Exam Preparation

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

November 20, 2025 6 min read

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:

Condition	Type of Lines	Solutions
a₁/a₂ ≠ b₁/b₂	Intersecting	Unique solution
a₁/a₂ = b₁/b₂ = c₁/c₂	Coincident	Infinite solutions
a₁/a₂ = b₁/b₂ ≠ c₁/c₂	Parallel	No solution

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:

Angle	0°	30°	45°	60°	90°
sin	0	1/2	1/√2	√3/2	1
cos	1	√3/2	1/√2	1/2	0
tan	0	1/√3	1	√3	∞

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

Practice NCERT exercises thoroughly
Memorize all formulas
Solve previous year board papers
Show all steps in calculations
Draw neat diagrams for geometry

Additional Key Formulas and Exam Tips

Polynomials (Chapter 2)

For a quadratic polynomial ax2 + bx + c: Sum of zeroes = -b/a; Product of zeroes = c/a
For a cubic polynomial ax3 + bx2 + cx + d: Sum of zeroes = -b/a; Sum of product of pairs = c/a; Product of zeroes = -d/a
Division algorithm: Dividend = Divisor × Quotient + Remainder

Linear Equations in Two Variables (Chapter 3)

For a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0:
Unique solution (lines intersect): a1/a2 ≠ b1/b2
No solution (parallel lines): a1/a2 = b1/b2 ≠ c1/c2
Infinite solutions (coincident lines): a1/a2 = b1/b2 = c1/c2

Quadratic Equations (Chapter 4)

Quadratic formula: x = [-b ± √(b2 – 4ac)] / 2a
Discriminant (D) = b2 – 4ac: D > 0 → two distinct real roots; D = 0 → two equal real roots; D less than 0 → no real roots

Arithmetic Progressions (Chapter 5)

General term: an = a + (n-1)d
Sum of n terms: Sn = n/2 × [2a + (n-1)d] = n/2 × [a + l], where l = last term
Relation: an = Sn – S(n-1)

Triangles (Chapter 6)

Basic Proportionality Theorem (Thales): If a line is parallel to one side of a triangle and intersects the other two sides, it divides them in the same ratio.
AA similarity, SSS similarity, SAS similarity criteria
In similar triangles: ratio of areas = square of ratio of corresponding sides
Pythagoras theorem: In a right triangle, (hypotenuse)2 = (base)2 + (perpendicular)2

Coordinate Geometry (Chapter 7)

Distance formula: d = √[(x2-x1)2 + (y2-y1)2]
Section formula (internal division): x = (m1x2 + m2x1)/(m1+m2), y = (m1y2 + m2y1)/(m1+m2)
Midpoint: x = (x1+x2)/2, y = (y1+y2)/2
Area of triangle: (1/2)|x1(y2-y3) + x2(y3-y1) + x3(y1-y2)|

Circles (Chapter 10)

Tangent is perpendicular to radius at the point of contact
Two tangents drawn from an external point are equal in length
Angle in a semicircle = 90 degrees

Surface Areas and Volumes (Chapter 13)

Cube: SA = 6a2; Volume = a3
Cuboid: SA = 2(lb + bh + lh); Volume = lbh
Cylinder: CSA = 2πrh; TSA = 2πr(r+h); Volume = πr2h
Cone: CSA = πrl (l = slant height = √(r2+h2)); TSA = πr(r+l); Volume = (1/3)πr2h
Sphere: SA = 4πr2; Volume = (4/3)πr3
Hemisphere: CSA = 2πr2; TSA = 3πr2; Volume = (2/3)πr3

Statistics (Chapter 14)

Mean by assumed mean method: x̄ = a + (Σfidi / Σfi), where di = xi – a
Median = l + [(n/2 – cf) / f] × h
Mode = l + [(f1 – f0) / (2f1 – f0 – f2)] × h
Empirical relation: Mode = 3 Median – 2 Mean

Probability (Chapter 15)

P(E) = Number of favourable outcomes / Total number of outcomes
P(E) + P(not E) = 1
0 ≤ P(E) ≤ 1; P(impossible event) = 0; P(certain event) = 1

Board Exam Strategy for Class 10 Maths

Chapters 1, 2, 4, 5 carry high weightage — practice all NCERT exercise problems
Chapter 13 (Mensuration) frequently appears in long-answer (5-mark) questions involving combinations of shapes
Always show all steps — partial credit is awarded even for partially correct solutions
Learn all formulas by heart — the exam paper does not provide a formula sheet
Practice previous 5 years question papers for pattern recognition

View all posts →