#### 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

#### definition

The exponential function with positive base b > 1 is the function y = f(x) = bx

The graph of y = 10x is given in the Fig.

It is advised that the reader plots this graph for particular values of b like 2, 3 and 4. Following are some of the salient features of the exponential functions:

(1) Domain of the exponential function is R, the set of all real numbers.

(2) Range of the exponential function is the set of all positive real numbers.

(3) The point (0, 1) is always on the graph of the exponential function (this is a restatement of the fact that b0 = 1 for any real b > 1).

(4) Exponential function is ever increasing; i.e., as we move from left to right, the graph rises above.

(5) For very large negative values of x, the exponential function is very close to 0. In other words, in the second quadrant, the graph approaches x-axis (but never meets it).

Exponential function with base 10 is called the common exponential function.

The sum of the series

`1+1/1! + 1/2!+...`

is a number between 2 and 3 and is denoted by e. Using this e as the base we obtain an extremely important exponential function y = ex. This is called natural exponential function.

#### definition

Let b > 1 be a real number. Then we say logarithm of a to base b is x if `b^x` = a. Logarithm of a to base b is denoted by `log_b` a. Thus logb a = x if bx = a. Let us work with a few explicit examples to get a feel for this.

A base b > 1, we may look at logarithm as a function from positive real numbers to all real numbers. This function, called the logarithmic function, is defined by

`log_b : R^+ -> R`

`x -> log_b x = y if b^y = x`

As before if the base b = 10, we say it is common logarithms and if b = e, then we say it is natural logarithms. Often natural logarithm is denoted by ln. In this chapter, log x denotes the logarithm function to base e, i.e., ln x will be written as simply log x. Fig.

the plots of logarithm function to base 2, e and 10.

Some of the important observations about the logarithm function to any base b > 1 are listed below:

(1) We cannot make a meaningful definition of logarithm of non-positive numbers and hence the domain of log function is `R^+`.

(2) The range of log function is the set of all real numbers.

(3) The point (1, 0) is always on the graph of the log function.

(4) The log function is ever increasing, i.e., as we move from left to right the graph rises above.

(5) For x very near to zero, the value of log x can be made lesser than any given real number. In other words in the fourth quadrant the graph approaches y-axis (but never meets it).

(6) The plot of y = ex and y = ln x. It is of interest to observe that the two curves are the mirror images of each other reflected in the line y = x. Fig.

Two properties of ‘log’ functions are proved below:

(1) There is a standard change of base rule to obtain `log_a p` in terms of `log_b`p. Let `log_a` p = α, `log_b` p = β and `log_b a` = γ. This means `a^α` = p, `b^β` = p and `b^γ` = a. Substituting the third equation in the first one, we have

`(b^γ)^α = b^(γα)` = p

Using this in the second equation, we get `b^β` = p = `b^(γα)`

which implies β = αβ or α = `β/γ` . But then

`log_a p = (log_bp)/(log_ba)`

(2) Another interesting property of the log function is its effect on products. Let `log_b` pq = α. Then `b^α` = pq.

If `log_b` p = β and `log_b` q = γ, then `b^β` = p and `b^γ` = q.

But then `b^α` = pq = `b^βb^γ` = `b^(β + γ)`

which implies α = β + γ, i.e.,

`log_b` pq = `log_b` p + `log_b` q

A particularly interesting and important consequence of this is when p = q.

In this case the above may be rewritten as `log_b` `p^2` = `log_b` p + `log_b` p = 2 log p

An easy generalisation of this (left as an exercise!) is

`log_b` `p^n` = n log p

for any positive integer n. In fact this is true for any real number n, but we will not attempt to prove this. On the similar lines the reader is invited to verify

`log _b x/y = log_b x - log _b y`