by Tiwari Academy
NCERT Textbook Solutions for Class 10 Maths Chapter 1 Updated for New Session 2025-26 Real Numbers in English and Hindi Medium revised and modified for Board Exams. NCERT Math Solutions for Class 10 Mathematics Book Chapter 1 Real Numbers, guide students about the basics of number theory. Grade 10th Maths 1st chapter is foundational and it is covering topics like Euclid’s division lemma and the Fundamental Theorem of Arithmetic.
Class 10 Maths Chapter 1 Solutions
Class 10 Maths Exercise 1.1
1. Express each number as product of its prime factors:
(i) 140 (ii) 156 (iii) 3825 (iv) 5005 (v) 7429
See Answer(i) 140 = 2 × 2 × 5 × 7 = 22 × 5 × 7
(ii) 156 = 2 × 2 × 3 × 13 = 22 × 3 × 13
(iii) 3825 = 3 × 3 × 5 × 5 × 17 = 32 × 52 × 17
(iv) 5005 = 5 × 7 × 11 × 13
(v) 7429 = 17 × 19 × 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
See Answer(i) 26 and 91
26 = 2 × 13
91 = 7 × 13
HCF = 13
LCM = 2 × 7 × 13 = 182
Product of the two numbers = 26 × 91 = 2366
HCF × LCM = 13 × 182 = 2366
Hence, product of two numbers = HCF × LCM
(ii) 510 and 92
See Answer(ii) 510 and 92
510 = 2 × 3 × 5 × 17
92 = 2 × 2 × 23
HCF = 2
LCM = 2 × 2 × 3 × 5 × 17 × 23 = 23460
Product of the two numbers = 510 × 92 = 46920
HCF × LCM = 2 × 23460 = 46920
Hence, product of two numbers = HCF × LCM
(iii) 336 and 54
See Answer(iii) 336 and 52
336 = 2 × 2 × 2 × 2 × 3 × 7 = 24 × 3 × 7
54 = 2 × 3 × 3 × 3 = 2 × 33
HCF = 2 × 3 = 6
LCM = 24 × 33 × 7 = 3024
Product of the two numbers= 336 × 54 = 18144
HCF × LCM = 6 × 3024 = 18144
Hence, product of two numbers = HCF × LCM
3. Find the LCM and HCF of the following integers by applying the prime factorisation method.
(i) 12, 15 and 21
See Answer(i) 12, 15 and 21
12 = 2 × 2 × 3 = 22 × 3
15 = 3 × 5
21 = 3 × 7
HCF = 3 and LCM = 22 × 3 × 5 × 7 = 420
(ii) 17, 23 and 29
See Answer(ii) 17, 23 and 29
17 = 1 × 17
23 = 1 × 23
29 = 1 × 29
HCF = 1
LCM = 17 × 23 × 29 = 11339
(iii) 8, 9 and 25
See Answer(iii) 8, 9 and 25
8 = 2 × 2 × 2 = 23
9 = 3 × 3 = 32
25 = 5 × 5 = 52
HCF = 1
LCM = 23 × 32 × 52 = 8 × 9 × 25 = 1800
4. Given that HCF (306, 657) = 9, find LCM (306, 657).
See AnswerHCF (306, 657) = 9
We know that,
LCM × HCF = Product of two numbers
Therefore, LCM = Product of two numbers/HCF
So, LCM = (306 × 657)/9 = 22338
Hence, LCM (306, 657) = 22338
5. Check whether 6ⁿ can end with the digit 0 for any natural number n.
See AnswerIf any number ends with the digit 0, it should be divisible by 10 or in other words, it will also be divisible by 2 and 5 as 10 = 2 × 5.
Prime factorisation of 6ⁿ = (2 × 3)ⁿ
It can be observed that 5 is not in the prime factorisation of 6ⁿ.
Hence, for any value of n, 6ⁿ will not be divisible by 5.
Therefore, 6ⁿ cannot 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.
See AnswerNumbers are of two types – prime and composite.
Prime numbers can be divided by 1 and only itself, whereas composite numbers have factors other than 1 and itself.
It can be observed that:
7 × 11 × 13 + 13 = 13 × (7 × 11 + 1)
= 13 × (77 + 1)
= 13 × 78
The given expression has 6 and 13 as its factors. Therefore, it is a composite number.
Now consider the next number:
7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 = 5 ×(7 × 6 × 4 × 3 × 2 × 1 + 1)
= 5 × (1008 + 1)
= 5 ×1009
1009 cannot be factorised further. Therefore, the given expression has 5 and 1009 as its factors. Hence, it is a composite number.
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?
See AnswerIt can be observed that Ravi takes lesser time than Sonia for completing 1 round of the circular path.
As they are going in the same direction, they will meet again at the same time when Ravi will have completed 1 round of that circular path with respect to Sonia.
The total time taken for completing this 1 round of circular path will be the LCM of time taken by Sonia and Ravi for completing 1 round of circular path respectively i.e., LCM of 18 minutes and 12 minutes.
18 = 2 × 3 × 3 and
12 = 2 × 2 × 3
LCM of 12 and 18 = 2 × 2 × 3 × 3 = 36
Therefore, Ravi and Sonia will meet together at the starting point after 36 minutes.
Class 10 Maths Exercise 1.2
1. Prove that √5 is irrational.
See AnswerLet √5 is a rational number.
Therefore, we can find two integers a, b (b ≠ 0) such that √5 = a/b. Let a and b have a common factor other than 1. Then we can divide them by the common factor and assume that a and b are co-prime.
a = √5b
⇒ a² = 5b²
Therefore, a² is divisible by 5 and it can be said that a is divisible by 5.
Let a = 5k, where k is an integer.
⇒ (5k)² = 5b²
⇒ 25k² = 5b²
⇒ 5k² = b²
This means that b² is divisible by 5 and hence, b is divisible by 5.
This implies that a and b have 5 as a common factor.
This is a contradiction to the fact that a and b are co-prime.
Hence, √5 cannot be expressed as p/q or it can be said that √5 is irrational.
2. Prove that is 3 + 2√5 irrational.
See AnswerLet 3 + 2√5 is rational.
Therefore, we can find two co-prime integers a, b (b ≠ 0) such that
3 + 2√5 = a/b
⇒ 2√5 = a/b – 3
⇒ √5 = 1/2(a/b – 3)
Since a and b are integers, 1/2(a/b – 3) will also be rational.
Therefore, √5 is rational.
This contradicts the fact that √5 is irrational.
Hence, our assumption that 3 + 2√5 is rational is false.
Therefore, 3 + 2√5 is irrational.
3. Prove that the following are irrationals:
(i) 1/√2
See Answer(i) 1/√2
Let 1/√2 is rational.
Therefore, we can assume two co-prime integers a, b (b ≠ 0) such that
1/√2 = a/b
⇒ √2 = b/a
b/a is rational as a and b are integers.
Therefore, √2 is also rational, which contradicts to the fact that √2 is irrational.
Hence, our assumption is false and 1/√2 is irrational.
(ii) 7√5
See Answer(ii) 7√5
Let 7√5 is rational.
Therefore, we assume find two co-prime integers a, b (b ≠ 0) such that
7√5 = a/b
⇒ √5 = a/7b
a/7b is rational as a and b are integers.
Therefore, √5 should be rational.
This contradicts the fact that √5 is irrational.
Therefore, our assumption that 7√5 is rational is false.
Hence, 7√5 is irrational.
(iii) 6 + √2
See Answer(iii) 6 + √2
Let 6 + √2 be rational.
Therefore, we can assume two co-prime integers a, b (b ≠ 0) such that
6 + √2 = a/b
⇒ √2 = a/b – 6
Since a and b are integers, a/b – 6 is also rational and hence, √2 should be rational.
This contradicts the fact that √2 is irrational.
Therefore, our assumption is false and hence, 6 + √2 is irrational.
Class 10 Maths Chapter 1 Previous Years Questions
Board Paper Questions From 10th Maths Chapter 1
Class 10 Maths Chapter 1 Board Questions and their answers with explanation are given here. These questions are asked in CBSE Board Exams in previous years.
1. Prove that √2 is an irrational number. [CBSE 2025]
See AnswerWe’ll use proof by contradiction.
Assume that √2 is rational. Then it can be written as √2 = p/q where p and q are integers with no common factors (in lowest form) and q ≠ 0.
This means: p² = 2q²
This equation shows that p² is even (as it equals 2q²), which means p must be even (since odd numbers squared remain odd).
So we can write p = 2k for some integer k.
Substituting: (2k)² = 2q²
4k² = 2q²
q² = 2k²
Now this shows that q² is even, which means q must be even.
But this contradicts our initial assumption that p and q have no common factors (since both would be divisible by 2).
Therefore, our initial assumption must be false and √2 cannot be expressed as a ratio of integers, making it irrational.
2. Show that 6ⁿ cannot end with digit 0 for any natural number n. [CBSE 2023]
See SolutionFor a number to end in 0, it must be divisible by 10.
10 = 2 × 5
Prime factorization of 6 = 2 × 3
Any power of 6ⁿ will only have 2 and 3 as prime factors.
Hence, it can never be divisible by 5.
Therefore, 6ⁿ cannot end in 0.
3. Find the HCF and LCM of 72 and 120. [CBSE 2023]
See SolutionPrime factorization:
72 = 2³ × 3²
120 = 2³ × 3 × 5
HCF = 2³ × 3 = 24
LCM = 2³ × 3² × 5 = 360
4. Using prime factorisation, find HCF and LCM of 96 and 120. [CBSE 2023]
See Solution96 = 2⁵ × 3
120 = 2³ × 3 × 5
HCF = 2³ × 3 = 24
LCM = 2⁵ × 3 × 5 = 480
5. Find the greatest 3-digit number which is divisible by 18, 24 and 36. [CBSE 2023]
See SolutionFind LCM of 18, 24, 36
= 2³ × 3² = 72
Greatest 3-digit number = 999
Divide: 999 ÷ 72 = 13.875
→ Take floor = 13
13 × 72 = 936
6. If √2 is an irrational number, prove (5 – 2√2) is also an irrational number. [CBSE 2023 Compt]
See SolutionAssume 5 – 2√2 is rational.
Then, 2√2 = 5 – (rational number) = rational
⇒ √2 = rational
⇒ Contradiction.
So, 5 – 2√2 is an irrational number.
7. Prove that √2 is an irrational number. [CBSE 2023, 2020 Compt]
See SolutionLet √2 = a/b in lowest terms.
⇒ 2 = a²/b²
⇒ a² = 2b²
⇒ a is even
⇒ a = 2k
⇒ (2k)² = 4k² = 2b²
⇒ b² = 2k²
⇒ b is even
⇒ Both a and b even
⇒ Contradiction.
So, √2 is irrational.
8. If p = ab³ and q = a²b, a and b are prime, find LCM(p, q). [CBSE 2020 Compt]
See Solutionp = ab³
q = a²b
LCM = a²b³
9. Prove that √3 is an irrational number. [CBSE 2020]
See SolutionSame method as √2 proof.
Assume √3 = a/b
⇒ 3 = a²/b²
⇒ a² = 3b²
⇒ a is divisible by 3, let a = 3k
⇒ (3k)² = 9k² = 3b²
⇒ b² = 3k²
⇒ b divisible by 3
⇒ Both divisible by 3
⇒ Contradiction
So, √3 is an irrational number.
10. The traffic lights at three different road crossings change after every 48s, 72s, and 138s. If they change simultaneously at 7am, at what time will they change together next? [CBSE 2020]
See SolutionLCM(48, 72, 138) = LCM = 432 sec
Therefore, after 432 s, they will change simultaneously. We know that 60s = 1 min
= 1 s = 1/60 min
= 432 s = 432/60 min = 7 min 12 s
Hence, the lights will change simultaneously at 7 : 07 : 12 am.
11. Find the HCF and LCM of 26, 65, 117 using prime factorisation. [CBSE 2020]
See Solution26 = 2 × 13
65 = 5 × 13
117 = 3 × 3 × 13
HCF = 13
LCM = 2 × 3² × 5 × 13 = 1170.
12. Prove that √5 is an irrational number. [CBSE 2020]
See SolutionSame process as √2 proof.
Let √5 = a/b
⇒ 5 = a²/b²
⇒ a² = 5b²
Then a and b both divisible by 5
⇒ contradiction
⇒ √5 is irrational number.
13. Prove 2 + 5√3 is an irrational number, given √3 is irrational. [CBSE 2019]
See SolutionAssume 2 + 5√3 is rational
⇒ Subtract 2: 5√3 = rational
⇒ √3 = rational, which is contradiction.
So, 2 + 5√3 is irrational.
14. Prove (2 + √3)/5 is an irrational, given √3 is an irrational number. [CBSE 2019]
See SolutionAssume (2 + √3)/5 is rational
⇒ 2 + √3 = rational
⇒ √3 = rational
⇒ Contradiction
So, (2 + √3)/5 is irrational.
15. Write the smallest number divisible by both 306 and 657. [CBSE 2019]
See SolutionFind LCM(306, 657)
Prime factors:
306 = 2 × 3² × 17
657 = 3 × 219 = 3 × 3 × 73 = 3² × 73
LCM = 2 × 3² × 17 × 73 = 22338
16. a = x³y² and b = xy³, where x and y are primes. Find LCM(a, b). [CBSE 2019]
See SolutionGiven two positive integers
a = x³y² = x × x × x × y × y
and b = xy³ = x × y × y × y
LCM (a,b) = Product of the greatest power of each prime factor involved in the numbers
= x³ × y³
LCM = x³ y³
17. If HCF(336, 54) = 6, find LCM(336, 54). [CBSE 2019]
See SolutionProduct of numbers = HCF × LCM
⇒ 336 × 54 = 6 × LCM
⇒ LCM = (336 × 54)/6 = 3024
18. LCM is 9 times HCF. LCM + HCF = 500. Find HCF. [CBSE 2019 Compt]
See SolutionLet HCF = x ⇒ LCM = 9x
x + 9x = 500
⇒ 10x = 500
⇒ x = 50
19. Show that (3 + √7)/5 is an irrational number, given √7 is irrational. [CBSE 2019 Compt]
See SolutionAssume (3 + √7)/5 is rational
⇒ Multiply both sides by 5: 3 + √7 = rational
⇒ √7 = rational
⇒ Contradiction
So, (3 + √7)/5 is irrational
Preparing for mathematics exams becomes simpler with NCERT Class 10 Maths Chapter 1 Important Questions and CBSE Previous Year Questions. These resources help students focus on frequently asked problems, enhancing their problem-solving skills. For instance, understanding Real Numbers in depth requires familiarity with proofs of irrational numbers and applications of the Fundamental Theorem of Arithmetic. Textbook Class 10 Maths Chapter 1 Notes and Summary act as quick references, saving time while revising key concepts.
Solving 10th Math Exercise 1.1 to Exercise 1.4 further aids in solidifying these concepts. For students looking for additional support, online video lectures and study materials provide step-by-step explanations of the chapter’s topics. Incorporating NCERT Complete Solutions for Real Numbers Class 10 into daily study routines can significantly boost confidence, ensuring students are well-prepared to perform in their board exams. Learning Class 10 Maths Chapter 1 Real Numbers effectively requires a systematic approach and a clear understanding of the concepts involved. Here are some steps and strategies to help you learn class 10 mathematics chapter 1.
Euclid’s Division Lemma states that for any two positive integers a and b, there exist unique integers q and r such that: a = bq + r, where 0 ≤ r < b.
The Fundamental Theorem of Arithmetic states that every positive integer greater than 1 can be expressed as a product of prime numbers in a unique way, up to the order of factors.
Key Highlights of Class 10 Maths Chapter 1 Real Numbers
Class 10 Maths Chapter 1, Real Numbers, focuses on Euclid’s Division Lemma, the Fundamental Theorem of Arithmetic, proving the irrationality of numbers, terminating and non-terminating decimals and revisiting HCF and LCM. These concepts form the basis of board exam questions, often involving direct proofs, applications and numerical problems. Mastery ensures strong foundational skills.
| Day | Topic | Tasks | Resources |
|---|---|---|---|
| Day 1 | Introduction to Real Numbers | Read theory and solve examples from Exercise 1.1. | NCERT textbook, video lectures |
| Day 2 | Euclid’s Division Lemma | Practice proving HCF using Euclid’s method. | NCERT examples, important questions |
| Day 3 | Fundamental Theorem of Arithmetic | Learn and solve problems from Exercise 1.2. | NCERT Textbook Solutions PDF |
| Day 4 | Irrational Numbers | Prove irrationality of √2, √3, and other numbers. | NCERT exercises, past papers |
| Day 5 | Terminating and Non-Terminating Decimals | Solve problems from Exercise 1.4. | Practice worksheets |
| Day 6 | Revision and Tests | Attempt a full-length chapter test. | Mock tests, online quizzes |
Class 10 Maths NCERT Book Chapter 1 Real Numbers is not only useful for CBSE board exams but also lays the groundwork for advanced mathematics. Tools like NCERT Class 10 Maths Chapter 1 Exercises Formula List and Practice Problems simplify understanding and application of theorems. Teachers often recommend using NCERT Simplified Solutions for Real Numbers Class 10 to enhance topics clarity. Students can also test their knowledge with online tests, MCQs and sample papers prepared for Chapter 1. For comprehensive preparation, the Class 10 Maths Chapter 1 Exercise 1.1, Exercise 1.2 and Exercise 1.3 Solutions are particularly helpful in mastering the topic step by step. Resources such as Class 10th NCERT Maths Textbook Chapter 1 Study Material ensure that learners have everything they need in one place. With consistent practice and a focus on important questions, students can confidently approach any problem from this critical chapter.
| Class: 10 | Mathematics |
| Chapter: 1 | Real Numbers |
| Number of Exercises: | 2 (Only Two) |
| Content: | Exercises Solutions and Tests |
| Academic Session: | Year 2025-26 |
| Medium: | Hindi and English Medium |
Methods for proving the irrationality of certain numbers, such as the square root of a prime number, are discussed. Understanding decimal expansions of real numbers. Terminating decimals and recurring decimals. Rationalization is the process of converting an irrational number into a rational one by multiplying or dividing by a suitable number.
According to new syllabus for academic session, only two exercises are in chapter 1. The exercise 1.1 deleted and exercise 1.2 become the exercise 1.1. Similarly exercise 1.3 become exercise 1.2 and exercises 1.4 is also deleted.
The course structure of class 10 mathematics chapter 1 is given below:
Fundamental Theorem of Arithmetic – statements after reviewing work done earlier and
after illustrating and motivating through examples, Proofs of irrationality of √2, √3, √5.
>> Total Number of Exercises: 2
>> Total Marks in Board Exam: 6
>> Number of Periods needed: 15
Class 10 Maths Chapter 1 Real Numbers, is an important chapter that lays the foundation for understanding various mathematical concepts. The main points covered in this chapter include introduction to Real Numbers. Real numbers are a set of numbers that includes all rational and irrational numbers. Real numbers can be represented on the number line.
Utilize online resources like video tutorials, educational websites like Tiwari Academy and interactive quizzes to reinforce your understanding of the chapter. As you progress, practice solving sample papers or past year question papers. This will help you become familiar with the exam pattern and time management. Periodically test your knowledge by attempting self-assessment quizzes or problems from the chapter. This will help you gauge your progress.
In class 10 Maths Chapter we will learn about understanding square numbers and their properties. Finding the square root of a number using prime factorization. Real numbers are used in various mathematical applications, such as finding the distance between two points in coordinate geometry.
Class 10 Maths Chapter Real Numbers
The chapter 1 provides numerous examples and practice problems to reinforce the concepts discussed. These are the main points covered in Class 10 Maths Chapter 1 Real Numbers. It serves as the building block for further mathematical concepts and is essential for a solid foundation in mathematics. Get here topic wise solutions for 10th Mathematics chapter 1 all exercises in Hindi and English medium. Students can take help of videos for solving questions of each exercise.
Class 10 Maths Chapter 1 Solutions for new Session
Class 10 Maths Chapter 1 Solutions for State Boards
10th Maths Chapter 1 Exercise 1.1 Solutions
Class 10 Maths Chapter 1 Exercise 1.1 Question 1
Express each number as a product of its prime factors: (i) 140 (ii) 156 (iii) 3825 (iv) 5005 (v) 7429
The step by step solutions of each part is given here with practice questions>. You can discuss with us Part (i), Part (ii), Part (iii), Part (iv), and Part (v) to learn more about the solution.
Class 10 Maths Chapter 1 Exercise 1.1 Question 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 (ii) 510 and 92 (iii) 336 and 54.
Get the solution of question 2 and discuss Part (i), Part (ii) and Part (iii) with us along with assignments and other doubts for more clarification.
Class 10 Maths Chapter 1 Exercise 1.1 Question 3
Find the LCM and HCF of the following integers by applying the prime factorisation method. (i) 12, 15 and 21 (ii) 17, 23 and 29 (iii) 8, 9 and 25.
Discuss with part one, two and three in detail and get worksheets to understand the solution in simple way.
Class 10 Maths Chapter 1 Exercise 1.1 Question 4
Given that HCF (306, 657) = 9, find LCM (306, 657).
The solution of this question is very simple and usually asked in MCQ type questions. For more revision, please write to us.
Class 10 Maths Chapter 1 Exercise 1.1 Question 5
Check whether 6ⁿ can end with the digit 0 for any natural number n.
The solution of question 5 is similar to the example given in textbook. Discuss with us and Practice here more questions based on question 5.
Class 10 Maths Chapter 1 Exercise 1.1 Question 6
Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers.
Extra questions for practice based on the solution of question 6 are given here to download. Put your view to discuss with us.
Class 10 Maths Chapter 1 Exercise 1.1 Question 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?
Question number 7 is important for exams. Get here the solution and practice assignments with answers for revision.
10th Maths Chapter 1 Exercise 1.2 Solutions
Class 10 Maths Chapter 1 Exercise 1.2 Question 1
Prove that √5 is irrational.
To understand the solution of question 1, student need to practice assignments and discuss the explanation in depth.
Class 10 Maths Chapter 1 Exercise 1.2 Question 2
Prove that 3 + 2√5 is irrational.
Get here the assignments and practice questions based on the solution of question 2. If still there is any doubt, please visit to discussion forum.
Class 10 Maths Chapter 1 Exercise 1.2 Question 3
Prove that the following are irrationals: (i) 1/√2 (ii) 7√5 (iii) 6 + √2
The solution of question 3 is similar to question 1 and 2. Get more practice to learn the concepts in better way. Discuss in detail here.
Use number lines and diagrams to visualize concepts like decimal expansions and the placement of real numbers on the number line. Visual aids can help make abstract concepts more concrete. Don’t hesitate to ask your teacher or a classmate for help if you encounter difficulties. Sometimes, discussing problems with others can provide valuable insights. Maintain a study schedule and allocate specific time slots for learning and practicing this chapter.
Download App for Class 10
The complete NCERT chapter 1 of 10th Maths 1 is divided in four main parts. The explanation of each part with video explanation is given below. Start by reading the chapter thoroughly and grasp the basic definitions and concepts. Ensure that you understand the definitions of real numbers, Euclid’s Division Lemma and the Fundamental Theorem of Arithmetic. Take concise notes while reading the chapter. Write down key formulas, definitions and important points. These notes will be helpful for quick revision.
Learn here how to find the prime factorization of numbers. This skill is essential for various aspects of number theory. Practice finding the prime factorization of different numbers. Practice working with terminating and recurring decimals. Solve NCERT textbook problems that involve decimal expansions of real numbers. Familiarize yourself with methods for rationalizing irrational numbers, especially square roots. Practice these techniques with different examples. Remember that learning mathematics often requires patience and persistence. Take your time to understand each concept thoroughly before moving on to the next. By following these steps and strategies, you can effectively learn Class 10 Maths NCERT Chapter 1 Real Numbers.
Class 10 Maths Chapter 1 Main Topics
Mathematics is best learned through practice. Solve a variety of problems related to real numbers, including examples from your textbook and additional practice questions. Pay attention to different types of questions, such as finding HCF, LCM, prime factorization and working with decimal expansions.
| Topics | Detail about the Topics |
|---|---|
| Topic: 1 | Euclid’s Division Lemma |
| Topic: 2 | The Fundamental Theorem of Arithmetic |
| Topic: 3 | Dealing with Irrational with Square Root |
| Topic: 4 | Decimal Representation of Rational Numbers |
10th Maths Chapter 1 Practice Test Papers
Euclid’s Division Lemma is a fundamental concept in this chapter. Make sure you understand how it works and can apply it to solve problems related to divisibility and remainders. There are four exercises in chapter 1 covering all the four topics on Real Numbers. Students can practice well using the 6 question papers given below. It will help to prepare the chapter for CBSE exams also.
Class 10 Maths Chapter 1 Practice Tests with Answers
Historical Facts on Real Numbers
The word algorithm come from the name of the name of the 9th century Persian mathematician al-Khwarizmi. The word ‘algebra’ is derived from a book, he wrote, called Hisab al-jabr w’al-muqabala.
An equivalent version of Fundamental theorem of Arithmetic was probably first recorded as Proposition 14 of Book IX in Euclid’s Elements, before it came to be known as the Fundamental Theorem of Arithmetic.
However, the first correct proof was given by Carl Friedrich Gauss in his al-Khwarizmi.
Carl Friedrich Gauss is often referred to as the ‘Prince of Mathematicians’ and is considered one of the three greatest mathematicians of all time, along with Archimedes and Newton. He has made fundamental contributions to both mathematics and science. Learn here how to make NCERT class 10 Maths chapter 1 easy to learn.
What is the best way to study Class 10 Maths Chapter 1 Real Numbers?
To study Class 10 Maths Chapter 1, start by understanding the basic concepts from your NCERT textbook, such as Euclid’s division lemma and the Fundamental Theorem of Arithmetic. Use NCERT Exercises Solutions for Class 10 Maths Chapter 1 to practice exercises like Exercise 1.1, 1.2 and 1.3, which focus on HCF, LCM and rational and irrational numbers. Make notes of key points and formulas for quick revision. Solve Important Questions, MCQs and worksheets to test your understanding. Watching video lectures can help clarify doubts, while practice problems and online tests provide additional preparation.
How many exercises are there in class 10 Maths chapter 1 Real numbers?
Now there are only 2 exercises in class 10 chapter 1 (Real numbers):
In first exercise (Ex 1.1), there are 5 questions.
In second exercise (Ex 1.2), there are 7 questions.
So, there are in all 18 questions in class 10 chapter 1 (Real numbers).
There are in all 11 examples in class 10 chapter 1 (Real numbers).
- Examples 1,2,3,4 are based on Ex 1.1.
- Examples 5,6,7,8 are based on Ex 1.2.
Where can I find reliable NCERT Solutions for Class 10 Maths Chapter 1?
Reliable NCERT Textbook Solutions for Class 10 Maths Chapter 1 can be found on educational websites, apps like Tiwari Academy and NCERT guides. Many platforms provide PDF versions of solutions that include step-by-step explanations for all exercises, including Exercises 1.1, 1.2 and 1.3. These solutions help students understand methods for finding the HCF and LCM using Euclid’s division lemma, among other concepts. Some websites also offer video lectures, assignments and study material tailored for this chapter. You can check for resources that include important questions, worksheets and previous year questions to reinforce your learning.
Are there any Theorems or Algorithm in 10th Maths chapter 1?
Yes, there are 7 Theorems and 1 Algorithm in class 10 chapter 1 (Real numbers).
First exercise (Ex 1.1) is based on Theorem 1.1 (Euclid’s Division Lemma) and Euclid’s Division algorithm.
Theorem 1.2 (Fundamental Theorem of Arithmetic) is used in second exercise (Ex 1.2).
What are the important topics in Class 10 Maths Chapter 1 Real Numbers?
The important topics in Class 10 Maths Chapter 1 include Euclid’s division lemma, the Fundamental Theorem of Arithmetic, finding HCF and LCM, properties of rational and irrational numbers and decimal expansions of real numbers. Exercises like Exercise 1.1 and 1.2 focus on applying these concepts. Students should pay special attention to problem-solving techniques and practice questions that are frequently asked in exams. Using Class 10 Maths Chapter 1 Notes, a formula sheet and important questions can help you focus on these critical topics during your preparation.
What are the Important questions from Maths class 10 chapter 1 Real numbers?
Important questions from Maths class 10 chapter 1:
- In first exercise (Ex 1.1), all questions are important.
- In second exercise (Ex 1.2), Q2, 4, 5, 7 are important.
How can I revise Class 10 Maths Chapter 1 effectively before exams?
To revise Class 10 Maths Chapter 1 effectively, start by reviewing the summary and key points from your notes or study material. Use a formula sheet to memorize essential formulas and revisit solved examples for clarity. Practice MCQs, worksheets and previous year questions to cover all problem types. Online resources often include assignments and online tests, which can help you identify weak areas. Watching short video lectures or tutorials can refresh concepts quickly. Focus on solving important questions from this chapter to ensure a strong understanding of real numbers and related applications.
Which topics students should recall before starting class 10 Maths chapter 1?
Students should recall some topics of class 9 chapter 1 (Number systems).
Topics:
- Types of numbers (Natural number, Whole number, Integers, Rational numbers, Irrational numbers and Real numbers).
- Decimal expansion of Rational numbers (Terminating and Non-terminating recurring) and Irrational numbers (Non-terminating non-recurring).
- HCF (Highest Common Factor) and LCM (Lowest/Least Common Multiple).
- And some other topics like Meaning of Dividend, Divisor, Quotient and Remainder, Prime numbers, Composite numbers, Factors.
How can I ensure full marks in Class 10 Maths Chapter 1 exercises?
To ensure full marks in Class 10 Maths Chapter 1 exercises, start by thoroughly understanding the concepts from the NCERT textbook. Practice all questions in Exercise 1.1, 1.2 and related practice problems from other sources. Use NCERT Solutions for Class 10 Maths Chapter 1 for step-by-step guidance and check your answers. Solve additional worksheets, MCQ and assignments to gain confidence. Revising with a formula sheet and solving previous year papers will prepare you for exam-style questions. Consistent practice, along with reviewing mistakes, will help you master the topic and score full marks.
