Class 9 Maths Chapter 1 Number Systems Ex 1.1

 

Class 9 Maths Chapter 1 Number Systems Ex 1.1

Ex 1.1 Class 9 Maths Question 1.
Is zero a rational number? Can you write it in the form pq,where p and q are integers and q ≠0?
Solution:
Yes, zero is a rational number it can be written in the form pq.
0 = 01 = 02 = 03 etc. denominator q can also be taken as negative integer.

Ex 1.1 Class 9 Maths Question 2.
Find six rational numbers between 3 and 4.
Solution:
Let q_i be the rational number between 3
and 4, where j = 1 to 6.
∴ Six rational numbers are as follows:

Thus, the six rational numbers between 3 and 4 are

Ex 1.1 Class 9 Maths Question 3.
Find five rational numbers between 35 and 45.
Solution:
Since, we need to find five rational numbers, therefore, multiply numerator and denominator by 6.

Ex 1.1 Class 9 Maths Question 4.
State whether the following statements are true or false. Give reasons for your answers.
(i) Every natural number is a whole number.
(ii) Every integer is a whole number.
(iii) Every rational number is a whole number.
Solution:
(i) True
∵ The collection of all natural numbers and 0 is called whole numbers.
(ii) False
∵ Negative integers are not whole numbers.
(iii) False
∵ Rational numbers are of the form p/q, q ≠ 0 and q does not divide p completely that are not whole numbers.

English Grammar: Future continuous tense

 

Future continuous tense:

Affirmative Sentences:

Subject + will be + verb(ing)

Examples:

• I will be working.

• She will be studying.

• They will be traveling.

Negative Sentences:

Subject + will not be + verb(ing)

OR

Subject + won't be + verb(ing)

Examples:

• I will not be working.

• She won't be studying.

• They will not be traveling.

Interrogative Sentences:

Formula:

Will + subject + be + verb(ing)?

Examples:

• Will I be working?

• Will she be studying?

• Will they be traveling?

Formula:

( Who/ what/ where/ when/ why/ how) + will + subject + be + verb(ing)?

1. What will you be doing tomorrow at 3 PM?

2. Where will she be staying during her vacation?

3. Why will they be moving to a new city?

4. Who will be using the conference room next?

5. When will we be meeting the new manager?

6. How will he be getting to work next week?

7. Which project will you be working on this afternoon?

8. What time will they be arriving at the airport?

Fill-in-the-blank questions focusing on interrogative sentences in the future continuous tense:

1. What _____ you _____ (do) tomorrow at noon?

2. Where _____ she _____ (live) next year?

3. Why _____ they _____ (move) to another country?

4. Who _____ he _____ (meet) at the conference?

5. When _____ we _____ (leave) for the trip?

6. How _____ she _____ (travel) to the office next week?

7. Which subject _____ he _____ (study) this semester?

8. What time _____ they _____ (arrive) at the party?

9. Where _____ you _____ (work) during the summer?

10. Why _____ you _____ (wear) a suit tomorrow?

Answer:

1. What will you be doing tomorrow at noon?

2. Where will she be living next year?

3. Why will they be moving to another country?

4. Who will he be meeting at the conference?

5. When will we be leaving for the trip?

6. How will she be traveling to the office next week?

7. Which subject will he be studying this semester?

8. What time will they be arriving at the party?

9. Where will you be working during the summer?

10. Why will you be wearing a suit tomorrow?

English Grammar: Simple future tense

 

Simple future tense:

Formula:

Subject+will/shall+Base Form of Verb+Rest of the Sentence

Structure:

1. Subject: The person or thing performing the action.

2. Will/Shall: The modal auxiliary verbs used to indicate the future tense. "Will" is more commonly used, while "shall" is used with "I" and "we" to indicate a stronger intention or promise.

3. Base Form of Verb: The infinitive form of the main verb without "to".

4. Rest of the Sentence: Additional information about the action.

Examples:

1. I (subject) will visit (will + visit) my parents tomorrow (rest of the sentence).

   • I will visit my parents tomorrow.

2. She (subject) shall call (shall + call) you later (rest of the sentence).

   • She shall call you later. (Note: "Shall" is less commonly used in modern English except in formal or specific contexts.)

3. We (subject) will go (will + go) to the beach next weekend (rest of the sentence).

   • We will go to the beach next weekend.

Fill-in-the-blank sentences in the simple future affirmative tense with verbs in brackets:

1. She _______ (attend) the conference next week.

2. They _______ (travel) to Paris in July.

3. He _______ (start) his new job on Monday.

4. We _______ (go) hiking this weekend.

5. The company _______ (launch) a new product next month.

6. I _______ (visit) my grandparents next Sunday.

7. They _______ (buy) a new car next year.

8. She _______ (graduate) from college in June.

9. He _______ (finish) his project by Friday.

10. We _______ (celebrate) our anniversary with a trip.

Answer:

1. She will attend the conference next week.

2. They will travel to Paris in July.

3. He will start his new job on Monday.

4. We shall go hiking this weekend.

5. The company will launch a new product next month.

6. I shall visit my grandparents next Sunday.

7. They will buy a new car next year.

8. She will graduate from college in June.

9. He will finish his project by Friday.

10. We shall celebrate our anniversary with a trip.

1. 
   

Negative Form:

To form the negative simple future tense, add "not" after "will" or "shall".

Subject+will/shall+not+Base Form of Verb+Rest of the Sentence

Examples:

1. They (subject) will not come (will + not + come) to the party (rest of the sentence).

   • They will not come to the party.

2. He (subject) shall not forget (shall + not + forget) your birthday (rest of the sentence).

   • He shall not forget your birthday.

Fill-in-the-blank sentences in the simple future negative tense, with verbs in brackets:

1. She ____________ (attend) the party tomorrow night.

2. They ____________ (visit) their relatives next weekend.

3. He ____________ (complete) the project by the end of this month.

4. We ____________ (go) on vacation next summer.

5. I ____________ (buy) a new car next year.

6. They ____________ (finish) their homework before dinner.

7. She ____________ (call) you later tonight.

8. He ____________ (start) his new job until next month.

9. We ____________ (see) each other next week.

10. They ____________ (invite) anyone to the party on Saturday.

Answer:

1. She will not attend the party tomorrow night.

2. They will not visit their relatives next weekend.

3. He will not complete the project by the end of this month.

4. We shall not go on vacation next summer.

5. I shall not buy a new car next year.

6. They will not finish their homework before dinner.

7. She will not call you later tonight.

8. He will not start his new job until next month.

9. We will not see each other next week.

10. They will not invite anyone to the party on Saturday.

Interrogative Form:

To form questions in the simple future tense, invert the subject and "will/shall".

Will/Shall+subject+Base Form of Verb+Rest of the Sentence ?

 

Examples:

1. Will (will) you (subject) come (base form of "come") to the meeting? (rest of the sentence)

   • Will you come to the meeting?

2. Shall (shall) we (subject) meet (base form of "meet") at 5 o'clock? (rest of the sentence)

   • Shall we meet at 5 o'clock?
     

When forming interrogative sentences in the simple future tense using a WH-word ( Who, what, where, when, why, or how), the formula is:

( Who/ what/ where/ when/ why/ how) + will + subject + base form of the verb + ?

Here's the breakdown:

- WH-word: Who, what, where, when, why, or how.

- Will: The auxiliary verb indicating future tense.

- Subject: The person or thing performing the action.

- Base form of the verb: The main verb in its infinitive form without "to."

- ?: A question mark to indicate the interrogative nature of the sentence.

Examples:

- What will you do tomorrow?

- Where will they go for vacation?

- When will she call me?

- Why will he leave early?

- How will we get to the airport?

- Who will attend the meeting?

Fill-in-the-blank questions focusing on interrogative sentences in the simple future tense using WH-words:

1. Where _____ you go for your vacation next year?

2. What _____ she cook for dinner tomorrow?

3. When _____ they arrive at the airport?

4. Why _____ he need a new computer?

5. How _____ we get to the museum from here?

6. Who _____ be the new manager of the team?

7. Which book _____ you read next?

8. What time _____ the meeting start?

9. Where _____ they hold the conference?

10. How long _____ it take to complete the project?

11. Why _____ she move to another city?

12. Who _____ you invite to the party?

13. What _____ you do if it rains tomorrow?

14. When _____ the new movie be released?

15. How _____ they celebrate their anniversary?

16. Which university _____ he apply to for his master’s degree?

17. Where _____ you buy your new car?

18. What _____ you say in your speech?

19. When _____ we know the results of the election?

20. Who _____ win the championship game?

Answers:

1. Where will you go for your vacation next year?

2. What will she cook for dinner tomorrow?

3. When will they arrive at the airport?

4. Why will he need a new computer?

5. How will we get to the museum from here?

6. Who will be the new manager of the team?

7. Which book will you read next?

8. What time will the meeting start?

9. Where will they hold the conference?

10. How long will it take to complete the project?

11. Why will she move to another city?

12. Who will you invite to the party?

13. What will you do if it rains tomorrow?

14. When will the new movie be released?

15. How will they celebrate their anniversary?

16. Which university will he apply to for his master’s degree?

17. Where will you buy your new car?

18. What will you say in your speech?

19. When will we know the results of the election?

20. Who will win the championship game?

 

 

 

English Grammar: Past perfect continuous tense

 

Past perfect continuous tense:

Formula:

Subject+had been+Verb(ing) +Rest of the Sentence

Structure:

1. Subject: The person or thing performing the action.

2. Had been: The past perfect form of "to be" (had been).

3. Verb-ing (Present Participle): The main verb in its present participle form (verb + -ing).

4. Rest of the Sentence: Additional information about the action.

Examples:

1. They (subject) had been waiting (had been + waiting) for hours (rest of the sentence).

   • They had been waiting for hours.

2. She (subject) had been studying (had been + studying) all night (rest of the sentence).

   • She had been studying all night.

3. We (subject) had been working (had been + working) on the project since morning (rest of the sentence).

   • We had been working on the project since morning.

Fill-in-the-blank sentences in the past perfect continuous affirmative form, with the answers provided after all questions:

1. By the time we arrived, she __________ (wait) for two hours.

2. They were tired because they __________ (work) all day.

3. He looked exhausted because he __________ (run) for an hour.

4. She had a sore throat because she __________ (sing) all night.

5. The garden was beautiful because they __________ (water) it regularly.

6. By the time I got there, they __________ (play) football for three hours.

7. She understood the topic well because she __________ (study) it extensively.

8. He was out of breath because he __________ (run) for so long.

9. The cake was delicious because she __________ (bake) it with care.

10. By the time they left, we __________ (discuss) the issue for hours.

Answers:

1. By the time we arrived, she had been waiting for two hours.

2. They were tired because they had been working all day.

3. He looked exhausted because he had been running for an hour.

4. She had a sore throat because she had been singing all night.

5. The garden was beautiful because they had been watering it regularly.

6. By the time I got there, they had been playing football for three hours.

7. She understood the topic well because she had been studying it extensively.

8. He was out of breath because he had been running for so long.

9. The cake was delicious because she had been baking it with care.

10. By the time they left, we had been discussing the issue for hours.

Negative Form:

Subject+had not been+Verb(ing) +Rest of the Sentence

Examples:

1. He (subject) had not been sleeping (had not been + sleeping) well (rest of the sentence).

   • He had not been sleeping well.

2. They (subject) hadn't been practicing (had not been + practicing) for long (rest of the sentence).

   • They hadn't been practicing for long.

Fill-in-the-blank sentences in the past perfect continuous negative form, with the answers provided after all questions:

1. She ________ (not study) for the exam before she failed it.

2. They ________ (not play) basketball before they broke their leg.

3. He ________ (not eat) before he felt sick.

4. We ________ (not wait) for her when she arrived.

5. I ________ (not work) on the project before it was due.

6. She ________ (not sleep) well before the exam.

7. They ________ (not practice) for the concert before they performed.

8. You ________ (not clean) the house before the guests came.

9. He ________ (not exercise) regularly before he got injured.

10. We ________ (not talk) about the issue before it became a problem.

Answers:

1. She had not been studying for the exam before she failed it.

2. They had not been playing basketball before they broke their leg.

3. He had not been eating before he felt sick.

4. We had not been waiting for her when she arrived.

5. I had not been working on the project before it was due.

6. She had not been sleeping well before the exam.

7. They had not been practicing for the concert before they performed.

8. You had not been cleaning the house before the guests came.

9. He had not been exercising regularly before he got injured.

10. We had not been talking about the issue before it became a problem.

Interrogative Form:

To form the interrogative past perfect continuous, place "had" before the subject.

Had+subject+been+Verb-ing (present participle)+Rest of the Sentence ?

Examples:

1. Had (had) she (subject) been waiting (been + waiting) for long? (rest of the sentence)

   • Had she been waiting for long?

2. Had (had) they (subject) been practicing (been + practicing) before the competition? (rest of the sentence)

   • Had they been practicing before the competition?

Fill-in-the-blank sentences in the past perfect continuous interrogative form, with the verbs provided in brackets:

1. ________ she ________ (wait) for a long time before they arrived?

2. ________ they ________ (study) before the exam?

3. ________ he ________ (work) on the project all day?

4. ________ you ________ (play) football before it started raining?

5. ________ we ________ (travel) for hours before we reached our destination?

6. ________ she ________ (practice) singing before the concert?

7. ________ he ________ (learn) Spanish before he moved to Spain?

8. ________ they ________ (live) in that house for many years before they moved?

9. ________ you ________ (wait) for me before you left?

10. ________ we ________ (talk) for a while before he called?

Answers:

1. Had she been waiting for a long time before they arrived?

2. Had they been studying before the exam?

3. Had he been working on the project all day?

4. Had you been playing football before it started raining?

5. Had we been traveling for hours before we reached our destination?

6. Had she been practicing singing before the concert?

7. Had he been learning Spanish before he moved to Spain?

8. Had they been living in that house for many years before they moved?

9. Had you been waiting for me before you left?

10. Had we been talking for a while before he called?

English Grammar: Past perfect tense

 

Past perfect tense:

Affirmative Form:

Formula:

Subject+had+Past Participle (Verb)+Rest of the Sentence

Breakdown:

1. Subject: The person or thing performing the action.

2. Had: The past tense of the verb "to have."

3. Past Participle: The third form of the main verb (often ending in -ed for regular verbs, or irregular forms).

4. Rest of the Sentence: Additional information about the action.

Examples:

1. She (subject) had finished (had + finished) her homework (rest of the sentence) before dinner.

   • She had finished her homework before dinner.

2. They (subject) had traveled (had + traveled) to Europe (rest of the sentence) before COVID-19.

   • They had traveled to Europe before COVID-19.

3. He (subject) had read (had + read) the book (rest of the sentence) before the movie started.

   • He had read the book before the movie started.

Fill-in-the-blank sentences in the past perfect affirmative form with the verbs provided in brackets:

1. She ________ (finish) her homework before dinner.

2. They ________ (arrive) at the airport before the flight departed.

3. He ________ (study) for the exam all night.

4. We ________ (clean) the house before the guests arrived.

5. I ________ (eat) lunch by the time he called.

6. She ________ (read) all the books on the shelf.

7. They ________ (fix) the car before it broke down.

8. He ________ (complete) the project before the deadline.

9. We ________ (watch) the movie before it started raining.

10. The teacher ________ (give) the students the test after they had studied.

Answers:

1. She had finished her homework before dinner.

2. They had arrived at the airport before the flight departed.

3. He had studied for the exam all night.

4. We had cleaned the house before the guests arrived.

5. I had eaten lunch by the time he called.

6. She had read all the books on the shelf.

7. They had fixed the car before it broke down.

8. He had completed the project before the deadline.

9. We had watched the movie before it started raining.

10. The teacher had given the students the test after they had studied.

Negative Form:

To form the negative past perfect, add "not" after "had."

Formula:

Subject+had not +Past Participle (Verb)+Rest of the Sentence

OR

Subject+hadn’t+Past Participle (Verb)+Rest of the Sentence

Examples:

1. She (subject) had not finished (hadn't + finished) her homework (rest of the sentence) before dinner.

   • She hadn't finished her homework before dinner.

2. They (subject) hadn't traveled (hadn't + traveled) to Europe (rest of the sentence) before COVID-19.

   • They hadn't traveled to Europe before COVID-19.

3. He (subject) had not read (hadn't + read) the book (rest of the sentence) before the movie started.

   • He hadn't read the book before the movie started.

Fill-in-the-blank sentences in the past perfect negative form, with the verbs provided in brackets:

1. She ________ (finish) her work before the meeting.

2. They ________ (leave) the house when it started raining.

3. He ________ (eat) lunch before he left.

4. We ________ (clean) the kitchen before guests arrived.

5. I ________ (read) the book before I watched the movie.

6. She ________ (buy) groceries before she went home.

7. They ________ (fix) the car before it broke down.

8. You ________ (study) for the exam before the teacher announced it.

9. He ________ (call) his friend before he went to bed.

10. We ________ (finish) dinner before the guests arrived.

Answers:

1. She had not finished her work before the meeting.

2. They had not left the house when it started raining.

3. He had not eaten lunch before he left.

4. We had not cleaned the kitchen before guests arrived.

5. I had not read the book before I watched the movie.

6. She had not bought groceries before she went home.

7. They had not fixed the car before it broke down.

8. You had not studied for the exam before the teacher announced it.

9. He had not called his friend before he went to bed.

10. We had not finished dinner before the guests arrived.

Interrogative Form:

To form the interrogative past perfect, place "had" before the subject.

Formula:

Had+Subject+Past Participle (Verb)+Rest of the Sentence?

Examples:

1. Had (had) she (subject) finished (finished) her homework (rest of the sentence) before dinner?

   • Had she finished her homework before dinner?

2. Had (had) they (subject) traveled (traveled) to Europe (rest of the sentence) before COVID-19?

   • Had they traveled to Europe before COVID-19?

3. Had (had) he (subject) read (read) the book (rest of the sentence) before the movie started?

   • Had he read the book before the movie started?

Fill-in-the-blank sentences in the past perfect interrogative form, with the answers provided after all questions:

1. ________ she ________ (finish) her work before you arrived?

2. ________ they ________ (leave) when the movie started?

3. ________ he ________ (eat) before he went to the gym?

4. ________ you ________ (study) for the test before the teacher announced it?

5. ________ we ________ (meet) before the conference began?

6. ________ she ________ (clean) the house before guests arrived?

7. ________ he ________ (fix) the car before it broke down?

8. ________ they ________ (pack) their bags before the trip?

9. ________ you ________ (read) the book before the movie came out?

10. ________ he ________ (complete) the project before the deadline?

Answers:

1. Had she finished her work before you arrived?

2. Had they left when the movie started?

3. Had he eaten before he went to the gym?

4. Had you studied for the test before the teacher announced it?

5. Had we met before the conference began?

6. Had she cleaned the house before guests arrived?

7. Had he fixed the car before it broke down?

8. Had they packed their bags before the trip?

9. Had you read the book before the movie came out?

10. Had he completed the project before the deadline?

English Grammar: Past continuous tense

 

Past continuous tense:

Affirmative Form:

Formula:

Subject + was/were + Verb(ing) + Rest of the Sentence

Breakdown:

1. Subject: The person or thing performing the action.

2. Was/Were: The past tense of the verb "to be." Use "was" with singular subjects (I, he, she, it) and "were" with plural subjects (you, we, they).

3. Verb(ing): The main verb in its present participle form.

4. Rest of the Sentence: Additional information about the action.

Examples:

1. I (subject) was reading (was + reading) a book (rest of the sentence).

   • I was reading a book.

2. They (subject) were playing (were + playing) soccer (rest of the sentence).

   • They were playing soccer.

3. She (subject) was cooking (was + cooking) dinner (rest of the sentence).

   • She was cooking dinner.

Fill-in-the-blank sentences in the past continuous affirmative form with the verbs provided in brackets:

1. I ________ (read) a book when you called.

2. They ________ (play) soccer in the park.

3. She ________ (cook) dinner when I arrived.

4. We ________ (watch) TV all evening.

5. He ________ (write) an email to his boss.

6. The children ________ (draw) pictures in their notebooks.

7. You ________ (work) on your project all night.

8. I ________ (study) for my exams last night.

9. The cat ________ (sleep) on the sofa.

10. We ________ (discuss) our plans for the weekend.

Answer:

1. I was reading a book when you called.

2. They were playing soccer in the park.

3. She was cooking dinner when I arrived.

4. We were watching TV all evening.

5. He was writing an email to his boss.

6. The children were drawing pictures in their notebooks.

7. You were working on your project all night.

8. I was studying for my exams last night.

9. The cat was sleeping on the sofa.

10. We were discussing our plans for the weekend.

Negative Form:

To form the negative past continuous, add "not" after "was" or "were."

Formula:

Subject+was/were+not+Verb(ing)+Rest of the Sentence

Examples:

1. I (subject) was not (was + not) reading (reading) a book (rest of the sentence).

   • I was not reading a book.

2. They (subject) were not (were + not) playing (playing) soccer (rest of the sentence).

   • They were not playing soccer.

Fill-in-the-blank sentences in the past continuous negative form with the verbs provided in brackets.

1. She ________ ________ (not watch) TV last night.

2. They ________ ________ (not play) soccer yesterday afternoon.

3. He ________ ________ (not study) for the exam.

4. We ________ ________ (not eat) dinner when you called.

5. I ________ ________ (not read) the book you gave me.

6. The cat ________ ________ (not sleep) on the couch.

7. You ________ ________ (not work) on your project.

8. She ________ ________ (not cook) when I arrived.

9. They ________ ________ (not drive) to the beach.

10. He ________ ________ (not write) a letter to his friend.

Answers:

1. She was not watching  TV last night.

2. They were not playing  soccer yesterday afternoon.

3. He was not studying  for the exam.

4. We were not eating  dinner when you called.

5. I was not reading  the book you gave me.

6. The cat was not sleeping  on the couch.

7. You were not working  on your project.

8. She was not cooking  when I arrived.

9. They were not driving  to the beach.

10. He was not writing  a letter to his friend.

Interrogative Form:

To form the interrogative past continuous, place "was" or "were" before the subject.

Formula:

Sentence Was/Were + Subject + Verb(ing) + Rest of the Sentence

Examples:

1. Was (was) I (subject) reading (reading) a book (rest of the sentence)?

   • Was I reading a book?

2. Were (were) they (subject) playing (playing) soccer (rest of the sentence)?

   • Were they playing soccer?

Fill-in-the-blank sentences in the past continuous interrogative form, with the verbs provided in brackets:

1. ________ she ________ (watch) TV when I called her?

2. ________ you ________ (study) when the lights went out?

3. ________ he ________ (play) basketball with his friends yesterday afternoon?

4. ________ they ________ (wait) for the bus when you saw them?

5. ________ I ________ (work) on the project all night?

6. ________ we ________ (talk) about our vacation plans when you arrived?

7. ________ the children ________ (play) outside when it started raining?

8. ________ she ________ (cook) dinner when you came home?

9. ________ you ________ (listen) to music during the party?

10. ________ he ________ (read) a book when the phone rang?

Answers:

1. Was she watching TV when I called her?

2. Were you studying when the lights went out?

3. Was he playing basketball with his friends yesterday afternoon?

4. Were they waiting for the bus when you saw them?

5. Was I working on the project all night?

6. Were we talking about our vacation plans when you arrived?

7. Were the children playing outside when it started raining?

8. Was she cooking dinner when you came home?

9. Were you listening to music during the party?

10. Was he reading a book when the phone rang?