MATHEMATICS

1 MATHEMATICS (860). CLASS XII. 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. 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. UNIT TOTAL WEIGHTAGE. SECTION A: 65 MARKS.

2 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 1. SECTION A. 1. Relations and Functions - Formulae for 2sin-1x, 2cos-1x, 2tan-1x, 3tan-1x etc. and application of these (i) Types of relations: reflexive, symmetric, formulae. transitive and equivalence relations. One to one and onto functions, composite functions. 2. Algebra Relations as: Matrices and Determinants - Relation on a set A (i) Matrices - Identity relation, empty relation, Concept, notation, order, equality, types of universal relation. matrices, zero and identity matrix, transpose - Types of Relations: reflexive, of a matrix, symmetric and skew symmetric symmetric, transitive and matrices.

3 Operation on matrices: Addition equivalence relation. and multiplication and multiplication with a scalar. Simple properties of addition, Functions: multiplication and scalar multiplication. Non- - As special relations, concept of commutativity of multiplication of matrices. writing y is a function of x as y = Invertible matrices, if it exists (here all f(x). matrices will have real entries). - Types: one to one, many to one, into, (ii) Determinants onto. Determinant of a square matrix (up to 3 x 3. - Real Valued function. matrices), properties of determinants, - Composite functions (algebraic minors, co-factors. Adjoint and inverse of a functions only). square matrix. Solving system of linear (ii) Inverse Trigonometric Functions equations in two or three variables (having unique solution) using inverse of a matrix. Definition, domain, range, principal value branch. Elementary properties of inverse - Types of matrices (m n; m, n 3), trigonometric functions.

4 Order; Identity matrix, Diagonal matrix. - Principal values. - Symmetric, Skew symmetric. - sin-1x, cos-1x, tan-1x etc. - Operation addition, subtraction, x multiplication of a matrix with scalar, - sin-1x = cos 1 1 x 2 = tan 1 . multiplication of two matrices 1 x2 (the compatibility). 1 . - sin-1x= cosec 1 ; sin-1x+cos-1x= and 1 1 . x 2 1 2 . 0 2 = AB( say ) but BA is similar relations for cot-1x, tan-1x, etc. 2 2 . 1 1 . not possible. - Singular and non-singular matrices. - Existence of two non-zero matrices whose product is a zero matrix. AdjA. - Inverse (2 2, 3 3) A 1 =. A. 2. Martin's Rule ( using matrices). - Derivatives of exponential functions. a1x + b1y + c1z = d1. - Derivatives of logarithmic functions. a2x + b2y + c2z = d2 - Derivatives of inverse trigonometric a3x + b3y + c3z = d3 functions - differentiation by means of substitution. a 1 b 1 c1 d1 x - Derivatives of implicit functions and A = a 2 b2 c 2 B = d 2 X = y.

5 Chain rule. - e for composite functions. a 3 b3 c3 d 3 z . - Derivatives of Parametric functions. AX = B X = A 1 B - Differentiation of a function with respect to another function Problems based on above. differentiation of sinx3 with respect Determinants to x3. - Order. - Logarithmic Differentiation - x . - Minors. Finding dy/dx when y = x x . - Cofactors. - Successive differentiation up to 2nd - Expansion. order. NOTE 1: Derivatives of composite functions 3. Calculus using chain rule. (i) Continuity, Differentiability and Differentiation. Continuity and L' Hospital's theorem. differentiability, derivative of composite , forms only. functions, chain rule, derivatives of inverse trigonometric functions, derivative of (ii) Applications of Derivatives implicit functions. Concept of exponential Applications of derivatives: and logarithmic functions. increasing/decreasing functions, tangents Derivatives of logarithmic and exponential and normals, maxima and minima (first functions.)

6 Logarithmic differentiation, derivative test motivated geometrically and derivative of functions expressed in second derivative test given as a provable parametric forms. Second order derivatives. tool). Simple problems (that illustrate basic Continuity principles and understanding of the subject as - Continuity of a function at a point well as real-life situations). x = a. Equation of Tangent and Normal - Continuity of a function in an Increasing and decreasing functions. interval. - Algebra of continues function. Maxima and minima. - Removable discontinuity. - Stationary/turning points. Differentiation - Absolute maxima/minima - Concept of continuity and - local maxima/minima differentiability of x , [x], etc. - First derivatives test and second - Derivatives of trigonometric derivatives test functions. - Application problems based on maxima and minima. 3. (iii) Integrals Integrals of the type: Integration as inverse process of differentiation.

7 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 Definite Integral on them. - Fundamental theorem of calculus (without proof). Fundamental Theorem of Calculus (without proof). Basic properties of - Properties of definite integrals. definite integrals and evaluation of definite - Problems based on the following integrals. properties of definite integrals are to be covered. Indefinite integral b b - Integration as the inverse of differentiation.. a f ( x)dx = f (t )dt a - Anti-derivatives of polynomials and b a functions (ax +b)n , sinx, cosx, sec2x, cosec2x etc .. a f ( x)dx = f ( x)dx b - Integrals of the type sin2x, sin3x, sin4x, cos2x, cos3x, cos4x. b c b - Integration of 1/x, ex. f ( x)dx = f ( x)dx + f ( x)dx - Integration by substitution. a a c where a < c < b - Integrals of the type f ' (x)[f (x)]n, b b f ( x).

8 F ( x).. a f ( x)dx = f (a + b x)dx a - Integration of tanx, cotx, secx, a a cosecx.. 0. f (=. x)dx f (a x)dx 0. - Integration by parts. a - Integration using partial fractions. 2a 2 f ( x)dx, if f (2a x) = f ( x). f ( x) f ( x)dx = 0. Expressions of the form when 0 0, f (2a x) = f ( x). g ( x) . degree of f(x) < degree of g(x) a . a 2 f ( x)dx,if f is an even function x+2. =. A. +. B f ( x)dx = 0.. ( x 3)( x + 1) x 3 x + 1 a 0,if f is an odd function (iv) Differential Equations x+2 A B C Definition, order and degree, general and = + +. ( x 2)( x 1) 2. x 1 ( x 1) 2. x 2 particular solutions of a differential equation. Solution of differential equations by method of separation of variables When degree of f (x) degree of g(x), solutions of homogeneous differential equations of first order and first degree. x2 +1 3x + 1 Solutions of linear differential equation of = 1 2 . 2. x + 3x + 2 x + 3x + 2 dy the type: +py= q, where p and q are dx dx functions of x or constants.

9 + px = q, dy 4. where p and q are functions of y or - Scalar (dot) product of vectors and its constants. geometrical significance. - Cross product - its properties - area of a - Differential equations, order and degree. triangle, area of parallelogram, collinear - Solution of differential equations. vectors. - Variable separable. NOTE: Proofs of geometrical theorems by - Homogeneous equations. using Vector algebra are excluded. dy 6. Three - dimensional Geometry - Linear form + Py = Q where P and Q. dx Direction cosines and direction ratios of a line are functions of x only. Similarly, for joining two points. Cartesian equation and vector dx/dy. equation of a line, coplanar and skew lines. NOTE : The second order differential Cartesian and vector equation of a plane. Angle equations are excluded. between two lines. Distance of a point from a plane. 4. Probability - Equation of x-axis, y-axis, z axis and lines parallel to them.

10 Conditional probability, multiplication theorem on probability, independent events, total - Equation of xy - plane, yz plane, probability, Bayes' theorem. zx plane. - Independent and dependent events - Direction cosines, direction ratios. conditional events. - Angle between two lines in terms of direction - Laws of Probability, addition theorem, cosines /direction ratios. multiplication theorem, conditional - Condition for lines to be perpendicular/. probability. parallel. - Theorem of Total Probability. Lines - Baye's theorem. - Cartesian and vector equations of a line through one and two points. SECTION B. - Coplanar and skew lines. 5. Vectors - Conditions for intersection of two lines. Vectors and scalars, magnitude and direction - Distance of a point from a line. of a vector. Direction cosines and direction ratios of a vector. Types of vectors (equal, unit, Planes zero, parallel and collinear vectors), position - Cartesian and vector equation of a vector of a point, negative of a vector, components of a vector, addition of vectors, plane.