TS Edcet Exam syllabus Mathematics

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TS Ed.CET -2020
Part – C
MATHEMATICS (SYLLABUS) (100 Marks)
DIFFERENTIAL CALCULUS:
Successive Differentiation – 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 – Lagrange‘s Method of multipliers – Asymptotes – Envelopes.
DIFFERENTIAL EQUATIONS:
Differential Equations of first order and first degree: Exact differential equations – Integrating
Factors – Change in variables – Total Differential Equations – Simultaneous Total Differential equations
– Equations of the form
dx dy dz
P Q R
 
. Differential Equations first order but not first degree: Equations
solvable for
y – Equations solvable for
x – Equations that do not contain
x
( or
y
) – Clairaut‘s Equation.
Higher order linear differential equations: 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
( ) ax Q x be  , b ax b ax Sin / Cos ,
k
bx ,
ax Ve .
Method of undetermined coefficients – Method of variation of parameters – Linear differential equations
with non constant coefficients – The Cauchy- Euler Equation.
Partial Differential equations: Formation and solution- Equations easily integrable – Linear equations
of first order – Non linear equations of first order – Charpit‘s method – Homogeneous linear partial
differential equations with constant coefficient – Non homogeneous linear partial differential equations –
Separation of variables.
REAL ANALYSIS:
Sequences: Limits of Sequences – A Discussion about Proofs – Limit Theorems for Sequences –
Monotone Sequences and Cauchy Sequences. Subsequences – Lim sup‘s and Lim inf‘s – Series –
Alternating Series and Integral Tests. Sequences and Series of Functions: Power Series – Uniform
Convergence – More on Uniform Convergence – Differentiation and Integration of Power Series.
Integration: The Riemann Integral – Properties of Riemann Integral – Fundamental Theorem of
Calculus.
ALGEBRA:
Groups: Definition and Examples of Groups- Elementary Properties of Groups – Finite Groups;
Subgroups -Terminology and Notation -Subgroup Tests – Examples of Subgroups Cyclic Groups:
Properties of Cyclic Groups – Classification of Subgroups Cyclic Groups – Permutation Groups:
Definition and Notation – Cycle Notation -Properties of Permutations – A Check Digit Scheme Based
on D5. Isomorphisms: Motivation – Definition and Examples – Cayley‘s Theorem Properties of
Isomorphisms – Automorphisms – Cosets and Lagrange‘s Theorem Properties of Cosets 138 –
Lagrange‘s Theorem and Consequences – An Application of Cosets to Permutation Groups – The
Rotation Group of a Cube and a Soccer Ball – Normal Subgroups and Factor Groups ; Normal
Subgroups – Factor Groups – Applications of Factor Groups – Group Homomorphisms – Definition and
Examples – Properties of Homomorphisms – The First Isomorphism Theorem.
Introduction to Rings: Motivation and Definition – Examples of Rings – Properties of Rings – Subrings
– Integral Domains: Definition and Examples –Characteristics of a Ring – Ideals and Factor Rings;
Ideals – Factor Rings – Prime Ideals and Maximal Ideals.
Ring Homomorphisms: Definition and Examples – Properties of Ring – Homomorphisms – The Field of
Quotients Polynomial Rings: Notation and Terminology.
LINEAR ALGEBRA:
Vector Spaces: Vector Spaces and Subspaces – Null Spaces, Column Spaces, and Linear
Transformations – Linearly Independent Sets; Bases – Coordinate Systems – The Dimension of a
Vector Space.
Rank-Change of Basis – Eigen values and Eigenvectors – The Characteristic Equation.
Diagonalization – Eigenvectors and Linear Transformations- Complex Eigen values- Applicationsto
Differential Equations- Orthogonality andLeast Squares: Inner Product, Length, and Orthogonality
– Orthogonal Sets.
NUMERICAL ANALYSIS:
Solutions of Equations in One Variable: The Bisection Method – Fixed-Point Iteration – Newton‘s
Method and Its Extensions – Error Analysis for Iterative Methods – Accelerating Convergence – Zeros of
Polynomials and Mu¨ller‘s Method – Survey of Methods and Software.
Interpolation and Polynomial Approximation: Interpolation and the Lagrange Polynomial – Data
Approximation and Neville‘s Method – Divided Differences – Hermite Interpolation – Cubic Spline
Interpolation.
Numerical Differentiation and Integration: Numerical Differentiation – Richardson‘s Extrapolation –
Elements of Numerical Integration – Composite Numerical Integration – Romberg Integration –
Adaptive Quadrature Methods – GaussianQuadrature.
Texts:
 Shanti Narayan and Mittal, Differential Calculus
 Zafar Ahsan, Differential Equations and Their Applications
 Kenneth A Ross, Elementary Analysis-The Theory of Calculus
 Joseph A Gallian, Contemporary Abstract algebra (9th edition)
 David C Lay, Linear Algebra and its Applications 4e
 Richard L. Burden and J. Douglas Faires, Numerical Analysis (9e)
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