**CBSE exams 2019:** The Central Board of Secondary Education is scheduled to conduct examinations from the month of February next year. The board has already released a list of vocational subjects for the examinations that will be conducted from February to March 2019, the schedule and date of the examinations will be released later.

Apart from the 40 different vocational subjects, the board will conduct exams for Typography and Computer Applications (English), Web applications, Graphics, Office Communication, etc in February as these subjects have larger practical component, and shorter theory papers.

**ALSO READ | **CBSE relaxes passing criteria for Class 10 students

## Here is the detail syllabus of CBSE Class 10, 12 Maths paper

**Class 10**

**Units I.** Number Systems

II. Algebra

III. Trigonometry

IV. Coordinate Geometry

**READ | CBSE relaxes passing criteria for Class 10 students**

V. Geometry

VI. Mensuration VII. Statistics and Probability

Appendix: 1. Proofs in Mathematics

2. Mathematical Modelling

Unit I: Number Systems Real Numbers (Periods 15) Euclid’s division lemma, Fundamental Theorem of Arithmetic – statements after reviewing work done earlier and after illustrating and motivating through examples. Proofs of results – irrationality of 2, 3, 5, decimal expansions of rational numbers in terms of terminating/non-terminating recurring decimals.

Unit II: Algebra 1. Polynomials (Periods 6) Zeros of a polynomial. Relationship between zeros and coefficients of a polynomial with particular reference to quadratic polynomials. Statement and simple problems on division algorithm for polynomials with real coefficients. 2. Pair of Linear Equations in Two Variables (Periods 15) Pair of linear equations in two variables. Geometric representation of different possibilities of solutions/inconsistency. Algebraic conditions for a number of solutions. The solution of a pair of linear equations in two variables algebraically – by substitution, by elimination and by cross multiplication. Simple situational problems must be included. Simple problems on equations reducible to linear equations may be included.

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3. Quadratic Equations (Periods 15) Standard form of a quadratic equation ax2 + bx + c = 0, (a ?0). Solution of quadratic equations (only real roots) by factorization and by completing the square, i.e., by using quadratic formula. The relationship between discriminant and the nature of roots. Problems related to day-to-day activities to be incorporated.

4. Arithmetic Progressions (AP) (Periods 8) Motivation for studying AP. Derivation of standard results of finding the nth term and sum of first n terms.

Unit III: Trigonometry 1. Introduction to Trigonometry (Periods 18) Trigonometric ratios of an acute angle of a right-angled triangle. Proof of their existence (well defined); motivate the ratios, whichever are defined at 0° and 90°. Values (with proofs) of the trigonometric ratios of 30°, 45° and 60°. Relationships between the ratios. Trigonometric Identities: Proof and applications of the identity sin2 A + cos2 A = 1. Only simple identities to be given. Trigonometric ratios of complementary angles. 2. Heights and Distances (Periods 8) Simple and believable problems on heights and distances. Problems should not involve more than two right triangles. Angles of elevation/depression should be only 300, 450, 600.

Unit IV: Coordinate Geometry Lines (In two-dimensions) (Periods 15) Review the concepts of coordinate geometry done earlier including graphs of linear equations. Awareness of geometrical representation of quadratic polynomials. The distance between two points and section formula (internal). Area of a triangle.

Unit V: Geometry 1. Triangles (Periods 15) Definitions, examples, counterexamples of similar triangles. 1. (Prove) If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio. 2. (Motivate) If a line divides two sides of a triangle in the same ratio, the line is parallel to the third side. 3. (Motivate) If in two triangles, the corresponding angles are equal, their corresponding sides are proportional and the triangles are similar.

4. (Motivate) If the corresponding sides of two triangles are proportional, their corresponding angles are equal and the two triangles are similar.

5. (Motivate) If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are proportional, the two triangles are similar.

6. (Motivate) If a perpendicular is drawn from the vertex of the right angle to the hypotenuse, the triangles on each side of the perpendicular are similar to the whole triangle and to each other.

7. (Prove) The ratio of the areas of two similar triangles is equal to the ratio of the squares on their corresponding sides.

8. (Prove) In a right triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides.

9. (Prove) In a triangle, if the square on one side is equal to sum of the squares on the other two sides, the angles opposite to the first side is a right triangle. 2. Circles (Periods 8) Tangents to a circle motivated by chords drawn from points coming closer and closer to the point. 1. (Prove) The tangent at any point of a circle is perpendicular to the radius through the point of contact. 2. (Prove) The lengths of tangents drawn from an external point to a circle are equal. 3. Constructions (Periods 8) 1. Division of a line segment in a given ratio (internally). 2. Tangent to a circle from a point outside it. 3. Construction of a triangle similar to a given triangle.

Unit VI: Mensuration 1. Areas Related to Circles (Periods 12) Motivate the area of a circle; area of sectors and segments of a circle. Problems based on areas and perimeter/circumference of the above said plane figures. (In calculating area of segment of a circle, problems should be restricted to central angle of 60°, 90° and 120° only. Plane figures involving triangles, simple quadrilaterals and circle should be taken.) 2. Surface Areas and Volumes (Periods 12) 1. Problems on finding surface areas and volumes of combinations of any two of the following: cubes, cuboids, spheres, hemispheres and right circular cylinders/cones. Frustum of a cone.

2. Problems involving converting one type of metallic solid into another and other mixed problems. (Problems with combination of not more than two different solids be taken.)

Unit VII: Statistics and Probability 1. Statistics (Periods 15) Mean, median and mode of grouped data (bimodal situation to be avoided). Cumulative frequency graph. 2. Probability (Periods 10) Classical definition of probability. Connection with probability as given in Class IX. Simple problems on single events, not using set notation.

Appendix 1. Proofs in Mathematics Further discussion on concept of ‘statement’, ‘proof’ and ‘argument’. Further illustrations of deductive proof with complete arguments using simple results from arithmetic, algebra and geometry. Simple theorems of the “Given ……… and assuming… prove ……..”. Training of using only the given facts (irrespective of their truths) to arrive at the required conclusion. Explanation of ‘converse’, ‘negation’, constructing converses and negations of given results/statements.

2. Mathematical Modelling Reinforcing the concept of mathematical modelling, using simple examples of models where some constraints are ignored. Estimating probability of occurrence of certain events and estimating averages may be considered. Modelling fair instalments payments, using only simple interest and future value (use of AP).

**Class 12**

Units I. Relations and Functions

II. Algebra

III. Calculus

IV. Vectors and Three-Dimensional Geometry

V. Linear Programming

VI. Probability

Appendix: 1. Proofs in Mathematics

2. Mathematical Modelling

Chapters with Time Allocation

Relations and Functions Periods

Inverse Trigonometric Functions Periods

Matrices Periods

Determinants Periods

Continuity and Differentiability Periods

Applications of Derivatives Periods

Integrals Periods

Applications of the Integrals Periods

Differential Equations Periods

Vectors Periods

Three-dimensional Geometry Periods

Linear Programming Periods

Probability Periods

Unit I: Relations and Functions 1. Relations and Functions Types of relations: Reflexive, symmetric, transitive and equivalence relations. One to one and onto functions, composite functions, inverse of a function. Binary operations. 2. Inverse Trigonometric Functions Definition, range, domain, principal value branches. Graphs of inverse trigonometric functions. Elementary properties of inverse trigonometric functions.

Unit II: Albegra 1. Matrices Concept, notation, order, equality, types of matrices, zero matrix, transpose of a matrix, symmetric and skew symmetric matrices. Addition, multiplication and scalar multiplication of matrices, 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 2). 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). 2. Determinants Determinant of a square matrix (up to 3 × 3 matrices), properties of determinants, minors, cofactors 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.

Unit III: Calculus

1. Continuity and Differentiability Continuity and differentiability, derivative of composite functions, chain rule, derivatives of inverse trigonometric functions, derivative of implicit function. Concept of exponential and logarithmic functions and their derivatives. Logarithmic differentiation. Derivative of functions expressed in parametric forms. Second order derivatives. Rolle’s and Lagrange’s Mean Value Theorems (without proof) and their geometric interpretations.

2. Applications of Derivatives Applications of derivatives: Rate of change, increasing/decreasing functions, tangents and normals, approximation, 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).

3. Integrals Integration as inverse process of differentiation. Integration of a variety of functions by substitution, by partial fractions and by parts, only simple integrals of the type ± + + ± – + + ? ? ? ? ? 2 2 2 2 2 2 2 2 , , , , , dx dx dx dx dx x a ax bx c x a a x ax bx c ++ ±- ++ ++ ? ? ? ? 2 2 2 2 2 2 ( ) ( ) ,, px q px q dx dx a x dx and x a dx ax bx c ax bx c to be evaluated. Definite integrals as a limit of a sum. Fundamental Theorem of Calculus (without proof). Basic properties of definite integrals and evaluation of definite integrals.

4. Applications of the Integrals Applications in finding the area under simple curves, especially lines, arcs of circles/parabolas/ ellipses (in standard form only), area between the two above said curves (the region should be clearly identifiable).

5. 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 method of separation of variables, homogeneous differential equations of first order and first degree. Solutions of linear differential equation of the type: P Q,dy y dx += where P and Q are functions of x.

Unit IV: Vectors and Three-Dimensional Geometry

1. Vectors Vectors and scalars, magnitude and direction of a vector. Direction cosines/ratios of vectors. 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. Scalar (dot) product of vectors, projection of a vector on a line. Vector (cross) product of vectors.

2. Three-dimensional Geometry Direction cosines/ratios of a line joining two points. Cartesian 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.

Unit V: 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, feasible and infeasible solutions, optimal feasible solutions (up to three non-trivial constrains).

Unit VI: Probability Multiplication theorem on probability. Conditional probability, independent events, total probability, Baye’s theorem. Random variable and its probability distribution, mean and variance of haphazard variable. Repeated independent (Bernoulli) trials and Binomial distribution.

Appendix 1. Proofs in Mathematics Through a variety of examples related to mathematics and already familiar to the learner, bring out different kinds of proofs: direct, contrapositive, by contradiction, by counter-example. 2. Mathematical Modelling Modelling real-life problems where many constraints may really need to be ignored (continuing from Class XI). However, now the models concerned would use techniques/results of matrices, calculus and linear programming.