Chapter Notes›Class 10 Mathematics

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Class 10 MathematicsChapter Notes

14 chapters · Definitions, key points, formulas & exam tips · Updated 2025-26

Ch 1 Ch 2 Ch 3 Ch 4 Ch 5 Ch 6 Ch 7 Ch 8 Ch 9 Ch 10 Ch 11 Ch 12 Ch 13 Ch 14

Ch 1

Real Numbers

Key Definitions

Euclid's Division Lemma: For any two positive integers a and b, there exist unique integers q and r such that a = bq + r, where 0 ≤ r < b.

HCF: Highest Common Factor — the largest number that divides two or more integers exactly.

LCM: Lowest Common Multiple — the smallest number divisible by two or more integers.

Irrational Number: A number that cannot be expressed as p/q where p and q are integers and q ≠ 0. E.g., √2, π.

Key Points to Remember

→HCF × LCM = Product of two numbers (only for two numbers).
→√2, √3, √5 are always irrational — prove using contradiction method.
→A rational number p/q has a terminating decimal if q has only 2 and 5 as prime factors.
→If q has any prime factor other than 2 or 5, the decimal is non-terminating repeating.
→Every composite number can be expressed as a product of primes in exactly one way (Fundamental Theorem of Arithmetic).

Formulas & Equations

a = bq + r (Euclid's Division Lemma)

HCF × LCM = a × b

For terminating decimal: q = 2ᵐ × 5ⁿ

Exam Tips

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HCF by Euclid's Algorithm: always start with the larger number.

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To prove √p is irrational, assume it's rational, then reach a contradiction.

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For 3-mark problems on HCF/LCM, show all steps of prime factorisation.

Ch 2

Polynomials

Key Definitions

Polynomial: An expression of the form aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀, where coefficients are real numbers.

Zero of a Polynomial: A value of x for which p(x) = 0. Also called root.

Degree: The highest power of x in the polynomial.

Key Points to Remember

→A polynomial of degree n has at most n zeroes.
→Quadratic polynomial ax² + bx + c has at most 2 zeroes.
→Sum of zeroes (α + β) = −b/a
→Product of zeroes (αβ) = c/a
→For cubic: α + β + γ = −b/a, αβ + βγ + γα = c/a, αβγ = −d/a
→Geometrically, zeroes are x-coordinates where graph cuts the x-axis.

Formulas & Equations

α + β = −b/a

αβ = c/a

Quadratic: x = [−b ± √(b²−4ac)] / 2a

p(x) = x² − (α+β)x + αβ

Exam Tips

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Form quadratic from zeroes using: x² − (sum)x + (product).

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Division Algorithm: Dividend = Divisor × Quotient + Remainder.

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Verify zeroes by substituting back into p(x) — shows working in exam.

Ch 3

Pair of Linear Equations in Two Variables

Key Definitions

Consistent System: A system with at least one solution (unique or infinite solutions).

Inconsistent System: A system with no solution. Lines are parallel.

Dependent Equations: Two equations representing the same line — infinitely many solutions.

Key Points to Remember

→Graphical method: plot both lines; intersection point is the solution.
→Substitution method: express one variable in terms of the other.
→Elimination method: multiply equations to make coefficients equal, then add/subtract.
→Cross-multiplication method: fastest for exams.
→Condition for unique solution: a₁/a₂ ≠ b₁/b₂
→Condition for no solution: a₁/a₂ = b₁/b₂ ≠ c₁/c₂
→Condition for infinite solutions: a₁/a₂ = b₁/b₂ = c₁/c₂

Formulas & Equations

Cross-multiplication: x/(b₁c₂−b₂c₁) = y/(c₁a₂−c₂a₁) = 1/(a₁b₂−a₂b₁)

Exam Tips

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For word problems: define variables clearly at the start.

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Cross-multiplication is fastest in MCQs and 3-mark questions.

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Always verify your answer by substituting back into both equations.

Ch 4

Quadratic Equations

Key Definitions

Quadratic Equation: An equation of the form ax² + bx + c = 0, where a ≠ 0.

Discriminant (D): D = b² − 4ac. Determines nature of roots.

Key Points to Remember

→If D > 0: two distinct real roots.
→If D = 0: two equal (coincident) real roots.
→If D < 0: no real roots.
→Methods to solve: factorisation, completing the square, quadratic formula.
→Completing the square: used to derive the quadratic formula.

Formulas & Equations

x = [−b ± √(b²−4ac)] / 2a

D = b² − 4ac

Sum of roots = −b/a, Product of roots = c/a

Exam Tips

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Always check if D ≥ 0 before solving — state nature of roots first in exam.

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Factorisation is fastest when roots are integers.

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For 'find k' problems: set D = 0 for equal roots or D > 0 for real roots.

Ch 5

Arithmetic Progressions

Key Definitions

AP: A sequence where each term differs from the previous by a constant amount called common difference (d).

Common Difference (d): d = a₂ − a₁ = a₃ − a₂. Can be positive, negative, or zero.

Key Points to Remember

→General term: aₙ = a + (n−1)d
→Middle term of finite AP = (first + last) / 2
→If sum Sₙ is given, nth term aₙ = Sₙ − Sₙ₋₁
→Three numbers in AP: take a−d, a, a+d
→Four numbers in AP: take a−3d, a−d, a+d, a+3d

Formulas & Equations

aₙ = a + (n−1)d

Sₙ = n/2 [2a + (n−1)d]

Sₙ = n/2 [a + l] where l = last term

d = (l − a) / (n − 1)

Exam Tips

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To find how many terms: use aₙ = a + (n−1)d and solve for n — n must be a positive integer.

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Sum problems: use Sₙ formula directly, don't find all terms.

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Word problems often involve 'first term' and 'common difference' — identify them first.

Ch 6

Triangles

Key Definitions

Similar Triangles: Triangles with same shape but not necessarily same size. Corresponding angles are equal, corresponding sides are proportional.

Basic Proportionality Theorem (BPT): If a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally.

Key Points to Remember

→BPT (Thales Theorem): DE ∥ BC ⟹ AD/DB = AE/EC
→Criteria for similarity: AA, SSS, SAS
→Ratio of areas of similar triangles = square of ratio of corresponding sides.
→Pythagoras Theorem: In a right triangle, AC² = AB² + BC²
→Converse of Pythagoras: If AC² = AB² + BC², then angle B = 90°.

Formulas & Equations

Area ratio: (Area of △ABC) / (Area of △DEF) = (AB/DE)²

Pythagoras: (Hypotenuse)² = (Base)² + (Perpendicular)²

Exam Tips

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For proof questions: always state the theorem name before using it.

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BPT and its converse both appear in 3-mark proofs.

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Draw and label diagrams — examiners give marks for correct diagrams.

Ch 7

Coordinate Geometry

Key Definitions

Coordinate Plane: A plane formed by x-axis and y-axis intersecting at origin (0,0).

Section Formula: Formula to find coordinates of a point dividing a line segment in a given ratio.

Key Points to Remember

→Distance formula: d = √[(x₂−x₁)² + (y₂−y₁)²]
→Midpoint formula: M = ((x₁+x₂)/2, (y₁+y₂)/2)
→Section formula (internal division): P = ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n))
→Centroid of triangle: G = ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3)
→Area of triangle with vertices: ½|x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|
→If area = 0, points are collinear.

Formulas & Equations

Distance = √[(x₂−x₁)² + (y₂−y₁)²]

Section (m:n) = ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n))

Area = ½|x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|

Exam Tips

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To show a quadrilateral is a rhombus: prove all sides equal but diagonals unequal.

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For collinearity: use area formula — if area = 0, points are collinear.

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The y-axis divides a segment where x = 0 — use section formula with that condition.

Ch 8

Introduction to Trigonometry

Key Definitions

Trigonometric Ratios: Ratios of sides of a right-angled triangle with respect to an acute angle.

Complementary Angles: Two angles whose sum is 90°. sin A = cos(90°−A).

Key Points to Remember

→sin θ = Opposite/Hypotenuse, cos θ = Adjacent/Hypotenuse, tan θ = Opposite/Adjacent
→cosec θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ
→sin²θ + cos²θ = 1
→1 + tan²θ = sec²θ
→1 + cot²θ = cosec²θ
→Values: sin 0°=0, sin 30°=½, sin 45°=1/√2, sin 60°=√3/2, sin 90°=1

Formulas & Equations

sin²θ + cos²θ = 1

sec²θ − tan²θ = 1

cosec²θ − cot²θ = 1

sin(90°−θ) = cosθ, cos(90°−θ) = sinθ

tan(90°−θ) = cotθ

Exam Tips

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Memorise the table of values for 0°, 30°, 45°, 60°, 90° — at least 2 MCQs come from it.

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For identities: start from the more complex side and simplify.

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Convert everything to sin and cos when stuck on an identity proof.

Ch 9

Applications of Trigonometry

Key Definitions

Angle of Elevation: The angle formed by the line of sight with the horizontal when looking up at an object.

Angle of Depression: The angle formed by the line of sight with the horizontal when looking down at an object.

Key Points to Remember

→Angle of elevation = angle of depression (alternate interior angles).
→Always draw a clear diagram before solving.
→Use tan θ = height/distance for most problems.
→For two-position problems (observer moves), form two equations.
→Shadow problems: use tan(angle) = object height / shadow length.

Formulas & Equations

tan θ = Perpendicular / Base

sin θ = Perpendicular / Hypotenuse

Height = Distance × tan(angle of elevation)

Exam Tips

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Draw the right-angled triangle — half the marks are for correct setup.

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Standard values: tan 30°=1/√3, tan 45°=1, tan 60°=√3.

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Rationalise the denominator in final answer — e.g., 20/√3 = 20√3/3.

Ch 10

Circles

Key Definitions

Tangent: A line that touches the circle at exactly one point (point of tangency).

Secant: A line that intersects the circle at two points.

Key Points to Remember

→Tangent is perpendicular to the radius at the point of contact.
→From an external point, exactly two tangents can be drawn to a circle.
→Lengths of two tangents from an external point are equal.
→Angle between two tangents from external point + angle at centre = 180°.
→Tangent-radius forms a right angle: use Pythagoras to find lengths.

Formulas & Equations

PA = PB (tangent lengths from external point P)

OA ⊥ PA (radius ⊥ tangent)

OP² = OA² + PA² (where O is centre, A is point of contact)

Exam Tips

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For proof: state 'tangent ⊥ radius' as a reason whenever you use it.

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Most circle problems use Pythagoras — identify the right angle first.

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PA = PB is used in almost every tangent problem involving external points.

Ch 11

Areas Related to Circles

Key Definitions

Sector: Region between two radii and the arc connecting them.

Segment: Region between a chord and the arc it cuts off.

Minor Arc: The smaller arc between two points on a circle.

Key Points to Remember

→Area of sector = (θ/360) × πr²
→Length of arc = (θ/360) × 2πr
→Area of minor segment = Area of sector − Area of triangle
→Area of major segment = Area of circle − Area of minor segment
→For semicircle: θ = 180°

Formulas & Equations

Area of circle = πr²

Circumference = 2πr

Area of sector = (θ/360°) × πr²

Arc length = (θ/360°) × 2πr

Area of segment = Area of sector − Area of △

Exam Tips

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Use π = 22/7 unless the problem says otherwise.

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Always find the area of the triangle separately using ½ × base × height or ½r²sinθ.

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Shaded region problems: identify what to add and what to subtract.

Ch 12

Surface Areas and Volumes

Key Definitions

Slant Height (l): The distance from the apex of a cone to any point on its circumference.

Frustum: The portion of a cone left when a smaller cone is cut off from the top parallel to the base.

Key Points to Remember

→When solids are combined, add their volumes but be careful about surface areas.
→For hollow objects: subtract volumes.
→Frustum: a truncated cone — has two circular faces of different radii.
→When a sphere is melted into cylinders: volume remains the same.
→Slant height of cone: l = √(r² + h²)

Formulas & Equations

Cylinder: V = πr²h, CSA = 2πrh, TSA = 2πr(r+h)

Cone: V = ⅓πr²h, CSA = πrl, TSA = πr(r+l), l = √(r²+h²)

Sphere: V = 4/3πr³, SA = 4πr²

Hemisphere: V = 2/3πr³, CSA = 2πr², TSA = 3πr²

Frustum: V = πh/3(r₁²+r₂²+r₁r₂), l = √[h²+(r₁−r₂)²]

Exam Tips

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Conversion problems: equate volumes of the two solids.

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For combined solids, draw and label all dimensions.

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CSA vs TSA: CSA excludes the flat circular faces; TSA includes them.

Ch 13

Statistics

Key Definitions

Mean: Average value. Sum of all values divided by number of values.

Median: Middle value when data is arranged in order. Divides data into two equal halves.

Mode: Value that appears most frequently in the data.

Key Points to Remember

→For grouped data, use assumed mean method for easy calculation.
→Median class: find n/2, then locate the class in cumulative frequency table.
→Modal class: the class with highest frequency.
→Empirical formula: Mode = 3 × Median − 2 × Mean
→Ogive: cumulative frequency curve used to find median graphically.

Formulas & Equations

Mean (Direct): x̄ = Σfx / Σf

Mean (Assumed): x̄ = a + Σfd / Σf

Median = l + [(n/2 − cf) / f] × h

Mode = l + [(f₁−f₀) / (2f₁−f₀−f₂)] × h

Mode = 3 Median − 2 Mean

Exam Tips

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In the median formula, cf = cumulative frequency of the class BEFORE the median class.

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Always construct the cumulative frequency table for median problems.

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The ogive question: draw smooth curve, find median at n/2 on y-axis.

Ch 14

Probability

Key Definitions

Probability: A measure of likelihood of an event occurring. P(E) = Number of favourable outcomes / Total outcomes.

Complementary Event: P(E) + P(Ē) = 1. If P(E) = 0.3, then P(not E) = 0.7.

Equally Likely Events: Events with the same chance of occurring.

Key Points to Remember

→0 ≤ P(E) ≤ 1 always.
→P(sure event) = 1, P(impossible event) = 0.
→Total outcomes for a die: 6. For two dice: 36. For a deck: 52.
→A deck: 52 cards = 4 suits × 13 cards. Face cards = 12 (J, Q, K of each suit).
→Bag problems: total outcomes = total number of items in bag.

Formulas & Equations

P(E) = Favourable outcomes / Total outcomes

P(Ē) = 1 − P(E)

P(E) + P(Ē) = 1

Exam Tips

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List the sample space for small problems — don't guess total outcomes.

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Two dice: list as ordered pairs (1,1), (1,2)... total = 36.

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Face cards ≠ honour cards. Ace is NOT a face card in CBSE exams.

Frequently Asked Questions

Are these notes based on 2025-26 CBSE syllabus for Class 10 Mathematics?

Yes. All chapter notes here are based on the latest 2025-26 CBSE syllabus for Class 10 Mathematics. Deleted topics are clearly marked so you focus only on what will be tested in your board exam.

How to study Class 10 Mathematics notes effectively for board exams?

Read each chapter's notes once to build understanding. Then close the notes and try to recall every key point, definition, and formula from memory. Anything you miss is your weak area — revisit only those points. This active recall method takes less time and retains far more than re-reading.

What is the difference between NCERT notes and chapter summaries?

Chapter notes contain detailed definitions, key terms, formulas, and concept breakdowns — they're for learning and understanding. Chapter summaries are shorter paragraph-style overviews — they're for quick revision. Use notes when you're studying a chapter for the first time; use summaries the night before the exam.

Do I need to memorise formulas for Class 10 Mathematics CBSE board exam?

Yes. Formulas listed in these notes must be memorised precisely — CBSE doesn't give formula sheets during exams. Write each formula 5–10 times, then recall it without looking. In the exam, write the formula first, then substitute values — this helps you earn partial marks even if the final answer has a calculation error.

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