BSEB Class 10 Mathematics Chapter 1: Real Numbers (वास्तविक संख्याएँ) – Practice Guide

Introduction:
Chapter 1 of the BSEB Class 10 Mathematics syllabus introduces students to the world of Real Numbers. Real numbers are the combination of rational and irrational numbers. In this chapter, we explore key concepts such as Euclid’s Division Lemma, the Fundamental Theorem of Arithmetic, and the nature of decimal expansions. Understanding these basics is crucial for scoring well in the Bihar Board matriculation exams.

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Section 1: Multiple Choice Questions (MCQs)

Select the correct option for the following questions:

1. Which of the following is an irrational number?
   A) 2/3
   B) √4
   C) √5
   D) 0.5
2. The HCF of two consecutive even numbers is:
   A) 1
   B) 2
   C) 4
   D) 0
3. If p and q are two prime numbers, then their HCF is:
   A) p
   B) q
   C) 1
   D) pq
4. The decimal expansion of 13/3125 is:
   A) Terminating
   B) Non-terminating repeating
   C) Non-terminating non-repeating
   D) None of these
5. π (Pi) is a/an:
   A) Rational number
   B) Irrational number
   C) Integer
   D) Whole number

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Section 2: Short Answer Type Questions (2 Marks)

Solve the following questions with steps:

• Question 1: Use Euclid’s division algorithm to find the HCF of 135 and 225.
• Question 2: Find the HCF and LCM of 12, 15, and 21 using the prime factorization method.
• Question 3: Explain why (7 × 11 × 13 + 13) is a composite number.
• Question 4: Show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5.
• Question 5: Check whether 6^n can end with the digit 0 for any natural number n.

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Section 3: Long Answer Type Questions (5 Marks)

Detailed explanation is required for these questions:

1. Question: Prove that √5 is an irrational number.
2. Question: Prove that 3 + 2√5 is irrational.
3. Question: State the Fundamental Theorem of Arithmetic. Use it to find the HCF and LCM of 96 and 404 and verify that HCF × LCM = Product of the two numbers.
4. Question: 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?

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Section 4: Frequently Asked Questions (FAQs)

1. What are Real Numbers?
   Real numbers include both rational numbers (like 1, 2.5, 3/4) and irrational numbers (like √2, π).
2. What is Euclid’s Division Lemma?
   It states that for any two positive integers a and b, there exist unique integers q and r such that a = bq + r, where 0 ≤ r < b.
3. How can we identify a terminating decimal?
   A rational number p/q has a terminating decimal expansion if the prime factorization of ‘q’ is of the form 2^n * 5^m.
4. Is 0 a rational number?
   Yes, 0 is a rational number because it can be written as 0/1.
5. What is the HCF of two prime numbers?
   The HCF of any two prime numbers is always 1.
6. What is the LCM of two prime numbers?
   The LCM of two prime numbers p and q is their product (p × q).
7. Can HCF be greater than LCM?
   No, the HCF of two or more numbers is always less than or equal to their LCM.
8. Is every irrational number a real number?
   Yes, every irrational number is a part of the real number system.
9. What is a composite number?
   A number that has more than two factors is called a composite number.
10. Is √9 an irrational number?
    No, √9 = 3, which is a rational number.

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End of Chapter 1 Practice Set. Good luck with your BSEB Exams!