Chapter Notes›Class 10 Mathematics
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Class 10 MathematicsChapter Notes

14 chapters · Definitions, key points, formulas & exam tips

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

More resources for Class 10 Mathematics

NCERT Solutions →Chapter Summaries →Important Questions →Revision Checklist →Sample Papers →