AIOU 207 Solved Assignment Autumn 2025

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ALLAMA IQBAL OPEN UNIVERSITY

(Department of English)

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WARNING

1. Plagiarism or hiring of ghost writer(s) for solving the assignment(s) will debar the student from award of degree/certificate if found at any stage.

2. Submitting assignment(s) borrowed or stolen from other(s) as one's own will be penalized as defined in the "Aiou Plagiarism Policy".

Assignment Submission Schedule
6 Credit Hours	Due Date	3 Credit Hours	Due Date
Assignment 1	15-12-2025	Assignment 1	08-01-2026
Assignment 2	08-01-2026
Assignment 3	30-01-2026	Assignment 2	20-02-2026
Assignment 4	20-02-2026

Course: Compulsory English-I (207)	Semester: Autumn,2025
Level: Matric / SSC

Please read the following instructions for writing your assignments. (SSC, HSSC & BA Programmes)
1. All questions are compulsory and carry equal marks but within a question the marks are distributed according to its requirements.
2. Read the question carefully and then answer it according to the requirements of the questions.
3. Late submission of assignments will not be accepted.
4. Your own analysis and synthesis will be appreciated.
5. Avoid irrelevant discussion/information and reproducing from books, study guide of allied material.

Total Marks: 100	Pass Marks: 40

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ASSIGNMENT No. 1

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Q1. Change each of the following statements into WH questions by using the question words given below. (what, where, who, whose, when, why, which)

i. He found a wallet on the street.
ii. The teacher explained the lesson clearly.
iii. We are having lunch at a new restaurant.
iv. The movie begins at 8 PM.
v. She is dancing with her partner.
vi. The books are in the library.
vii. I watched a comedy show last night.
viii. He owns a black motorcycle.
ix. The backpack belongs to the student.
x. The children played in the garden. ▶

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i. He found a wallet on the street.
What did he find on the street?

ii. The teacher explained the lesson clearly.
Who explained the lesson clearly?

iii. We are having lunch at a new restaurant.
Where are we having lunch?

iv. The movie begins at 8 PM.
When does the movie begin?

v. She is dancing with her partner.
Who is she dancing with?

vi. The books are in the library.
Where are the books?

vii. I watched a comedy show last night.
What did you watch last night?

viii. He owns a black motorcycle.
Which motorcycle does he own?

ix. The backpack belongs to the student.
Whose backpack is this?

x. The children played in the garden.
Why did the children play in the garden?

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Q2. In Unit 2, you learned about words that are used to express contrast. Now imagine you are deciding between two sports to join with your friends. Write five sentences explaining your choice by comparing the two sports, using the words although, however, but, yet, and nevertheless to highlight the contrasts. ▶

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1. Although basketball is fun indoors, I prefer football because I enjoy playing outside.

2. I like basketball, however, football gives me more excitement.

3. Basketball is fast, but football feels more challenging to me.

4. Basketball has many rules, yet football is easier for me to understand.

5. I get tired in basketball practice, nevertheless, I still choose football because I love it.

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Q3. What are your thoughts on the topics given below? Provide your personal judgment in the form of an easy of around 250-300 words.

1. Is social media doing more harm than good to society?
2. Should students have access to their phones during school hours? ▶

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1. Is social media doing more harm than good to society?

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Social media has become a big part of our lives. People use it to stay connected, share ideas, and learn about the world. While it offers many benefits, I believe that in today’s society it is doing more harm than good.

One of the biggest problems is how social media affects mental health. Many young people spend hours scrolling through posts, comparing themselves to others, and feeling insecure. This constant comparison can lead to anxiety and low self-esteem. Another issue is the spread of misinformation. Fake news can go viral in minutes, creating confusion and even conflict in society. For example, during health crises or elections, wrong information shared online can harm people’s decisions.

Social media also encourages addiction. People spend so much time on it that they ignore real-life relationships and responsibilities. Family dinners, study time, and even sleep get disturbed because of endless scrolling. While it is true that social media helps us stay informed and connected, the negative effects are becoming stronger every day.

In my view, social media itself is not the enemy, but the way it is being used makes it harmful. If people used it responsibly, with limits on screen time and more focus on positive content, it could become a tool for good. For now, though, I think social media is causing more harm than good to society.

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2. Should students have access to their phones during school hours?

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Phones are a common part of life, and students depend on them for communication, learning, and even safety. However, the question is whether they should be allowed during school hours. In my opinion, students should not have free access to their phones in class, but there can be limited use under guidance.

One strong reason against phones is distraction. Many students use them to play games, chat, or scroll through social media during lessons. This makes it hard to focus on studies and affects overall learning. Teachers also face challenges in keeping classrooms disciplined when phones are allowed freely.

On the other hand, phones can be useful. They can help students research quickly, use learning apps, or call parents in case of emergencies. In some schools, teachers integrate phones into lessons for activities like online quizzes, which shows that controlled use can be helpful.

The best solution, in my opinion, is balance. Students should not have their phones in their hands all the time, but schools can set rules where phones are used only for learning purposes under supervision. For emergencies, students can keep phones switched off in their bags.

To conclude, while phones are useful tools, free access during school hours is not a good idea. With proper limits, students can benefit from technology without losing focus on their education.

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Q4. In Unit 1, you learned about introducing yourself. Imagine you have joined a new community club. How would you introduce yourself to the other members, sharing some of your hobbies and what you hope to achieve as part of the club? ▶

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Hello everyone, my name is Ahmed Ali. I recently joined this club and I am very excited to be here. I enjoy spending my free time reading books, playing badminton, and doing a bit of gardening. These hobbies help me relax and also keep me active.

One of the main reasons I joined this community club is to meet new people, make friends, and learn from others. I also hope to take part in different activities that can improve my skills and give me a chance to contribute something useful to the group.

I believe that working together as a community makes us stronger, and I look forward to sharing ideas and supporting each other. Thank you for welcoming me, and I am really looking forward to being part of this club.

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Q5. There is a mother and her daughter having a conversation at a food stall. Read the answers given below and write five appropriate questions using the words what, how, does, will, and are.

i. The hotdog is too big for me to finish.
ii. I’m enjoying the match while eating.
iii. The sauce makes the food taste even better.
iv. I think this is the best stall at the stadium.
v. The popcorn is perfect for snacking. ▶

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i. The hotdog is too big for me to finish.
What do you think about the hotdog?

ii. I’m enjoying the match while eating.
How are you feeling while eating?

iii. The sauce makes the food taste even better.
Does the sauce change the taste of the food?

iv. I think this is the best stall at the stadium.
Will you say this is the best stall at the stadium?

v. The popcorn is perfect for snacking.
Are the popcorn good for snacking?

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Q6. Using the provided list of sequence markers (first, firstly, second, secondly, later, next, after that, meanwhile, in the meantime, finally), rewrite Mr. Ali's Planning a Birthday Party by constructing a paragraph.

i. Decide on the date and guest list.
ii. Choose a suitable venue and book it.
iii. Select a theme and gather decorations.
iv. Plan the menu and choose food/drinks.
v. Organize entertainment (DJ, activities).
vi. Send out invitations to friends and family.
vii. Buy supplies and check RSVP responses.
viii. Set up decorations and prepare the party space.
ix. Celebrate on the day with loved ones.
x. Clean up after the party. ▶

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First, Mr. Ali decided on the date and made a guest list. Firstly, he also discussed the plan with his family to make sure everyone agreed. Second, he chose a suitable venue and booked it for the special day. Secondly, he selected a theme and gathered decorations to match it. Later, he planned the menu carefully and chose the food and drinks. Next, he organized entertainment such as a DJ and fun activities to keep everyone engaged. After that, he sent out invitations to friends and family. Meanwhile, he bought supplies and checked the RSVP responses. In the meantime, he set up the decorations and prepared the party space. Finally, Mr. Ali celebrated the day with his loved ones and, after enjoying every moment, he cleaned up and wrapped up the event successfully.

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Q7. In Unit 3, you learned about career aspirations. Now talk to a family member about their dream job and the steps they plan to take to achieve it. Write a paragraph describing their goals, motivations, and strategies for success. ▶

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My brother shared with me that his dream job is to become a software engineer. He is motivated by his love for technology and the way computers can solve real-world problems. His goal is to work at a well-known tech company where he can design useful applications that make life easier for people. To achieve this, he plans to focus on his studies, practice coding daily, and take part in online courses to improve his skills. He also wants to join internships to gain experience and learn from professionals in the field. His strategy is to stay consistent, never stop learning, and build a strong portfolio of projects that will impress employers. What inspires him most is the idea that with hard work and dedication, he can turn his passion for technology into a successful career.

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Q8. Read the following sentences and fill in the columns. Here each sentence contains a difficult word which is written in bold. First, guess the meaning of the difficult word based on the context of the sentence. Then, look up the word in a dictionary and write down its actual meaning. Finally, find a synonym for the difficult word from the dictionary.

1. The melancholic artist, who painted scenes of solitude, captured deep emotions in every brushstroke.
2. The tenacious mountaineer, who refused to give up, conquered the treacherous peak after weeks of climbing.
3. The Coincidental discovery, which occurred during a routine experiment, led to a groundbreaking innovation.
4. The effervescent fountain, which sparkled under the sunlight, delighted the children playing nearby.
5. The meticulous architect, who paid attention to every detail, designed a building that became an iconic landmark.

Difficult Words	Guess Meaning	Actual Meaning	Synonym
-	-	-	-

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Difficult Words	Guess Meaning	Actual Meaning	Synonym
melancholic	Sad or gloomy	Feeling or expressing deep sadness	Depressed
tenacious	Determined, not giving up	Holding firmly, very determined or persistent	Persistent
Coincidental	Happening by chance	Occurring by chance at the same time without planning	Accidental
effervescent	Sparkling or lively	Giving off bubbles; lively and enthusiastic	Bubbly
meticulous	Very careful, detailed	Showing great attention to detail; very precise	Precise

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Q9. Use the following words to make five sentences of your own: (Therefore, Finaly, Hence, Similarly, Because). ▶

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1. I was feeling tired, therefore I went to bed early.

2. We finished all the decorations and finally the room looked beautiful.

3. It was raining heavily, hence the football match was cancelled.

4. My friend loves reading novels and similarly I also enjoy spending time with books.

5. She stayed home because she was not feeling well.

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ASSIGNMENT No. 2

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Q1. Think about your favorite hobby. Write down five things you enjoy about it and five challenges you face while pursuing it. ▶

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Five things I enjoy about gardening:

1. Watching seeds grow into healthy plants feels very rewarding.

2. It makes me feel peaceful and relaxed.

3. I get fresh vegetables and herbs from my own garden.

4. Spending time outside in fresh air is refreshing.

5. It sparks creativity in planning the layout and choosing plants.

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Five challenges I face in gardening:

1. Plants sometimes die despite proper care.

2. Pests and insects damage crops.

3. Weather changes can ruin growth.

4. It requires a lot of patience to see results.

5. Regular watering and maintenance take time and effort.

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Q2. Read the given sentences and underline the contractions in each sentence. Then rewrite the sentences using the full, uncontracted forms.

Example:
Contracted form: It's a beautiful day.
Uncontracted form: It is a beautiful day.

1. I’m going to the store later.
2. You’ve already finished your homework.
3. She’s excited about the party tonight.
4. They’re coming over for dinner.
5. It’s a beautiful day outside.
6. We’re meeting at 6 PM.
7. He’s been waiting for hours.
8. I’d like to grab a coffee.
9. You’ll need to bring your ID.
10. They’ve never been to that museum. ▶

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1. I’m going to the store later.
I am going to the store later.

2. You’ve already finished your homework.
You have already finished your homework.

3. She’s excited about the party tonight.
She is excited about the party tonight.

4. They’re coming over for dinner.
They are coming over for dinner.

5. It’s a beautiful day outside.
It is a beautiful day outside.

6. We’re meeting at 6 PM.
We are meeting at 6 PM.

7. He’s been waiting for hours.
He has been waiting for hours.

8. I’d like to grab a coffee.
I would like to grab a coffee.

9. You’ll need to bring your ID.
You will need to bring your ID.

10. They’ve never been to that museum.
They have never been to that museum.

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Q3. In Unit 4, you explored the traits of different personalities. Now, write ten sentences explaining if you prefer social situations or solitude, and provide reasons for your preference. ▶

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1. I prefer solitude because it gives me peace of mind.

2. Being alone helps me recharge my energy.

3. I can focus better on my hobbies when I am by myself.

4. Solitude allows me to think deeply and reflect on life.

5. I enjoy reading books quietly without distractions.

6. Spending time alone helps me become more creative.

7. I do not feel pressured to meet others’ expectations when I am alone.

8. Solitude makes me feel independent and strong.

9. I can organize my thoughts clearly when I am not in a crowd.

10. Although I sometimes enjoy social events, I find true comfort in being alone.

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Q4. In the following paragraph about a visit to Gilgit, identify and label the different parts of speech. Specifically, look for nouns, verbs, adjectives, pronouns, prepositions, conjunctions, and interjections.

Gilgit, located in northern Pakistan, is a picturesque valley surrounded by towering mountains and crystal-clear rivers. The vibrant culture blends traditional music, local crafts, and mouth-watering foods. Every year, tourists from around the world visit to explore the breathtaking landscapes. In Gilgit, adventure seekers can go trekking through lush green forests or camp near the serene lakes. The warm hospitality of the locals adds to the charm of this beautiful place. The fresh air and peaceful surroundings make it a perfect destination for those looking to escape the chaos of city life. ▶

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Nouns

Proper Nouns: Gilgit, Pakistan

Common Nouns: valley, mountains, rivers, culture, music, crafts, foods, year, tourists, world, landscapes, forests, lakes, hospitality, locals, charm, place, air, surroundings, destination, chaos, life

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Verbs

is, located, blends, visit, explore, can go, camp, adds, make, looking, to escape

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Adjectives

northern, picturesque, towering, crystal-clear, vibrant, traditional, local, mouth-watering, breathtaking, green, serene, warm, beautiful, fresh, peaceful, perfect

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Pronouns

who, it, those

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Prepositions

in, of, by, from, to, through, near, for

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Conjunctions

and, or, which

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Interjections

There are no interjections in this paragraph

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Q5. Write the comparative and superlative forms of the following adjectives. And then, write a paragraph by using either the comparative or superlative form of these adjectives.

Positive Degree	Comparative Degree	Superlative Degree
Big
Small
Fat
Smart
Friendly
High
Delicious
Old
Clean
Cold

▶

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Positive Degree	Comparative Degree	Superlative Degree
Big	Bigger	Biggest
Small	Smaller	Smallest
Fat	Fatter	Fattest
Smart	Smarter	Smartest
Friendly	Friendlier	Friendliest
High	Higher	Highest
Delicious	More delicious	Most delicious
Old	Older	Oldest
Clean	Cleaner	Cleanest
Cold	Colder	Coldest

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Q6. Read the following scenarios and write both formal and informal invitation based on the context.

• Job Interview
You are inviting a potential candidate to an interview for a job position at your company.

• Team Building Activity
You want to invite your team to a fun and collaborative team-building activity.

• Cooking Class
You are inviting friends or family to join you for a cooking class at a local culinary school.

• Volunteer Training
You are organizing a volunteer training session and inviting participants to join.

• Summer Camp
You are inviting children to sign up for a fun and educational summer camp program. ▶

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Job Interview - Formal Invitation

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Subject: Interview Invitation – Marketing Executive Role

Dear Haider Ali,

We are pleased to invite you for an interview for the Marketing Executive position at ABC Solutions.

Date: Monday, 6th October 2025
Time: 10:00 AM
Venue: ABC Solutions Office, 123 Main Street, Karachi.

Please confirm your availability by replying to this email.

Sincerely,
Sara Khan
HR Manager
ABC Solutions

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Team Building Activity - Informal Invitation

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Hey Team,

We’re planning a fun team-building activity and would love for everyone to join!

Date: Friday, 10th October 2025
Time: 3:00 PM
Place: Green Park Adventure Center, Karachi.

There will be games, challenges, and lots of laughs. Don’t miss out—come and enjoy some quality time with the team!

See you all there!
Alex
Team Coordinator

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Cooking Class - Formal Invitation

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Dear Friends and Family,

I am pleased to invite you to join a cooking class at Gourmet Culinary School. This will be a wonderful opportunity to learn new recipes and enjoy a culinary experience together.

Date: Saturday, 11th October 2025
Time: 2:00 PM
Venue: Gourmet Culinary School, 45 Foodie Lane, Karachi.

Please RSVP by 5th October 2025 to confirm your attendance.

Warm regards,
Maria Khan

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Volunteer Training - Formal Invitation

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Dear Participant,

We are pleased to invite you to participate in the upcoming Volunteer Training Session organized by Helping Hands Organization. This session will provide essential skills and guidance for our volunteer programs.

Date: Wednesday, 15th October 2025
Time: 10:00 AM to 2:00 PM
Venue: Helping Hands Community Center, 78 Civic Street, Karachi.

Please confirm your participation by replying to this email at your earliest convenience.

Sincerely,
Ahmed Raza
Volunteer Coordinator
Helping Hands Organization

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Summer Camp - Informal Invitation

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Hey Kids,

We’re excited to announce our Summer Camp 2025! It’s going to be full of fun, games, and educational activities designed to make this summer unforgettable.

Dates: Monday, 1st June 2025 – Friday, 30th June 2025
Time: 9:00 AM – 3:00 PM
Place: Sunshine Activity Center, 22 Happy Street, Karachi,

Sign up now and join us for a summer full of adventure and learning! We can’t wait to see you there.

Cheers,
Lily Ahmed
Summer Camp Coordinator

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Q7. You have received an invitation from a colleague to attend an important business conference. Write a letter to your colleague either accepting or declining the invitation. ▶

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Sara Ahmed
Business Development Manager
TechNova Solutions
45 Shahrah-e-Faisal
Karachi, Pakistan

Dear Sara Ahmed,

I hope this message finds you well. I sincerely appreciate your kind invitation to attend the upcoming business conference scheduled for 25th September 2025 at the Karachi Expo Center. It would have been a valuable opportunity to connect and gain insights.

However, I regret to inform you that I will be unable to attend due to a prior commitment that requires my presence during the same period. I deeply value our professional relationship and hope to participate in future events.

Thank you once again for considering me, and I wish the conference great success.

Sincerely,
Ali Khan
Marketing Manager
Sunrise Enterprises

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Read More

AIOU 247 Solved Assignment Autumn 2025

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ALLAMA IQBAL OPEN UNIVERSITY

(Department of Mathematics)

---

WARNING

1. Plagiarism or hiring of ghost writer(s) for solving the assignment(s) will debar the student from award of degree/certificate if found at any stage.

2. Submitting assignment(s) borrowed or stolen from other(s) as one's own will be penalized as defined in the "Aiou Plagiarism Policy".

Assignment Submission Schedule
6 Credit Hours	Due Date	3 Credit Hours	Due Date
Assignment 1	15-12-2025	Assignment 1	08-01-2026
Assignment 2	08-01-2026
Assignment 3	30-01-2026	Assignment 2	20-02-2026
Assignment 4	20-02-2026

Course: Mathematics-I (247)	Semester: Autumn,2025
Level: Matric / SSC

Please read the following instructions for writing your assignments. (SSC, HSSC & BA Programmes)
1. All questions are compulsory and carry equal marks but within a question the marks are distributed according to its requirements.
2. Read the question carefully and then answer it according to the requirements of the questions.
3. Late submission of assignments will not be accepted.
4. Your own analysis and synthesis will be appreciated.
5. Avoid irrelevant discussion/information and reproducing from books, study guide of allied material.

Total Marks: 100	Pass Marks: 40

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ASSIGNMENT No. 1

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Q1(a). Two angles in a triangle are 50° and 80°. Find the ratio of the third angle to the sum of the first two. ▶

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1. The sum of the angles in a triangle is always:

$$ 180^\circ $$

2. The first two angles are:

$$ 50^\circ \text{ and } 80^\circ $$

3. Add them:

$$ 50^\circ + 80^\circ = 130^\circ $$

4. Find the third angle:

$$ 180^\circ - 130^\circ = 50^\circ $$

5. Ratio = third angle : sum of first two

$$ 50 : 130 $$

6. Simplify:

$$ 5 : 13 $$

7. As a fraction:

$$ \frac{50}{130} = \frac{5}{13} $$

Final Answer:
The ratio is 5 : 13 or the fraction is $$ \frac{5}{13} $$.

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Q1(b). If the stay of 12 men for 7 days in a hostel costs Rs. 21200, find the cost for the stay of 10 men for 8 days. ▶

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1. Find the total man-days in the given case:

$$ 12 \times 7 = 84 \ \text{man-days} $$

2. Cost for 84 man-days is:

$$ 21200 \ \text{rupees} $$

3. Cost for 1 man-day:

$$ \frac{21200}{84} = 252.38 \ \text{rupees (approx)} $$

4. Now, find the required man-days:

$$ 10 \times 8 = 80 \ \text{man-days} $$

5. Cost for 80 man-days:

$$ 80 \times 252.38 = 20190.4 \ \text{rupees} $$

Final Answer:
The cost for 10 men for 8 days is Rs. 20,190.40.

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Q2(a). Calculate Ushr on a wheat crop amounting to Rs. 15,00,000 produced by artificial resources. ▶

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1. Rule of Ushr:

If irrigated naturally → Ushr = 10%
If irrigated artificially → Ushr = 5%

2. Given data:

Crop value = Rs. 15,00,000
Irrigation = Artificial resources
Ushr rate = 5%

3. Calculation:

$$ \text{Ushr} = \frac{5}{100} \times 15,00,000 $$

$$ \text{Ushr} = 75,000 $$

Final Answer:
The Ushr on the wheat crop is Rs. 75,000.

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Q2(b). Calculate Zakat on gold amounting to Rs. 11,75,000. ▶

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1. Rule of Zakat on Gold:

Zakat is payable at the rate of 2.5% of the total value of gold (if it reaches or exceeds nisab).

2. Given data:

Value of gold = Rs. 11,75,000
Zakat rate = 2.5%

3. Calculation:

$$ \text{Zakat} = \frac{2.5}{100} \times 11,75,000 $$

$$ \text{Zakat} = 29,375 $$

Final Answer:
The Zakat on gold worth Rs. 11,75,000 is Rs. 29,375.

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Q3(a). Distribute Rs. 19600 as profit among three friends, so that their shares are in the ratio 2:4:7. ▶

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1. The given ratio is:

$$ 2 : 4 : 7 $$

2. Total parts:

$$ 2 + 4 + 7 = 13 $$

3. Value of one part:

$$ \frac{19600}{13} \approx 1507.69 $$

4. Now calculate each friend’s share:

Friend A (2 parts):

$$ \frac{2}{13} \times 19600 = 3015.38 $$

Friend B (4 parts):

$$ \frac{4}{13} \times 19600 = 6030.77 $$

Friend C (7 parts):

$$ \frac{7}{13} \times 19600 = 10553.85 $$

Final Answer:
Friend A’s share ≈ Rs. 3015.38
Friend B’s share ≈ Rs. 6030.77
Friend C’s share ≈ Rs. 10553.85

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Q3(b). The marked price of a clock is Rs. 800. The shopkeeper offers a discount of 6% and still gains 12%. Find the price at which the shopkeeper purchased it. ▶

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1. Marked Price (MP): 800

2. Discount = 6% of 800:

$$ \frac{6}{100} \times 800 = 48 $$

3. Selling Price (SP):

$$ 800 - 48 = 752 $$

4. Formula:

$$ SP = CP \times \left(1 + \frac{\text{Gain%}}{100}\right) $$

Substitute values:

$$ 752 = CP \times \left(1 + \frac{12}{100}\right) $$

$$ 752 = CP \times 1.12 $$

5. Find CP:

$$ CP = \frac{752}{1.12} $$

$$ CP = 671.43 $$

Final Answer:
The shopkeeper purchased the clock at Rs. 671.43.

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Q4(a). Find the compound profit on Rs.2500 for 2 1/2 years at 4% per annum. ▶

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1. Formula for Compound Amount:

$$ A = P \times \left(1 + \frac{r}{100}\right)^n $$

Where:

• Principal (P) = 2500
• Rate (r) = 4%
• Time (n) = 2 \tfrac{1}{2} years = 2 years + \tfrac{1}{2} year

2. After 2 years:

$$ A_2 = 2500 \times \left(1 + \frac{4}{100}\right)^2 $$

$$ A_2 = 2500 \times (1.04)^2 $$

$$ A_2 = 2500 \times 1.0816 = 2704 $$

3. For next half year (rate = 2%):

$$ A = 2704 \times \left(1 + \frac{2}{100}\right) $$

$$ A = 2704 \times 1.02 $$

$$ A = 2758.08 $$

4. Compound Profit (CI):

$$ CI = A - P $$

$$ CI = 2758.08 - 2500 $$

$$ CI = 258.08 $$

Final Answer:
The compound profit is Rs. 258.08.

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Q4(b). At what annual rate of profit would a sum of Rs. 840 will increase to Rs. 1050 in 1 year 8 months? ▶

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1. Given Data:

• Principal (P) = 840
• Amount (A) = 1050
• Time (T) = 1 year 8 months = \( 1 \tfrac{8}{12} = \tfrac{5}{3} \, \text{years} \)

2. Formula for Simple Interest:

$$ A = P \left(1 + \frac{r \times T}{100}\right) $$

3. Substitution:

$$ 1050 = 840 \left(1 + \frac{r \times \tfrac{5}{3}}{100}\right) $$

$$ \frac{1050}{840} = 1 + \frac{5r}{300} $$

$$ 1.25 = 1 + \frac{5r}{300} $$

4. Solve for r:

$$ 1.25 - 1 = \frac{5r}{300} $$

$$ 0.25 = \frac{5r}{300} $$

$$ 0.25 \times 300 = 5r $$

$$ 75 = 5r $$

$$ r = 15 $$

Final Answer:
The annual rate of profit is 15%.

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Q5(a). If the gross pay of a person is Rs.70,000, then calculate his net take-home salary, after deductions of Rs.350 as income tax, Rs.900 as a benevolent fund, Rs.1600 s G.P fund and Rs.300 as group insurance. ▶

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1. Gross Pay: 70,000

2. Deductions:

$$ \text{Total Deductions} = 350 + 900 + 1600 + 300 $$

$$ \text{Total Deductions} = 3150 $$

3. Net Salary:

$$ \text{Net Salary} = \text{Gross Pay} - \text{Deductions} $$

$$ \text{Net Salary} = 70000 - 3150 $$

$$ \text{Net Salary} = 66850 $$

Final Answer:
The net take-home salary is Rs. 66,850.

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Q5(b). The price of a motorcycle is Rs.65,000. If 18% sales tax is charged, then calculate the amount of sales tax on 30 such motorcycles. ▶

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1. Price of one motorcycle: 65,000

2. Sales Tax on one motorcycle:

$$ \text{Sales Tax} = \frac{18}{100} \times 65,000 $$

$$ \text{Sales Tax} = 11,700 $$

3. Sales Tax on 30 motorcycles:

$$ \text{Total Sales Tax} = 11,700 \times 30 $$

$$ \text{Total Sales Tax} = 351,000 $$

Final Answer:
The amount of sales tax on 30 motorcycles is Rs. 351,000.

---

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ASSIGNMENT No. 2

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Q1(a). If A.M between a and 10 is 40, then find the value of a. ▶

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1. Formula of Arithmetic Mean (A.M):

$$ A.M = \frac{a + b}{2} $$

2. Given values: A.M = 40, b = 10

$$ 40 = \frac{a + 10}{2} $$

3. Multiply both sides by 2:

$$ 80 = a + 10 $$

4. Subtract 10 from both sides:

$$ a = 70 $$

The value of a is 70.

---

Q1(b). The positive G.M between two numbers is 6 and the A.M between them is 10. Find the numbers. ▶

---

1. Formula of Arithmetic Mean (A.M):

$$ A.M = \frac{x + y}{2} $$

Given A.M = 10:

$$ \frac{x + y}{2} = 10 $$

$$ x + y = 20 $$

2. Formula of Geometric Mean (G.M):

$$ G.M = \sqrt{xy} $$

Given G.M = 6:

$$ \sqrt{xy} = 6 $$

$$ xy = 36 $$

3. Form the quadratic equation using sum and product of roots:

$$ t^2 - (x+y)t + xy = 0 $$

$$ t^2 - 20t + 36 = 0 $$

4. Solve the quadratic:

$$ t = \frac{20 \pm \sqrt{20^2 - 4(36)}}{2} $$

$$ t = \frac{20 \pm \sqrt{400 - 144}}{2} $$

$$ t = \frac{20 \pm \sqrt{256}}{2} $$

$$ t = \frac{20 \pm 16}{2} $$

5. Values of t:

$$ t = \frac{20 + 16}{2} = 18 $$

$$ t = \frac{20 - 16}{2} = 2 $$

The two numbers are 18 and 2.

---

Q2(a). If logx = 0.1597, find x. ▶

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1. Given:

$$ \log x = 0.1597 $$

2. Rewrite in exponential form:

$$ x = 10^{0.1597} $$

3. Calculate the value:

$$ x \approx 1.445 $$

The value of x is approximately 1.445.

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Q2(b). Prove that log(2 x 4 x 6) = log2 + log4 + log6. ▶

---

1. Start with the left-hand side:

$$ \log(2 \times 4 \times 6) $$

2. Group the terms:

$$ \log((2 \times 4) \times 6) $$

3. Apply product rule of logarithms:

$$ \log(2 \times 4) + \log(6) $$

4. Apply the rule again:

$$ \log(2) + \log(4) + \log(6) $$

$$ \log(2 \times 4 \times 6) = \log 2 + \log 4 + \log 6 $$

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Q3(a). If A = {1,2,3,4,5,6,7,8,9}, B = {2,3,4,5} and C = {3,4,6}. Verify that (A∩B)∩C = An(B∩C). ▶

---

Given:

$$ A = \{1,2,3,4,5,6,7,8,9\}, \; B = \{2,3,4,5\}, \; C = \{3,4,6\} $$

We need to verify:

$$ (A \cap B) \cap C = A \cap (B \cap C) $$

1. Find intersection of A and B:

$$ A \cap B = \{2,3,4,5\} $$

2. Now, (A ∩ B) ∩ C:

$$ (A \cap B) \cap C = \{2,3,4,5\} \cap \{3,4,6\} = \{3,4\} $$

3. Find intersection of B and C:

$$ B \cap C = \{2,3,4,5\} \cap \{3,4,6\} = \{3,4\} $$

4. Now, A ∩ (B ∩ C):

$$ A \cap (B \cap C) = A \cap \{3,4\} = \{3,4\} $$

$$ (A \cap B) \cap C = A \cap (B \cap C) = \{3,4\} $$

Hence, verified.

---

Q3(b). If X = {3,5,7} and Y = {3,6}. Find X×Y and Y×X and also the domains and ranges of the two binary relations established on our own for each case. ▶

---

Given:

$$ X = \{3,5,7\}, \quad Y = \{3,6\} $$

1. Cartesian Product X × Y:

$$ X \times Y = \{(3,3), (3,6), (5,3), (5,6), (7,3), (7,6)\} $$

2. Cartesian Product Y × X:

$$ Y \times X = \{(3,3), (3,5), (3,7), (6,3), (6,5), (6,7)\} $$

3. Binary Relation from X × Y:

Let us define a relation R from X to Y as:

$$ R = \{(x,y) \in X \times Y \;|\; x < y\} $$

So:

$$ R = \{(3,6), (5,6)\} $$

Domain of R:

$$ \text{Dom}(R) = \{3,5\} $$

Range of R:

$$ \text{Ran}(R) = \{6\} $$

4. Binary Relation from Y × X:

Let us define a relation S from Y to X as:

$$ S = \{(y,x) \in Y \times X \;|\; y > x\} $$

So:

$$ S = \{(6,3), (6,5)\} $$

Domain of S:

$$ \text{Dom}(S) = \{6\} $$

Range of S:

$$ \text{Ran}(S) = \{3,5\} $$

---

$$ X \times Y = \{(3,3),(3,6),(5,3),(5,6),(7,3),(7,6)\} $$

$$ Y \times X = \{(3,3),(3,5),(3,7),(6,3),(6,5),(6,7)\} $$

$$ \text{For relation R: Domain = {3,5}, Range = {6}} $$

$$ \text{For relation S: Domain = {6}, Range = {3,5}} $$

---

Q4(a). Draw the graph of: $$ \frac{y}{5} = \frac{x}{3} - \frac{7}{4} $$ ▶

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---

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Q4(b). Draw the graph of: $$ \frac{3y}{2} = \frac{2x}{3} - \frac{7}{2} $$ ▶

---

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Q5(a). Find the harmonic mean of the values 5, 25 and 75. ▶

---

1. Formula of Harmonic Mean (H.M):

$$ H.M = \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + \frac{1}{x_3} + \cdots + \frac{1}{x_n}} $$

2. Given values: \(x_1 = 5, \; x_2 = 25, \; x_3 = 75, \; n = 3\)

$$ H.M = \frac{3}{\frac{1}{5} + \frac{1}{25} + \frac{1}{75}} $$

3. Simplify the denominators:

$$ \frac{1}{5} = \frac{15}{75}, \quad \frac{1}{25} = \frac{3}{75}, \quad \frac{1}{75} = \frac{1}{75} $$

$$ \frac{1}{5} + \frac{1}{25} + \frac{1}{75} = \frac{15+3+1}{75} = \frac{19}{75} $$

4. Substitute back:

$$ H.M = \frac{3}{\tfrac{19}{75}} = \frac{3 \times 75}{19} $$

5. Final value:

$$ H.M = \frac{225}{19} = 11.84 $$

The harmonic mean of 5, 25 and 75 is
$$ \frac{225}{19} \; = 11.84 $$

---

Q5(b). Find the median for the set of values: 8,6,5,2,3,1,7,4. ▶

---

1. Arrange the data in ascending order:

$$ 1, 2, 3, 4, 5, 6, 7, 8 $$

2. Number of values:

$$ n = 8 \quad (\text{even}) $$

3. Formula for median when n is even:

$$ \text{Median} = \frac{\text{Value at } \tfrac{n}{2} \text{ position} + \text{Value at } (\tfrac{n}{2}+1) \text{ position}}{2} $$

4. Identify the two middle values:

Value at position $$ \tfrac{8}{2} = 4 $$ is $$ 4 $$

Value at position $$ \tfrac{8}{2} + 1 = 5 $$ is $$ 5 $$

5. Calculate the median:

$$ \text{Median} = \frac{4 + 5}{2} = \frac{9}{2} = 4.5 $$

The median of the given data is 4.5.

---

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AIOU 1349 Past Paper Spring 2021

ALLAMA IQBAL OPEN UNIVERSITY

Level: I.Com

Semester: Spring 2021

Course Code: 1349 - Introduction to Business Mathematics

Maximum Marks: 100

Time Allowed: 03 Hours

Pass Marks: 40

---

Note: Attempt any Five Questions, Question No. 1 is compulsory.

---

Q1.(a) During a sale, a store offers a discount of 25% off any purchase. What is the regular price of a dress that a customer purchased for $73.50?

---

Step 1. Let P be the regular price. That is 100% of the price before the discount.

Step 2. After a 25% discount the customer pays 75% of the regular price.

\[ 0.75P = 73.50 \]

Step 3. Solve for P.

\[ P = \frac{73.50}{0.75} = \frac{7350}{75} = 98 \]

Step 4. State the answer.

\[ P = \$98 \]

Optional. Amount the customer saved.

\[ \text{Savings} = P \times 0.25 = 98 \times 0.25 = 24.50 \]

\[ \text{Customer saved } \$24.50 \]

---

Q1.(b) Two numbers have the ratio 2:3. The larger is 30 more than 1/2 of the smaller. Find the numbers.

---

Step 1. Assume the numbers.

Let the smaller number be \(2x\) and the larger number be \(3x\).

Step 2. Translate the condition.

\[ 3x = \tfrac{1}{2}(2x) + 30 \]

Step 3. Simplify.

\[ 3x = x + 30 \]

Step 4. Solve for \(x\).

\[ 3x - x = 30 \quad \Rightarrow \quad 2x = 30 \quad \Rightarrow \quad x = 15 \]

Step 5. Find the numbers.

\[ \text{Smaller} = 2x = 2(15) = 30 \] \[ \text{Larger} = 3x = 3(15) = 45 \]

Final Answer:

\[ \text{The two numbers are } 30 \text{ and } 45 \]

---

Q1.(c) Jeremy said that if the means and extremes of a proportion are interchanged, the resulting ratios form a proportion. Do you agree with Jeremy? Explain why or why not.

---

A proportion means that two ratios are equal. Suppose we have:

\[ \frac{a}{b} = \frac{c}{d} \]

Here, \(a\) and \(d\) are the extremes, and \(b\) and \(c\) are the means.

If we interchange the means, we get:

\[ \frac{c}{b} = \frac{a}{d} \]

If we interchange the extremes, we get:

\[ \frac{d}{b} = \frac{c}{a} \]

Both of these are still true proportions because in every case the cross multiplication gives:

\[ ad = bc \]

Conclusion: Yes, Jeremy is correct. Interchanging either the means or the extremes in a proportion still results in a true proportion.

---

Q2.(a) If $1,000 is invested at 8% compounded. (i) annually
(ii) semiannually
(iii) quarterly
(iv) monthly
What is the amount after 5 years? Write answers to the nearest cent.

---

\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]

where \(P = 1000\), \(r = 0.08\), \(t = 5\) years, and \(n\) is the number of compounding periods per year.

---

(i) Compounded Annually (n = 1)

\[ A = 1000 \left(1 + \frac{0.08}{1}\right)^{1 \times 5} \]

\[ A = 1000 (1.08)^5 = 1000 \times 1.4693 \approx 1469.33 \]

---

(ii) Compounded Semiannually (n = 2)

\[ A = 1000 \left(1 + \frac{0.08}{2}\right)^{2 \times 5} \]

\[ A = 1000 (1.04)^{10} = 1000 \times 1.4802 \approx 1480.24 \]

---

(iii) Compounded Quarterly (n = 4)

\[ A = 1000 \left(1 + \frac{0.08}{4}\right)^{4 \times 5} \]

\[ A = 1000 (1.02)^{20} = 1000 \times 1.4859 \approx 1485.95 \]

---

(iv) Compounded Monthly (n = 12)

\[ A = 1000 \left(1 + \frac{0.08}{12}\right)^{12 \times 5} \]

\[ A = 1000 (1.0067)^{60} = 1000 \times 1.4899 \approx 1489.85 \]

---

Final Answers:

• (i) Annually: \$1469.33
• (ii) Semiannually: \$1480.24
• (iii) Quarterly: \$1485.95
• (iv) Monthly: \$1489.85

---

Q2.(b) Which is the better investment and why: 9% compounded quarterly or 9.25% compounded annually? Explain your answer.

---

We compare the two options by finding the Effective Annual Rate (EAR).

Option 1: 9% compounded quarterly

\[ EAR = \left(1 + \frac{0.09}{4}\right)^4 - 1 \]

\[ EAR = (1 + 0.0225)^4 - 1 \]

\[ EAR = (1.0225)^4 - 1 \approx 0.09308 \]

\[ EAR \approx 9.31\% \]

Option 2: 9.25% compounded annually

\[ EAR = 9.25\% = 0.0925 \]

The better investment is 9% compounded quarterly because it gives a slightly higher effective annual yield than 9.25% compounded annually.

---

Q3. Determine the solution of the system of equations by (i) Matrix Method (ii) Cramer's rule
10x₁ + 4x₂ = 46
-5x₁ + 6x₂ = 9

---

Let x₁ = x and x₂=y

So, equation is:
10x + 4y = 46
-5x + 6y = 9

---

\[ 10x + 4y = 46 \]

\[ -5x + 6y = 9 \]

---

(i) Matrix Method

---

Write the system as \(A \mathbf{x} = \mathbf{b}\):

\[ A = \begin{pmatrix}10 & 4 \\ -5 & 6\end{pmatrix}, \quad \mathbf{x} = \begin{pmatrix}x \\ y\end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix}46 \\ 9\end{pmatrix} \]

Determinant:

\[ \det(A) = 10 \cdot 6 - (-5)\cdot 4 = 60 + 20 = 80 \]

Inverse:

\[ A^{-1} = \frac{1}{\det(A)} \begin{pmatrix}6 & -4 \\ 5 & 10\end{pmatrix} = \frac{1}{80}\begin{pmatrix}6 & -4 \\ 5 & 10\end{pmatrix} \]

Now:

\[ \mathbf{x} = A^{-1}\mathbf{b} = \frac{1}{80}\begin{pmatrix}6 & -4 \\ 5 & 10\end{pmatrix} \begin{pmatrix}46 \\ 9\end{pmatrix} \]

\[ = \frac{1}{80}\begin{pmatrix}6\cdot46 - 4\cdot9 \\ 5\cdot46 + 10\cdot9\end{pmatrix} = \frac{1}{80}\begin{pmatrix}240 \\ 320\end{pmatrix} = \begin{pmatrix}3 \\ 4\end{pmatrix} \]

\[ x = 3, \quad y = 4 \]

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(ii) Cramer's Rule

---

\[ D = \det(A) = 80 \]

Replace the first column with \(\mathbf{b}\):

\[ D_x = \det\begin{pmatrix}46 & 4 \\ 9 & 6\end{pmatrix} = 46\cdot6 - 9\cdot4 = 276 - 36 = 240 \]

Replace the second column with \(\mathbf{b}\):

\[ D_y = \det\begin{pmatrix}10 & 46 \\ -5 & 9\end{pmatrix} = 10\cdot9 - (-5)\cdot46 = 90 + 230 = 320 \]

So:

\[ x = \frac{D_x}{D} = \frac{240}{80} = 3, \quad y = \frac{D_y}{D} = \frac{320}{80} = 4 \]

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Final Answer:

\[ x = 3, \quad y = 4 \]

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Q4.(a) Simplify the following in binary system.
{(10001101)₂ × (235)₁₀} - (27)₁₀

---

Step 1: Convert the binary number to decimal

---

(10001101)₂ = 1×2⁷ + 0×2⁶ + 0×2⁵ + 0×2⁴ + 1×2³ + 1×2² + 0×2¹ + 1×2⁰

= 128 + 8 + 4 + 1 = 141

---

Step 2: Multiply by 235

---

141 × 235 = 141 × (200 + 30 + 5)

= 141×200 + 141×30 + 141×5

= 28200 + 4230 + 705 = 33135

---

Step 3: Subtract 27

---

33135 - 27 = 33108

---

Step 4: Convert 33108 to binary

---

Divide by 2 repeatedly and record remainders:

Division	Quotient	Remainder
33108 ÷ 2	16554	0
16554 ÷ 2	8277	0
8277 ÷ 2	4138	1
4138 ÷ 2	2069	0
2069 ÷ 2	1034	1
1034 ÷ 2	517	0
517 ÷ 2	258	1
258 ÷ 2	129	0
129 ÷ 2	64	1
64 ÷ 2	32	0
32 ÷ 2	16	0
16 ÷ 2	8	0
8 ÷ 2	4	0
4 ÷ 2	2	0
2 ÷ 2	1	0
1 ÷ 2	0	1

---

Reading the remainders from bottom to top:

33108₁₀ = (1000000101010100)₂

---

Final Answer:

{ (10001101)₂ × (235)₁₀ } - (27)₁₀ = (1000000101010100)₂

---

---

Q4.(b) Solve the equation:

\( \sqrt{2x^2 + 7x + 3} + \sqrt{x^2 + 5x + 6} = \sqrt{x + 3} \)

---

Solving a Square Root Equation

We solve the equation:

\[ \sqrt{2x^{2} + 7x + 3} + \sqrt{x^{2} + 5x + 6} = \sqrt{x + 3} \]

---

Step 1. Domain

Inside square roots must be non-negative:

\(x + 3 \ge 0\) and \(x^{2} + 5x + 6 \ge 0\).

So possible values are \(x = -3\) or \(x \ge -2\).

---

Step 2. Try \(x = -3\)

Substitute \(x = -3\):

\[ \sqrt{2(-3)^{2} + 7(-3) + 3} + \sqrt{(-3)^{2} + 5(-3) + 6} = \sqrt{0} + \sqrt{0} = 0 \]

Right side: \(\sqrt{-3+3} = 0\). So \(x = -3\) is correct.

---

Step 3. Other values

For \(x \ge -2\), the left side (two square roots added) is always bigger than the right side (one square root). So no other solution exists.

---

Final Answer

\[ {x = -3} \]

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Q5.(a)

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Q5.(b)

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Q6.(a) Find six Arithmetic Means between 5 and 12.

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Step 1: Formula

The general term of an arithmetic progression (AP) is: \[ a_n = a_1 + (n-1)d \]

Step 2: Known values

Here, \[ a_1 = 5, \quad a_8 = 12, \quad n = 8 \]

Step 3: Put values into the formula

\[ a_8 = a_1 + (8-1)d \] \[ 12 = 5 + 7d \]

Step 4: Solve for \(d\)

\[ 12 - 5 = 7d \] \[ 7 = 7d \] \[ d = 1 \]

Step 5: Write all terms

\[ a_1 = 5 \] \[ a_2 = a_1 + d = 5 + 1 = 6 \] \[ a_3 = a_1 + 2d = 5 + 2(1) = 7 \] \[ a_4 = a_1 + 3d = 5 + 3(1) = 8 \] \[ a_5 = a_1 + 4d = 5 + 4(1) = 9 \] \[ a_6 = a_1 + 5d = 5 + 5(1) = 10 \] \[ a_7 = a_1 + 6d = 5 + 6(1) = 11 \] \[ a_8 = a_1 + 7d = 5 + 7(1) = 12 \]

Final Answer

The six arithmetic means are: \[ 6, \; 7, \; 8, \; 9, \; 10, \; 11 \]

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Q6.(b) Find the n^th term of a Geometric Progression,

\[ \text{If } \frac{a_5}{a_3} = \frac{16}{9} \quad \text{and} \quad a_2 = \frac{6}{7} \]

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\[ \text{Given: } \quad \frac{a_5}{a_3} = \frac{16}{9}, \quad a_2 = \frac{6}{7} \] \[ \text{In a G.P., } \quad a_n = a \cdot r^{\,n-1} \] \[ a_5 = a \cdot r^4, \quad a_3 = a \cdot r^2 \] \[ \frac{a_5}{a_3} = \frac{a \cdot r^4}{a \cdot r^2} = r^2 \] \[ r^2 = \frac{16}{9} \quad \Rightarrow \quad r = \pm \frac{4}{3} \] \[ a_2 = a \cdot r \] \[ \frac{6}{7} = a \cdot r \quad \Rightarrow \quad a = \frac{6}{7r} \] \[ \text{Case 1: } r = \frac{4}{3} \] \[ a = \frac{6}{7 \cdot \tfrac{4}{3}} = \frac{6}{\tfrac{28}{3}} = \frac{18}{28} = \frac{9}{14} \] \[ \text{So, } a = \frac{9}{14}, \quad r = \frac{4}{3} \] \[ \text{Case 2: } r = -\frac{4}{3} \] \[ a = \frac{6}{7 \cdot \left(-\tfrac{4}{3}\right)} = \frac{6}{-\tfrac{28}{3}} = \frac{18}{-28} = -\frac{9}{14} \] \[ \text{So, } a = -\frac{9}{14}, \quad r = -\frac{4}{3} \] \[ \text{Final Answer: } \quad (a, r) = \left(\tfrac{9}{14}, \tfrac{4}{3}\right) \quad \text{or} \quad (a, r) = \left(-\tfrac{9}{14}, -\tfrac{4}{3}\right) \]

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Q7.(a) \(\text{If } U=\{1,2,3,\dots,20\},\ A=\{2,4,6,\dots,20\} \text{ and } B=\{1,3,5,\dots,19\} \text{ then verify } (A \cup B)^{c} = A^{c} \cap B^{c}\)

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\(U = \{1,2,3,\dots,20\}\)

\(A = \{2,4,6,\dots,20\}\)

\(B = \{1,3,5,\dots,19\}\)

\(A^{c} = U - A = \{1,3,5,\dots,19\} = B\)

\(B^{c} = U - B = \{2,4,6,\dots,20\} = A\)

\(A \cup B = U\)

\((A \cup B)^{c} = U^{c} = \varnothing\)

\(A^{c} \cap B^{c} = B \cap A = \varnothing\)

\(\therefore (A \cup B)^{c} = A^{c} \cap B^{c}\)

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Q7.(b)

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Q8.(a)

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Q8.(b)

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AIOU 247 Past Paper Spring 2017

ALLAMA IQBAL OPEN UNIVERSITY

Level: Matric

Semester: Spring 2017

Course Code: General Mathematics-1 (247)

Maximum Marks: 100

Time Allowed: 03 Hours

Pass Marks: 40

---

Note: Attempt any Five Questions, Question No. 1 is compulsory.

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Q1. Fill in the blanks.

i. 0.28 as percentage is 28%
ii. Zakat on an amount of Rs. 4,00,000 is 10,000.
iii. When SP < CP then Loss% is positive value.
iv. The company undertaking the act of insurance is called the insurer or insurance company.
v. If the annual value of a flat is Rs. 8,00,000. Then the tax payable at a rate of 20% is 160,000.
vi. In 4⁶, 6 is called exponent or power.
vii. If a_n=3n+5 then a₁ is 8.
viii. If R₂={2,1}, {3,2}, {4,3}, then range of R₂ is {1,2,3}
ix. The pair of numbers (4,5) is called an ordered pair.
x. Σ(y_i-Y)=0 is one of the properties of arithmetic mean.

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Q2.(a) Sadia scored 40 out of 70 in a English test, 70 out of 80 in a Mathematics test and 67 out of 75 in a Chemistry test. In which subject did she perform best?

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\[ \text{English: } \frac{40}{70} \times 100 = 57.14\% \] \[ \text{Mathematics: } \frac{70}{80} \times 100 = 87.5\% \] \[ \text{Chemistry: } \frac{67}{75} \times 100 \approx 89.33\% \]

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Sadia performed best in Chemistry 89.33%.

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Q2.(b) 5 persons can do a job in 15 days. If 2 more persons are employed, how many days will they take to complete the job?

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\[ \text{Total work} = 5 \times 15 = 75 \text{ man-days} \]

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If 2 more persons are added, total persons = 5 + 2 = 7

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\[ \text{Time required} = \frac{75}{7} \approx 10.71 \text{ days} \]

---

They will complete the job in about 10.7 days.

---

Q3.(a) Aslam left a property of worth Rs. 48,00,000. Calculate the amount of share of his wife, two sons and two daughters.

---

\[ \text{Total property} = 48,00,000 \] \[ \text{Wife's share} = \tfrac{1}{8} \times 48,00,000 = 6,00,000 \] \[ \text{Remaining property} = 48,00,000 - 6,00,000 = 42,00,000 \] \[ \text{Children's shares: Sons : Daughters = 2 : 1} \] \[ \text{For 2 sons and 2 daughters, ratio } = 2 + 2 + 1 + 1 = 6 \text{ parts} \] \[ \text{One part} = \tfrac{42,00,000}{6} = 7,00,000 \] \[ \text{Each son} = 2 \times 7,00,000 = 14,00,000 \] \[ \text{Each daughter} = 1 \times 7,00,000 = 7,00,000 \] \[ \therefore \text{Wife } = 6,00,000, \quad \text{Each Son } = 14,00,000, \quad \text{Each Daughter } = 7,00,000 \]

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Q3.(b) Three chairs are purchased at Rs. 500 each. One of these is sold at a loss of 10%, at what price should the other two be sold so as to gain 20% on the whole transaction?

---

\[ \text{Total cost of 3 chairs} = 3 \times 500 = 1500 \] \[ \text{Selling price of 1 chair at 10\% loss} = 500 - \tfrac{10}{100}\times 500 = 450 \] \[ \text{Required total selling price for 20\% gain} = 1500 + \tfrac{20}{100}\times 1500 = 1800 \] \[ \text{Amount to be obtained from 2 chairs} = 1800 - 450 = 1350 \] \[ \text{Selling price of each of the 2 chairs} = \tfrac{1350}{2} = 675 \] \[ \therefore \text{Each of the other two chairs should be sold at Rs. 675.} \]

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Q4.(a) Farhan purchases a car in Saudi Arabia for 20,000 Riyals. Delivery was to be made after two months and payment is also to be made at the time of delivery. At the time of contract, the rate was 1 Riyal = Rs. 21.5, while at the time of delivery the rate was 1 Riyal = Rs. 21 02. Determine the loss in rupees due to change in the rate.

---

Car cost = 20,000 Riyals
Rate at contract time: 1 Riyal = 21.5 Rs
Rate at delivery time: 1 Riyal = 21.02 Rs

Cost in Rs at contract time: \[ 20,000 \times 21.5 = 430,000 \text{ Rs} \]

Cost in Rs at delivery time: \[ 20,000 \times 21.02 = 420,400 \text{ Rs} \]

Loss due to exchange rate change: \[ 430,000 - 420,400 = 9,600 \text{ Rs} \]

\[ \therefore \text{Loss} = 9,600 \text{ Rs} \]

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Q4.(b) If gross pay of a person is Rs. 50,000, then calculate his net take home salary, after deductions of Rs. 500 as income tax. Rs. 1400 as benevolent fund, Rs 1800 as G.P fund and Rs. 300 as group insurance.

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Gross pay = Rs. 50,000
Deductions:
Income tax = Rs. 500
Benevolent fund = Rs. 1,400
G.P. fund = Rs. 1,800
Group insurance = Rs. 300

Total deductions: \[ 500 + 1400 + 1800 + 300 = 4000 \text{ Rs} \]

Net take-home salary: \[ \text{Net salary} = \text{Gross pay} - \text{Total deductions} = 50,000 - 4,000 = 46,000 \text{ Rs} \]

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Q5.(a) Prove that log_a ab = log_aa + log_ab

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\[ \text{LHS} = \log_a(ab) \] \[ \text{Using the logarithm property } \log_a(xy) = \log_a x + \log_a y \] \[ \log_a(ab) = \log_a a + \log_a b \] \[ \text{RHS} = \log_a a + \log_a b \] \[ \therefore \log_a(ab) = \log_a a + \log_a b \]

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Q5.(b) Using logarithmic table evaluate.

\[ \dfrac{(0.0537)^{\tfrac{1}{2}} \times (1.5)^3}{(0.0014)^{\tfrac{1}{3}} \times (1.435)^{\tfrac{1}{6}}} \]

\[ N \;=\; \dfrac{(0.0537)^{\tfrac{1}{2}} \times (1.5)^3}{(0.0014)^{\tfrac{1}{3}} \times (1.435)^{\tfrac{1}{6}}} \]

Take base ten logarithms

\[ \log N \;=\; \tfrac{1}{2}\log(0.0537) \;+\; 3\log(1.5) \;-\; \tfrac{1}{3}\log(0.0014) \;-\; \tfrac{1}{6}\log(1.435) \]

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Using logarithm table values rounded to four decimal places

Number	Logarithm
\(0.0537\)	\(\log(0.0537)= -1.2700\)
\(1.5\)	\(\log(1.5)= 0.1761\)
\(0.0014\)	\(\log(0.0014)= -2.8539\)
\(1.435\)	\(\log(1.435)= 0.1569\)

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Multiply by the powers

\[ \tfrac{1}{2}\log(0.0537) = \tfrac{1}{2}\times(-1.2700) = -0.6350 \]

\[ 3\log(1.5) = 3\times 0.1761 = 0.5283 \]

\[ \tfrac{1}{3}\log(0.0014) = \tfrac{1}{3}\times(-2.8539) = -0.9513 \]

\[ \tfrac{1}{6}\log(1.435) = \tfrac{1}{6}\times 0.1569 \approx 0.0262 \]

Now assemble the terms

\[ \log N = -0.6350 + 0.5283 - (-0.9513) - 0.0262 \]

\[ \log N = -0.6350 + 0.5283 + 0.9513 - 0.0262 = 0.8184 \]

Antilog gives the final result

\[ N = 10^{0.8184} \approx 6.5828 \]

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Q6.(a) Find six Arithmetic Means between 5 and 12.

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Step 1: Formula

The general term of an arithmetic progression (AP) is: \[ a_n = a_1 + (n-1)d \]

Step 2: Known values

Here, \[ a_1 = 5, \quad a_8 = 12, \quad n = 8 \]

Step 3: Put values into the formula

\[ a_8 = a_1 + (8-1)d \] \[ 12 = 5 + 7d \]

Step 4: Solve for \(d\)

\[ 12 - 5 = 7d \] \[ 7 = 7d \] \[ d = 1 \]

Step 5: Write all terms

\[ a_1 = 5 \] \[ a_2 = a_1 + d = 5 + 1 = 6 \] \[ a_3 = a_1 + 2d = 5 + 2(1) = 7 \] \[ a_4 = a_1 + 3d = 5 + 3(1) = 8 \] \[ a_5 = a_1 + 4d = 5 + 4(1) = 9 \] \[ a_6 = a_1 + 5d = 5 + 5(1) = 10 \] \[ a_7 = a_1 + 6d = 5 + 6(1) = 11 \] \[ a_8 = a_1 + 7d = 5 + 7(1) = 12 \]

Final Answer

The six arithmetic means are: \[ 6, \; 7, \; 8, \; 9, \; 10, \; 11 \]

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Q6.(b) Find the n^th term of a Geometric Progression,

\[ \text{If } \frac{a_5}{a_3} = \frac{16}{9} \quad \text{and} \quad a_2 = \frac{6}{7} \]

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\[ \text{Given: } \quad \frac{a_5}{a_3} = \frac{16}{9}, \quad a_2 = \frac{6}{7} \] \[ \text{In a G.P., } \quad a_n = a \cdot r^{\,n-1} \] \[ a_5 = a \cdot r^4, \quad a_3 = a \cdot r^2 \] \[ \frac{a_5}{a_3} = \frac{a \cdot r^4}{a \cdot r^2} = r^2 \] \[ r^2 = \frac{16}{9} \quad \Rightarrow \quad r = \pm \frac{4}{3} \] \[ a_2 = a \cdot r \] \[ \frac{6}{7} = a \cdot r \quad \Rightarrow \quad a = \frac{6}{7r} \] \[ \text{Case 1: } r = \frac{4}{3} \] \[ a = \frac{6}{7 \cdot \tfrac{4}{3}} = \frac{6}{\tfrac{28}{3}} = \frac{18}{28} = \frac{9}{14} \] \[ \text{So, } a = \frac{9}{14}, \quad r = \frac{4}{3} \] \[ \text{Case 2: } r = -\frac{4}{3} \] \[ a = \frac{6}{7 \cdot \left(-\tfrac{4}{3}\right)} = \frac{6}{-\tfrac{28}{3}} = \frac{18}{-28} = -\frac{9}{14} \] \[ \text{So, } a = -\frac{9}{14}, \quad r = -\frac{4}{3} \] \[ \text{Final Answer: } \quad (a, r) = \left(\tfrac{9}{14}, \tfrac{4}{3}\right) \quad \text{or} \quad (a, r) = \left(-\tfrac{9}{14}, -\tfrac{4}{3}\right) \]

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Q7.(a) \(\text{If } U=\{1,2,3,\dots,20\},\ A=\{2,4,6,\dots,20\} \text{ and } B=\{1,3,5,\dots,19\} \text{ then verify } (A \cup B)^{c} = A^{c} \cap B^{c}\)

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\(U = \{1,2,3,\dots,20\}\)

\(A = \{2,4,6,\dots,20\}\)

\(B = \{1,3,5,\dots,19\}\)

\(A^{c} = U - A = \{1,3,5,\dots,19\} = B\)

\(B^{c} = U - B = \{2,4,6,\dots,20\} = A\)

\(A \cup B = U\)

\((A \cup B)^{c} = U^{c} = \varnothing\)

\(A^{c} \cap B^{c} = B \cap A = \varnothing\)

\(\therefore (A \cup B)^{c} = A^{c} \cap B^{c}\)

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Q7.(b)

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Q8.(a)

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Q8.(b)

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پاکستان میں بولی جانے والی زبانوں کے بارے میں آپ کیا جانتے ہیں؟

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اردو، سندھی، پشتو، بلوچی اور پنجابی زبان کے بارے میں آپ کیا جانتے ہیں؟

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تعارف

زبان کسی قوم کی شناخت اور ثقافتی ورثے کی امین ہوتی ہے۔ پاکستان مختلف زبانوں کا گہوارہ ہے جہاں ہر زبان اپنی تاریخ، ادب اور ثقافت کے ساتھ ایک منفرد رنگ لے کر آتی ہے۔ اردو، سندھی، پشتو، بلوچی اور پنجابی نہ صرف عوامی رابطے کے ذرائع ہیں بلکہ یہ قومی یکجہتی، علاقائی تہذیب اور ادبی تنوع کی نمائندہ بھی ہیں۔ ہر زبان نے مقامی معاشرت، صوفیانہ روایت اور عوامی اظہار کو مستحکم کیا ہے اور یہ زبانیں تعلیمی، ثقافتی اور غیر رسمی سطح پر لوگوں کو جوڑنے کا کام کرتی ہیں۔

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اردو زبان

اردو پاکستان کی قومی زبان ہے اور مختلف صوبوں کے درمیان رابطے کا اہم ذریعہ سمجھی جاتی ہے۔ اس کی تشکیل فارسی، عربی، ترکی اور مقامی بولیوں کے امتزاج سے ہوئی، جس نے اسے ادبی امتیاز اور اظہار کی قوت بخشی۔ اردو نے غزل، نظم اور افسانے کے ذریعے عظیم ادبی موجد دیے؛ غالب، اقبال اور فیض جیسے شعرا نے اس زبان کو عالمی مقام دیا۔ تحریکِ پاکستان میں اردو نے قومی احساسات کو پروان چڑھانے میں کلیدی کردار ادا کیا۔ آج اردو ذرائع ابلاغ، تعلیم، عدلیہ اور سرکاری معاملات میں وسیع پیمانے پر استعمال ہوتی ہے اور لاکھوں افراد کے درمیان ثقافتی ہم آہنگی کی علامت بنی ہوئی ہے۔

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سندھی زبان

سندھی زبان سندھ کے تہذیبی ورثے کی ترجمان ہے اور اس کا تاریخی رشتہ بہت قدیم تہذیبوں سے جڑا ہوا ہے۔ سندھی کی مٹھاس اور لوک ادبیات نے اسے نمایاں مقام دیا ہے، خاص طور پر صوفی شاعری میں شاہ عبداللطیف بھٹائی اور سچل سرمست کے کلام نے اس زبان کی شان بڑھائی۔ سندھی رسم الخط عربی حروف سے مشابہت رکھتا ہے مگر اس میں کچھ مخصوص حروف شامل ہیں۔ سندھ کی عوامی زندگی، روایات اور تہوار سندھی زبان میں واضح طور پر اظہار پاتے ہیں، اور یہ صوبائی پہچان کے ساتھ ساتھ قومی ثقافتی تنوع میں نمایاں حصہ ڈالتی ہے۔

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پشتو زبان

پشتو زبان خیبرپختونخوا اور پشتون علاقوں کی ثقافتی علامت ہے اور ایرانی زبانوں کے خاندان سے تعلق رکھتی ہے۔ پشتو میں بہادری، غیرت اور مہمان نوازی کے جذبے نمایاں ہیں اور اس کی شاعری نے پشتون شناخت کو مضبوط کیا ہے۔ رحمان بابا اور خوشحال خان خٹک جیسے شعرا نے پشتو ادب کو بلند پایہ عطا کیا۔ آج پشتو ریڈیو، ٹی وی اور تعلیمی سطح پر وسیع پیمانے پر مستعمل ہے، اور یہ زبان پشتون معاشرتی اقدار، تاریخی واقعات اور لوک روایات کو نئی نسل تک منتقل کرنے میں کلیدی کردار ادا کرتی ہے۔

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بلوچی زبان

بلوچی زبان بلوچستان کے لوگوں کی مادری زبان ہے اور یہ ایرانی لسانی گروہ سے متعلقہ ہے۔ بلوچی نے لوک کہانیوں، داستانوں اور گیتوں کے ذریعے بلوچ ثقافت کا بھرپور اظہار کیا ہے۔ اس زبان کے مختلف لہجے اور ذیلی بولیاں موجود ہیں جو علاقائی تنوع کو ظاہر کرتی ہیں۔ بلوچی شاعری اور افسانوی روایت نے بلوچ قوم کو اجتماعی شناخت دی ہے، اور علاقے کی روایات، بہادری اور مہمان نوازی اسی زبان کے ذریعے زندہ رکھی جاتی ہیں۔ ایران، افغانستان اور خلیجی علاقوں میں بھی بلوچی بولنے والے ملتے ہیں، جو اسے علاقائی سطح پر بھی اہم بناتا ہے۔

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پنجابی زبان

پنجابی پاکستان کی سب سے زیادہ بولی جانے والی لسانی روایت ہے اور پنجاب کے عوام کی روزمرہ زندگی، تہوار اور صوفیانہ رنگ اسی زبان میں جھلکتے ہیں۔ پنجابی کا ادبی خزانہ بابا فرید، وارث شاہ اور بلھے شاہ جیسے شعرا کی بدولت بے مثال ہے، جن کے کلام نے معاشرتی علامات اور روحانی تعلیمات کو عوام تک پہنچایا۔ پنجابی کے مختلف لہجے—ماجھی، پوٹوہاری، سرائیکی وغیرہ—اس کی لچک اور وسعت کو ظاہر کرتے ہیں۔ تھیٹر، موسیقی اور لوک کہانیاں پنجابی کو عوامی ثقافت میں زندہ رکھتی ہیں۔

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خلاصہ

اردو، سندھی، پشتو، بلوچی اور پنجابی زبانیں پاکستان کی ثقافتی، ادبی اور تاریخی شناخت کا قیمتی سرمایہ ہیں۔ ہر زبان نے اپنی منفرد صوفیانہ، ادبی اور لوک روایت کے ذریعے معاشرتی اقدار اور علاقائی شناخت کو پروان چڑھایا ہے۔ یہ زبانیں نہ صرف علاقائی رنگ اور تنوع کو ظاہر کرتی ہیں بلکہ قومی یکجہتی اور ثقافتی ہم آہنگی میں بھی کلیدی کردار ادا کرتی ہیں۔

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