NCERT Solutions For Class 10 Math

NCERT Solutions for Class 10 Math provide accurate and easy-to-understand answers for all exercises from Chapters 1 to 15. Prepared by subject experts, these solutions help students understand concepts clearly and improve their problem-solving skills.

With these NCERT Solutions for Class 10 Math, students can learn step-by-step methods to solve textbook questions, clear their doubts quickly, and strengthen their preparation for school exams as well as the CBSE Board Exam. Regular practice of these solutions helps in building confidence, improving accuracy, and achieving better marks in Mathematics.

Students can access chapter-wise solutions and download PDFs for convenient learning and revision anytime.

2. HCF and LCM Relationship

For any two positive integers a and b:

$$HCF(a,b) \times LCM(a,b) = a \times b$$

or

$$LCM = \frac{a \times b}{HCF}$$

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NCERT Solutions for Class 10 Math (2026–27 Updated Edition)

Preparing for CBSE Board Exams becomes much easier with NCERT Solutions for Class 10 Math. These chapter-wise solutions are designed to help students understand concepts clearly, improve problem-solving skills, and score better marks in examinations. All answers are explained in a simple and step-by-step manner, making them useful for both Hindi and English medium students.

Chapter No.	Chapter Name
Chapter 1	Real Numbers
Chapter 2	Polynomials
Chapter 3	Pair of Linear Equations in Two Variables
Chapter 4	Quadratic Equations
Chapter 5	Arithmetic Progressions
Chapter 6	Triangles
Chapter 7	Coordinate Geometry
Chapter 8	Introduction to Trigonometry
Chapter 9	Applications of Trigonometry
Chapter 10	Circles
Chapter 11	Areas Related to Circles
Chapter 12	Surface Areas and Volumes
Chapter 13	Statistics
Chapter 14	Probability

NCERT Solutions Class 10 Maths Chapter 1 Real Numbers

Chapter 1 – Real Numbers

Real Numbers is the very first chapter in Class 10 Mathematics, and it lays the foundation for several concepts you’ll use throughout the rest of the syllabus — and even in Class 11 and 12. This chapter mainly deals with the properties of positive integers, focusing on divisibility, prime factorization, and the nature of rational and irrational numbers. Let’s break down the key concepts in a simple and easy-to-understand way.

1. Euclid’s Division Lemma

The chapter begins with Euclid’s Division Lemma, which states that for any two positive integers a and b, there exist unique whole numbers q (quotient) and r (remainder) such that:

a = bq + r, where 0 ≤ r < b

This simple statement is incredibly powerful. It forms the basis of the Euclid’s Division Algorithm, which is used to find the HCF (Highest Common Factor) of two positive integers. Instead of listing out factors, you repeatedly apply the division lemma until the remainder becomes zero. The last non-zero divisor is the HCF.

Example: Find the HCF of 420 and 130 using Euclid’s Division Algorithm.

• 420 = 130 × 3 + 30
• 130 = 30 × 4 + 10
• 30 = 10 × 3 + 0

Since the remainder is now 0, the HCF of 420 and 130 is 10.

2. The Fundamental Theorem of Arithmetic

This is one of the most important concepts in the entire chapter. The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a product of prime numbers, and this factorization is unique, except for the order in which the prime factors occur.

For example, 156 can be written as 2 × 2 × 3 × 13, and no matter how you approach the factorization, you’ll always arrive at the same set of prime factors.

This theorem is extremely useful for:

• Finding the HCF and LCM of numbers using prime factorization
• Proving numbers as rational or irrational
• Understanding the relationship: HCF × LCM = Product of the two numbers

Example: Find the HCF and LCM of 96 and 404 using prime factorization.

• 96 = 2^5 × 3
• 404 = 2^2 × 101

HCF = 2^2 = 4
LCM = 2^5 × 3 × 101 = 9696

Check: HCF × LCM = 4 × 9696 = 38784, and 96 × 404 = 38784. ✅ The relationship holds true.

3. Irrational Numbers

A major portion of this chapter focuses on proving whether numbers like √2, √3, √5, and similar values are irrational. The standard method used is proof by contradiction:

1. Assume the number is rational, meaning it can be written as p/q (where p and q are co-prime integers and q ≠ 0).
2. Using algebraic manipulation, show that both p and q must share a common factor.
3. Since this contradicts our assumption that p and q are co-prime, the original assumption is false.
4. Hence, the number must be irrational.

Example: Prove that √2 is irrational.

• Assume √2 = p/q, where p and q are co-prime integers and q ≠ 0.
• Squaring both sides: 2 = p²/q², so p² = 2q².
• This means p² is even, so p must also be even. Let p = 2k.
• Substituting: (2k)² = 2q² → 4k² = 2q² → q² = 2k².
• This means q² is also even, so q must be even too.
• But if both p and q are even, they share a common factor of 2 — contradicting our assumption that they are co-prime.
• Hence, our assumption is wrong, and √2 is irrational.

This proof technique appears repeatedly in exams, so practicing it for different numbers is essential.

4. Decimal Expansion of Rational Numbers

The chapter also explains how to determine whether the decimal expansion of a rational number p/q will be terminating or non-terminating repeating, simply by looking at the prime factors of the denominator (q).

• If q can be expressed in the form 2^m × 5^n, the decimal expansion terminates.
• If q has any other prime factor besides 2 and 5, the decimal expansion is non-terminating and repeating.

Example: Without actual division, determine if 13/3125 has a terminating decimal expansion.

• 3125 = 5^5
• Since the denominator is only made up of the prime factor 5 (in the form 2^0 × 5^5), the decimal expansion of 13/3125 terminates.

NCERT Solutions For Class 10 Math Real Numbers Frequently Asked Questions (Q&A)

Q1. What is Euclid’s Division Lemma?
A. It 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.

Q2. Is Euclid’s Division Algorithm applicable to all integers, or only positive integers?
A. It is stated for positive integers, although it can be extended to all integers with some modification. In the NCERT syllabus, it is applied only to positive integers.

Q3. Can the Fundamental Theorem of Arithmetic be applied to 1?
A. No. The theorem applies to composite numbers greater than 1. The number 1 is neither prime nor composite, so it has no prime factorization.

Q4. Why is √2 irrational but 4/2 (=2) rational?
A. 4/2 simplifies to a whole number, which can be written as p/q (2/1), satisfying the definition of a rational number. √2, however, cannot be expressed as p/q in its simplest form without contradiction, as proven above.

Q5. How can I quickly check if a decimal is terminating without long division?
A. Factorize the denominator of the fraction (in its lowest form). If the only prime factors are 2 and/or 5, the decimal terminates. Otherwise, it is non-terminating and repeating.

Q6. Is HCF × LCM = Product of numbers true for three or more numbers?
A. No, this relationship only holds true for exactly two numbers. For three or more numbers, this formula does not apply directly.

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