country is a strange land on this earth.

No Men Are Foreign Summary

James Falconer Kirkup reminds us that all the people who belong to the earth are similar and share the brotherhood of men. Though the colours of uniform are different, it is the same human body. All walk on the same earth, see the same sun, inhale air and use water the same way. They feed by peaceful harvests or starve during war. Their eyes see the same things while awake and shut when asleep. Everywhere strength is overcome by true love, shown towards each other. When one is taught hatred it means hating oneself, cheat ourselves. When we hold weapons against each other, we are ruining the earth. The painful hell of fire and anger and dust spoils the pure air that is inhaled by all of us on earth. No one on earth is a stranger to anyone else and no country an enemy or foreign land.

No Men Are Foreign  Explanation of Poetic Lines

Line No. 1 – 4
Remember, no men are strange, no countries foreign
Beneath all uniforms, a single body breathes
Like ours: the land our brothers walk upon
Is earth like this, in which we all shall lie.
Explanation:
The poet tells that men are not different from each other simply on the basis that they come from separate countries. They may fight for the army of their country, but underneath the uniform, they are all similar. Our opponent army’s soldiers live and breathe just as we do. Most fights between two countries result from disputes over a piece of land. However, all the land on earth is the same. Moreover, every man will have to lie in the earth at one time or the other. That is, all men have to die and all pieces of land are potential graveyards in which they fight.

Line No. 5 – 8
They, too, aware of sun and air and water,
Are fed by peaceful harvests, by war’s long winter starv’d.
Their hands are ours, and in their lines we read
A labour not different from our own.
Explanation:
The poet gives further evidence of the unity of man. He says that those who live in other countries also depend on the same sun, air and water for their survival. They too have seen peace and war. During a peaceful period, they have a good harvest and food products. Similarly during the war period, just like a long winter, availability of food will be in scarce, resulting in famine. They look just like us and are descended from the same ancestors as we are.

Line No. 9 – 12
Remember they have eyes like ours that wake
Or sleep, and strength that can be won
By love. In every land is common life
That all can recognise and understand.
Explanation:
The poet urges us to keep in mind the fact that our opponents sleep and wake like we do, and follow a similar type of lifestyle. Every human being has an inner resource of strength that he can use to help his fellow beings. However, that resource can only be unlocked by following the path of love. When we love someone then only we stand up for them. That is why it is necessary for every man to love his fellow beings without creating divisions amongst them. Despite the differences between various nations, there is one thing that they all have in common. The common people live the same kind of life everywhere. Therefore, we can recognise the presence of that life no matter where we travel in this wide world.

Line No. 13 – 15
Let us remember, whenever we are told
To hate our brothers, it is ourselves
That we shall dispossess, betray, condemn.
Explanation:
The poet disagrees against the efforts of military and religious groups that encourage us to wage wars. Those groups are misguiding their own people. They are asking their followers to hate their opponents without realising that they are our own brothers. It is a form of self-destruction. Any hatred that we show on any member of the human race is equivalent to our betrayal towards the people. If we kill people of any other country, we are endangering human beings as a whole and their survival on earth.

Line No. 16 – 17
Remember, we who take arms against
each other
It is the human earth that we defile.
Explanation:
The poet specifies the impacts of war. If we, the guardians of the earth, were to die out as a result of the war, then there would no one left to take care of the home that our ancestors had passed down to us. We would then have nothing to pass down to our future generations.

Line No. 18 – 20
Our hells of fire and dust outrage the innocence
Of air that is everywhere our own,
Remember, no men are foreign, and no
countries strange.
Explanation:
The poet compares wartime with hell. In war as in hell, there is fire and smoke everywhere. As a result of this, the earth is becoming poisoned. The very air that we breathe is becoming impure and will not be able to sustain human life for much longer. That is why the poet encourages us not to wage war on our fellow men thinking they are foreigners and that their countries are unlike our own.

No Men Are Foreign  Glossary

betray – disloyal
condemn – express complete disapproval
defile – damage the purity or appearance
dispossess – deprive someone of a possession
foreign – a country or language other than one’s own
innocence – not guilty of a crime
labour – hardwork
outrage – extremely strong reaction of anger, shock
recognise – know someone having come across earlier

Tamilnadu Board Class 10 English Solutions

Tamilnadu State Board Class 10 Maths Solutions Chapter 2 Numbers and Sequences Ex 2.3

Question 1.
Find the least positive value of x such that
(i) 71 = x (mod 8)
(ii) 78 + x = 3 (mod 5)
(iii) 89 = (x + 3) (mod 4)
(iv) 96 = \(\frac { x }{ 7 } \) (mod 5)
(v) 5x = 4 (mod 6)
Solution:
To find the least value of x such that
(i) 71 = x (mod 8)
71 = 7 (mod 8)
∴ x = 7.[ ∵ 71 – 7 = 64 which is divisible by 8]

(ii) 78 + x = 3 (mod 5)
⇒ 78 + x – 3 = 5M for some integer n.
75 + x = 5 n 75 + x is a multiple of 5.
75 + 5 = 80. 80 is a multiple of 5.
Therefore, the least value of x must be 5.

(iii) 89 = (x + 3) (mod 4)
89 – (x + 3) = 4n for some integer n.
86 – x = 4 n
86 – x is a multiple of 4.
∴ The least value of x must be 2 then
86 – 2 = 84.
84 is a multiple of 4.
∴ x value must be 2.

(iv) 96 ≡ \(\frac { x }{ 7 } \) (mod 5)
96 – \(\frac { x }{ 7 } \) = 5n for some integer n.
\(\frac { 672-x }{ 7 } \) = 5n
672 – x = 35n.
672 – x is a multiple of 35.
∴ The least value of x must be 7 i.e. 665 is a multiple of 35.

(v) 5x = 4 (mod 6)
5x – 4 = 6M for some integer n.
5x = 6n + 4
x = \(\frac { 6n+4 }{ 5 } \)
When we put 1, 6, 11, … as n values in x = \(\frac { 6n+4 }{ 5 } \) which is divisible by 5.
When n = 1, x = \(\frac { 10 }{ 5 } \) = 2
When n = 6, x = \(\frac { 36+4 }{ 5 } \) = \(\frac { 40 }{ 5 } \) = 8 and so on.
∴ The solutions are 2,8,14…..
∴ Least value is 2.

Question 2.
If x is congruent to 13 modulo 17 then 7x -3 is congruent to which number modulo 17?
Solution:
x ≡ 13 (mod 17)
Let p be the required number …(1)
7x – 3 = p (mod 17) …(2)
From (1),
x – 13 = 17n for some integer M.
x – 13 is a multiple of 17.
x must be 30.
∴ 30 – 13 = 17
which is a multiple of 17.
From (2),
7 × 30 – 3 ≡ p (mod 17)
210 – 3 ≡ p (mod 17)
207 ≡ p (mod 17)
207 ≡ 3 (mod 17)
∴ P ≡ 3

Question 3.
Solve 5x ≡ 4 (mod 6)
Solution:
5x ≡ 4 (mod 6)
5x – 4 = 6M for some integer n.
5x = 6n + 4
x = \(\frac { 6n+4 }{ 5 } \) where n = 1,6,11,…..
∴ x = 2, 8, 14,…

Question 4.
Solve 3x – 2 = 0 (mod 11)
Solution:
3x – 2 = 0 (mod 11)
3x – 2 = 11 n for some integer n.
3x = 11n + 2

Question 5.
What is the time 100 hours after 7 a.m.?
Solution:
100 ≡ x (mod 12) (∵7 comes in every 12 hrs)
100 ≡ 4 (mod 12) (∵ Least value of x is 4)
∴ The time 100 hrs after 7 O’ clock is 7 + 4 = 11 O’ clock i.e. 11 a.m

Question 6.
What is the time 15 hours before 11 p.m.?
Solution:
15 ≡ x (mod 12)
15 – x = 12n
15 – x is a multiple of 12 x must be 3.
∴ The time 15 hrs before 11 O’clock is 11 – 3 = 8 O’ clock i.e. 8 p.m

Question 7.
Today is Tuesday. My uncle will come after 45 days. In which day my uncle will be coming?
Solution:
No. of days in a week = 7 days.
45 ≡ x (mod 7)
45 – x = In
45 – x is a multiple of 7.
∴ Value of x must be 3.
∴ Three days after Tuesday is Friday. Uncle will come on Friday.

Question 8.
Prove that 2ⁿ + 6 × 9ⁿ is always divisible by 7 for any positive integer n.
Solution:
2¹ + 6 × 9¹ = 2 + 54 = 56 is divisible by 7
When n = k,
2^k + 6 × 9^k = 7 m [where m is a scalar]
⇒ 6 × 9^k = 7 m – 2^k …(1)
Let us prove for n = k + 1
Consider 2^k+1 + 6 × 9^k+1 = 2^k+1 + 6 × 9^k × 9
= 2^k+1 + (7m – 2^k)9 (using (1))
= 2^k+1 + 63m – 9.2^k = 63m + 2^k.2¹ – 9.2^k
= 63m – 2^k (9 – 2) = 63m – 7.2^k
= 7 (9m – 2^k) which is divisible by 7
∴ 2ⁿ + 6 × 9ⁿ is divisible by 7 for any positive integer n

Question 9.
Find the remainder when 2⁸¹ is divided by 17.
Solution:
2⁸¹ = x (mod 17)
2⁴⁰ × 2⁴⁰ × 2⁴¹ = x (mod 17)
(24)¹⁰ × (24)¹⁰ × 2¹ = x (mod 17)
(16)¹⁰ × (16)¹⁰ × 2 = x(mod 17)
(16⁵)² × (16⁵)² × 2
(16⁵) = 16 (mod 17)
(16⁵)² = 16² (mod 17)
(16⁵)² = 256 (mod 17)
= 1 (mod 17)
[∵ 255 is divisible by 17]
(16⁵)² × (16⁵)² × 2 ≡ 1 × 1 × 2 (mod 17)
∴ 2⁸¹ = 2(mod 17)
∴ x = 2

Question 10.
The duration of flight travel from Chennai to London through British Airlines is approximately 11 hours. The airplane begins its journey on Sunday at 23:30 hours. If the time at Chennai is four and half hours ahead to that of London’s time, then find the time at London, when will the flight lands at London Airport.
Solution:
The duration of flight from Chennai to London is 11 hours.
Starting time at Chennai is 23.30 hrs. = 11.30 p.m.
Travelling time = 11.00 hrs. / = 22.30 hrs = 10.30 a.m.
Chennai is 4 \(\frac { 1 }{ 2 } \) hrs ahead to London.
= 10.30 – 4.30 = 6.00
∴ At 6 a.m. on Monday the flight will reach at London Airport.

Samacheer Kalvi 10th Maths Book Solutions

Tamilnadu State Board Class 10 English Vocabulary Constructing Sentences using different Parts of Speech

Words have different forms like noun, verb, adjective and adverb, etc. This exercise contains three of a word, in the options. The student is expected to use any one of the form in a sentence.

Examples:
1. (a) sympathy (n) : I show sympathy to the poor.
(b) sympathise (y) : We sympathise for the poor.
(c) sympathetically (adv) : He looked sympathetically at the poor.

2. (a) music (n) : Music and astronomy are the two sides of the same coin.
(b) musical (adj) : Ravi gets his musical talent from his father.
(c) musically (adv) : Suresh is musically talented.

3. (a) energy (n) : The children are full of energy
(b) energetic (adj) : Alex is the energetic player in the team.
(c) energetically (adv) . : Prabu danced energetically.

4. (a) vision (n) : His vision blurred with tears.
(b) visual (adj) : There was no visual evidence to support his claim.
(c) visually (adv) : Visually challenged persons demanded reservation.

5. (a) habit (n) : Reading books is a good habit
(b) habitual (adj) : He is a habitual thief.
(c) habitually (adv) : The artist habitually drew the sketch.

6. (a) spirit (n) : The human spirit is boundless.
(b) spiritual (adj) : Thomas became a spiritual leader.
(c) spirituallÿ (adv) : God must be worshipped spiritually by prayer.

7. (a) conclusion (n) : The meeting came to a conclusion.
(b) conclusive (adj) : There is no conclusive evidence for the corruption case.
(c) conclusively (adv) : The CID officer conclusively proved a murder case.

Exercises
Construct a sentence using one of the following words given below.

Question 1.
(a) obedient
(b) obediently
(c) obedience
Answer:
(a) He is an obedient student.
(b) He obediently followed his teacher’s instructions.
(c) Obedience is essential in any army.

Question 2.
(a) renovate
(b) renovation
(c) renovative
Answer:
(a) I need money to renovate my house.
(b) The hotel is closed for renovation.
(c) The civil contractor has a renovative talent.

Question 3.
(a) high
(b) highly
(c) height
Answer:
(a) The rent is very high.
(b) She is highly intelligent.
(c) The height of Mount Everest is 8848 meters.

Question 4.
(a) migrating
(b) migrant
(c) migration
Answer:
(a) Migrating birds seek shelter and food.
(b) Migrant birds are brave little voyagers.
(c) Migration to cities is a problem.

Question 5.
(a) innocent
(b) innocence
(c) innocently
Answer:
(a) I am innocent.
(b) I like his innocence.
(c) I speak innocently.

Question 6.
(a) earnest
(b) earnestly
(c) earnestness
Answer:
(a) He is earnest in his studies.
(b) He studies earnestly for the examination.
(c) The teacher found his earnestness in sports also.

Question 7.
(a) life
(b) live
(c) lively
Answer:
(a) All people want a good life.
(b) We live in Tamilnadu.
(c) We had a lively discussion about the match.

Question 8.
(a) happy
(b) happiness
(c) happily
Answer:
(a) He is happy.
(b) I need happiness in life.
(c) They lived happily.

Question 9.
(a) penalty
(b) penalised
(c) penal
Answer:
(a) He paid penalty for disobedience.
(b) He was penalised for his disobedience.
(c) The penal penalised him for his disobedience.

Question 10.
(a) popular
(b) popularly
(c) popularity
Answer:
(a) J. Nehru was a popular leader.
(b) He was popularly known as uncle Nehru.
(c) His popularity was well known.

Question 11.
(a) beautify
(b) beautifully
(c) beautiful
Answer:
(a) Rani knows how to beautify her skin.
(b) The building is beautifully decorated.
(c) The artist has drawn a beautiful picture.

Question 12.
(a) life
(b) living
(c) lively
Answer:
(a) Life is very precious.
(b) I am living in Chennai.
(c) The match was lively telecasted.

Question 13.
(a) Courage
(b) Courageous
(c) Courageously
Answer:
(a) Shivaji was known for his courage
(b) He was courageous like a lion.
(c) He fought courageously against the enemies.

Question 14.
(a) diminish
(b) diminishing
(c) diminished
Answer:
(a) In summerwater levels in dams diminish.
(b) Diminishing water supply in Chennai makes us suffer.
(c) The corporation has diminished water supply in many parts of the city.

Question 15.
(a) angry
(b) anger
(c) angrily
Answer:
(a) We should not get angry.
(b) Anger is unwelcome to all.
(c) He beat the dog angrily.

Question 16.
(a) kind
(b) kindly
(c) kindness
Answer:
(a) Kumar is a kind man.
(b) He talks kindly to all.
(c) His kindness won him many friends.

Question 17.
(a) free
(b) freedom
(c) freely
Answer:
(a) Nothing is free nowadays.
(b) We got freedom in 1947.
(c) He talks freely.

Question 18.
(a) decision
(b) decide
(c) decisive
Answer:
(a) He makes a right decision.
(b) He decides to visit Delhi.
(c) He is very decisive

Question 19.
(a) calm
(b) calmly
(c) calmness
Answer:
(a) Be calm when praying.
(b) Pray calmly.
(c) Feel calmness when praying.

Question 20.
(a) intelligence
(b) intelligent
(c) intelligently
Answer:
(a) His intelligence is appreciated.
(b) I am an intelligent boy.
(c) I talk intelligently.

Tamilnadu Board Class 10 English Solutions

Tamilnadu State Board Class 10 English Vocabulary Expanded form of Abbreviation, Acronyms & Contractions

♦ Expanded Form Of Abbreviation, Acronyms & Contractions (TextBook Page No. : 131-133)

Abbreviations and acronyms are shortened forms of words or phrases. An abbreviation is typically a shortened form of words used to represent the whole (such as Dr. or Prof.) while an acronym contains a set of initial letters from a phrase that usually form another word (such as radar or scuba).

Abbreviations and acronyms are often interchanged, yet the two are quite distinct. The main point of reference is that abbreviations are merely a series of letters while acronyms form new words.

We use contractions (I’m, we’re) in everyday speech and informal writing. Contractions, which are sometimes called ‘short forms’, commonly combine a pronoun or noun and a verb, or a verb and not, in a shorter form.

Contractions with I, you, he, she, it, we, and they
‘m = am (I’m)
’re = are (you’re, we’re, they’re)
’s = is and has (he’s, she’s, it’s)
’ve = have (’ve, you’ve, we’ve, they’ve)
’ll = will (I’ll, you’ll, he’ll, she’ll, it’ll, we’ll, they’ll)
’d = had and would (I’d, you’d, he’d, she’d, it’d, we’d, they’d)

Contractions with auxiliary verb and not The contraction for not is n’t:
aren’t = are not (we aren’t, you aren’t)
can’t = cannot
couldn’t = could not
didn’t = did not (I didn’t, they didn’t)

We can abbreviate the following:

Titles before names:
Mr. = Mister (for men) (plural – Misters) Mrs. = Mistress (for women)
Prof. = Professor (plural – Profs.)
St. = Saint (plural – Sts.)
Rev. = Reverend (plural – Revs.)
Hon. = Honourable (plural – Hons.)
Jr. =unior
Pres. = President

Names of a Few Objects:

VCR = Videocassette Recorder
CD = Compact Disc
DVD = Digital Video/Versatile Disk
GPS = Global Positioning System
VR = Virtual reality
AR = Augmented Reality
TV = Television

Words used with numbers:

a.m. = ante meridiem (before noon)
p.m. = post meridiem (after noon)
A.D. = anno domini
B.C.E. = before common era
C.E. = common era

Common Latin terms:

etc. = et cetera (and*so forth)
i.e. = id est(that is)
e.g. = exempligratia (for example)
et al. = et alii (and others)
vs. = versus

Terms of mathematical units:

ft = feet
ft² = square feet
kg = kilogram
km = kilometre
mm = millimetre
ml = millilitre
°F = degrees Fahrenheit
°C = degrees Celsius

Names of Familiar Institutions:

CBI = Central Bureau of Investigation
IB = Intelligence Bureau
IMF = International Monetary Fund
UN = United Nations
EC = Election Commission
EU = European Union
IIT = Indian Institute of Technology

Names of Countries:

USA = United Stated of America
UK = United Kingdom
UAE = United Arab Emirates

Long, common phrases

IQ = Intelligence Quotient
mph = miles per hour
mpg = miles per gallon

Abbreviation

• Abbreviation refers to any shortened or contracted form of a word or phrase.
• Some abbreviations are not pronounced as new words.
• But not every abbreviation is an acronym.

Abbreviation	Expansion
AICTE	All India Council for Technical Education
ATM	Automated Teller Machine
B.Tech	Bachelor of Technology
B.E.	Bachelor of Engineering
B.Sc	Bachelor of Science
BA	Bachelor of Arts
BBC	British Broadcasting Corporation
BPO	Business Process Outsourcing
BSNL	Bharat Sanchar Nigam Limited
CA	Chartered Accountant
CBI	Central Bureau of Investigation
CCTV	Closed-Circuit Television
DTH	Direct To Home
EMI	Equated Monthly Installment
FIR	First Information Report
HIV	Human Immunodeficiency Virus
IAS	Indian Administrative Service
ICU	Intensive Care Unit
IIT	Indian Institute of Technology
ILO	International Labour Organization
ISI	Indian Standard Institute
1ST	Indian Standard Time
KPO	Knowledge Process Outsourcing
LCD	Liquid Crystal Display
LED	Light Emitting Diode
L1C	Life Insurance Corporation of India
LPG	Liquefied Petroleum Gas
M.B.B.S.	Bachelor of Medicine and Bachelor of Surgery
MNC	Multi-National Corporation / Company
NGO	Non-Governmental Organization
NRI	Non-Resident Indian
PA	Personal Assistant / Postal Assistant
POW	Prisoners of War
PS	Personal Secretary / PostScript
RPF	Railway Protection Force
RRB	Railway Recruitment Board
SBI	State Bank of India
SHG	Self Help Group (for women)
SMS	Short Message Service
ssc	Staff Selection Commission
STD	Subscriber Trunk Dialling
SR	Southern Railways
TNPSC	Tamil Nadu Public Service Commission
TRB	Teachers Recruitment Board
UAE	United Arab Emirates
UFO	Unidentified Flying Object
UGC	University Grants Commission
UPS	Uninterrupted Power Supply
USA	United State of America
WTC	World Trade Centre
WWW	World Wide Web

Exercises

Choose the expanded form of the abbreviation :

Question 1.
The common expansion of CCTV is :
(a) Charged Channel Television
(b) Closed Channel Television
(c) Closed-Circuit Television
(d) Checked-Circuit Television
Answer:
(c) Closed-Circuit Television

Question 2.
The common expansion of SSC is:
(a) State Subordinate Commission
(b) Staff Selection Commission
(c) State School Certificate
(d) School System Common
Answer:
(b) Staff Selection Commission

Question 3.
The common expansion of TNPSC is:
(a) Tamil Nadu Police Service Commission
(b) Tamil Nadu Public Service Committee
(c) Tamil Nadu People Service Commission
(d) Tamil Nadu Public Service Commission
Answer:
(d) Tamil Nadu Public Service Commission

Question 4.
The common expansion of SBI is:
(a) Selection Board of India
(b) Selection Bureau of India
(c) State Bank of India
(d) Security Board of India
Answer:
(c) State Bank of India

Question 5.
The common expansion of CA is:
(a) Chief Accountant
(b) Central Accountant
(c) Central Army
(d) Chartered Accountant
Answer:
(d) Chartered Accountant

Question 6.
The common expansion of IAS is:
(a) Indian Administrative Service
(b) Indian Armed Service
(c) Indian Army Service
(d) Indian Agricultural Service
Answer:
(a) Indian Administrative Service

Question 7.
The common expansion of TPC is:
(a) Integrated Police Council
(b) Indian Penal Code
(‘) Independent Police Corps
(d) International Public Census
Answer:
(b) Indian Penal Code

Question 8.
The common expansion of BPT is:
(a) Bachelor of Pure Therapy
(b) Bachelor of Physiotherapy
(c) Bachelor of Post Treatment
(d) Bachelor of Poverty Treatment
Answer:
(b) Bachelor of Physiotherapy

Question 9.
The common expansion of TC is:
(a) Training Card
(b) Train Concession
(c) Transfer Certificate
(d) Transfer Code
Answer:
(c) Transfer Certificate

Question 10.
The common expansion of SR is:
(a) Suburban Railways
(b) Standard Railways
(c) Southern Railways
(d) Steam Railways
Answer:
(c) Southern Railways

Question 11.
The common expansion of KPO is:
(a) Kinetic Potential Output
(b) Knowledge Process Outsourcing
(c) Keen Programme Observation
(d) Knowledge Performance Outcome
Answer:
(b) Knowledge Process Outsourcing

Question 12.
The common expansion of ATM is:
(a) Automatic Teller Machine
(b) Automated Teller Machine
(c) Automatic Typing Machine
(d) Automated Telling Man
Answer:
(b) Automated Teller Machine

Question 13.
The common expansion of RRB is:
(a) Rails and Reservation Board
(b) Record and Revenue Bank
(c) Railway Recruitment Board
(d) Roadways and Railways Bureau
Answer:
(c) Railway Recruitment Board

Question 14.
The common expansion of BPO is:
(a) Bank Process Outsourcing
(b) Better Process Outsourcing
(c) Business Process Outsourcing
(d) Boyle Process Outsourcing
Answer:
(c) Business Process Outsourcing

Question 15.
The common expansion of ILO is:
(a) International Law and Order
(b) International Labour Organisation
(c) Indian Legal Orientation
(d) Inter-state Lawful Ordinance
Answer:
(b) International Labour Organisation

Question 16.
The common expansion of PA?
(a) Personal Ambassador
(b) Private Affair
(c) Personal Assistant
(d) Public Apartment
Answer:
(c) Personal Assistant

Question 17.
The common expansion of SMS?
(a) Simple Message Service
(b) Simple and Minimum Service
(c) Short Message Service
(d) Short Membership Service
Answer:
(c) Short Message Service

Question 18.
The common expansion of IOB is:
(a) International Overseas Banking
(b) Indian Organisation of Banks
(c) Indian Overseas Bank
(d) Indian Origin of Banks
Answer:
(c) Indian Overseas Bank

Question 19.
The common expansion of WHO is:
(a) Western Health Organisation
(b) Western Hemisphere Ocean
(c) World Health Organisation
(d) World Heritage Organisation
Answer:
(c) World Health Organisation

Question 20.
The common expansion of UFO is:
(a) Unidentified Foreign Object
(b) Unidentified Flying Object
(c) Universal Force Organisation
(d) United Federal Organisation
Answer:
(b) Unidentified Flying Object

Acronym

• Acronym is a type of abbreviation where a new word is formed from the first letters of a series of words.
• Pronounced as a separate word,
• All acronyms are abbreviations.

AIDS – Acquired Immune Deficiency Syndrome
AIR – All India Radio
BHEL – Bharat Heavy Electricals Limited
CAT – Common Admission Test
FIFA – Federation of International Football Association
IGNOU – Indira Gandhi National Open University
INSAT – Indian National Satellite
ISRO – Indian Space Research Organisation
LAN – Local Area Network
NATO – North Atlantic Treaty Organisation
OTP – One Time Password
PAN – Permanent Account Number
PIN – Postal Index Number
RADAR – Radio Detecting and Ranging
RAM – Random Access Memory
ROM – Read Only Memory
SAI – Sports Authority of India
SAIL – Steel Authority of India Limited
SAT – Standard Assessment Task (British), Scholastic Aptitude Test (USA)
SIM – Subscriber Identification Module
TAPE – Tractor and Farm Equipment Limited
TANCET – Tamil Nadu Common Entrance Test
TANSI – Tamil Nadu Small Industries Corporation
TET – Teacher Eligibility Test
TOEFL – Test of English as a Foreign Language
VAT – Value Added Tax
WHO – World Health Organization

Exercises

Choose the expanded form of the abbreviation:

Question 1.
The common expansion of INSAT is:
(a) International Satellite
(b) Integrated Space and Technology Work
(c) Indian National Satellite
(d) Independent Satellite Track
Answer:
(c) Indian National Satellite

Question 2.
The common expansion of AIR is:
(a) All India Railways
(b) All India Rivers
(c) All India Radio
(d) All Indian Religions
Answer:
(c) All India Radio

Question 3.
The common expansion of RAM is:
(a) Read Only Memory
(b) Read Access Memory
(c) Rapid Accurate Memory
(d) Random Access Memory
Answer:
(d) Random Access Memory

Question 4.
The common expansion of ISRO is
(a) International Satellite Research Organisation
(b) Indian Space Research Organisation
(c) Indian Social Reforms Organisation
(d) Indian Southern Railway Organisation
Answer:
(b) Indian Space Research Organisation

Question 5.
The common expansion of CAT is:
(a) Common Aptitude Test
(b) Computer Aptitude Test
(c) Computer Accuracy Test
(d) Computer Accessories Test
Answer:
(a) Common Aptitude Test

Question 7.
The common expansion of PIN is:
(a) Population Index Number
(b) Postal Index Number
(c) Poor Identification Notice
(d) Physics Internal Note
Answer:
(b) Postal Index Number

Question 8.
The common expansion of LAN is:
(a) Local Available Network
(b) Local Area Node
(c) Local Area Network
(d) Large Area Network
Answer:
(c) Local Area Network

Question 9.
The common expansion of RADAR is:
(a) Radio Digital Accurate Report
(b) Radio Defect And Rectification
(c) Radio Digital And Range
(d) Radio Deduction And Ranging
Answer:
(d) Radio Deduction And Ranging

Question 10.
The common expansion of VAT is:
(a) Value Audit Trade
(b) Valves And Tubes
(ç) Value Added Tax
(d) Value Access Track
Answer:
(c) Value Added Tax

Question 11.
The common expansion of GATE is:
(a) General Aptitude Test in Electronics
(b) Graduate Aptitude Test in Electronics
(c) Graduate Aptitude Test in Engineering
(d) General Aptitude Test in Engineering
Answer:
(c) Graduate Aptitude Test in Engineering

Question 12.
The common expansion of FERA is:
(a) Federal Railway Act
(b) Foreign Exchange Regulation Act
(c) Federal Eastern Railways Administration
(d) Foreign Exchange Regional Act
Answer:
(b) Foreign Exchange Regulation Act

Contractions

Exercises

Choose the correct expansion of the following contractions.

Question 1.
Won’t
(a) Will not
(b) Would not
(c) Woo not
Answer:
(a) will not

Question 2.
Must’ve
(a) Must give
(b) Must have
(c) Must live
Answer:
(b) must have

Question 3.
I’d
(a) I would
(b) I do
(c) I did
Answer:
(a) I would

Question 4.
She’s
(a) She sees
(b) She does
(c) She is
Answer:
(c) She is

Question 5.
Who’s
(a) Whose
(b) Who has
(c) Who does
Answer:
(b) Who has

Tamilnadu Board Class 10 English Solutions

Tamilnadu State Board Class 9 Maths Solutions Chapter 1 Set Language Ex 1.5

Question 1.
Using the adjacent venn diagram, find the following sets:
(i) A – B
(ii) B – C
(iii) A’ ∪ B’
(vi) A’ ∩ B’
(v) (B ∪ C)’
(vi) A – (B ∪ C)
(vii) A – (B ∩ C)

Solution:
(i) A – B = {3, 4, 6}
(ii) B – C = {-1, 5, 7}
(iii) A’ ∪ B’
A’ = {1, 2, 0, -3, 5, 7, 8}
B’ = {-3, 0, 1, 2, 3, 4, 6)
A’ ∪ B’ = {-3, 0, 1, 2, 3, 4, 5, 6, 7, 8)

(iv) A’ ∩ B’
A’ ∩ B’ = {-3, 0, 1, 2}

(v) B ∪ C = {-3, -2, -1, 0, 3, 5, 7, 8}
(B ∪ C)’ = U – (B ∪ C)
= {-3, -2, -1, o, 1,2, 3, 4, 5, 6, 7, 8} – {-3, -2, -1, o, 3, 5, 7, 8}
(B ∪ C)’ = {1, 2, 4, 6}

(vi) A – (B ∪ C) = {-2, -1, 3, 4,6} – {-3, -2, -1, 0, 3, 5, 7, 8} = {4, 6}
A – (B ∩ C)
B ∩ C = {-2, 8}
A- (B ∩ C) = {-2, -1, 3, 4, 6} – {-2, 8} = {-1, 3,4, 6}

Question 2.
If K = {a, b, d, e,f}, L = {b, c, d, g} and M {a, b, c, d, h} then find the following:
(i) K ∪ (L ∩ M)
(ii) K ∩ (L ∪ M)
(iii) (K ∪ L) ∩ (K ∪ M)
(iv) (K ∩ L) ∪ (K ∩ M) and verify distributive laws.
Solution:
K = {a, b, d, e,f}, L = {b, c, d, g} and M {a, b, c, d, h}
(i) K ∪ (L ∩ M)
L ∩ M = {b, c, d, g} ∩ {a, b, c, d, h} = {b, c, d}
K ∪ (L ∩ M) = {a, b,d,e,f } ∪ {b, c, d) = {a, b, c, d, e,f}

(ii) K ∩(L ∪ M)
L ∪ M = {a, b, c, d,g, h }
K ∩ (L ∪ M) = {a, b, d,e,f} ∩ {a, b, c, d, g, h} = {a, b, d}

(iii) (K ∪ L) ∩ (K ∪ M)
K ∪ L = {a, b, c, d, e, f, g}
K ∪ M = {a, b, c, d, e, f, h}
(K ∪ L) ∩ (K ∪ M) = {a, b, c, d, e,f}

(iv) (K ∩ L) ∪ (K ∩ M)
(K ∩ L) = {b, d)
(K ∩ M) = {a,b,d}
(K ∩ L) ∪ (K ∩ M) = {b, d} ∪ {a, b, d} = {a, b, d}

Distributive laws
K ∪ (L ∩ M) = (K ∪ L) ∩ (K ∪ M)
{a, b, c, d, e,f) = {a, b, c, d, e,f g} ∩ {a, b, c, d, e,f h}
= {a, b, c, d, e,f}
Thus Verified.
K ∩ (L ∪ M) = (K ∩ L) ∪ (K ∩ M)
{a, b, d} = {a, b, c, d, e,f g} ∪ {a, b, c, d, e,f h}
= {a, b, d}
Thus Verified.

Question 3.
If A = {x : x ∈ Z, -2 < x ≤ 4}, B = {x : x ∈ W, x ≤ 5}, C ={-4, -1,0, 2, 3, 4}, then verify A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C).
Solution:
A = {x : x ∈ Z, -2 < x ≤ 4} = {-1, 0, 1, 2, 3, 4}
B = {x : x ∈ W, x ≤ 5} = {0, 1, 2, 3, 4, 5}
C = {-4, -1, 0, 2, 3, 4}
A ∪ (B ∩ C)
B ∩ C = {0, 1, 2, 3, 4, 5} ∩ {-4, -1, 0, 2, 3, 4} = {0, 2, 3, 4}
A ∪ (B ∩ C) = {-1, 0, 1, 2, 3, 4} ∪ (0, 2, 3, 4} ={-1, 0, 1, 2, 3, 4} … (1)
(A ∪ B) ∩ (A ∪ C)
A ∩ B = {0, 1, 2, 3, 4}
A ∩ C = {-1, 0, 2, 3, 4}
(A ∩ B) ∪ (A ∩ C) = {0, 1,2, 3,4} ∪ {-1, 0, 2, 3, 4}= {-1, 0, 1, 2, 3, 4} … (2)
From (1) and (2), it is verified that
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

Question 4.
Verify A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) using Venn diagrams.
Solution:
L.H.S A ∪ (B ∩ C)

From (2) and (5), it is verified that A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

Question 5.
If A = {b, c, e, g, h}, B = {a, c, d, g, i} and C = {a, d, e, g, h}, then show that A – (B ∩ C) = (A – B) ∪ (A – C).
Solution:
A = {b, c, e, g, h}
B = {a, c, d, g, i}
C = {a, d,e, g, h}
B ∩ C = {a, d, g}
A – (B ∩ C) = {b, c, e, g, h} – {a, d, g} = {b, c, e, h} … (1)
A- B = {b, c, e, g, h} – {a, c, d, g, i} = {b, e, h}
A – C = {b, c, e, g, h} – {a, d, e, g, h} = {b, c}
(A – B) ∪ (A – C) = {b, c, e, h} … (2)
From (1) and (2) it is verified that
A – (B ∩ C) = (A – B) ∪ (A – C)

Question 6.
If A = {x : x = 6 n ∈ W and n < 6}, B = {x : x = 2n, n ∈ N and 2 < n ≤ 9} and C = {x : x = 3n, n ∈ N and 4 ≤ n < 10}, then show that A – (B ∩ C) = (A – B) ∪ (A – C)
Solution:
A = {x : x = 6n, n ∈ W, n < 6}
x = 6n
n = {0,1, 2, 3, 4, 5}
⇒ x = 6 x 0 = 0
x = 6 x 1=6
x = 6 x 2 = 12
x = 6 x 3 = 18
x = 6 x 4 = 24
x = 6 x 5 = 30
∴ A = {0, 6, 12, 18, 24, 30}

B = { x : x = 2n, n ∈ N, 2 < n ≤ 9}
n = {3, 4, 5, 6, 7, 8,9}
x = 2 n
⇒ x = 2 x 3 = 6
2 x 4 = 8
2 x 5 = 10
2 x 6 = 12
2 x 7 = 14
2 x 8 = 16
2 x 9 = 18
∴ B {6, 8, 10, 12, 14, 16, 18}

C = { x : x = 3n, n ∈ N, 4 ≤ n < 10}
N = { 4, 5, 6, 7, 8, 9}
x = 3 x4 = 12
⇒ x = 3 x 5 = 15
x = 3 x 6 = 18
x = 3 x 7 = 21
x = 3 x 8 = 24
x = 3 x 9 = 27
x = 2 x 9 = 18
∴ C = {12, 15, 18, 21, 24, 27}

A – (B ∩ C) = (A – B) ∪ (A – C)
L.H.S R.H.S
B ∩ C = {12,18}
A – (B ∩ C) = {0, 6, 12, 18, 24, 30} – {12, 18} = {0, 6, 24, 30} ….(1)
(A – B) = {0,24,30}
(A – C) = {0, 6, 30}
(A – B) ∪ (A – C) = {0, 6, 24, 30} …..(2)
From (1) and (2), it is verified that
A – (B ∩ C) = (A – B) ∪ (A – C).

Question 7.
If A = {-2, 0,1, 3, 5}, B = {-1, 0, 2, 5, 6} and C = {-1, 2, 5, 6, 7}, then show that A – (B ∪ C) = (A – B) ∩ (A – C).
Solution:
A = {-2,0, 1,3,5},
B = {-1,0, 2, 5, 6}
C ={-1,2, 5, 6, 7}
B ∪ C = {-1,0, 2, 5, 6, 7}
A – (B ∪ C) = {-2, 1, 3} …(1)
(A – B) = {-2, 1, 3}
(A – C) = {-2, 0, 1, 3}
(A – B) ∩ (A – C) = {-2,1,3} … (2)
From (1) and (2), it is verified that . A – (B ∪ C) = (A – B) ∩ (A – C)

Question 8.
if \(\mathbf{A}=\left\{y : y=\frac{a+1}{2}, a \in \mathbb{W} \text { and } a \leq 5\right\}, \mathbf{B}=\left\{y : y=\frac{2 n-1}{2}, n \in \mathbb{W} \text { and } n<5\right\}\) and \(\mathbf{C}=\left\{-1,-\frac{1}{2}, \mathbf{1}, \frac{3}{2}, 2\right\}\) then show that A – (B ∪ C) = (A – B) ∩ (A – C).
Solution:


(A – B) ∩ (A – C) = {3} …(2)
From (1) and (2), it is verified that A – (B ∪ C) = (A – B) ∩ (A – C).

Question 9.
Verify A – (B ∩ C) = (A – B) ∪ (A – C) using Venn diagrams.
Solution:

∴ A – (B ∩ C) = (A – B) ∪ (A – C) Hence it is proved.

Question 10.
If U = {4, 7, 8, 10, 11, 12, 15, 16}, A = {7, 8, 11, 12} and B = {4, 8, 12, 15}, then verify De Morgan’s Laws for complementation.
U = {4, 7, 8,10, 11, 12, 15, 16}
A = {7, 8, 11, 12}, B = {4, 8, 12, 15}
De Morgan’s Laws for complementation.
(A ∪ B)’ = A’ ∩ B’
A ∪ B = {4,7,8, 11, 12, 15}
(A ∪ B)’ = {4, 7, 8,10, 11, 12, 15, 16} – {4, 7, 8,11, 12, 15}
= {10, 16} … (1)
A’ = {4, 10, 15, 16}
B’ = {7, 10, 11, 16}
A’ ∩ B’ = {10, 16} …(2)
From (1) and (2) it is verified that (A ∪ B)’ = A’ ∩ B’.

Question 11.
Verify (A ∩ B)’ = A’ ∪ B’ using Venn diagrams.
Solution:
(A ∩ B)’ = A’ ∪ B’

(2) = (5)
∴ (A ∩ B)’ = A’ ∪ B’

Samacheer Kalvi 9th Maths Book Solutions

Tamilnadu State Board Class 10 English Solutions Poem Chapter 4 The Ant and the Cricket

The Ant and the Cricket Textual Questions

A. Based on your understanding of the poem, read the following lines and answer the questions given below.

1. A silly young cricket accustomed to sing
Through the warm, sunny months of gay summer and spring.
(a) What was the routine of the cricket?
(b) Name the seasons mentioned here.
Answer:
(a) The routine of the cricket was to sing and dance all through the warm sunny months.
(b) The seasons mentioned here are summer and spring.

2. Began to complain when he found that, at home,
His cupboard was empty, and winter was come.
(a) Who does he refer to?
(b) Why was his cupboard empty?
Answer:
(a) ‘He’ refers to the cricket.
(b) His cupboard was empty because the cricket didn’t lay aside anything for winter.

3. Not a crumb to be found
On the snow-covered ground;
(a) What couldn’t he find on the ground?
(b) Why was the ground covered with snow?
Answer:
(a) He couldn’t find any piece of bread on the ground.
(b) The ground was covered with snow because it was winter season.

4. At last by starvation and famine made bold,
All dripping with wet, and all trembling with cold,
(a) What made the cricket bold?
(b) Why did the cricket drip and tremble?
Answer:
(a) Starvation and famine made the cricket bold.
(b) The cricket dripped and trembled as it was wet and cold.

5. Away he set off to a miserly ant,
To keep if, to keep him alive, he would grant
Him shelter from rain,
And a mouthful of grain.
(a) Whom did the cricket want to meet? Why?
(b) What would keep him alive?
Answer:
(a) The cricket wanted to meet the miserly ant for food and shelter.
(b) A mouthful of grain and shelter from rain would keep him alive.

6. But we ants never borrow; we ants never lend.
(a) Why do you think ants neither borrow nor lend?
(b) Who says these lines to whom?
Answer:
(a) Ants neither borrow nor lend because they work hard and plan for their future.
(b) The ant says these lines to the cricket.

7. “Not I!
My heart was so light
That I sang day and night,
For all nature looked gay.
(a) Who does ‘I refer to?
(b) What was the nature of the cricket? How do you know?
Answer:
(a) I refers to the cricket.
(b) The cricket was lazy and silly. We know this, since it had not saved anything for its future.

8. Thus ending, he hastily lifted the wicket,
And out of the door turned the poor little cricket,
(a) The ant refused to help the cricket. Why?
(b) Explain the second line.
Answer:
(a) The ant refused to help the cricket because the cricket was lazy and silly. It was not a hard worker.
(b) The ant opened its door and sent away the poor little cricket without giving him anything.

9. He wished only to borrow;
He’d repay it tomorrow;
(a) Pick out the rhyming words in the above lines.
(b) Give more examples of rhyming words from the poem.
Answer:
(a) “Borrow – tomorrow” are the rhyming words in the above lines.
(b) The other examples of rhyming words are sing-spring; home – come; found – ground, see – tree, bold-cold; ant – grant; rain – grain; friend – lend; light – might and. wicket – cricket.

10. My heart was so light
that I sang day and night,
For all nature looked gay.
“You sang, Sir, you say”?
(a) Mention the rhyme scheme employed in the above lines.
Answer:
The rhyming scheme is “a a b b”.

Poetic Comprehension – Additional

1. Not a flower could he see,
Not a leaf on a tree.
“Oh! what will become,” says cricket, of me?”
(a) Which season is mentioned here?
(b) Why is the cricket sad?
Answer:
(a) The winter season is mentioned here.
(b) The cricket is sad because there was nothing found to eat or shelter to rest.

2. Says the ant to the cricket, “I’m your servant and friend,
But we ants never borrow; we ants never lend.
(a) What does the ant say to the cricket?
(b) What rule do the ants follow?
Answer:
(a) The ant says that it is his servant and friend.
(b) They never lend or borrow.

3. But tell me, dear cricket,
Did you lay anything by
When the weather was warm?”
(a) What did the ant ask the cricket?
(b) Which season is specified here?
Answer:
(a) The ant questioned the cricket whether it laid anything aside when the weather was warm.
(b) Summer season is specified here.

4. “You sang, Sir, you say?
Go then”, says the ant, “and dance the winter away”.
(a) To whom does the ant say this?
(b) What does this mean, “and dance the counter away?”
(c) Was the ant angry with the cricket?
Answer:
(a) The ant says this to the cricket.
(b) Dance and sing once again all throughout the winter season.
(c) Yes, the ant was angry with the cricket.

Poetic Devices – Additional

1. Not a flower could he see,
Not a leaf on a tree.
“Oh! what will become,” says cricket, of me?”
(a) Pick out the rhyming words.
Answer:
“see – tree” are the rhyming words.

2. Says the ant to the cricket, “I’m your servant and friend,
But we ants never borrow; we ants never lend.
(a) Pick out the words in alliteration.
Answer:
“the, the”; “says, servant”; are words in alliteration.

3. When the weather was warm?
(a) Pick out the alliterated words in this line.
Answer:
when, weather, was, warm are the alliterated words.

B. Based on your understanding of the poem, complete the summary using the phrases given below.

In this narrative poem, the poet brings out the idea that is essential for every creature. He conveys this message to the readers through a story of (1)_______ The ant spends all its summer saving (2)_______. The cricket (3)_______happily in the summer. He (4) ______ anything for the winter. When winter comes, he is worried that his (5) ______ is empty. So, he seeks the help of the ant to have (6)______ and a (7)________ to stay. The cricket was even prepared to repay it in the future. The ant made it clear that ants (8) _________. He also enquired the cricket if it had saved anything when the weather was fine. The cricket answered that it had sung day and night enjoying (9) ______ threw the cricket out and stated in a stern voice it should dance in the winter season too. In his concluding lines, the poet affirms that this is not (10) ________but it is true and applicable to (11)______also.

the pleasant nature, human beings, doesn’t save, warm place, kitchen cupboard, just a fable, saving for future, some grains, never borrow or lend, an ant and a cricket, sings and dances.
Answer:

1. an ant and a cricket
2. for future
3. sings and dances
4. doesn’t save
5. kitchen cupboard
6. some grains
7. warm place
8. never borrow or lend
9. the pleasant nature
10. just a fable
11. human beings

C. Answer each of the following questions in a paragraph of 120-150 words.

Question 1.
“Some crickets have four legs and some have two? Elucidate this statement from the poet’s point of view:
Answer:
Some cricket have four legs and some have two, this last line means that the fable is true for human beings as well. Though two-legged, some of us behave like that four-legged crickets. They are not wise like the ant. They don’t save for their future. They are lazy and are not hard working. They spend all their earnings when they are capable of earning. In the moments of crisis, or in their old age, they find that they don’t have any money to depend upon. Such people are like the cricket of the story. The moral of the poem is “Save something for bad days”.
So we should always plan ahead and save money for our future.

Question 2.
Compare and contrast the attitude of the ant and the cricket.
Answer:
The cricket in the poem is a silly creature who sings and dances all summer and does not save
any food for its future. It used to enjoy its good times during the summer season. He didn’t plan anything for the future. Whereas the ant is clever and saves food during summer so that he doesn’t have to do any work during winter. The ant gives a very important lesson of life during its conversation with the cricket. The ant says that ants never borrow from anyone. They never lend to anyone. They are hard working creatures who save for their future. Though it says that it is a friend and servant to the cricket, it gets angry with what the cricket has done during the summer. It turns him out of his house saying to sing and dance once again during winter. This shows that the ant is hardworking and clever.

Question 3.
If given a chance, who would you want to be- the ant or the cricket? Justify your answer.
Answer:
If given a chance, I would like to be an ant forever. The ant is clever and hard working. It plans for its future well ahead and avoids the problem of starvation and famine during winter. Likewise, we human beings should earn well in our young age and save money for ourselves and for our future generations. But some people are careless and lazy like the cricket. They don’t plan out for their future or for the bad times. They just enjoy the present moment. The poet is thus calling such people as mindless as the cricket. So we must always be foresighted enough to see our future and make plans accordingly. One should be ready for the good as well as the bad times ahead.

Paragraph Question & Answer – Additional

Question 1.
What is the theme of the poem ‘The Ant and the Cricket?
Answer:
Hard work versus laziness. Through the story of the lazy cricket and the hard working ant, the poet teaches us the virtue of hard work. The cricket was foolish enough not to see the future. He made no plans or provisions for the future. He lived in the present moment. The ant, on the other hand, was wise and hard working. She worked hard so that she could enjoy the future. The poet wants us to learn an important lesson in life. One must always save for the future. As you sow, so shall you reap’ is a popular proverb that fits this story. We must be foresighted and make plans accordingly, we should always be ready for good as well as the bad times ahead.

The Ant and the Cricket By Aesop

Aesop is the name of the man credited with the authorship of a collection Hlor book of fables. Aesop was a slave believed to have lived in ancient Greece between 620 and 564 BCE. He was an ugly, deformed dwarf, and the famous marble statue in Rome depicts Aesop accordingly. Each one of Aesop’s fables has a lesson, or moral, just like a parable or allegory. The Morals, Sayings and Proverbs featured in the Book of Aesop’s fables ensure that they still have meaning for us today.

The Ant and the Cricket Key Points

• This poem teaches the importance of hard work and planning.
• It is about a silly young cricket and an ant.
• The cricket spends its time wasting by singing and dancing all summer.
• During winter, its cupboard is empty. Nothing can be found at all.
• So it goes to the miserly ant and begs to give him shelter and food.
• The ant says that they never borrow from anyone and they never lend to anyone.
• The ant questions the cricket what it was doing during the summer.
• The cricket replies that since his heart was so light and happy, it sang all day and night.
• Hearing this, the ant gets angry and tells the cricket to sing and dance once again all winter.
• He hurriedly sends away the cricket and shuts his door.
• We should always plan carefully and ensure a secure future.

The Ant and the Cricket Summary

The poem “The Ant and the Cricket” teaches us the importance of hard work and planning. It is a narrative poem. The poet brings out the idea that is essential. This poem is about a silly young cricket and an ant. The cricket was only singing all day long and used to enjoy his good times during the summer season. He didn’t plan anything for the future. When winter arrives, the cricket couldn’t find a small amount of food to eat. So, the cricket thinks of going to the miser ant to borrow food and to get shelter. Then, the cricket knocks on the ant’s door asking for help.

The ant gives a very important lesson of life during its conversation with the cricket. The ant says that they never lend to anyone and they never borrow from anyone. Ants are hard working creatures who save for their future. The ant questioned the cricket what it was doing during the happier times. On hearing that the cricket indulged in dancing, singing and making merry, the ant gets angry. It tells the cricket to dance and sing once again in winter. The moral of this poem is that we should plan carefully and ensure a secure future.

The Ant and the Cricket Glossary

accustomed to (v) – be used to
bold – brave
crumb (n) – piece of bread
fable – a short story
famine (n) – extreme scarcity of food
folks – people
gay (adj.) – glad, joyful
grant – give
hastily (adv.) – hurriedly
miserly (adj.) – hesitant to spend money
quoth (v) – said (old English usage, used only in first and third person singular before the subject)
starvation – extreme hunger
warrant ( v) – guarantee, promise
wicket – door

The Ant and the Cricket Explanation of poetic lines

Line No. 1 – 4
A silly young cricket, accustomed to sing Through the warm, sunny months of gay summer and spring, Began to complain when he found that, at home, His cupboard was empty, and winter was come.
Explanation:
A silly young cricket used to sing and dance during the warm happy days of summer and spring. When winter came, he began to complain that there was nothing left in his cupboard at home. It was empty.

Line No. 5 – 8
Not a crumb to be found
On the snow-covered ground;
Not a flower could he see,
Not a leaf on a tree.
Explanation:
There Was not a piece of bread to be found on the snow-covered ground. Not a flower could be seen on any tree. It could also not see a leaf on any tree.

Line No. 9 – 13
“Oh! what will become,” says cricket, “of me?”
At last by starvation and famine made bold,
All dripping with wet, and all trembling with cold,
Away he set off to a miserly ant,
To see if, to keep him alive, he would grant
Explanation:
It felt unhappy. He started worrying about itself. What will become of the cricket? Finally, the extreme hunger, famine made him bold. It was dripping with wet and trembling with cold. It went to the ant to see if he could give anything to keep him alive.

Line No. 14 – 17
Him shelter from rain.
And a mouthful of grain.
He wished only to borrow;
He’d repay it tomorrow;
Explanation:
The cricket hoped that the ant would shelter him from rain and give him a mouthful of grain. He wished only to borrow from him and repay it the next day

Line No. 18 – 24
If not, he must die of starvation and sorrow.
Says the ant to the
cricket, “I’m your servant
and friend,
But we ants never
borrow; we ants never
lend.
Explanation:
If the ant doesn’t give what he needs, he had to die of starvation and sorrow. The ant says to the cricket that it is the servant and friend of the cricket. But ants never borrow and they never lend.

Line No. 25 – 29
But tell me, dear cricket,
Did you lay anything by
When the weather was
warm?” Quoth the cricket,
“Not I!”
Explanation:
The ant questions the cricket whether he kept aside anything for the future when the weather was warm. The cricket said that he did not do so.

Line No. 30 – 34
My heart was so light –
That I sang day and night,
For all nature looked gay.”
“For all nature looked gay”.
“You sang, Sir, you say?
Explanation:
The cricket said that its heart was so light and happy, that it sang all day and night. The cricket said that nature looked happy and fresh. The ant gets angry. He asks the cricket whether he sang all day since nature looked happy.

Line No. 35 – 39
Go then”, says the ant, “and dance the winter away”.
Thus ending, he hastily lifted the wicket,
And out of the door turned the poor little cricket.
Folks call this a fable. I’ll warrant it true:
Some crickets have four legs, and some have two.
Explanation:
The ant shouts at the cricket saying go and dance once again during the winter. Thus ending his talk, he hurriedly lifted the door and turned the poor little cricket. People call this a short story. I promise it’s true. Some crickets have four legs and some have two. (He means the human beings who are like the cricket waste their earnings.)

Tamilnadu Board Class 10 English Solutions