#### Topics

##### Complex Numbers Old

##### Expansion of Sinn θ,Cosn θ in Terms of Sines and Cosines Of Multiples Of θ And Expansion of Sinnθ, Cosnθ In Powers of Sinθ, Cosθ

##### Separation of Real and Imaginary Parts of All Types of Functions

##### Circular Functions of Complex Number and Hyperbolic Functions.Inverse Circular and Inverse Hyperbolic Functions. Logarithmic Functions.

##### Powers and Roots of Exponential and Trigonometric Functions

##### Matrices and Numerical Methods Old

##### Solution of System Of Linear Algebraic Equations

##### Types of Matrices and Rank of a Matrix

##### Differential Calculus Old

##### Euler’S Theorem on Homogeneous Functions with Two and Three Independent Variables (With Proof)

##### Partial Differentiation

##### Successive Differentiation

##### Application of Partial Differentiation, Expansion of Functions , Indeterminate Forms and Curve Fitting Old

##### Fitting of Curves by Least Square Method for Linear, Parabolic, And Exponential

##### Maxima and Minima of a Function of Two Independent Variables

##### Taylor’S Theorem and Taylor’S Series, Maclaurin’S Series

##### Complex Numbers

- Review of Complex Numbers‐Algebra of Complex Number
- Different Representations of a Complex Number and Other Definitions
- D’Moivre’S Theorem
- Powers and Roots of Exponential Function
- Powers and Roots of Trigonometric Functions
- Expansion of sinn θ, cosn θ in terms of sines and cosines of multiples of θ
- Expansion of sinnθ, cosnθ in powers of sinθ, cosθ
- .Circular Functions of Complex Number
- Hyperbolic functions of complex number
- Inverse Circular Functions
- Inverse Hyperbolic Functions
- Separation of Real and Imaginary Parts of All Types of Functions

##### Logarithm of Complex Numbers , Successive Differentiation

##### Successive Differentiation

- Successive Differentiation
- nth Derivative of Standard Functions
- Leibnitz’S Theorem (Without Proof) and Problems

##### Logarithm of Complex Numbers

- Logarithmic Functions
- Separation of Real and Imaginary Parts of Logarithmic Functions

##### Matrices

- Inverse of a Matrix
- Addition of a Matrix
- Multiplication of a Matrix
- Transpose of a Matrix
- Types of Matrices
- Rank of a Matrix Using Echelon Forms
- Reduction to Normal Form
- PAQ in normal form
- System of Homogeneous and Non – Homogeneous Equations
- consistency and solutions of homogeneous and non – homogeneous equations
- Linear Dependent and Independent Vectors
- Application of Inverse of a Matrix to Coding Theory

##### Partial Differentiation

- Partial Derivatives of First and Higher Order
- Total Differentials
- Differentiation of Composite Functions
- Differentiation of Implicit Functions
- Euler’s Theorem on Homogeneous functions with two and three independent variables (with proof)
- Deductions from Euler’S Theorem

##### Applications of Partial Differentiation , Expansion of Functions

- Maxima and Minima of a Function of Two Independent Variables
- Jacobian
- Taylor’S Theorem (Statement Only)
- Taylor’S Series Method
- Maclaurin’s series (Statement only)
- Expansion of 𝑒^𝑥 , sin(x), cos(x), tan(x), sinh(x), cosh(x), tanh(x), log(1+x), 𝑠𝑖𝑛−1 (𝑥),𝑐𝑜𝑠−1 (𝑥),𝑡𝑎𝑛−1 (𝑥)
- Binomial Series

##### Indeterminate Forms, Numerical Solutions of Transcendental Equations and System of Linear Equations

- Indeterminate Forms
- L‐ Hospital Rule
- Problems Involving Series
- Solution of Transcendental Equations
- Solution by Newton Raphson Method
- Regula – Falsi Equation
- Solution of System of Linear Algebraic Equations by Gauss Elimination Method
- Gauss Jacobi Iteration Method
- Gauss Seidal Iteration Method

- Row Matrix
- Column Matrix
- Zero or Null matrix
- Square Matrix
- Diagonal Matrix
- Scalar Matrix
- Unit or Identity Matrix
- Upper Triangular Matrix
- Lower Triangular Matrix
- Triangular Matrix
- Symmetric Matrix
- Skew-Symmetric Matrix
- Determinant of a Matrix
- Singular Matrix
- Transpose of a Matrix

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