NCERT Class 10 Maths – Chapter-wise Summary & Key Formulas

NCERT Class 10 Mathematics is crucial for board exams and forms the foundation for competitive exams. This guide provides chapter-wise summaries and key formulas for quick revision.

Chapter 1: Real Numbers

Key Concepts:

• Euclid’s Division Lemma: a = bq + r, where 0 ≤ r < b
• HCF: Use Euclid’s Division Algorithm
• Fundamental Theorem of Arithmetic: Every composite number can be expressed as product of primes uniquely
• LCM × HCF = Product of two numbers
• Irrational Numbers: √2, √3, √5 are irrational
• Terminating decimal: Denominator has only 2 and 5 as factors

Chapter 2: Polynomials

Key Formulas:

• For quadratic polynomial ax² + bx + c:
• Sum of zeroes (α + β) = -b/a
• Product of zeroes (αβ) = c/a
• For cubic polynomial ax³ + bx² + cx + d:
• α + β + γ = -b/a
• αβ + βγ + γα = c/a
• αβγ = -d/a

Chapter 3: Pair of Linear Equations

Methods of Solving:

• Graphical Method: Plot lines and find intersection
• Substitution Method
• Elimination Method
• Cross-Multiplication Method

Conditions for Solutions:

Chapter 4: Quadratic Equations

Standard Form:

ax² + bx + c = 0

Quadratic Formula:

x = (-b ± √(b² – 4ac)) / 2a

Nature of Roots (Discriminant D = b² – 4ac):

• D > 0: Two distinct real roots
• D = 0: Two equal real roots
• D < 0: No real roots

Chapter 5: Arithmetic Progressions

Key Formulas:

• nth term: aₙ = a + (n-1)d
• Sum of n terms: Sₙ = n/2 [2a + (n-1)d]
• Sum of n terms: Sₙ = n/2 (a + l) [l = last term]
• Common difference: d = aₙ – aₙ₋₁
• Middle term: When n is odd, middle term = a + ((n-1)/2)d

Chapter 6: Triangles

Similarity Criteria:

• AAA: All angles equal
• AA: Two angles equal
• SSS: All sides proportional
• SAS: Two sides proportional with included angle equal

Important Theorems:

• Basic Proportionality Theorem (BPT): If a line is drawn parallel to one side of a triangle, it divides other two sides proportionally
• Pythagoras Theorem: In right triangle, (Hypotenuse)² = (Base)² + (Perpendicular)²
• Area ratio of similar triangles = (Ratio of corresponding sides)²

Chapter 7: Coordinate Geometry

Key Formulas:

• Distance Formula: d = √[(x₂-x₁)² + (y₂-y₁)²]
• Section Formula (Internal): ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n))
• Midpoint Formula: ((x₁+x₂)/2, (y₁+y₂)/2)
• Area of Triangle: ½|x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂)|
• If area = 0, points are collinear

Chapter 8: Introduction to Trigonometry

Trigonometric Ratios:

• sin θ = Opposite/Hypotenuse = P/H
• cos θ = Adjacent/Hypotenuse = B/H
• tan θ = Opposite/Adjacent = P/B
• cosec θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ

Standard Values:

Identities:

• sin²θ + cos²θ = 1
• 1 + tan²θ = sec²θ
• 1 + cot²θ = cosec²θ

Chapter 9: Applications of Trigonometry

Key Terms:

• Angle of Elevation: Angle above horizontal line of sight
• Angle of Depression: Angle below horizontal line of sight

Chapter 10: Circles

Key Theorems:

• Tangent at any point is perpendicular to radius
• Tangents drawn from external point are equal
• If two tangents are drawn from external point, angle between them = 2 × angle at center subtended by line segment joining points of contact

Chapter 11: Constructions

Important Constructions:

• Division of line segment in given ratio
• Similar triangle construction (scale factor > 1 or < 1)
• Tangent to circle from external point

Chapter 12: Areas Related to Circles

Key Formulas:

• Area of Circle: πr²
• Circumference: 2πr
• Area of Sector: (θ/360°) × πr²
• Length of Arc: (θ/360°) × 2πr
• Area of Segment: Area of Sector – Area of Triangle

Chapter 13: Surface Areas and Volumes

Key Formulas:

Cube (side = a):

• Volume = a³
• TSA = 6a²
• LSA = 4a²

Cuboid (l × b × h):

• Volume = lbh
• TSA = 2(lb + bh + hl)
• LSA = 2h(l + b)

Cylinder (radius r, height h):

• Volume = πr²h
• CSA = 2πrh
• TSA = 2πr(r + h)

Cone (radius r, height h, slant l):

• Volume = (1/3)πr²h
• CSA = πrl
• TSA = πr(l + r)
• l = √(r² + h²)

Sphere (radius r):

• Volume = (4/3)πr³
• Surface Area = 4πr²

Hemisphere:

• Volume = (2/3)πr³
• CSA = 2πr²
• TSA = 3πr²

Chapter 14: Statistics

Measures of Central Tendency:

• Mean (Direct): x̄ = Σfᵢxᵢ / Σfᵢ
• Mean (Assumed Mean): x̄ = a + (Σfᵢdᵢ / Σfᵢ)
• Mean (Step Deviation): x̄ = a + (Σfᵢuᵢ / Σfᵢ) × h
• Median: l + ((n/2 – cf)/f) × h
• Mode: l + ((f₁ – f₀)/(2f₁ – f₀ – f₂)) × h

Relationship:

Mode = 3 Median – 2 Mean

Chapter 15: Probability

Key Concepts:

• P(E) = Number of favorable outcomes / Total outcomes
• 0 ≤ P(E) ≤ 1
• P(E) + P(not E) = 1
• P(sure event) = 1
• P(impossible event) = 0

Important Examples:

• Coin: P(Head) = P(Tail) = 1/2
• Dice: P(any number) = 1/6
• Cards: Total = 52, Each suit = 13

Tips for Board Exam

1. Practice NCERT exercises thoroughly
2. Memorize all formulas
3. Solve previous year board papers
4. Show all steps in calculations
5. Draw neat diagrams for geometry