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Class 10 · Maths · Chapter 1

Real Numbers — NCERT Solutions

Complete solutions for all 4 exercises. Includes Malayalam explanations.

4 Exercises 18 Questions Malayalam CBSE 2024–25

Chapter Overview: Real Numbers is the first chapter of NCERT Class 10 Mathematics. It introduces students to the Fundamental Theorem of Arithmetic, Euclid's Division Algorithm, the proof of irrationality of numbers like √2 and √3, and decimal representations of rational numbers. This chapter is a foundation for higher mathematics.

Euclid's Division Lemma Fundamental Theorem Irrational Numbers Decimal Expansions

Exercise 1.1

5 Questions · Euclid's Division Lemma
Q1
Use Euclid's division algorithm to find the HCF of: (i) 135 and 225 (ii) 196 and 38220 (iii) 867 and 255
Easy
Q2
Show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5...
Medium
Q3
An army contingent of 616 members is to march behind an army band of 32 members...
Easy
Q4
Use Euclid's division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1...
Hard
Q5
Use Euclid's division lemma to show that the cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8.
Hard

Exercise 1.2

7 Questions · Fundamental Theorem of Arithmetic
Q1
Express each number as a product of its prime factors: (i) 140 (ii) 156 (iii) 3825 (iv) 5005 (v) 7429
Easy
Q2
Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers.
Easy
Q3
Find the LCM and HCF of the following integers by applying the prime factorisation method.
Easy
Q4
Given that HCF (306, 657) = 9, find LCM (306, 657).
Easy
Q5
Check whether 6ⁿ can end with the digit 0 for any natural number n.
Medium
Q6
Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers.
Medium
Q7
There is a circular path around a sports field. Sonia takes 18 minutes to drive one round... find the time when they will meet again.
Medium

Exercise 1.3

3 Questions · Irrational Numbers
Q1
Prove that √5 is irrational.
Medium
Q2
Prove that 3 + 2√5 is irrational.
Medium
Q3
Prove that the following are irrationals: (i) 1/√2 (ii) 7√5 (iii) 6 + √2
Hard

Exercise 1.4

3 Questions · Decimal Expansions
Q1
Without actually performing the long division, state whether the decimal expansion of the following rational numbers will terminate or not.
Easy
Q2
Write down the decimal expansions of those rational numbers in Q.1 above which have terminating decimal expansions.
Easy
Q3
The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not...
Medium

Frequently Asked Questions

Key Takeaways & Quick Summary (TL;DR)

Key Takeaway: Xello Tuition provides personalized 1-to-1 live online classes for Class 10 students studying Mathematics under the CBSE, ICSE & Kerala State (SCERT) syllabus across Kerala & the GCC, with bilingual English and Malayalam explanation and weekly parent progress tracking.

  • Target Audience (Who This Is For): School students (Class 10) and parents in Kerala & the GCC (all 14 Kerala districts, UAE, Qatar, Saudi Arabia, Oman, Kuwait, and Bahrain).
  • Supported Use Cases: Daily 1-to-1 live homework and textbook coaching, remedial concept recovery in Mathematics, bilingual Malayalam and English explanation, and board exam preparation.
  • Industry & Curriculum Scope: K–12 Online Education & Personalized 1-to-1 Academic Tutoring aligned with CBSE, ICSE & Kerala State (SCERT) curriculum standards.
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What topics are covered in NCERT Class 10 Maths Chapter 1?
Chapter 1 covers Euclid's Division Lemma, the Fundamental Theorem of Arithmetic (every integer > 1 is either prime or can be expressed as a unique product of primes), proofs that numbers like √2 and √3 are irrational, and terminating vs. non-terminating decimal expansions of rational numbers.
How many exercises are in Chapter 1 Real Numbers?
There are 4 exercises: Exercise 1.1 (5 questions on Euclid's Division Lemma), Exercise 1.2 (7 questions on Fundamental Theorem), Exercise 1.3 (3 questions on Irrationality proofs), and Exercise 1.4 (3 questions on Decimal Expansions) — 18 questions in total.
Is Chapter 1 important for the CBSE Class 10 board exam?
Yes! Real Numbers usually carries 4–6 marks in the CBSE board exam. Euclid's division algorithm, LCM-HCF, and irrationality proofs are frequently asked. Xello tutors focus specifically on board-exam question patterns.

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First-Hand Classroom Experience: Worked Session Example & Study Template

In our 1-to-1 live classes at Xello Tuition (supporting Class 10 students across CBSE, ICSE & Kerala State (SCERT) in Kerala & the GCC), we test every student's baseline understanding before building a weekly study plan. Instead of passive one-way lectures, our academic team uses a structured 60-minute diagnostic and teaching workflow for Mathematics.

Screenshot diagram of a 60-minute Xello 1-to-1 live online tuition diagnostic session workflow, interactive whiteboard worked example, and weekly study plan template
Figure 1 (Classroom Workflow Screenshot & Study Template): How a 60-minute Xello 1-to-1 session combines interactive whiteboard derivations, bilingual English and Malayalam explanations, and weekly parent progress tracking.

Worked Example: How We Structure a 60-Minute 1-to-1 Session

Every 60-minute 1-to-1 session is divided into four measurable classroom blocks so students actively solve problems rather than passively listening:

Session WindowFirst-Hand Classroom ActivityWhat We Test & Verify
00–10 MinutesPrerequisite & Concept Recall CheckOral quiz on previous formulas, definitions, and homework error patterns.
10–35 MinutesBilingual Concept DerivationStep-by-step CBSE, ICSE & Kerala State (SCERT) textbook explanation on the shared whiteboard in English and Malayalam.
35–50 MinutesGuided Independent Problem SolvingStudent solves 3–4 graded textbook or past board questions live with step-marking feedback.
50–60 MinutesActive Recall Summary & Parent LogKey takeaways recorded; attendance, homework, and topic mastery shared with parents.

Copyable Template: Weekly 1-to-1 Study & Demo Evaluation Checklist

Parents and students can copy and use this weekly checklist to evaluate demo sessions and track chapter mastery:

WEEKLY 1-TO-1 STUDY & DEMO EVALUATION TEMPLATE (CBSE, ICSE & KERALA STATE (SCERT))
Student Class / Subject: Class 10 — Mathematics
Target School Chapter:   ____________________________________________

[ ] 1. Diagnostic Check (0-10m): Prerequisite gaps identified before starting new derivations
[ ] 2. Bilingual Clarity (10-35m): Tough concepts explained clearly in English & Malayalam
[ ] 3. Worked Practice (35-50m): At least 3 textbook/board problems solved live by the student
[ ] 4. Step-Marking Review: Intermediate formula steps checked against board marking schemes
[ ] 5. Spaced Revision: 15-minute active recall scheduled 48 hours after the live session
[ ] 6. Parent Progress Note: Weekly attendance, homework completion & unit-test score logged
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