Class 12 · Mathematics · English medium · Part 2
Class 12 Mathematics Part 2 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 7: Integrals
Undoing the derivative, and measuring area - ENHinglish
Chapter 8: Application of Integrals
Areas under curves, one thin strip at a time - ENHinglish
Chapter 9: Differential Equations
Equations about rates of change, and the functions that satisfy them - ENHinglish
Chapter 10: Vector Algebra
Arrows you can add, stretch and multiply - ENHinglish
Chapter 11: Three Dimensional Geometry
Lines in space: which way they point, how to write them, and how far apart they pass - ENHinglish
Chapter 12: Linear Programming
The best plan inside straight-line limits - ENHinglish
Chapter 13: Probability
Updating chances when you learn something new
Also in Hinglish
- Chapter 7: Integrals (Hinglish)
- Chapter 8: Application of Integrals (Hinglish)
- Chapter 9: Differential Equations (Hinglish)
- Chapter 10: Vector Algebra (Hinglish)
- Chapter 11: Three Dimensional Geometry (Hinglish)
- Chapter 12: Linear Programming (Hinglish)
- Chapter 13: Probability (Hinglish)
A
antiderivativeप्रतिअवकलजA function F whose derivative is f on an interval; also called a primitive of f.Ch 7 · Integralsarbitrary constantस्वेच्छ अचरA constant, such as C, that may take any value in a general solution.Ch 9 · Differential Equationsarea functionक्षेत्रफल फलनA(x) = ∫a x f(t) dt, the area under the graph from a fixed a up to a moving x.Ch 7 · Integralsarea under a curveवक्र के अंतर्गत क्षेत्रफलThe area of the region between a curve, the x -axis and two ordinates; it equals ∫a b f(x) dx when f ≥ 0.Ch 8 · Application of IntegralsB
Bayes' theoremबेज़-प्रमेयThe rule that the posterior P(Ei mid A) equals P(Ei) P(A mid Ei) divided by P(A): the chance of a cause once its effect is seen.Ch 13 · Probabilitybounded regionपरिबद्ध क्षेत्रA feasible region that fits inside some circle; one that runs on without end is unbounded.Ch 12 · Linear ProgrammingC
Cartesian equation of a lineरेखा का कार्तीय समीकरणThe line written in coordinates, x - x1 a = y - y1 b = z - z1 c, through (x1, y1, z1) with direction ratios a, b, c.Ch 11 · Three Dimensional Geometrycoinitial vectorsसह-आदिम सदिशVectors with the same initial point.Ch 10 · Vector AlgebracollinearसंरेखLying on one straight line.Ch 11 · Three Dimensional Geometrycollinear vectorsसंरेख सदिशVectors parallel to the same line; one is a scalar multiple of the other.Ch 10 · Vector Algebracommon perpendicularउभयनिष्ठ लंबThe one segment that meets two skew lines and is perpendicular to both; its length is their shortest distance.Ch 11 · Three Dimensional Geometrycompleting the squareपूर्ण वर्ग बनानाRewriting x² + bx + c as (x + b 2)² + c - b² 4, a square plus a constant.Ch 7 · Integralsconditional probabilityसप्रतिबंध प्रायिकताThe probability of E once we know F has happened: P(E mid F) = P(E ∩ F)/P(F), with P(F) ≠ 0.Ch 13 · Probabilityconstant of integrationसमाकलन अचरThe arbitrary real number C added to an antiderivative; one extra condition fixes it.Ch 7 · Integralsconstraintव्यवरोधA linear inequality or equation that limits the choices of the decision variables in an LPP.Ch 12 · Linear Programmingconvex regionउत्तल क्षेत्रA region that contains the whole segment joining any two of its points.Ch 12 · Linear ProgrammingcoplanarसमतलीयLying in one plane. Two lines are coplanar exactly when they meet or are parallel.Ch 11 · Three Dimensional Geometrycorner pointकोणीय बिंदुA point of the feasible region where two of its boundary lines meet.Ch 12 · Linear ProgrammingD
decision variablesनिर्णायक चरThe unknowns we choose in an LPP, such as the numbers x and y of two products to make.Ch 12 · Linear Programmingdefinite integralनिश्चित समाकलनThe number ∫a b f(x) dx: the signed area between y = f(x) and the x -axis from x = a to x = b.Ch 7 · Integralsdefinite integralनिश्चित समाकलनThe limit of sums of thin rectangles, ∫a b f(x) dx, evaluated as F(b) - F(a) with an antiderivative F.Ch 8 · Application of IntegralsdegreeघातThe exponent on the highest derivative, once the equation is a polynomial in its derivatives; otherwise not defined.Ch 9 · Differential Equationsdifferential equationअवकल समीकरणAn equation involving derivatives of an unknown function.Ch 9 · Differential Equationsdirected lineदिष्ट रेखाA line together with a chosen one of its two directions, shown by an arrow.Ch 11 · Three Dimensional Geometrydirection anglesदिक्-कोणThe angles α, β, γ (each from 0 ° to 180 °) that a directed line makes with the positive x -, y - and z -axes.Ch 11 · Three Dimensional Geometrydirection cosinesदिक्-कोसाइनThe cosines l, m, n of the angles a vector makes with the positive axes; l² + m² + n² = 1.Ch 10 · Vector Algebradirection cosinesदिक्-कोसाइनThe cosines l, m, n of the direction angles; they satisfy l² + m² + n² = 1.Ch 11 · Three Dimensional Geometrydirection ratiosदिक्-अनुपातAny three numbers proportional to the direction cosines of a vector.Ch 10 · Vector Algebradirection ratiosदिक्-अनुपातAny three numbers, not all zero, in the same ratio as the direction cosines of a line.Ch 11 · Three Dimensional GeometryE
elementary areaप्रारंभिक क्षेत्रफलThe area of one thin strip of a region: y dx for a vertical strip, x dy for a horizontal one.Ch 8 · Application of Integralsellipseदीर्घवृत्तThe curve x² a² + y² b² = 1 in standard form, a circle stretched or squashed along one axis; it encloses π ab.Ch 8 · Application of Integralsequal vectorsसमान सदिशVectors with the same magnitude and direction, wherever they start.Ch 10 · Vector Algebraeven and odd functionsसम और विषम फलनf is even if f(-x) = f(x) and odd if f(-x) = -f(x) for every x in its domain.Ch 7 · IntegralsF
feasible regionसुसंगत क्षेत्रThe set of points satisfying all the constraints of an LPP, non-negativity included.Ch 12 · Linear Programmingfeasible solutionसुसंगत हलAny point of the feasible region, inside it or on its boundary.Ch 12 · Linear Programmingfundamental theorem of calculusकलन की आधारभूत प्रमेयThe link between the two halves of calculus: A'(x) = f(x), and hence ∫a b f(x) dx = F(b) - F(a).Ch 7 · IntegralsG
general solutionव्यापक हलA solution with free constants, one for each integration: n of them when the equation has order n.Ch 9 · Differential EquationsH
half-planeअर्धतलThe part of the plane on one side of a straight line; a linear inequality in x and y describes one.Ch 12 · Linear Programminghomogeneous differential equationसमघातीय अवकल समीकरणdy dx = F(x, y) with F homogeneous of degree zero.Ch 9 · Differential Equationshomogeneous functionसमघातीय फलनF with F(λ x, λ y) = λⁿF(x, y) for every non-zero λ; n is its degree.Ch 9 · Differential Equationshypothesisपरिकल्पनाIn Bayes' theorem, one of the events Ei of the partition: a possible cause of what was observed.Ch 13 · ProbabilityI
indefinite integralअनिश्चित समाकलनThe family ∫ f(x) dx = F(x) + C of all antiderivatives of f on an interval.Ch 7 · Integralsindependent eventsस्वतंत्र घटनाएँEvents with P(E ∩ F) = P(E) P(F), so knowing one does not change the chance of the other. Otherwise they are dependent.Ch 13 · Probabilityintegrandसमाकल्यThe function f(x) that stands inside ∫ f(x) dx.Ch 7 · Integralsintegrating factorसमाकलन गुणकe ∫ P dx, which makes the left side of a linear equation an exact derivative.Ch 9 · Differential EquationsintegrationसमाकलनThe process of finding an integral; for indefinite integrals, finding antiderivatives.Ch 7 · Integralsintegration by partsखंडशः समाकलनThe rule ∫ u v' dx = uv - ∫ u' v dx, the product rule read backwards.Ch 7 · Integralsintegration by substitutionप्रतिस्थापन द्वारा समाकलनChanging the variable, u = g(x), so that ∫ f(g(x))g'(x) dx becomes ∫ f(u) du.Ch 7 · Integralsiso-profit lineA line ax + by = k along which the objective has the constant value k; such lines are parallel.Ch 12 · Linear ProgrammingL
limits of integrationसमाकलन की सीमाएँThe end points a (lower) and b (upper) in ∫a b f(x) dx.Ch 7 · Integralslinear differential equationरैखिक अवकल समीकरणA first order equation in which y and y' appear only to the first power, so it reads y' + Py = Q with P, Q depending on x alone.Ch 9 · Differential Equationslinear programming problemरैखिक प्रोग्रामन समस्याAn optimisation problem whose objective function and constraints are all linear, with non-negative decision variables.Ch 12 · Linear ProgrammingM
magnitudeपरिमाणThe length of a vector, written | a |; never negative.Ch 10 · Vector Algebramethod of exhaustionनिःशेषण विधिThe Greek way of finding curved areas by filling a region with ever smaller polygons until the leftover is negligible.Ch 8 · Application of Integralsmultiplication ruleप्रायिकता का गुणन नियमP(E ∩ F) = P(E) P(F mid E): the chance of both is the chance of the first times the chance of the second given the first.Ch 13 · Probabilitymutually exclusive eventsपरस्पर अपवर्जी घटनाएँEvents with no outcome in common, E ∩ F = varnothing; they cannot happen together.Ch 13 · ProbabilityN
non-negativity restrictionsऋणेतर व्यवरोधThe constraints x ≥ 0, y ≥ 0.Ch 12 · Linear ProgrammingO
objective functionउद्देश्य फलनThe linear function Z = ax + by that is to be maximised or minimised.Ch 12 · Linear Programmingoptimal solutionइष्टतम हलA feasible solution at which the objective function takes its maximum or minimum value.Ch 12 · Linear Programmingoptimisation problemइष्टतमीकरण समस्याA problem that asks for the largest or smallest value of a quantity, such as profit or cost, subject to limits.Ch 12 · Linear ProgrammingorderकोटिThe highest number of times the unknown function is differentiated anywhere in the equation.Ch 9 · Differential Equationsordinary differential equationसाधारण अवकल समीकरणA differential equation with only one independent variable.Ch 9 · Differential EquationsordinateकोटिHere, a vertical line such as x = a that closes off a region under a curve at one side.Ch 8 · Application of IntegralsP
parabolaपरवलयA curve such as y = x² or y² = 4ax; Archimedes showed a parabolic region fills 2 3 of its bounding box.Ch 8 · Application of Integralsparameterप्राचलThe real number λ in r = a + λ b; each value picks out one point of the line.Ch 11 · Three Dimensional Geometrypartial fractionsआंशिक भिन्नSimpler fractions whose sum equals a given proper rational function.Ch 7 · Integralsparticular solutionविशिष्ट हलA solution free of arbitrary constants, fixed by given conditions.Ch 9 · Differential Equationspartition of a sample spaceप्रतिदर्श समष्टि का विभाजनEvents E1, …, En that are pairwise disjoint, together make up S, and each have non-zero probability.Ch 13 · Probabilityposition vectorस्थिति सदिशThe vector OP from the origin to a point P.Ch 10 · Vector Algebraposterior probabilityP(Ei mid A), the probability of a hypothesis after the evidence A is taken into account.Ch 13 · Probabilityprior probabilityP(Ei), the probability of a hypothesis before the evidence is taken into account.Ch 13 · Probabilityprojectionप्रक्षेपThe signed length | a | cos θ of the shadow of a on a directed line.Ch 10 · Vector AlgebraQ
quadrantचतुर्थांशOne of the four regions into which the axes divide the plane; in the first quadrant x ≥ 0 and y ≥ 0.Ch 8 · Application of IntegralsR
random variableयादृच्छिक चरA rule that gives a real number for each outcome of a random experiment: a function from S to the real numbers.Ch 13 · Probabilityrational functionपरिमेय फलनA quotient P(x) Q(x) of polynomials; it is proper when deg P < deg Q.Ch 7 · IntegralsresultantपरिणामीThe sum of two or more vectors.Ch 10 · Vector AlgebraS
scalarअदिशA quantity described completely by one real number, such as mass or temperature.Ch 10 · Vector Algebrascalar componentsअदिश घटकThe numbers x, y, z in x i + y j + z k.Ch 10 · Vector Algebrascalar productअदिश गुणनफलa · b = | a || b | cos θ, a number.Ch 10 · Vector Algebrasemi-axisअर्ध-अक्षHalf of an axis of an ellipse: the distances a and b from the centre to the curve along the two axes.Ch 8 · Application of Integralsshortest distanceन्यूनतम दूरीThe length of the shortest segment joining a point of one line to a point of the other.Ch 11 · Three Dimensional Geometryskew linesविषमतलीय रेखाएँA pair of lines in space that never meet and are not parallel; no plane holds both.Ch 11 · Three Dimensional GeometrysolutionहलA function that turns the differential equation into an identity when substituted.Ch 9 · Differential Equationssolution curveहल वक्रThe graph of a solution; it follows the slope the equation demands at each point.Ch 9 · Differential EquationssymmetryसममितिA curve is symmetric about an axis when reflecting it in that axis gives the same curve; then the two sides have equal areas.Ch 8 · Application of IntegralsT
theorem of total probabilityसंपूर्ण प्रायिकता प्रमेयFor a partition E1, …, En: P(A) = ∑j P(Ej) P(A mid Ej).Ch 13 · Probabilitytree diagramवृक्ष आरेखA branching picture of a staged experiment; multiplying along a path gives the probability of that path.Ch 13 · ProbabilityU
unit vectorमात्रक सदिशA vector of magnitude 1; a = a/| a |.Ch 10 · Vector AlgebraV
variables separableचर पृथक्करणीयOf the form dy dx = g(x) h(y), so each variable can be moved to its own side.Ch 9 · Differential EquationsvectorसदिशA quantity with both magnitude and direction, drawn as a directed line segment.Ch 10 · Vector Algebravector productसदिश गुणनफलa × b = | a || b | sin θ n, a vector perpendicular to both.Ch 10 · Vector AlgebraZ
zero vectorशून्य सदिशThe vector 0 of length 0, whose initial and terminal points coincide.Ch 10 · Vector Algebrancert.ai is an independent learning companion aligned to the NCERT curriculum. It is not affiliated with or endorsed by NCERT.