Class 10 Maths Chapter 1 Real Numbers NCERT Solutions – Ex 1.1, 1.2, 1.3, 1.4 | narrowit.in

Welcome to narrowit.in’s NCERT Solutions for Class 10 Maths Chapter 1 – Real Numbers. This page covers complete step-by-step solutions for Exercise 1.1, Exercise 1.2, Exercise 1.3, and Exercise 1.4. Topics include Euclid’s Division Lemma, Euclid’s Division Algorithm, HCF and LCM using Prime Factorisation, Fundamental Theorem of Arithmetic, Irrational Numbers, and Decimal Expansions of Rational Numbers. All solutions are written by narrowit.in in simple language – no login required, no app download, completely free. Official NCERT textbooks are available at https://ncert.nic.in

Class 10 Maths Chapter 1 is one of the most important chapters in the CBSE curriculum. Real Numbers form the foundation of higher mathematics and are essential for scoring well in board examinations. This chapter builds upon the concepts introduced in Class 9 and takes students deeper into the world of numbers – from integers and rational numbers to irrational numbers and their decimal expansions.

Notes PDF :

Class 10 Maths Ch2 NCERT Solutions –>Click Here

Exercise 1.1 – Euclid’s Division Lemma

Exercise 1.1 is based on Euclid’s Division Lemma, which 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. This lemma is the backbone of Euclid’s Division Algorithm, which is used to find the Highest Common Factor (HCF) of two or more numbers. The questions in this exercise require students to apply the lemma step by step until the remainder becomes zero. The last non-zero remainder at that point is the HCF. This method is efficient and works for very large numbers where prime factorisation would take longer.

Ex 1.1 PDF:

Exercise 1.2 – Fundamental Theorem of Arithmetic

Exercise 1.2 is based on the Fundamental Theorem of Arithmetic, which states that every composite number can be expressed as a product of prime numbers in a unique way, regardless of the order. This is also called prime factorisation. In this exercise, students learn how to find the HCF and LCM of numbers using prime factorisation. An important formula used here is HCF × LCM = Product of the two numbers. This exercise also includes word problems where students must find HCF or LCM in real-life situations such as finding when two events will occur together or dividing items into equal groups.

Ex 1.2 PDF:

Exercise 1.3 – Irrational Numbers

Exercise 1.3 deals with proving whether a given number is irrational or not. Students are required to prove that numbers like √2, √3, √5, and √7 are irrational using the method of contradiction. This method assumes that the number is rational, expresses it as p/q in lowest terms, and then arrives at a contradiction, thereby proving it must be irrational. This exercise also covers proving that the sum or product of a rational and an irrational number is irrational. These proofs require logical thinking and are frequently asked in CBSE board exams.

Ex 1.3 PDF:

Exercise 1.4 – Decimal Expansions of Rational Numbers

Exercise 1.4 focuses on the decimal expansion of rational numbers. Students learn to determine whether the decimal expansion of a rational number will be terminating or non-terminating repeating without actually performing the division. A rational number p/q in its simplest form has a terminating decimal expansion if and only if the prime factorisation of the denominator q contains only the primes 2 and 5. If the denominator has any other prime factor, the decimal expansion will be non-terminating and repeating. This is a very scoring topic in board exams as questions are straightforward once the concept is clear.

Ex 1.4 PDF:

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At narrowit.in, every solution is written in clear and simple Hindi-English language so that students from all backgrounds can understand easily. Each step is explained in detail so that students not only get the answer but also understand the method. All solutions follow the latest CBSE syllabus and NCERT guidelines. There are no shortcuts that bypass understanding – every solution is complete and exam-ready.

Students preparing for CBSE Class 10 board exams will find these solutions extremely helpful for last-minute revision as well as for clearing doubts during regular study. Teachers and parents can also use these solutions to guide students at home.

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══════════════════════════════════════════
CLASS 10 MATHEMATICS — COMPLETE BOARD EXAM ROADMAP
CBSE | 2024-25 | New + Old NCERT Both
══════════════════════════════════════════

📊 UNIT-WISE WEIGHTAGE (Board Exam — 80 Marks)

Unit 1 — Number Systems          →  06 Marks
Unit 2 — Algebra                 →  20 Marks  ★ HIGHEST
Unit 3 — Coordinate Geometry     →  06 Marks
Unit 4 — Geometry                →  15 Marks
Unit 5 — Trigonometry            →  12 Marks
Unit 6 — Mensuration             →  10 Marks
Unit 7 — Statistics & Probability→  11 Marks
─────────────────────────────────────────
TOTAL                            →  80 Marks
Internal Assessment              →  20 Marks
GRAND TOTAL                      → 100 Marks

══════════════════════════════════════════
CHAPTER-WISE COMPLETE ROADMAP
══════════════════════════════════════════

📘 CHAPTER 1 — REAL NUMBERS

Unit          : Number Systems
Marks         : 06
Difficulty    : Easy-Medium
Time to Study : 3-4 days

Topics :
→ Euclid’s Division Lemma & Algorithm (Old NCERT)
→ Fundamental Theorem of Arithmetic ★
→ HCF and LCM by Prime Factorisation ★
→ Proving Irrationality of numbers ★★★
→ Decimal Expansions — Terminating vs Non-Terminating ★★

Most Asked in Board :
→ Prove √2, √3, √5 irrational — 3 marks
→ HCF × LCM = product — 2 marks
→ Terminating/Non-terminating check — 1 mark MCQ
→ 6ⁿ ending in zero type — 2 marks

Extra Page Content :
→ Proof that √p is irrational for any prime p
→ Prove sum of rational + irrational = irrational
→ Prove product of nonzero rational × irrational = irrational
→ Why HCF × LCM formula fails for 3 numbers
→ Difference between Old and New NCERT syllabus of this chapter

Status : ✅ DONE (All 4 Exercises Complete)

📘 CHAPTER 2 — POLYNOMIALS

Unit          : Algebra
Marks         : 20 (shared with Ch 3,4,5)
Difficulty    : Easy
Time to Study : 3-4 days

Topics :
→ Geometrical meaning of zeroes ★
→ Relationship between zeroes and coefficients ★★★
→ Division Algorithm for polynomials (Old NCERT)
→ Zeroes of Linear, Quadratic, Cubic polynomials

Most Asked in Board :
→ Find zeroes and verify relationship — 2-3 marks ★
→ Form quadratic polynomial given sum/product of zeroes — 2 marks
→ Division algorithm based questions — 3 marks

Extra Page Content :
→ Graph shapes of linear, quadratic, cubic polynomials
→ Maximum number of zeroes possible for degree n polynomial
→ If α, β are zeroes → find α²+β², α³+β³, 1/α+1/β type identities
→ Cubic polynomial — relationship between zeroes and coefficients
→ Common mistakes in sign of sum and product of zeroes

📘 CHAPTER 3 — PAIR OF LINEAR EQUATIONS IN TWO VARIABLES

Unit          : Algebra
Marks         : 20 (shared)
Difficulty    : Medium
Time to Study : 5-6 days

Topics :
→ Graphical method of solution ★
→ Substitution method ★★
→ Elimination method ★★★
→ Cross multiplication method ★★
→ Consistency — consistent/inconsistent/dependent ★★★
→ Reducible equations (equations reducible to linear form)

Most Asked in Board :
→ Solve pair of equations — 3 marks (elimination/substitution)
→ Word problems — 3-5 marks ★★★
→ Consistency condition (a1/a2, b1/b2, c1/c2) — 1-2 marks
→ Reducible equations — 3 marks

Extra Page Content :
→ Full consistency condition table with all 3 cases
→ Graphical interpretation of each case
→ Word problem types — Age, Speed-Distance, Numbers, Fractions, Coins
→ Cross multiplication method — full derivation
→ Reducible to linear form — all standard types

📘 CHAPTER 4 — QUADRATIC EQUATIONS

Unit          : Algebra
Marks         : 20 (shared)
Difficulty    : Medium
Time to Study : 4-5 days

Topics :
→ Standard form of quadratic equation ★
→ Factorisation method ★★★
→ Completing the square method ★★
→ Quadratic formula (Sridharacharya formula) ★★★
→ Nature of roots — Discriminant ★★★
→ Word problems ★★

Most Asked in Board :
→ Solve by factorisation — 2 marks
→ Solve by quadratic formula — 3 marks
→ Nature of roots (D = b²-4ac) — 2 marks ★
→ Word problems — 3-4 marks

Extra Page Content :
→ Derivation of quadratic formula by completing the square
→ Nature of roots — all 3 cases with examples
→ D > 0 → real distinct, D = 0 → real equal, D < 0 → no real roots
→ Word problem types — consecutive integers, areas, speeds, ages
→ Relation between roots — if α,β are roots → sum = -b/a, product = c/a

📘 CHAPTER 5 — ARITHMETIC PROGRESSIONS

Unit          : Algebra
Marks         : 20 (shared)
Difficulty    : Easy-Medium
Time to Study : 4-5 days

Topics :
→ Introduction to AP — definition, common difference ★
→ General term (nth term) formula ★★★
→ Sum of first n terms formula ★★★
→ Finding AP when conditions given ★★
→ Word problems ★★

Most Asked in Board :
→ Find nth term — 2 marks
→ Find sum of n terms — 2-3 marks ★★★
→ Find AP / find values — 3 marks
→ Word problems on AP — 3-4 marks

Extra Page Content :
→ Derivation of Sn formula (both methods)
→ If last term l is given → Sn = n/2(a+l)
→ Relation between a, d, n, Sn, an — all formulas
→ Three numbers in AP → take as a-d, a, a+d ★ (exam trick)
→ Four numbers in AP → take as a-3d, a-d, a+d, a+3d ★

📘 CHAPTER 6 — TRIANGLES

Unit          : Geometry
Marks         : 15 (shared with Ch 10)
Difficulty    : Medium-Hard
Time to Study : 6-7 days

Topics :
→ Similar figures — definition ★
→ Basic Proportionality Theorem (BPT) / Thales Theorem ★★★
→ Converse of BPT ★★
→ Criteria for similarity — AA, SAS, SSS ★★★
→ Areas of similar triangles theorem ★★★
→ Pythagoras Theorem ★★★
→ Converse of Pythagoras Theorem ★★

Most Asked in Board :
→ Prove BPT — 3 marks ★★★
→ Prove Pythagoras Theorem — 5 marks ★★★
→ Areas of similar triangles — 3 marks
→ Find sides using similarity — 2-3 marks

Extra Page Content :
→ Full proof of BPT with converse
→ Full proof of Pythagoras theorem (board exact proof)
→ Ratio of areas = square of ratio of sides — full proof
→ All 3 similarity criteria with examples and differences
→ Mid Point Theorem — statement and use

📘 CHAPTER 7 — COORDINATE GEOMETRY

Unit          : Coordinate Geometry
Marks         : 06
Difficulty    : Easy
Time to Study : 3-4 days

Topics :
→ Distance Formula ★★★
→ Section Formula ★★★
→ Mid-point Formula ★★
→ Area of Triangle using coordinates ★★

Most Asked in Board :
→ Distance formula — 2 marks
→ Section formula — 2-3 marks ★★★
→ Area of triangle — 2-3 marks
→ Collinearity of points — 2 marks

Extra Page Content :
→ Derivation of distance formula
→ Derivation of section formula (internal division)
→ External division using section formula
→ Centroid of triangle formula → G = ((x1+x2+x3)/3, (y1+y2+y3)/3)
→ Condition for collinearity — area = 0

📘 CHAPTER 8 — INTRODUCTION TO TRIGONOMETRY

Unit          : Trigonometry
Marks         : 12 (shared with Ch 9)
Difficulty    : Medium
Time to Study : 5-6 days

Topics :
→ Trigonometric ratios — sin, cos, tan, cosec, sec, cot ★★★
→ Trigonometric ratios of specific angles (0°,30°,45°,60°,90°) ★★★
→ Trigonometric ratios of complementary angles ★★★
→ Trigonometric identities ★★★

Most Asked in Board :
→ Prove trigonometric identity — 3 marks ★★★
→ Evaluate using specific angles — 2 marks
→ Complementary angles based — 2 marks
→ Find other ratios given one ratio — 2 marks

Extra Page Content :
→ Complete table of all trig ratios for 0°,30°,45°,60°,90°
→ All reciprocal relations and quotient relations
→ Three main identities — sin²θ+cos²θ=1 and its variations ★
→ Proof strategy for identity questions — which side to start from
→ 10 most common identity proof patterns in board exam

📘 CHAPTER 9 — SOME APPLICATIONS OF TRIGONOMETRY

Unit          : Trigonometry
Marks         : 12 (shared with Ch 8)
Difficulty    : Medium
Time to Study : 3-4 days

Topics :
→ Angle of Elevation ★★★
→ Angle of Depression ★★★
→ Word problems using trigonometry ★★★

Most Asked in Board :
→ Height and distance word problems — 4-5 marks ★★★
→ Two observer / two angle problems — 5 marks
→ Tower, building, lighthouse problems

Extra Page Content :
→ Difference between angle of elevation and depression
→ Standard diagram templates for all problem types
→ Problems with two angles from same point
→ Problems with two points and one angle
→ Mixed problems — both elevation and depression together

📘 CHAPTER 10 — CIRCLES

Unit          : Geometry
Marks         : 15 (shared with Ch 6)
Difficulty    : Medium
Time to Study : 4-5 days

Topics :
→ Tangent to a circle — definition ★
→ Theorem 1 : Tangent ⊥ radius at point of contact ★★★
→ Theorem 2 : Tangents from external point are equal ★★★
→ Number of tangents from a point ★

Most Asked in Board :
→ Prove tangent ⊥ radius — 3 marks ★★★
→ Prove tangents from external point equal — 3 marks ★★★
→ Find lengths using tangent properties — 2-3 marks
→ Angle based problems — 2-3 marks

Extra Page Content :
→ Full proof of both theorems — board exam exact version
→ Angle in alternate segment theorem
→ Problems combining circles + triangles
→ Direct and transverse common tangents (concept only)
→ PA × PB = PC × PD type problems

📘 CHAPTER 11 — AREAS RELATED TO CIRCLES

Unit          : Mensuration
Marks         : 10 (shared with Ch 12)
Difficulty    : Easy-Medium
Time to Study : 3-4 days

Topics :
→ Perimeter and Area of Circle ★★★
→ Area of Sector ★★★
→ Area of Segment ★★★
→ Areas of combinations of figures ★★★

Most Asked in Board :
→ Area of sector + segment — 3 marks ★★★
→ Combination figures — 4 marks
→ Shaded region problems — 3-4 marks

Extra Page Content :
→ All formulas — sector, segment, ring in one place
→ Minor vs Major sector/segment difference
→ Standard combination figures — square+circle, triangle+circle
→ Shaded and unshaded region strategy
→ Use of π = 22/7 vs 3.14 — when to use which

📘 CHAPTER 12 — SURFACE AREAS AND VOLUMES

Unit          : Mensuration
Marks         : 10 (shared with Ch 11)
Difficulty    : Easy-Medium
Time to Study : 4-5 days

Topics :
→ Surface area and volume — Cube, Cuboid ★
→ Cylinder, Cone, Sphere, Hemisphere ★★★
→ Combination of solids ★★★
→ Conversion of solids ★★★
→ Frustum of a cone ★★

Most Asked in Board :
→ Combination of solids SA/Volume — 4 marks ★★★
→ Conversion of solid — sphere melted into cones type — 3 marks
→ Frustum — 3-4 marks

Extra Page Content :
→ All formulas in one master table
→ Conversion problems — step by step strategy
→ Combination solids — how to identify which formula to use
→ Frustum derivation and formula explanation
→ Real life application problems

📘 CHAPTER 13 — STATISTICS

Unit          : Statistics & Probability
Marks         : 11 (shared with Ch 14)
Difficulty    : Easy-Medium
Time to Study : 4-5 days

Topics :
→ Mean — Direct method, Assumed mean, Step deviation ★★★
→ Mode ★★★
→ Median ★★★
→ Ogive — Less than and More than ★★
→ Relation : Mode = 3Median − 2Mean ★★

Most Asked in Board :
→ Find Mean by assumed mean / step deviation — 3 marks ★★★
→ Find Median — 3 marks ★★★
→ Find Mode — 2 marks
→ Draw ogive — 4 marks

Extra Page Content :
→ All 3 mean methods with when to use which
→ Median formula derivation explanation
→ Mode formula — identifying modal class
→ Ogive drawing step by step
→ Converting less than to more than ogive

📘 CHAPTER 14 — PROBABILITY

Unit          : Statistics & Probability
Marks         : 11 (shared with Ch 13)
Difficulty    : Easy
Time to Study : 2-3 days

Topics :
→ Classical definition of probability ★★★
→ Complementary events ★★★
→ Problems on cards, dice, coins, bags ★★★

Most Asked in Board :
→ Card probability — 2 marks ★★★
→ Dice probability — 2 marks ★★★
→ Bag/balls probability — 2 marks
→ Complementary event based — 1-2 marks

Extra Page Content :
→ Complete deck of cards breakdown — all 52 cards listed
→ Sample space for 1 die, 2 dice
→ Impossible event (P=0) vs Sure event (P=1)
→ P(E) + P(not E) = 1 — all problems based on this
→ Geometric probability (if in syllabus)

══════════════════════════════════════════
★ PRIORITY ORDER FOR BOARD EXAM PREPARATION
══════════════════════════════════════════

TIER 1 — DO FIRST (Highest marks + Easiest)
★ Ch 5  — AP              (20 marks unit, easy)
★ Ch 8  — Trigonometry    (12 marks, formula based)
★ Ch 13 — Statistics      (11 marks, formula based)
★ Ch 14 — Probability     (11 marks, very easy)

TIER 2 — DO SECOND (High marks, medium effort)
★ Ch 3  — Linear Equations (20 marks unit)
★ Ch 4  — Quadratic Eq    (20 marks unit)
★ Ch 6  — Triangles       (15 marks, proof heavy)
★ Ch 9  — Applications    (12 marks)

TIER 3 — DO LAST (Conceptual, proof heavy)
★ Ch 2  — Polynomials
★ Ch 7  — Coordinate Geometry
★ Ch 10 — Circles
★ Ch 11 — Areas Circles
★ Ch 12 — SA & Volumes
★ Ch 1  — Real Numbers (already done ✅)

══════════════════════════════════════════
★ EXTRA PAGE CONTENT — MASTER LIST
(Topics beyond textbook that fetch extra marks)
══════════════════════════════════════════

For Proofs :
→ Always write Given, To Prove, Construction, Proof — 4 parts ★
→ Never skip Construction in geometry proofs
→ End every proof with “Hence Proved” ★

For Word Problems :
→ Always write Given, To Find, Solution, Answer separately
→ Draw diagram wherever possible — examiners give marks for it
→ Write units in final answer — marks cut if units missing ★

For MCQs (1 mark) :
→ No steps needed but write formula used
→ If answer comes wrong → cross and rewrite, don’t overwrite

For Case Study (4-5 marks) :
→ New pattern in CBSE — 2 questions compulsory from case study
→ Read the passage carefully — all data is given in it
→ Usually from : Mensuration, Coordinate Geometry, Statistics

For Assertion-Reason (1 mark) :
→ New pattern — Assertion and Reason both given
→ 4 options — Both true + Reason explains / Both true + doesn’t explain / A true R false / A false
→ Usually from : Real Numbers, Polynomials, Trigonometry

══════════════════════════════════════════
★ BOARD EXAM DAY STRATEGY
══════════════════════════════════════════

First 15 min  → Read paper fully, mark easy ones
Next 25 min   → Attempt Section A (MCQ + A/R) — 20 marks
Next 30 min   → Attempt Section B (2 mark) — 20 marks
Next 40 min   → Attempt Section C (3 mark) — 24 marks
Next 30 min   → Attempt Section D (5 mark) — 20 marks
Last 20 min   → Case Study + check paper

★ Paper Pattern :

Section A → 20 MCQ/A-R        → 1 mark each = 20 marks
Section B → 5 Short Answer    → 2 marks each = 10 marks
Section C → 6 Short Answer    → 3 marks each = 18 marks
Section D → 4 Long Answer     → 5 marks each = 20 marks
Section E → 3 Case Studies    → 4 marks each = 12 marks
─────────────────────────────────────────
TOTAL                                    = 80 marks

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