Complete Solutions and Summary of Real Numbers – NCERT Class 10, Mathematics, Chapter 1 – Summary, Questions, Answers, Extra Questions

Comprehensive summary and explanation of Chapter 1 'Real Numbers', covering Euclid’s division algorithm, Fundamental Theorem of Arithmetic, prime factorisation, irrational numbers, proving irrationality of 2 , 3 , 5 2 , 3 , 5 , and the nature of decimal expansion of rational numbers—with all question answers and extra questions from NCERT Class X Mathematics.

Updated: 9 months ago

Categories: NCERT, Class X, Mathematics, Summary, Extra Questions, Real Numbers, Euclid Algorithm, Fundamental Theorem of Arithmetic, Irrational Numbers, Prime Factorisation, Chapter 1

Tags: Real Numbers, Euclid Division Algorithm, Fundamental Theorem of Arithmetic, Prime Factorisation, Irrational Numbers, Decimal Expansion, HCF, Rational Numbers, NCERT, Class 10, Mathematics, Chapter 1, Answers, Extra Questions

Real Numbers - Complete Study Guide

Comprehensive Chapter Summary

1. Introduction

In Class IX, you began your exploration of the world of real numbers and encountered irrational numbers. We continue our discussion on real numbers in this chapter. We begin with two very important properties of positive integers, namely the Euclid’s division algorithm and the Fundamental Theorem of Arithmetic. The chapter explores divisibility, prime factorization, irrationality proofs, and decimal expansions of rationals.

Real numbers include both rational and irrational numbers. Rational numbers are those expressible as p/q where p and q are integers, q ≠ 0. Irrational numbers cannot be expressed this way, like √2, √3. The real number line represents all reals, with rationals dense but countable, irrationals uncountable. Historical context: Discovery of irrationals by Hippasus, Gauss on FTA.

Importance: Real numbers form the basis for calculus, analysis. Applications in physics (measurements), engineering (precise calculations).

2. Euclid’s Division Algorithm

Euclid’s division algorithm deals with divisibility of integers. Any positive integer a can be divided by another positive integer b leaving a remainder r smaller than b: a = bq + r, 0 ≤ r < b. This is used to find HCF using repeated division until remainder is 0.

Applications include computing HCF, which is the last non-zero remainder in the process. Also used in cryptography (Euclidean algorithm for gcd), number theory proofs.

Extended: For negative integers, take absolute values. Relation to modular arithmetic: r = a mod b.

3. The Fundamental Theorem of Arithmetic

Prime Factorization

Every composite number can be expressed as a product of primes uniquely (up to order). For example, 32760 = 2^3 × 3^2 × 5 × 7 × 13. This uniqueness is key for many proofs.

Composite numbers are natural numbers greater than 1 that are not prime, i.e., have factors other than 1 and itself. Primes are infinite, proof by Euclid: Assume finite, product+1 not divisible by any, new prime.

Method: Factor tree, trial division up to sqrt(n).

Applications

Proving irrationality: Used to show numbers like √2 are irrational by contradiction. Also, determines if decimal expansion of 1/p is terminating (if p has only 2 and 5 as prime factors) or repeating.

Infinitely many primes: If finite, product +1 would be new prime, contradiction. Used in RSA encryption, number of divisors formula d(n) = (e1+1)(e2+1)...

Radical simplification, rationalizing denominators.

HCF and LCM

HCF: Product of smallest powers of common primes. LCM: Product of highest powers. Relation: HCF(a,b) × LCM(a,b) = a × b. Extended to three numbers with formulas provided.

For example, HCF(6,20)=2, LCM=60, product=120=6*20. Applications: Simplifying fractions, solving Diophantine equations, scheduling problems.

Note for >2 numbers: Product ≠ HCF * LCM, use pairwise methods or formulas.

4. Irrational Numbers

Proof of Irrationality

Proof by contradiction for √p where p prime: Assume rational a/b coprime, square to a^2 = p b^2, p divides a^2 hence a, substitute, p divides b, contradiction.

Examples: √2, √3, √5 irrational. Properties: Rational + irrational = irrational, etc. Transcendental irrationals like π, e beyond scope but mentioned.

Historical: Pythagoreans, crisis in mathematics.

Properties

Sum/difference/product/quotient of rational and irrational is irrational (non-zero rational). Examples: 5 - √3, 3√2 irrational.

Locating irrationals on number line from Class IX recalled. Density: Between any two reals, infinite rationals and irrationals.

Irrational + irrational can be rational (√2 + (-√2)=0), product too (√2 * √2=2).

5. Decimal Expansions

For rational p/q in lowest terms, terminating if q = 2^m * 5^n, else non-terminating repeating. Prime factorization of q reveals this.

Examples: 1/2=0.5 terminating, 1/3=0.333... repeating. Converting repeating to fraction: Let x=0.333..., 10x=3.333..., 9x=3, x=1/3.

Non-repeating infinite decimals for irrationals like π=3.14159..., e=2.71828.... Applications in computing precision, financial calculations.

Key Concepts and Definitions

Real Numbers

Union of rational and irrational numbers. Rationals: p/q, q≠0. Irrationals: Not p/q, e.g., √2, π.

Every real number has a unique decimal expansion, possibly infinite. Completeness axiom: Every non-empty upper bounded set has least upper bound.

Countability: Rationals countable, reals uncountable (Cantor's diagonal).

Euclid's Division Lemma

For positive integers a, b, unique q, r: a = bq + r, 0 ≤ r < b. Basis for Euclidean algorithm for HCF.

Algorithm: Replace a by b, b by r, repeat until r=0, last b is HCF. Time complexity O(log min(a,b)).

Extended Euclidean: Finds x,y such that ax + by = gcd.

Fundamental Theorem of Arithmetic

Every composite has unique prime factorization (up to order). Primes are building blocks of numbers.

Composite: Number >1 not prime. Prime: Only factors 1 and itself. Twin primes, Goldbach conjecture (unproven: even >2 = sum two primes).

Sieve of Eratosthenes for finding primes up to n.

Irrational Number

Cannot be p/q. Proofs use contradiction and FTA. Examples: √p for prime p, e, π.

Properties: Operations with rationals yield irrationals. Algebraic vs transcendental: √2 algebraic, π transcendental.

Approximations: Continued fractions for best rational approximations.

HCF (Highest Common Factor)

Largest divisor of both numbers. From prime factors: Min powers of commons.

Also called GCD. Used in simplifying fractions. Coprime if HCF=1, Bezout's identity: gcd= linear combo.

Properties: gcd(a,b)=gcd(a,b+ka), etc.

LCM (Least Common Multiple)

Smallest multiple of both. From primes: Max powers of all.

Applications: Time cycles, like meeting times in circular path. lcm(a,b)= |a b|/gcd(a,b) for nonzero a,b.

For multiples: lcm(a,b,c)=lcm(lcm(a,b),c).

Important Facts

HCF × LCM = a × b

Irrational √2, √3, π, e

Terminating Decimal Denom: 2^m * 5^n

Prime Factorization Unique

Infinitely Many Primes

Proof Technique Contradiction

Reals Uncountable Cantor

GCD Linear Combo Bezout

Theorems and Proofs

Theorem 1.1: Fundamental Theorem of Arithmetic

Every composite number can be expressed as a product of primes uniquely (up to order). Proof based on Euclid's work, formalized by Gauss.

Detailed: Existence by repeated division by primes. Uniqueness: Assume two factorizations, use smallest prime to divide, contradict minimality. Lemma: If p prime divides ab, p divides a or b (Euclid's lemma).

Proof of lemma: If p divides ab, not a, then gcd(p,a)=1, so px + ay=1, multiply by b: p xb + a y b = b, p divides left, hence b.

Theorem 1.2: If p (prime) divides a², then p divides a

Proof: a = p1 p2 ... pn primes. a² has even powers. p divides a², so p one of pi by uniqueness. Thus p divides a.

Corollary: p divides a^k implies p divides a. General: For any k.

Proof for cube: Similar, p divides a^3, from FTA.

Theorem 1.3: √2 is irrational

Proof by contradiction: Assume √2 = a/b coprime. 2b² = a², 2 divides a² hence a (Thm 1.2). a=2c, 2b²=4c², b²=2c², 2 divides b. Contradiction.

Similar for √3: 3b²=a², 3 divides a, a=3c, 9c²=3b², b²=3c², 3 divides b.

For √5: 5b²=a², 5 divides a, a=5c, 25c²=5b², b²=5c², 5 divides b.

Theorem: √p irrational for prime p

General proof: Assume a/b coprime, a² = p b², p divides a² hence a, a= p c, p² c² = p b², b² = p c², p divides b. Contradiction.

Extension: For non-square-free n, if n not perfect square, √n irrational.

Theorem: Sum of rational and irrational is irrational

Proof: Assume r + i = s rational. Then i = s - r rational, contradiction unless i=0 but i irrational.

Similar for difference, product (nonzero r), quotient.

Counter: Irrational + irrational can be rational, e.g., √2 + (2 - √2) = 2.

Theorem: There are infinitely many primes

Proof: Assume finite p1...pk. Let n = p1...pk +1. n >1, has prime factor q not among pi (since n mod pi =1). Contradiction.

Variants: Dirichlet's theorem on primes in arithmetic progression.

Additional Proofs

√6 irrational: Assume a/b, a²=6b², 2 and 3 divide a² hence a, substitute, divide b.

log2 3 irrational: Assume p/q, 3^q = 2^p, odd=even contradiction for p,q>0.

π irrational: Advanced, but known since 1761 by Lambert.

1/√2 = √2/2 irrational since √2 irrational, divide by rational.

Solved Examples from Chapter

Example 1: Check if 4^n ends with 0 for some natural n.

Solution: \( 4^n = (2^2)^n = 2^{2n} \). To end with 0, divisible by 10=2*5, but no 5 in factors. By FTA uniqueness, impossible. Examples: \( 4^1=4 \), \( 4^2=16 \), \( 4^3=64 \), none end 0.

Example 2: HCF and LCM of 6 and 20 by prime factorization.

Solution: \( 6=2 \times 3 \), \( 20=2^2 \times 5 \). HCF=\( 2^1=2 \), LCM=\( 2^2 \times 3 \times 5=60 \). Verify \( 2 \times 60=120=6 \times 20 \). Application: Simplify 6/20 = 3/10.

Example 3: HCF of 96 and 404, then LCM.

Solution: \( 96=2^5 \times 3 \), \( 404=2^2 \times 101 \). HCF=\( 2^2=4 \). LCM=(96*404)/4=9696. Euclidean: 404=4*96 +20, 96=4*20+16, 20=1*16+4, 16=4*4+0, HCF=4.

Example 4: HCF and LCM of 6,72,120.

Solution: \( 6=2 \times 3 \), \( 72=2^3 \times 3^2 \), \( 120=2^3 \times 3 \times 5 \). HCF=\( 2 \times 3=6 \), LCM=\( 2^3 \times 3^2 \times 5=360 \). Note 6*72*120 = 51840 ≠ 6*360=2160. Use pairwise LCM.

Example 5: Prove √3 irrational.

Solution: Assume \( \sqrt{3} = \frac{a}{b} \) coprime. \( a^2=3b^2 \), 3 divides a, a=3c, 9c^2=3b^2, b^2=3c^2, 3 divides b. Contradiction. Approximation: 1.7320508...

Example 6: Show 5 - √3 irrational.

Solution: Assume rational \( \frac{a}{b} \). Then \( \sqrt{3} = 5 - \frac{a}{b} \) rational, contradiction. Similarly for 5 + √3.

Example 7: Show 3√2 irrational.

Solution: Assume \( \frac{a}{b} \). Then \( \sqrt{2} = \frac{a}{3b} \) rational, contradiction. General: k√m irrational if √m irrational, k rational nonzero.

Example 8: Decimal of 1/7.

Solution: 7 has prime 7, so repeating: 0.142857 repeating. Period 6.

Example 9: Factorize 123456789.

Solution: \( 3^2 \times 3803 \times 3607 \), where 3803 and 3607 primes.

Example 10: HCF(306,657).

Solution: Euclidean: 657=2*306+45, 306=6*45+36, 45=1*36+9, 36=4*9+0, HCF=9.

Questions and Answers

Exercise 1.1

1. Express each number as a product of its prime factors:

(i) 140

Solution: \( 140 = 2^{2} \times 5 \times 7 \)

(ii) 156

Solution: \( 156 = 2^{2} \times 3 \times 13 \)

(iii) 3825

Solution: \( 3825 = 3^{2} \times 5^{2} \times 17 \)

(iv) 5005

Solution: \( 5005 = 5 \times 7 \times 11 \times 13 \)

(v) 7429

Solution: \( 7429 = 17 \times 19 \times 23 \)

2. Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers.

(i) 26 and 91

Solution: \( 26 = 2 \times 13 \), \( 91 = 7 \times 13 \), HCF = 13, LCM = \( 2 \times 7 \times 13 = 182 \), Verify: \( 13 \times 182 = 2366 = 26 \times 91 \)

(ii) 510 and 92

Solution: \( 510 = 2 \times 3 \times 5 \times 17 \), \( 92 = 2^{2} \times 23 \), HCF = 2, LCM = \( 2^{2} \times 3 \times 5 \times 17 \times 23 = 23460 \), Verify: \( 2 \times 23460 = 46920 = 510 \times 92 \)

(iii) 336 and 54

Solution: \( 336 = 2^{4} \times 3 \times 7 \), \( 54 = 2 \times 3^{3} \), HCF = \( 2 \times 3 = 6 \), LCM = \( 2^{4} \times 3^{3} \times 7 = 3024 \), Verify: \( 6 \times 3024 = 18144 = 336 \times 54 \)

3. Find the LCM and HCF of the following integers by applying the prime factorisation method.

(i) 12, 15 and 21

Solution: \( 12 = 2^{2} \times 3 \), \( 15 = 3 \times 5 \), \( 21 = 3 \times 7 \), HCF = 3, LCM = \( 2^{2} \times 3 \times 5 \times 7 = 420 \)

(ii) 17, 23 and 29

Solution: All primes, HCF = 1, LCM = \( 17 \times 23 \times 29 = 11339 \)

(iii) 8, 9 and 25

Solution: \( 8 = 2^{3} \), \( 9 = 3^{2} \), \( 25 = 5^{2} \), HCF = 1, LCM = \( 2^{3} \times 3^{2} \times 5^{2} = 1800 \)

4. Given that HCF (306, 657) = 9, find LCM (306, 657).

Solution: LCM = \( \frac{306 \times 657}{9} = 22338 \)

5. Check whether 6n can end with the digit 0 for any natural number n.

Solution: \( 6^{n} = (2 \times 3)^{n} = 2^{n} \times 3^{n} \), to end with 0 need 2 and 5, no 5, impossible.

6. Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers.

Solution: \( 7 \times 11 \times 13 + 13 = 13(7 \times 11 + 1) = 13 \times 78 \) composite. \( 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5 = 5(7 \times 6 \times 4 \times 3 \times 2 \times 1 + 1) = 5 \times 1009 \) composite.

7. There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. After how many minutes will they meet again at the starting point?

Solution: LCM(18,12) = 36 minutes.

Exercise 1.2

1. Prove that √5 is irrational.

Solution: Assume \( \sqrt{5} = \frac{a}{b} \) coprime. \( a^{2} = 5b^{2} \), 5 divides a, a=5c, \( 25c^{2} = 5b^{2} \), \( b^{2} = 5c^{2} \), 5 divides b, contradiction.

2. Prove that 3 + 2√5 is irrational.

Solution: Assume rational r, then \( 2\sqrt{5} = r - 3 \) rational, \( \sqrt{5} = \frac{r-3}{2} \) rational, contradiction.

3. Prove that the following are irrationals:

(i) \( \frac{1}{\sqrt{2}} \)

Solution: Assume \( \frac{a}{b} \), then \( \sqrt{2} = \frac{b}{a} \) rational, contradiction.

(ii) 7√5

Solution: Assume \( \frac{a}{b} \), then \( \sqrt{5} = \frac{a}{7b} \) rational, contradiction.

(iii) 6 + √2

Solution: Assume \( \frac{a}{b} \), then \( \sqrt{2} = \frac{a}{b} - 6 \) rational, contradiction.

Short Questions

Q1. Prime factors of 12.

Answer: \( 2^{2} \times 3 \).

Q2. HCF(8,12).

Answer: 4.

Q3. LCM(4,6).

Answer: 12.

Q4. Is √4 rational?

Answer: Yes, 2.

Q5. Terminating? 1/4.

Answer: Yes, 0.25.

Q6. Prime factors of 30.

Answer: \( 2 \times 3 \times 5 \).

Q7. HCF(15,25).

Answer: 5.

Q8. LCM(5,7).

Answer: 35.

Q9. Is π rational?

Answer: No.

Q10. Terminating? 1/6.

Answer: No, 0.1666...

Q11. Factors of 42.

Answer: \( 2 \times 3 \times 7 \).

Q12. HCF(18,24).

Answer: 6.

Q13. LCM(8,12).

Answer: 24.

Q14. Rational? 0.5.

Answer: Yes, \( \frac{1}{2} \).

Q15. Repeating? 1/9.

Answer: Yes, 0.111...

Q16. Factors of 100.

Answer: \( 2^{2} \times 5^{2} \).

Q17. HCF(20,30).

Answer: 10.

Q18. LCM(9,15).

Answer: 45.

Q19. Irrational? e.

Answer: Yes.

Q20. Terminating? 3/8.

Answer: Yes, 0.375.

Medium Questions

Q1. Prove √7 irrational.

Solution: Assume \( \sqrt{7} = \frac{a}{b} \) coprime. \( a^{2} = 7b^{2} \), 7 divides a, a=7c, 49c^{2}=7b^{2}, b^{2}=7c^{2}, 7 divides b. Contradiction.

Q2. HCF(48,60,72).

Solution: \( 48=2^{4} \times 3 \), \( 60=2^{2} \times 3 \times 5 \), \( 72=2^{3} \times 3^{2} \). HCF=\( 2^{2} \times 3=12 \).

Q3. LCM(10,15,25).

Solution: \( 10=2 \times 5 \), \( 15=3 \times 5 \), \( 25=5^{2} \). LCM=\( 2 \times 3 \times 5^{2}=150 \).

Q4. Show 2 + √5 irrational.

Solution: Assume rational, \( \sqrt{5} = \) rational -2 rational, contradiction.

Q5. Decimal type 1/11.

Solution: 11 prime not 2 or 5, repeating 0.0909...

Q6. Factorize 980.

Solution: \( 2^{2} \times 5 \times 7^{2} \).

Q7. HCF(105,175).

Solution: Euclidean: 175=1*105+70, 105=1*70+35, 70=2*35+0, HCF=35.

Q8. LCM(12,18,24).

Solution: \( 12=2^{2} \times 3 \), \( 18=2 \times 3^{2} \), \( 24=2^{3} \times 3 \). LCM=\( 2^{3} \times 3^{2}=72 \).

Q9. Prove 4√2 irrational.

Solution: Assume rational, \( \sqrt{2} = \) rational/4 rational, contradiction.

Q10. Terminating? 5/16.

Solution: 16=\( 2^{4} \), yes, 0.3125.

Q11. Factors of 315.

Solution: \( 3^{2} \times 5 \times 7 \).

Q12. HCF(84,96).

Answer: 12.

Q13. LCM(14,21).

Answer: 42.

Q14. Show √3 + √5 irrational.

Solution: Assume rational r. Square: 8 + 2√15 = r^2 rational, 2√15 rational, √15 rational, contradiction.

Q15. Decimal 2/5.

Solution: 0.4 terminating.

Q16. Factorize 1232.

Solution: \( 2^{4} \times 7 \times 11 \).

Q17. HCF(225,300).

Answer: 75.

Q18. LCM(20,25,30).

Answer: 300.

Q19. Prove √10 irrational.

Solution: 10=2*5, not square, assume a/b, a^2=10b^2, 2 and 5 divide a, then b, contradiction.

Q20. Repeating? 1/12.

Solution: 12=\( 2^{2} \times 3 \), repeating 0.08333...

Long Questions

Q1. Use Euclid algorithm to find HCF(1234,5678), show steps.

Solution: 5678=4*1234+742, 1234=1*742+492, 742=1*492+250, 492=1*250+242, 250=1*242+8, 242=30*8+2, 8=4*2+0, HCF=2.

Q2. Prove that if p prime divides ab, then p divides a or b.

Solution: Assume not a, gcd(p,a)=1, extended Euclidean px + ay=1, multiply b: p xb + a b y = b, p divides ab hence left, so b.

Q3. Find LCM(12,18,24,30) using prime factors, verify with formula.

Solution: \( 12=2^{2} \times 3 \), \( 18=2 \times 3^{2} \), \( 24=2^{3} \times 3 \), \( 30=2 \times 3 \times 5 \). LCM=\( 2^{3} \times 3^{2} \times 5=360 \). Pairwise verify.

Q4. Prove √6 irrational using FTA.

Solution: Assume a/b, a^2=6b^2, 2*3 b^2 = a^2, 2 and 3 divide a^2 hence a, a=6c, 36c^2=6b^2, 6c^2=b^2, 2 and 3 divide b, contradiction.

Q5. Determine decimal type for 7/40, compute.

Solution: 40=\( 2^{3} \times 5 \), terminating. 7/40=0.175.

Q6. Factorize 987654321, check if prime factors unique.

Solution: \( 3^{2} \times 37 \times 333667 \), unique by FTA.

Q7. Show that √2 + √3 irrational.

Solution: Assume r rational. Square: 5 + 2√6 = r^2 rational, 2√6 rational, √6 rational, contradiction as 6 not square.

Q8. Find HCF(1001,1000) using Euclidean, explain steps.

Solution: 1001=1*1000+1, 1000=1000*1+0, HCF=1. Coprime.

Q9. Prove there are infinitely many primes of form 4k+3.

Solution: Assume finite p1..pn all 4k+3. Let n=4(p1..pn)-1=4m+3. n odd >1, prime factors 4k+1 or 4k+3. If all 4k+1, product 4k+1 mod 4, but n=3 mod 4, contradiction. So new 4k+3 prime.

Q10. Convert 0.123123... to fraction.

Solution: x=0.123123..., 1000x=123.123..., 999x=123, x=\( \frac{123}{999} = \frac{41}{333} = \frac{13}{111} \).

Q11. Factorize 2025.

Solution: \( 45^{2} = (9 \times 5)^{2} = 3^{4} \times 5^{2} \).

Q12. HCF(385,462).

Solution: 462=1*385+77, 385=5*77+0, HCF=77.

Q13. LCM(16,24,36,48).

Solution: Prime max: \( 2^{4} \times 3^{2} = 144 \).

Q14. Prove √2 * √3 = √6 irrational.

Solution: Product of irrationals can be irrational. Assume rational, square 6 rational, but √6 irrational.

Q15. Decimal of 3/7.

Solution: 0.428571 repeating, period 6.

Q16. Factorize 5544.

Solution: \( 2^{3} \times 3^{2} \times 7 \times 11 \).

Q17. Show 1/√3 irrational.

Solution: Assume \( \frac{a}{b} \), \( \sqrt{3} = \frac{b}{a} \) rational, contradiction.

Q18. HCF(999,1001).

Solution: 1001=1*999+2, 999=499*2+1, 2=2*1+0, HCF=1.

Q19. LCM(35,49,63).

Solution: \( 35=5 \times 7 \), \( 49=7^{2} \), \( 63=3^{2} \times 7 \). LCM=\( 3^{2} \times 5 \times 7^{2}=2205 \).

Q20. Prove log10 2 irrational.

Solution: Assume p/q, \( 10^{p/q}=2 \), \( 10^{p} = 2^{q} \), left 2 or 5 factors only, right 2 only, contradiction unless p=q=0.

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10 Qs · ~10 min

#13

कक्षा 10 संस्कृत — धातुरूप परिचय (अभ्यास प्रश्नोत्तरी)

10 Qs · ~10 min

#14

कक्षा 10 संस्कृत — शब्दरूप परिचय (अभ्यास प्रश्नोत्तरी)

10 Qs · ~10 min

#15

कक्षा 10 संस्कृत — सन्धिः (अभ्यास प्रश्नोत्तरी)

10 Qs · ~10 min

#16

कक्षा 10 संस्कृत — संज्ञा एवं परिभाषा (अभ्यास प्रश्नोत्तरी)

10 Qs · ~10 min

#17

कक्षा 10 संस्कृत — वर्णविचारः (अभ्यास प्रश्नोत्तरी)

10 Qs · ~10 min

#18

कक्षा 10 संस्कृत — भूकंपविभीषिका (अभ्यास प्रश्नोत्तरी)

10 Qs · ~10 min

#19

कक्षा 10 संस्कृत — सूक्तयः (अभ्यास प्रश्नोत्तरी)

10 Qs · ~10 min

#20

कक्षा 10 संस्कृत — विचित्रः साक्षी (अभ्यास प्रश्नोत्तरी)

10 Qs · ~10 min

#21

कक्षा 10 संस्कृत — सौहार्दं प्रकृतेः शोभा (अभ्यास प्रश्नोत्तरी)

10 Qs · ~10 min

#22

कक्षा 10 संस्कृत — सुभाषितानि (अभ्यास प्रश्नोत्तरी)

10 Qs · ~10 min

#23

कक्षा 10 संस्कृत — जननी तुल्यवत्सला (अभ्यास प्रश्नोत्तरी)

10 Qs · ~10 min

#24

कक्षा 10 संस्कृत — शिशुलालनम् (अभ्यास प्रश्नोत्तरी)

10 Qs · ~10 min

#25

कक्षा 10 संस्कृत — व्यायामः सर्वदा पथ्यः (अभ्यास प्रश्नोत्तरी)

10 Qs · ~10 min

#26

कक्षा 10 संस्कृत — बुद्धिर्बलवती सदा (अभ्यास प्रश्नोत्तरी)

10 Qs · ~10 min

#27

कक्षा 10 संस्कृत — शुचिपर्यावरणम् (अभ्यास प्रश्नोत्तरी)

10 Qs · ~10 min

#28

कक्षा 10 हिंदी — टोपी शुक्ला (अभ्यास प्रश्नोत्तरी)

10 Qs · ~10 min

#29

कक्षा 10 हिंदी — सपनों के-से दिन (अभ्यास प्रश्नोत्तरी)

10 Qs · ~10 min

#30

कक्षा 10 हिंदी — हरिहर काका (अभ्यास प्रश्नोत्तरी)

10 Qs · ~10 min

#31

कक्षा 10 हिंदी — पतझर में टूटी पत्तियाँ, कारतूस (अभ्यास प्रश्नोत्तरी)

10 Qs · ~10 min

#32

कक्षा 10 हिंदी — अब कहाँ दूसरे के दुख से दुखी होने वाले (अभ्यास प्रश्नोत्तरी)

10 Qs · ~10 min

#33

कक्षा 10 हिंदी — गिरगिट (अभ्यास प्रश्नोत्तरी)

10 Qs · ~10 min

#34

कक्षा 10 हिंदी — तीसरी कसम के शिल्पकार शैलेंद्र (अभ्यास प्रश्नोत्तरी)

10 Qs · ~10 min

#35

कक्षा 10 हिंदी — तताँरा-वामीरो कथा (अभ्यास प्रश्नोत्तरी)

10 Qs · ~10 min

#36

कक्षा 10 हिंदी — डायरी का एक पन्ना (अभ्यास प्रश्नोत्तरी)

10 Qs · ~10 min

#37

कक्षा 10 हिंदी — बड़े भाई साहब (अभ्यास प्रश्नोत्तरी)

10 Qs · ~10 min

#38

कक्षा 10 हिंदी — तोप, कर चले हम फ़िदा, आत्मत्राण (अभ्यास प्रश्नोत्तरी)

10 Qs · ~10 min

#39

कक्षा 10 हिंदी — मधुर-मधुर मेरे दीपक जल (अभ्यास प्रश्नोत्तरी)

10 Qs · ~10 min

#40

कक्षा 10 हिंदी — पर्वत प्रदेश में पावस (अभ्यास प्रश्नोत्तरी)

10 Qs · ~10 min

#41

कक्षा 10 हिंदी — मनुष्यता (अभ्यास प्रश्नोत्तरी)

10 Qs · ~10 min

#42

कक्षा 10 हिंदी — बिहारी के दोहे (अभ्यास प्रश्नोत्तरी)

10 Qs · ~10 min

#43

कक्षा 10 हिंदी — मीरा के पद (अभ्यास प्रश्नोत्तरी)

10 Qs · ~10 min

#44

कक्षा 10 हिंदी — कबीर की साखी (अभ्यास प्रश्नोत्तरी)

10 Qs · ~10 min

#45

कक्षा 10 हिंदी — मैं क्यों लिखता हूँ (अभ्यास प्रश्नोत्तरी)

10 Qs · ~10 min

#46

कक्षा 10 हिंदी — साना-साना हाथ जोड़ि (अभ्यास प्रश्नोत्तरी)

10 Qs · ~10 min

#47

कक्षा 10 हिंदी — माता का अँचल (अभ्यास प्रश्नोत्तरी)

10 Qs · ~10 min

#48

कक्षा 10 हिंदी — संस्कृति (अभ्यास प्रश्नोत्तरी)

10 Qs · ~10 min

#49

कक्षा 10 हिंदी — नौबतखाने में इबादत (अभ्यास प्रश्नोत्तरी)

10 Qs · ~10 min

#50

कक्षा 10 हिंदी — एक कहानी यह भी (अभ्यास प्रश्नोत्तरी)

10 Qs · ~10 min

#51

कक्षा 10 हिंदी — लखनवी अंदाज़ (अभ्यास प्रश्नोत्तरी)

10 Qs · ~10 min

#52

कक्षा 10 हिंदी — बालगोबिन भगत (अभ्यास प्रश्नोत्तरी)

10 Qs · ~10 min

#53

कक्षा 10 हिंदी — नेताजी का चश्मा (अभ्यास प्रश्नोत्तरी)

10 Qs · ~10 min

#54

कक्षा 10 हिंदी — संगतकार (अभ्यास प्रश्नोत्तरी)

10 Qs · ~10 min

#55

कक्षा 10 हिंदी — यह दंतुरित मुसकान, फसल (अभ्यास प्रश्नोत्तरी)

10 Qs · ~10 min

#56

कक्षा 10 हिंदी — उत्साह, अट नहीं रही है (अभ्यास प्रश्नोत्तरी)

10 Qs · ~10 min

#57

कक्षा 10 हिंदी — आत्मकथ्य (अभ्यास प्रश्नोत्तरी)

10 Qs · ~10 min

#58

कक्षा 10 हिंदी — राम-लक्ष्मण-परशुराम संवाद (अभ्यास प्रश्नोत्तरी)

10 Qs · ~10 min

#59

कक्षा 10 हिंदी — सूरदास के पद (अभ्यास प्रश्नोत्तरी)

10 Qs · ~10 min

#60

Class 10 English — For Anne Gregory (Practice Quiz)

10 Qs · ~10 min

#61

Class 10 English — The Tale of Custard the Dragon (Practice Quiz)

10 Qs · ~10 min

#62

Class 10 English — Fog (Practice Quiz)

10 Qs · ~10 min

#63

Class 10 English — The Trees (Practice Quiz)

10 Qs · ~10 min

#64

Class 10 English — Amanda! (Practice Quiz)

10 Qs · ~10 min

#65

Class 10 English — The Ball Poem (Practice Quiz)

10 Qs · ~10 min

#66

Class 10 English — How to Tell Wild Animals (Practice Quiz)

10 Qs · ~10 min

#67

Class 10 English — A Tiger in the Zoo (Practice Quiz)

10 Qs · ~10 min

#68

Class 10 English — Fire and Ice (Practice Quiz)

10 Qs · ~10 min

#69

Class 10 English — Dust of Snow (Practice Quiz)

10 Qs · ~10 min

#70

Class 10 Health & PE — Doping and Ethics in Sports (Practice Quiz)

10 Qs · ~10 min

#71

Class 10 Health & PE — Adolescent Health and Wellbeing (Practice Quiz)

10 Qs · ~10 min

#72

Class 10 Health & PE — Diseases: Communicable and Non-communicable (Practice Quiz)

10 Qs · ~10 min

#73

Class 10 Health & PE — The Olympic Movement (Practice Quiz)

10 Qs · ~10 min

#74

Class 10 Health & PE — Common Sports Injuries and Prevention (Practice Quiz)

10 Qs · ~10 min

#75

Class 10 Health & PE — First Aid and Safety Education (Practice Quiz)

10 Qs · ~10 min

#76

Class 10 Health & PE — Yoga and Healthy Living (Practice Quiz)

10 Qs · ~10 min

#77

Class 10 Health & PE — Posture and Postural Deformities (Practice Quiz)

10 Qs · ~10 min

#78

Class 10 Health & PE — Food and Nutrition for Health (Practice Quiz)

10 Qs · ~10 min

#79

Class 10 Health & PE — Effects of Exercise on the Body Systems (Practice Quiz)

10 Qs · ~10 min

#80

Class 10 Health & PE — Physical Fitness, Wellness and Lifestyle (Practice Quiz)

10 Qs · ~10 min

#81

Class 10 Health & PE — Physical Education: Meaning and Importance (Practice Quiz)

10 Qs · ~10 min

#82

Class 10 English — The Book That Saved the Earth (Practice Quiz)

10 Qs · ~10 min

#83

Class 10 English — Bholi (Practice Quiz)

10 Qs · ~10 min

#84

Class 10 English — The Necklace (Practice Quiz)

10 Qs · ~10 min

#85

Class 10 English — The Making of a Scientist (Practice Quiz)

10 Qs · ~10 min

#86

Class 10 English — Footprints Without Feet (Practice Quiz)

10 Qs · ~10 min

#87

Class 10 English — A Question of Trust (Practice Quiz)

10 Qs · ~10 min

#88

Class 10 English — The Midnight Visitor (Practice Quiz)

10 Qs · ~10 min

#89

Class 10 English — The Thief’s Story (Practice Quiz)

10 Qs · ~10 min

#90

Class 10 English — The Proposal (Practice Quiz)

10 Qs · ~10 min

#91

Class 10 English — The Sermon at Benares (Practice Quiz)

10 Qs · ~10 min

#92

Class 10 English — Madam Rides the Bus (Practice Quiz)

10 Qs · ~10 min

#93

Class 10 English — Mijbil the Otter (Practice Quiz)

10 Qs · ~10 min

#94

Class 10 English — Glimpses of India (Practice Quiz)

10 Qs · ~10 min

#95

Class 10 English — From the Diary of Anne Frank (Practice Quiz)

10 Qs · ~10 min

#96

Class 10 English — Two Stories about Flying (Practice Quiz)

10 Qs · ~10 min

#97

Class 10 Social Science — Consumer Rights (Practice Quiz)

10 Qs · ~10 min

#98

Class 10 Social Science — Globalisation and the Indian Economy (Practice Quiz)

10 Qs · ~10 min

#99

Class 10 Social Science — Money and Credit (Practice Quiz)

10 Qs · ~10 min

#100

Class 10 Social Science — Sectors of the Indian Economy (Practice Quiz)

10 Qs · ~10 min

#101

Class 10 Social Science — Development (Practice Quiz)

10 Qs · ~10 min

#102

Class 10 Social Science — Outcomes of Democracy (Practice Quiz)

10 Qs · ~10 min

#103

Class 10 Social Science — Political Parties (Practice Quiz)

10 Qs · ~10 min

#104

Class 10 Social Science — Gender, Religion and Caste (Practice Quiz)

10 Qs · ~10 min

#105

Class 10 Social Science — Federalism (Practice Quiz)

10 Qs · ~10 min

#106

Class 10 Social Science — Power-sharing (Practice Quiz)

10 Qs · ~10 min

#107

Class 10 Social Science — Lifelines of National Economy (Practice Quiz)

10 Qs · ~10 min

#108

Class 10 Social Science — Manufacturing Industries (Practice Quiz)

10 Qs · ~10 min

#109

Class 10 Social Science — Minerals and Energy Resources (Practice Quiz)

10 Qs · ~10 min

#110

Class 10 Social Science — Agriculture (Practice Quiz)

10 Qs · ~10 min

#111

Class 10 Social Science — Water Resources (Practice Quiz)

10 Qs · ~10 min

#112

Class 10 Social Science — Forest and Wildlife Resources (Practice Quiz)

10 Qs · ~10 min

#113

Class 10 Social Science — Resources and Development (Practice Quiz)

10 Qs · ~10 min

#114

Class 10 Social Science — Print Culture and the Modern World (Practice Quiz)

10 Qs · ~10 min

#115

Class 10 Social Science — The Age of Industrialisation (Practice Quiz)

10 Qs · ~10 min

#116

Class 10 Social Science — The Making of a Global World (Practice Quiz)

10 Qs · ~10 min

#117

Class 10 Social Science — Nationalism in India (Practice Quiz)

10 Qs · ~10 min

#118

Class 10 Social Science — The Rise of Nationalism in Europe (Practice Quiz)

10 Qs · ~10 min

#119

Class 10 Science — Our Environment (Practice Quiz)

10 Qs · ~10 min

#120

Class 10 Science — Magnetic Effects of Electric Current (Practice Quiz)

10 Qs · ~10 min

#121

Class 10 Science — Electricity (Practice Quiz)

10 Qs · ~10 min

#122

Class 10 Science — The Human Eye and the Colourful World (Practice Quiz)

10 Qs · ~10 min

#123

Class 10 Science — Light – Reflection and Refraction (Practice Quiz)

10 Qs · ~10 min

#124

Class 10 Science — Heredity (Practice Quiz)

10 Qs · ~10 min

#125

Class 10 Science — How do Organisms Reproduce? (Practice Quiz)

10 Qs · ~10 min

#126

Class 10 Science — Control and Coordination (Practice Quiz)

10 Qs · ~10 min

#127

Class 10 Science — Life Processes (Practice Quiz)

10 Qs · ~10 min

#128

Class 10 Science — Carbon and its Compounds (Practice Quiz)

10 Qs · ~10 min

#129

Class 10 Science — Metals and Non-metals (Practice Quiz)

10 Qs · ~10 min

#130

Class 10 Science — Acids, Bases and Salts (Practice Quiz)

10 Qs · ~10 min

#131

Class 10 Maths — Probability (Practice Quiz)

10 Qs · ~10 min

#132

Class 10 Maths — Statistics (Practice Quiz)

10 Qs · ~10 min

#133

Class 10 Maths — Surface Areas and Volumes (Practice Quiz)

10 Qs · ~10 min

#134

Class 10 Maths — Areas Related to Circles (Practice Quiz)

10 Qs · ~10 min

#135

Class 10 Maths — Circles (Practice Quiz)

10 Qs · ~10 min

#136

Class 10 Maths — Some Applications of Trigonometry (Practice Quiz)

10 Qs · ~10 min

#137

Class 10 Maths — Introduction to Trigonometry (Practice Quiz)

10 Qs · ~10 min

#138

Class 10 Maths — Coordinate Geometry (Practice Quiz)

10 Qs · ~10 min

#139

Class 10 Maths — Triangles (Practice Quiz)

10 Qs · ~10 min

#140

Class 10 Maths — Arithmetic Progressions (Practice Quiz)

10 Qs · ~10 min

#141

Class 10 Maths — Quadratic Equations (Practice Quiz)

10 Qs · ~10 min

#142

Class 10 Maths — Pair of Linear Equations in Two Variables (Practice Quiz)

10 Qs · ~10 min

#143

Class 10 Maths — Polynomials (Practice Quiz)

10 Qs · ~10 min

#144

Class 10 Maths — Real Numbers (Practice Quiz)

10 Qs · ~10 min

#145

Class 10 Science — Chemical Reactions and Equations (Practice Quiz)

10 Qs · ~10 min

#146

Political Science: Democratic Politics II Practice Quiz | CBSE Class 10 Board Examination

10 Qs · ~10 min

#147

Economics: Understanding Economic Development Practice Quiz | CBSE Class 10 Board Examination

10 Qs · ~10 min

#148

Hindi A/B (Kshitij II / Sparsh II / Kritika II / Sanchayan II) Practice Quiz | CBSE Class 10 Board Examination

10 Qs · ~10 min

#149

Carbon and Its Compounds Advanced Challenge | CBSE Class 10 Board Examination

10 Qs · ~10 min

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Complete Solutions and Summary of Real Numbers – NCERT Class 10, Mathematics, Chapter 1 – Summary, Questions, Answers, Extra Questions

Real Numbers

Comprehensive Chapter Summary

1. Introduction

2. Euclid’s Division Algorithm

3. The Fundamental Theorem of Arithmetic

Prime Factorization

Applications

HCF and LCM

4. Irrational Numbers

Proof of Irrationality

Properties

5. Decimal Expansions

Key Concepts and Definitions

Real Numbers

Euclid's Division Lemma

Fundamental Theorem of Arithmetic

Irrational Number

HCF (Highest Common Factor)

LCM (Least Common Multiple)

Important Facts

All Formulas

Euclid's Division Algorithm

HCF and LCM Relation

For Three Numbers

Prime Factorization

Decimal Expansion

HCF from Primes

Theorems and Proofs

Theorem 1.1: Fundamental Theorem of Arithmetic

Theorem 1.2: If p (prime) divides a², then p divides a

Theorem 1.3: √2 is irrational

Theorem: √p irrational for prime p

Theorem: Sum of rational and irrational is irrational

Theorem: There are infinitely many primes

Additional Proofs

Solved Examples from Chapter

Example 1: Check if 4^n ends with 0 for some natural n.

Example 2: HCF and LCM of 6 and 20 by prime factorization.

Example 3: HCF of 96 and 404, then LCM.

Example 4: HCF and LCM of 6,72,120.

Example 5: Prove √3 irrational.

Example 6: Show 5 - √3 irrational.

Example 7: Show 3√2 irrational.

Example 8: Decimal of 1/7.

Example 9: Factorize 123456789.

Example 10: HCF(306,657).

Questions and Answers

Exercise 1.1

1. Express each number as a product of its prime factors:

2. Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers.

3. Find the LCM and HCF of the following integers by applying the prime factorisation method.

4. Given that HCF (306, 657) = 9, find LCM (306, 657).

5. Check whether 6n can end with the digit 0 for any natural number n.

6. Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers.

Exercise 1.2

1. Prove that √5 is irrational.

2. Prove that 3 + 2√5 is irrational.

3. Prove that the following are irrationals:

Short Questions

Q1. Prime factors of 12.

Q2. HCF(8,12).

Q3. LCM(4,6).

Q4. Is √4 rational?

Q5. Terminating? 1/4.

Q6. Prime factors of 30.

Q7. HCF(15,25).

Q8. LCM(5,7).

Q9. Is π rational?

Q10. Terminating? 1/6.

Q11. Factors of 42.

Q12. HCF(18,24).

Q13. LCM(8,12).

Q14. Rational? 0.5.

Q15. Repeating? 1/9.

Q16. Factors of 100.

Q17. HCF(20,30).

Q18. LCM(9,15).

Q19. Irrational? e.

Q20. Terminating? 3/8.