**TS ECET-2020 SYLLABUS: MATHEMATICS (Common for Diploma)**

**Unit-I: Matrices: **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.

**Unit–II: Trigonometry: **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.

**Unit–III: Analytical Geometry: **Straight Lines–different forms of Straight Lines, distance of a point from a line, acute angle between two lines, intersection of two non-parallel lines and distance between two parallel lines. Circles-Equation of circle given center and radius, given ends of diameter-General equation-finding center and radius, center and a point on the circumference, 3 non-collinear points, center and tangent, equation of tangent and normal at a point on the circle.

**Unit–IV: Differentiation and its Applications: **Functions and limits – Standard limits – Differentiation from the First Principle – Differentiation of sum, product, quotient of functions, function of function, trigonometric, inverse trigonometric, exponential, logarithmic, Hyperbolic functions, implicit, explicit and parametric functions–Derivative of a function with respect to another function-Second order derivatives – Geometrical applications of the derivative(angle between curves, tangent and normal)–Increasing and decreasing functions–Maxima and Minima(single variable functions) using second order derivative only – Partial Differentiation– Partial derivatives up to second order–Euler‘s theorem.

**Unit–V: Integration and its Applications: **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

**Unit–VI: Differential Equations: **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 eax, sin ax, cos ax, xm ( a polynomial of m-th degree m=1,2).

**Unit–VII: Laplace Transforms: **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.

**Unit–VIII: Fourier Series: **Define Fourier series, Euler’s formulae over the interval (C, C+2π) for determining the Fourier coefficients. Fourier series of simple functions in (0, 2π) and (–π, π).Fourier series for even and odd functions in the interval (–π, π).