Mathematics Class 12 Science (English Medium) CBSE Topics and Syllabus

• Introduction of Relations and Functions 
• Concept of Binary Operations 
  • Commutative Binary Operations
  • Associative Binary Operations
  • Identity Binary Operation,
  • Invertible Binary Operation
• Inverse of a Function 
• Composition of Functions and Invertible Function 
• Types of Functions 
  • One-one (or injective)
  • Many-one
  • Onto (or surjective)
  • One-one and onto (or bijective)
• Types of Relations 
  • Empty Relation
  • Universal Relation
  • Trivial Relations
  • Equivalence Relation - Reflexive, symmetric, transitive, not reflexive, not symmetric and not transitive.
  • One-One Relation (Injective)
  • Many-one relation
  • Into relation
  • Onto relation (Surjective)

• Types of relations: reflexive, symmetric, transitive and equivalence relations.
• One to one and onto functions, composite functions, inverse of a function. Binary operations.

• Inverse Trigonometric Functions (Simplification and Examples) 
• Properties of Inverse Trigonometric Functions 

  Inverse of Sin, Inverse of cosin, Inverse of tan, Inverse of cot, Inverse of Sec, Inverse of Cosec

• Inverse Trigonometric Functions 
  • Graphs of Inverse Trigonometric Functions 
  • Inverse Trigonometric Functions - Principal Value Branch 
• Basic Concepts of Inverse Trigonometric Functions 
  • sine, cosine, tangent, cotangent, secant, cosecant function

1. Definition, range, domain, principal value branch.
2. Graphs of inverse trigonometric functions.
3. Elementary properties of inverse trigonometric functions.

• Introduction of Operations on Matrices 
• Inverse of Matrix 
  • Inverse of a Matrix by Elementary Transformation 
• Multiplication of Two Matrices 
• Negative of Matrix 
• Operations on Matrices 
  • Properties of Matrix Addition 
    • Commutative Law
    • Associative Law
    • Existence of additive identity
    • The existence of additive inverse
  • Addition of Matrices 
  • Multiplication of Matrices 
    • Non-commutativity of multiplication of matrices
    • Zero matrix as the product of two non zero matrices
  • Properties of Multiplication of Matrices 
    • The associative law
    • The distributive law
    • The existence of multiplicative identity
  • Multiplication of a Matrix by a Scalar 
  • Properties of Scalar Multiplication of a Matrix 
• Transpose of a Matrix 
  • Write transpose of given matrix
• Subtraction of Matrices 
• Symmetric and Skew Symmetric Matrices 
  • Define symmetric and skew symmetric matrix
• Types of Matrices 
  • Types of Matrices:

  1. Column matrix
  2. Row matrix
  3. Square matrix
  4. Diagonal matrix
  5. Scalar matrix
  6. Identity matrix
  7. Zero matrix
• Matrices 
  • Proof of the Uniqueness of Inverse 
  • Matrices Notation 

    Matrices Notation

• Invertible Matrices 
• Elementary Transformations 
  • Interchange of any two rows or any two columns
  • Multiplication of the elements of any row or column by a non-zero scalar
  • Adding the scalar multiples of all the elements of any row (column) to corresponding elements of any other row (column)
• Equality of Matrices 
  • Determine equality of two matrices
• Order of a Matrix 
• Introduction of Matrices 
• Properties of Transpose of the Matrices 

• Concept, notation, order, equality, types of matrices, zero andidentity 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).
• 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).

• Applications of Determinants and Matrices 
  • Consistent System
  • Inconsistent System
  • Solution of a system of linear equations using the inverse of a matrix
• Elementary Transformations 
  • Interchange of any two rows or any two columns
  • Multiplication of the elements of any row or column by a non-zero scalar
  • Adding the scalar multiples of all the elements of any row (column) to corresponding elements of any other row (column)
• Inverse of Matrix 
  • Inverse of a Square Matrix by the Adjoint Method 
• Properties of Determinants 
  • Property 1 - The value of the determinant remains unchanged if its rows are turned into columns and columns are turned into rows.
  • Property 2 -  If any two rows  (or columns)  of a determinant are interchanged then the value of the determinant changes only in sign.
  • Property 3 - If any two rows ( or columns) of a  determinant are identical then the value of the determinant is zero.
  • Property  4  -  If each element of a row (or column)  of a determinant is multiplied by a  constant k then the value of the new determinant is k times the value of the original determinant.
  • Property  5  -  If each element of a row (or column) is expressed as the sum of two numbers then the determinant can be expressed as the sum of two determinants
  • Property  6  -  If a constant multiple of all elements of any row  (or column)  is added to the corresponding elements of any other row  (or column  )  then the value of the new determinant so obtained is the same as that of the original determinant. 
  • Property 7 -  (Triangle property) - If all the elements of a  determinant above or below the diagonal are zero then the value of the determinant is equal to the product of its diagonal elements.
• Determinant of a Square Matrix 

  up to 3 x 3 matrices

• Determinants of Matrix of Order One and Two 
• Introduction of Determinant 
• Area of a Triangle 
• Minors and Co-factors 
• Determinant of a Matrix of Order 3 × 3 
  • 1st, 2nd and 3rd Row
  • 1st, 2nd and 3rd Columns
  • Expansion along the first Row (R₁)
  • Expansion along the second row (R₂)
  • Expansion along the first Column (C₁)
• Rule A=KB 

• 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.
• 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.

• Derivative - Exponential and Log 
• Concept of Differentiability 
• Proof Derivative X^n Sin Cos Tan 

  f(x) = xn

  f(x) = sin x

  f(x) = cos x

  f(x) = tan x

• Infinite Series 
• Higher Order Derivative 
  • Derivative of Functions Which Expressed in Higher Order Derivative Form
• Algebra of Continuous Functions 
• Continuous Function of Point 

  Continuous left hand limit

  Continuous right hand limit
  

• Mean Value Theorem 
  • Rolle’s and Lagrange’s Mean Value Theorems (without proof) and their geometric interpretations
• Second Order Derivative 
• Derivatives of Functions in Parametric Forms 
• Logarithmic Differentiation 
• Exponential and Logarithmic Functions 
• Derivatives of Implicit Functions 
• Derivatives of Inverse Trigonometric Functions 
• Derivatives of Composite Functions - Chain Rule 
• Concept of Continuity 

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

• Maximum and Minimum Values of a Function in a Closed Interval 
• Maxima and Minima 
  • First and Second Derivative test
  • Determine critical points of the function
  • Find the point(s) of local maxima and local minima and corresponding local maximum and local minimum values
  • Find the absolute maximum and absolute minimum value of a function
• Simple Problems on Applications of Derivatives 

  Simple problems (that illustrate basic principles and understanding of the subject as well as real-life situations)

• Graph of Maxima and Minima 
• Approximations 
• Tangents and Normals 
• Increasing and Decreasing Functions 
• Rate of Change of Bodies or Quantities 
• Introduction to Applications of Derivatives 

• Applications of derivatives: rate of change of bodies, increasing/decreasing functions, tangents and normals, use of derivatives in 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).

• Definite Integrals Problems 
• Indefinite Integral Problems 
• Comparison Between Differentiation and Integration 
• Integration 
  • Geometrical Interpretation of Indefinite Integrals 
• Integrals of Some Particular Functions 

  1) `int (dx)/(x^2 - a^2) = 1/(2a) log |(x - a)/(x + a)| + C`

  2) `int (dx)/(a^2 - x^2) = 1/(2a) log |(a + x)/(a - x)| + C`

  3) `int (dx)/(x^2 - a^2) = 1/a  tan^(-1) (x/a) + C`

  4) `int (dx)/sqrt (x^2 - a^2) = log |x + sqrt (x^2-a^2)| + C`

  5) `int (dx)/sqrt (a^2 - x^2) = sin ^(-1) (x/a) +C`

  6)  `int (dx)/sqrt (x^2 + a^2) = log |x + sqrt (x^2 + a^2)| + C`

  7) To find the integral `int (dx)/(ax^2 + bx + c)`

  8) To find the integral of the type `int (dx)/sqrt(ax^2 + bx + c)`

  9) To find the integral of the type `int (px + q)/(ax^2 + bx + c) dx`

  10) For the evaluation of the integral of the type `int (px + q)/sqrt(ax^2 + bx + c) dx`

• Indefinite Integral by Inspection 
• Some Properties of Indefinite Integral 
• Integration Using Trigonometric Identities 
• Introduction of Integrals 
• Evaluation of Definite Integrals by Substitution 
• Properties of Definite Integrals 
  1. `int_a^a f(x) dx = 0`
  2. `int_a^b f(x) dx = - int_b^a f(x) dx`
  3. `int_a^b f(x) dx = int_a^b f(t) dt`
  4. `int_a^b f(x) dx = int_a^c f(x) dx + int_c^b f(x) dx`
     where a < c < b, i.e., c ∈ [a, b]
  5. `int_a^b f(x) dx = int_a^b f(a + b - x) dx`
  6. `int_0^a f(x) dx = int_0^a f(a - x) dx`
  7. `int_0^(2a) f(x) dx = int_0^a f(x) dx + int_0^a f(2a - x) dx`
  8. `int_(-a)^a f(x) dx = 2. int_0^a f(x) dx`, if f(x) even function
     = 0,  if f(x) is odd function
• Fundamental Theorem of Calculus 

  Area function, First fundamental theorem of integral calculus and Second fundamental theorem of integral calculus

• Definite Integral as the Limit of a Sum 
• Evaluation of Simple Integrals of the Following Types and Problems 
• Methods of Integration: Integration by Parts 
  • `int(u.v) dx = u intv dx - int((du)/(dx)).(intvdx) dx`
  • Integral of the type ∫ e^x [ f(x) + f'(x)] dx = e^xf(x) + C
  • Integrals of some more types

  1. `I = int sqrt(x^2 - a^2) dx = x/2 sqrt(x^2 - a^2) - a^2/2 log | x + sqrt(x^2 - a^2 )| + C`
  2. `I = int sqrt(x^2 - a^2) dx = x/2 sqrt(x^2 - a^2) - a^2/2 log | x + sqrt(x^2 - a^2 )| + C`
  3. `I = int sqrt(x^2 - a^2) dx = x/2 sqrt(x^2 - a^2) - a^2/2 log | x + sqrt(x^2 - a^2 )| + C`
• Methods of Integration: Integration Using Partial Fractions 

  No	From of the rational function	Form of the partial fraction
  1	`(px + q )/((x-a)(x-b))`a ≠ b	`A/(x-a) + B/(x-b)`
  2	`(px+q)/(x-a)^2`	`A/(x-a) + B/(x-a)^2`
  3	`((px)^2 + qx +r)/((x-a)(x-b)(x-c))`	`A/(x-a)+B/(x-b) + C /(x-c)`
  4	`((px)^2 + qx + r)/((x-a)^2 (x-b))`	` A/(x-a) + B/(x-a)^2 +C/(x-b)`
  5	`((px)^2 + qx +r)/((x-a)(x^2 + bx +c))`	`A/(x-a) + (Bx + C)/ (x^2 + bx +c)`,

• Methods of Integration: Integration by Substitution 
  • ∫ tan x dx = log | sec x |  + C
  • ∫ cot x dx = log | sin x | + C
  • ∫ sec x dx = log | sec x + tan x | + C
  • ∫ cosec x dx = log | cosec x – cot x | + C
• Integration as an Inverse Process of Differentiation 

  Derivatives	Integrals (Anti derivatives)
  `d/(dx) (x^(n+1)/(n+1)) = x^n`	`int x^n dx = x^(n+1)/(n+1) + "C"`, n ≠ –1
  `d/(dx)`(x) = 1	`int dx` = x + C
  `d/(dx)`(sin x) = cos x	`int` cos x dx = sin x +C
  `d/(dx)` (-cos x) = sin x	`int`sin x dx = -cos x +C
  `d/(dx)` (tan x) = sec2x	`int sec^2 x` dx = tanx + C
  `d/(dx)`(-cot x) = `cosec^2x `	`int cosec^2x` dx = -cot x +C
  `d/(dx)` (sec x) = sec x tan x	`int` sec x tan x dx = sec x +C
  `d/(dx)` (-cosecx) = cosec x cot x	`int` cosec x cot x dx = -cosec x +C
  `d/(dx) (sin^-1) = 1/(sqrt(1-x^2))`	`int (dx)/(sqrt(1-x^2))= sin^(-1) x +C `
  `d/(dx) (-cos^(-1)) = 1/(sqrt (1-x^2))`	`int (dx)/(sqrt (1-x^2))= -cos^(-1) x + C `
  `d/(dx) (tan^(-1) x) = 1/(1+x^2)`	`int (dx)/(1+x^2)= tan^(-1) x + C `
  `d/(dx) (-cot^(-1) x) = 1/(1+x^2)`	`int (dx)/(1+x^2)= -cot^(-1) x + C `
  `d/(dx) (sec^(-1) x) = 1/(x sqrt (x^2 - 1))`	`int (dx)/(x sqrt (x^2 - 1))`= `sec^(-1)` x + C
  `d/(dx) (-cosec^(-1) x) = 1/(x sqrt (x^2 - 1))`	`int (dx)/(x sqrt (x^2 - 1))=-cosec^(-1) x + C `
  `d/(dx)(e^x) = e^x`	`int e^x dx = e^x + C`
  `d/(dx) (log|x|) = 1/x`	`int 1/x dx = log|x| +C`
  `d/(dx) ((a^x)/(log a)) = a^x`	`int a^x dx = a^x/log a` +C

• 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.
• Definite integrals as a limit of a sum, Fundamental Theorem of Calculus (without proof).
• Basic properties of definite integrals and evaluation of definite integrals.

• Area of the Region Bounded by a Curve and a Line 
  • Circle-line, elipse-line, parabola-line
• Area Between Two Curves 
• Area Under Simple Curves 
  • Simple curves: lines, parabolas, polynomial functions

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

• Methods of Solving First Order, First Degree Differential Equations 
  • Linear Differential Equations 
  • Homogeneous Differential Equations 
  • Differential Equations with Variables Separable Method 
• Solutions of Linear Differential Equation 
  • Solutions of linear differential equation of the type:

  1. `dy/dx` + py = q, where p and q are functions of x or constants.
  2. `dx/dy` + px = q, where p and q are functions of y or constants.

  • Differential equations, order and degree.
  • 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`.
• Formation of a Differential Equation Whose General Solution is Given 
• General and Particular Solutions of a Differential Equation 
• Order and Degree of a Differential Equation 
• Basic Concepts of Differential Equation 
• Procedure to Form a Differential Equation that Will Represent a Given Family of Curves 

• 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 solutions of homogeneous differential equations of 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.

• Direction Cosines 
• Properties of Vector Addition 
• Geometrical Interpretation of Scalar 
• Scalar Triple Product of Vectors 
• Product of Two Vectors 
  • Vector (Or Cross) Product of Two Vectors 
  • Scalar (Or Dot) Product of Two Vectors 
  • Projection of a Vector on a Line 
• Position Vector of a Point Dividing a Line Segment in a Given Ratio 
• Multiplication of a Vector by a Scalar 
• Addition of Vectors 
• Introduction of Vector 
• Magnitude and Direction of a Vector 
• Basic Concepts of Vector Algebra 
  • Position Vector
  • Direction Cosines and Direction Ratios of a Vector
• Vectors and Their Types 
  • Zero Vector
  • Unit Vector
  • Co-initial and Co-terminus Vectors
  • Equal Vectors
  • Negative of a Vector
  • Collinear Vectors
  • Free Vectors
  • Localised Vectors
• Components of a Vector 
• Section Formula 
  • Section formula for internal division
  • Midpoint formula
  • Section formula for external division
• Vector Joining Two Points 
• Vectors Examples and Solutions 
• Introduction of Product of Two 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, scalar triple product of vectors.

• Three - Dimensional Geometry Examples and Solutions 
• Introduction of Three Dimensional Geometry 
• Plane 
  • Equation of a Plane Passing Through Three Non Collinear Points 
  • Intercept Form of the Equation of a Plane 
  • Equation of a Plane in Normal Form 
  • Equation of a Plane Perpendicular to a Given Vector and Passing Through a Given Point 
  • Plane Passing Through the Intersection of Two Given Planes 
• Relation Between Direction Ratio and Direction Cosines 
• Coplanarity of Two Lines 
• Distance of a Point from a Plane 
• Angle Between Line and a Plane 
• Angle Between Two Planes 
• Angle Between Two Lines 
• Vector and Cartesian Equation of a Plane 
• Shortest Distance Between Two Lines 
  • Coplanar
  • Skew lines
  • Distance between two skew lines
  • Distance between parallel lines
• Equation of a Line in Space 
  • Equation of a line through a given point and parallel to a given vector `vec b`
  • Equation of a line passing through two given points
• Direction Cosines and Direction Ratios of a Line 
  • Relation between the direction cosines of a line
  • Direction cosines of a line passing through two points
  • Direction cosines/ratios of a line joining two points

• Direction cosines and direction ratios of a line joining two points.
• Cartesian equation 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.

• Introduction of Linear Programming 
  • Definition of related terminology such as constraints, objective function, optimization.
• Mathematical Formulation of Linear Programming Problem 
• Different Types of Linear Programming Problems 
  • Different types of linear programming (L.P.) problems

  1. Manufacturing problem
  2. Diet Problem
  3. Transportation problem
• Graphical Method of Solving Linear Programming Problems 
  • Graphical method of solution for problems in two variables
  • Feasible and infeasible regions and bounded regions
  • Feasible and infeasible solutions
  • Optimum feasible solution
• Linear Programming Problem and Its Mathematical Formulation 

• Introduction, related terminology such as constraints, objective function, optimization, different types of linear programming (LP.) problems, mathematical formulation of LP. problems, graphical method of solution for problems in two variables, feasible and infeasible regions (bounded or unbounded), feasible and infeasible solutions, optimal feasible solutions (up to three non-trivial constraints).

• Variance of a Random Variable 
• Probability Examples and Solutions 
• Conditional Probability 
• Multiplication Theorem on Probability 
• Independent Events 
• Bayes’ Theorem 
  • Partition of a sample space
  • Theorem of total probability
• Random Variables and Its Probability Distributions 
  • Probability distribution of a random variable
• Mean of a Random Variable 
• Bernoulli Trials and Binomial Distribution 
• Introduction of Probability 
  • Random experiment
  • Outcome
  • Equally likely outcomes
  • Sample space
  • Event
• Properties of Conditional 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. Repeated independent (Bernoulli) trials and Binomial distribution.