Mathematics HSC Science Class 12 Tamil Nadu Board of Secondary Education Syllabus 2024-25

• Adjoint of a Square Matrix
• Definition of inverse matrix of a square matrix
• Properties of inverses of matrices
• Application of matrices to Geometry
• Application of matrices to Cryptography

• Elementary row and column operations
• Row-Echelon form
• Rank of a Matrix
• Gauss-Jordan Method

• Formation of a System of Linear Equations
• System of Linear Equations in Matrix Form
• Solution to a System of Linear equations

1. Matrix Inversion Method
2. Cramer’s Rule
3. Gaussian Elimination Method

• Non-homogeneous Linear Equations 
• Homogeneous system of linear equations 

• Powers of imaginary unit i

• Rectangular form
• Argand plane A complex number 
• Algebraic operations on complex numbers

• Properties of complex numbers

• Geometrical representation of conjugate of a complex number
• Properties of Complex Conjugates

• Properties of Modulus of a complex number
• Square roots of a complex number 

• Polar form of a complex number
• Euler’s Form of the complex number

• de Moivre's Theorem
• Finding nth roots of a complex number
• The nth roots of unity

• Different types of Polynomial Equations
• Quadratic Equations

• Vieta’s formula for Quadratic Equations
• Vieta’s formula for Polynomial Equations

1. The Fundamental Theorem of Algebra
2. Vieta’s Formula 
3. Formation of Polynomial Equations with given Roots

• Imaginary Roots
• Irrational Roots
• Rational Roots

• Imaginary or Surds Roots
• Polynomial equations with Even Powers Only
• Zero Sum of all Coefficients
• Equal Sums of Coefficients of Odd and Even Powers
• Roots in Progressions
• Partly Factored Polynomials

• Rational Root Theorem
• Reciprocal Equations
• Non-polynomial Equations

• Statement of Descartes Rule
• Attainment of bounds

1. Bounds for the number of real roots
2. Bounds for the number of Imaginary (Nonreal Complex)roots

• Introduction of Inverse Trigonometric Functions

• Domain and Range of trigonometric functions
• Graphs of functions
• Amplitude and Period of a graph
• Inverse functions
• Graphs of inverse functions

• The graph of sine function
• Properties of the sine function
• The inverse sine function and its properties
• Graph of the inverse sine function

• Graph of cosine function
• Properties of the cosine function
• The inverse cosine function and its properties
• Graph of the inverse cosine function

• The graph of tangent function
• Properties of the tangent function
• The inverse tangent function and its properties
• Graph of the inverse tangent function

• Graph of the cosecant function
• The inverse cosecant function
• Graph of the inverse cosecant function

• The graph of the secant function
• Inverse secant function
• Graph of the inverse secant function

• The graph of the cotangent function
• Inverse cotangent function
• Graph of the inverse cotangent function

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

3.4.1 The equation of a circle when the centre and radius are given

3.4.2 Equation of a circle when the end points of a diameter are given

3.4.3 General equation of a circle

3.4.4 Parametric form of a circle 

3.4.5 Tangents 

• Length of the tangent to the circle

3.5.1 Parabola

3.5.2 Definitions regarding a parabola: y² = 4ax

3.5.3 Other standard parabolas

3.5.4 General form of the standard equation of a parabola, which is open rightward 

• Geometric description of conic section
• Degenerate Forms
•  Identifying the conics from the general equation of the conic 

• Parametric equations

•  Equation of tangent and normal to the parabola y²=4ax
• Equations of tangent and normal to Ellipse and Hyperbola
• Condition for the line y=mx+c to be a tangent to the conic sections 

• Parabola
• Ellipse
• Hyperbola
• Reflective property of parabola
• Reflective Property of an Ellipse
• Reflective Property of a Hyperbola

• Geometrical interpretation
• Application of dot and cross products in plane Trigonometry
• Application of dot and cross products in Geometry
• Application of dot and cross product in Physics

• Different forms of equation of a straight line
• A point on the straight line and the direction of the straight line are given 
• Straight Line passing through two given points
• Angle between two straight lines 
• Point of intersection of two straight lines
• Shortest distance between two straight lines

• Equation of a plane when a normal to the plane and the distance of the plane from the origin are given
• Equation of a plane perpendicular to a vector and passing through a given point
• Intercept form of the equation of a plane
• Equation of a plane passing through three given non-collinear points
• Equation of a plane passing through a given point and parallel to two given non-parallel vectors
• Equation of a plane passing through two given distinct points and is parallel to a non-zero vector
• Condition for a line to lie in a plane
• Condition for coplanarity of two lines
• Equation of plane containing two non-parallel coplanar lines
• Angle between two planes
• Angle between a line and a plane
• Distance of a point from a plane
• Distance between two parallel planes
• Equation of line of intersection of two planes
• Equation of a plane passing through the line of intersection of two given planes

• The coordinates of the image of a point in a plane

• Early Developments

• Derivative as slope
• Derivative as rate of change
• Related rates
• Equations of Tangent and Normal
• Angle between two curves

• Rolle’s Theorem
• Lagrange’s Mean Value Theorem
• Applications

• A Limit Process
• The l’Hôpital’s Rule
• Indeterminate forms
•  Indeterminate forms 0⁰,1∞ and ∞⁰

• Monotonicity of functions
• Absolute maxima and minima
• Relative Extrema on an Interval
• Extrema using First Derivative Test

• Concavity, Convexity, and Points of Inflection
• Extrema using Second Derivative Test

• Symmetry
• Asymptotes

• Linear Approximation
• Errors: Absolute Error, Relative Error, and Percentage Error

• Recall of Limit and Continuity of Functions of One Variable

• Successive partial derivatives
• Euler’s theorem and its applications

• Function of Function Rule
• Homogeneous Functions and Euler’s Theorem

• Riemann Integral
• Limit Formula to Evaluate

• Area of the region bounded by a curve, x – axis and the lines x = a and x = b
• Area of the region bounded by a curve, y–axis and the lines y = cand y = d
• Area of the region bounded between two curves

• Formation of Differential equations from Physical Situations
• Formation of Differential Equations from Geometrical Problems

• Variables Separable Method 
• Substitution Method 
• Homogeneous Form or Homogeneous Differential Equation
• General solution and particular solution
• Differential Equation in which variables are separable
• Linear differential equations of first order

• Population growth
• Radioactive decay
• Newton’s Law of cooling/warming
• Mixture problems

• Definition of a random variable
• Types of Random Variable
• Discrete random variable
• Continuous random variable

• Discrete random variable
• Continuous random variable
• Probability Mass Function
• Cumulative Distribution Function or Distribution Function
• Cumulative Distribution Function from Probability Mass function
• Probability Mass Function from Cumulative Distribution Function 

• The definition of continuous random variables
• Probability density function
• Distribution function (Cumulative distribution function)
• Distribution function from Probability density function
• Probability density function from Probability distribution function

• Mean
• Variance
• Expected value and Variance
• Properties of Mathematical expectation

• The One point distribution
• The Two point distribution
• The Bernoulli distribution
• The Binomial Distribution

• Commutative Binary Operations
• Associative Binary Operations
• Identity Binary Operation,
• Invertible Binary Operation

• Statement and its truth value
• Compound Statements, Logical Connectives, and Truth Tables
• Tautology, Contradiction, and Contingency
• Duality
• Logical Equivalence