Latest Update 2024

Exams for Class 12 will be administered between February 12 to April 03, 2024. The Mathematics Exam Will be held on February 20, 2024 Tuesday from 02:00 PM to 05:00 PM. For Detailed Time Table 2024, students are advised to check the ISC CLASS XII 2024, Time Table on its official website.

Official Syllabus

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

The syllabus is divided into three sections A, B, and C.

Section A is compulsory for all candidates. Candidates will have a choice of attempting questions from EITHER Section B OR Section C.

There will be one paper of three hours duration of 80 marks.

Section A (65 Marks): Candidates will be required to attempt all questions. The internal choice will be provided in two questions of two marks, two questions of four marks, and two questions of six marks each.

Section B/ Section C (15 Marks): Candidates will be required 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.

DISTRIBUTION OF MARKS FOR THE THEORY PAPER

SECTION A

1. Relations and Functions

(i) Types of relations: reflexive, symmetric, transitive, and equivalence relations. One-to-one and functions, composite functions.

• 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, the 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.

- Composite functions (algebraic functions only).

(ii) Inverse Trigonometric Functions:

Definition, domain, range, principal value branch. Elementary properties of inverse trigonometric functions.

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. Invertible matrices, if it exists (Here all matrices will have real entries).

(ii) Determinants: Determinants of a square matrix (up to 3 x 3 matrices), properties of determinants, minors, and 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 less than or equal to 3), order; Identity matrix, Diagonal matrix.

- Symmetric, Skew symmetric.

Latest Update 2024

Exams for Class 12 will be administered between February 12 to April 03, 2024. The Mathematics Exam Will be held on February 20, 2024 Tuesday from 02:00 PM to 05:00 PM. For Detailed Time Table 2024, students are advised to check the ISC CLASS XII 2024, Time Table on its official website.

Official Syllabus

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

The syllabus is divided into three sections A, B, and C.

Section A is compulsory for all candidates. Candidates will have a choice of attempting questions from EITHER Section B OR Section C.

There will be one paper of three hours duration of 80 marks.

Section A (65 Marks): Candidates will be required to attempt all questions. The internal choice will be provided in two questions of two marks, two questions of four marks, and two questions of six marks each.

Section B/ Section C (15 Marks): Candidates will be required 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.

DISTRIBUTION OF MARKS FOR THE THEORY PAPER

SECTION A

1. Relations and Functions

(i) Types of relations: reflexive, symmetric, transitive, and equivalence relations. One-to-one and functions, composite functions.

• 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, the 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.

- Composite functions (algebraic functions only).

(ii) Inverse Trigonometric Functions:

Definition, domain, range, principal value branch. Elementary properties of inverse trigonometric functions.

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. Invertible matrices, if it exists (Here all matrices will have real entries).

(ii) Determinants: Determinants of a square matrix (up to 3 x 3 matrices), properties of determinants, minors, and 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 less than or equal to 3), order; Identity matrix, Diagonal matrix.

- Symmetric, Skew symmetric.

- Operation – addition, subtraction, multiplication of a matrix with scalar, multiplication of two matrices (the compatibility).

- Singular and non-singular matrices.

- Existence of two non-zero matrices whose product is a zero matrix.

• Martin’s Rule (i.e. using matrices)

a₁x + b₂y + c₁z = d₁

a₂x +b₂y + c₂z = d₂

a₃x + b₃y + c₃z = d₃

• Determinants

- Order.
- Minors.
- Cofactors.
- Expansion.

3. Calculus

(i) Continuity, Differentiability, and Differentiation. Continuity and differentiability, a 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 continuous 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.

- e for composite functions.

- 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 = y = x^x

- Successive differentiation up to 2nd order

NOTE: Derivatives of composite functions using the chain rule.

• L'Hospital's theorem.

(ii) Applications of Derivatives

Applications of derivatives: increasing/decreasing functions, tangents and normals, maxima and minima (first derivative test motivated geometrically and second derivative test given as approvable tool). Simple problems (that illustrate basic principles and understanding of the subject as well as real-life situations).

• Equation of Tangent and Normal
• Increasing and decreasing functions.
• Maxima and minima.
- Stationary/turning points.
- Absolutemaxima/minima
- local maxima/minima
- First derivatives test and second derivatives test

- Application problems maxima and minima.

(iii) Integrals

Integration is the 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 is the inverse of differentiation.
- Anti-derivatives of polynomials and functions (ax +b)ⁿ, sinx, cosx, sec²x, cosec²x etc.
- Integrals of the type sin²x, sin³x, sin⁴x, cos²x, cos³x, cos⁴x.
- Integration of 1/x, e^x.
- Integration by substitution.
- Integrals of the type f '(x) [f (x)]n,

- Integration of tanx, cotx, secx, cosecx.
- Integration by parts. When the degree of f(x) is greater than or equal to the degree of g(x), e.g.

• 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. Formation of differential equation whose general solution is given. 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 equation of the type: (dy/dx) + py = q, where p and q are functions of x or constants. (dx/dy) +px = q, where p and q are 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.
- Linear form (dy/dx) + Py = Q where P and Q are functions of x only. Similarly, for dx/dy.

NOTE 1: The second-order differential equations are excluded.

4. Probability

Conditional probability, multiplication theorem on probability, independent events, total probability, Bayes’ theorem.

- Independent and dependent events conditional events.
- Laws of Probability, addition theorem, multiplication theorem, conditional probability.
- Theorem of Total Probability.
- Baye’s theorem.

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, 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.

- 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.
- iˆ, jˆ, kˆas 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 their geometrical significance.
- Cross product - its properties - the area of a triangle, area of the 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. Cartesian and vector equation of a plane. Distance of a point from a plane.

- Equation of the 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 the 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.

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 - is abounded by curves, lines, and coordinate axes is required to be covered.
- Simple curves: lines, parabolas, and polynomial functions.

SECTION C

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.

9. Linear Regression

- Lines of regression of x on y and y on x.
- Lines of best fit
- Regression coefficient of x on y and y on x.
- b_xy × b_yx = r², 0 less than or equal b_xy × b_yx less than or equal 1.
- Identification of regression equations.
- Estimation of the value of one variable using the value of another variable from the appropriate line of regression.

10. 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, the 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.

PAPER II – PROJECT WORK – 20 Marks

Candidates will be expected to have completed two projects, one from Section A and one from either Section B or Section C.

The project work will be assessed by the subject teacher and a Visiting Examiner appointed locally and approved by the Council.

Mark allocation for each Project [10 marks] :

Topics for Project Work :

Section A

1. Using a graph, demonstrate a function that is one-one but not onto.

2. Using a graph demonstrates a function that is invertible.

3. Construct a composition table using a binary function addition/multiplication modulo up to 5 and verify the existence of the properties of the binary operation.

4. Draw the graph of y = sin-1 x (or any other inverse trigonometric function), using the graph of y = sin x (or any other relevant trigonometric function). Demonstrate the concept of mirror line (about y = x) and find its domain and range.

5. Explore the principal value of the function sin-1 x (or any other inverse trigonometric function) using a unit circle.

6. Find the derivatives of a determinant of the order of 3 x 3 and verify the same by other methods.

7. Verify the consistency of the system of three linear equations of two variables and verify the same graphically. Give its geometrical interpretation.

8. For a dependent system (non-homogeneous) of three linear equations of three variables, identify an infinite number of solutions.

9. For a given function, give the geometrical interpretation of Mean Value theorems. Explain the significance of closed and open intervals for continuity and differentiability properties of the theorems.

10. Explain the concepts of increasing and decreasing functions, using the geometrical significance of dy/dx. Illustrate with proper examples.

11. Explain the geometrical significance of the point of inflexion with examples and illustrate it using graphs.

12. Explain and illustrate (with suitable examples) the concept of local maxima and local minima using a graph.

13. Explain and illustrate (with suitable examples) the concept of absolute maxima and absolute minima using a graph.

14. Illustrate the concept of definite integral, expressed as the limit of a sum, and verify it by actual integration.

15. Demonstrate the application of differential equations to solve a given problem (for example, population increase or decrease, bacteria count in culture, etc.).

16. Explain the conditional probability, the theorem of total probability, and the concept of Bayes’ theorem with suitable examples.

17. Explain the types of probability distributions and derive the mean and variance of the binomial probability distribution for a given function.

Section B

18. Using vector algebra, find the area of a parallelogram/triangle. Also, derive the area analytically and verify the same.

19. Using Vector algebra, prove the formulae of properties of triangles (sine/cosine rule, etc.)

20. Using Vector algebra, prove the formulae of compound angles, e.g. sin (A + B) = Sin A Cos B + Sin B Cos A, etc.

21. Describe the geometrical interpretation of the scalar triple product and for a given data, find the scalar triple product.

22. Find the image of a line with respect to a given plane.

23. Find the distance of a point from a given plane measured parallel to a given line.

24. Find the distance of a point from a line measured parallel to a given plane.

25. Find the area bounded by a parabola and an oblique line.

26. Find the area bounded by a circle and an oblique line.

27. Find the area bounded by an ellipse and an oblique line.

28. Find the area bounded by a circle and a circle.

29. Find the area bounded by a parabola and a parabola.

30. Find the area bounded by a circle and a parabola.

(Any other pair of curves which are specified in the syllabus may also be taken.)

Section C

31. Draw a rough sketch of Cost (C), Average Cost (AC), and Marginal Cost (MC)

                                                            Or

Revenue (R), Average Revenue (AR), and Marginal Revenue (MR).

Give their mathematical interpretation using the concept of increasing-decreasing functions and maxima-minima.

32. For a given data, find regression equations by the method of least squares. Also, find angles between regression lines.

33. Draw the scatter diagram for a given data. Use it to draw the lines of best fit and estimate the value of Y when X is given and vice-versa.

34. Using any suitable data, find the minimum cost by applying the concept of the Transportation problem.

35. Using any suitable data, find the minimum cost and maximum nutritional value by applying the concept of the Diet problem. 36. Using any suitable data, find the Optimum cost in the manufacturing problem by formulating a linear programming problem (LPP).

NOTE: No question paper for Project Work will be set by the Council.

To Know More: Click Here