ISC Class 12 Mathematics Syllabus 2026-27
CISCE has released the ISC Class 12 Mathematics Syllabus for the 2026–27 academic session. The syllabus lists all chapters and topics prescribed for the ISC board examination.
Students and teachers should review the latest ISC Mathematics syllabus for Class 12 before starting exam preparation. The syllabus explains the chapter structure, topic coverage, and exam scope defined by the CISCE board.
The ISC 12th Maths Reduced syllabus helps students plan their study schedule and focus on important concepts. A clear understanding of the syllabus improves exam preparation and strengthens subject knowledge.
Students can download the ISC 12th Mathematics Syllabus PDF from the link below and review all topics prescribed by CISCE.
Oswal Publishers has also updated its books according to the latest ISC board paper pattern and syllabus guidelines. These books support effective preparation for the ISC board examination.
Download the ISC Class 12 Mathematics Syllabus 2026–27 PDF using the link below.
ISC Mathematics Class 12 Latest Syllabus 2026-27
There will be two papers in the subject:
Paper I : Theory (3 hours) ……80 marks
Paper II: Project Work ……20 marks
Paper I (Theory)
80 Marks
Section A
1. Relations and Functions
(i) Types of relations: reflexive, symmetric, transitive and equivalence relations. One to one and onto functions, inverse of a function.
- Relations as:
- Relation on a set A
- Identity relation, empty relation, universal relation.
- Types of Relations: reflexive, symmetric, transitive and equivalence relation.
- Functions:
- As special relations, concept of writing “y is a function of x” as y = f(x).
- Types: one to one, many to one, into, onto.
- Real Valued function.
- Domain and range of a function.
- Conditions of invertibility.
- and invertible functions (algebraic functions only).
(ii) Inverse Trigonometric Functions
Definition, domain, range, principal value branch. Elementary properties of inverse trigonometric functions.
- Principal values.
- sin-1x, cos-1 x, tan-1 x etc.
$$-\space \text{sin}^{-1} x = \text{cos}^{-1}\sqrt{1 - x^{2}}\\=\text{tan}^{\normalsize-1}\frac{x}{\sqrt{1 - x^{2}}}.\\-\space \text{sin}^{-1}x = \text{cosec}^{\normalsize-1}\frac{1}{x};\\\text{sin}^{\normalsize-1}x + \text{cos}^{\normalsize-1} x =\frac{\pi}{2}$$ and similar relations for cot-1x, tan-1 x, etc.
$$\text{sin}^{\normalsize-1}x\pm\text{sin}^{\normalsize-1}y\\ =\text{sin}^{\normalsize-1}\bigg(x\sqrt{1 - y^{2}}\pm y\sqrt{1-x^{2}}\bigg)\\\text{cos}^{-1}x\pm\text{cos}^{-1}y =\\\text{cos}^{\normalsize-1}\bigg(xy\mp\sqrt{1 - y^{2}}\sqrt{1 - x^{2}}\bigg)\\\text{similarly}\space\text{tan}^{-1}x + \text{tan}^{-1}y=\\\text{tan}^{\normalsize-1}\frac{x+y}{1 - xy},\\xy\lt 1\\\text{tan}^{-1}x - \text{tan}^{-1}y\\= \text{tan}^{\normalsize-1}\frac{x-y}{1 + xy},\\ xy\gt-1$$
- Formulae for 2sin-1 x, 2cos-1 x, 2tan-1 x, 3tan-1 x etc. and application of these formulae.
2. Algebra
Matrices and Determinants
(i) Matrices
Concept, notation, order, equality, types of matrices, zero and identity matrix, transpose of a matrix, symmetric and skew symmetric matrices. Operation on matrices: Addition and multiplication and multiplication with a scalar. Simple properties of addition, multiplication and scalar multiplication. Noncommutativity of multiplication of matrices and existence of non-zero matrices whose product is the zero matrix (restrict to square matrices of order upto 3). Concept of elementary row and column operations. Invertible matrices and proof of the uniqueness of inverse, if it exists (here all matrices will have real entries).
(ii) Determinants
Determinant of a square matrix (up to 3 x 3 matrices), properties of determinants, minors, co-factors and applications of determinants in finding the area of a triangle. Adjoint and inverse of a square matrix. Consistency, inconsistency and number of solutions of system of linear equations by examples, solving system of linear equations in two or three variables (having unique solution) using inverse of a matrix.
-Types of matrices (m × n; m, n ≤ 3), order; Identity matrix, Diagonal matrix.
- Symmetric, Skew symmetric.
-Operation – addition, subtraction, multiplication of a matrix with scalar, multiplication of two matrices
(the compatibility).
$$\text{E.g.}\begin{bmatrix}1 & 1 \\ 0 & 2 \\1 & 1\end{bmatrix}\begin{bmatrix}1 & 2 \\2 & 2\end{bmatrix} =\text{AB}(say)$$
but BA is not possible.
- Singular and non-singular matrices.
- Existence of two non-zero matrices whose product is a zero matrix.
- Inverse (2×2, 3×3)
$$\text{A}^{\normalsize-1}\frac{\text{Adj A}}{|\text{A}|}$$
- Martin’s Rule (i.e. using matrices)
a1x + b1y + c1z = d1
a2x + b2y + c2z = d2
a3x + b3y + c3z = d3
$$\text{A} = \begin{bmatrix}a_{1} &b_{1} &c_{1}\\a_{2} &b_{2} &c_{2}\\a_{3} &b_{3} &c_{3}\end{bmatrix}\space\text{B} = \begin{bmatrix}d_{1}\\d_{2} \\d_{3}\end{bmatrix}\\\text{X} = \begin{bmatrix}x\\y\\z\end{bmatrix}\\\text{AX} = B\\\Rarr \text{X} =\text{A}^{\normalsize-1}\text{B}$$
Problems based on above.
NOTE 1: The conditions for consistency of equations in two and three variables, using matrices, are to be covered.
- Determinants
- Order.
- Minors.
- Cofactors.
- Expansion.
- Applications of determinants in finding the area of triangle and collinearity.
- Properties of determinants. Problems based on properties of determinants.
3. Calculus
(i) Continuity, Differentiability and Differentiation. Continuity and differentiability, derivative of composite functions, chain rule, derivatives of inverse trigonometric functions, derivative of implicit functions. Concept of exponential and logarithmic functions.
Derivatives of logarithmic and exponential functions. Logarithmic differentiation, derivative of functions expressed in parametric forms. Second order derivatives.
- Continuity
- Continuity of a function at a point x = a.
- Continuity of a function in an interval.
- Algebra of continues function.
- Removable discontinuity
- Differentiation
-Concept of continuity and differentiability of x , [x], etc.
- Derivatives of trigonometric functions.
- Derivatives of exponential functions.
- Derivatives of logarithmic functions.
- Derivatives of inverse trigonometric functions - differentiation by means of substitution.
- Derivatives of implicit functions and chain rule.
- Derivatives of Parametric functions.
- Differentiation of a function with respect to another function e.g. differentiation of sinx3 with respect to x3.
- Logarithmic Differentiation - Finding dy/dx y = x x x .
- Successive differentiation up to 2nd order.
NOTE 1: Derivatives of composite functions using chain rule.
(ii) Applications of Derivatives
Applications of derivatives: rate of change of bodies, increasing/decreasing functions, tangents and normals, maxima and minima (first derivative test motivated geometrically and second derivative test given as a provable tool). Simple problems (that illustrate basic principles and understanding of the subject as well as real-life situations).
- Equation of Tangent and Normal
- Rate measure
- Increasing and decreasing functions.
- Maxima and minima.
-Stationary/turning points.
- Absolute maxima/minima
- local maxima/minima
- First derivatives test and second derivatives test
- Application problems based on maxima and minima.
(iii) Integrals
Integration as inverse process of differentiation. Integration of a variety of functions by substitution, by partial fractions and by parts, Evaluation of simple integrals of the following types and problems based on them.
Fundamental Theorem of Calculus (without proof). Basic properties of definite integrals and evaluation of definiteintegrals.
- Indefinite integral
- Integration as the inverse of differentiation.
- Anti-derivatives of polynomials and functions (ax +b)n , sinx, cosx, sec2 x,cosec2x etc .
- Integrals of the type sin2x, sin3x, sin4 x, cos2 x, cos3x, cos4x.
- Integration of 1/x, ex.
- Integration by substitution.
- Integrals of the type f ' (x)[f (x)]n,
$$\frac{f'(x)}{f(x)}.$$
- Integration of tanx, cotx, secx, cosecx.
- Integration by parts.
- Integration using partial fractions.
Expressions of the form
$$\frac{f(x)}{g(x)}\space\text{when degree of}\\\text{ f(x) < degree of g(x)}$$
$$\text{E.g.}\space\frac{x+2}{(x-3)(x+1)} =\\\frac{\text{A}}{\text{x-3}} +\frac{\text{B}}{x+1}\\\frac{x + 2}{(x-2)(x-1)^{2}} =\\\frac{\text{A}}{x-1} +\frac{\text{B}}{(x-1)^{2}} +\frac{\text{C}}{x-2}\\\frac{x+1}{(x^{2} + 3)(x-1)} \\=\frac{Ax + B}{x^{2} +3} + \frac{\text{C}}{x-1}$$
When degree of f (x) ≥ degree of g(x),
$$\text{e.g.}\frac{x^{2} +1}{x^{2} + 3x+ 2}\\= 1 - \bigg(\frac{3x+1}{x^{2} + 3x +2}\bigg)$$
- Integrals of the type:
$$\int\frac{dx}{x^{2} + a^{2}},\int\frac{dx}{\sqrt{x^{2}\pm a^{2}}},\\\int\frac{px+q}{ax^{2} +bx +c}dx,\\\int\frac{px +q}{\sqrt{ax^{2} + bx +c}}dx\\\text{and}\space\int\sqrt{a^{2}\pm x^{2}}dx,\\\int\sqrt{x^{2} - a^{2}}dx,$$
$$\int\sqrt{ax^{2} + bx +c}\space dx,\\\int(px+q)\sqrt{ax^{2} + bx + c}\space dx,$$
integrations reducible to the above forms.
$$\int\frac{dx}{a cos x + bsin x},\\\int\frac{dx}{a + b cos x},\int\frac{dx}{a + b sin x}\\\int\frac{dx}{a cos x +b sin x + c},\\\int\frac{(a cos x + bsin x)dx}{c cos x + d sin x},\\\int\frac{dx}{a cos^{2}x + b sin^{2}x + c}\\\int\frac{1\pm x^{2}}{1 + x^{4}}dx,\\\int\frac{dx}{1 + x^{4}},\int\sqrt{tan x}\space dx,\\\int\sqrt{cot x }dx etc$$
- Definite Integral
-Definite integral as a limit of the sum.
- Fundamental theorem of calculus (without proof)
- Properties of definite integrals.
- Problems based on the following properties of definite integrals are to be covered.
$$\int^{b}_{a}f(x)dx = \int^{b}_{a}f(t)dt\\\int^{b}_{a}\text{f(x)}dx =-\int^{a}_{b}\text{f(x)}dx\\\int^{b}_{a}\text{f(x)dx} = \int^{c}_{a}\text{f(x)dx} +\\ \int^{b}_{c}f(x)dx\\\text{where a < c < b}\\\int^{b}_{a}\text{f(x)dx} = \int^{b}_{a}f(a + b-x)dx\\\int^{a}_{0}\text{f(x)dx} = \int^{a}_{0}f(a-x)dx$$
$$\int^{2a}_{0}\text{f(x)dx} = \\\begin{cases}2\int^{a}_{0}\text{f(x)}dx, \text{if}\space f(2a- x) = f(x)\\0, f(2a-x) = -f(x)\end{cases}$$
$$\int^{a}_{-a}\text{f(x)dx} =\\\begin{cases} 2\int^{a}_{0}f(x)dx,\text{if}\space f(2a-x) = f(x)\\0,\space \text{if f is an odd function}\end{cases}$$
(iv) Differential Equations
Definition, order and degree, general and particular solutions of a differential equation. Solution of differential equations by method of separation of variables solutions of homogeneous differential equations of first order and first degree. Solutions of linear differential equation of the type:
$$\frac{dy}{dx} +py = q,$$
where p and q are functions of x or constants.
$$\frac{dx}{dy} +px = q,$$
functions of y or constants.
- Differential equations, order and degree.
- Formation of differential equation by eliminating arbitrary constant(s).
- Solution of differential equations.
- Variable separable.
- Homogeneous equations.
$$-\text{Linear form}\space \frac{dy}{dx} + \text{Py} = Q$$
where P and Q are functions of x only. Similarly, for dx/dy.
NOTE 1: Equations reducible to variable separable type are included.
NOTE 2: The second order differential equations are excluded.
4. Probability
Conditional probability, multiplication theorem on probability, independent events, total probability, Bayes’ theorem, Random variable and its probability distribution, mean of random variable.
- Independent and dependent events conditional events.
- Laws of Probability, addition theorem, multiplication theorem, conditional probability.
- Theorem of Total Probability.
- Baye’s theorem.
- Theoretical probability distribution, probability distribution function; mean and of random variable.
Section B
5. Vectors
Vectors and scalars, magnitude and direction of a vector. Direction cosines and direction ratios of a vector. Types of vectors (equal, unit, zero, parallel and collinear vectors), position vector of a point, negative of a vector, components of a vector, addition of vectors, multiplication of a vector by a scalar, position vector of a point dividing a line segment in a given ratio. Definition, Geometrical Interpretation, properties and application of scalar (dot) product of vectors, vector (cross) product of vectors.
- As directed line segments.
- Magnitude and direction of a vector.
- Types: equal vectors, unit vectors, zero vector.
- Position vector.
- Components of a vector.
- Vectors in two and three dimensions.
- $$-\space \hat{i},\hat{j},\hat{k}$$ as unit vectors along the x, y and the z axes; expressing a vector in terms of the unit vectors.
- Operations: Sum and Difference of vectors; scalar multiplication of a vector.
- Section formula.
- Scalar (dot) product of vectors and its geometrical significance.
- Cross product - its properties - area of a triangle, area of parallelogram, collinear vectors.
NOTE: Proofs of geometrical theorems by using Vector algebra are excluded.
6. Three - dimensional Geometry
Direction cosines and direction ratios of a line joining two points. Cartesian equation and vector equation of a line, coplanar and skew lines, shortest distance between two lines. Cartesian and vector equation of a plane. Angle between (i) two lines, (ii) two planes, (iii) a line and a plane. Distance of a point from a plane.
Equation of x-axis, y-axis, z axis and lines parallel to them.
- Equation of xy - plane, yz – plane, zx – plane.
- Direction cosines, direction ratios.
- Angle between two lines in terms of direction cosines /direction ratios.
- Condition for lines to be perpendicular/ parallel.
- Lines
- Cartesian and vector equations of a line through one and two points.
- Coplanar and skew lines.
- Conditions for intersection of two lines.
- Distance of a point from a line.
- Shortest distance between two lines.
- Planes
- Cartesian and vector equation of a plane.
- Direction ratios of the normal to the plane.
- One point form.
- Normal form.
- Intercept form.
- Distance of a point from a plane.
- Intersection of the line and plane.
- Angle between two planes, a line and a plane.
7. Linear Programming
Introduction, related terminology such as constraints, objective function, optimization, different types of linear programming (L.P.) problems, mathematical formulation of L.P. problems, graphical method of solution for problems in two variables, feasible and infeasible regions (bounded and unbounded), feasible and infeasible solutions, optimal feasible solutions (up to three non-trivialconstraints).
Introduction, definition of related terminology such as constraints, objective function, optimization, advantages of linear programming; limitations of linear programming; application areas of linear programming; different types of linear programming (L.P.) problems, mathematical formulation of L.P problems, graphical method of solution for problems in two variables, feasible and infeasible regions, feasible and infeasible solutions, optimum feasible solution.
