#### Topics

##### Mathematical Logic

- Statements - Introduction in Logic
- Sentences and Statement in Logic
- Truth Value of Statement in Logic
- Open Sentences in Logic
- Compound Statement in Logic
- Quantifier and Quantified Statements in Logic
- Logical Connectives
- Truth Tables of 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, Switching Table.

##### Mathematical Logic

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

##### 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
- Dot/Scalar Product of Vectors
- Cross/Vector Product of Vectors
- Scalar Triple Product of Vectors
- Vector Triple Product
- Addition of Vectors

##### Line and Plane

##### Linear Programming

##### Matrices

- Elementary Operation (Transformation) of a Matrix
- 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
- Solution of a 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

Let `L_1` and `L_2` be two lines passing through the origin and with direction ratios `a_1, b_1, c_1` and `a_2, b_2, c_2`, respectively. Let P be a point on `L_1` and Q be a point on `L_2`. Consider the directed lines OP and OQ as given in following fig.

Let θ be the acute angle between OP and OQ. Now recall that the directed line segments OP and OQ are vectors with components `a_1, b_1, c_1` and `a_2, b_2, c_2`, respectively. Therefore, the angle θ between them is given by

` cos θ = |(a_1a_2 + b_1b_2 + c_1c_2)/(sqrt(a_1^2 + b_1^2 + c_1^2) sqrt (a_2^2 + b_2^2 + c_2^2))| ` ...(1)

The angle between the lines in terms of sin θ is given by

`sin θ = sqrt (1- cos^2 θ) `

`= sqrt (1- (a_1a_2 + b_1b_2 + c_1c_2)^2 / ((a_1^2 + b_1^2 + c_1^2) (a_2^2 + b_2^2 + c_2^2)))`

`= sqrt ((a_1^2 + b_1^2 + c_1^2) (a_2^2 + b_2^2 + c_2^2) - (a_1a_2 + b_1b_2 + c_1c_2)^2 ) / (sqrt(a_1^2 + b_1^2 + c_1^2 ) sqrt(a_2^2 + b_2^2 +c_2^2))`

`= sqrt ((a_1 b_2 - a_2 b_1) ^2 + (b_1 c_2 - b_2 c_1)^ 2 + (c_1 a_2 - c_2 a_1) ^2 )/ (sqrt (a_1^2 + b_1^2 + c_1^2) sqrt (a_2^2 + b_2^2 + c_2^2))` ...(2)`

If instead of direction ratios for the lines `L_1` and `L_2`, direction cosines, namely, `l_1, m_1, n_1` for `L_1` and `l_2, m_2, n_2` for `L_2` are given, then (1) and (2) takes the following form:

cos θ = `|l_1 l_2 + m_1m_2 + n_1n_2|` `("as" l_1^2 + m_1^2 + n_1^2 = 1 = l_2^2 + m_2^2 + n_2^2)` ...(3)

and sin θ = `sqrt( (l_1m_2 - l_2m_1)^2 - (m_1n_2 - m_2n_1)^2 + (n_1l_2 - n_2 l_1)^2)` ..(4)

Two lines with direction ratios `a_1, b_1, c_1` and `a_2, b_2, c_2` are

(i) perpendicular i.e. if θ = 90° by (1)

`a_1a_2 + b_1b_2 + c_1c_2 `= 0

(ii) parallel i.e. if θ = 0 by (2)

`a_1/a_2 = b_1/b_2 = c_1/c_2`

Now, we find the angle between two lines when their equations are given. If θ is acute the angle between the lines

`vec r = vec a_1 + lambda vec b_1` and `vec r = vec a_2 + mu vec b _2`

then `cos theta = |(vec b _1 . vec b_2)/(|vec b _1|| vec b _2|)|`