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Real Numbers Chapter 1 is an important topic in class 10 Mathematics. It is a foundational chapter that lays the groundwork for many other topics in higher mathematics. Here are some important questions that students should focus on while studying this chapter:

I. What are Real Numbers?
- Define Real Numbers
- What is the difference between Rational and Irrational Numbers?
- Prove that the square root of 2 is an irrational number.

II. Euclid's Division Lemma
- Explain Euclid's Division Lemma
- Prove that for any two positive integers a and b, there exist unique integers q and r such that a = bq + r, where 0 ≤ r < />

III. Fundamental Theorem of Arithmetic
- Define the Fundamental Theorem of Arithmetic
- Prove that every composite number can be expressed as a product of primes.

IV. Rational and Irrational Numbers
- Prove that √3 + √5 is an irrational number.
- Prove that the sum of a rational number and an irrational number is always an irrational number.
- Prove that the product of a nonzero rational number and an irrational number is always an irrational number.

V. Real Numbers and their Decimal Expansions
- Define the decimal expansion of a number
- Prove that every terminating decimal is a rational number.
- Prove that every non-terminating, repeating decimal is a rational number.

VI. Operations on Real Numbers
- Prove that the sum of two irrational numbers can be a rational number.
- Prove that the product of two irrational numbers can be a rational number.

VII. Square Roots
- Find the square root of a given number using the long division method.
- Prove that √2 + √3 is an irrational number.

VIII. HCF and LCM
- Define HCF and LCM
- Prove that for any two positive integers a and b, HCF(a, b) × LCM(a, b) = a × b.

IX. Euclid's Algorithm
- Explain Euclid's Algorithm
- Use Euclid's Algorithm to find the HCF of two or more numbers.

X. Squares and Square Roots
- Prove that if n is a positive integer, then n² + (n + 1)² is an odd integer.
- Find the square root of a given number using the prime factorization method.

These are some of the important questions that students should focus on while studying the Real Numbers Chapter 1 in class 10 Mathematics.