## Topics with syllabus and resources

- Mathematical Logic
- Logical Connectives
Logical connectives : conjunction, disjunction, negation, implication/ conditional, biconditional

- Examples Related to Real Life and Mathematics
Examples related to real life and mathematics

- Statement Patterns and Logical Equivalence
tautology, contradiction, contingency, duality, negation of compound statement, contrapositive, converse, inverse

- Algebra of Statements
idempotent law, associative law, commutative law, distributive law, identity law, complement law, involution law, DeMorgan’s laws

- Application - Introduction to Switching Circuits
simple examples

- Logical Connectives

- Statements - Introduction
- Sentences and statement
- Truth value of statement
- Open sentences
- Compound statement
- Quantifier and quantified statements
- Logical connectives : conjunction, disjunction, negation, implication/ conditional, biconditional
- Truth tables of compound statements
- Examples related to real life and mathematics
- Statement patterns and logical equivalence - tautology, contradiction, contingency, duality, negation of compound statement, contrapositive, converse, inverse
- Algebra of statements - idempotent law, associative law, commutative law, distributive law, identity law, complement law, involution law, DeMorgan’s laws
- Difference between converse, contrapositive, contradiction
- Application - introduction to switching circuits (simple examples).

- Elementary Operation (Transformation) of a Matrix
- Matrices
- Determinants
- Operations on Matrices
- Solution of System of Linear Equations by – Inversion Method

- Elementary transformation of a matrix revision of cofactor and minor
- Elementary row transformation
- Elementary column transformation
- Inverse of a matrix existance and uniqueness of inverse of a matrix
- Inverse by elementary transformation
- Adjoint method
- Application - solution of system of linear equations by – reduction method, inversion method.

- Trigonometric Functions
- General Solution of Trigonometric Equation of the Type
sinθ, = 0, cosθ = 0, tanθ = 0, sinθ = sinα, cosθ = cosα, tanθ = tanα, sin2 θ = sin2 α, cos2 θ = cos2 α, tan2 θ = tan2 α, acosθ + bsinθ = C

- Solution of a Triangle
Polar coordinates, Sine rule, Cosine rule, Projection rule, Area of a triangle, Application

- General Solution of Trigonometric Equation of the Type
- Basic Concepts of Trigonometric Functions
sine, cosine, tangent, cotangent, secant, cosecant function

- Inverse Trigonometric Functions
- Properties of Inverse Trigonometric Functions
Inverse of Sin, Inverse of cosin, Inverse of tan, Inverse of cot, Inverse of Sec, Inverse of Cosec

- Trigonometric equations -
- General solution of trigonometric equation of the type : sinθ, = 0, cosθ = 0, tanθ = 0, sinθ = sinα, cosθ = cosα, tanθ = tanα, sin2 θ = sin2 α, cos2 θ = cos2 α, tan2 θ = tan2 α, acosθ + bsinθ = C solution of a triangle :
- Polar coordinates
- Sine rule
- Cosine rule
- Projection rule
- Area of a triangle
- Application
- Hero’s formula
- Napier Analogues
- Inverse trigonometric functions - definitions, domain, range, principle values, graphs of inverse trigonometric function, properties of inverse functions.

- Pair of Straight Lines
- Acute Angle Between the Lines
Acute Angle Between the Lines represented by ax

^{2}+2hxy+by^{2}=0

- Pair of lines passing through origin - Combined equation, Homogenous equation
- Theorem - the joint equation of a pair of lines passing through origin and its converse
- Acute angle between the lines represented by ax
^{2}+2hxy+by^{2}=0 - Condition for parallel lines
- Condition for perpendicular lines
- Pair of lines not passing through origin-combined equation of any two lines
- Condition that the equation ax2 +2hxy+by2 +2gx+2fy+c=0 should represent a pair of lines (without proof)
- Acute angle between the lines (without proof)
- Condition of parallel and perpendicular lines
- Point of intersection of two lines.

- Circle
- Condition of tangency
only for line y = mx + c to the circle x

^{2}+ y^{2}= a^{2}

- Condition of tangency

- Tangent of a circle - equation of a tangent at a point to 1) standard circle,2) general circle
- Condition of tangency only for line y = mx + c to the circle x
^{2}+ y^{2}= a^{2} - Tangents to a circle from a point outside the circle
- Director circle
- Length of tangent segments
- Normal to a circle - equation of normal at a point.

- Conics
- Tangents and normals - equations of tangent and normal at a point
for parabola, ellipse, hyperbola

- Condition of tangency
for parabola, ellipse, hyperbola

- Tangents in terms of slope
for parabola, ellipse, hyperbola

- Properties of Tangents and Normals to Conics
without proof

- Tangents and normals - equations of tangent and normal at a point

- Tangents and normals - equations of tangent and normal at a point for parabola, ellipse, hyperbola
- Condition of tangency for parabola,ellipse, hyperbola
- Tangents in terms of slope for parabola, ellipse, hyperbola
- Tangents from a point outside conics
- Locus of points from which two tangents are mutually perpendicular
- Properties of tangents and normals to conics (without proof).

- Vectors
- Section formula
for internal and external division

- Basic Concepts of Vector Algebra
- Position Vector
- Direction Cosines and Direction Ratios of a Vector

- Scalar Triple Product of Vectors
volume of a parallelepiped, co-planarity

- Geometrical Interpretation of Scalar Triple Product

- Revision
- Collinearity and coplanarity of vectors :
- Linear combination of vectors
- Condition of collinearity of two vectors
- Conditions of coplanarity of three vectors
- Section formula : section formula for internal and external division
- Midpoint formula
- Centroid formula
- Scaler triple product : definition, formula, properties, geometrical interpretation of scalar triple product
- Application of vectors to geometry medians of a triangle are concurrent
- Altitudes of a triangle are concurrent
- Angle bisectors of a triangle are concurrent
- Diagonals of a parallelogram bisect each other and converse
- Median of trapezium is parallel to the parallel sides and its length is half the sum of parallel sides
- Angle subtended on a semicircle is right angle.

- Introduction of Three Dimensional Geometry
- Direction Cosines and Direction Ratios of a Line
- Angle Between Two Lines
- Three - Dimensional Geometry
- Relation Between Direction Ratio and Direction Cosines
- Three Dimensional Geometry - Problems

- Direction cosines and direction ratios: direction angles, direction cosines, direction ratios
- Relation between direction ratio and direction cosines
- Angle between two lines
- Condition of perpendicular lines.

- Line
- Shortest Distance Between Two Lines
- Distance between two skew lines
- Distance between parallel lines

- Equation of a Line in Space

- Equation of line passing through given point and parallel to given vector
- Equation of line passing through two given points
- Distance of a point from a line
- Distance between two skew lines
- Distance between two parallel lines (vector approach).

- Plane
- Vector and Cartesian Equation of a Plane
- Angle Between Two Planes
- Angle Between Line and a Plane
- Coplanarity of Two Lines
- Distance of a Point from a Plane

- Equation of plane in normal form
- Equation of plane passing through the given point and perpendicular to given vector
- Equation of plane passing through the given point and parallel to two given vectors
- Equation of plane passing through three noncollinear points
- Equation of plane passing through the intersection of two given planes
- Angle between two planes
- Angle between line and plane
- Condition for the coplanarity of two lines
- Distance of a point from a plane (vector approach).

- Introduction of Linear Programming
- Mathematical Formulation of Linear Programming Problem
- Linear Programming
- Different Types of Linear Programming Problems
Different types of linear programming (L.P.) problems:-

- Manufacturing problem
- Diet Problem
- Transportation problem

- Graphical Method of Solving Linear Programming Problems

- Introduction of L.P.P.
- Definition of constraints
- Objective function
- Optimization
- Constraint equations
- Nonnegativity restrictions
- Feasible and infeasible region
- Feasible solutions
- Mathematical formulation-mathematical formulation of L.P.P
- Different types of L.P.P. problems
- Graphical solutions for problem in two variables
- Optimum feasible solution.

- Introduction of Continuity
- Continuity
- Continuity of a Function at a Point
left hand limit, right hand limit

- Continuity of a Function at a Point
- Concept of Continuity
- Algebra of Continuous Functions
- Exponential and Logarithmic Functions
- Continuity of Some Standard Functions - Polynomial Function
- Continuity of Some Standard Functions - Rational Function
- Continuity of Some Standard Functions - Trigonometric Function
- Continuity - Problems

- Continuity of a function at a point : left hand limit, right hand limit
- Definition of continuity of a function at a point
- Discontinuity of a function
- Types of discontinuity
- Algebra of continuous functions
- Continuity in interval - definition
- Continuity of some standard functions - polynomial, rational, trigonometric, exponential and logarithmic function.

- Derivative
- Derivative of Functions in Product of Function Form
Derivative of Functions Which Are Expressed in Product of Function Form

- Derivative of Functions in Quotient of Functions Form
Derivative of Functions Which Are Expressed in Quotient of Function Form

- Derivative of Functions in Product of Function Form
- Relationship Between Continuity and Differentiability
left hand derivative and right hand derivative (need and concept)

- Derivatives of Composite Functions - Chain Rule
- Derivatives of Inverse Trigonometric Functions
- Derivatives of Implicit Functions
- Exponential and Logarithmic Functions
- Derivatives of Functions in Parametric Forms
- Higher Order Derivative
Derivative of Functions Which Expressed in Higher Order Derivative Form

- Second Order Derivative

- Revision- revision of derivative
- Relationship between continuity and differentiability-left hand derivative and right hand derivative (need and concept)
- Every differentiable function is continuous but converse is not true
- Derivative of composite function-chain rule
- Derivative of inverse function
- Derivative of inverse trigonometric function :
- Derivative of implicit function definition and examples
- Derivative of parametric function – definition of parametric function
- Exponential and logarithmic function
- Derivative of functions which are expressed in one of the following form a) product of functions, b) quotient of functions, c) higher order derivative, second order derivative d) [f
_{(x)}]^{[g(x)]}

- Mean Value Theorem
- Rate of Change of Bodies Or Quantities
- Increasing and Decreasing Functions
- Tangents and Normals
- Approximations
- Maxima and Minima - Introduction of Extrema and Extreme Values
- Maxima and Minima in Closed Interval
- Maxima and Minima

- Geometrical application-tangent and normal at a point
- Rolle's theorem, and Mean value theorem and their geometrical interpretation (without proof)
- Derivative as a rate measure-introduction
- Increasing and decreasing function
- Approximation (without proof)
- Maxima and minima-introduction of extrema and extreme values
- Maxima and minima in a closed interval
- First derivative test
- Second derivative test.

- Methods of Integration - Integration by Substitution
- Methods of Integration - Integration Using Partial Fractions
- Methods of Integration - Integration by Parts
- Definite Integral as the Limit of a Sum
- Fundamental Theorem of Calculus
Area function, First fundamental theorem of integral calculus and Second fundamental theorem of integral calculus

- Properties of Definite Integrals
- Evaluation of Definite Integrals by Substitution
- Integration

- Indefinite integrals-methods of integration
- Substitution method
- Integrals of the various types
- Integration by parts (reduction formulae are not expected)
- Integration by partial fraction-factors involving repeated and non-repeated linear factors
- Non-repeated quadratic factors
- Definite integral-definite integral as a limit of sum
- Fundamental theorem of integral calculus (without proof)
- Evaluation of definite integral 1) by substitution, 2) integration by parts, properties of definite integrals.

- Area of the Region Bounded by a Curve and a Line
circle-line, elipse-ine, parabola-line

- Area Between Two Curves
- Applications of the Integrations
- Volume of Solid of Revolution
volume of solid obtained by revolving the area under the curve about the axis (simple problems)

- Volume of Solid of Revolution

- Area under the curve : area bounded by curve and axis (simple problems)
- Area bounded by two curves
- volume of solid of revolution-volume of solid obtained by revolving the area under the curve about the axis (simple problems).

- Basic Concepts of Differential Equation
- Order and Degree of a Differential Equation
- General and Particular Solutions of a Differential Equation
- Formation of Differential Equation by Eliminating Arbitary Constant
- Methods of Solving First Order, First Degree Differential Equations
- Differential Equations
- Applications of Differential Equation
Population growth, Bacterial colony growth, Surface area, Newton’s laws of cooling, Radioactive decay

- Applications of Differential Equation

- Definition-differential equation
- Order
- Degree
- General solution
- Particular solution of differential equation
- Formation of differential equation-formation of differential equation by eliminating arbitary constants (at most two constants)
- Solution of first order and first degree differential equation-variable separable method
- Homogeneous differential equation (equation reducible to homogeneous form are not expected)
- Linear differential equation

Applications :-

- Population growth
- Bacterial colony growth
- Surface area
- Newton’s laws of cooling
- Radioactive decay.

- Statistics
- Bivariate Frequency Distribution
bivariate data, tabulation of bivariate data

- Bivariate Frequency Distribution

- Bivariate frequency distribution - bivariate data, tabulation of bivariate data
- Scatter diagram
- Covariance of ungrouped data
- Covariance for bivariate frequency distribution
- Karl Pearson’s coefficient of correlation.

- Conditional Probability
- Random Variables and Its Probability Distributions
- Probability Distribution
- Probability Distribution of a Discrete Random Variable

- Probability distribution of a random variable-definition of a random variable
- Discrete and continuous random variable
- Probability mass function (p.m.f.)
- Probability distribution of a discrete random variable
- Cumulative probability distribution of a discrete random variable
- Expected value, Variance and standard deviation of a discrete random variable
- Probability density function (p.d.f.)
- Distribution function of a continuous random variable.

- Bernoulli Trials and Binomial Distribution
- Bernoulli Trials and Binomial Distribution
- Normal Distribution (P.D.F)
mean, variance and standard deviation, standard normal variable, simple problems

- Normal Distribution (P.D.F)
- Mean of Binomial Distribution (P.M.F.)
- Variance of Binomial Distribution (P.M.F.)
- Standard Deviation of Binomial Distribution (P.M.F.)

- Definition of Bernoulli trial
- Conditions for Binomial distribution
- Binomial distribution (p.m.f.)
- Mean, Variance and standard deviation
- Calculation of probabilities (without proof)
- Normal distribution : p.d.f., mean, variance and standard deviation, standard normal variable, simple problems (without proof).