Author : Paakhi Jain

August 31, 2024

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Overview: With 3 sections and 10 units, the ISC Class 12 Maths syllabus contains several important topics relevant to entrance exams after 12th too. To score 80/80 on the theory paper in the class 12th maths exam, go through the detailed syllabus overview, the weightage of each topic, and the best books.

ISC board's Maths is a little tough compared to the CBSE board. A thorough understanding of the ISC Class 12 Maths Syllabus is essential:

• To create a well-structured study plan with time allotted to each section and revision.
• To solve relevant topic-wise and unit-wise sample papers and practice tests.

## ISC Class 12th Maths Syllabus (Unit-wise)

The Maths theory exam has a weightage of 80 marks, and 20 marks are allotted to Project Work.

The class 12 ISC Maths syllabus has three sections: A, B, and C.

• Section A is compulsory for all candidates. Section A (65 marks) will consist of six questions.
• You will be required to attempt all questions. The internal choice will be provided in two questions of two marks, two of four marks and two of six marks each.
• You can attempt questions from either Section B or Section C. Section B / C (15 marks).
• You have to attempt all questions EITHER from Section B or Section C.
• Internal choice will be provided in one question of two marks and one question of four marks.

The unit-wise marking scheme as per the updated ISC Class 12th Maths syllabus is listed below:

 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 SECTION C: 15 MARKS 8 Application of Calculus 5 Marks 9 Linear Regression 6 Marks 10 Linear Programming 4 Marks Total 80 Marks

## Detailed ISC Class 12 Maths Syllabus: Section A

### Unit 1. Relations and Functions

(i) Types of relations:

• Reflexive, symmetric, transitive, and equivalence relations.
• One-to-one and onto functions, composite functions, the 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: Conditions of invertibility.
• Composite functions 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-1x, tan-1x , etc. and their graphs.
• Sin-1x=cos-11 x;sin-1x+cos-1x=2 and similar relations for cot-1x, tan-1x, etc.
• sin-1x+sin-y=sin-1(x1-y2 -y1-x2)
• sin-1x-sin-y=sin-1(x1-y2 +y1-x2)
• cos-1x+cos-1y=cos-1(xy-1-y2 1-x2)
• cos-1x-cos-1y=cos-1(xy+1-y2 1-x2)
• Similarly, tan-1x+tan`-1y=tan-1x-y1+xy, xy>-1
• Formulae for 2sin-1x, 2cos-1x, 2tan-1x, 3tan-1x etc, and application of these formulae.

### Unit 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.
• Non Commutativity of multiplication of matrices and existence of non-zero matrices whose product is the zero matrices (restricted to square matrices of order up to 3).
• Invertible matrices and proof of the uniqueness of inverse, if it exists (Here, all matrices will have real entries).

(ii) Determinants

• The determinant of a square matrix (up to 3 x 3 matrices), properties of determinants, minors, co-factors.
• Adjoint and inverse of a square matrix.
• Solving a system of linear equations in two or three variables (having unique solutions) using the 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).
• Above =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

• Martin’s Rule (i.e. using matrices)

a1x+b1y+c1z=d1

a2x+b2y+c2z=d2

a3x+b3y+c3z=d3

Problems based on the above equations.

• Determinants, Order, Minors, Cofactors, Expansion
• Properties of determinants & problems based on the properties of determinants

### Unit 3. Calculus

(i) Differentiation

• The derivative of composite functions, chain rule, inverse trigonometric functions, and derivative of implicit functions.
• Concept of exponential and logarithmic functions.
• Derivatives of logarithmic and exponential functions.
• Logarithmic differentiation, the derivative of functions expressed in parametric forms.
• Second-order derivatives.

Rolle's and Lagrange's Mean Value Theorems (without proof) and their geometric interpretation.

• Differentiation
• Derivatives of trigonometric functions
• Derivatives of exponential functions
• Derivatives of logarithmic functions
• Derivatives of inverse trigonometric functions
• Differentiation using substitution
• Derivatives of implicit functions and chain rule
• e for composite functions.
• Derivatives of Parametric functions.
• Differentiation of a function to another function, e.g. differentiation of Sinx3 for x3
• Logarithmic Differentiation
• Finding dy/dx when y=xxx
• Successive differentiation up to 2nd order

Note: Derivatives of composite functions using the chain rule.

• L' Hospital's theorem.

00 form and ∞∞ form

(ii) Applications of Derivatives

Applications of derivatives: rate of change of bodies, increasing/decreasing functions, tangents and normals, maxima, and minima

• Simple problems (that illustrate basic principles and understanding of the subject and real-life situations).
• Equation of Tangent and Normal
• 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 the inverse process of differentiation
• Integration of a variety of functions by substitution, partial fractions and 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, sec2x, cosec2x, etc
• Integrals of the type sin2x, cos2x, cos3x, cos4x
• Integration of 1x, ex, etc.
• Integration by substitution
• Integrals of the type f' (x) [f(x)]n, f('x)f(x),
• Integration of tanx, cotx, secx, cosec x
• Integration by parts.

When degree of f (x) ≥ degree of g(x), e.g.

x2+1x2+3x+1 =1-(3x+1x2+3x+2)

Definite Integral

• The fundamental theorem of calculus (without proof)
• Properties of definite integrals.

Problems based on the following properties of definite integrals are to be covered.

• abf(x)dx= abf(t)dt
• abf(x)dx= -abf(x)dx
• abf(x)dx= acf(x)dx+bcf(x)dx, where a<c<b
• abf(x)dx= abf(a+b-x)dx
• 0af(x)dx= 0af(a-x)dx
• -aaf(x)dx= 20af(x)dx, if f is an even function
• 0, if f is an odd function

(iv) Differential Equations

• Definition, order, and degree, general, and particular solutions of a differential equation.
• Formation of differential equations whose general solution is given.
• The solution of differential equations by the method of separation of variables solutions of homogeneous differential equations of the first order and first degree.
• Solutions of linear differential equations of the type: dydx +py=q, where p and q are functions of x or constants.

dxdy +px=q, where p and q are functions of y or constants.

• Differential equations, order, and degree
• Formation of differential equations by eliminating arbitrary constant(s)
• The solution of differential equations
• Variable separable
• Homogeneous equations
• Linear form Py Q dx dy + = 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.

### Unit 4. Probability

• Conditional probability, multiplication theorem on probability, independent events, total probability
• Independent and dependent events conditional events
• Laws of Probability, addition theorem, multiplication theorem, conditional probability
• The theorem of Total Probability
• Bayes theorem

## Detailed ISC Class 12 Maths Syllabus: Section B

### Unit 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, the addition of vectors, multiplication of a vector by a scalar.
• Definition, Geometrical Interpretation, properties, and application of scalar (dot) product of vectors, vector (cross) product of vectors, and the scalar triple product of vectors.

As directed line segments.

• Magnitude and direction of a vector
• Types: equal vectors, unit vectors, zero vectors
• 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
• Scalar (dot) product of vectors and its geometrical significance
• Cross product - its properties
• Area of a triangle, area of the parallelogram, collinear vectors
• Scalar triple product
• The volume of a parallelepiped, coplanarity

NOTE: Proofs of geometrical theorems using vector algebra are excluded.

### Unit 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.
• Cartesian and vector equation of a plane.
• The angle between (i) two lines, (ii) two planes, (iii) a line and a plane.
• The 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
• The 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 the intersection of two lines
• The 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
• The distance of a point from a plane
• The intersection of the line and plane
• The angle between two planes, a line, and a plane

### Unit 7. Application of Integrals

• Application in finding the area bounded by simple curves and coordinate axes.
• The area enclosed between two curves.
• Application of definite integrals
• The area bounded by curves, lines, and coordinate axes is required to be covered
• Simple curves: lines, parabolas, and polynomial functions

## Detailed ISC Class 12 Maths Syllabus: Section C

### Unit 8. 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 break-even point
• Increasing-decreasing functions

NOTE: Application involving differentiation, increasing, and decreasing function to be covered.

### Unit 9. Linear Regression

• Lines of regression of x on y and y on x.
• Lines of best fit.
• The regression coefficient of x on y and y on x.
• bxy byx=r2, 0 bxy byx 1
• Identification of regression equations
• Estimation of the value of one variable using the value of another variable from the appropriate line of regression.

### Unit 10. Linear Programming

• Introduction, related terminology such as constraints, objective function, optimization
• Different types of linear programming (L.P.) problems, the 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)

## ISC Class 12 Maths Syllabus Blueprint 2025

The number of questions of different types in each part is mentioned below:

 Section Topics Question Type Marking Number of Questions Total Marks Section A Relation and Functions, Algebra, Calculus, and Probability MCQ & Question with one-liner Answers 1 mark 15 questions 15 marks Questions with short answer I 2 marks 5 questions 10 marks Questions with short answer II 4 marks 4 questions 16 marks Questions with Long answers 6 marks 4 questions 24 marks Total 65 marks Section B Vectors, Three-dimensional Geometry, and application of integrals MCQ & Question with one-liner Answers 1 mark 5 questions 5 marks Questions with short answer I 2 marks 1 question 2 marks Questions with short answer II 4 marks 2 questions 10 marks Total 15 Marks OR Section C Application of Calculus, Linear Programming, and Linear regression MCQ & Question with one-liner Answers 1 marks 5 questions 5 marks Questions with short answer I 2 marks 1 question 2 marks Questions with short answer II 4 marks 2 questions 10 marks Total 15 Marks Total 80 Marks

## Best Books for ISC Class 12 Maths Syllabus

The table below shows the best Preparation Books for the class 12 ISC maths syllabus prescribed by the CICSE Board.

 Author 12th Maths books O.P. Malhotra, S.K. Gupta, Anubhuti Gangal S. Chand's ISC Mathematics Book II for Class XII Paperback M.L. Aggarwal Understanding I.S.C. Mathematics (Vol. I & II) Class- XII Paperback C.B. Gupta S. Chand's ISC Commerce for Class XII Paperback Arihant Experts All In One ISC Mathematics Class 12 Paperback R.D Sharma Mathematics for Class 12 by R D Sharma (set of 2 volumes) Oswaal Oswaal Sample Question Paper Class 12 Mathematics

## ISC Class 12 Maths Syllabus: Preparation Tips

Here are a few crucial tips and tricks that will help you prepare for your ISC Maths Board Exam:

• First of all, take a thorough look at the detailed ISC Class 12 Maths Syllabus PDF and exam pattern.
• Study each topic of the prescribed books.
• Make sure you take notes during your study that include important points. Those notes will help during the revision period.
• Strategize and prepare the long-form questions (4-6 marks), usually from the Calculus or Probability section, and try to perfect them by practising them regularly.
• Try as many mock tests as you can. After completing each section, attempt the section-wise tests to evaluate your preparation.

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