NCERT Solutions for Class 10 Maths Chapter 1 - Real Numbers
Introduction to Real Numbers
Chapter 1 of Class 10 Maths deals with Real Numbers. This chapter covers important concepts like Euclid's division lemma, the Fundamental Theorem of Arithmetic, irrational numbers, and operations on real numbers.
These NCERT solutions will help you understand the concepts better and prepare for your exams effectively.
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Exercise-wise Solutions
Exercise 1.1
Question 1: Use Euclid's division algorithm to find the HCF of 135 and 225.
Solution:
Step 1: Apply Euclid's division lemma to get
225 = 135 × 1 + 90
Step 2: Apply the division lemma to 135 and 90 to get
135 = 90 × 1 + 45
Step 3: Apply the division lemma to 90 and 45 to get
90 = 45 × 2 + 0
Since the remainder is zero, the divisor 45 is the HCF of 135 and 225.
Therefore, HCF (135, 225) = 45
Question 2: Show that any positive odd integer is of the form 4q + 1 or 4q + 3, where q is some integer.
Solution:
Let's consider a positive odd integer 'a'.
When we divide 'a' by 4, the possible remainders are 0, 1, 2, or 3.
So, a = 4q + r, where r = 0, 1, 2, or 3 and q is the quotient.
Since 'a' is odd, r cannot be 0 or 2 (as 4q, 4q + 2 are even).
Therefore, r can only be 1 or 3.
Hence, any positive odd integer is of the form 4q + 1 or 4q + 3, where q is some integer.
Exercise 1.2
Question 1: Express each number as a product of its prime factors: (i) 140 (ii) 156 (iii) 3825
Solution:
(i) 140 = 2² × 5 × 7
(ii) 156 = 2² × 3 × 13
(iii) 3825 = 3² × 5² × 17
Important Concepts
- Euclid's Division Lemma: For positive integers a and b, there exist unique integers q and r such that a = bq + r, where 0 ≤ r < b.
- Fundamental Theorem of Arithmetic: Every composite number can be expressed as a product of primes, and this factorization is unique apart from the order of factors.
- HCF and LCM: For two positive integers a and b, HCF(a, b) × LCM(a, b) = a × b
- Irrational Numbers: Numbers that cannot be expressed in the form p/q, where p and q are integers and q ≠ 0.
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