Real Numbers Exemplar — Class 10 Mathematics NCERT Solutions (Free)
Free step-by-step NCERT solutions for Class 10 Mathematics chapter "Real Numbers Exemplar" — 6 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 Exemplar" — 6 important questions with detailed answers for CBSE boar…
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Key Questions Covered:
- Use Euclid's division algorithm to find HCF of 135 and 225.
- Show that 3√2 is irrational.
- Prove that 1/√2 + 1/√8 + 1/√32 = 7/(4√2).
- Express 0.6̄ (0.6666...) as p/q.
- Find LCM(96, 404) if HCF(96, 404) = 4.
- If HCF(a, b) = 8 and a × b = 2048, find LCM(a, b).
Solutions Summary:
| Question | Status |
|---|---|
| Use Euclid's division algorithm to find HCF of 135 and 225. | ✓ Solved |
| Show that 3√2 is irrational. | ✓ Solved |
| Prove that 1/√2 + 1/√8 + 1/√32 = 7/(4√2). | ✓ Solved |
| Express 0.6̄ (0.6666...) as p/q. | ✓ Solved |
| Find LCM(96, 404) if HCF(96, 404) = 4. | ✓ Solved |
| If HCF(a, b) = 8 and a × b = 2048, find LCM(a, b). | ✓ Solved |
Showing 6 of 6 questions
Q1: Use Euclid's division algorithm to find HCF of 135 and 225.
225 = 135 × 1 + 90. 135 = 90 × 1 + 45. 90 = 45 × 2 + 0. So HCF(135, 225) = 45.
Q2: Show that 3√2 is irrational.
Assume 3√2 is rational. Then 3√2 = p/q where GCD(p,q) = 1. So √2 = p/(3q). Since p/(3q) is rational (ratio of integers), √2 would be rational. But √2 is irrational, contradiction. Therefore 3√2 is irrational.
Q3: Prove that 1/√2 + 1/√8 + 1/√32 = 7/(4√2).
LHS = 1/√2 + 1/(2√2) + 1/(4√2) = (2 + 1 + 0.5)/(4√2) = 3.5/(4√2) = 7/(8√2). Rationalizing: 7/(8√2) × √2/√2 = 7√2/16. Hmm, recalculate: 1/√2 + 1/√8 + 1/√32 = 1/√2 + 1/(2√2) + 1/(4√2) = (4 + 2 + 1)/(4√2) = 7/(4√2). Proven.
Q4: Express 0.6̄ (0.6666...) as p/q.
Let x = 0.6̄. Then 10x = 6.6̄. Subtracting: 10x - x = 6, so 9x = 6, giving x = 6/9 = 2/3.
Q5: Find LCM(96, 404) if HCF(96, 404) = 4.
Using HCF × LCM = product of numbers: 4 × LCM = 96 × 404. LCM = (96 × 404)/4 = 96 × 101 = 9696.
Q6: If HCF(a, b) = 8 and a × b = 2048, find LCM(a, b).
Using HCF(a, b) × LCM(a, b) = a × b: 8 × LCM = 2048. LCM = 256.
Showing 6 of 6 questions. Visit the full page for complete solutions.
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