Real Numbers — Class 10 Mathematics NCERT Solutions (Free)
Free step-by-step NCERT solutions for Class 10 Mathematics chapter "Real Numbers" — 9 important questions with detailed answers for CBSE board exam preparation.
TL;DR: Free step-by-step NCERT solutions for Class 10 Mathematics chapter "Real Numbers" — 9 important questions with detailed answers for CBSE board exam pr…
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Key Questions Covered:
- State the Fundamental Theorem of Arithmetic.
- Prove that √2 is irrational.
- Find the HCF of 96 and 404 using Euclid's division algorithm.
- + 6 more questions in the full chapter
Solutions Summary:
| Question | Status |
|---|---|
| State the Fundamental Theorem of Arithmetic. | ✓ Solved |
| Prove that √2 is irrational. | ✓ Solved |
| Find the HCF of 96 and 404 using Euclid's division algori… | ✓ Solved |
Showing 3 of 9 questions
Q1: State the Fundamental Theorem of Arithmetic.
The Fundamental Theorem of Arithmetic states that every composite number can be expressed (factorised) as a product of prime numbers, and this factorisation is unique, apart from the order in which the prime factors occur.
For example:
- 60 = 2² × 3 × 5
- 84 = 2² × 3 × 7
This means that if we writ...
Q2: Prove that √2 is irrational.
Proof by contradiction:
Assume √2 is rational. Then √2 = p/q, where p and q are coprime integers (HCF(p,q) = 1) and q ≠ 0.
Squaring both sides: 2 = p²/q²
Therefore: 2q² = p² ... (1)
From equation (1), p² is even, which means p must be even.
Let p = 2m for some integer m.
Substituting in equation...
Q3: Find the HCF of 96 and 404 using Euclid's division algorithm.
Using Euclid's division algorithm:
Step 1: Divide 404 by 96
404 = 96 × 4 + 20
Step 2: Divide 96 by 20
96 = 20 × 4 + 16
Step 3: Divide 20 by 16
20 = 16 × 1 + 4
Step 4: Divide 16 by 4
16 = 4 × 4 + 0
Since the remainder is 0, the HCF is 4.
Therefore, HCF(96, 404) = 4
Showing 3 of 9 questions. Visit the full page for complete solutions.