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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NCERT Solutions for Class 10 Mathematics — Real Numbers Exemplar. Step-by-step answers to all 6 textbook questions from this chapter, written for CBSE board preparation and free to use.
Real Numbers Exemplar — All 6 Questions Solved
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.
How to Use These Solutions
Attempt each question yourself first and only then compare with the worked answer. Marks in the CBSE board exam are awarded for the METHOD as much as the final result, so reproduce the steps rather than memorising the last line. Where a solution states a law, a formula or a definition, learn that wording — examiners look for it.
Related practice for this chapter: chapter MCQs, previous-year questions, revision notes and sample papers.