#### 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

- Elementry Transformations
- Inverse of Matrix
- Application of Matrices
- Applications of Determinants and Matrices

##### Trigonometric Functions

- Trigonometric Equations and Their Solutions
- Solutions of Triangle
- Inverse Trigonometric Functions

##### Pair of Straight Lines

- Combined Equation of a Pair Lines
- Homogeneous Equation of Degree Two
- Angle between lines represented by ax2 + 2hxy + by2 = 0
- General Second Degree Equation in x and y
- Equation of a Line in Space

##### Vectors

- Representation of Vector
- Vectors and Their Types
- Algebra of Vectors
- Coplaner Vector
- 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

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

##### Linear Programming

- Linear Inequations in Two Variables
- Linear Programming Problem (L.P.P.)
- Lines of Regression of X on Y and Y on X Or Equation of Line of Regression
- Graphical Method of Solving Linear Programming Problems
- Linear Programming Problem in Management Mathematics

##### 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

- Differentiation
- Derivatives of Composite Functions - Chain Rule
- Geometrical Meaning of Derivative
- Derivatives of Inverse Functions
- Logarithmic Differentiation
- Derivatives of Implicit Functions
- Derivatives of Parametric Functions
- Higher Order Derivatives

##### Applications of Derivatives

- Applications of Derivatives in Geometry
- Derivatives as a Rate Measure
- Approximations
- Rolle's Theorem
- Lagrange's Mean Value Theorem (LMVT)
- Increasing and Decreasing Functions
- Maxima and Minima

##### Indefinite Integration

##### Definite Integration

- Definite Integral as Limit of Sum
- Fundamental Theorem of Integral Calculus
- Methods of Evaluation and Properties of Definite Integral

##### Application of Definite Integration

- Application of Definite Integration
- Area Bounded by the Curve, Axis and Line
- Area Between Two Curves

##### Differential Equations

- Differential Equations
- Order and Degree of a Differential Equation
- Formation of Differential Equations
- Homogeneous Differential Equations
- Linear Differential Equations
- Application of Differential Equations
- Solution of a Differential Equation

##### Probability Distributions

- Random Variables and Its Probability Distributions
- Types of Random Variables
- Probability Distribution of Discrete Random Variables
- Probability Distribution of a Continuous Random Variable
- Variance of a Random Variable
- Expected Value and Variance of a Random Variable

##### Binomial Distribution

- Bernoulli Trial
- Binomial Distribution
- Mean of Binomial Distribution (P.M.F.)
- Variance of Binomial Distribution (P.M.F.)
- Bernoulli Trials and 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 Inverse 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

- 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

##### 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

- Bernoulli Trials and Binomial Distribution
- Conditions for Binomial Distribution
- Mean of Binomial Distribution (P.M.F.)
- Variance of Binomial Distribution (P.M.F.)
- Standard Deviation of Binomial Distribution (P.M.F.)
- Calculation of Probabilities
- Normal Distribution (P.D.F)

- Vector addition using components
- Components of a vector in two dimensions space
- Components of a vector in three-dimensional space

## Notes

Let us take the points A(1, 0, 0), B(0, 1, 0) and C(0, 0, 1) on the x-axis, y-axis and z-axis, respectively. Then, clearly

`|vec (OA)| = 1, |vec (OB)| = 1` and `|vec (OC)| = 1 `

The vectors `vec (OA)` , `vec (OB)` and `vec (OC)`, each having magnitude 1, are called unit vectors along the axes OX, OY and OZ, respectively, and denoted by `hat i , hat j ,`and `hat k` respectively.

Now, consider the position vector `vec (OP)` of a point P(x, y, z) as in following Fig . Let `P_1` be the foot of the perpendicular from P on the plane XOY.

We, thus, see that `P_1P` is parallel to z-axis. As `hat i,hat j` and `hat k` are the unit vectors along the x, y and z-axes, respectively, and by the definition of the coordinates of P, we have `vec (P_1P) = vec (OR) = z hatk `. Similarly, `vec (QP_1) = vec (OS) = y hat j` and `vec (OQ) = x hat i .`

Therefore, it follows that `vec (OP_1) = vec (OQ )+ vec (QP_1) = x hat i + y hat j`

and `vec (OP) = vec (OP_1) + vec (P_1P) = x hat i + y hat j + z hat k`

Hence, the position vector of P with reference to O is given by

`vec (OP) (or vec r) = x hat i + y hat j + zhat k`

This form of any vector is called its component form. Here, x, y and z are called as the scalar components of `vec r`, and `x hat i, y hat j` and `z hat k` are called the vector components of `vec r` along the respective axes. Sometimes x, y and z are also termed as rectangular components.

The length of the vector `vec r = x hat i + y hat j + z hat k ,` is readily determined by applying the Pythagoras theorem twice. We note that in the right angle triangle `OQP_1` in above fig.

`|vec (OP_1)| = sqrt | vec (OQ)|^2 + |vec (QP_1)|^2 = sqrt (x^2 + y^2),`

and in the right angle triangle `OP_1P`, we have

`vec (OP) = sqrt | vec (OP_1)|^2 + |vec (P_1P)|^2 = (sqrt (x^2 + y^2)+z_2),`

Hence, the length of any vector `vec r = x hat i + y hat j + z hat k` is given by

`|vec r| = |xhat i + y hat j +z hat k| = sqrt (x^2 + y^2 + z^2)`

If `vec a` and `vec b ` are any two vectors given in the component form `a_1hat i + a_2hat j + a_3hat k` and `b_1hat i + b_2hat j + b_3hat k` respectively , then

(i) the sum (or resultant) of the vectors `vec a "and" vec b` is given by

`vec a + vec b = (a_1 + b_1) hat i + (a_2 + b_2) hat j + (a_3 + b_3) hat k`

(ii) the difference of the vector `vec a` and `vec b` is given by

`vec a - vec b = (a_1 - b_1) hat i + (a_2 - b_2) hat j + (a_3 - b_3) hat k`

(iii) the vectors `vec a` and `vec b` are equal if and only if

`a_1 =b_1, a_2 = b_2 and a_3 = b_3`

(iv) the multiplication of vector `vec a` by any scalar λ is given by

`lambda vec a = (lambda a_1)hat i + (lambda a_2) hat j + (lambda a_3) hat k`

The addition of vectors and the multiplication of a vector by a scalar together give the following distributive laws:

Let `vec a` and `vec b` be any two vectors, and k and m be any scalars. Then

i) `k vec a + m vec a = (k+m)vec a`

ii) `k(m vec a) = (km) vec a`

iii) `k(vec a + vec b) = k vec a + k vec b`