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Class 11 MathematicsChapter Summaries

14 chapters · Quick revision in under 3 minutes per chapter · Updated 2025-26

Ch 1

Sets

Introduces the concept of a set as a well-defined collection of objects. Students learn set representations (roster and set-builder form), types of sets, and operations such as union, intersection, difference, and complement. Venn diagrams are used to visualise relationships between sets. The chapter also covers De Morgan's laws and practical applications involving counting using the inclusion-exclusion principle.

Topics covered

Set representations: roster form and set-builder formTypes: empty set, finite/infinite sets, universal set, power setSubsets and proper subsets: A ⊆ B iff every element of A is in BSet operations: A ∪ B, A ∩ B, A − B, A′ (complement)De Morgan's laws: (A ∪ B)′ = A′ ∩ B′; (A ∩ B)′ = A′ ∪ B′Number of elements in power set P(A): n(P(A)) = 2ⁿ where n = n(A)Inclusion-exclusion principle: n(A ∪ B) = n(A) + n(B) − n(A ∩ B)Venn diagrams for visualising union, intersection, and difference

Ch 2

Relations and Functions

Builds on sets to define ordered pairs, Cartesian products, and relations. Students learn to distinguish between relations and functions, identify domain and range, and work with different types of functions (one-one, onto, into). The chapter also covers the algebra of real functions including sum, difference, product, and quotient.

Topics covered

Cartesian product: A × B = {(a, b) : a ∈ A, b ∈ B}n(A × B) = n(A) × n(B)Relation R from A to B: R ⊆ A × B; domain = set of first elementsFunction: every element of domain has exactly one imageTypes of functions: one-one (injective), onto (surjective), bijectiveAlgebra of functions: (f + g)(x) = f(x) + g(x); (fg)(x) = f(x)·g(x)Domain of f/g: {x : g(x) ≠ 0}; domain of √f: {x : f(x) ≥ 0}Range of a function: set of all actual output values

Ch 3

Trigonometric Functions

Extends trigonometric ratios to all real numbers via the unit circle, introducing radian measure and relationships between degree and radian. Students learn values of trigonometric functions for standard angles, fundamental identities, and formulas for compound, multiple, and sub-multiple angles. Finding the general solution of trigonometric equations is an important board exam skill.

Topics covered

Radian-degree conversion: 1 radian = 180°/π; arc length l = rθSigns of trig functions in all four quadrants (ASTC rule)Compound angle formulas: sin(A ± B), cos(A ± B), tan(A ± B)Double angle formulas: sin 2A = 2 sin A cos A; cos 2A = cos²A − sin²A = 1 − 2sin²AProduct-to-sum formulas: 2 sin A cos B = sin(A+B) + sin(A−B)General solutions: sin θ = sin α ⟹ θ = nπ + (−1)ⁿα; cos θ = cos α ⟹ θ = 2nπ ± αSine rule: a/sin A = b/sin B = c/sin C = 2RFundamental identities: sin²x + cos²x = 1; 1 + tan²x = sec²x; 1 + cot²x = cosec²x

Ch 4

Complex Numbers and Quadratic Equations

Introduces complex numbers as numbers of the form a + ib where i = √(−1). Students learn algebraic operations on complex numbers, the modulus-argument (polar) form, and the Argand plane representation. The chapter culminates in the fundamental theorem of algebra — every polynomial equation has a root in C — and solving quadratic equations with complex roots using the discriminant.

Topics covered

Imaginary unit: i = √(−1); i² = −1; i³ = −i; i⁴ = 1; i⁴ⁿ = 1Complex number z = a + ib; Re(z) = a, Im(z) = bModulus: |z| = √(a² + b²); Argument: arg(z) = θ = tan⁻¹(b/a)Conjugate: z̄ = a − ib; |z|² = z · z̄Polar form: z = r(cos θ + i sin θ) where r = |z|Algebra: (a+ib)(c+id) = (ac−bd) + i(ad+bc)Division: z₁/z₂ = z₁ · z̄₂ / |z₂|²Quadratic with D < 0: x = (−b ± i√|D|) / 2a

Ch 5

Linear Inequalities

Covers one-variable and two-variable linear inequalities and their graphical solutions. Students learn rules for manipulating inequalities (sign reversal on multiplication/division by a negative number) and solve systems of linear inequalities to find the feasible region. Word problems involving inequalities from daily life are important for exams.

Topics covered

Properties of inequalities: if a > b and c < 0 then ac < bcSolution set of a linear inequality in one variable on the number lineIntersection and union of solution sets for a system of inequalitiesGraphical solution: half-plane representation of ax + by ≤ cFeasible region: intersection of half-planes for a system of inequalitiesSolving word problems using linear inequalities

Ch 6

Permutations and Combinations

Covers the fundamental counting principle and factorial notation, then builds up to permutations (arrangements) and combinations (selections). Students learn when to use nPr versus nCr and handle problems involving repetition, identical objects, and circular arrangements. This chapter is heavily tested in board and competitive exams.

Topics covered

Fundamental counting principle: m × n ways for two independent choicesFactorial: n! = n × (n−1) × ... × 1; 0! = 1Permutations: ⁿPᵣ = n! / (n−r)!Combinations: ⁿCᵣ = n! / (r!(n−r)!); ⁿCᵣ = ⁿCₙ₋ᵣPermutations with identical objects: n! / (p! q! r!)Circular permutations: (n−1)! arrangementsProperty: ⁿCᵣ + ⁿCᵣ₋₁ = ⁿ⁺¹Cᵣ (Pascal's identity)

Ch 7

Binomial Theorem

States and proves the Binomial Theorem for positive integral indices. Students learn to expand (a + b)ⁿ using the general term Tᵣ₊₁, find the middle term(s), and identify specific terms in binomial expansions. Properties of binomial coefficients, including Pascal's triangle and the sum of coefficients, are important exam topics.

Topics covered

Binomial Theorem: (a + b)ⁿ = Σ(r=0 to n) ⁿCᵣ aⁿ⁻ʳ bʳGeneral term: Tᵣ₊₁ = ⁿCᵣ aⁿ⁻ʳ bʳMiddle term: T(n/2 + 1) for even n; T((n+1)/2) and T((n+3)/2) for odd nSum of all binomial coefficients: 2ⁿ (put a = b = 1)Pascal's triangle: each entry is sum of two entries aboveTerm independent of x: find r such that power of x = 0Greatest binomial coefficient: ⁿCₙ/₂ for even n

Ch 8

Sequences and Series

Covers arithmetic progressions (AP), geometric progressions (GP), and the special sums Σn, Σn², and Σn³. Students learn to find the nth term and sum of n terms of both AP and GP, including the sum of infinite GP when |r| < 1. Arithmetic-geometric mean inequality and insertion of means between two numbers are also tested.

Topics covered

AP: nth term aₙ = a + (n−1)d; sum Sₙ = n/2 [2a + (n−1)d]GP: nth term aₙ = arⁿ⁻¹; sum Sₙ = a(rⁿ − 1)/(r − 1) for r ≠ 1Infinite GP sum: S∞ = a/(1 − r) when |r| < 1Arithmetic mean: AM = (a + b)/2; Geometric mean: GM = √(ab)AM ≥ GM for positive realsInsertion of n AMs: d = (b − a)/(n + 1)Insertion of n GMs: r = (b/a)^(1/(n+1))Special sums: Σn = n(n+1)/2; Σn² = n(n+1)(2n+1)/6; Σn³ = [n(n+1)/2]²

Ch 9

Straight Lines

Provides a thorough treatment of lines in the Cartesian plane, covering slope, various forms of the equation of a line (slope-point, slope-intercept, two-point, intercept, and normal forms), distance of a point from a line, and the angle between two lines. Students also learn to find the family of lines passing through the intersection of two given lines.

Topics covered

Slope of a line: m = tan θ; m = (y₂ − y₁)/(x₂ − x₁)Condition for parallel lines: m₁ = m₂; perpendicular: m₁ × m₂ = −1Slope-intercept form: y = mx + cTwo-point form: (y − y₁)/(y₂ − y₁) = (x − x₁)/(x₂ − x₁)Normal form: x cos α + y sin α = pGeneral form Ax + By + C = 0; slope = −A/BDistance of point (x₁, y₁) from line Ax + By + C = 0: |Ax₁ + By₁ + C|/√(A² + B²)Distance between parallel lines Ax+By+C₁=0 and Ax+By+C₂=0: |C₁−C₂|/√(A²+B²)

Ch 10

Conic Sections

Introduces the four conic sections — circle, parabola, ellipse, and hyperbola — as intersections of a plane with a double-napped cone. Students derive and apply standard equations for each conic, identify key parameters (centre, radius, foci, vertices, eccentricity, latus rectum), and solve problems involving equations of conics given geometric conditions.

Topics covered

Circle: (x − h)² + (y − k)² = r²; general equation x² + y² + 2gx + 2fy + c = 0Parabola: y² = 4ax (focus: (a,0), directrix: x = −a, latus rectum: 4a)Ellipse: x²/a² + y²/b² = 1 (a > b > 0); b² = a²(1 − e²); sum of focal radii = 2aEllipse: foci: (±ae, 0); vertices: (±a, 0); latus rectum: 2b²/aHyperbola: x²/a² − y²/b² = 1; b² = a²(e² − 1); e > 1; asymptotes: y = ±(b/a)xEccentricity e: circle e = 0; parabola e = 1; ellipse 0 < e < 1; hyperbola e > 1Focal chord: chord passing through the focusStandard equations for horizontal and vertical conics

Ch 11

Introduction to Three-Dimensional Geometry

Extends coordinate geometry to three dimensions, introducing the three coordinate axes, octants, and three-dimensional coordinates. Students learn to calculate the distance between two points in 3D and apply the section formula to find coordinates of a point dividing a line segment in a given ratio. The midpoint formula is a special case of the section formula.

Topics covered

3D coordinate system: x, y, z axes; eight octantsDistance formula: PQ = √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²]Section formula (internal): P = ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n), (mz₂+nz₁)/(m+n))Midpoint formula: M = ((x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2)Centroid of a triangle: G = ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3, (z₁+z₂+z₃)/3)Section formula (external): m:n externally — use (mx₂−nx₁)/(m−n)

Ch 12

Limits and Derivatives

Introduces the concept of a limit intuitively and develops the algebra of limits (sum, product, quotient rules). Students evaluate standard limits including lim(x→0) sin x/x = 1 and lim(x→a) (xⁿ − aⁿ)/(x − a) = naⁿ⁻¹. The derivative is defined as the first principle (limit of the difference quotient), and differentiation rules (product rule, quotient rule) are developed.

Topics covered

Limit: lim(x→a) f(x); left-hand limit, right-hand limit; existence condition: LHL = RHLAlgebra of limits: lim(fg) = lim f · lim g; lim(f/g) = lim f/lim g (g ≠ 0)Standard limit: lim(x→0) sin x / x = 1; lim(x→0) tan x / x = 1Standard limit: lim(x→a) (xⁿ − aⁿ)/(x − a) = naⁿ⁻¹Derivative from first principles: f′(x) = lim(h→0) [f(x+h) − f(x)] / hStandard derivatives: d/dx(xⁿ) = nxⁿ⁻¹; d/dx(sin x) = cos x; d/dx(cos x) = −sin xProduct rule: (uv)′ = u′v + uv′; Quotient rule: (u/v)′ = (u′v − uv′)/v²Derivatives of tan x, cot x, sec x, cosec x using quotient rule

Ch 13

Statistics

Extends the study of statistics to measures of dispersion — range, mean deviation, variance, and standard deviation — for both ungrouped and grouped data. Students learn to calculate mean deviation about the mean and median, and standard deviation using the direct and assumed mean methods. Coefficient of variation is used to compare variability of two series.

Topics covered

Range = Maximum value − Minimum valueMean Deviation (MD) about mean: MD = Σ|xᵢ − x̄| / n (ungrouped)Mean Deviation for grouped data: MD = Σfᵢ|xᵢ − x̄| / ΣfᵢVariance: σ² = Σ(xᵢ − x̄)² / n; Standard Deviation: σ = √(Σ(xᵢ − x̄)²/n)Shortcut for variance: σ² = (Σxᵢ²)/n − (x̄)²Coefficient of Variation (CV) = (σ/x̄) × 100; lower CV ⟹ more consistentVariance of grouped data: σ² = (1/N) Σfᵢ(xᵢ − x̄)²

Ch 14

Probability

Introduces the axiomatic approach to probability alongside the classical definition. Students learn sample space, events, algebra of events (complementary, mutually exclusive, exhaustive), and basic laws including the addition rule. The chapter contrasts empirical (experimental) probability with theoretical (classical) probability and applies them to dice, cards, and general sample spaces.

Topics covered

Sample space S and events as subsets of SAxiomatic definition: 0 ≤ P(A) ≤ 1; P(S) = 1; P(A ∪ B) = P(A) + P(B) for mutually exclusive eventsClassical probability: P(A) = n(A)/n(S)Complementary event: P(A′) = 1 − P(A)Addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)Mutually exclusive events: P(A ∩ B) = 0 ⟹ P(A ∪ B) = P(A) + P(B)Exhaustive events: P(A₁ ∪ A₂ ∪ ... ∪ Aₙ) = 1

⚠️ Removed from 2025-26 syllabus

Principle of Mathematical Induction (separate chapter removed from 2025-26 board exam syllabus)

Mathematical Reasoning (entire chapter removed from 2025-26 board exam syllabus)

Frequently Asked Questions

Can I revise Class 11 Mathematics in one day using summaries?

Yes. Each chapter summary here takes under 3 minutes to read. With 14 chapters, you can cover all of Class 11 Mathematics in a focused 2–3 hour session. Use these summaries to identify gaps — then revisit only those chapters in detail.

Are chapter summaries enough for CBSE Class 11 Mathematics board exam?

Summaries are for revision, not first learning. Use them after you've already studied the chapter — they quickly confirm what you remember and flag what you don't. For first-time study, read the NCERT textbook and work through important questions chapter-by-chapter.

What is covered in Class 11 Mathematics chapter summaries?

Each summary here covers the main concepts of the chapter, key topics that CBSE tests, and important points for the board exam. Deleted topics (removed from the 2025-26 CBSE syllabus) are clearly marked so you don't waste time on content that won't be tested.

What is the fastest way to revise Class 11 Mathematics for CBSE boards?

Read the chapter summary, then immediately close it and try to recall the key topics listed — without looking. Anything you miss, mark for one more read. This active recall method is proven to be 3× more effective than re-reading the textbook, and takes a fraction of the time.

Complete Study Pack — Class 11 Mathematics

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