#### Topics

##### Mathematical Logic

- Statements - Introduction in Logic
- Sentences and Statement in Logic
- Truth Value of Statement
- Open Sentences in Logic
- Compound Statement in Logic
- Quantifier and Quantified Statements in Logic
- Logical Connective, Simple and Compound Statements
- Logical Connective, Simple and Compound Statements
- Examples Related to Real Life and Mathematics
- Statement Patterns and Logical Equivalence
- Algebra of Statements
- Difference Between Converse, Contrapositive, Contradiction
- Application of Logic to Switching Circuits

##### Mathematical Logic

- Concept of Statements
- Truth Value of Statement
- Logical Connective, Simple and Compound Statements
- Statement Patterns and Logical Equivalence
- Tautology, Contradiction, and Contingency
- Duality
- Quantifier and Quantified Statements in Logic
- Negations of Compound Statements
- Converse, Inverse, and Contrapositive
- Algebra of Statements
- Application of Logic to Switching Circuits

##### Matrics

##### Trigonometric Functions

##### Pair of Straight Lines

##### Vectors

- Representation of Vector
- Vectors and Their Types
- Algebra of Vectors
- Coplanar Vectors
- Vector in Two Dimensions (2-D)
- Three Dimensional (3-D) Coordinate System
- Components of Vector
- Position Vector of a Point P(X, Y, Z) in Space
- Component Form of a Position Vector
- Vector Joining Two Points
- Section Formula
- Scalar Product of Vectors (Dot)
- Vector Product of Vectors (Cross)
- Scalar Triple Product of Vectors
- Vector Triple Product
- Addition of Vectors

##### Line and Plane

##### Linear Programming

##### Matrices

- Elementary Transformations
- Inverse by Elementary Transformation
- Elementary Transformation of a Matrix Revision of Cofactor and Minor
- Inverse of a Matrix Existance
- Adjoint Method
- Addition of Matrices
- Solving System of Linear Equations in Two Or Three Variables Using Reduction of a Matrix Or Reduction Method
- Solution of System of Linear Equations by – Inversion Method

##### Differentiation

##### Applications of Derivatives

##### Indefinite Integration

##### Definite Integration

##### Application of Definite Integration

##### Differential Equations

##### Probability Distributions

##### Binomial Distribution

##### Trigonometric Functions

- Trigonometric equations
- General Solution of Trigonometric Equation of the Type
- Solutions of Triangle
- Hero’s Formula in Trigonometric Functions
- Napier Analogues in Trigonometric Functions
- Basic Concepts of Trigonometric Functions
- Inverse Trigonometric Functions - Principal Value Branch
- Graphs of Inverse Trigonometric Functions
- Properties of Inverse Trigonometric Functions

##### Pair of Straight Lines

- Pair of Lines Passing Through Origin - Combined Equation
- Pair of Lines Passing Through Origin - Homogenous Equation
- Theorem - the Joint Equation of a Pair of Lines Passing Through Origin and Its Converse
- Acute Angle Between the Lines
- Condition for Parallel Lines
- Condition for Perpendicular Lines
- Pair of Lines Not Passing Through Origin-combined Equation of Any Two Lines
- Point of Intersection of Two Lines

##### Circle

- Tangent of a Circle - Equation of a Tangent at a Point to Standard Circle
- Tangent of a Circle - Equation of a Tangent at a Point to General Circle
- Condition of tangency
- Tangents to a Circle from a Point Outside the Circle
- Director circle
- Length of Tangent Segments to Circle
- Normal to a Circle - Equation of Normal at a Point

##### Conics

##### Vectors

- Vectors Revision
- Collinearity and Coplanarity of Vectors
- Linear Combination of Vectors
- Condition of collinearity of two vectors
- Conditions of Coplanarity of Three Vectors
- Section Formula
- Midpoint Formula for Vector
- Centroid Formula for Vector
- Basic Concepts of Vector Algebra
- Scalar Triple Product of Vectors
- 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

##### Three Dimensional Geometry

##### Line

##### 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 a Plane Passing Through Three Non Collinear Points
- Equation of Plane Passing Through the Intersection of Two Given Planes
- 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

##### Linear Programming Problems

##### Continuity

- Introduction of Continuity
- Continuity of a Function at a Point
- Defination of Continuity of a Function at a Point
- Discontinuity of a Function
- Types of Discontinuity
- Concept of Continuity
- Algebra of Continuous Functions
- Continuity in Interval - Definition
- 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

##### Differentiation

- Revision of Derivative
- Relationship Between Continuity and Differentiability
- Every Differentiable Function is Continuous but Converse is Not True
- Derivatives of Composite Functions - Chain Rule
- Derivative of Inverse Function
- Derivatives of Inverse Trigonometric Functions
- Derivatives of Implicit Functions
- Exponential and Logarithmic Functions
- Derivatives of Functions in Parametric Forms
- Derivative of Functions in Product of Function Form
- Derivative of Functions in Quotient of Functions Form
- Higher Order Derivative
- Second Order Derivative

##### Applications of Derivative

##### Integration

- 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
- Properties of Definite Integrals
- Evaluation of Definite Integrals by Substitution
- Integration by Non-repeated Quadratic Factors

##### Applications of Definite Integral

##### Differential Equation

- 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
- Differential Equations with Variables Separable Method
- Homogeneous Differential Equations
- Linear Differential Equation
- Applications of Differential Equation

##### Statistics

##### Probability Distribution

- Conditional Probability
- Random Variables and Its Probability Distributions
- 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

#### notes

A rational function is defined as the ratio of two polynomials in the form `(P(x))/(Q(x))` , where P (x) and Q(x) are polynomials in x and Q(x) ≠ 0. If the degree of P(x) is less than the degree of Q(x), then the rational function is called proper, otherwise, it is called improper. The improper rational functions can be reduced to the proper rational functions by long division process. Thus, if `(P(x))/(Q(x))` is improper ,then `(P(x))/(Q(x))` = T(x) + `(P_1(x))/(Q(x))`

where T(x) is a polynomial in x and `(P_1(x))/(Q(x))` is a proper rational function. As we know how to integrate polynomials, the integration of any rational function is reduced to the integration of a proper rational function. The rational functions which we shall consider here for integration purposes will be those whose denominators can be factorised into linear and quadratic factors. Assume that we want to evaluate `int (P(x))/(Q(x))` dx , where `(P(x))/(Q(x))` is proper rational function. It is always possible to write the integrand as a sum of simpler rational functions by a method called partial fraction decomposition. The above table shows that the types of simpler partial fractions that are to be associated with various kind of rational functions.

Sr.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)`, |

Point (5) `x^2 + bx + c` cannot be factorised further

#### description

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)`, |