ISC Class 12th Maths Syllabus 2023: Maths (code: 860) is an elective subject in ISC Class 12 and is one of the most preferred choices of courses by students. Maths is a challenging but important subject that plays a huge role even after school. Most competitive examinations nowadays feature a mathematics section. Maths is especially important for the commerce and science stream due to its application in subjects like physics and accountancy. The ISC class 12 Maths syllabus covers various advanced topics like algebra, calculus and vectors along with project work to test students’ ability to put theoretical knowledge to practical use. Calculus makes up the majority of the ISC board Class 12 Maths syllabus. Read here and download the latest and revised ISC Class 12 Maths syllabus 2023 in pdf format here.
ISC Class 12th Datesheet 2023: Check the full date sheet with the guidelines here
ISC Board Class 12 Maths Syllabus
The ISC class 12 Maths subject is divided into two papers: theory and project work. The Paper 1: Theory carries 80 marks and the duration will be 3 hours. The Project Work will carry 20 marks. Check here the ISC Board Class 12 Maths Syllabus along with unit-wise marks distribution below. There are three sections in ISC class 12 exams out of which, Section A of 65 marks is compulsory and a choice will be given between Sections B and C, of 15 marks each.
DISTRIBUTION OF MARKS FOR THE THEORY PAPER
S.No. |
UNIT |
TOTAL WEIGHTAGE |
SECTION A: 65 MARKS |
||
1. |
Relations and Functions |
10 Marks |
2. |
Algebra |
10 Marks |
3. |
Calculus |
32 Marks |
4. |
Probability |
13 Marks |
SECTION B: 15 MARKS |
||
5. |
Vectors |
5 Marks |
6. |
Three - Dimensional Geometry |
6 Marks |
7. |
Applications of Integrals |
4 Marks |
OR SECTION C: 15 MARKS |
||
8. |
Application of Calculus |
5 Marks |
9. |
Linear Regression |
6 Marks |
10. |
Linear Programming |
4 Marks |
TOTAL |
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.
- Invertible functions (algebraic functions only).
(ii) Inverse Trigonometric Functions
Definition, domain, range, principal value branch. Elementary properties of inverse trigonometric functions.
- Principal values.
- sin^{-1}x, cos^{-1}x, tan^{-1}x etc. and their graphs.
- sin^{-1}x = cos^{-1}√1-x^{2 }= tan^{-1}x/√1-x^{2}
- sin^{-1}x= cosec^{-1}1/x sin^{-1}x+cos^{-1}x = π/2 and similar relations for cot^{-1}x, tan^{-1}x, etc.
- Formulae for 2sin^{-1}x, 2cos^{-1}x, 2tan^{-1}x, 3tan^{-1}x etc. and application of these formulae.
- 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. Non- commutativity of multiplication of matrices and existence of non-zero matrices whose product is the zero matrix (restrict to square matrices of order upto 3). 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. 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).
- Determinants
- Order.
- Minors.
- Cofactors.
- Expansion.
- Properties of determinants. Problems based on properties of determinants.
- 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. x
- 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.
- Derivatives of functions using 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 when y = x^{x…}
- Successive differentiation up to 2nd order.
(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 definite integrals.
- Indefinite integral
- Integration as the inverse of differentiation.
- Anti-derivatives of polynomials and functions (ax +b)^{n} , sinx, cosx, sec^{2}x, cosec^{2}x etc .
- Integrals of the type sin^{2}x, sin^{3}x, sin^{4}x, cos^{2}x, cos^{3}x, cos^{4}x.
- Integration of 1/x, e^{x}.
- Integration by substitution.
- Integrals of the type:
- Definite Integral
- Fundamental theorem of calculus (without proof)
- Properties of definite integrals.
- Problems based on the following properties of definite integrals are to be covered.
(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: +py= q, where p and q are functions of x or constants. + px = q, where p and q are functions of y or constants. dydxdxdy
- Differential equations, order and degree.
- Solution of differential equations.
- Variable separable.
- Homogeneous equations.
- Linear form dx/dy + 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 and variance 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 variance of random variable.
SECTION B
- 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.
- ^i, ^j, ^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.
- Three - dimensional Geometry
Direction cosines and direction ratios of a line joining two points. Cartesian equation and vector equation of a line. 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.
- Conditions for intersection of two lines.
- Distance of a point from a line.
- 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.
- Application of Integrals
Application in finding the area bounded by simple curves and coordinate axes. Area enclosed between two curves.
- Application of definite integrals - area bounded by curves, lines and coordinate axes is required to be covered.
- Simple curves: lines, circles/ parabolas/ ellipses (only standard forms), simple polynomial functions, modulus function.
SECTION C
- Application of Calculus
Application of Calculus in Commerce and Economics in the following:
- Cost function,
- average cost,
- marginal cost and its interpretation
- demand function,
- revenue function,
- marginal revenue function and its interpretation,
- Profit function and breakeven point.
- Rough sketching of the following curves: AR, MR, R, C, AC, MC and their mathematical interpretation using the concept of maxima & minima and increasing- decreasing functions.
Self-explanatory
NOTE: Application involving differentiation, increasing and decreasing function and maxima and minima to be covered.
- Linear Regression
- Lines of regression of x on y and y on x.
- Scatter diagrams
- The method of least squares.
- Lines of best fit.
- Regression coefficient of x on y and y on x.
- b_{xy} x b_{yx} = r2, 0≤ b_{xy} x b_{yx} ≤ 1
- Identification of regression equations
- Properties of regression lines.
- Estimation of the value of one variable using the value of other variable from appropriate line of regression.
Self-explanatory
- 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-trivial constraints).
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.
Note: Transportation problem is excluded.
Download and read the full ISC Class 12th Maths Syllabus 2022-23 below.
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The ISC class 12 final exams are on the horizon and the date sheet has also been released. Check the ISC Class 12 mock tests here to better prepare for the ISC class 12 Maths exams.
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