PSEB 10th Class English Message Writing

Punjab State Board PSEB 10th Class English Book Solutions English Message Writing Exercise Questions and Answers, Notes.

PSEB 10th Class English Message Writing

A message is some information, news, request, etc. sent to somebody in writing, speech, by telephone, etc. Telephone is the quickest means for sending a message. Sometimes it so happens that a message is received by telephone, but the person for whom the message is meant is not at hand. So someone else has to receive it. And if he himself has to go out, he has to leave in writing the message for the person for whom it is meant.

(1) Read the telephonic conversation given below and write the message in about 50 words:

Sanjeev : Hello ! Is it 13557799?
Sandeep : Yes, may I know who is calling?
Sanjeev : This is Sanjeev. I want to talk to Mr. Mittal.
Sandeep : Papa is not at home. Would you like me to convey him any message?
Sanjeev : Tell him that the meeting scheduled for 30th March has been
postponed to 4th April.
Sandeep : I’ll tell him.
Sanjeev : Thank you.

MESSAGE

22nd March
Dear Papa

5:00 p.m.

There was a phone call from Mr. Sanjeev. He aids that the meeting scheduled for 30th March had been postponed to 4th April.

Sandeep

PSEB 10th Class English Message Writing

(2) Read the telephonic conversation given below :

Ramesh : Hello Ria, where is Mama ?
Ria : She has gone to the kitty party.
Ramesh : Please tell her that I will bring two of my friends for dinner tonight.
Ria : Fine. Is there anything else she should know ?
Ramesh : Yes, tell her to prepare fish-curry.
Ria : Is that all ?
Ramesh : Yes. Bye.

Ria has to go to her friend’s house. So she writes the message for her mother. Write the message for Ria.

MESSAGE

6.00 Pm

8th June
Mama
Ramesh told me on the phone that he bringing two of his friend for dinner tonight. His want you to prepare fish-curry for the dinner.
Ria

(3) Read the following telephonic conversation :

Neha : Hello ! Is it 4607875?
Seema : Yes.
Neha : I am Neha. May I speak to Kriti, please?
Seema : She has gone to market and won’t be back till 9 o’clock.
Neha : Please tell her that she is invited to my birthday party on 15 April at 6 p.m. at my house.
Seema : I’ll convey your message.
Neha : Thank you.

Write this message on behalf of Seema.

MESSAGE

7.00 Pm

3 April
Kriti
There was a phone call from Neha. she has invited you to her birthday party on September at 6 pm at her house.
Seema

PSEB 10th Class English Message Writing

(4) Given below is a telephonic conversation between Nandan and Nikita.

Nandan : I am Nandan. May I talk to Arun?
Nikita : I’m sorry, he is not at home.
Nandan : Why didn’t he come to school today?
Nik ita . : He has gone to visit a relative.
Nandan : Will you please convey my message to him?
Nikita : What is it?
Nan dan : Tell him to bring my English notebook as I need it for correction tomorrow.
Nikita : Anything else?
Nandan : Yes, tell him that our science test has been postponed. Our teacher
will announce the new date tomorrow.
Nikita : I will tell him.
Nandan : Thank you.

Nikita has to go to the market, so she writes the message for Arun. Write this message for her.

MESSAGE

4.30 pm

17 Sept,
Arun
There was a phone call from Nandan He said that you must take his English notebook to school as he needs it for correction He also said that your science test had been postponed The new date will he announced tomorrow.
Nikita

(5) Read the telephonic conversation given below :

Anoushka : Hello, is it 2295098?
Ritu : Yes, may I know who is calling?
Anoushka : I am Anoushka. Can I talk to Ishaan?
Ritu : He has gone to the library.
Anoushka : Please tell him that his interview for ‘Tata Indicom’ has been fixed for tomorrow 9 a.m. sharp. He should reach the office in time and also bring his certificates along with him. Don’t forget. It is urgent.
Ritu : I’ll convey the message.
Anoushka : Thank you.

Ritu has to go to visit her sister, so she writes the message for Ishaan. write this message.

MESSAGE

6.00 Pm

11 Sept,
Ishaan
There was a telephone cell from Anoushka. She said that your interview for Tata indicom has been fixed for tomorrow 9 a.m sharp. You should reach the office in time along with your certificates.
Ritu.

PSEB 10th Class English Message Writing

(6) Read the following telephonic conversation between Gulzar and Sonam :

Gulzar – Is it 8287135 ?
Sonam – Yes, who is calling ?
Gulzar – I am Gulzar from JCT, Phagwara. I would like to speak to Dhruv.
Sonam – He is not at home. I am his sister and you can give me the message.
Gulzar – Please tell him that he has been selected for the post of junior engineer and he must join duty tomorrow at 10 a.m.
Sonam – Anything else ?
Gulzar – Yes, tell him to bring along the photocopies of his certificates.
Sonam – Sure, I’ll convey your message.
Gulzar – Thank you. Bye.

Sonam writes the message for Dhruv as she has to go out for dinner. Write this message for her.

MESSAGE

5.00 PM

7th March
Dhruv
One Mr Gulzar from JCT, Phagwara. rang up to inform you that you have been selected for the post of junior engineer. You must join duty tomorrow at 10 a.m. You should also take with you the photocopies of you certificates’.

Sonam

(7) Read the following telephonic conversation :

Ravi : Hello ! Is it 2895634 ?
Tina : Yes, may I know who is on the line ?
Ravi : This is Ravi. May I talk to Suraj ?
Tina : He is not at home.
Ravi : Do me a favour. Tell him to pick me up while going to office tomorrow as my car has been sent for servicing.
Tina : Anything else ?
Ravi : Tell him to bring the file I gave him.
Tina : I’ll convey your message.
Ravi : Thank you. Bye.

Tina writes the message for Suraj as she has to go to her kitty party. Write this message on her behalf.

MESSAGE

3:00 p.m

22 March
Suraj
There was a phone call from Ravi. He want, you to pick him up while going to office tomorrow as his car has been sent for servicing. Also take along the file he gave you.
Tina

(8) Read the telephonic conversation given below :

Rajiv : Hello, is that Anu ?
Rajni : No, it’s Rajni speaking. Anu is not at home at the moment. What is the matter ?
Rajiv : We had planned to see ‘Power Cut’ running at Arora Palace today evening. Please tell her to wait for me at the main gate.
Rajni : Do not worry. I shall leave a message for her.
Rajiv : Thank you, Rajni. Bye.

Rajni has to go to market and she will not be able to meet Anu. Write a message on her behalf in about 50 words.

MESSAGE

10:00 a.m

10th October
Anu
You had planned to see ‘Power Cut’ runtime at Arora Palace today evening with Rajiv. He rang up to tell you that you should wait for him at the main gate.
Rajni

(9) Read the following telephonic conversation :

Ajay : Hello, is it 2376589 ?
Anu : Yes, may I know who is speaking ?
Ajay : This is Ajay. I want to speak to Mr. Goyal.
Anu : Papa is not at home. Would you like me to convey any message, to him ?
Ajay : Tell him that the meeting scheduled for 9th March has been postponed to 11th March.
Anu : I’ll tell him.

Anu has to go to school. Write the message on her behalf.

MESSAGE

8:00 a.m

7th March
Papa
Mr. Ajay rang upto inform you that the meeting scheduled for 9th March has been ostpàied to 11th March.
Anu

PSEB 10th Class English Message Writing

(10) Read the telephonic conversation given below :

Mrs. Sharma: Hello, is it 2984567?
Ria: Yes, who would you like to speak to?
Mrs. Sharma: I want to talk to Sunita. I am Megha Sharma.
Ria: Hello auntie, mummy has gone to market.
Mrs. Sharma: Tell her that Nina’s father-in-law is not well and we are going
to see him at Kapoor Hospital where he has been admitted.
Tell your mother to be ready by 6 p.m. as I’ll pick her up on the way.
Ria: I’ll give her your message, auntie.
Mrs. Sharma: Thank you, dear. Bye.

Ria has to go to her friend’s house. Write the message for her mother on lier behalf.

MESSAGE

2:00 a.m

15th March
Mummy
Sharma auntie rang up to say that Nina aunties father-in-law is not well. They are going to see him at Kapoor Hospital Sharma aùntIwill pick you up out. the way You shrink be ready by 6 pm.
Ria

(11) Read the telephonic conversation given below :

Kishore : Hello ! Could I speak to Raghu ?
Rani : Raghu has gone out. This is his sister Rani speaking.
Kishore : Will you please tell him to reach the Extension Library by 10 a.m.
tomorrow for the renewal of our membership ?
Rani : Of course, I will tell him. Anything else ?
Kishore : Yes, tell him to carry his membership card.
Rani : Sure. Bye.

Rani has to go to club for a meeting. Write down the message she would write for Raghu.

MESSAGE

4:00 p.m

4th March
Raghu

There was a phone call from Kishore. He said that vim should reach the extensive Library by 10 a.m. tomorrow for the renewal of membership. You should also carry your membership card.
Rani

PSEB 10th Class English Message Writing

(12) Read the following telephonic conversation :

Rajesh: Hello, could I speak to Hemant, please?
Harish: Sorry, Hemant has not yet arrived.
Rajesh: Could you please convey to him that Mr. Verma has arrived at the company head office and he would like to meet Hemant tomorrow morning.
Harish: Sure, I’ll tell him?
Rajesh: Tell him to reach the office by 9 a.m.
Harish: I’ll coñvey the message.
Rajesh: Thank you. Bye.

Harish: has to go to Delhi, so he leaves the message for Hemant. Write the message on his behalf.

MESSAGE

2:00 p.m

11 March
Hemant
Rajesh rang up to say that Mr. Verma has arrived at the company head office. He would like to meet you tomorrow morning. Reach the office by 9 a.m.
Harish

13. Read the following telephonic conversation between Asha and Kiran.

Kiran is about to leave home and will not be able to meet Arun. She leaves a message for him. Write this message in not more than 50 words. Put the message in a box.
Asha : Hello, is that Kiran ?
Kiran: Speaking.
Asha: It’s Asha here. You know there’s a good movie on. It’s ‘My Left Foot’.
It’s running at Alpana. How about it tomorrow?
Kiran: Yes, not a bad idea.
Asha: Let’s go to the evening show. Where should we meet?
Kiran: Let’s meet at Wimpy’s Corner, at 5.30. Is that okay?
Asha: Fine. Look, Vijay’s coming along too. Why don’t you ask Arun to join us?
Kiran: I’ll do that. He’s not at home at the moment, but I’ll ask him to ring you up this evening.
Please don’t buy the tickets until he calls.

MESSAGE

5:30 p.m

8 March.
Arun
There was a phone from Asha She and Vijay are going to see the movie ‘My Left Foot tomorrow evening I, too, have agreed to accompany them We shall he meeting at Wimpy’s Cornerrhat 5:30 Asha wants you also to accompany. Please ring her up this evening and confirm She’ll be waiting.
Kiran

PSEB 10th Class English Message Writing

(14) Read the following telephonic conversation which took place when Suresh was staying with his uncle. Write the message from Suresh to his uncle, using not more than 50 words. Put the message in a box.

Seshu : Hello ! Hello ! This is Seshu from Lucknow. Can I speak to Mr. Rao, please? I am a friend of his son, Matlhav Rao.
Suresh : This is Suresh Rao My uncle’s not here the moment. We heard about the earthquake. Is Madhav all right?
Seshu : Yes, yes. He’s okay now. But he had a bad fall during the earthquake and he broke his left leg. It was a multiple fracture, but there’s nothing to worry about now.
Suresh : Is he in hospital?
Seshu : Yes, he’s at the Tata Memorial Hospital here. Would you please inform his family?
Suresh : Of course, I will. Thank you for calling.

MESSAGE

9.15 p.m

25 April
Dear Uncle
There Was a telephone call for you from one Mr. Seshu, our Madhav’s friend, from Lucknow. He said that Madhav had a fall during the earthquake and he broke his leg. He got a multiple fracture and has been admitted to the Tata Memorial Hospital there. But he added that there was nothing to worry.
Suresh.

PSEB 10th Class Maths Solutions Chapter 2 Polynomials Ex 2.4

Punjab State Board PSEB 10th Class Maths Book Solutions Chapter 2 Polynomials Ex 2.4 Textbook Exercise Questions and Answers.

PSEB Solutions for Class 10 Maths Chapter 2 Polynomials Ex 2.4

Question 1.
Verify that the number given alongside of the cubic polynomials below are their zeroes. Also verify the relationship between the zeroes and the coefficients in each case :
(i) 2x3 + x2 – 5x + 2; \(\frac{1}{2}\), 1, -2
(ii) x3 – 4x2 + 5x – 2; 2, 1, 1
Solution:
(i) Let p (x) = 2x3 + x2 – 5x + 2
Compare it with ax3 + bx2 + cx + d
a = 2, b = 1, c = -5, d = 2
Now, P(\(\frac{1}{2}\)) = 2\(\left(\frac{1}{2}\right)^{3}\) + \(\left(\frac{1}{2}\right)^{2}\) – 5\(\frac{1}{2}\) + 2

= \(\frac{1}{4}+\frac{1}{4}-\frac{5}{2}\)

= \(\frac{1+1-10+8}{4}=\frac{0}{4}\) = 0
∴ \(\frac{1}{2}\) is zero of p (x)
and p(1) = 2(1)3 + (1)2 – 5 (1) + 2
= 2 + 1 – 5 + 2 = 5 – 5 = 0
∴ 1 is zero of p (x).
Also, p(-2) = 2(-2)3 + (- 2)2 -5(-2) + 2
= -16 + 4 + 10 + 2 = – 16+ 16 = 0
∴ – 2 is zero of p (x).
From above discussion, it is clear that \(\frac{1}{2}\), 1, – 2 are the zeroes of given polynomial.
Let these zeroes are
α = \(\frac{1}{2}\), β = 1, γ = – 2
Now, α + β + γ = \(\frac{1}{2}\) + 1 + (-2)
= \(\frac{1}{2}\) + 1 – 2
= \(\frac{1+2-4}{2}\)
= –\(\frac{1}{2}\) = –\(\frac{b}{a}\)

PSEB Solutions PSEB 10th Class Maths Solutions Chapter 2 Polynomials Ex 2.4

αβ + βγ + γα = (\(\frac{1}{2}\)) (1) + (-1) (-2) + (-2) (\(\frac{1}{2}\))
= \(\frac{1}{2}\) – 2 – 1
= \(\frac{1-4-2}{2}\)
= \(-\frac{5}{2}\) = \(\frac{c}{a}\)

αβγ = (\(\frac{1}{2}\)) (1) (-2)
= (\(-\frac{2}{2}\))
= (\(-\frac{d}{a}\))
From above discussion, it is clear that there is a relationship between zeroes and coefficients.

(ii) Let p (x) = x3 – 4x2 + 5x – 2.
Compare it with ax3 + bx2 + cx + d
∴ a = 1, b = – 4, c = 5, d = – 2
Now p(2) = (2)3 – 4 (2)2 + 5 (2) – 2
= 8 – 16 + 10 – 2
= 18 – 18 = 0 2 is zero of p (2).
and p( 1) = (1)3 – 4(1)2 + 5(1) – 2
= 1 – 4 + 5 – 2 = 6 – 6 = 0
From above discussion it is clear that 2, 1, 1 are the zeroes of given polynomial. Ans.
Let these zeroes are
α = 2, β = 1, γ = 1
Now, α + β + γ = 2 + 1 + 1 = 4
= \(\frac{-(-4)}{1 \cdot}=\frac{-b}{a}\)
αβ + βγ + γα = (2) (1) + (1) (1) + (1) (2) = 2 + 1 + 2 = 5
= \(\frac{5}{1}=\frac{c}{a}\)

αβγ = (2) (1) (1) = 2
= \(\frac{-(-2)}{1}=\frac{-d}{a}\)
From above discussion, it is clear that there is a relationship between zeroes and coefficient.

PSEB Solutions PSEB 10th Class Maths Solutions Chapter 2 Polynomials Ex 2.4

Question 2.
Find a cubic polynomial with the sum, sum of the product of its zeroes taken two at a time, and the product of its zeroes as 2, – 7, – 14 respectively.
Solution:
The general expression of cubic polynomial are
ax3 + bx2 + cx + d.
Let α, β, γ be its zeroes
α + β + γ = Sum of zeroes = 2
αβ + βγ + γα = Sum of product of zeroes = – 7
αβγ = Product of zeroes = – 14
∴ ax3 + bx2 + cx + d
= k [(x – α) (x – β) (x – γ)] where k is any constant.
= k [x3 – (α + β + γ) x2 + (αβ + βγ + γα)x – αβγ]
= k [x3 – 2x2 – 7x + 14] [Using (1)]
For different values of k, we get different cubic polynomial.

PSEB Solutions PSEB 10th Class Maths Solutions Chapter 2 Polynomials Ex 2.4

Question 3.
If the zeroes of the polynomial x3 – 3x2 + x +1 are a – b, a, a + b, find a and b.
Solution:
Let p (x) = x3 – 3x2 + x + 1 and its zeroes are a – b, a, a + b.
a – b is zero of p (x) ………(Given)
∴ p (a – b) = 0
or (a – b)3 – 3 (a – b)2 + (a – b) + 1 = 0 or
[a3 – b3 – 3a2b + 3ab2] – 3 [a2 + b2 – 2ab] + a – b + 1 = 0
or a3 – b3 – 3a2b + 3ab2 – 3a2 – 3b2
+ 6ab + a – b + 1 = 0 ….(1)
and a is zero of p (x) …………(Given)
∴ p (a) = 0
or a3 – 3a3 + a + 1 = 0 ………….(2)
Also, a + b is zero of p (x) …(Given)
∴ p (a + b) = 0
or (a + b)3 – 3 (a + b)2 + (a + b) + 1 = 0
or (a3 + b3 + 3a2b + 3ab2) – 3 (a2 + b3 + 2ab) + a + b – 1 = 0 or
a3 + b3 + 3a2b + 3ab2 – 3a2 – 3b3 – 6ab + a + b + 1 = 0 …………….(3)
Adding (1) and (3), we get :
2a3 + 6ab2 – 6a2 – 6b2 + 2a + 2 = 0 or
a3+ 3ab2 – 3a2 – 3b2 + a + 1 = 0 or
(a3 – 3a3 + a + 1) + (3ab2 – 3b2) = 0 or
0 + 3b2 (a – 1) = 0 [Using (2)]
or a – 1 = 0
or a = 1 ………(4)
From (3) and (4), we get:
(1)3 + b3 + 3(1)2b + 3(1 )b2 – 3(1)2 – 3b2 – 6 (1) b + 1 + b + 1 = 0 or
1 + b3+ 3b + 3b2 – 3 – 3b2 – 6b + b + 2 = 0
or b3 – 2b = 0 or
b (b2 – 2) = 0 or
b2 – 2 = 0 or
b2 = 2
or b = ±√2
Hence a = 1, b = ±√2

PSEB Solutions PSEB 10th Class Maths Solutions Chapter 2 Polynomials Ex 2.4

Another Solution:
Given that three zeroes of polynomial x3 – 3x2 + x + 1 are a – b, a, a + b respectively
Now, sum of zeroes = (a – b) + a + (a + b) = a – b + a + a + b = 3a
But, sum of zeroes using coefficient = \(\frac{-\text { Coefficient of } x^{2}}{\text { Coefficient of } x^{3}}\)
= \(\frac{-(-3)}{1}\) = 3
∴ 3a = 3 or a = 1
Also, product of zeroes = (a – b) . a . (a + b) = (a2 – b2) a
Putting the value of a, we get :
= (12 – b2) . 1
= 1 – b2
But, product of zeroes using coefficient = \(=\frac{-\text { Constant term }}{\text { Coefficient of } x^{3}}\)
= \(\frac{-1}{1}\) = -1
1 – b2 = – 1
– b2 = – 1 – 1
– b2 = – 2 or
b2 = 2
b = ±√2
Hence a = 1 and b = ±√2

Question 4.
If two zeroes of the polynomial x4 – 6x3 – 26x2 + 138x – 35 are 2 ± √3, find other zeroes.
Solution:
Given that two zeroes are (2 + √3) and (2 – √3)
∴ [x – (2 + √3)] [x – (2 – √3)] are factors of given polynomial.
Now, [x – (2 + √3)] [x – (2 – √3)]
= x2 – [2 – √3 + 2 + √3]x + [(2 + √3) + (2 – √3)]
= x2 – 4x + [(2)2 – (√3)2]
= x2 – 4x + 1
∴ (x2 – 4x + 1) is factor of given polynomial. Now, apply division algorithm to given polynomial and (x2 – 4x + 1)

PSEB 10th Class Maths Solutions Chapter 2 Polynomials Ex 2.4 1

∴ x4 – 6x3 – 26x2 + 138x – 35
= (x2 – 4x + 1) (x2 – 2x – 35)
= (x2 – 4x + 1) [x2 + 5x – 7x – 35]
S = – 2, P = – 35
= (x2 – 4x + 1) [x (x + 5) – 7 (x + 5)]
= (x2 – 4x + 1) (x + 5).(x – 7)
Now, other zeroes of polynomials are given by :
x + 5 = 0 Or x – 7 = 0
x = – 5 Or x = 7
∴ the zeroes of the given fourth degree polynomial are :
2 + √3, 2 – √3, -5, 7.

PSEB Solutions PSEB 10th Class Maths Solutions Chapter 2 Polynomials Ex 2.4

Question 5.
If the polynomial x4 – 6x3 + 16x2 – 25x + 10 is divided by another polynomial x2 – 2x + k, the remainder comes out to be x + a, find k and a.
Solution:
Given that, polynomial x4 – 6x3 + 16x2 – 25x + 10 is divided by another polynomial x2 – 2x + k then remainder comes out to be x + a
So, first of all we divide,
x4 – 6x3 + 16x2 – 25x + 10 with x2 – 2x + k and find quotient and remainder.

PSEB 10th Class Maths Solutions Chapter 2 Polynomials Ex 2.4 2

∴ by division algorithm for Polynomials x4 – 6x3 + 16x2 – 25x +10
= (x2 – 2x + k) [x2 – 4x + (8 – k) + [(- 9 + 2k) x + (10 – 8k + k2]
Quotient = x2 – 4x + (8 – k) and
Remainder = (- 9 + 2k) x + (10 – 8k + k2)
But, remainder = x + a ….(Given)
∴ (- 9 + 2k) x + (10 -8k + k2) = x + a
Compare the like coefficients, we get :
-9 + 2k = 1 Or
2k = 1 + 9
2k = 10
k = \(\frac{10}{2}\) = 5 or

10 – 8k + k2 = a
Putting the value of k, we get
10 – 8 × 5 + (5)2 = a
10 – 40 + 25 = a
-5 = a
a = -5
Hence k = 5 and a = -5

PSEB 10th Class Maths Solutions Chapter 2 Polynomials Ex 2.2

Punjab State Board PSEB 10th Class Maths Book Solutions Chapter 2 Polynomials Ex 2.2 Textbook Exercise Questions and Answers.

PSEB Solutions for Class 10 Maths Chapter 2 Polynomials Ex 2.2

Question 1.
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
(i) x2 – 2x – 8
(ii) 4s2 – 4s + 1
(iii) 6x2 – 3 – 7x
(iv) 4u2 + 8u
(v) t2 – 15
(vi) 3x2 – x -4
Solution:
(i) Given Quadratic polynomial,
x2 – 2x – 8
∵ [S = -2, P = -8]
= x2 – 4x + 2x – 8
= x (x – 4) + 2 (x – 4)
= (x – 4) (x + 2)
The value of x2 – 2x – 8 is zero,
iff (x – 4) = 0 or (x + 2) = 0
iff x = 4 or x = – 2
Therefore, zeroes of x2 – 2x – 8 are – 2 and 4.
Now, Sum of zeroes = (- 2) + (4) = 2
= \(\frac{-(-2)}{1}=-\frac{(\text { Coefficient of } x)}{\text { Coefficient of } x^{2}}\)
Product of zeroes = (- 2) (4) = – 8
= \(\frac{-8}{1}=\frac{\text { Constant term }}{\text { Coefficient of } x^{2}}\)
Hence, relationship between the zeroes and the coefficient are verified.

PSEB Solutions PSEB 10th Class Maths Solutions Chapter 2 Polynomials Ex 2.2

(ii) Given quadratic polynomial,
4s2 – 4s + 1
= 4s2 – 2s – 2s + 1
∵ [S = -4, P = 4 × 1]
= 2s (2s – 1) – 1 (2s – 1)
= (2s – 1) (2s – 1)
The value of 4s2 – 4s + 1 is zero
iff (2s – 1) = 0 or (2s – 1) = 0
iff s = \(\frac{1}{2}\) or s = \(\frac{1}{2}\)
Therefore, zeroes of 4s2 – 4s + 1 are \(\frac{1}{2}\) and \(\frac{1}{2}\)

Now, sum of zeroes = \(\frac{1}{2}\) + \(\frac{1}{2}\) = 1
= \(\frac{-(-4)}{4}=\frac{-(\text { Coefficient of } s)}{\left(\text { Coefficient of } s^{2}\right)}\)

Product of Zeroes = \(\left(\frac{1}{2}\right)\left(\frac{1}{2}\right)=\frac{1}{4}\)

= \(\frac{\text { Constant term }}{\text { Coefficient of } s^{2}}\)
Hence, relationship between the zeroes and the coefficients are verified.

(iii) Given quadratic polynomial,
6x2 -3 – 7x
= 6x2 – 7x – 3
∵ [S = – 7, P = 6x-3=-18]
= 6x2 -9x + 2x-3
= 3x (2x – 3) + 1 (2x – 3)
= (2x – 3) (3x + 1)
The value of 6x2 – 3 – 7x is zero
iff (2x – 3) = 0 or 3x + 1 = 0
iff x = \(\frac{3}{2}\) or x = –\(\frac{1}{3}\)
Therefore, zeroes of 6x2 – 3 – 7x are \(\frac{3}{2}\) and –\(\frac{1}{3}\)
Now, Sum of zeroes = \(\frac{1}{3}\)

= \(\frac{3}{2}+\left(\frac{-1}{3}\right)\)

= \(\frac{3}{2}-\frac{1}{3}=\frac{9-2}{6}\)

= \(\frac{7}{6}=\frac{-(-7)}{6}\)

= \(\frac{-(\text { Coefficient of } x)}{\text { Coefficient of } x^{2}}\)

Product of zeroes = \(\left(\frac{3}{2}\right)\left(\frac{-1}{3}\right)\)

= \(\frac{-3}{6}=\frac{\text { Constant term }}{\text { Coefficient of } x^{2}}\)

Hence, relationship between the zeroes and the coefficients are verified.

PSEB Solutions PSEB 10th Class Maths Solutions Chapter 2 Polynomials Ex 2.2

(iv) Given quadratic polynomial,
4u2 + 8u = 4u(u + 2)
The value of 4u2 + 8 u is zero
iff 4u = 0 or u + 2 = 0
iff u = 0 or u = – 2
Therefore, zeroes of 4u2 + 8M are 0 and – 2
Now, Sum of zeroes = 0 + (- 2) = -2
= \(\frac{-8}{4}\)
= \(-\frac{(\text { Coefficient of } u)}{\text { Coefficient of } u^{2}}\)

Product of zeroes = (0) (- 2) = 0
= \(\frac{0}{4}=\frac{\text { Constant term }}{\text { Coefficient of } u^{2}}\)
Hence, relationship between the zeroes and the coefficients are verified.

(v) Given quadratic polynomial,
t2 – 15 = t2 – (√15)2
= (t – √15) (t + √15)
The value of t2 – 15 is zero
iff t – √15 = 0 or t + √15 = 0
iff t = √15 or t = – √15
Therefore, zeroes of t2 – 15 are – √15 and √15.
Now, Sum of zeroes = -√15 + √15 = 0 = \(\frac{0}{1}\)
= \(\frac{-(\text { Coefficient of } t)}{\text { Coefficient of } t^{2}}\)

Product of zeroes = (-√15) (√15) = – 15 = \(-\frac{15}{1}\)

= \(\frac{0}{1}\)
= \(=\frac{\text { Constant term }}{\text { Coefficient of } t^{2}}\)
Hence, relationship between the zeroes and the coefficients are verified.

PSEB Solutions PSEB 10th Class Maths Solutions Chapter 2 Polynomials Ex 2.2

(vi) Given quadratic polynomial,
3x2 – x – 4
= 3x2 + 3x – 4x – 4
= 3x (x + 1) – 4 (x + 1)
∵ [S = – 1, P = 3 x – 4 = – 12]
= (x + 1) (3x – 4)
The value of 3x2 – x – 4 is zero
iff (x + 1) = 0 or 3x – 4 = 0
iff x = -1 or x = \(\frac{4}{3}\)
Therefore, zeroes of 3x2 – x – 4 are – 1 and \(\frac{4}{3}\)

Now, Sum of zeroes = – 1 + \(\frac{4}{3}\)
= \(\frac{-3+4}{3}=\frac{1}{3}\)
= \(\frac{-(-1)}{3}=\frac{-\text { Coefficient of } x}{\text { Coefficient of } x^{2}}\)
Product of zeroes = (- 1) \(\frac{4}{3}\)
= \(\frac{-4}{3}=\frac{\text { Constant term }}{\text { Coefficient of } x^{2}}\)
Hence, relationship between the zeroes and the coefficients are verified.

PSEB Solutions PSEB 10th Class Maths Solutions Chapter 2 Polynomials Ex 2.2

Question 2.
Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.
(i) \(\frac{1}{4}\), -1
(ii) √2, \(\frac{1}{3}\)
(iii) 0, √5
(iv) 1, 1
(v) –\(\frac{1}{4}\), \(\frac{1}{4}\)
(vi) 4, 1. [MQP 2015]
Solution:
(i) Given that, sum of zeroes and products of zeroes of given polynomial \(\frac{1}{4}\) are -1 respectively.
Let the quadratic polynomial be ax2 + bx + c and its zeroes be α and β
α + β = Sum of zeroes = \(\frac{1}{4}\)
and αβ = Product of zeroes = – 1
Now, ax2 + bx + c = k (x – α) (x – β) where k is any constant
= k [x2 – (α + β)x + αβ]
= k[x2 – \(\frac{1}{4}\)x + (-1)]
= k[x2 – \(\frac{1}{4}\)x – 1]
for different value of k, we get different quadratic polynomials.

(ii) Given that, sum of zeroes and product of zeroes of given quadratic polynomial are √2 and \(\frac{1}{3}\) respectively.
Let the quadratic polynomial be ax2 + bx + c and its zeroes be α and β
α + β = Sum of zeroes = √2
and αβ = Product of zeroes = \(\frac{1}{3}\)
Now, ax2 + bx + c = k (x – α) (x – β) where k is any constant
= k[x2 – (α + β) x + αβ]
= k[x2 – √2x + \(\frac{1}{3}\)]of k, we get different quadratic polynomial.
for different values of k, we get different quadratic polynomial.

PSEB Solutions PSEB 10th Class Maths Solutions Chapter 2 Polynomials Ex 2.2

(iii) Given that, sum of zeroes and products of zeroes of given quadratic polynomial are 0 and √5 respectively.
Let the quadratic polynomial be ax2 + bx + c and its zeroes be α and β
α + β = Sum of zeroes = 0
and αβ = Product of zeroes = √5
Now, ax2 + bx + c = k(x – α) (x – β)
where k is any constant
= k [x2 – (a + (α+ β)x + αβ)
= k[x2 – 0x + √5]
= k[x2 + √5]
for different values of k, we get different quadratic polynomial.

(iv) Given that, sum of zeroes and product of zeroes of given quadratic polynomial are 1 and 1 respectively.
Let the quadratic polynomial be ax2 + bx + c and its zeroes be α and β
α + β = Sum of zeroes = 1 and
αβ = Product of zeroes = 1
Now, ax2 + bx + c = k(x – α) (x – β)
where k is any constant.
= k [x2 – (α + β)x + αβ]
= k [x2 – 0x + √5]
= it [x2 – x + √5]
for different values of k, we get different quadratic polynomial.

PSEB Solutions PSEB 10th Class Maths Solutions Chapter 2 Polynomials Ex 2.2

(v) Given that, sum of zeroes and product of zeroes of given quadratic polynomial are \(-\frac{1}{4}\) and \(\frac{1}{4}\) respectively.
Let the quadratic polynomial be ax2 + bx + c and its zeroes be α and β
α + β = Sum of zeroes = –\(\frac{1}{4}\)
and αβ = Product of zeroes = \(\frac{1}{4}\)
Now ax2 + bx + c = k (x – α) (x – β) where k is any constant
= k[x2 – (α + β) x + αβ]
= k[x2 – (-\(\frac{1}{4}\))x + \(\frac{1}{4}\)]
= k[x2 + \(\frac{1}{4}\)x + \(\frac{1}{4}\)]
For different values of k, we get different quadratic polynomial.

PSEB Solutions PSEB 10th Class Maths Solutions Chapter 2 Polynomials Ex 2.2

(vi) Given that, sum of zeroes and product of zeroes of given quadratic polynomial are 4 and 1 respectively. Let the quadratic polynomial be
ax2 + bx + c and its zeroes be α and β
α + β = sum of zeroes = 4 and
αβ = Product of zeroes = 1
Now, ax2 + bx + c = k(x – α) (x – β) where k is any constant
= k [x2 – (α + β) x + αβ]
= k [x2 – 4x + 1]
For different values of k, we get different quadratic polynomials.

PSEB 10th Class English Paragraph Writing

Punjab State Board PSEB 10th Class English Book Solutions English Paragraph Writing Exercise Questions and Answers, Notes.

PSEB 10th Class English Paragraph Writing

नोट-परीक्षा-पत्र में विद्यार्थियों को तीन अथवा चार विषय दिए जाएंगे जिनमें से किसी एक पर उन्हें लगभग 100 शब्दों में एक पैराग्राफ लिखना होगा। ये पैराग्राफ किसी व्यक्ति, घटना अथवा अनुभव, आदि के बारे में पूछे जा सकते हैं। किसी विषय पर एक अच्छा पैराग्राफ लिखने के लिए निम्नलिखित बातों का ध्यान रखिए –

1. दिए हुए विषय पर चिन्तन कीजिए और इसके सम्बन्ध में जो बातें (points) आप लिखना चाहते हों, उन्हें क्रमबद्ध कर लीजिए।

2. पैराग्राफ लिखते समय इस बात का ध्यान रखिए कि आपके पैराग्राफ में एक सहज प्रवाह हो – अर्थात् पैराग्राफ का प्रत्येक वाक्य अपने पूर्व के वाक्य में से निकलता हुआ प्रतीत होना चाहिए।

3. अपने पैराग्राफ में केवल उन्हीं शब्दों का प्रयोग कीजिए जिनका अर्थ और प्रयोग आप को भली प्रकार आता हो। अपनी योग्यता से अधिक योग्य दिखलाई देने का यत्न कभी मत कीजिए।

4. अपने वाक्यों की लम्बाई इतनी रखिए कि वे आपके काबू में रहें। अधिक लम्बे वाक्य सुन्दर लेख का लक्षण नहीं होते हैं।

5. आपके पैराग्राफ का पहला वाक्य विषय का उल्लेख करने वाला होना चाहिए और अन्तिम वाक्य निष्कर्ष-पूर्ण होना चाहिए। आरम्भिक और अन्तिम वाक्य लिखने में विशेष ध्यान दिया जाना चाहिए।

6. आपका पैराग्राफ दिए हुए विषय पर ही केन्द्रित होना चाहिए। व्यर्थ की बातें लिखकर अपने पैराग्राफ को लम्बा करने का यत्न कभी मत कीजिए।

7. आपका पैराग्राफ संक्षिप्त तथा अर्थ-पूर्ण होना चाहिए। पुस्तक के इस भाग में अनेक विषयों पर पैराग्राफ दिए गए हैं। विद्यार्थियों को परामर्श दिया जाता है कि वे इन्हें ध्यानपूर्वक पढ़ें और देखें कि किस प्रकार से किसी विषय को एक सारगर्भित पैराग्राफ का रूप दिया जा सकता है।

1. An Indian Festival

Outline: Diwali…… twenty days…… Dussehra…… Shri Ram…… Ayodhya…… set free…… Mughal Emperor …… festival…… village and town…… houses and shops…… painted…… new colours…… candles and electric lights …… gifts …… fireworks…… worship goddess Lakshmi…… gamble…… evil…… ended.

Diwali is an important Indian festival. It generally falls in the month of November. It comes twenty days after Dussehra. Shri Ram came back to Ayodhya on this day. Shri Guru Hargobind was set free by the Mughal Emperor on this day. This festival is celebrated in every village and town. Houses and shops are painted in new colours. People light their homes with candles and electric lights. They buy sweets and toys. They distribute gifts among friends and relatives. Children enjoy fireworks at night. On this day, people worship goddess Lakshmi. Some people gamble on this day. It is an evil. It should be ended.

PSEB 10th Class English Paragraph Writing

2. Good Manners

Outline : Good manners oil the machine of life ……… earn us respect ……… ‘please’and ‘thank you’settle bitter quarrels ……… should be observed respect elders ……… be polite to the younger ……… talk less and listen more ……… take turn in queue ……… civility of behaviour is a moral obligation ……… bad manners create bitterness.

Good manners oil the machine of life and keep it healthy. They are necessary in every walk of life. They are a double blessing. They give us inner joy and also keep others happy. They earn us respect also. So we should be courteous in our behaviour. We should respect our elders and be polite to younger ones. We should talk less and listen more. We should take turn in queue. We should never forget that civility of behaviour is a moral obligation. While bad manners create bitterness, good manners win us friends. Even simple words like ‘Please’ and ‘Thank you’ settle bitter quarrels. If we have good manners we can make journey of life successful not only for others, but also for ourselves.

PSEB 10th Class English Paragraph Writing

3. My Motherland

Outline : India my motherland …… most people live in villages …… many big cities …… land of saints …… big country …… Himalayas …… Indian Ocean …… fertile land …… brave and wise people …… proud of my motherland.

India is my motherland. Most of the people here live in villages. There are many big cities also. Mumbai, Delhi, Kolkata and Chennai are some of them. India is a land of saints. It is the land of Kalidas, Kabir and Guru Nanak. India is a very big country. In its north, there are the Himalayas. They are the highest mountains in the world. The Indian Ocean is in its south. India is a land of big forests, lakes, valleys and rivers. They add to its beauty. The soil of India is very fertile. It gives us many kinds of crops. India is a land of lions, tigers and elephants. And Indians are brave and wise people. They are peace-loving people. I am proud of my country and its people.

4. Fashion Among Students

Outline : Students …… most of the time in front of the mirror …… very particular about clothes …… hairstyles …… show off…… inwardly hollow …… no attention to inner development…… a foolish trend.

Most of the time of students is spent in front of the mirror. They want to decorate themselves in such a way as to attract every eye. In their desire to look modern and attractive, they go crazy for new fashions. They dress themselves in the latest styles. With every change in fashion, their hairstyles also change. To be in the latest fashion seems to be the only care of our young boys and girls. They care so much for their exteriors that they have no time to attend to their interiors. They frequent the beautician for facials but never visit the library for books, journals or periodicals. If ever they carry some book or journal in their hand, that, too, is never meant for reading but merely for the sake of fashion. I hate these modern clowns.

5. Discipline

Outline : Discipline necessary …… house…… classroom …… assembly …… office ……playground …… without discipline confusion, disorder and chaos…… no work or progress possible without discipline …… training of mind.

Discipline is necessary in all walks of life. Any house without discipline will surely go to the dogs. A teacher can’t teach if his students don’t keep discipline in the class. The head of a family won’t be able to run his family if its members don’t maintain discipline. The members of an assembly must observe discipline if they are to do any good job. No office can function without discipline. Even in a playground, the players have to obey the rules of the game. They have to play under discipline. Without discipline, there is bound to be confusion, disorder and chaos. No work or progress is possible without discipline. It is discipline which enables a nation to march on the way to progress. It must be kept in mind that discipline can’t be enforced by law alone. It is training of the mind. It has to come from within.

PSEB 10th Class English Paragraph Writing

6. A Small Family Is a Happy Family

Outline : Small family enjoys life …… food and clothes of their choice …… expenses limited …… enough money …… enough time …… enjoy luxuries of life.

A small family enjoys life to the full. The couple may enjoy the food of their choice. They can wear clothes of their liking. They have only one or two children. So their expenses are limited. They can enjoy life in any way they like. They have enough money to spend on the education and clothes, etc. of their children. They have a lot of time to look after their children. They can work hard and buy their own house to live in. They can enjoy luxuries of life. It is rightly said that a small family is a happy family.

7. Fast Food — a Health Hazard

Outline : Present-day life affects eating habits …… taken to fast foods …… such foods lose value and taste …… oily and spicy …… cause various diseases …… should not be eaten.

The present-day fast life has affected our eating habits also. We have taken to foods that can be cooked fast and eaten fast. But such food loses much of its value and taste. Fast foods are generally prepared at wayside eating stalls and cheap restaurants. They are ill-cooked. They are very oily and spicy. They can cause a number of ailments in our body. They can- give us ulcers and even cancer. Habit of taking fast foods is suicidal. It is in our own interest to give it up. Fast foods are junk foods and junk should be thrown away rather than be eaten.

8. A Scene at the Railway Station

Outline : I spend some time at the railway station …… like watching trains …… before the arrival of a train …… growing excitement and activity …… some in a great hurry …… some unhappy at the departure of loved ones s coolies run …… vendor boys move in and out of the train jostling and elbowing …… after the train has left platform gives a deserted look.

Last Sunday, I went to the railway station to receive a friend of mine who was coming from Delhi. I bought a platform ticket and since there was time for the train to arrive I thought of spending some time there. There was a lot of excitement and activity on the platform. People were in a great hurry to catch the train. After the arrival of the train some people, who had come to see off, were unhappy at the departure of their loved ones. The coolies were running. The vendor boys were moving in and out of the train. There was a great rush because of which there was a lot of jostling and elbowing. But after the train left, it was all quiet at the platform and the platform gave a deserted look.

9. The Scene at an Election Booth

Outline : The election day full of excitement …… my school …… polling centre …… lavish activity …… rush of voters …… cars, minibuses used to bring candidates from homes …… free drinks offered to voters …… polling officer enquired the names of voters verified identity cards …… after casting vote, ink mark put on, voter’s left forefinger …… booth closed at 4 p.m.

In our city municipal elections were held last month. Our school was chosen as the polling centre. There was a feverish activity on the day of polling. Everybody was full of excitement. Cars and mini buses were used to bring the candidates from homes. There was a great rush of voters. Free drinks were offered to the voters. Then the voters entered polling booth from one side. The polling officer enquired the names of the voters. He verified each voter by his/her identity card. Before casting the vote, the indelible ink mark was put on every voter’s fore-finger. After that, they went behind the curtain. There they pressed the button of the Electronic Voting Machine according to their choice. They came out from the other side after casting their vote.

PSEB 10th Class English Paragraph Writing

10. Inauguration of the New Computer Wing of Our School

Outline: New Computer Wing…… inaugurated by Principal, Govt. Polytechnic College…… interesting and informative talk on computers …… new computer wing going to help the students.

The new Computer Wing of the school was inaugurated by Dr. Ravi Sharma, Principal of Govt. Polytechnic, Faridabad. On this occasion, a talk on computers and their use was organised. The talk was given by Dr. Arvind Sharma. His talk was very interesting and informative. He told the students about different kinds of computers and their use in our present-day life. He impressed upon the students to take maximum advantage of their new computer wing and gain some elementary knowledge about the working of computers. The new computer wing is going to help the students a great deal.

11. A Visit to a Book Fair

Outline : Organised by the local publishers …… held in Town Hall…… rich display of books …… large number of people …… series of competitions …… fair a great success.

A three-day book fair was organised by the local publishers and distributors of the city. It was held in the Town Hall from the 17th to the 19th of the month. There was a rich display of books on almost all subjects for school and college students. The fair attracted a large number of students, parents and other book-lovers. In order to make the fair attractive to students, a series of competitions were held for school students on each day of the fair. Students of all local schools could take part in these competitions. Prizes were given to the winners in painting, cursive writing and intelligence tests in science, maths and general knowledge. The fair was a ‘great success in so far as it aroused the interest of the students to read good and useful books.

12. Annual Inspection of Our School

Outline : Annual inspection last month …… few days before Principal called meeting …… students asked to decorate the school…… whitewashed and cleaned the classrooms …… won first prize.
Answer:
The annual inspection of our school took place last month. A few days before the inspection, our Principal called a meeting. All the teachers and the students took part in it. The Principal asked the students to lend a hand in decorating the school. We were asked to clean and whitewash our classrooms ourselves. The students gladly agreed to it. First of all, we removed all the benches and desks from our classroom. We took a broom and swept the room. Then we prepared some buckets of whitewash. We took a brush and whitewashed the walls. A ladder was used to whitewash the ceiling. Then we took some enamel paint. Two coatings of this paint were given to the doors and windows. After whitewashing and painting, we cleaned the floor. Lastly, we arranged the furniture. On the day of inspection, we won the first prize.

13. A Visit to a Hill Station

Outline : During…… last summer…… Mussoorie…… friends…… new experience…. number of visitors. …… foreigners…… atmosphere…… hustle and bustle…… very pleasant…… ‘Queen of Hills’. ……material life…… visit is still fresh….

During the last summer vacation, I visited Mussoorie with some of my friends. I had never been to Mussoorie before. So it was a new experience for me. There were a large number of visitors. The presence of foreigners and honeymooners lent an additional charm to the atmosphere. Everywhere, there was a great hustle and bustle. The weather was very pleasant. The scenery all round was very charming. The fir-clad hills presented a fine view. Rightly has Mussoorie been called the ‘Queen of Hills’. It transports one into a heaven of joy. It makes one forget about the worries of material life. We visited all the beauty spots in the neighbourhood of Mussoorie. The memory of this visit is still fresh in my mind.

PSEB 10th Class English Paragraph Writing

14. Value of Newspapers

Outline : Newspapers …… our life …… well-informed …… give us confidence …… government…… wrong policies…… social purpose…… social evils…… help job-seekers…… suitable matches…… sons and daughters …… part and parcel….

Newspapers have an important place in our life. They keep us well-informed. They increase our general knowledge. They give us confidence. They tell us what is happening around us. They are the world’s mirrors. Newspapers guard our rights. They keep a check on the government. They bring to light the wrong policies of the government. Newspapers serve a social purpose also. They educate us against social evils. Thus, they help in reforming society. Newspapers help job-seekers also by advertising the vacant posts. They help parents in finding suitable matches for their sons and daughters. Truly, newspapers have become a part and parcel of modern life.

15. Science Fair in Our School 

Outline : Inaugurated by the District Education Officer …… 120 items on display…… most of them working models …… showed newness …… everyone fascinated ?.. students explained with confidence …… teachers had worked hard …… appreciated.

A science fair was organized in our school auditorium on the 7th of this month. It was inaugurated by the District Education Officer. There were more than 120 items on display. Most of them were working models. Each model showed newness and innovation in the practical application of some scientific law. Everyone was fascinated to see the exhibits. Students explained with confidence the mechanism of each model. Both, the students and teachers, had worked very hard to make this fair a success. Their efforts were appreciated by all present.

PSEB 10th Class English Paragraph Writing

16. A Scene of a Flooded Village

Outline : There…… heavy rains…… July …… got flooded…… three days…… shelter…… roofs…… crops were all washed …… cattle died …… water level kept rising …… would get drowned …… army men came…… boats to safety…… villagers …… homes.

There were heavy rains in the month of July this year. The village in which I live got flooded after it had been raining for three days and nights. There was knee-deep water in all the houses of the village. The villagers had nowhere to go to. They took shelter on the roofs of their houses. Their crops were all washed away. Many of their cattle died. It was a horrible sight. To make it worse, the water level kept rising hour by hour. The villagers began to fear that all of them would get drowned. But luckily, the army men came to our help. They ferried the villagers in their boats to safety. The flood subsided after five days. The villagers came back to their homes.

17. A Rainy Day 

Outline : India ……….. land of varying seasons ……….. summer very hot ……….. rain brings relief dark clouds gather in the sky cool breeze ………..downpour children dance in rain ……….. streets flooded with water ……….. rain stopped cool weather……….. rainbow in the sky……….. people enjoy.

India is a land of various seasons. Here summer is very hot. Only rain brings relief. It was the month of July last year. One day, it was very hot. Men and animals were panting. All were perspiring. We longed for a shower of rain. In the afternoon, some dark clouds appeared in the sky. A cool breeze began to blow. Soon the whole sky was overcast with dark clouds and there was a heavy downpour. Streets and bazaars were flooded with water. Little children came out and played in the rain. The rain stopped after two hours. The weather became cool. A rainbow appeared in the sky after the rain. People enjoyed the cool weather.

18. Drug Addiction

Outline: Drug addiction very common among the youth …… causes…… neglect and harsh treatment …… frustration…… educational system.

Drug addiction is becoming very common among our youth. Perhaps the youth find this life too burdensome. As an escape from it, they take to drugs. Sometimes parents themselves become the cause of it. They are either too strict or have no time to attend to their children. The young ones feel neglected. Our purposeless system of education is also one of the big causes. When the youth fail to get any employment after 15-20 years of education, they get frustrated and take to drugs. Drug addiction has become a big social problem. There is no short-cut cure for drug abuse. We often complain that our youth are going wrong. But, we hardly do anything to understand their problems. The best way to keep our youth away from drugs is to use their energies in constructive ways. An employment-based educational system should be evolved.

19. Environmental Pollution

Outline : Living in polluted world …… air, water, food, etc. …… everything polluted ……scientists listed the dangers of environment pollution …… all governments worried…… man himself responsible …… use of fertilizers …… nuclear testing contaminating water …… several deadly diseases ….. an immediate control must.

Environment pollution is the greatest problem facing a man. We are living in a polluted world; where air, water, food, etc. everything upon which we survive is polluted. Scientists have listed down the dangers of environment pollution and the governments around the globe are worried about the same. In fact, man himself is responsible for such a situation. In his ignorance he has so far been destroying the wealth of nature, trees and forests. An immediate control is required to keep the environment free from pollution. More attention is being paid to afforestation. The unauthorized felling of trees is being checked. Scientists are developing methods to minimize the effect of smoke let out by our autos and chimneys.

PSEB 10th Class English Paragraph Writing

20. Wonders of Science

Outline: Science…… this world…… marvellous changes…… life easy…… blessing for man…… our dreams into realities…… reduced human suffering…… medicine and surgery…… fatal…… many articles of daily use …… made…… comfortable.

Science has made man the master of this world. It has brought marvellous changes in man’s life. It has made our life easy and comfortable. It has, in fact, proved a blessing for man. It has translated our dreams into realities. It has brought revolution in the fields of agriculture, industry, transport and communication. Science has greatly reduced human suffering. Great progress has been made in the fields of medicine and surgery. Diseases once thought fatal can now be cured very easily. Science has given us many articles of daily use. These articles have made our life easy and comfortable.

21. Planting Trees

Outline: Trees…… in this world…… without trees…… man to live…… necessities of life…… purify the air …… beautify our surroundings…… the land safe from …… pleasant shade …… sun…… in his greed …… wealth of nature …… problems for mankind…… man’s own …… grow trees ….

Trees do more good to man than any other thing in this world. In fact, without trees, it would be impossible for man to live. They give us all the necessities of life. They give food for man. They purify the air. They help to bring rain. They beautify our surroundings. They give shelter to countless wild animals and birds. They keep the land safe from soil erosion. They give us pleasant shade from the heat of the sun. But it is very sad that man, in his greed, has destroyed this wealth of nature. It has resulted in many grave problems for mankind. It will be in man’s own interest to grow trees and look after them.

22. Life in a Village

Outline : The three words …… Simple, Pure and Fresh …… simple-hearted people …… thoughts and actions…… hospitable…. no anxiety…… calm and peaceful…… free from…… din of cities…… Says Leo Tolstoy …… A villager…… grow rich…… enough to eat…… worth-living.

The three words that can amply describe the life in a village are Simple, Pure and Fresh. The villagers are very simple-hearted people. They know no cunning. They are pure in their thoughts and actions. They are very hospitable. They live simply and happily. They have no anxiety. Life in a village is very calm and peaceful. It is free from the noise and din of cities. The air is fresh and health-giving. Says Leo Tolstoy in one of his stories, “A villager’s life is not a.fat one, but it is a long one.” A villager may never grow rich, but he has always enough to eat. In short, we can say that life in a village is worth-living.

PSEB 10th Class English Paragraph Writing

23. An Ideal Citizen

Outline : Pride of nation …… duty comes first…… law-abiding …… pays taxes …… national outlook …… above the considerations of caste and creed…… works for national unity…… patriotism highest religion.

An ideal citizen is the pride of his nation. He is conscious of his rights and his duties. For him duty comes first. He knows that it is impossible to enjoy one’s rights without first doing one’s duty. He never forgets the duties that he owes to society. He is law-abiding. He obeys all the laws of the land. He pays all the taxes that are due to him. He has a truly national outlook. He is not narrow-minded. He is above the considerations of caste and creed. He works for national unity. Patriotism is the highest religion for him.

24. A House on Fire

Outline: It was Sunday…… with my friend…… saw clouds of smoke…… big fire…… running…… buckets of water…… sand and water …… after half an hour …… suffered a great loss …… grain and money…… People felt sorry …… gave him food, clothes …… carpenter thanked …… folded hands.

It was Sunday. I was sitting in my room with my friend. Suddenly, we saw clouds of smoke rising in the sky. There was a big fire. People were running to the site of fire. Children were shouting for help. People brought buckets of water. We also joined them. We threw sand and water on the flames. The fire was put out after half an hour. It was the house of a carpenter. The poor man suffered a great loss. All his wood, grain and money were gone. The house was reduced to ashes. He was very sad at his loss. People felt sorry for him. They gave him food, clothes and some money. The poor carpenter thanked them with folded hands.

25. My Hobby

Outline: My hobby…… reading books…… good and useful books…… knowledge ……pleasure…… great writers …… noble things of life…… best friends…… endless store …… food for the mind…… not very expensive…… borrow books from libraries.

Reading books is my hobby. I read only good and useful books. I read some good journals also. I get much useful knowledge from them. Reading gives me much pleasure. I can enjoy the company of great writers by reading their works. I can learn from them good and noble things of life. Good books are the best friends. They are an endless store of knowledge. We can learn much from them. They give us food for the mind. Reading books is the best of all hobbies. It is not very expensive. We can borrow books from libraries.

26. My Favourite Book

Outline: Many books…… “Gita’…… liked the most…… religious book…… philosophy…… work hard…… worry …… result…… certainly rewards…… leave the rest to God get peace of mind.

I have read many books. But I have liked one book the most of all. It is the ‘Gita’. It is a religious book of the Hindus. It has a very rich philosophy. It teaches us that man should work hard. He should not worry about the restful. God certainly rewards us for our work. Good work always leads to good results. So an man should only mind his work. He should leave the rest to God. I like this philosophy very much. I get peace of mind by reading this book.

27. My Best Friend

Outline: My best friend…… class-fellow…… lives next door tome ……go to school together…… intelligent…… captain of the tennisteam…… teachers love…… honest and sincere…… respects his elders…… polite to all…… poor and needy students…… true friend ……in the real sense…… friendly with him.

The person I like most is my best friend, Amit Kapoor. He is my class-fellow. He lives next door to me. We go to school together. We play and study together. Amit is very intelligent. He always stands first in the class. Amit is also good at sports. He is the captain of the tennis team of our school. All teachers love him. I like Amit for his good and simple habits. He always speaks the truth. He is honest and sincere. He respects his elders. He is polite to all. He helps the poor and needy students. He is a true friend in the real sense. Everybody wants to be friendly with him. I am proud of my friend. May he live long !

PSEB 10th Class English Paragraph Writing

28. My Favourite Teacher

Outline: My favourite teacher…… active and smart…… M.A. B.Ed…. English…… method of teaching…… simple and effective…… ideal teacher…… real sense…… helps his students…… source of inspiration…… for the students…… proud of him…… live long

Mr. Ram Paul is my favourite teacher. He is my class teacher. He is forty years old. He is very active and smart. He is an M.A. B.Ed. He teaches us English. He is the master of this subject. His method of teaching is very simple and effective. His results are always good. Mr. Ram Paul is an ideal teacher in the real sense. He is ever ready to help his students. He helps the poor students with money and books. He is very kind to his1 students. He loves the students and the students respect him. He is a source of inspiration for his students. We are proud of him. May he live long !

29. Our School Library

Outline : A house of books …… big library …… big hall…… divided into four sections…… reading room …… 10,000 books …… newspapers and magazines …… students …… vacant periods …… read …… librarian …… issue books…… guides the students how to choose…… students can take…… after reading them.

A library is a house of books. There is a library in every school. There is a big library in our school also. It is housed in a big hall. The hall is divided into different sections. There is one section for each subject. There is a reading room also. There are about 10,000 books in our school library. The books are arranged on shelves. The newspapers and magazines are placed on the tables. The students go to the library in their vacant periods. They read books, newspapers and magazines there. The students love sitting in it. There is a librarian. He issues books. The students can take books to their homes also. They return the books after reading them.

30. The Morning Assembly of Our School

Outline: Students and teachers…… morning assembly…… stand in rows ……stand facing the assembly …… school compound …… head boy of school…… main news …… thought of the day …… explains it …… some important announcement…… go to classrooms.

Our school begins at half past eight in the morning. All the students and teachers reach the school in time. The morning assembly begins. All the students stand in rows in the school compound. They stand facing the school building. Four boys stand facing the assembly. First of all, there is the morning prayer. Then the head boy of the school reads out the main news of the day. A teacher gives the thought of the day and explains it. Sometimes our Headmaster makes some important announcement during the assembly. Then we go to our classrooms and the first period begins.

31. An Indian Farmer

Outline: An Indian farmer…… fields…… plough…… waters them …… sows seeds…… looks after crops…… no easy job…… very difficult and tiring…… remains poor…… hand to mouth …… lot is pitiable…… lives happily…… simple in habits ……pure in thoughts…… evils of city life.

The life of an Indian farmer is very hard. He has to work hard in his fields. He has to work in sun and rain. He has to plough his fields. He has to water them. Then he has to sow seeds in his fields. He looks after his crops day and night. Harvesting is also no easy job. It is very difficult and tiring. Even after so much labour, the Indian farmer remains poor. He has to live from hand to mouth. His lot is pitiable. But though he is poor, he lives happily. He is simple in his habits. He is pure in his thoughts. He lives away from the evils of city life.

PSEB 10th Class English Paragraph Writing

32. An Indian Beggar

Outline: An Indian beggar…… common sight…… streets and bazaars…… railway stations…… in rags…… looks very dirty…… pretend to be lame…… painful voices ……pity …… money, clothes, flour…… don’t like…… poor homes…… work to do…… allowed to beg…… very ugly.

The Indian beggar is a common sight. We can see him anywhere and everywhere. We can see him in streets and bazaars. We can see him at bus stops and railway stations. He is always in rags. He looks very dirty. Some beggars pretend to be lame or blind. They cry in painful voices. People are moved to pity. They give them money, clothes, flour, etc. But I don’t like it. Beggars should be kept in poor homes. There they should be given work to do. Nobody should be allowed to beg in public places. It looks very ugly.

33. The Independence Day

Outline: India became free…… August 15,1947…… Indepdendence Day…… British rule…… red-letter day …… history of the country…… celebrated with great enthusiasm…… national holiday ……public meetings…… big function…… Prime Minister…… unfurls the National flag…… Red Fort…… hard-won freedom ……protect it.

India became a free country on August 15, 1947. So, August 15 is called the Independence Day of India. The British rule came to an end on this day. It is a red- letter day in the history of the country. It is celebrated all over the country with great enthusiasm. On this day all schools, colleges and offices remain closed. It is a national holiday. Public meetings are held in all towns and cities. A big function is held in Delhi. The Prime Minister unfurls the national flag at the Red Fort. Our freedom is a hard-won freedom. We should protect it.

34. The Republic Day

Outline : India became Republic…… Constitution came into force…… secular democratic …… power of government…… common people…… castes, creeds and religions…… equal…… red-letter day…… celebrated all over the country…… enthusiasm…… national flag unfurled…… public buildings…… big function…… Delhi…… President …… presides…… worth seeing…. national holiday.

India became a Republic on January 26, 1950. The Constitution of the country came into force on this day. India became a secular democratic country. The power of government passed into the hands of the common people. All castes, creeds and religions are to be equal in the eyes of the law. It is a red-letter day in the history of the country. It is celebrated all over the country with great enthusiasm. The national flag is unfurled at all the public buildings. A big function is held in Delhi. The President of the country presides over this function. It is worth seeing. This day is a national holiday.

35. A Football Match

Outline: Our school…… Khalsa School…… football match…… Sh. Jaswant Singh referee…… toss…… chose our side…… game slow ……played well…… our defence…… no goal…… blew the whistle…… interval…… Vinod passed …… the ball…… ran into the D…… kicked it hard…… goal…… loud cheers…… referee blew the whistle…… game was over…… won the match.

Last Monday, a football match was played between our school and Khalsa School. It was played on our school grounds. Sh. Jaswant Singh was the referee. He blew the whistle. There was a toss. We won the toss. We chose our side. The match began. At first the game was slow, but soon it became brisk. All the players played well. Our defence was very strong. There was no goal. The referee blew the whistle for interval. In the second half, Vinod passed the ball on to me. I ran with it into the D. Then I kicked it hard. It went through the poles. It was a goal. There were loud cheers. The referee blew the whistle. The game was over. We won the match by one goal.

36. Morning Walk

Outline: Morning walk…… best exercise…… costs nothing…… health…… refreshes …… saves from diseases…… develops…… habit…… pure thoughts…… dewdrops…… fresh flowers…… charm…… mind…… loving…… natural beauty …… not only…… body…… mind also.

Morning walk is the best form of exercise. It costs nothing. It is very useful for our health. It refreshes our mind. It strengthens our body. It saves us from many diseases. Morning walk keeps us fresh for the whole day. It develops in us the habit of rising early. It brings pure thoughts in our mind. The dewdrops, the fresh flowers, the chirping birds and the rustling leaves charm our mind. We start loving these objects of natural beauty. Thus, morning walk is useful not only for our body, but for our mind also.

37. A Visit to a Zoo 

Outline :A zoo…… visited …… parents…… tickets…… saw birds…… rare kinds of…… herons, gulls…… wild beasts …… lioness and her cub…… basking…… wolves…… rhinos…… muddy pond…… big snakes…… fearful…… stayed there …… three hours…… came back home.

There is a zoo in our city. I visited it last Sunday. I went with my parents. We bought tickets and went in. First of all we saw birds. There were many beautiful and rare kinds of birds. We saw parrots, canaries, swallows, peacocks, ducks, cranes, herons, gulls and geese. Then we saw some wild beasts. A lioness and her cub were basking in the sun. They roared now and then. We also saw wolves, tigers, elephants and rhinos. When we were coming back, we saw a muddy pond. There were many big snakes in it. It was fearful to look at them. We stayed in the zoo for about three hours. Then we came back home.

PSEB 10th Class English Paragraph Writing

38. Leisure 

Outline : Life is not worth living if we remain busy in work ….. lot of energy wasted ….. restored by leisure ….. enjoy sights of nature develop new tastes and creative hobbies ….. busy men miss all this ….. overwork causes ailments….. make right use of leisure.

Leisure is the free or spare time one gets after one’s busy schedule. One must have some leisure to enjoy the sights of nature and to look after one’s physical, mental, moral, emotional and spiritual needs. A lot of energy is wasted in doing the day’s work. That energy is restored by leisure. All work and no play makes Jack a dull boy. It has been rightly said. During the leisure hours, we can develop new tastes as well as creative hobbies. All this is possible only if we have some leisure hours at our disposal. Busy men miss all this. Overwork and no rest causes ailments of various kinds. We should make the right use of leisure. This life is not worth living, if we always remain busy in work.

PSEB 10th Class English Notice Writing

Punjab State Board PSEB 10th Class English Book Solutions English Notice Writing Exercise Questions and Answers, Notes.

PSEB 10th Class English Notice Writing

A notice is a written or printed news, announcement or information. It is usually displayed publicly as on a school / college notice-board. A notice can also be given for insertion in a newspaper like an advertisement. But there is one main difference between an advertisement and a notice. An advertisement is chiefly commercial (or matrimonial) in nature; but a notice is a general piece of information for a particular group. A notice should be in complete sentences or even in the form of a short paragraph, whereas an advertisement can be in the form of merely catchy phrases and slogans.

PSEB 10th Class English Notice Writing

Question 1.
A notice is to be prepared for putting up on the students’ noticeboard, informing them about the school inspection to be held on 16th May, 2018. It is from Reema Sharma, The Principal, Government High School, Mohali.
Answer:

Govt. High School, Mohali
Notice

10 May, 2018
This is to inform all the students and the school staff that the school inspection will be held on 16 May, 2018 by the District Education Officer. Therefore, all the students, teachers and the school staff are requested to maintain discipline, cleanliness and punctuality to keep up the good reputation of the school during the inspection.

Reema Sharma
(Principal)

Question 2.
You are Manav Shukla, the Secretary, the Help All Club of the Navodaya Vidyalaya, Kashmiri Gate, Delhi. Prepare a notice for the school students to help the earthquake victims with money, medicine, food and clothes.
Answer:

Navodaya Vidyalaya
16 October 20_ _
Kashmiri Gate, Delhi
NOTICE

The recent earthquake in the hilly areas has caused heavy loss of life and property. The affected people are having a horrible time. They are without food, clothes, money and medicine. The Help All Club of the school appeals to all students to help the victims with money, medicine, food and clothes. For your donations, contact the undersigned.

Manav Shukla
Secretary

PSEB 10th Class English Notice Writing

Question 3.
As the Secretary of Sai Baba Society, Somesh Vihar, Delhi, prepare a notice for the residents of the colony, giving hints for protection against Dengue fever. Your name is Sheetal Jain.
Answer:

SAI BABA SOCIETY
10 August 20_ _ Somesh
Somesh Vihar, Delhi
NOTICE
Hints for Protection against Dengue

Many cases of dengue have been reported in the city. The escape from this disease lies more in prevention than in cure. All residents of the colony are advised to take the following steps to protect themselves against this dangerous disease.

  • Disinfect all places where mosquitoes can breed.
  • Use mosquito repellents.
  • Do not let water stagnate around your houses.
  • Wear clothes that cover arms and legs.
  • In case of fever, contact your doctor or the nearest hospital.

Sheetal Jain Secretary

Question 4.
Draft a notice on behalf of the Principal, Delhi Public School, Ludhiana, announcing elections for the posts of President and Secretary of English Literarary Society of the school.
Answer:

 DELHI PUBLIC SCHOOL
LUDHIANA
NOTICE

20 October 20–

The school is going to hold elections for the posts of President and Secretary of the English Literary Society of the school. Those who want to give their nominations for the same posts should give their names to the undersigned by the 13th of this month.

Sanjeev Jain
(Principal)

Question 5.
You are Mohan Kumar, Sports Secretary, Govt High School, Jalandhar. Some old sports goods have to be put on sale to collect money for donation to the poor cancer patients. Draft a notice, inviting students to help by buying these goods.
Answer:

GOVT. HIGH SCHOOL, JALANDHAR
NOTICE

10 March 20 – –

The school has decided to sell some of the old sports goods. The sale will be organized in the school hall on 15 March, 20 _ _. These goods include cricket bats, balls, hockey sticks, footballs, volleyballs and basketballs. All the goods are in a fairly good condition. The amount collected from the sale will be donated for the treatment of poor cancer patients. Please come, buy and help in the good cause.

Mohan Kumar
(Sports Secretary)

PSEB 10th Class English Notice Writing

Question 6.
You are Nimisha, Student Editor, School Magazine, Radha Vatika School, Khanna. Draft a notice, inviting entries for the magazine from students. The last date is 10th of October.
Answer:

RADHA VATIKA SCHOOL, KHANNA
NOTICE
Contributions For The School Magazine

1 October 20 – –

We are going to bring out the annual issue of our school magazine next month. Those who want to have their articles, stories, poems, etc. published in this issue should hand over their articles to the undersigned by the 10th of this month. The articles should be original and written neatly on one side of the paper only.

Nimisha Paul
(Student Editor)

Question 7.
You wish to arrange a charity show in your school. Write a notice in this regard, specifying the purpose, date and time of the show.
Answer:

P.S.N. PUBLIC SCHOOL, LUDHIANA

10 December 20_ _

The Cultural Committee of the school has decided to arrange a charity show to extend help to the victims of recent earthquake in Uttarakhand. Funds f collected from this charity show will be spent for purchasing clothes, food, blankets, etc. for the earthquake victims. Skits, folk dances, songs and a lot of fun is included in this charity show. So, all students are invited to this show on 15 December, 2017 at 11.00 a.m. at the school auditorium.

Tamanna Bhatia
(Cultural President)

Question 8.
You are Sonal, Cultural Secretary of the City Public School, Lucknow. Write a notice, inviting students to give their names for the Fancy Dress Competition being held in the School.
Answer:

CITY PUBLIC SCHOOL, LUCKNOW
NOTICE
FANCY DRESS COMPETITION

1 October 20 – –

A Fancy Dress Competition is going to be held at our school on 10 October, 2017. The competition will be open to the students from Nursery to Class-V only. Famous dress designer, Ritu Beri, has consented to be the judge for this competition. So those desirous of taking part in this event, should give their names to the undersigned by
5 October, 20

Sonal
(Cultural Secretary)

PSEB 10th Class English Notice Writing

Question 9.
You are Munish, the Head Boy of Kamal Public School, Kamal. Write a notice ‘ for the school notice-board, inviting the students to participate in the Annual Sports Day.
Answer:

KARNAL PUBLIC SCHOOL, KARNAL
NOTICE

10 March 20_ _

The school is holding its Annual Sports Day on 21st March. Those who want to participate in the various events should give their names to the undersigned by the 13th of this month. The list of various events is available with the undersigned. No student can take part in more than three events, excluding the relay.

Munish Lai
(Head Boy)

Question 10.
You are Nikhil Kumar, the Head Boy of Govt. Sr. Sec. School, Ambala. Draft a notice regarding a book that has been found in the school compound.
Answer:

GOVT. SR. SEC. SCHOOL, AMBALA
NOTICE
LOST AND FOUND

22 March 20_ _

A book on English Grammar was found lying in the school compound last evening. It is of brown colour with a name written on it. The owner is requested to collect it from the school office after showing his / her identity card.
Nikhil Kumar
(Head Boy)

PSEB 10th Class English Notice Writing

Question 11.
Rakesh Sharma is a student of class X-A in Govt. Sen. Sec. School, Amritsar. He has lost his English Reader Book during the lunch .break. Draft a notice on his behalf for the school notice-board.
Answer:

GOVT. SEN. SEC. SCHOOL, AMRITSAR
NOTICE
LOST ! LOST! LOST!

15 October 20

I have lost my English Reader Book somewhere in the school. The book bears my name and class roll number. I have put in it some notes and hints in my own hand. The finder is requested to return the book to me or deposit it with the school office. It will be an act of great favour.

Rakesh Sharma
Roll No. 5

PSEB 10th Class Maths Solutions Chapter 2 Polynomials Ex 2.1

Punjab State Board PSEB 10th Class Maths Book Solutions Chapter 2 Polynomials Ex 2.1 Textbook Exercise Questions and Answers.

PSEB Solutions for Class 10 Maths Chapter 2 Polynomials Ex 2.1

Question 1.
The graphs of y = p (x) are given in Fig. below, for some polynomials p(x). Find the number of zeroes of p (x), in each case.

PSEB 10th Class Maths Solutions Chapter 2 Polynomials Ex 2.1 1

Solution:
The graphs of y = p (x) are given in figure above, for some polynomials p (x). The number of zeroes of p (x) in each case are given below:
(i) From the graph, it is clear that it does not meet x-axis at any point.
Therefore, it has NIL; no. of zeroes.

(ii) From the graph, it is clear that it meets x- axis at only one point.
Therefore, it has only one no. of zeroes.

(iii) From the graph, it is clear that it meets x-axis at three points.
Therefore, it has three no. of zeroes.

(iv) From the graph, it is clear that it meets x- axis at two points.
Therefore, it has two no. of zeroes.

(v) From the graph, it is clear that it meets x- axis at four points.
Therefore, it has four no. of zeroes.

(vi) From the graph, it is clear that it meets pr-axis at three points.
Therefore, it has three no. of zeroes.

PSEB 10th Class Maths Solutions Chapter 1 Real Numbers Ex 1.4

Punjab State Board PSEB 10th Class Maths Book Solutions Chapter 1 Real Numbers Ex 1.4 Textbook Exercise Questions and Answers.

PSEB Solutions for Class 10 Maths Chapter 1 Real Numbers Ex 1.4

Question 1.
Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion :
(i) \(\frac{13}{3125}\)

(ii) \(\frac{17}{8}\)

(iii) \(\frac{64}{455}\)

(iv) \(\frac{15}{1600}\)

(v) \(\frac{29}{343}\)

(vi) \(\frac{23}{2^{3} 5^{2}}\)

(vii) \(\frac{129}{2^{5} 5^{7} 7^{5}}\)

(viii) \(\frac{6}{15}\)

(ix) \(\frac{35}{50}\)

(x) \(\frac{77}{210}\)
Solution:
(i)Let x = \(\frac{13}{3125}\) ………..(1)
Compare (1) with x = \(\frac{p}{q}\)
Here p = 13 and q = 3125
Prime factors of q = 3125 = 5 × 5 × 5 × 5 × 5 = 55 × 20
which are of the form 2n × 5m here n = 0, m = 5
which are non negative integers.
∴ x = \(\frac{13}{3125}\) have a terminating decimal expansion.

PSEB Solutions PSEB 10th Class Maths Solutions Chapter 1 Real Numbers Ex 1.4

(ii) Let x = \(\frac{17}{8}\) …………………(1)
Compare (1) with x = \(\frac{p}{q}\)
Here p = 17 and q = 8
Prime factors of q = 8 = 2 × 2 × 2 = 23 = 23 × 50

which are of the form 2n × 5m here n = 3, m = 0
and these are non negative integers.
∴ x = \(\frac{17}{8}\) have a terminating decimal expansion.

(iii) Let x = \(\frac{64}{455}\) …………(1)
Compare (1) with x = \(\frac{p}{q}\)
Here p = 64, q = 455
Prime factors of q = 455 = 5 × 7 × 13 which are not of the form 2n × 5m
∴ x = \(\frac{64}{455}\) has a non – terminating decimal expansion.

PSEB Solutions PSEB 10th Class Maths Solutions Chapter 1 Real Numbers Ex 1.4

(iv) Let x = \(\frac{15}{1600}\) ……….(1)
Compare (1) with x = \(\frac{p}{q}\)
Here p = 15 and q = 1600
Prime factors of q = 1600
= 2 × 2 × 2 × 2 × 2 × 2 × 5 × 5 = 26 × 52
which are of the form 2n × 5m, here n = 6, m = 2 and these are non negative integers.
∴ x = having terminating decimal expansion.

(v) Let x = \(\frac{29}{343}\) ………… (1)
Compare (1) with x = \(\frac{p}{q}\)
Here p = 29 and q = 343
prime factors of q = 343
= 7 × 7 × 7 = 73
which are not of the form 2n × 5m, here n = 0, m = 0
∴ x = \(\frac{29}{343}\) will have a non – terminating decimal expansion.

(vi) Let x = \(\frac{23}{2^{3} 5^{2}}\) ……….(1)
Compare (1) with x = \(\frac{p}{q}\)
Here p = 23, q = 2352
Prime factors of q = 2352
which are of the form 2n × 5m, here n = 3, m = 2 and these are non negative integers.
∴ x = \(\frac{23}{2^{3} 5^{2}}\) will have a terminating decimal expansion.

(vii) Let x = \(\frac{129}{2^{5} 5^{7} 7^{5}}\) ……………….(1)
Compare (1) with x = \(\frac{p}{q}\)
Here p = 129 and q = 25 57 75
Prime factors of q = 25 57 75
which are not of the form 2n × 5m,
∴ x = \(\frac{129}{2^{5} 5^{7} 7^{5}}\) have a non – terminating decimal expansion.

PSEB Solutions PSEB 10th Class Maths Solutions Chapter 1 Real Numbers Ex 1.4

(ix) Let x = \(\frac{35}{50}=\frac{7}{10}\) …………. (1)
Compare (1) with x = \(\frac{p}{q}\)
Here p = 7, q = 10
Prime factors of q = 10 = 2 × 5 = 21 × 51
Which is of the form 2n × 5m here n = 1, m = 1
both n and m are non negative integer.
∴ x = \(\frac{35}{50}\) have a terminating decimal expansion.

(x) Let x = \(\frac{77}{210}=\frac{11}{30}\) ……………..(1)
Compare (1) with x = \(\frac{p}{q}\)
Here p = 11, q = 30
Prime factors of q = 30 = 2 × 5 × 3
which are not of the form 2n × 5m,
∴ x = \(\frac{77}{210}\) have a non – terminating decimal expansion.

PSEB Solutions PSEB 10th Class Maths Solutions Chapter 1 Real Numbers Ex 1.4

Question 2.
Write down the decimal expansions of those rational numbers in Question 1 above which have terminating decimal expansions.
Solution:
(i) Let x = \(\frac{13}{3125}\)
Compare (1) with x = \(\frac{p}{q}\)
Here p = 13,q = 3125
Prime factors of q = 3125 = 5 × 5 × 5 × 5 × 5 = 55 × 20
Which are of the form 2n × 5m, where n = 0, m = 5 and these are non negative integers
∴ x = have a terminating decimal expansion.
To Express in Decimal form
x = \(\frac{13}{3125}=\frac{13}{5^{5} \times 2^{0}}\)

x = \(\frac{13 \times 2^{5}}{5^{5} \times 2^{5}}\)
[∵ we are to make 10 in the denominator so multiply and divide by 25]

x = \(\frac{13 \times 32}{(2 \times 5)^{5}}\)

x = \(\frac{416}{(10)^{5}}=\frac{416}{100000}\)

x = 0.00416

(ii) Let x = \(\frac{17}{8}\) ………………..(1)
Compare (1) with x = \(\frac{p}{q}\)
Here p = 17, q = 8
Prime factors of q = 8 = 2 × 2 × 2 = 23 × 50
Which are of the form 2n × 5m, where n = 3, m = 0 and these are non negative integers
∴ x = \(\frac{17}{8}\) can be expressed as a terminating decimal expansion.
To Express in Decimal form

x = \(\frac{17}{8}=\frac{17}{2^{3} \times 5^{0}}\)

x = \(\frac{17 \times 5^{3}}{2^{3} \times 5^{3}}\)
[Multiply and divide with 53 to make the denominator 10]

x = \(\frac{17 \times 125}{(2 \times 5)^{3}}\)

x = \(\frac{2125}{(10)^{3}}=\frac{2125}{1000}\)

x = 2.125

⇒ \(\frac{17}{8}\) = 2.125

PSEB Solutions PSEB 10th Class Maths Solutions Chapter 1 Real Numbers Ex 1.4

(iii) Let x = \(\frac{15}{1600}\) …………….(1)
Compare (1) with x = \(\frac{p}{q}\)
Here p = 15, q = 1600
Prime factors of q = 1600
= 2 × 2 × 2 × 2 × 2 × 2 × 5 × 5
= 26 × 52
Which are of the form 2n × 5m, where n = 6, m = 2 and these are non negative integers
∴ x = \(\frac{15}{1600}\) can be expressed as a terminating decimal expansion.
To Express in Decimal form

x = \(\frac{15}{1600}\)

x = \(\frac{15 \times 5^{4}}{2^{6} \times 5^{2} \times 5^{4}}\)
[To make denominator a power of 10 multiply and divide by 54]

x = \(\frac{15 \times 625}{2^{6} \times 5^{6}}\)

x = \(\frac{9375}{(2 \times 5)^{6}}\)

x = \(\frac{9375}{(10)^{6}}=\frac{9375}{1000000}\) = 0.009375

In Decimal form, x = \(\frac{15}{1600}\) = 0.009375

(iv) Let x = \(\frac{23}{2^{3} 5^{2}}\) …………….(1)
Compare (1) with x = \(\frac{p}{q}\)
Here p = 15, q = 23 52
Prime factors of q = 23 52
Which are of the form 2n × 5m, where n = 3, m = 2 and these are non negative integers
∴ x = \(\frac{23}{2^{3} 5^{2}}\) have a terminating decimal expansion.
To Express in Decimal form

x = \(\frac{23}{2^{3} 5^{2}}=\frac{23 \times 5}{2^{3} \times 5^{2} \times 5}=\frac{115}{2^{3} \times 5^{3}}\)

x = \(\frac{115}{(2 \times 5)^{3}}=\frac{115}{1000}\) = 0.115

In Decimal form,
x = \(\frac{23}{2^{3} 5^{2}}\) = 0.115

PSEB Solutions PSEB 10th Class Maths Solutions Chapter 1 Real Numbers Ex 1.4

(v) Let x = \(\frac{6}{15}=\frac{2}{5}\)
Compare (1) with x = \(\frac{p}{q}\)
Here p = 2, q = 5
Prime factors of q = 5 = 20 × 51
Which are of the form 2n × 5m, where n = 0, m = 1 and these are non negative integers
∴ x = \(\frac{6}{15}\) have a terminating decimal expansion.
To Express in Decimal form

x = \(\frac{6}{15}=\frac{2}{5}\)

x = \(\frac{2 \times 2^{1}}{2^{1} \times 5^{1}}=\frac{4}{10}\) = 0.4

In Decimal form,
x = \(\frac{6}{15}\) = 0.4

(vi) Let x = \(\frac{35}{50}=\frac{7}{10}\)
Compare (1) with x = \(\frac{p}{q}\)
Here p = 7, q = 10
Prime factors of q = 5 = 21 × 51
Which are of the form 2n × 5m, where n = 1, m = 1 and these are non negative integers
∴ x = \(\frac{7}{10}\) have a terminating decimal expansion.
To Express in Decimal form

x = \(\frac{35}{50}\)

x = \(\frac{7}{10}\)

x = \(\frac{7}{2^{1} \times 5^{1}}\)

x = \(\frac{7}{(2 \times 5)^{1}}=\frac{7}{(10)^{1}}\) = 0.7
Hence in decimal form, x = 0.7

PSEB Solutions PSEB 10th Class Maths Solutions Chapter 1 Real Numbers Ex 1.4

Question 3.
The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form f, what can you say about the prime factors of q?
(i) 43.123456789
(ii) O.120120012000120000……
(iii) 4.3.123456789
Solution:
(i) Let x= 43.123456789 ……….. (1)
It is clear from the number that x is rational number.
Now remove the decimal from the number

∴ x = \(\frac{43123456789}{1000000000}\)

= \(\frac{43123456789}{10^{9}}\) …………….(2)
From (2) x is a rational number and of the \(\frac{p}{q}\).

Where p = 43123456789 and q = 109
Now, Prime factors of q = 100 = (2 × 5)9
⇒ Prime factors of q are 29 × 59

PSEB Solutions PSEB 10th Class Maths Solutions Chapter 1 Real Numbers Ex 1.4

(ii) Let x = 0.120120012000120000
It is clear from the number that it is an irrational number.

(iii) Let x = 43.123456789 …. (1)
It is clear that the given number is a rational number because it is non-terminating and repeating decimal.
To show that (i) is of the form \(\frac{p}{q}\)
Multiply (1) with 109 on both sides,
109 x = 43123456789.123456789 …………….(2)
Subtract (1) from (2), we get:

PSEB 10th Class Maths Solutions Chapter 1 Real Numbers Ex 1.4 6

which is rational number of the form \(\frac{p}{q}\)
x = \(\frac{4791495194}{111111111}\)
Here p = 4791495194, q = 111111111
x = \(\frac{4791495194}{3^{2}(12345679)}\)
Hence, prime factors of q are 32 (123456789)

PSEB 10th Class English Letter Writing

Punjab State Board PSEB 10th Class English Book Solutions English Letter Writing Exercise Questions and Answers, Notes.

PSEB 10th Class English Letter Writing

परीक्षा-पत्र में निम्नलिखित प्रकार के पत्र पूछे जा सकते हैं
1. Personal letters to friends and relatives.
2. Letters of complaint, enquiry, request; applications.
3. Letters to Editor.
पत्र-लेखन के सम्बन्ध में कुछ आवश्यक जानकारी
1. आजकल पत्र-लेखन के सम्बन्ध में एक नई विधि को अपनाना शुरू कर दिया गया है, यद्यपि भाषा-विज्ञान की दृष्टि से इसका कोई आधार नहीं है।

PSEB 10th Class English Letter Writing

2. नई विधि के अनुसार पत्र-प्रेषक का पता तथा पत्र लिखने की तिथि कई बार ऊपरी दाएं कोने में लिखने की बजाए बाएं कोने में लिख दिए जाते हैं, और प्राप्तकर्ता का नाम, पता तथा सम्बोधन करने के शब्द, आदि भी इसी के साथ नीचे लिख दिए जाते हैं तथा इनके लिखने में विराम-चिन्हों का प्रयोग भी लुप्त कर दिया जाता है। पत्र के अन्त में पत्र-प्रेषक अपना नाम या हस्ताक्षर भी बाईं तरफ़ ही डालता है। यह विधि यद्यपि कुछ अटपटी सी प्रतीत होती है, फिर भी बहुधा लोग इसे अपनाना बेहतर मानते हैं।

3. परम्परागत पुरानी विधि के अनुसार पत्र-प्रेषक का पता तथा पत्र लिखने की तिथि दोनों ही दाईं तरफ़ के ऊपरी कोने में लिखे जाते हैं और प्राप्तकर्ता को सम्बोधित करने सम्बन्धी शब्द बाईं तरफ़ नीचे को हटा कर लिखे जाते हैं। पत्र-प्रेषक अपना नाम या हस्ताक्षर, आदि भी पत्र के अन्त में दाईं सरफ़ हटा कर डालता है। तथा इन सब के लिखने में विराम-चिन्हों का विशेष ध्यान रखा जाता है।

4. विद्यार्थी को पत्र लिखते समय इस बात का विशेष ध्यान रखना चाहिए कि वह नई या पुरानी, किसा भा वाध का प्रयोग कर सकता है। किन्तु इन दोनों को किसी भी हालत में एक-दूसरी से मिला कर न लिखे। ऐसा करने से अंक काटे जा सकते हैं।

5. इस पृष्ठ पर ऊपर दिए गए उदाहरणों में पुरानी विधि के अनुसार विराम-चिन्हों का प्रयोग किया गया है, किन्तु . आजकल अधिकतर पत्र नई विधि के अनुसार ही लिखे जाते हैं, जैसा कि अगले पृष्ठों में दिखाया गया है।

Important Personal Letters

Advising Brother To Improve His Health

Question 1.
You are Rohit Verma, living at 27, Gandhi Nagar, Delhi. Your younger brother is a bookworm. He is good at studies but weak in health. Write him a letter, advising him to take part in games.
Answer:
27 Gandhi Nagar
Delhi
12 April 20… …
Dear Kapil
Your friend, Ashok, met me a few days ago. He told me that you have become a bookworm. You do not care at all about your health. You do not take part in games. It is not good on your part. Games are a part and parcel of education. Health is wealth. You can’t have a sound mind without a sound body. No doubt, you are good at studies, but you have a very poor health.
I advise you to be careful about your health. You should go out for a walk daily. Take part in games also. You must improve your health. One can’t enjoy one’s life without good health. If health is lost, everything is lost. Be a good boy. I hope you will act upon my advice.
With love
Yours affectionately
Rohit

Permission To Join An Educational Tour

Question 2.
You are Rahul Duggal, studying in class X at Govt. Sr. Secondary School, Maud Kalan, Bathinda. Write a letter to your father, requesting him to allow you to join an educational tour conducted by your school.
Answer:
Queen’s Hostel
Govt. Sr. Secondary School
Maud Kalan
Bathinda
10 December 20– —
My dear Father
Our school is breaking up for Christmas holidays on the 24th of this month. Some students of our school are going on an educational tour. The tour has been organised by our Principal. It will be a seven-day tour, from December 25 to December 31. The students shall visit Bengaluru, Mysore, Chennai and Madurai. I request you to allow me to go on this tour. Each student has to deposit ₹5000/ for this tour. I shall do with ₹ 1000/- only for my pocket expenses. Kindly send me ₹ 6000/- as soon as possible. I hope you will not disappoint me.
With regards
Yours affectionately
Rahul Duggal

PSEB 10th Class English Letter Writing

Inviting Cousin To Spend Winter Break With You

Question 3.
Write a letter to your cousin Pulkit, inviting him to spend his winter break with you. You are Rohan and you live at 24, Mall Road, Shimla.
Answer:
24 Mall Road Shimla
16 December 20– —
Dear Pulkit
I invite you to come to Shimla in your winter vacation. We will have a nice time together. Snowfall in Shimla is a beautiful sight. Many tourists come to see it. We will also visit all the beauty spots near Shimla. I hope you will definitely come. Pay my deep regards to dear uncle and aunt.
Yours affectionately
Rohan

To Father About Home News

Question 4.
You are Monu and you live at 51, Central Town, Nangal. Write a letter to your father, who is away on a long tour, giving him home news.
Answer:
51 Central Town Nangal
6 September 20
Dear Father
Many things have happened since I wrote you last. Bittoo fell down the stairs and broke her leg. She became senseless. She was moved to hospital. She is quite fit now. You need not worry about her. Mohan’s result Was out yesterday. He got 700 marks. He is sure to get a scholarship. He is very happy. Uncle Kartar Singh also visited us last week. He stayed with us for two days. We all miss you badly. Please come back home soon.
Yours affectionately
Monu.

Giving details of a medical camp

Question 5.
Write a letter to your sister, Mohini, giving her the details of a free medical camp that your grandmother arranged recently. You are Sudhir Singh and live at 111, Chandigarh Road, Rajpura.
Answer:
111 Chandigarh Road Rajpura
20 November 20– —
Dear Mohini
You know our grandmother is a social activist. Last week, she organized a free medical camp for poor patients in the local Town Hall. Eleven eminent doctors provided their services. Patients were given free consultation and medicines. More than 1000 patients benefited from this camp. Such camps are really a true service towards mankind.
Yours affectionately
Sudhir Singh.

A Letter Of Condolence

Question 6.
You are Tanbir Singh, living at G-312, Adarsh Colony, Moga. Write a letter of condolence to your friend Amrit on the death of her mother.
Answer:
G-312 Adarsh Colony Moga
5th January 20
Dear Amrit
I was shocked to learn about the untimely death of your mother. She was quite healthy. I met her the other day. I never thought that her end was so near.
She was a very noble lady. She was kind to all. She was a lovable mother. She took keen interest in your studies. She looked upon me as her own daughter.
Dear Amrit your loss is great. But it was God’s will. His ways are strange. We must bow before Him. May God grant peace to the departed soul !
With deep sympathies …
Yours sincerely
Tanbir Singh

PSEB 10th Class English Letter Writing

Thanking For Gift

Question 7.
Write a letter to your uncle, thanking him for the birthday gift he has sent you. You are Bhushan, living at 37, Civil Lines, Ludhiana.
Answer:
37 Civil Lines Ludhiana
5 August 20_ _
My dear Uncle
Thank you for the lovely gift you have sent me on my birthday. Your gift is truly after my heart. The wristwatch sent by you looks very beautiful. All my friends have liked it very much.
Dear uncle, I was badly in need of a watch. I was often late for school. I had to pay some fine every month. This watch will now prove useful for me. It will help me during the examinations. It will also help me to regulate my daily work. It is a token of your love for me. I shall keep it with great care. I once again thank you for this nice present.
Yours affectionately
Bhushan

Describing A Visit To A Historical Place

Question 8.
Write a letter to your younger brother Hitesh, describing your visit to some historical building. You are Naresh, living at Pratibha Hostel, Bal Bharti, Delhi.
Answer:
Pratibha Hostel
Bal Bharti
Delhi
10 October 20_ _
Dear Hitesh
You will be glad to know that I went to Agra during the last Christmas holidays. Our history teacher arranged this tour. There we visited the Taj. It has a majestic look. It is a large and beautiful building. It stands on a raised platform. In the middle of the platform, there is a splendid white dome. At its four corners, there are four stately towers. Underneath the white dome are the marble tombs of Mumtaz Mahal and Shah Jahan. The Taj was built by Shah Jahan in the sweet memory of his beloved wife. It is made of pure white marble. The beauty of the Taj beggars description. It looks like a fairy dressed in white. We stayed there for about two hours. I left the place most unwillingly. Its memory is still fresh in my mind.
With love
Yours affectionately
Naresh

About Your Stay With Your Uncle And Aunt

Question 9.
Write a letter to your mother, telling her about your stay with your uncle and aunt. You are Rajni and you live at Mansarovar Hostel, Vanketeshwara College, Karnal.
Answer:
Mansarovar Hostel
Vanketeshwara College
Karnal
20 June 20 _ _
Dear Mother
During my last holidays, I visited Uncle Rajan in Delhi. I spent there a couple of days. Aunt Neeru was very happy to have me there. She took me daily for shopping. We visited a number of historical places also. I came back on the fourth day with lots of gifts. How nice these people are !
With love
Rajni

PSEB 10th Class English Letter Writing

Congratulating Friend On Success In Examination

Question 10.
You are Vishal and you live at New Janta Nagar, Jalandhar. Write a letter to your friend, Tanuj, congratulating him on his grand success in the Matriculation Examination.
Answer:
415 New Janta Nagar
Jalandhar 20 March
20- –
My dear Tanuj
The result of the Matriculation Examination was declared today. I looked up your roll number in the gazette. You are getting a high first division. It has given me great pleasure. Please accept my heartiest congratulations on your grand success. It is really a big achievement for you. And it is all the result of your hard work.
All of us are full of joy at your success. Please convey my congratulations to your dear parents also. They must be proud of your success. I hope you will be giving a party to celebrate your success. Don’t forget to keep my share. What are your future plans ? Write to me.
Your loving friend
Vishal

Scolding Brother For Neglecting Studies

Question 11.
You are Deepak Gupta. You live with your parents at 61, G.T., Road, Rajpura. Write a letter to your younger brother, Keshav, scolding him for neglecting his studies.
Answer:
61 G.T. Road
Rajpura
20 March 20_ _
My dear Keshav
I received your result card yesterday. Your result is very poor. You fail in all the subjects. This is a matter of shame for us. Manav, you should mend your ways. You should work very hard. Engage some tutor. Leave bad company. Make friends with good boys. Stop seeing movies. Start playing games in the evening. All this will pay you in the long run.
If you do not mend your ways now, you will repent later on.
I hope you will act upon my advice.
Yours affectionately
Deepak

Important Bussiness Letters

Wrong Supply Of Books

Question 1.
Write a letter to a bookseller, complaining about the wrong supply of a books. You are Jatinder Singh, living at 21, Model Town, Nakodar.
Answer:
21 Model Town
Nakodar
26 March 20 _ _
Messrs India Book Depot Mai Heera Gate Jalandhar
Subject : Wrong Supply of Books Dear Sirs
On 15th March, I ordered 10 copies each of books, ‘Background of Music’ by H.S. Narang and ‘History of Punjab’, by A.C. Arora, remitting ₹ 2000/-, the entire amount of their price, in advance.
On opening the parcel, received this morning, I found that it contained copies of the different titles by the same authors. I regret to say that I did not place the order for these titles. I am, therefore, returning them by parcel and requesting the replacement.  I hope the replacement will be despatched at the earliest date. I also hope that I shall not be charged any additional packing or postal expenses.
Yours faithfully
Jatinder Singh

PSEB 10th Class English Letter Writing

Ordering Sports Goods

Question 2.
Write a letter ordering some sports goods to the firm, Messrs Avtar Singh and Sons at Jalandhar. You are Prabhjot, studying at Government High School, Nawan Shehar.
Answer:
Govt. High School Nawan Shehar
28 November 20_ _
Messrs Avtar Singh and Sons ’ Sports Goods Suppliers
Nakodar Road Jalandhar
Subject : Order for sports goods Dear Sirs
Kindly send the following sports goods at your earliest.
Footballs  Cosmo                      6 pcs.
Cricket      Bats English Willow One dozen
Cricket      Pads Antel               One dozen
Please check the above goods before packing. Send these goods through Satluj Transport Company. An a/c payee cheque for ₹2500/- is enclosed as advance payment.
Yours faithfully
Prabhjot Kaur

Complaint About A Tv Set

Question 3.
Ravi Sundaram of 28, Civil Lines, Ludhiana, bought a colour TV set from Messrs Ram Electronics, 14, Bazaar Marg, Ludhiana, a month ago. Now he finds that the sound is not clear and the picture changes to black and white from time to time. He writes a letter to the dealer, complaining about the same and requesting him to attend to it at the earliest. Write the letter.
Answer:
28 Civil Lines Ludhiana
28 March 20_ _
Messrs Ram Electronics
14 Bazaar Marg
Ludhiana
Subject: A recently purchased colour TV Set Dear Sirs
I bought a Venus Colour TV set from your shop against cash memo no. 1786 dated 17.3.20 and I regret to inform you that the set is not working properly, to the great dismay of each one in the house. The sound is not clear and the volume keeps fluctuating automatically. The picture also changes to black and white from time to time. The set is still under warranty, and I request you to attend to it at your earliest.
I hope you’ll enable us to watch on our TV the one-day Singer Cup cricket matches going to commence next Monday.
Thanking you
Yours faithfully
Ravi Sundram.

Cancelling An Order

Question 4.
You placed an order with Messrs Readymade Woollens, Ludhiana, for the supply of ladies cardigans, but they have delayed the execution of the order. Write a letter to them, cancelling the order. You are Nirmal Jain, Proprietor, Messrs Nirmal and Sons, Sangrur.
Answer:
Messrs Nirmal and Sons
Sangrur
5 October 20_ _
Messrs Readymade Woollens Ludhiana
Subject : Cancelling the order Sirs
We placed an order with you on 5th September for 200 pieces of ladies cardigans to be delivered by 20th September. But till now, we have received neither the goods nor any letter from you. The time of delivery has long expired and we are compelled to cancel the order, and should the goods arrive, they will now be refused.
Yours faithfully
Nirmal Jain
Proprietor

Placing An Order For Books

Question 6.
Imagine you are Rahul. You live at Muktsar. You live in Gandhi Nagar. Your house number is 765. You want to buy same books. Write a letter to Manager, Layal Book Depot, Ludhiana, ordering books of your choice.
Answer:
765, Gandhi Nagar
Muktsar
21 August 20_ _
The Manager
Layai Book Depot
Ludhiana
Subject: Placing an order for books
Dear Sir
Kindly send me one copy each of the following books by V.P.P. AU the books should be of the latest edition. I need them very urgently. Please send them at your earliest.
1. Modern English Grammar by Prof. Dharam Pal Bhanot (B.A. I)
2. History of Punjab by M.S. Mann (B.A.I)
3. Indian Political System by R.N. Duggal (B.A.I)
4. Essentials of Macro Economics by V.K. Srivastava (B.A.I)
Yours truly
Rahul Kushwaha

Complaint About A Beauty Product

Question 7.
You are Simran, living at Old Court Road, Sunam. Write a letter to Rama Herbal Cosmetics, Bengaluru, complaining about the body lotion they sent you.
36 Old Court Road Sunam
20 June 20_ _
Messrs Rama Herbal Cosmetics Bengaluru
Subject : Complaint about the body lotion sent by you Dear Sirs
Taken in by your much advertised body lotion, I had ordered three vials of it, the supply of which was received last Monday. But now I find that the lotion is very sticky and gives a very bad smell. Since this lotion is of no use to me, I want my money back. Please arrange to take back the vials and make good all the costs that have been borne by me.
Yours faithfully Simran
(Prop. Shiva Cosmetics)

PSEB 10th Class English Letter Writing

Complaint Against The Postman

Question 7.
The postman of your street is rude and irregular. Write a letter to the postmaster complaining against the conduct of the postman. You are Gurpreet Singh, living at 11, Sector 25, Model Town, Jalandhar.
11, Sector 25
Model Town ‘
Jalandhar 19 April 20 _ _
The Postmaster General Post Office Jalandhar Sir
I am very sorry to report that my letters are not properly delivered to me. Ram Lai, the postman of our area, is very careless. Often he comes very late. He does not do his duty honestly. On the outer wall of my house, I have put up a letter box. It bears my name. But the postman never cares to put my letters into this box. He often throws them in at the gate. Sometimes, he hands them to the children playing in the street. Many important letters are thus lost. Kindly look into the matter and take suitable action.
Yours faithfully
Gurpreet Singh

Important Official Letters

Suggesting New Programmes For Children

Question 1.
During summer vacation, children stick to the television most of the time. Write a letter to the Director, Doordarshan, suggesting some new programmes you would like to have for children. You are Nikhil Gupta, living at 12, Patiala Road, Nabha.
Answer:
12 Patiala Road Nabha
28 May 20- –
The Director
Doordarshan
New Delhi
Subject : Request for showing certain programmes for children
Sir
As you know, these are the leisurely days of summer vacation. The school-going children can’t play outdoors due to scorching heat. They spend most of the time in front of the television. Therefore, I request you to telecast programmes on cartoons, general knowledge and new discoveries. It would help to increase their knowledge and also use their time in a constructive manner.
I hope you will do the needful.
Thanking you in anticipation
Yours faithfully
Nikhil Gupta.

Request For Installation Of Traffic Lights

Question 2.
Write a letter to the Superintendent of Police (Traffic), Chandigarh, requesting him to install traffic lights at the crossing near your school. You are Raj an Sethi, a student of class Xllth, DA.V. School, Chandigarh.
Answer:
D.A.V. School Chandigarh
28 September 20
The Superintendent of Police (Traffic)
Chandigarh
Subject : Installation of traffic lights at the crossing Sir .
There are no traffic lights at the crossing near our school. The traffic is very fast there. There is a grave danger to the life and limb of the students.
I, therefore, request you to kindly arrange traffic lights at the crossing.
Hoping for an immediate action
Yours faithfully
Rajan Sethi
Class XII

PSEB 10th Class English Letter Writing

Request For A Study Loan

Question 3.
Write a letter to the manager of a bank, requesting for a loan for higher studies. You are Nikhil, living at 34, Krishna Nagar, Patiala.
Answer:
34 Krishna Nagar Patiala
28 November 20
The Manager Punjab National Bank Patiala
Subject : Request for a study loan Dear Sir
I read your advertisement in Hindustan Times yesterday. I have cleared the CAT examination and’have been selected for IIM, Ahmedabad. My father is a government teacher. The entire family banks upon him alone. So it is quite impossible for him to pay the fees for my course which is about ₹5 lac. I request you to sanction me a plan of this much amount. I undertake you to return the amount as per the conditions given in the advertisement.
Yours faithfully
Nikhil

Complaint About Insanitation

Question 4.
Imagine you are Raman. You live in Gali Ram Nath, Malerkotla. Write a letter to the Health Officer of your town complaining about the insanitary condition of your street.
Answer:
Gali Ram Nath
Malerkotla
16 March 20 _ _
The Health Officer Malerkotla
Subject : Insanitary condition of our street Sir
I beg to draw your attention to the insanitary condition of Gali Ram Nath in this city. Mainly poor people live in this area. Perhaps that is why no Sanitary Inspector
has ever visited it. There are very few proper drains here. The drains are not cleaned regularly. These are never flushed with water. No dustbins have been provided. People throw all their refuse here and there. Flies and mosquitoes buzz about. All this gives this area a very dirty look.
I hope you will take suitable measures to improve the sanitary condition of this area.
Yours faithfully
Raman

About Environmental Pollution

Question 5.
You are Raghu, staying at 12, Balmik Colony, Gurdaspur. Write a letter to the Editor of a newspaper to create awareness among the masses about pollution hazards.
Answer:
Balmik Colony Gurdaspur
March 20- –
The Editor The Tribune
Chandigarh
Subject : Environmental pollution
Sir
Environmental pollution is the biggest problem facing the modern man. Only recently has our government become aware of the gravity of this situation. More attention is being paid to afforestation. The unauthorised felling of trees is being checked. Scientists are developing methods to minimise the effect of smoke let out by our autos and chimneys. In fact, environmental awareness is a social necessity. It is not only the duty of the government, but also a social responsibility of the masses to keep the environment free from pollution.
Yours truly
Raghu

About The Problem Of Begging

Question 6.
You are Ramesh, living at 86, Sunam Road, Mansa. Write a letter to the Editor of a newspaper suggesting how the problem of begging can be solved.
Answer:
Adarsh Vihar Dwarka 27 April 20-
The Editor
The Indian Express
Delhi
Subject : The problem of begging Sir
The problem of beggars has assumed an alarming proportion in the city. Some beggars, no doubt, deserve our sympathy. They are handicapped. But the pity is that most of the beggars are able-bodied. Begging has become a profession for them. Some of them are criminals. They beg only to hide their crimes.
Begging should be abolished by law. Able-bodied beggars should be forced to work. If they go without work, they must also go without food. We should not encourage them by giving alms. People can thus play a big role in ending this evil.
Yours truly
Ramesh

PSEB 10th Class English Letter Writing

Application For The Job Of A Computer Operator

Question 7.
Write an application for employment in Hindustan Limited, New Delhi, as a computer operator. You are Manu Gupta, living at 811, Sector 15, Rohini, New Delhi.
Answer:
811 Sector 15
Rohini New Delhi
25 July 20_ _
The General Manager Hindustan Limited
New Delhi
Subject : Application for the job of a computer operator
Sir
With reference to your advertisement in The Times of India, dated 20 July 20
I want to apply for the job of a computer operator in your company. As regards my qualifications, I have done Diploma in Computer Applications (DCA) after doing 10+2 from the CBSE. I have served for three years with a travel agency and am fairly good at all basic operations related to computer. If given a chance, I am sure you will feel satisfied with my services.
Yours faithfully
Manu Gupta

PSEB 10th Class Maths Solutions Chapter 3 Pair of Linear Equations in Two Variables Ex 3.7

Punjab State Board PSEB 10th Class Maths Book Solutions Chapter 3 Pair of Linear Equations in Two Variables Ex 3.7 Textbook Exercise Questions and Answers.

PSEB Solutions for Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables Ex 3.7

Question 1.
The ages of two friends Ani and Biju differ by 3 years. Ani’s father Dharam is twice as old as Am and Biju Is twice as old as his sister Cathy. The ages of Cathy and Dharam differ by 30 years. Find the ages of Ani and Biju.
Solution:
Let Ani’s age = x years
and Biju’s age = y years
Dharam’s age = 2x years
Cathy’s age = years
According to 1st condition,
(Ani’s age) (Biju’s age) = 3
x – y = 3 ……………(1)
According to 2nd condition,
(Dharam’ age) – (Cathy’s age) = 30
2x – \(\frac{y}{2}\) = 30
or \(\frac{4 x-y}{2}\) = 30
or 4x – y = 60 ………….(2)
Now (2) – (1) gives,

PSEB 10th Class Maths Solutions Chapter 3 Pair of Linear Equations in Two Variables Ex 3.7 1

Substitute this value of x in (1), we get:
19 – y = 3
or -y = 3 – 19
or -y = -16
or y = 16
Hence, Ani’s age = 19 years
Biju’s age =16 years.

PSEB Solutions PSEB 10th Class Maths Solutions Chapter 3 Pair of Linear Equations in Two Variables Ex 3.7

Question 2.
One says, “Give me a hundred, friend! I shall then become twice as rich as you”. The other replies, “if you give me ten, I shall be six times as rich as you.” Tell me what is the amount of their (respective) capital ? [From the Bijaganita of Bhaskara II] [Hint: x + 100 = 2(y – 100), y + 10 = 6 (x – 10)].
Solution:
Let Capital of one friend = ₹ x
and capital of 2nd friend = ₹ y
According to 1st condition
x + 100 = 2(y – 100)
or x + 100 = 2y – 200
or x – 2y = -200 – 100
or x – 2y = -300 …………..(1)
According to 2nd condition
y + 10 = 6(x – 10)
or v-f 10 = 6x – 60
or 6x – y = 10 + 60
or 6x – y = 70 ……………(2)
Multiplying (1) by 6, we get
6x – 12y = – 1800 …………….(3)
Now, (3) – (2) gives

PSEB 10th Class Maths Solutions Chapter 3 Pair of Linear Equations in Two Variables Ex 3.7 2

Substitute this value of y in (2), we get:
6x – 170 = 70
or 6x = 70 + 170
or 6x = 240
or x = \(\frac{240}{6}\) = 40
Hence, amount of their capital are 40 and 170 respectively.

PSEB Solutions PSEB 10th Class Maths Solutions Chapter 3 Pair of Linear Equations in Two Variables Ex 3.7

Question 3.
A train covered a certain distance at a uniform speed. If the train would have been 10 km/b faster, It would have taken 2 hours less than the scheduled time. And, if the train were slower by 10 km/h; it would have taken 3 hours more than the scheduled time. Find the distance covered by the train.
Solution:
Let speed of train =x km/hour
and time taken by train =y hour
∴ Distance covered by train = (Speed) (Time) = (xy) km
According to 1st condition,
(x + 10)(y – 2) = xy
or xy – 2x + 10y – 20 = y
or -2x + 10y – 20 = 0
or x – 5y + 10 = 0
According to 2nd condition,
(x – 10) (y + 3) = xy
or xy + 3x – 10y – 30 = xy
or 3x – 10y – 30 = 0
Multiplying (1) by 3, we get:
3x – 15y + 30 = 0
Now, (3) – (2) gives

PSEB 10th Class Maths Solutions Chapter 3 Pair of Linear Equations in Two Variables Ex 3.7 3

Substitute this value of y in (1), we get:
x – 5 × 12 + 10 = 0
or x – 60 + 10 = 0
or x – 50 = 0
or x = 50
Speed of train = 50 km/hour
Time taken by train = 12 hour
Hence, distance covered by train = (50 × 12) km = 600 km.

PSEB Solutions PSEB 10th Class Maths Solutions Chapter 3 Pair of Linear Equations in Two Variables Ex 3.7

Question 4.
The students of a class are made to stand in rows. If 3 students are extra in a row, there would be 1 row less. If 3 students are less in a row, there would be 2 rows more. Find the number of students in the class.
Solution:
Let number of students in each row = x
and number of rows = y
Total number of students in the class = xy
According to 1st condition,
(x + 3) (y – 1) = xy
or xy – x + 3y – 3 = xy
or -x + 3y – 3 = 0
or x – 3y + 3 = 0
According to 2nd condition,
(x -3) (y + 2) = xy
or xy + 2x – 3y – 6= xy
or 2x – 3y – 6 = 0
Now, (2) – (1) gives

PSEB 10th Class Maths Solutions Chapter 3 Pair of Linear Equations in Two Variables Ex 3.7 4

x = 9
Substitute this value of x in (1), we get:
9 – 3y + 3 = 0
or -3y + 12 = 0
or -3y = -12
or y = \(\frac{12}{3}\) = 4
∴ Number of students in each row = 9 and number of rows = 4
Hence, total number of students in the class = 9 × 4 = 36.

PSEB Solutions PSEB 10th Class Maths Solutions Chapter 3 Pair of Linear Equations in Two Variables Ex 3.7

Question 5.
In a ∆ABC, ∠C = 3 ∠B = 2(∠A +∠B) find the three angles.
Solution:
In ∆ABC,
Given that, ∠C = 3∠B = 2(∠A + ∠B)
I II III
From II and III, we get:
3∠B =2(∠A+∠B)
or 3∠B = 2∠A + 2∠B
or 3∠B – 2∠B = 2∠A
or ∠B = 2∠A ……………..(1)
From I and II, we get:
∠C = 3∠B
or ∠C = 3(2∠A) [using (1)]
or ∠C = 6∠A …………….(2)
Sum of three angles of a triangle is 180°
∠A + ∠B + ∠C = 180°
or ∠A + 2∠A + 6∠A = 180°
or 9∠A = 180°
∠A = \(\frac{180^{\circ}}{9}\) = 20°
Hence, ∠A = 20°: ∠B =2 x 20° = 40°; ∠C = 6 x 20° = 120°.

PSEB Solutions PSEB 10th Class Maths Solutions Chapter 3 Pair of Linear Equations in Two Variables Ex 3.7

Question 6.
Draw the graphs of the equations 5x – y = 5 and 3x – y = 3. Determine the co-ordinates of the vertices of the triangle
formed by these lines and the y axis.
Solution:
Given pair of linear equation are 5x – y = 5 and 3x – y = 3
Consider,
5x – y = 5
or 5x = 5 + y
Putting y = 0 in (1), we get:
x = \(\frac{5+0}{5}=\frac{5}{5}\)
Putting y = – 5 in (1), we get:
x = \(\frac{5-5}{5}=\frac{0}{5}\) = 0
Putting y = 5 in (1), we get:
x = \(\frac{5+5}{5}=\frac{10}{5}\) = 2

PSEB 10th Class Maths Solutions Chapter 3 Pair of Linear Equations in Two Variables Ex 3.7 5

Plotting A (1, 0); B (0, — 5); C (2, 5) on the graph, we get the equation of line 5x — y = 5
and 3x – y = 3
or 3x = 3 + y
or x = \(\frac{3+y}{3}\) ……………(2)
Putting y = 0 in (2), we get:
x = \(\frac{3+0}{3}=\frac{3}{3}\) = 1
Putting y = 3 in (2), we get:
x = \(\frac{3-3}{3}=\frac{0}{3}\) = 0
Putting y = 3 in (2), we get:
x = \(\frac{3+3}{3}=\frac{6}{3}\) = 2
Table:

PSEB 10th Class Maths Solutions Chapter 3 Pair of Linear Equations in Two Variables Ex 3.7 6

PSEB 10th Class Maths Solutions Chapter 3 Pair of Linear Equations in Two Variables Ex 3.7 7

Plotting A (1, 0); D (0, -3); E (2, 3) on the graph. we get the equation of line 3x – y = 3. From the graph, it is clear that given lines intersect at A (1, 0). Triangle formed by these lines and y axis are shaded in the graph i.e. ∆ABD. Coordinates of the vertices of ∆ABD are A(1, 0); B(0, -5) and D(0, -3).

PSEB Solutions PSEB 10th Class Maths Solutions Chapter 3 Pair of Linear Equations in Two Variables Ex 3.7

Question 7.
Solve the following pair of linear equations:
(i) px + qy = p – q
qx – py = p + q

(ii) ax + by = c
bx + ay = 1 + c

(iii) \(\frac{x}{a}-\frac{y}{b}\) = 0
ax + by = a2 + b2

(iv) (a – b)x + (a + b)y = a2 – 2ab – b2
(a + b) (x + y) = a2 + b2

(v) 152x – 378y = 74
-378x + 152y = -604
Solution:
(i) Given pair of linear equation are
px + qy = p – q …………(1)
and qx – py = p + q ………….(2)
Multiplying (1) by q and (2) by p, we get:

PSEB 10th Class Maths Solutions Chapter 3 Pair of Linear Equations in Two Variables Ex 3.7 8

or y = -1
Substitute this value of y in (1), we get:
px + q(-1) = p – q
or px – q = p – q
or px = p – q + q
or px = p
or x = 1
Hence, x = 1 and y = -1.

PSEB Solutions PSEB 10th Class Maths Solutions Chapter 3 Pair of Linear Equations in Two Variables Ex 3.7

(ii) Given pair of linear equation are
ax + by = c
bx + ay = 1 + c
ax + by – c = 0
bx + ay – (1 + c) = 0

PSEB 10th Class Maths Solutions Chapter 3 Pair of Linear Equations in Two Variables Ex 3.7 9

From I and III, we get:
\(\frac{x}{-b-b c+a c}=\frac{1}{a^{2}-b^{2}}\)
or x = \(\frac{a c-b c-b}{a^{2}-b^{2}}\)

From II and III, we get:
\(\frac{y}{-b c+a+a c}=\frac{1}{a^{2}-b^{2}}\)
or y = \(\frac{a+a c-b c}{a^{2}-b^{2}}\)
Hence, x = \(\frac{a c-b c-b}{a^{2}-b^{2}}\) and y = \(\frac{a+a c-b c}{a^{2}-b^{2}}\).

(iii) Given pair of linear equation are
\(\frac{x}{a}-\frac{y}{b}\) = 0
or \(\frac{b x-a y}{a b}\) = 0
or bx – ay = 0
or bx—ay = O …(1)
and ax + by = a2 + b2
or ax + by – (a2 + b2) = 0 …………..(2)

PSEB 10th Class Maths Solutions Chapter 3 Pair of Linear Equations in Two Variables Ex 3.7 10

\(\frac{x}{a\left(a^{2}+b^{2}\right)-0}=\frac{y}{0+b\left(a^{2}+b^{2}\right)}=\frac{1}{b^{2}+a^{2}}\)

or \(\frac{x}{a\left(a^{2}+b^{2}\right)}=\frac{y}{b\left(a^{2}+b^{2}\right)}=\frac{1}{a^{2}+b^{2}}\)

or \(\frac{x}{a}=\frac{y}{b}=\frac{1}{1}\)
I II III
From I and III, we get:
\(\frac{x}{a}=\frac{1}{1}\)
⇒ x = a
From II and III, we get:
\(\frac{y}{b}=\frac{1}{1}\)
⇒ y = b
Hence, x = a, y = b.

PSEB Solutions PSEB 10th Class Maths Solutions Chapter 3 Pair of Linear Equations in Two Variables Ex 3.7

(iv) Given pair of linear equation are
(a – b)x + (a + b)y = a2 – 2ab – b2
or ax – bx + ay + by = a2 – 2ab – b2 …………….(1)
and (a + b) (x + y) = a2 + b2
or ax + bx + ay + by = a2 + b2
Now, (1) – (2) gives

PSEB 10th Class Maths Solutions Chapter 3 Pair of Linear Equations in Two Variables Ex 3.7 11

Substitute this value of x in (1), we get:
(a – b) (a + b) + (a + b) y = a2 – 2ab – b2
or a2 – b2 + (a + b) y = a2 – 2ab – b2
or (a + b) y = a2 – 2ab – b2 – a2 + b2
or (a + b)y = -2ab
or y = \(\frac{-2 a b}{a+b}\)
Hence, x = a + b and y = a + b

(v) Given pair of linear equation are
152x – 378y = – 74
and -378x + 152y = -604
or 76x – 189y + 37 = 0
and -189x + 76y + 302 = 0

PSEB 10th Class Maths Solutions Chapter 3 Pair of Linear Equations in Two Variables Ex 3.7 12

From I and III, we get:
\(\frac{x}{59890}=\frac{1}{29945}\)
⇒ x = \(\frac{59890}{29945}\)
⇒ x = 2

From II and III, we get:
\(\frac{y}{29945}=\frac{1}{29945}\)
⇒ y = \(\frac{29945}{29945}\)
⇒ y = 1
Hence x = 2 and y = 1.

PSEB Solutions PSEB 10th Class Maths Solutions Chapter 3 Pair of Linear Equations in Two Variables Ex 3.7

Question 8.
ABCD is a cyclic quadrilateral (see Fig.). Find the angles of the cyclic quadrilateral.

PSEB 10th Class Maths Solutions Chapter 3 Pair of Linear Equations in Two Variables Ex 3.7 13

Solution:
In cyclic quadrilateral ABCD,
∠A = (4y + 20); ∠B = 3y – 5; ∠C = 4x and ∠D = 7x + 5
Sum of opposite angles of a cyclic quadrilateral are of measure 180°.
∴ ∠A + ∠C = 180°
or 4y + 20 + (4x) = 180°
or 4x + 4y = 180° – 20
or 4x + 4y = 160
or x + y = 40
or y = 40 – x ……………..(1)
and ∠B + ∠D = 180°
or 3y – 5 + (7x + 5)= 180°
or 3y – 5 + 7x + 5 = 180°
or 7x + 3y = 180° …………….(2)
Substitute the value of y from (1) in (2). we get:
7x + 3(40 – x)= 180°
or 7x + 120 – 3x = 180°
or 4x = 180 – 120
or 4x = 60
x = \(\frac{60}{4}\) = 15
Substitute this value of x in (1 ), we get:
y = 40 – 15 = 25
∴ ∠A = 4y + 20 = 4 × 25 + 20 = 120°
∠B = 3y – 5 = 3 × 25 – 5 = 70°
∠C = 4x =4 × 15 = 60°
∠D = 7x + 5 = 7 × 15 + 5 = 110°
Hence, ∠A = 120°, ∠B = 70°; ∠C = 60° and ∠D = 110°.

PSEB 10th Class Maths Solutions Chapter 1 Real Numbers Ex 1.2

Punjab State Board PSEB 10th Class Maths Book Solutions Chapter 1 Real Numbers Ex 1.1 Textbook Exercise Questions and Answers.

PSEB Solutions for Class 10 Maths Chapter 1 Real Numbers Ex 1.2

Question 1.
Express each number as a product of its prime factors :
(i) 140 (Pb, 2019)
(ii) 156
(iii) 3825
(iv) 5005 (Pb. 2019, Set-1, II, III)
(v) 7429.
Solution:
(i) Prime factorisation of 140 = (2)2 (35) = (2)2 (5) (7)

(ii) Prime factorisation of 156 = (2)2 (39) = (2)2 (3) (13)

(iii) Prime factorisation of 3825 = (3)2 (425)
= (3)2 (5) (85)
= (3)2 (5)2 (17)

(iv) Prime factorisation of 5005
= (5) (1001)
= (5) (7) (143)
= (5) (7) (11) (13)

(v) Prime factorisation of 7429
= (17) (437)
= (17) (19) (23)

PSEB Solutions PSEB 10th Class Maths Solutions Chapter 1 Real Numbers Ex 1.2

Question 2.
Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = Product of the two numbers.
(i) 26 and 91 [Pb. 2017 Set-C]
(ii) 510 and 92
(iii) 336 and 54.
Solution:
(i) Given numbers are 26 and 91 Prime factorisation of 26 and 91 are
26 = (2) (13) and
91 = (7) (13)
HCF (26, 91)
Product of least powers of common factors
∴ HCF (26, 91) = 13
and LCM (26, 91) = Product of highest powers of all the factors
= (2) (7) (13) = 182
Verification :
LCM (26, 91) × HCF (26, 91)
= (13) × (182) = (13) × (2) × (91)
= (26) × (91)
= Product of given numbers.

(ii) Given numbers are 510 and 92 Prime factorisation of 510 and 92 are
510 = (2) (255)
= (2) (3) (85)
= (2) (3) (5) (17)
and 92 = (2) (46) = (2)2 (23)
HCF (510, 92) = Product of least powers of common factors = 2
LCM (510, 92) = Product of highest Powers of all the factors
= (2)2 (3) (5) (17) (23) = 23460

Verification:
LCM (510, 92) × HCF (510, 92)
= (2) (23460)
= (2) × (2)2 (3) (5) (17) (23)
= (2) (3) (5) (17) × (2)2 (23)
= 510 × 92 = Product of given numbers.

PSEB Solutions PSEB 10th Class Maths Solutions Chapter 1 Real Numbers Ex 1.2

(iii) Given numbers are 336 and 54
Prime factorisation of 336 and 54 are
336 = (2) (168)
= (2) (2) (84)
= (2) (2) (2) (42)
= (2) (2) (2) (2) (21)
= (2)4 (3) (7)

and 54 = (2) (27)
= (2) (3) (9)
= (2) (3) (3) (3)
= (2) (3)3
HCF (336, 54) = Product of least powers of common factors = (2) (3) = (6)
LCM (336, 54) = Product of highest powers of all the factors
= (2)4 (3)3 (7) = 3024

Verification :
LCM (336, 54) × H.C.F. (336, 54)
= 6 × 3024
= (2) (3) × (2)4 (3)3 (7)
= (2)4 (3) (7) × (2) (3)3
= 336 × 54
= Product of given numbers.

Question 3.
Find the LCM and HCF of the following integers by applying the prime factorisation method.
(i) 12, 15 and 21 [MQP 2015, Pb. 2015 Set A, Set B, Set C]
(ii) 17, 23 and 29
(iii) 8, 9 and 25
Solution:
(i) Given numbers are 12, 15 and 21
Prime factorisation of 12, 15 and 21 are
12 = (2) (6)
= (2) (2) (3)
= (2)2 (3)
15 = (3) (5)
21 = (3) (7)
HCF (12, 15 and 21) = 3
LCM (12, 15 and 21) = (2)2 (3) (5) (7) = 420

PSEB Solutions PSEB 10th Class Maths Solutions Chapter 1 Real Numbers Ex 1.2

(ii) Given numbers are 17, 23 and 29
Prime factorisation of 17, 23 and 29 are
17 = (17) (1)
23 = (23) (1)
29 = (29) (1)
HCF (17, 23 and 29) = 1
LCM (17, 23 and 29)
= 17 × 23 × 29 = 11339

(iii) Given numbers are 8, 9 and 25
Prime factorisation of 8, 9 and 25 are
8 = (2) (4)
= (2) (2) (2)
= (2)3 (1)
9 = (3) (3) = (3)2 (1)
25 = (5) (5) = (5)2 (1)
HCF (8, 9 and 25) = 1
LCM (8, 9 and 25) = (2)3 (3)2 (5)2 = 1800

Question 4.
Prove that HCF (306,657) = 9, find LCM (306,657). [Pb. 2016 Set-B, 2017 Set-B]
Solution:
Given numbers are 306 and 657 Prime factorisation of 306 and 657 are
306 = (2) (153)
= (2) (3) (51)
= (2) (3) (3) (17)
= (2) (3)2 (17)

657 = (3) (219)
= (3) (3) (73)
= (3)2 (73)

HCF (306, 657) = (3)2 = 9
∵HCF × LCM = Product of given number
∵9 × LCM (306, 657) = 306 × 657

PSEB 10th Class Maths Solutions Chapter 1 Real Numbers Ex 1.2 1

= 34 × 657 = 22338

PSEB Solutions PSEB 10th Class Maths Solutions Chapter 1 Real Numbers Ex 1.2

Question 5.
Check whether 6n can end with the digit 0 for any natural number n:
Solution:
Let us suppose that 6n ends with the digit 0 for some n ∈ N.
6n is divisible by 5.
But, prime factor of 6 are 2 and 3 Prime factor of (6)n are (2 × 3)n
⇒ It is clear that in prime factorisation of 6n there is no place for 5.
∵ By Fundamental Theorem of Arithmetic, Every composite number can be expressed as a product of primes and this factorisation is unique, apart from the order in which the prime factors occur.
∴ Our supposition is wrong.
Hence, there exists no natural number n for which 6n ends with the digit zero.

Question 6.
Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers.
Solution:
Consider, 7 × 11 × 13 + 13 = 13 [7 × 11 + 1]
which is not a prime number because it has a factor 13. So, it is a composite number.
Also, 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5
= 5[7 × 6 × 5 × 4 × 3 × 2 × 1 + 1], which is not a prime number because it has a factor 5. So it is a composite number.

PSEB Solutions PSEB 10th Class Maths Solutions Chapter 1 Real Numbers Ex 1.2

Question 7.
There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for die same. Suppose they both start at the same point and at file same time, and go in the same direction. After how many minutes will they meet again at the starting point?
Solution:
Time taken by Sonia to drive one round of the field =18 minutes
Time taken by Ravi to drive one round of same field = 12 minutes
They meet again at the starting point = LCM (18, 12)
Now, Prime factorisation of 18 and 12 are 18 = (2) (9)
= (2) (3) (3)
= (2) (3)2

12 = (2) (6)
= (2) (2) (3)
= (2)2 (3)
LCM (18, 12) = (2)2 (3)2 = 4 × 9 = 36
Hence, After 36 minutes Sonia and Ravi will meet again at the starting point.