Strengthen your preparation for the CBSE Class 10 Board Exams with this guide on CBSE Class 10 Mathematics Chapter 1 Important Questions - Real Numbers. Explore key concepts like physical and chemical properties, reactivity series, and practical applications to excel in exams confidently.
Cbse Class 10 Mathematics Chapter 1 Important Questions - Real Numbers
Here are 50 important questions on Cbse Class 10 Mathematics Chapter 1 Important Questions - Real Numbers
- What are real numbers? Define them with examples.
- Explain the difference between rational and irrational numbers.
- How can you express a number as a product of its prime factors? Provide an example.
- Find the prime factorization of 140.
- Find the prime factorization of 156.
- Find the prime factorization of 3825.
- Find the prime factorization of 5005.
- Find the prime factorization of 7429.
- What is the definition of HCF (Highest Common Factor)?
- What is the definition of LCM (Lowest Common Multiple)?
- Calculate the HCF and LCM of 26 and 91.
- Calculate the HCF and LCM of 510 and 92.
- Calculate the HCF and LCM of 336 and 54.
- How do you verify that HCF × LCM = product of two numbers?
- Find the HCF and LCM of 12, 15, and 21 using prime factorization.
- Find the HCF and LCM of 17, 23, and 29.
- Find the HCF and LCM of 8, 9, and 25.
- Given that HCF(306, 657) = 9, find LCM(306, 657).
- Explain why 6n cannot end with the digit zero for any natural number n.
- Prove that 5 is an irrational number.
- Prove that 3 + √5 is an irrational number.
- Prove that √7 is an irrational number.
- What are composite numbers? Provide examples from real numbers.
- Explain how to check if a number is composite or prime.
- What is a circular path problem in relation to real numbers?
- If Sonia takes 18 minutes to complete one round and Ravi takes 12 minutes, how do you determine when they will meet again at the starting point?
- Explain how to find the least common multiple using prime factorization.
- Describe how to find the highest common factor using division method.
- How can you express a rational number in decimal form?
- What is a terminating decimal? Provide examples.
- What is a non-terminating decimal? Provide examples.
- Define integers within the context of real numbers.
- What role do real numbers play in geometry?
- How can real numbers be represented on a number line?
- Discuss the properties of real numbers under addition and multiplication.
- How do you convert a fraction into a decimal?
- Explain how to simplify square roots in real numbers.
- What are some common misconceptions about irrational numbers?
- How can you demonstrate that √2 is irrational using contradiction?
- Give an example of a rational number that can be expressed as a repeating decimal.
- What is the significance of real numbers in algebra?
- Discuss how to find square roots using prime factorization.
- Explain why every integer is also a rational number but not vice versa.
- How do you determine if two fractions are equivalent?
- Describe how to add and subtract rational numbers with different denominators.
- What is meant by 'co-prime' numbers? Give examples.
- Explain how to find multiples of a given number systematically.
- Discuss any applications of real numbers in everyday life scenarios.
- How can we use real numbers to solve problems involving ratios?
- Why is it important for students to understand real numbers in mathematics?
These questions cover various aspects of Real Numbers, including their properties, reactions, uses, and applications as presented in the document provided, ensuring comprehensive coverage of key topics within this chapter on Real Numbers.
Class 10 Real Numbers Notes
The chapter “Real Numbers” in Class 10 Science explores the fundamental properties, reactivity, and applications of Real Numbers. Below is a detailed explanation of the key topics covered in this chapter based on class 10 science syllabus:
1. Real Numbers:
- A real number is any number that can be represented on the number line. Real numbers include rational numbers (numbers that can be expressed as a fraction of two integers) and irrational numbers (numbers that cannot be expressed as a fraction of two integers, such as √2, π).
- Rational Numbers: Numbers that can be written in the form of p/q, where p and q are integers, and q ≠ 0.
- Irrational Numbers: Numbers that cannot be written as p/q (e.g., √2, π). These numbers have non-terminating, non-repeating decimal expansions.
2. The Fundamental Theorem of Arithmetic:
- The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a product of primes in a unique way, apart from the order of factors.
- Example: 30 = 2 × 3 × 5 (this is the unique prime factorization of 30).
3. Properties of Real Numbers:
- Closure Property: If a and b are real numbers, then their sum (a + b) and product (a × b) are also real numbers.
- Commutative Property: For real numbers a and b, a + b = b + a and a × b = b × a.
- Associative Property: For real numbers a, b, and c, (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c).
- Distributive Property: a × (b + c) = a × b + a × c for all real numbers a, b, and c.
4. Irrational Numbers and Proof of Irrationality:
- Proof of √2 being irrational: To prove that √2 is irrational, we assume the opposite — that √2 is rational and can be expressed as p/q, where p and q are coprime integers. By squaring both sides and simplifying, we reach a contradiction, thus proving √2 is irrational.
- Proof of √3 being irrational: Similar to the proof for √2, the irrationality of √3 can be shown by assuming it is rational (i.e., it can be written as p/q) and arriving at a contradiction.
- Proof of π being irrational: This proof involves more advanced concepts and is usually introduced in higher-level mathematics. The irrationality of π was proven by Johann Lambert in 1768.
Theorem:
Fundamental Theorem of Arithmetic:
- Every composite number can be factored uniquely as a product of prime numbers, and this factorization is unique except for the order of the factors.
Example: 56 can be written as 2 × 2 × 2 × 7. This is the unique prime factorization of 56.
Important Definitions:
- Prime Number: A natural number greater than 1 that has no divisors other than 1 and itself.some text
- Example: 2, 3, 5, 7, 11 are prime numbers.
- Composite Number: A natural number greater than 1 that is not prime, i.e., it has divisors other than 1 and itself.some text
- Example: 4, 6, 8, 9 are composite numbers.
- Rational Numbers: Numbers that can be expressed as the ratio of two integers, i.e., in the form p/q, where p and q are integers and q ≠ 0.some text
- Example: 1/2, 3, -5/7, etc.
- Irrational Numbers: Numbers that cannot be expressed as the ratio of two integers. They have non-terminating, non-repeating decimal expansions.some text
- Co-prime Numbers: Two numbers are co-prime if their greatest common divisor (GCD) is 1.some text
- Example: 15 and 28 are co-prime because their GCD is 1.
Key Proofs:
1. Proof of the Irrationality of √2:
- Assumption: Assume √2 is a rational number, i.e., √2 = p/q, where p and q are integers with no common factors other than 1 (they are co-prime).
- Squaring both sides: 2 = p²/q² → p² = 2q².
- This implies that p² is even, so p must also be even (since the square of an odd number is odd).
- Let p = 2k. Substituting this into the equation: (2k)² = 2q² → 4k² = 2q² → q² = 2k².
- This implies q² is even, so q must also be even.
- But if both p and q are even, this contradicts the assumption that they are co-prime. Therefore, √2 cannot be a rational number, and is thus irrational.
2. Proof of the Irrationality of √3:
- The proof follows the same pattern as for √2. The assumption that √3 is rational leads to a contradiction, proving that √3 is irrational.
Applications:
- Real-Life Applications of Rational and Irrational Numbers:
- Rational Numbers: Used in measurements, fractions of time, money, etc.
- Irrational Numbers: Appear in various fields like geometry, where π and √2 are commonly used in calculations related to circles and right triangles.
- Prime Factorization in Cryptography:
- The Fundamental Theorem of Arithmetic is used in cryptography, especially in public-key cryptography, where the difficulty of factoring large composite numbers is crucial for security.
Important Formulas:
- Prime Factorization of a Number: Any composite number can be broken down into prime factors.
Example: 60 = 2 × 2 × 3 × 5.
Conclusion:
In Metals and Non-Metals, students learn to distinguish between the physical and chemical properties of metals and non-metals, along with their reactivity series.
Mastering these concepts is essential for tackling questions in the CBSE Class 10 Board Exams.
Focusing on CBSE Class 10 Science Chapter 3 Important Questions - Metals and Non-Metals and reviewing related sample papers will enhance understanding and exam performance. Consistent revision and well-organized notes are key to acing this chapter.
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