Class 12 · Mathematics · English medium · Part 1
Class 12 Mathematics Part 1 companion
An original companion to Mathematics, NCERT Class 12 Mathematics (2026–27 edition). Official textbook ↗ ncert.nic.in
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Chapters
- ENHinglish
Chapter 1: Relations and Functions
The chapter classifies relations and functions. - ENHinglish
Chapter 2: Inverse Trigonometric Functions
Class 12 Mathematics Part I, Chapter 2, builds the inverse trigonometric functions. - ENHinglish
Chapter 3: Matrices
Class 12 Matrices: what a matrix is (rectangular array, order m × n, entry a_ij), the types (row, column, square, diagonal, scalar, identity - ENHinglish
Chapter 4: Determinants
One number that decides areas, inverses and solutions - ENHinglish
Chapter 5: Continuity and Differentiability
Class 12 Mathematics, Chapter 5. - ENHinglish
Chapter 6: Application of Derivatives
Rates, rising and falling, and the best possible value
Also in Hinglish
- Chapter 1: Relations and Functions (Hinglish)
- Chapter 2: Inverse Trigonometric Functions (Hinglish)
- Chapter 3: Matrices (Hinglish)
- Chapter 4: Determinants (Hinglish)
- Chapter 5: Continuity aur Differentiability (Hinglish)
- Chapter 6: Application of Derivatives (Hinglish)
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absolute maximumनिरपेक्ष उच्चिष्ठThe greatest value a function takes anywhere on the given interval.Ch 6 · Application of Derivativesabsolute minimumनिरपेक्ष निम्निष्ठThe least value a function takes anywhere on the given interval.Ch 6 · Application of DerivativesadjointसहखंडजThe transpose of the matrix of cofactors of a square matrix A, written adj A.Ch 4 · Determinantsasymptoteअनंतस्पर्शीA line that a graph approaches ever more closely without reaching it, such as y = π 2 for y = tan⁻¹x.Ch 2 · Inverse Trigonometric FunctionsB
bijective functionएकैकी आच्छादक फलनA function that is both one-one and onto.Ch 1 · Relations and FunctionsbranchशाखाOne interval on which a trigonometric function is one-one and onto its range, or the inverse it gives.Ch 2 · Inverse Trigonometric FunctionsC
chain ruleशृंखला नियमThe derivative of a composite is the product of the rates along the chain: dy dx = dy dt · dt dx.Ch 5 · Continuity and DifferentiabilitycofactorसहखंडAij = (-1) i+jMij: the minor of aij with its checkerboard sign attached.Ch 4 · Determinantscollinear pointsसंरेख बिंदुPoints that lie on one straight line.Ch 4 · Determinantscompositionफलनों का संयोजनFor f: A → B and g: B → C, the function g ° f: A → C, (g ° f)(x) = g(f(x)).Ch 1 · Relations and Functionsconsistent systemसंगत निकायA system of equations that has at least one solution.Ch 4 · Determinantscontinuous (at a point)संततf is continuous at a point c of its domain when limx→c f(x) exists and equals f(c).Ch 5 · Continuity and Differentiabilitycontinuous functionसंतत फलनA function that is continuous at every point of its domain.Ch 5 · Continuity and Differentiabilitycritical pointक्रांतिक बिंदुA point c in the domain where f'(c) = 0 or where f is not differentiable.Ch 6 · Application of DerivativesD
decreasing functionह्रासमान फलनA function with f(x1) ≥ f(x2) whenever x1 < x2 in the interval; strictly decreasing if f(x1) > f(x2).Ch 6 · Application of DerivativesdeterminantसारणिकThe number assigned to a square matrix A, written |A| or det A. For a 2 × 2 matrix it is ad - bc.Ch 4 · Determinantsdiagonal matrixविकर्ण आव्यूहA square matrix whose entries off the main diagonal are all 0.Ch 3 · Matricesdifferentiableअवकलनीयf is differentiable at c when its left-hand and right-hand derivatives at c are finite and equal.Ch 5 · Continuity and DifferentiabilityE
elementअवयवOne number or function in a matrix; aij sits in row i and column j.Ch 3 · Matricesempty relationरिक्त संबंधThe relation varnothing, in which no element is related to any element.Ch 1 · Relations and Functionsequal matricesसमान आव्यूहMatrices of the same order whose corresponding entries are all equal.Ch 3 · Matricesequivalence classतुल्यता वर्गFor an equivalence relation, the set [a] of all elements related to a.Ch 1 · Relations and Functionsequivalence relationतुल्यता संबंधA relation that is reflexive, symmetric and transitive.Ch 1 · Relations and Functionsexpansion of a determinantसारणिक का प्रसारWriting a determinant as the sum of the entries of one row or column, each multiplied by its cofactor.Ch 4 · Determinantsexponential functionचरघातांकी फलनy = b x for a fixed base b > 1; with base e it is the natural exponential e x.Ch 5 · Continuity and Differentiabilityextreme valueचरम मानEither the greatest or the least value that a function takes on an interval.Ch 6 · Application of DerivativesF
functionफलनA relation from X to Y that relates each element of X to exactly one element of Y.Ch 1 · Relations and FunctionsG
greatest integer functionमहत्तम पूर्णांक फलन[x], the largest integer less than or equal to x; for example [-1.2] = -2.Ch 5 · Continuity and DifferentiabilityI
identity functionतत्समक फलनThe function IX: X → X with IX(x) = x for every x.Ch 1 · Relations and Functionsidentity matrixतत्समक आव्यूहThe square matrix I with 1 on the diagonal and 0 elsewhere; AI = IA = A.Ch 3 · Matricesimplicit functionअस्पष्ट फलनA function of x fixed by a relation between x and y that is not written in the form y = f(x).Ch 5 · Continuity and Differentiabilityinconsistent systemअसंगत निकायA system of equations that has no solution.Ch 4 · Determinantsincreasing functionवर्धमान फलनA function with f(x1) ≤ f(x2) whenever x1 < x2 in the interval; strictly increasing if f(x1) < f(x2).Ch 6 · Application of Derivativesinverse functionप्रतिलोम फलनFor a bijection f: X → Y, the function f⁻¹: Y → X that sends each f(x) back to x.Ch 2 · Inverse Trigonometric Functionsinverse of a matrixव्युत्क्रमThe matrix A⁻¹ with AA⁻¹ = A⁻¹A = I. When |A| ≠ 0 it equals 1 |A| adj A.Ch 4 · Determinantsinverse trigonometric functionप्रतिलोम त्रिकोणमितीय फलनThe inverse of a trigonometric function restricted to one branch; for example sin⁻¹x is the angle in [- π 2, π 2] whose sine is x.Ch 2 · Inverse Trigonometric Functionsinvertible functionव्युत्क्रमणीय फलनA function f: X → Y with a function f⁻¹: Y → X such that f⁻¹ ° f = IX and f ° f⁻¹ = IY.Ch 1 · Relations and Functionsinvertible matrixव्युत्क्रमणीय आव्यूहA square matrix A with some B such that AB = BA = I; then B = A⁻¹.Ch 3 · MatricesL
local maximumस्थानीय उच्चिष्ठAn interior point c with f(c) ≥ f(x) for every x in some open interval around c; f(c) is the local maximum value.Ch 6 · Application of Derivativeslocal minimumस्थानीय निम्निष्ठAn interior point c with f(c) ≤ f(x) for every x in some open interval around c; f(c) is the local minimum value.Ch 6 · Application of Derivativeslogarithmलघुगणकlogb a = x means b x = a. In this chapter log x is the natural logarithm, base e.Ch 5 · Continuity and Differentiabilitylogarithmic differentiationलघुगणकीय अवकलनDifferentiating log y instead of y, then multiplying by y; used for [u(x)] v(x) and long products.Ch 5 · Continuity and DifferentiabilityM
many-one functionबहुएक फलनA function that sends two or more different inputs to the same output, so it is not one-one.Ch 2 · Inverse Trigonometric Functionsmarginal costसीमांत लागतHow fast total cost grows as output grows, C'(x); roughly the extra cost of making one more unit.Ch 6 · Application of Derivativesmarginal revenueसीमांत आयHow fast total revenue grows as more units are sold, R'(x); roughly the money one more sale brings in.Ch 6 · Application of Derivativesmatrixआव्यूहA rectangle of numbers or functions, set out in rows and columns and handled as one object.Ch 3 · Matricesmatrix methodआव्यूह विधिSolving a system written as AX = B by computing X = A⁻¹B, which works when |A| ≠ 0.Ch 4 · Determinantsmatrix productआव्यूहों का गुणनफलFor A of order m × n and B of order n × p, the m × p matrix whose (i, k) entry is row i of A times column k of B.Ch 3 · MatricesminorउपसारणिकThe minor Mij of aij is the determinant left after deleting row i and column j.Ch 4 · Determinantsmonotonic functionएकदिष्ट फलनA function that is increasing throughout an interval or decreasing throughout it.Ch 6 · Application of DerivativesN
non-singular matrixव्युत्क्रमणीय आव्यूहA square matrix whose determinant is not 0. It always has an inverse.Ch 4 · DeterminantsO
one-one and onto functionएकैकी आच्छादक फलनA function that is both one-one and onto; exactly these functions have inverses.Ch 2 · Inverse Trigonometric Functionsone-one functionएकैकी फलनA function with f(x1) = f(x2) ⇒ x1 = x2: different inputs give different outputs.Ch 1 · Relations and Functionsone-sided limitएकपक्षीय सीमाThe value f(x) approaches as x → c from the left only (x < c) or from the right only (x > c).Ch 5 · Continuity and Differentiabilityonto functionआच्छादक फलनA function whose range equals its codomain: every element of the codomain is an image.Ch 1 · Relations and FunctionsorderकोटिThe size m × n of a matrix: m rows and n columns.Ch 3 · MatricesP
parameterप्राचलA third variable t through which x = f(t) and y = g(t) are both given.Ch 5 · Continuity and Differentiabilitypoint of discontinuityअसांतत्य का बिंदुA point of the domain at which the function is not continuous.Ch 5 · Continuity and Differentiabilitypoint of inflectionनति परिवर्तन बिंदुHere, a critical point where f' keeps the same sign on both sides, so the graph flattens for a moment but is neither a peak nor a dip.Ch 6 · Application of Derivativesprincipal valueमुख्य मानThe angle an inverse trigonometric function returns from its principal value branch, such as π 6 for sin⁻¹ 12.Ch 2 · Inverse Trigonometric Functionsprincipal value branchमुख्य मान शाखाThe branch fixed by convention as the range of an inverse trigonometric function, such as [0, π] for cos⁻¹.Ch 2 · Inverse Trigonometric FunctionsR
rate of changeपरिवर्तन की दरHow fast one quantity changes compared with another; for y = f(x) it is dy dx, measured in units of y per unit of x.Ch 6 · Application of Derivativesreflexive relationस्वतुल्य संबंधA relation with (a, a) ∈ R for every a in the set.Ch 1 · Relations and FunctionsrelationसंबंधA subset of A × B; a relation in A is a subset of A × A.Ch 1 · Relations and Functionsrestricted domainप्रतिबंधित प्रांतA part of a function's natural domain, chosen so that the function becomes one-one and onto its range.Ch 2 · Inverse Trigonometric FunctionsS
scalar matrixअदिश आव्यूहA diagonal matrix whose diagonal entries all equal one number k.Ch 3 · Matricessecond order derivativeद्वितीय कोटि का अवकलजThe derivative of dy dx, written d²y dx² or y''.Ch 5 · Continuity and Differentiabilitysingular matrixअव्युत्क्रमणीय आव्यूहA square matrix whose determinant is 0. It has no inverse.Ch 4 · Determinantsskew symmetric matrixविषम सममित आव्यूहA square matrix with A' = -A; its diagonal entries are 0.Ch 3 · Matricessquare matrixवर्ग आव्यूहA matrix with as many rows as columns.Ch 3 · Matricessymmetric matrixसममित आव्यूहA square matrix equal to its transpose: A' = A.Ch 3 · Matricessymmetric relationसममित संबंधA relation in which (a, b) ∈ R always implies (b, a) ∈ R.Ch 1 · Relations and FunctionsT
transitive relationसंक्रामक संबंधA relation in which (a, b) ∈ R and (b, c) ∈ R always imply (a, c) ∈ R.Ch 1 · Relations and Functionstransposeपरिवर्तThe matrix A' (or A T) whose rows are the columns of A.Ch 3 · MatricesU
universal relationसार्वत्रिक संबंधThe relation A × A, in which every element is related to every element.Ch 1 · Relations and FunctionsZ
zero matrixशून्य आव्यूहA matrix, written O, with every entry 0.Ch 3 · Matricesncert.ai is an independent learning companion aligned to the NCERT curriculum. It is not affiliated with or endorsed by NCERT.