Complete TS ECET 2023 Syllabus: Topics, Marking Scheme, and Preparation Tips

- Expansions of Functions

- Mean value theorems

- Indeterminate forms

- Curvature and Evolutes

- Partial differentiation

- Homogeneous functions

- Total derivative
- Maxima and Minima of functions of two variables, Asymptotes

- Envelopes

- Exact differential equations

- Integrating Factors

- Change in variables

- Total Differential Equations of the form
- Differential Equations first order but not of the first degree

- Equations Solvable for y

- Equations Solvable for x

- Equations that do not contain x or y

- Clairaut’s equation

- Solution of homogeneous linear differential
equations with constant coefficients

- Solution of non-homogeneous differential equations P (D)y = Q(x) with constant coefficients by means of polynomial operators when Q(x) = be ax, b sin ax/b cos ax, bxk, V e ax

- Partial Differential equations

- Formation and solution

- Equations easily integrable

- Linear equations of first order

- Limits of Sequences

- Limit Theorems for Sequences

- Monotonic Sequences

- Cauchy Sequences – Subsequences

- Lim sup’s and Lim inf’s – Series – Test for convergence

-Alternating Series

- Power Series

- Uniform Convergence

- More on Uniform Convergence

- Differentiation and Integration of Power Series. The Riemann Integral

- Properties of Riemann Integral

- Fundamental Theorem of Calculus

- Definition and Examples of Groups

- Elementary Properties of Groups

- Finite Groups

- Subgroups

- Subgroup Tests

- Cyclic Groups

- Properties of Cyclic Groups

- Permutation Groups – Homomorphism – Isomorphism of Groups and their properties – Auto
morphism – Cosets – Normal Sub Groups

- Examples of Rings

- Properties of Rings – Subrings

- Integral Domains – Field

- Characteristics of a Ring

- Ideals and Factor Rings

- Prime Ideals and Maximal Ideals

- Ring Homomorphisms – Isomorphism of Rings and their properties

- The Field of Quotients Polynomial Rings – Fundamental Theorem of Homomorphism

- Vector Spaces and Subspaces

- Null Spaces

- Column Spaces

- Linear Transformations

- Linearly Independent Sets

- Bases - Coordinate Systems

-The Dimension of a Vector Space

- Rank-Change of Basis

- The Characteristic Equation

- Diagonalization

- Eigenvectors and LinearTransformations

- Complex Eigenvalues – Orthogonality

- Inner Product – Length

- Orthogonality - Orthogonal Sets

- The Bisection Method

- Fixed-Point Iteration

- Newton’s Method and Its Extensions

- Error Analysis for Iterative Methods

- Accelerating Convergence

- Zeros of Polynomials

- Interpolation

- Lagrange Polynomial

- Divided Differences

-Numerical Differentiation and Integration

Definition of Matrix, Types of matrices-Algebra of matrices-Transpose of a matrix-Symmetric, skew-symmetric matrices-Minor, cofactor of an element-Determinant of a square matrix-Properties-Laplace‘s expansion-singular and nonsingular matrices-Adjoint and multiplicative inverse of a square matrix-System of linear equations in 3 variables-Solutions by Crammer‘s rule, Matrix inversion method-Gauss-Jordan method.

Partial Fractions: Resolving a given rational function into partial fractions.

Logarithms: Definition of logarithm and its properties, meaning of ‘e’ exponential function and logarithmic function. 

Properties of Trigonometric functions– Ratios of Compound angles, multiple angles, submultiple angles – Transformations of Products into sum or difference and vice versa- Simple trigonometric equations Properties of triangles–Inverse Trigonometric
functions, Hyperbolic functions.

Complex Numbers: Properties of Modulus, amplitude and conjugate of complex numbers, arithmetic operations on complex numbers—Modulus-Amplitude form (Polar form) - Euler form (exponential form) Properties.

Indefinite Integral – Standard forms – Integration by decomposition of the integrand, integration of trigonometric, algebraic, exponential, logarithmic and Hyperbolic functions– Integration by substitution –Integration of reducible and irreducible quadratic factors – Integration by parts– Definite Integrals and properties, Definite Integral as the limit of a sum – Application of Integration to find areas under plane curves and volumes of Solids of revolution– Mean and RMS values, Trapezoidal rule and Simpson’s 1/3 Rule for approximation integrals

Definition of a differential equation-order and degree of a differential equation- formation of differential equations-solution of differential equation of the type first order, first degree, variable-separable, homogeneous equations, exact, linear differential equation of the form dy/dx+Py=Q, Bernoulli‘s equation, nth order linear differential equation with constant coefficients both homogeneous and non-homogeneous and finding the Particular Integrals for the functions e ax, sin ax, cos ax, x m ( a polynomial of m-th degree m=1,2)

Laplace Transforms (LT) of elementary functions-Linearity property, first shifting property, change of scale property multiplication and division by t - LT of derivatives and integrals, Unit step function, LT of unit step function, second shifting property, evaluation of improper integrals, Inverse Laplace transform (ILT)-shifting theorem, change of scale property, multiplication and division by s, ILT by using partial fractions and convolution theorem. Applications of LT to solve ordinary differential equations up to second order only. 

Their classifications with examples - their relative applications, Familiarisation with new drug delivery systems.

Introduction to Pharmacopoeias with special reference to the Indian Pharmacopoeia.

Metrology-Systems of weights and measures. Calculations including conversion from one to another system. Percentage calculations and adjustments of products. Use of allegation method in calculations, Isotonic solutions.

Packaging of pharmaceuticals-Desirable features of a container- types of containers, study of glass and plastics as materials for containers and rubber as a material for closures - their merits and demerits. Introduction to aerosol packaging. 

Study of Hammer mill, ball mill, Fluid Energy Mill and Disintegrator.

Size separation - Size separation by sifting, Official Standard for powders. Sedimentation methods of size separation. Construction and working of cyclone separator.

Mixing and Homogenization – Liquid mixing and powder mixing. Mixing of semisolids,

Study of Silverson Mixer - Homogeniser, planetary Mixer, Agitated powder mixer. Triple Roller Mill, Propeller Mixer, Colloid Mill and Hand Homogeniser. Double cone mixer.

Clarification and Filtration - Theory of filtration, Filter media; Filter aids and selection of filters. Study of the following filtration equipments - Filter Press, Sintered Filter, candles, Metafilter.

Extraction and Galenicals- (a) Study of percolation and maceration and their modification, continuous hot extraction-Applications in the preparation of tinctures and extracts.

(b) Introduction to Ayurvedic dosage forms.

Heat process – Evaporation – Definition, factors affecting evaporation. Study of evaporating still and Evaporating pan.

Distillation – Simple distillation and Fractional distillation; Steam distillation and vacuum distillation, Study of Vacuum still, preparation of purified water I.P. and water of Injection I.P.

Construction and working of the still used for the same.

Introduction to drying process- Study of Tray Dryers; Fluidized Bed Dryer, Vacuum Dryer and Freeze Dryer.

Sterilization

Processing of tablets