Class 10 Maths Chapter 1 – Real Numbers | Full NCERT Questions & Detailed Solutions
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Chapter Introduction
Chapter 1, Real Numbers, is the gateway to number theory and algebraic reasoning. It’s not just the first chapter—it’s the foundation for everything that follows in Class 10 Maths. At Convex Classes Jaipur, we provide complete coverage of this chapter with:
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Exercise 1.1 – Euclid’s Division Lemma (4 Questions)
Q1. Use Euclid’s division algorithm to find the HCF of:
(i) 135 and 225
Solution: 225 = 135 × 1 + 90 135 = 90 × 1 + 45 90 = 45 × 2 + 0 ✅ HCF = 45
(ii) 196 and 38220
38220 = 196 × 195 + 0 ✅ HCF = 196
(iii) 867 and 255
867 = 255 × 3 + 102 255 = 102 × 2 + 51 102 = 51 × 2 + 0 ✅ HCF = 51
Q2. Show that any positive odd integer is of the form 6q + 1, 6q + 3, or 6q + 5
Solution: Any integer n can be written as: n = 6q + r, where r = 0, 1, 2, 3, 4, 5 Odd integers cannot be divisible by 2 → r ≠ 0, 2, 4 ✅ So odd integers are of the form: 6q + 1, 6q + 3, 6q + 5
Q3. Army contingent of 616 and band of 32 members. Max columns?
Solution: Find HCF of 616 and 32: 616 = 32 × 19 + 8 32 = 8 × 4 + 0
✅ HCF = 8 👉 Max columns = 8
Q4. Show that square of any positive integer is of the form 3m or 3m + 1
Solution: Let n be any integer. n = 3q, 3q + 1, or 3q + 2 → n² = 9q², 9q² + 6q + 1, or 9q² + 12q + 4 → Forms: 3m or 3m + 1
✅ Hence proved.
Exercise 1.2 – Fundamental Theorem of Arithmetic (7 Questions)
Q1. Express as product of prime factors:
(i) 140 = 2 × 2 × 5 × 7
(ii) 156 = 2 × 2 × 3 × 13
(iii) 3825 = 3 × 5² × 17
(iv) 5005 = 5 × 7 × 11 × 13
(v) 7429 = 17 × 19 × 23
Q2. Find HCF and LCM and verify:
(i) 12 and 15
HCF = 3, LCM = 60 → 3 × 60 = 180 = 12 × 15 ✅
(ii) 17 and 23
HCF = 1, LCM = 391✅
(iii) 336 and 54
336 = 2⁴ × 3 × 7 54 = 2 × 3³ HCF = 6, LCM = 3024 ✅
Q3. Prime factor method:
(i) 6, 72, 120
HCF = 6, LCM = 720
(ii) 306 and 657
306 = 2 × 3² × 17 657 = 3² × 73 HCF = 9, LCM = 22338
(iii) 455 and 42
455 = 5 × 7 × 13 42 = 2 × 3 × 7 HCF = 7, LCM = 2730
Q4. Can 6ⁿ end with 0?
Solution: 6ⁿ = (2 × 3)ⁿ → no factor of 5 ✅ So it cannot end with 0
Q5. Show 7 × 11 × 13 + 13 is composite
= 1001 + 13 = 1014 → divisible by 2 ✅ Composite
Q6. Show 15³ + 15 and 15³ + 45 are composite
15³ + 15 = 3375 + 15 = 3390 → divisible by 3 ✅ Composite 15³ + 45 = 3375 + 45 = 3420 → divisible by 5 ✅ Composite
Q7. Sonia (18 min), Ravi (12 min) → LCM(18,12) = 36
✅ They meet again after 36 minutes
Exercise 1.3 – Irrational Numbers (3 Questions)
Q1. Prove √2 is irrational
Assume √2 = p/q → p² = 2q² → p divisible by 2 Let p = 2k → contradiction ✅ √2 is irrational
Q2. Prove √3 is irrational
Same method → p² = 3q² → p divisible by 3 Contradiction ✅ √3 is irrational
Q3. Prove √5 is irrational
Same method → p² = 5q² → p divisible by 5 Contradiction ✅ √5 is irrational
Exercise 1.4 – Decimal Expansions (3 Questions)
Q1. Terminating or non-terminating?
(i) 13/3125 → Terminating
(ii) 17/8 → Terminating
(iii) 64/455 → Non-terminating
(iv) 15/1600 → Terminating \
(v) 29/343 → Non-terminating
(vi) 23/8 → Terminating
(vii) 129/(2² × 5⁷ × 7⁵) → Non-terminating
(viii) 6/15 → Terminating
Q2. Decimal expansions (Terminating only):
(i) 13/3125 = 0.00416
(ii) 17/8 = 2.125
(iii) 15/1600 = 0.009375
(vi) 23/8 = 2.875
(v) 6/15 = 0.4
Q3. Identify rational or irrational:
(i) 1/7 = 0.142857… (Repeating) → Rational
(ii) 1/3 = 0.333… → Rational
(iii) √2 = Non-repeating → Irrational
(iv) √5 = Non-repeating → Irrational
(v) π = Non-repeating → Irrational
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