How to Make Students Fear of Mathematics Disappear

* Welcome to Gyaan Kunja Tuition Centre podcast. I'm your host, Rohan, and today we are tackling a topic that haunts the nightmares of students and parents alike: Math Anxiety. Why do so many kids fear Mathematics, and more importantly, how can we make that fear disappear? Joining me today is Dr. Ananya Sharma, a renowned child education specialist and psychologist. Welcome to the show, Dr. Sharma.

Dr. Sharma: Thank you, Rohan. It’s a pleasure to be here. This is such a critical topic because math anxiety is incredibly common, yet highly preventable.

Anchor: Let’s dive right in. I remember sweating before my algebra exams when I was a kid. Why is math, out of all subjects, the one that causes the most panic?

Dr. Sharma: Well, Rohan, unlike a subject like literature where there can be multiple interpretations, math is often viewed as strictly black and white. You are either right or you are wrong. This creates a high-pressure environment. Plus, math is highly cumulative. If a student misses a core concept in third grade, they will inevitably struggle in fourth grade. That confusion snowballs into frustration, and eventually, a deep-seated fear.

Anchor: That makes a lot of sense. The pressure to be 'perfect' can be paralyzing. So, if a parent or teacher notices a child starting to develop this fear, what is the very first step they should take?

Dr. Sharma: The first step is validation. Don't say, "Oh, math is easy, you're just not trying hard enough." That dismisses their feelings and makes them feel isolated. Instead, say, "I can see this is really frustrating for you. Math can be tough, but we are going to figure it out together." Creating a safe space to make mistakes is crucial. Kids need to know that making a mistake isn't a failure; it is a required stepping stone to understanding.

Anchor: "A stepping stone to understanding." I love that phrasing. But practically speaking, how do we change the way math is taught or practiced at home to make it less intimidating?

Dr. Sharma: Gamification and real-world application are your best friends here. Math on a black-and-white worksheet can look like a foreign language. But math in the kitchen? That's baking. Math in the grocery store? That's budgeting. When you ask a child to help you double a recipe, calculate the discount on a toy, or figure out the change at a shop, they are doing math without the pressure of a ticking clock or a red pen.

Anchor: So, taking it out of the textbook and bringing it into reality. 

Dr. Sharma: Exactly. Another incredibly powerful tool is changing the way we praise our children. We often praise the result: "Wow, you got an A!" or "You answered that so fast!" Instead, we need to praise the effort and the process: "I noticed how hard you worked on that difficult problem," or "I'm proud of you for not giving up when that division got tricky." This builds what we call a 'growth mindset'.

Anchor: What about the teachers? What role do they play in the classroom to alleviate this anxiety?

Dr. Sharma: Teachers have a monumental role. The best math classrooms I've seen are loud and collaborative. Students are discussing problems, drawing visual models, and using physical blocks or manipulatives. When math becomes a hands-on, collaborative puzzle rather than a silent, solitary test, the fear naturally dissipates.

Anchor: It sounds like we need a massive cultural shift in how we view the subject as a society. Less about finding the 'right' answer quickly, and more about the journey of problem-solving.

Dr. Sharma: Spot on, Rohan. We need to normalize being stuck. The greatest mathematicians in the world spend most of their time being stuck. If we can teach kids to be comfortable with the struggle, the fear of math will disappear, replaced by genuine curiosity.

Anchor: Dr. Sharma, thank you so much for these invaluable insights. For our listeners at home, remember: patience, real-world practice, and praising the effort can transform your child's relationship with math. This brings us to the end of today's episode. Thank you for tuning in to the Gyaan Kunja Tuition Centre podcast. Keep learning, keep growing!*

Title: LLP Registration in India – Pros, Cons, Features & Complete Guide

Title: LLP Registration in India – Pros, Cons, Features & Complete Guide

[INTRO – Hook]

Thinking of starting a business in India but confused between Proprietorship, LLP, and Private Limited?

Then this video is for you.

Today we will understand LLP — Limited Liability Partnership.

What it is, its features, advantages, disadvantages, taxation, compliance, and whether it is the right choice for your business.

Let’s begin.

---

[SECTION 1 – What is LLP?]

LLP stands for Limited Liability Partnership.

It is a business structure introduced under the Limited Liability Partnership Act, 2008 in India.

Relevant authority:
Ministry of Corporate Affairs

It combines two things:

Partnership flexibility
plus
Company-like limited liability.

This means:

Partners can manage the business directly,
but their personal assets remain protected.

For example:

If your LLP has a debt of 10 lakh rupees and the business fails, creditors generally cannot take your personal house or car.

That is the major benefit.

---

[SECTION 2 – Main Features of LLP]

Now let’s understand all important features.

Feature 1: Separate Legal Entity

An LLP is separate from its owners.

It can:

Own assets
Sign contracts
Sue or be sued

in its own name.

---

Feature 2: Limited Liability

Each partner’s liability is limited to their agreed contribution.

Personal risk stays limited.

---

Feature 3: Minimum Two Partners

To start LLP:

Minimum 2 partners required.

No maximum limit.

You can have many partners.

---

Feature 4: No Minimum Capital

There is no minimum capital requirement.

You can start with even 1,000 rupees.

---

Feature 5: Flexible Management

Unlike companies, LLP has fewer management restrictions.

Partners decide everything through LLP Agreement.

---

Feature 6: Lower Compliance

Compared to Private Limited companies, LLP has less paperwork.

Less stress.

Less compliance cost.

---

Feature 7: Perpetual Succession

Even if one partner leaves or dies, LLP continues.

Business does not stop.

---

Feature 8: Better Credibility

Compared to Proprietorship, LLP gives better professional image.

Clients trust it more.

---

[SECTION 3 – Advantages of LLP]

Now let’s talk about the pros.

Advantage 1: Personal Asset Protection

Biggest advantage.

Your personal property remains safer.

---

Advantage 2: Low Setup Cost

Cheaper than Private Limited.

Usually between:

8,000 to 20,000 rupees depending on professionals and state.

---

Advantage 3: Less Compliance Cost

No board meetings.

No heavy secretarial compliance.

Lower annual filing burden.

---

Advantage 4: Tax Simplicity

Flat taxation.

No dividend tax complications.

Simple profit sharing.

---

Advantage 5: Easy Ownership Distribution

You can distribute profit ratios as per agreement.

No strict shareholding rules.

---

Advantage 6: Suitable for Service Businesses

Best for:

HR firms
Consulting
IT services
Law firms
Agencies
CA firms
Freelancers

---

[SECTION 4 – Disadvantages of LLP]

Now the cons.

Disadvantage 1: Minimum Two Partners Required

You cannot start alone.

For solo founders, proprietorship is easier.

---

Disadvantage 2: Hard to Raise Investors

Investors usually prefer Private Limited companies.

LLP is weak for venture capital.

---

Disadvantage 3: Ownership Transfer is Complex

Transferring partnership rights is not as easy as shares.

---

Disadvantage 4: Higher Closure Complexity

Closing LLP is more complex than closing proprietorship.

---

Disadvantage 5: Public Disclosure

Financial filings become visible through government records.

Less privacy.

---

[SECTION 5 – LLP Compliance in India]

Many people think LLP has no compliance.

That is wrong.

Important compliance:

Annual Return Filing
Statement of Accounts
Income Tax Return
GST Return if applicable
TDS Return if applicable

Government portal:
MCA LLP Services

Late filing can result in penalties.

---

[SECTION 6 – Taxation of LLP]

LLP tax rate:

30% plus cess and surcharge as applicable.

No dividend distribution tax.

Partners can withdraw profits.

This makes LLP tax-efficient in many service businesses.

---

[SECTION 7 – LLP vs Proprietorship]

Proprietorship:

Easy to start
Low cost
High personal risk

LLP:

Better protection
Better trust
Better structure

If you are serious about growth, LLP is stronger.

---

[SECTION 8 – LLP vs Private Limited]

Private Limited:

Best for funding
Best for scaling
High compliance

Relevant entity:
Startup India

LLP:

Best for service businesses
Lower compliance
Less investor friendly

---

[SECTION 9 – Who Should Choose LLP?]

Choose LLP if:

You have 2 or more founders
You want limited liability
You run service business
You do not need investors immediately
You want low compliance

Examples:

Recruitment agency
Software company
Digital marketing agency
Accounting firm
Consultancy firm

---

[OUTRO]

So that was the complete guide to LLP registration in India.

If you are starting small but want legal protection, LLP can be an excellent choice.

But if you plan to raise investment in future, Private Limited may be better.

Choose based on your business model.

Thank you for watching.

Where should I start learning coding if I want to become a full-stack developer one day?

If you want to become a full-stack developer, here’s a structured roadmap for beginners:

1. Start with Basic Programming

• Learn JavaScript (essential for full-stack) or Python (easy for beginners).
• Focus on variables, loops, conditionals, functions, arrays/objects, and basic problem-solving.
• Resources: freeCodeCamp, Codecademy, Python.org

2. Learn Frontend Development

• HTML: Structure of webpages
• CSS: Styling and layout
• JavaScript: Interactivity
• Frameworks/Libraries: React.js, Vue.js, or Angular (React recommended)
• Resources: MDN Web Docs, freeCodeCamp Frontend

3. Learn Backend Development

• Languages: Node.js (JavaScript), Python (Django/Flask), PHP
• Learn APIs (REST, JSON), databases, authentication, and routing
• Resources: Node.js Docs, Django Tutorial

4. Learn Databases

• SQL: MySQL, PostgreSQL
• NoSQL: MongoDB
• Practice CRUD operations (Create, Read, Update, Delete)

5. Version Control & Deployment

• Learn Git & GitHub for version control
• Deploy projects using Netlify, Vercel, or Heroku

6. Build Projects

• Start small: Personal portfolio website
• Medium projects: To-do list app (frontend + backend)
• Larger projects: Blog platform, e-commerce site

7. Keep Learning

• Learn about security, testing, and optimization
• Explore cloud services (AWS, Firebase)
• Follow trends in full-stack frameworks

Tip: Start with HTML, CSS, JavaScript first, then gradually move to backend. Building projects along the way helps you learn faster.

Learning English as Second Language

Learning a spoken language isn't just an academic exercise—it is a physiological process. You aren't just memorizing rules; you are physically rewiring your brain to process new sounds and coordinate your vocal tract to produce them.

The scientific study of this process is called Second Language Acquisition (SLA). If you want to achieve spoken fluency, science points to a few core mechanisms you need to activate.

1. The Brain Split: Understanding vs. Speaking

Many language learners get stuck in a frustrating phase where they can understand the target language perfectly, but freeze up when trying to speak. Neuroscience explains exactly why this happens.

 

Language is processed in two distinct parts of the brain:

• Wernicke’s Area: This handles comprehension (listening and reading). It decodes the meaning of the words you hear.

• Broca’s Area: This handles production (speaking). It is located near the motor cortex and literally commands your mouth, tongue, and vocal cords to move.

When you read textbooks or listen to podcasts, you are only training Wernicke’s area. To speak, you must physically train Broca’s area through the actual act of moving your mouth.

2. Comprehensible Input (The Foundation)

Before you can produce output, you need high-quality input. Linguist Stephen Krashen’s widely accepted Input Hypothesis states that we acquire language in only one way: by understanding messages.

You should expose yourself to massive amounts of Comprehensible Input ($i+1$):

• $i$ represents your current level of understanding.

• $+1$ represents language that is just one step slightly above your current level.

If the content is too easy ($i-1$), you don't learn anything new. If it's too hard ($i+10$), it sounds like noise and your brain ignores it. You need content where you understand the general context, allowing your brain to naturally deduce the meaning of the new, unknown words.

3. Phonemic Mapping and "Shadowing"

Every mother tongue has a specific set of sounds (phonemes). When you learn a new language, your brain naturally tries to map the new sounds onto the sounds you already know.

This often leads to mispronunciations. For example, if your mother tongue lacks the "th" sound, your brain might substitute an "s" or a "t"—making a word like "both" sound like "boat".

To fix this scientifically, you have to bypass your mother tongue's phonemic map. The best method for this is Shadowing:

1. Listen to a short audio clip of a native speaker (preferably 10-15 seconds).

2. Repeat what they say immediately after they say it—almost speaking over them.

3. Mimic everything: Don't just match the words; copy the rhythm, the pitch, and the exact physical mouth movements.

Shadowing forces Broca's area to build new muscle memory independent of your mother tongue.

4. Spaced Repetition to Fight the "Forgetting Curve"

In the late 19th century, Hermann Ebbinghaus discovered the Forgetting Curve, demonstrating that our brains are programmed to forget new information exponentially fast unless it is reviewed.

To move vocabulary from short-term to long-term memory, use Spaced Repetition Systems (SRS). Instead of reviewing a new word every day, an SRS algorithm tests you at increasing intervals: 1 day, then 3 days, then 10 days, then a month. You review the word right at the mathematical moment you are about to forget it, which signals to your brain that the information is crucial for survival.

The Scientific Action Plan

If you want to apply this research to your daily routine, structure your study like this:

• 80% Input: Spend the vast majority of your time listening to or reading content that is highly interesting to you, but just slightly above your level.

• 10% Shadowing: Spend a few minutes every day physically speaking out loud, mimicking native audio to build motor skills in your vocal tract and correct native-language interference.

• 10% Output with Feedback: Have conversations where you receive immediate, gentle corrections when you make a mistake, allowing your brain to adjust its patterns in real-time.

NCERT Class X Mathematics Chapter 1: Real Numbers

Chapter 1: Real Numbers

Comprehensive Study Notes, Day-by-Day Explanations, and Question Bank

Part 1: Day-to-Day Study Notes & Step-by-Step Explanations

Day 1: The Fundamental Theorem of Arithmetic

Objective: Understand prime factorization of composite numbers and its uniqueness.

Theorem 1.1 (Fundamental Theorem of Arithmetic): Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.

Step-by-Step Explanation:

• Any natural number can be written as a product of its prime factors.
• We typically arrange prime factors in ascending order (e.g., $p_1 \le p_2 \le \dots \le p_n$).
• Example: Let's factorise 32760 using a factor tree. We get $32760 = 2 \times 2 \times 2 \times 3 \times 3 \times 5 \times 7 \times 13 = 2^3 \times 3^2 \times 5 \times 7 \times 13$.

Day 2: Finding HCF and LCM Using Prime Factorisation

Objective: Apply prime factorization to find Highest Common Factor (HCF) and Least Common Multiple (LCM), and understand their relationship.

Step-by-Step Explanation:

• HCF: Product of the smallest power of each common prime factor in the numbers.
• LCM: Product of the greatest power of each prime factor, involved in the numbers.
• Crucial Formula: For any two positive integers $a$ and $b$, $HCF(a, b) \times LCM(a, b) = a \times b$.
• Note for 3 numbers: $HCF(p, q, r) \times LCM(p, q, r) \ne p \times q \times r$.

Application Example: Check whether $4^n$ can end with the digit zero. For a number to end with zero, its prime factorisation must contain the prime 5. Since $4^n = (2)^{2n}$, the only prime factor is 2[cite: 102, 103]. Thus, it can never end with zero.

Day 3: Revisiting Irrational Numbers & Theorem 1.2

Objective: Prove the irrationality of numbers like $\sqrt{2}$ and $\sqrt{3}$ using proof by contradiction.

Theorem 1.2: Let $p$ be a prime number. If $p$ divides $a^2$, then $p$ divides $a$, where $a$ is a positive integer.

Step-by-Step Proof that $\sqrt{2}$ is irrational:

1. Assume the contrary: Let $\sqrt{2}$ be a rational number.
2. Write it in simplest form: $\sqrt{2} = \frac{a}{b}$ where $a$ and $b$ are coprime integers ($b \ne 0$)[cite: 191].
3. Rearrange and square: $b\sqrt{2} = a$ becomes $2b^2 = a^2$[cite: 192, 193].
4. This means 2 divides $a^2$, so by Theorem 1.2, 2 divides $a$. Let $a = 2c$.
5. Substitute back: $2b^2 = (2c)^2 = 4c^2 \implies b^2 = 2c^2$.
6. This means 2 divides $b^2$, so 2 divides $b$.
7. Contradiction: Both $a$ and $b$ are divisible by 2, contradicting that they are coprime[cite: 197, 199]. Thus, $\sqrt{2}$ is irrational.

Day 4: Combinations of Rational and Irrational Numbers

Objective: Prove expressions like $5 - \sqrt{3}$ are irrational.

Key Facts:

• The sum or difference of a rational and an irrational number is irrational.
• The product and quotient of a non-zero rational and irrational number is irrational.

Step-by-Step Proof for $5 - \sqrt{3}$:

1. Assume $5 - \sqrt{3}$ is rational, meaning $5 - \sqrt{3} = \frac{a}{b}$ (coprime integers, $b \ne 0$)[cite: 231, 232, 233].
2. Rearrange to isolate the root: $\sqrt{3} = 5 - \frac{a}{b} = \frac{5b - a}{b}$.
3. Since $a$ and $b$ are integers, $\frac{5b - a}{b}$ is rational, implying $\sqrt{3}$ is rational[cite: 237, 238].
4. This contradicts the known fact that $\sqrt{3}$ is irrational. Hence, $5 - \sqrt{3}$ is irrational[cite: 239, 240].

---

Part 2: Multiple Choice Questions (MCQs)

Ideal for quick conceptual testing.

1. If $HCF(306, 657) = 9$, what is the $LCM(306, 657)$?

   • 22338
   • 23328
   • 28332
   • 32238

   Answer: (a) Use formula $LCM = \frac{a \times b}{HCF}$.

2. The prime factorisation of 140 is:

   • $2 \times 7 \times 10$
   • $2^2 \times 5 \times 7$
   • $2 \times 5^2 \times 7$
   • $2^2 \times 3 \times 7$

   Answer: (b)

3. According to Theorem 1.2, if a prime $p$ divides $a^2$, then:

   • $p$ divides $a$
   • $a$ divides $p$
   • $p^2$ divides $a$
   • None of the above

   Answer: (a)

4. The HCF of 96 and 404 is:

   • 2
   • 4
   • 8
   • 12

   Answer: (b)

5. The LCM of 24 and 36 is:

   • 48
   • 60
   • 72
   • 96

   Answer: (c)

6. The prime factorisation of 180 is:

   • $2^2 \times 3^2 \times 5$
   • $2 \times 3^2 \times 5$
   • $2^2 \times 3 \times 5$
   • $2^3 \times 3^2 \times 5$

   Answer: (a)

7. If $HCF(72, 120) = 24$, then $LCM(72,120)$ is:

   • 240
   • 360
   • 480
   • 600

   Answer: (b) Use $LCM = \frac{a \times b}{HCF}$.

8. Which of the following is irrational?

   • $\sqrt{16}$
   • $\sqrt{25}$
   • $\sqrt{7}$
   • $\frac{3}{4}$

   Answer: (c)

9. The decimal expansion of $\frac{7}{8}$ is:

   • Terminating
   • Non-terminating repeating
   • Non-terminating non-repeating
   • None of these

   Answer: (a)

10. The HCF of two consecutive integers is:

    • 0
    • 1
    • 2
    • The smaller number

    Answer: (b)

11. If $p$ is a prime number and $p$ divides the product $ab$, then:

    • $p$ divides $a$ or $p$ divides $b$
    • $p$ divides $a$ and $p$ divides $b$
    • $a$ divides $p$
    • $b$ divides $p$

    Answer: (a)

12. The LCM of two coprime numbers is equal to:

    • 1
    • Their difference
    • Their product
    • Their sum

    Answer: (c)

13. The HCF of 45 and 75 is:

    • 5
    • 10
    • 15
    • 25

    Answer: (c)

14. Which of the following has a terminating decimal expansion?

    • $\frac{13}{30}$
    • $\frac{7}{64}$
    • $\frac{11}{45}$
    • $\frac{17}{27}$

    Answer: (b)

15. The prime factorisation of 225 is:

    • $3^2 \times 5^2$
    • $3^3 \times 5$
    • $9 \times 25$
    • $15^2$

    Answer: (a)

16. Euclid’s Division Lemma states that for given positive integers $a$ and $b$, there exist unique integers $q$ and $r$ such that:

    • $a = bq + r$, $0 \le r < b$
    • $a = br + q$
    • $b = aq + r$
    • $a = b + q + r$

    Answer: (a)

17. The HCF of 17 and 23 is:

    • 1
    • 17
    • 23
    • 391

    Answer: (a)

18. If $LCM(a,b)=180$ and $HCF(a,b)=6$, and $a=30$, then $b$ is:

    • 18
    • 24
    • 36
    • 54

    Answer: (c)

19. The decimal expansion of $\frac{2}{11}$ is:

    • Terminating
    • Non-terminating repeating
    • Non-terminating non-repeating
    • Whole number

    Answer: (b)

20. Which of the following is a rational number?

    • $\sqrt{3}$
    • $\sqrt{2}$
    • $\frac{22}{7}$
    • $\pi$

    Answer: (c)

21. The HCF of 8, 12 and 16 is:

    • 2
    • 4
    • 8
    • 16

    Answer: (b)

22. The LCM of 8, 12 and 16 is:

    • 24
    • 32
    • 48
    • 96

    Answer: (c)

23. Which of the following fractions has a non-terminating repeating decimal?

    • $\frac{3}{25}$
    • $\frac{5}{16}$
    • $\frac{7}{15}$
    • $\frac{9}{40}$

    Answer: (c)

24. The prime factorisation of 84 is:

    • $2^2 \times 3 \times 7$
    • $2 \times 3 \times 14$
    • $4 \times 21$
    • $2^3 \times 3 \times 7$

    Answer: (a)

25. If $a$ and $b$ are coprime numbers, then $HCF(a,b)$ is:

    • 0
    • 1
    • $a$
    • $b$

    Answer: (b)

26. The LCM of 15 and 20 is:

    • 45
    • 60
    • 75
    • 90

    Answer: (b)

27. If $\sqrt{2}$ is irrational, then $5\sqrt{2}$ is:

    • Rational
    • Irrational
    • Integer
    • Whole number

    Answer: (b)

28. The HCF of 144 and 180 is:

    • 12
    • 18
    • 24
    • 36

    Answer: (d)

29. If a rational number has denominator of the form $2^m5^n$ (where $m,n$ are non-negative integers), then its decimal expansion is:

    • Terminating
    • Non-terminating repeating
    • Non-terminating non-repeating
    • Irrational

    Answer: (a)

30. The HCF of 27 and 36 using prime factorisation is:

    • 3
    • 6
    • 9
    • 12

    Answer: (c)

---

Part 3: Graded Sample Question Bank

Very Short Answer Type (1 Mark)

1. Express 156 as a product of its prime factors. [1 Mark]
2. Express 84 as a product of its prime factors. [1 Mark]
3. Express 210 as a product of its prime factors. [1 Mark]
4. Express 360 as a product of its prime factors. [1 Mark]
5. Express 128 as a product of its prime factors. [1 Mark]
6. Express 945 as a product of its prime factors. [1 Mark]
7. Express 231 as a product of its prime factors. [1 Mark]
8. Express 144 as a product of its prime factors. [1 Mark]
9. Express 225 as a product of its prime factors. [1 Mark]
10. Express 396 as a product of its prime factors. [1 Mark]
11. Express 504 as a product of its prime factors. [1 Mark]
12. If $a = 2^2 \times 3^1$ and $b = 2 \times 2 \times 5$, find the $HCF(a, b)$. [1 Mark]
13. If $a = 2^3 \times 5$ and $b = 2^2 \times 3$, find the $HCF(a, b)$. [1 Mark]
14. If $a = 3^2 \times 7$ and $b = 3 \times 5^2$, find the $HCF(a, b)$. [1 Mark]
15. If $a = 2^4 \times 3$ and $b = 2^2 \times 3^2$, find the $HCF(a, b)$. [1 Mark]
16. If $a = 5^2 \times 11$ and $b = 5 \times 7$, find the $HCF(a, b)$. [1 Mark]
17. If $a = 2^2 \times 7^2$ and $b = 2 \times 7 \times 3$, find the $HCF(a, b)$. [1 Mark]
18. If $a = 3^3 \times 5$ and $b = 3^2 \times 2$, find the $HCF(a, b)$. [1 Mark]
19. If $a = 2^3 \times 11$ and $b = 2 \times 11^2$, find the $HCF(a, b)$. [1 Mark]
20. If $a = 7^2 \times 13$ and $b = 7 \times 13^2$, find the $HCF(a, b)$. [1 Mark]
21. If $a = 2^5 \times 3^2$ and $b = 2^3 \times 3$, find the $HCF(a, b)$. [1 Mark]
22. If $a = 3^2 \times 5^2$ and $b = 3 \times 5^3$, find the $HCF(a, b)$. [1 Mark]

Short Answer Type I (2 Marks)

23. Check whether $6^n$ can end with the digit 0 for any natural number $n$. [2 Marks]
24. Check whether $4^n$ can end with the digit 0 for any natural number $n$. [2 Marks]
25. Check whether $5^n$ can end with the digit 5 for any natural number $n$. [2 Marks]
26. Check whether $2^n$ can end with the digit 0 for any natural number $n$. [2 Marks]
27. Check whether $3^n$ can end with the digit 1 for any natural number $n$. [2 Marks]
28. Check whether $7^n$ can end with the digit 3 for any natural number $n$. [2 Marks]
29. Check whether $8^n$ can end with the digit 2 for any natural number $n$. [2 Marks]
30. Check whether $9^n$ can end with the digit 7 for any natural number $n$. [2 Marks]
31. Check whether $10^n$ can end with the digit 0 for any natural number $n$. [2 Marks]
32. Check whether $12^n$ can end with the digit 5 for any natural number $n$. [2 Marks]
33. Check whether $15^n$ can end with the digit 0 for any natural number $n$. [2 Marks]
34. Prove that $3\sqrt{2}$ is irrational, provided that $\sqrt{2}$ is irrational. [2 Marks]
35. Prove that $2\sqrt{3}$ is irrational, provided that $\sqrt{3}$ is irrational. [2 Marks]
36. Prove that $5\sqrt{7}$ is irrational, provided that $\sqrt{7}$ is irrational. [2 Marks]
37. Prove that $4\sqrt{5}$ is irrational, provided that $\sqrt{5}$ is irrational. [2 Marks]
38. Prove that $7\sqrt{2}$ is irrational, provided that $\sqrt{2}$ is irrational. [2 Marks]
39. Prove that $6\sqrt{3}$ is irrational, provided that $\sqrt{3}$ is irrational. [2 Marks]
40. Prove that $9\sqrt{11}$ is irrational, provided that $\sqrt{11}$ is irrational. [2 Marks]
41. Prove that $8\sqrt{13}$ is irrational, provided that $\sqrt{13}$ is irrational. [2 Marks]
42. Prove that $10\sqrt{5}$ is irrational, provided that $\sqrt{5}$ is irrational. [2 Marks]
43. Prove that $11\sqrt{3}$ is irrational, provided that $\sqrt{3}$ is irrational. [2 Marks]
44. Prove that $12\sqrt{7}$ is irrational, provided that $\sqrt{7}$ is irrational. [2 Marks]
45. Prove that $3 + 2\sqrt{5}$ is irrational. Provided that $\sqrt{5}$ is irrational.[2 Marks]
46. Prove that $4 + 3\sqrt{7}$ is irrational, provided that $\sqrt{7}$ is irrational. [2 Marks]
47. Prove that $5 - 2\sqrt{3}$ is irrational, provided that $\sqrt{3}$ is irrational. [2 Marks]
48. Prove that $7 + \sqrt{11}$ is irrational, provided that $\sqrt{11}$ is irrational. [2 Marks]
49. Prove that $6 - 4\sqrt{2}$ is irrational, provided that $\sqrt{2}$ is irrational. [2 Marks]
50. Prove that $8 + 5\sqrt{13}$ is irrational, provided that $\sqrt{13}$ is irrational. [2 Marks]
51. Prove that $9 - 3\sqrt{5}$ is irrational, provided that $\sqrt{5}$ is irrational. [2 Marks]
52. Prove that $2 + 7\sqrt{3}$ is irrational, provided that $\sqrt{3}$ is irrational. [2 Marks]
53. Prove that $10 - 6\sqrt{7}$ is irrational, provided that $\sqrt{7}$ is irrational. [2 Marks]
54. Prove that $1 + 4\sqrt{6}$ is irrational, provided that $\sqrt{6}$ is irrational. [2 Marks]
55. Prove that $12 - 5\sqrt{2}$ is irrational, provided that $\sqrt{2}$ is irrational. [2 Marks]
56. Prove that $\frac{5 + 3\sqrt{5}}{4}$ is irrational, provided that $\sqrt{5}$ is irrational. [2 Marks]
57. Prove that $\frac{7 + 2\sqrt{3}}{5}$ is irrational, provided that $\sqrt{3}$ is irrational. [2 Marks]
58. Prove that $\frac{9 - 4\sqrt{2}}{3}$ is irrational, provided that $\sqrt{2}$ is irrational. [2 Marks]
59. Prove that $\frac{6 + 5\sqrt{7}}{8}$ is irrational, provided that $\sqrt{7}$ is irrational. [2 Marks]
60. Prove that $\frac{11 - 3\sqrt{5}}{6}$ is irrational, provided that $\sqrt{5}$ is irrational. [2 Marks]
61. Prove that $\frac{4 + 7\sqrt{11}}{9}$ is irrational, provided that $\sqrt{11}$ is irrational. [2 Marks]
62. Prove that $\frac{8 - 2\sqrt{13}}{7}$ is irrational, provided that $\sqrt{13}$ is irrational. [2 Marks]
63. Prove that $\frac{3 + 4\sqrt{6}}{5}$ is irrational, provided that $\sqrt{6}$ is irrational. [2 Marks]
64. Prove that $\frac{10 - 5\sqrt{3}}{12}$ is irrational, provided that $\sqrt{3}$ is irrational. [2 Marks]
65. Prove that $\frac{2 + 9\sqrt{2}}{11}$ is irrational, provided that $\sqrt{2}$ is irrational. [2 Marks]
66. Prove that $\frac{13 - 6\sqrt{7}}{10}$ is irrational, provided that $\sqrt{7}$ is irrational. [2 Marks]
67. Explain why $7 \times 11 \times 13 + 13$ is a composite number. [2 Marks]
68. Explain why $5 \times 9 \times 11 + 11$ is a composite number. [2 Marks]
69. Explain why $4 \times 7 \times 15 + 15$ is a composite number. [2 Marks]
70. Explain why $6 \times 13 \times 17 + 17$ is a composite number. [2 Marks]
71. Explain why $8 \times 9 \times 19 + 19$ is a composite number. [2 Marks]
72. Explain why $3 \times 10 \times 21 + 21$ is a composite number. [2 Marks]
73. Explain why $2 \times 5 \times 23 + 23$ is a composite number. [2 Marks]
74. Explain why $7 \times 8 \times 25 + 25$ is a composite number. [2 Marks]
75. Explain why $9 \times 14 \times 29 + 29$ is a composite number. [2 Marks]
76. Explain why $11 \times 12 \times 31 + 31$ is a composite number. [2 Marks]
77. Explain why $4 \times 6 \times 27 + 27$ is a composite number. [2 Marks]

Short Answer Type II (3 Marks)

1. Prove that $\sqrt{5}$ is an irrational number. [3 Marks]
2. Prove that $\sqrt{7}$ is an irrational number. [3 Marks]
3. Prove that $\sqrt{11}$ is an irrational number. [3 Marks]
4. Prove that $\sqrt{13}$ is an irrational number. [3 Marks]
5. Prove that $\sqrt{17}$ is an irrational number. [3 Marks]
6. Prove that $\sqrt{19}$ is an irrational number. [3 Marks]
7. Prove that $\sqrt{23}$ is an irrational number. [3 Marks]
8. Prove that $\sqrt{29}$ is an irrational number. [3 Marks]
9. Prove that $\sqrt{31}$ is an irrational number. [3 Marks]
10. Prove that $\sqrt{37}$ is an irrational number. [3 Marks]
11. Prove that $\sqrt{41}$ is an irrational number. [3 Marks]
12. Find the LCM and HCF of 240 and 96 by the prime factorisation method and verify that $LCM \times HCF = \text{product of the two numbers}$. [3 Marks]
13. Find the LCM and HCF of 180 and 144 by the prime factorisation method and verify that $LCM \times HCF = \text{product of the two numbers}$. [3 Marks]
14. Find the LCM and HCF of 210 and 84 by the prime factorisation method and verify that $LCM \times HCF = \text{product of the two numbers}$. [3 Marks]
15. Find the LCM and HCF of 324 and 90 by the prime factorisation method and verify that $LCM \times HCF = \text{product of the two numbers}$. [3 Marks]
16. Find the LCM and HCF of 150 and 225 by the prime factorisation method and verify that $LCM \times HCF = \text{product of the two numbers}$. [3 Marks]
17. Find the LCM and HCF of 288 and 108 by the prime factorisation method and verify that $LCM \times HCF = \text{product of the two numbers}$. [3 Marks]
18. Find the LCM and HCF of 420 and 168 by the prime factorisation method and verify that $LCM \times HCF = \text{product of the two numbers}$. [3 Marks]
19. Find the LCM and HCF of 360 and 72 by the prime factorisation method and verify that $LCM \times HCF = \text{product of the two numbers}$. [3 Marks]
20. Find the LCM and HCF of 252 and 198 by the prime factorisation method and verify that $LCM \times HCF = \text{product of the two numbers}$. [3 Marks]
21. Find the LCM and HCF of 270 and 192 by the prime factorisation method and verify that $LCM \times HCF = \text{product of the two numbers}$. [3 Marks]
22. Find the LCM and HCF of 336 and 54 by the prime factorisation method and verify that $LCM \times HCF =\text{product of the two numbers}$. [3 Marks]

Long Answer / Application Based (4 Marks)

1. 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 the same. Suppose they both start at the same point and at the same time, and go in the same direction. After how many minutes will they meet again at the starting point? Explain your mathematical reasoning. [4 Marks]
2. Two runners are running around a circular track. Aman completes one round in 15 minutes, while Rohan completes one round in 20 minutes. If they start together from the same point and run in the same direction, after how many minutes will they meet again at the starting point? Explain your reasoning. [4 Marks]
3. Neha and Priya walk around a circular park. Neha takes 24 minutes to complete one round, and Priya takes 16 minutes. If they start at the same time from the same point in the same direction, after how many minutes will they meet again at the starting point? Explain your reasoning. [4 Marks]
4. Two cyclists take 14 minutes and 21 minutes respectively to complete one round of a circular track. If they start together and move in the same direction, after how many minutes will they be together again at the starting point? Justify your answer. [4 Marks]
5. Rahul takes 28 minutes to complete one round of a circular ground, while Mohan takes 35 minutes. If both start at the same time from the same point and move in the same direction, after how many minutes will they meet again at the starting point? Explain clearly. [4 Marks]
6. Two athletes complete one round of a stadium in 18 minutes and 27 minutes respectively. If they begin together from the same point in the same direction, after how many minutes will they again reach the starting point together? Give mathematical reasoning. [4 Marks]
7. Anita and Kavya jog around a circular path. Anita takes 32 minutes for one round and Kavya takes 48 minutes. If they start simultaneously from the same point and move in the same direction, after how many minutes will they meet again at the starting point? Explain your reasoning. [4 Marks]
8. Two friends complete one round of a circular park in 9 minutes and 15 minutes respectively. If they start together and move in the same direction, after how many minutes will they be together again at the starting point? Explain your method. [4 Marks]
9. Arjun and Vivek take 20 minutes and 30 minutes respectively to complete one round of a circular field. If they start at the same time from the same point and go in the same direction, after how many minutes will they meet again at the starting point? Justify your answer. [4 Marks]
10. Two walkers take 25 minutes and 40 minutes respectively to go once around a circular garden. If they begin together and walk in the same direction, after how many minutes will they reach the starting point together again? Explain clearly. [4 Marks]
11. Seema and Ritu take 36 minutes and 45 minutes respectively to complete one round of a circular path. If they start together from the same point and move in the same direction, after how many minutes will they meet again at the starting point? Explain the mathematical reasoning. [4 Marks]
12. Find the HCF and LCM of 6, 72 and 120, using the prime factorisation method. Show that the product of these three numbers is not equal to the product of their HCF and LCM. [4 Marks]
13. Find the HCF and LCM of 8, 48 and 160, using the prime factorisation method. Show that the product of these three numbers is not equal to the product of their HCF and LCM. [4 Marks]
14. Find the HCF and LCM of 9, 54 and 81, using the prime factorisation method. Show that the product of these three numbers is not equal to the product of their HCF and LCM. [4 Marks]
15. Find the HCF and LCM of 12, 90 and 150, using the prime factorisation method. Show that the product of these three numbers is not equal to the product of their HCF and LCM. [4 Marks]
16. Find the HCF and LCM of 15, 60 and 105, using the prime factorisation method. Show that the product of these three numbers is not equal to the product of their HCF and LCM. [4 Marks]
17. Find the HCF and LCM of 10, 45 and 75, using the prime factorisation method. Show that the product of these three numbers is not equal to the product of their HCF and LCM. [4 Marks]
18. Find the HCF and LCM of 14, 84 and 196, using the prime factorisation method. Show that the product of these three numbers is not equal to the product of their HCF and LCM. [4 Marks]
19. Find the HCF and LCM of 16, 64 and 96, using the prime factorisation method. Show that the product of these three numbers is not equal to the product of their HCF and LCM. [4 Marks]
20. Find the HCF and LCM of 18, 108 and 144, using the prime factorisation method. Show that the product of these three numbers is not equal to the product of their HCF and LCM. [4 Marks]
21. Find the HCF and LCM of 20, 100 and 140, using the prime factorisation method. Show that the product of these three numbers is not equal to the product of their HCF and LCM. [4 Marks]
22. Find the HCF and LCM of 21, 63 and 126, using the prime factorisation method. Show that the product of these three numbers is not equal to the product of their HCF and LCM. [4 Marks]