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- 1. MTMH1
**Inequalities****Complex number****Number theory****Set theory****Group theory****Permutation****Abstract algebra-2****Ring, Filed, Integral Domain****Vector algebra**

AM>=GM>=HM. Their generalization like the theorem of weighted mean and m-th power theorem. Statement of Cauchy-Schwartz inequality, Weierstrass inequality and their applications. De Moivres theorem and its applications. Exponential sine, cosine & logarithm of complex number. Direct & inverse circular & hyperbolic functions.Expansion of trigonometrical functions. Gregorys series. Summation of series. Statements of well ordering principle , first principle of mathematical induction , second principle of mathematical induction . Proofs of some simple mathematical results by induction. The division algorithm, The greatest common divisor (g.c.d.) of two integers a and b . Relatively prime integers. The equation ax+by=c has integral solution iff (a,b) divides c. ( a,b,c are integers ). Prime Numbers. Euclid’s first theorem: If some prime p divides ab, then p divides either a or b. Euclid’s second theorem. There are infinitely many prime integer. Unique factorization theorem. Statement of Chinese Remainder Theorem and simple problem. Euler phi function. Revision of set theory and algebra, relation and mapping. Order relations, equivalence relations and partitions. Congruence modulo n. Further theory of sets and mapping, Cardinality of sets, countable and uncountable sets. Binary operation Semi-group ,Definition, examples and simple properties of group , some special groups like Zn, U(n), dihedral groups, etc., Abelian group. Subgroup, the necessary and sufficient condition of a non-empty subset of a group is a subgroup, intersection and union of two subgroups. Cyclic groups and its various properties. Order of a group and order of an element of a group Cycle, transposition, Statement of the result that every permutation can be expressed as a product of disjoint cycles. Even and odd permutations, permutation group. Symmetric group. Alternating Group. Order of an alternating group. Group Homomorphism, Automorphism, Endomorphism and Isomorphism. Cosets & their various properties, index of a subgroup, Lagrange’s theorem & its applications, Normal subgroups: Definition, examples and properties. Rings and Fields: Properties of Rings directly following from the definition, Divisors of zero, Integral domain, Every field is an integral domain, every finite integrals domain is a field.

Definition of sub - ring and sub - field. Necessary of sufficient condition for a sub set of a ring (field) to be sub ring (resp. subfield). Characteristic of ring & integral domain. Ring and field Homomorphism, Isomorphism, Quotient –ring.

1. Vector ( directed line segment ) Equality of two free vectors. Addition of Vectors. Multiplication by a Scalar. Position vector, Point of division, conditions of co-linearity of three points and co-planarity of four points. Rectangular components of a

vector in two and three dimensions. Product of two or more vectors. Scalar and vector products, scalar triple products and Vector triple products. Product of four vectors.

2. Direct application of Vector (i) Algebra in Geometrical and Trigonometrical problems (ii) Work done by a force, Moment of a force about a point.

3. Vector equations of straight lines planes. Volume of a tetrahedron. Shortest distance between two skew lines.

- 2. MTMH2
**Matrix****Row and Column Rank****Eigen values and Eigen Vectors****Vector space****Solution of linear equations****Linear transformation****Translation & Rotation****Canonical form****Pair of Straight lines****Polar coordinates****Three Dimension basics****Plane****Straight lines in three dimension****Sphere****Cone****Cylinder**

Matrices of real and complex numbers. Algebra of matrices. Symmetric and skew-symmetric matrices. Hermitian and skew-hermitian matrices. Orthogonal matrices. Inverse of a matrices, Solution of linear equation with not more than three unknown by matrix method. Rank of a matrix. Row space and Column space of matrix. Row rank, Column rank, Determination of rank either by considering minor or sweep out process. Row rank=Column rank= Rank of the matrix. Rank(A+B) Characteristics polynomial & Minimal polynomials.Characteristic equation, Eigen value & Eigen vector. Cayley Hamilton theorem (Statement only). Simple properties of Eigen value & Eigen vectors. Vector / Linear space: Definition and examples.Subspace, Union and Intersection of Subspace. Linear sum of two subspaces. Linear combination, Independence and dependence. Linear span, Basis of vector space. Finite dimensional vector space. Replacement Theorem, Extension Theorem. Statement of the result that any two bases of a finite dimensional vector space same number of element. Dimension of a vector space. Extraction of basis, Formation of basis with special emphasis on Rn (n≤4). Linear homogeneous system of equations: Solution space. For a homogeneous system AX=O in n unknowns. Rank X(A)+ Rank (A)=n, AX=O contains non trivial solution if Rank A< n. AX=O contains non trivial solution if Rank A< n. Necessary and sufficient condition for consistency of a linear non homogeneous system of equations. Solution of system of equation (Matrix method). Linear transformations and their representation as matrices. The algebra of linear transformations. Rank and nullity theorem. Transformation of rectangular axis, translation, rotation, and their combination. Theory of invariants General equation of second degree in two variables, Reduction into canonical form, length and position of the axis. Pair of straight lines:- Condition that the general equation of second degree in two variables may represent a pair of straight lines, Point of intersection of two intersecting straight lines. Angle between two lines given by ax2 +2hxy+by2, equation of bisectors of the angle between the pair of straight line, Equation of two lines joining the origin to the point in which two curves meet. Polar co-ordinates, polar equation of the straight lines,circle and conic referred to a focus as pole. Equation of chord, Tangent and normal. Rectangular Cartesian co-ordinates in space. Concept of geometry vector(directed line segment). Projection of vector on a co-ordinate axis. Inclination of a vector with an axis. Co-ordinates of a vector. Direction ratio and direction cosine of a vector. Distance between two points. Division of directed line segment in given ratio. Equation of a plane in general form,intercept and normal form. Signed distance of a point from a plane. Equation plane passing through the intersection of two planes. Angle between two intersecting planes. Parallel and perpendicularity of two planes. Straight line in space. Equation in symmetric and para symmetric form. Canonical equation of line of intersection of two intersecting planes. Angle between two lines, Distance of a point from a line, Condition of coplanarity of two lines, Shortest distance between two skew lines. General equation of sphere, circle, sphere through the intersection of two sphere, Radical plane, tangent, normal General equation of cone, right circular cone. General equation of cylinder, right circular cylinder. - 3. MTMH3
**Limit & Continuity****Uniform continuity****Infinite series****Differentiation****Successive derivative****Indeterminate form****Functions of several variables****Maximum and Minimum****Jacobian****Maximum and Minimum for functions of two variables****Tangent and Normal****Pedal equations****Differential of arc length****Curvature****Asymptotes****Envelopes****Reduction formulae****Vector analysis****Expansion of functions**

Limit and continuity of a real valued function at a point (the point must be a limit point of the domain set of a function). Algebra of limit, sandwich rule. Continuity of composite functions. Bounded functions. Neighbourhood properties of continuous functions regarding boundedness and maintenance of same sign Continuous functions on [a b] is bounded and attains its bounds Intermediate value theorem.

Discontinuity of function type of discontinuity. Step function. Piece wise continuity. Monotone function .Monotone function can have only jump discontinuity. Set of points of discontinuity of a monotone function is at most countable.Definition of uniform continuity and examples. Lipschitz condition and uniform continuity. Functions condition on a closed and bounded interval is uniformly continuous. Infinite series of real numbers: Convergence. Cauchy’s criterion of convergence. Series of non-negative real numbers: Test of convergence – Cauchy’s condensation test. Comparison test (ordinary form and upper limit and lower limit criteria), Ratio test, Root test, Raabe’s test, Bertrand’s test, Logarithmic test and Gauss’s test. Alternative series, Leibnitz,s test. Absolute and conditional convergent series. Rearrangement of series through examples. Definition of differentiability. Meaning of sign of derivative. Chain rule. Successive differentiation: Leibnitz theorem and its application. Statement of L. Hospital’s rule and its application. Limit and continuity (definition and example only). Statement of Taylor’s theorem for functions of two variables.

Point of local extremum (maximum, minimum) of a function in an interval. Sufficient condition for the existence of a local maximum/minimum of a function at a poi nt (statement only). Determination of local extremum using first order derivative. Application of the principle of maximum/minimum in geometrical problems. Jacobian Maxima, minima, saddle points of functions of two variables (example only) Tangent, normal, sub-tangent and sub-normal. Length of tangent and normal. Angle of intersection of curves. Pedal equation of a curve, pedal of a curve. Differential of arc length. Curvature-Radius of curvature, center of curvature, chord of curvature, evolute of a curve Rectilinear asymptotes for Cartesian and polar curves. Envelopes of families of straight lines and curves (Cartesian an parametric equations only). Reduction formulae 1. Vector function, Iimit and continuity, derivative of vector. derivative of sum and product

of vector functions. A necessary and sufficient condition that a proper vector a

(i) has constant length is that a . da/dt = 0, (ii) always remains parallel is that a x da/dt =0.

2 Vector integration. scalar and vector fields. directional derivatives. gradient of a scalar

point function, nable operator, divergence. curl and Laplacian.

3 Line, surface and volume integral. Gauss's, Stoke's theorem and problem based on these.Darboux theorem, Rolle's theorem, Mean value theorems of Lagrange and Cauchy - as an application of Rolle's theorem.

Taylor's theorem on closed and bounded interval with Lagrange's and Cauchy's form of remainder deducted from Lagrange's and Cauchy's mean value theorem respectively. Maclaurin's theorem as a consequence of Taylor's theorem. Statement of Maclaurin's theorem on infinite series expansion. Expansion of $e^x, \log (1+x), (1+x)^m, \sin x, \cos x$ with their range of validity. - 4. MTMH4
**Differential Equations-I****Differential Equations-II****Linear Programming Problem-I****Linear Programming Problem-II****Transportation Problem****Travelling Salesman Problem****Simplex / Big M Method****Two phase method****Degeneracy****Duality****Dual simplex method**

1. Significance of ordinary differential equation. Geometrical and physical consideration. Formation of differential equation by elimination of arbitrary constant. Meaning of the solution of ordinary differential equation . Concept of linear and non-linear differential equations. Equations of first order and first degree : Statement of existence theorem. Separable, Homogeneous and Exact equation . Condition of exactness, Integrating factor. Rules of finding integrating factor, (statement of relevant results only), Equations reducible to first order linear equations.

2. Equations of first order but not of first degree, Clairauts equation. Singular solution. Applications : Geometric applications, Orthogonal trajectories. Higher order linear equations with constant coefficients : Complementary function, Particular Integral. Symbolic operator D.

3. Method of undetermined coefficients, Method of variation of parameters. Eulers homogeneous equation and Reduction to an equation of constant coefficients.

1. Exact differential equations of higher order, method of solution. Non-linear exact equations. linear equations of some special forms

2. Second order linear equation with variable coefficients, reduction of order when one solution of the homogeneous part is known . Complete solution, method of variation of parameters.

3. Reduction to Normal form. Change of independent variable, Operational factors, Simple eigen value problems, Simultaneous linear differential equations.1. What is LPP? Mathematical form of LPP formulation. LPP in matrix notation. Graphical solution of LPP. Basic solution, Basic feasible solution, degenerate and non-degenerate BFS.

2. Euclidean space, hyperplane, convex set, extreme points, convex functions and concave functions, the hyperplane in convex set. Intersection of two convex sets is convex set, the collection of all feasible solution of a LPP constitutes a convex set. A BFS to a LPP corresponds to an extreme point of convex set of feasible solutions .

3. Slack, surplus and artificial variables, standard form of LPP, Fundamental theorem pf LPP

and their applications.

Assignment problem. Solution of AP [(Maximization, unbalanced, negative cost and impossible assignment. Transportation problem. TP in LPP form, Balanced TP. Optimality test of BFS. Traveling salesman problem.

(Problem should be set on simplex and Charne's method, two phase method in such a way that it may contain at most three or four tableau with approximate marks.)

Theory and application of the simplex method of solution of LPP. Charne's M-technique.

The two phase method. Degeneracy

Duality theory. The dual of the dual is primal, relation between the objective function value of dual and primal problems. Relation between their optimal values. Statement of fundamental theorem of duality.Dual simplex method.

- 5. MTMH5
**Analysis-I****Analysis-II****Analysis-III****Sequence of functions****Power series****Fourier series****Multiple integrals****Differentiation under the sign of integration****Volume and Surface area of solid**

1. Bounded subset of R, L.U.B. (supremum) and G.L.B. (infimum) of a set. Least upper bound axiom . Characterization of R as a complete ordered field. Definition of an

Archimedean ordered field. Archimedean property of R. Q is Archimedean ordered field but nor ordered complete.

Neighbourhood of a point. Interior point. Open set. Union, intersection of open sets. Limit point and isolated point of a set. Bolzano-Weierstrass. Complement of open set and closed set. Dense set in R

2. Covering and compactness, Heine Borel theorem. Sequences of real number : Bounded sequence. Convergence and divergence. Examples. Every convergent sequence is bounded and limit is unique.

3. Monotone sequences and their convergence. Sandwich rule. Nested interval theorem . Cauchy's first and second limit theorems. Subsequence. Subsequential limits. Lim sup upper (limit) and lim inf (lower limit) of a sequence using inequalities. Balzano-Weierstrass theorem for sequence. Cauchy's general principle of convergence.

1. Riemann integration on [ a,b]. Riemann approach Riemann sum and Riemann integrability. Darboux's approach: upper sum U(P,f) and lower sum L(P,f), upper and lower integral, Darboux's theorem, necessary and sufficient condition of Riemann integrability. Equality of Riemann and Darboux's approach.

2 R-integrability of sum, product and quotient. R-integrability of f => R-integrability of | f |

Integrability of monotone functions, continuous functions, piece wise continuous functions,

function having (i) finite number of point of discontinuities, (ii) having finite number of limit points of discontinuities.

3 Function defined by the definite integral int_{a_x} f (t) dt and its properties. Primitives or indefinite integrals. First mean value theorem of integral calculus. Second mean value theorem of integral calculus (both Bonet's and Weierstrass's forms).

1. Improper integrals and their convergence, absolute and non-absolute convergence. Tests of convergence: Comparison test, m-test. Abel's and Dirichlet's test for convergence of integral of a product.

2. Beta and Gamma functions and their convergence, their properties and inter relation.

3. Geometric interpretation of definite integral, Fundamental theorem of integral area enclosed by plane curves. Rectification of plane curves.

Sequence and sequence of functions, point wise and uniform convergence, boundness and continuity, integrability and differentiability of limit functions in case of uniform convergence, Weierstrass M-test of uniform and absolute convergence. Power series, radius of convergence using upper limit, uniform convergence of power series, properties, term by term integration and differentiation, uniqueness of power series. Fourier series, Dirichlet's condition of convergence, calculation of Fourier coefficients, Fourier theorem, half range series, sine series, cosine series, Fourier series in arbitrary interval, Parseval's identity, basic theorems. Evaluation of double and triple integrals, Dirichlet's integrals, change of order of integration in double integrals. Differentiability and integrability of an integral of a function of a parameter, Differentiation under the sign of integration. Volume and surface area of solid formed by revolution of plane curves and areas about x-axis and y-axis.

- 6. MTMH6
**Probability****Statistics****Tensor Calculus****Dynamics**

1.1 Frequency and Axiomatic definition of probability. Random Variable, distribution function, discrete and continuous distribution. Binomial, Poisson, Beta, Gama, Uniform and normal distribution. Poisson process. Transformation of random variables.

1.2 Two dimensional probability distributions, discrete and continuous distribution in two dimensions, Uniform distribution and two dimensional normal distribution. Conditional distribution. Transformation of random variables in two dimensions.

i .3 Mathematical expectation, mean, variance, moment, central moments, measures of dispersion, skewness and curtosis, median, mode, quartiles. Moment generating function, characteristic function, statement of their uniqueness. Two dimensional expectation,

covariance, correlation coefficient, joint characteristic function, multiplication rule for expectation, conditional expectation.

3.1 Random sample, concept of sampling and various types of sampling, sample and population. Collection, tabulation and graphical representation, grouping of data, sample characteristic and their computation, sampling distribution of statistic.

3.2 Estimates of population characteristic or parameter, point estimation and interval estimation, criterion of a good point estimate, maximum likelihood estimate. lnterval estimation of population proportion, interval estimation of a Normal population parameters, estimate of population parameters with large samples when distribution of the population is unknown.

3.3 Testing of hypothesis, null hypothesis and alternative hypothesis, Type one and type two error, testing of hypothesis for a population proportion and Normal population parameters and large sample test for population with unknown distribution, Chi-square test of goodness of fit.

1. Summation Convention. Kronecker symbol. n-dimensional space, Transformation of coordinates in S_n. Invariants. covariant and contravariant vectors. Covariant contravariant and mixed tensor . Algebra of tensors. Symmetric and skew-symmetric tensor, Contraction, outer and inner product of tensors. Quotient law, reciprocal tensor. Riemann space, the line element and metric tensor, raising and lowering of indices, associate tensor. magnitude of a

vector, inclination of two vectors, orthogonal vectors. ChristoffeI symbols and their properties, transformation law of Christoffel symbols.

2 Covariant differentiation of tensors, covariant differentiation of sum, difference and product of tensors. Gradient, divergence, curl and Laplacian. Curvilinear coordinate system in E_3: line element, length of vector, angle between two vectors in E_3 in a curvilinear coordinate system. Basis in a curvilinear coordinate system, reciprocal base, covariant and contravariant components of a vector in E_3, partial derivative of a vector. Spherical and cylindrical coordinate system.

3 Curves in E_3. Parallel vector fields along a curve in E_3, parallel vector field in E_3, parallel

vector space in a Riemannian space, parallel vector field in a surface of a Riemannian space. Serret-Frenet formulas.

1 Simple Harmonic Motion. Tangent and normal acceleration. Velocity and acceleration along radial and transverse directions.

2 Central orbits, central forces, motion of a particle under central force. Differential equation

in polar and pedal coordinates, velocity under central force. Apse, apsidal distance and apsidal angle.

3 Kepler's laws of planetary motion, artificial satellites, Escape velocity, Geo stationary satellite, Disturbed orbits.

- 7. MTMH7
**Numerical analysis-1****Numerical analysis-2****C Programming-1****C Programming-2****Numerical Solution of non-linear equations**

1. Error in numerical analysis, Gross error, rounding off error, truncation error, approximate

numbers, significant figure. Absolute, relative and percentage error, General formula for

error, Delta, nabla, E, delta, mu operators, their properties and interrelation. Equispaced argument. difference table, propagation of error in difference table.

2. Interpolation: Statement of Weierstrass' approximation theorem, polynomial interpolation and error term in polynomial interpolation, deduction of Lagrange's interpolation formula, inverse interpolation. finding root of a equation by interpolation method. Deduction of Newton 's forward and backward interpolation formula. Statement of Gauss· forward and backward interpolation formula. Starling's and Bessel's interpolation formulae. Error terms. Divided difference , General interpolation formulae, deduction of Lagrange's, Newton's forward and backward's interpolation formula.

3. Numerical Differentiation based on Newtons's forward, Newton's backward and Lagrange's interpolation formula. Error terms. Numerical integration: Integration of Newton's interpolation formula.Newton-Cotes formula. Deduction of Trapezoidal rule and Simpson's 1/3 rule. statement of Weddle's rule. Statements of error terms.

Euler Maclaurin's sum formula.2.2 Numerical solution of system of linear equations: Gauss elimination , Gauss-Jordan method, Pivoting strategy in Gauss elimination. LU-Decomposition. Inversion of 3-3 non singular matrices by Gauss elimination and Gauss-Jordan method. Gauss-Siedal iteration method for system of linear equation.

2.3 Numerical solution of ordinary differential equation of first order: Euler's method, modified· Euler's method, Picard's method, Taylor's series method , Runge-Kutta method, Milne's method.

Algorithm and flow charts with simple examples, Branching and looping, Introduction to ANSI-C: Character set in ANSI-C. Key words: int, char, float, white etc. Constant and variables, expressions, assignment statements, formatting source files, Header files, Data types, declarations, different types of integers, different kinds of integer constants, floating-point types, initialization, mixing types, the void data type, Type defs. standard input outputs, finding address of an object, Operations and expressions, precedence and associativity, unary minus and plus operators, binary arithmetic operators, arithmetic assignment operators, increment and decrement operators, comma operators, relational operators, logical operators. Control flow, conditional and unconditional branching, looping, nested loops, if-else, do-while, for, switch, break, continue, goto statements etc, Infinite loops, Functions, Arrays and pointers. Location of a real roots by tabular method , Bisection method, secant / Regula-Falsi, fixed point iteration and Newton-Raphson method, their geometrical significance and convergency, order of convergence. Newtons method for multiple roots.

- 8. MTMH8
**Numerical analysis practical****C Programming practical**

1. Problems on Newton's forward and Backward interpolation, Lagrange interpolation

formula. Inverse interpolation. Finding root of a equation by interpolation method.

2. Differentiation formula based on Newton's forward and backward interpolation formula.

3. Numerical integration by Trapezoidal, Simpson's l /3 rule and Weddle's rule.

4. Finding roots of an equation by Bisection method, Regula Falsi method, fixed point iteration method, Newton-Raphson method.

5. Solution of linear equation by Gauss elimination method, Gauss-Jordan method and Gauss Siedel method.

6. Finding inverse of a third order matrix without finding its determinant.

7. Runge-Kutta Method

1. Ascending / Descending order. Finding Largest / smallest.

2. Sum of finite series.

3. Sum of Convergent series.

4. Bisection method.

5. Checking whether a number is prime or not. Generation of prime numbers.

6. Solution of Quadratic equation

7. Newton's forward and Backward interpolation. Lagrange interpolation.

8. Bisection method. Newton-Raphson method. Regula-Falsi method.

9. Trapezoidal Rule. Simpson 's 1/3 rule.

10. Value of Determinant.

1 I . Matrix sum, subtraction, product, transposition.

12. Cramer's Rule (upto three variable).

13. Solution of linear equation by Gauss elimination method, Gauss-Jordan method.

14. Runge-Kutta Method.

15. Mean, variance, correlation coefficient, equation of regression line.

- 1. MTMG1
**Inequalities****Complex number****Vector algebra****Set theory****Abstract algebra****Matrix****Linear algebra****Linear transformation****Eigen values and eigen vectors**

I. Inequalities : A. M. >= G.M >= H.M. Their generalization like the theorem of weighted mean and m-th power theorem. Statement of Cauchy-Schwartz inequality. Weierstrass inequality and their application. De-Moivre's theorem and its application.

Exponential sine, cosine and logarithm of complex number. Direct and inverse circular and hyperbolic functions. Expansion of trigonometry functions. Gregory's series. Summation of series.

Revision of definition of vectors and its algebra. Rectangular solution of vector. linear dependent and independent of vectors. Two vectors are linear dependent iff one is scalar multiple of other. Every super set of linearly dependent set of vectors is linearly dependent. The set of non-zero vectors are linearly iff one of them is scalar combination of others.

Scalar and vector product of two vectors. Scalar and vector triple product. Product of four vectors. Reciprocal vectors. Simple applications to geometry. Vector equations of straight line, plane and circle. Applications to mechanics: work done, torque.1 . Revision of set theory, relation and mapping. Equivalence relation , partition of a set,

·equivalence classes, composition of functions. Congruence modulo n.Binary operation. Group Theory: Group, Abelian group, identity and inverse element in a group is unique.

Subgroups, necessary and sufficient condition of a non-empty subset of a group is a subgroup, cyclic group, order of a group and order of an element.

Rings and Fields: Properties of Rings directly following from the definition, unitary and commutative rings. Divisors of zero, Integral domain, Every field is an integral domain , every finite integrals domain is a field. Definitions of Sub-ring and sub-field. Statement of Necessary of sufficient condition for a subset of a ring (field) to be sub-ring (resp. subfield).

Matrix: Matrices of real and complex numbers : Algebra of matrices. Symmetric and skew-symmetric matrices, Solution of linear equation with not more than three unknown by matrix method. Rank of a matrix. Characteristics polynomial, characteristics equations, Eigen value & Eigen Vector. Cayley Hamilton theorem (statement only).

Vector space/Linear space (Def. and examples), Linear combination , independence and dependence, linear span, basis and dimension (Def. and examples). Subspace (Def. and examples), intersection and union of subspaces, linear sum of two subspaces, direct sum of subspaces, dimension of sum and subspaces. Linear transformation and their representation as matrices, kernal and range of a linear transformation , the algebra of linear transformation s, the rank nullity theorem (statement only). Characteristics polynomial, characteristics equations, Eigen value & Eigen Vector. Cayley Hamilton theorem (statement only).

- 2. MTMG2
**Limit & Continuity****Indeterminate form****Successive derivative****Expansion of functions****Sequence****Infinite series****Functions of several variables****Partial derivative****Jacobian****Extreme values****Tangent and Normal****Curvature****Asymptotes****Improper integral****Beta & Gamma functions****Reduction formulae****Length of plane curves****Volume and Surface area of revolution****Multiple integrals****Determination of plane area****Differentiation under the sign of integration****Centroid**

Idea of epsilon-delta definition of limit and continuity of a function.

Indeterminate forms. statement of L'Hospital rule and its applications. Successive differentiation, Leibnitz's theorem and its applications. Rolle's theorem and its geometric interpretation . Mean value theorem of Lagrange and Cauchy. Geometric interpretation of Lagrange's mean value theorem, Statement of Taylor's and Maclaurin's theorem with Lagrange 's and Cauchy's form of remainder. Taylor's and Maclaurin's series (Statement only). Expansions of functions in finite and infinite series like sin(X), Cos(X), exp(X), a,, (1+x)^n, log(l +x) (with restrictions whenever necessary). Limit of sequence. Convergent and non convergent Cauchy sequence. Convergence of infinite. Statement and use of different tests for convergence of series of non- negative terms. Functions of several variables: Limits and continuity (definition and examples only), Partial derivative. Total differentials . Statement of Schwartz's and Young's theorem on commutative property of mixed derivative. Euler's theorem of homogeneous functions of two variables. Statement of Taylor's theorem for functions of two variables. Jacobian Maxima, minima, saddle points of functions of two points (examples only). Tangent normal sub tangent and sub normal. Length of tangent and normal. Differential of arc length. Curvature for Cartesian and polar curve. Rectilinear asymptote for Cartesian and polar curve. Definition of improper integrals, example. Definition and simple properties of beta & Gamma functions & their uses (convergence and important relations being assumed)

Reduction formulae such as sin^n x dx , cos^n xdx ,tan^n xdx , sec^n xdx , sin^n x cos^m xdx ,

sin^n x cos mxdx etc where m and n are non-negative integers.Rectification of plane curves.

Volume and surface ·area of solid formed by revolution of plane curves and areas about x-axis and y-axis.

Working knowledge of double and triple integrals, change of order of integration.

Plane area Differentiability and integrability of an integral of a function of a parameter. Differentiability under the sign of integration, statements of necessary theorem. Centroid, Centroid of arc. - 3. MTMG3
**2D-Transformation of axes****General equation of 2nd degree****Pair of Straight lines****2D-Polar coordinate****3D-Basics****Plane****Straight lines in three dimension****Sphere****Differential Equations**

Transformation of rectangular axes, translation. rotation and their combinations. theory of invariants. General equation of second degree in two variables. reduction into canonical form. lengths and position of the axes. Pair of straight lines: Condition that the general equation of second degree i n two variables may represent a pair of straight lines. Point of intersection of two intersecting straight lines, angle between two lines given by ax2+2hxy+by2. equation of bisectors of the angle between the pair of straight

lines, equation of two lines joining the origin to the point in which two curves meet.Polar coordinates, polar equation of straight lines, circles and conic referred to a focus as pole, equation of chord, tangent and normal.

Rectangular Cartesian co-ordinates in space, concept of geometric vector (directed line segment), projection of vector on a co-ordinate axis, inclination of a vector with an axis, coordinates of a vector, direction ratio and direction cosine of a vector. Distance between two points, division of directed line segment in given ratio.

Equation of a plane in general form, intercept and normal form, signed distance of a point from a plane, equation plane passing

through the intersection of two planes, angle between two intersecting planes, parallel and perpendicularity of two planes.Straight lines in space, equation in symmetric and par\metric form, canonical equation of line of intersection of two intersecting planes, angle between two lines, distance of a point from a line, condition of coplanarity of two lines. General equation of sphere, circle, sphere through the intersection of two sphere, radical plane, tangent, normal.

1. Significance of ordinary differential equation. Geometrical and physical consideration. Formation of differential equation by elimination of arbitrary constant. Meaning of the solution of ordinary differential equation. Concept of linear and non-linear differential equations. Equations of first order and first degree: Statement of existence theorem. Separable, Homogeneous and Exact equation. Condition of exactness, Integrating factor. Rules of finding integrating factor, (statement of relevant results only), Equations reducible to first order linear equations.

2. Equations of first order but not of first degree, Clairaut's equation, Singular solution. Applications : Geometric applications, Orthogonal trajectories. Higher order linear equations with constant coefficients : Complementary function, Particular Integral, Symbolic operator D.

3. Method of undetermined coefficients, Method of variation of parameters. Euler's homogeneous equation and Reduction to an equation of constant coefficients. Ordinary simultaneous differential equations.

- 4. MTMG4
**Linear Programming Problem****Simplex / Big M****Duality****Transportation Problem****Assignment Problem****Travelling Salesman Problem****Probability****Vector analysis**

1. What is LPP ? Mathematical form of LPP formulation. LPP in matrix notation. Graphical

solution of LPP. Basic solution. Basic feasible solution. degenerate and non-degenerate BFS. Euclidean space. hyperplane, convex set, extreme points. convex functions and concave functions. the hyperplane in convex set. Intersection of two convex sets is convex set, the collection of all feasible solution of a LPP constitutes a convex set. A BFS to a LPP corresponds to an extreme point of convex set of feasible solution.

2. Slack, surplus and artificial variables, standard form of LPP, Fundamental theorem of LPP and their applications,Theory and application of the simplex method of solution of LPP. Charne's M-technique.

Duality. Transportation problem. TP in LPP form, Balanced TP. Optimality test of BFS. Assignment problem. Solution of AP (Maximization , unbalanced, negative cost and impossible assignment). Traveling salesman problem.

(Problem should be set on simplex and Charne's method, two phase method in such a way that it may contain at most three or four tableau with approximate marks.)

Frequency and Axiomatic definition of probability. Random variables. Probability Distribution function. Discrete and continuous random variable, probability mass function and probability density function, mathematical expectation, mean and variance (simple problems only). Binomial, Poission, uniform , Normal , Beta and Gamma Distributions. Moments of a probability distribution, skewness and kurtosis of a probability distribution , moment generating function. Transformation of one dimensional random variable (simple applications). Vector function, limit and continuity , derivative of vector, derivative of sums and product of vector functions. A necessary and sufficient condition that a proper vector a (i) has a constant length that a . da/dt = 0. (ii) always remains parallel is that a x da/dt=0.

Vector integration, scalar and vector fields, directional derivatives, gradient of a scalar point function, nebla operator, divergence, curl and Laplacian .

Line, surface and volume integral. Statement of Gauss's, Stoke's theorem and problem based on these.

- 5. MTMG5
**Numerical analysis****C Programming****C Programming practical****Numerical solution of algebraic and transcendental equations**

1. Approximate numbers and significant figures, rounding off numbers. Error and Absolute, relative and percentage errors. Linear operation, Difference, finite difference interpolation. Lagrange interpolation. Newton's forward and backward difference formula. Differentiation

formula based on Newton's forward and backward difference formula, Numerical integration, deduction of Trapezoidal, Simpson 's 1/3 rule from Newton 's forward difference formula.

2. Solution of linear equations: Gauss elimination, Gauss-Jordan method. LU Decomposition.

Inversion of 3 x 3 non-singular matrices by Gauss elimination and Gauss-Jordan method.

1. Introduction to ANSI-C : Character set in ANSI-C. Key words: int, char, float, while etc.

.Constant and Variables, expressions, assignment statements, formatting source files. Header files.Data types, declarations, different types of integers, different kinds of integer constants, floating-point types, initialization, standard input/output. finding address of an object.

2. Operations and expressions, precedence and associatively, unary plus and minus operators, binary arithmetic operators, arithmetic assignment operators, increment and decrement operators, comma operator, relational operators, logical operators.

3. Control flow, conditional and unconditional bracing, looping, nested loops. if-else, do

while, for, switch, break, continue, goto statements etc., Infinite loops. Arrays and Pointers

1. Ascending / Descending order. Finding Largest / smallest.

2. Sum of finite series. Mean and variance .

3. Conversion of binary to decimal and decimal to bi nary.

4. Checking whether a number i s prime or not. Generation prime numbers.

5. Solution of Quadratic equation . Newtone-Raphson's method. Lagrange interpolation.

6. Bisection method. Newton-Raphson method.

7. Trapezoidal Rule. Simpson 's 1/3 rule.

8. Value of Determinant.

9. Cramer's Rule ( for two variables).

10. Matrix addition, subtraction, transposition.

Bisection, Secant / Regula Falsi, Newton's-Raphson method, iteration method.

MTMH1 | 1 | Inequalities | AM>=GM>=HM. Their generalization like the theorem of weighted mean and m-th power theorem. Statement of Cauchy-Schwartz inequality, Weierstrass inequality and their applications. |

MTMH1 | 1 | Complex number | De Moivres theorem and its applications. Exponential sine, cosine & logarithm of complex number. Direct & inverse circular & hyperbolic functions.Expansion of trigonometrical functions. Gregorys series. Summation of series. |

MTMH1 | 1 | Number theory | Statements of well ordering principle , first principle of mathematical induction , second principle of mathematical induction . Proofs of some simple mathematical results by induction. The division algorithm, The greatest common divisor (g.c.d.) of two integers a and b . Relatively prime integers. The equation ax+by=c has integral solution iff (a,b) divides c. ( a,b,c are integers ). Prime Numbers. Euclid’s first theorem: If some prime p divides ab, then p divides either a or b. Euclid’s second theorem. There are infinitely many prime integer. Unique factorization theorem. Statement of Chinese Remainder Theorem and simple problem. Euler phi function. |

MTMH1 | 2 | Set theory | Revision of set theory and algebra, relation and mapping. Order relations, equivalence relations and partitions. Congruence modulo n. Further theory of sets and mapping, Cardinality of sets, countable and uncountable sets. Binary operation |

MTMH1 | 2 | Group theory | Semi-group ,Definition, examples and simple properties of group , some special groups like Zn, U(n), dihedral groups, etc., Abelian group. Subgroup, the necessary and sufficient condition of a non-empty subset of a group is a subgroup, intersection and union of two subgroups. Cyclic groups and its various properties. Order of a group and order of an element of a group |

MTMH1 | 2 | Permutation | Cycle, transposition, Statement of the result that every permutation can be expressed as a product of disjoint cycles. Even and odd permutations, permutation group. Symmetric group. Alternating Group. Order of an alternating group. |

MTMH1 | 3 | Abstract algebra-2 | Group Homomorphism, Automorphism, Endomorphism and Isomorphism. Cosets & their various properties, index of a subgroup, Lagrange’s theorem & its applications, Normal subgroups: Definition, examples and properties. |

MTMH1 | 3 | Ring, Filed, Integral Domain | Rings and Fields: Properties of Rings directly following from the definition, Divisors of zero, Integral domain, Every field is an integral domain, every finite integrals domain is a field. Definition of sub - ring and sub - field. Necessary of sufficient condition for a sub set of a ring (field) to be sub ring (resp. subfield). Characteristic of ring & integral domain. Ring and field Homomorphism, Isomorphism, Quotient –ring. |

MTMH1 | 4 | Vector algebra | 1. Vector ( directed line segment ) Equality of two free vectors. Addition of Vectors. Multiplication by a Scalar. Position vector, Point of division, conditions of co-linearity of three points and co-planarity of four points. Rectangular components of a vector in two and three dimensions. Product of two or more vectors. Scalar and vector products, scalar triple products and Vector triple products. Product of four vectors. 2. Direct application of Vector (i) Algebra in Geometrical and Trigonometrical problems (ii) Work done by a force, Moment of a force about a point. 3. Vector equations of straight lines planes. Volume of a tetrahedron. Shortest distance between two skew lines. |

MTMH2 | 1 | Matrix | Matrices of real and complex numbers. Algebra of matrices. Symmetric and skew-symmetric matrices. Hermitian and skew-hermitian matrices. Orthogonal matrices. Inverse of a matrices, Solution of linear equation with not more than three unknown by matrix method. Rank of a matrix. |

MTMH2 | 1 | Row and Column Rank | Row space and Column space of matrix. Row rank, Column rank, Determination of rank either by considering minor or sweep out process. Row rank=Column rank= Rank of the matrix. Rank(A+B) |

MTMH2 | 1 | Eigen values and Eigen Vectors | Characteristics polynomial & Minimal polynomials.Characteristic equation, Eigen value & Eigen vector. Cayley Hamilton theorem (Statement only). Simple properties of Eigen value & Eigen vectors. |

MTMH2 | 2 | Vector space | Vector / Linear space: Definition and examples.Subspace, Union and Intersection of Subspace. Linear sum of two subspaces. Linear combination, Independence and dependence. Linear span, Basis of vector space. Finite dimensional vector space. Replacement Theorem, Extension Theorem. Statement of the result that any two bases of a finite dimensional vector space same number of element. Dimension of a vector space. Extraction of basis, Formation of basis with special emphasis on Rn (n≤4). |

MTMH2 | 2 | Solution of linear equations | Linear homogeneous system of equations: Solution space. For a homogeneous system AX=O in n unknowns. Rank X(A)+ Rank (A)=n, AX=O contains non trivial solution if Rank A< n. AX=O contains non trivial solution if Rank A< n. Necessary and sufficient condition for consistency of a linear non homogeneous system of equations. Solution of system of equation (Matrix method). |

MTMH2 | 2 | Linear transformation | Linear transformations and their representation as matrices. The algebra of linear transformations. Rank and nullity theorem. |

MTMH2 | 3 | Translation & Rotation | Transformation of rectangular axis, translation, rotation, and their combination. Theory of invariants |

MTMH2 | 3 | Canonical form | General equation of second degree in two variables, Reduction into canonical form, length and position of the axis. |

MTMH2 | 3 | Pair of Straight lines | Pair of straight lines:- Condition that the general equation of second degree in two variables may represent a pair of straight lines, Point of intersection of two intersecting straight lines. Angle between two lines given by ax2 +2hxy+by2, equation of bisectors of the angle between the pair of straight line, Equation of two lines joining the origin to the point in which two curves meet. |

MTMH2 | 3 | Polar coordinates | Polar co-ordinates, polar equation of the straight lines,circle and conic referred to a focus as pole. Equation of chord, Tangent and normal. |

MTMH2 | 4 | Three Dimension basics | Rectangular Cartesian co-ordinates in space. Concept of geometry vector(directed line segment). Projection of vector on a co-ordinate axis. Inclination of a vector with an axis. Co-ordinates of a vector. Direction ratio and direction cosine of a vector. Distance between two points. Division of directed line segment in given ratio. |

MTMH2 | 4 | Plane | Equation of a plane in general form,intercept and normal form. Signed distance of a point from a plane. Equation plane passing through the intersection of two planes. Angle between two intersecting planes. Parallel and perpendicularity of two planes. |

MTMH2 | 4 | Straight lines in three dimension | Straight line in space. Equation in symmetric and para symmetric form. Canonical equation of line of intersection of two intersecting planes. Angle between two lines, Distance of a point from a line, Condition of coplanarity of two lines, Shortest distance between two skew lines. |

MTMH2 | 4 | Sphere | General equation of sphere, circle, sphere through the intersection of two sphere, Radical plane, tangent, normal |

MTMH2 | 4 | Cone | General equation of cone, right circular cone. |

MTMH2 | 4 | Cylinder | General equation of cylinder, right circular cylinder. |

MTMH3 | 1 | Limit & Continuity | Limit and continuity of a real valued function at a point (the point must be a limit point of the domain set of a function). Algebra of limit, sandwich rule. Continuity of composite functions. Bounded functions. Neighbourhood properties of continuous functions regarding boundedness and maintenance of same sign Continuous functions on [a b] is bounded and attains its bounds Intermediate value theorem. Discontinuity of function type of discontinuity. Step function. Piece wise continuity. Monotone function .Monotone function can have only jump discontinuity. Set of points of discontinuity of a monotone function is at most countable. |

MTMH3 | 1 | Uniform continuity | Definition of uniform continuity and examples. Lipschitz condition and uniform continuity. Functions condition on a closed and bounded interval is uniformly continuous. |

MTMH3 | 1 | Infinite series | Infinite series of real numbers: Convergence. Cauchy’s criterion of convergence. Series of non-negative real numbers: Test of convergence – Cauchy’s condensation test. Comparison test (ordinary form and upper limit and lower limit criteria), Ratio test, Root test, Raabe’s test, Bertrand’s test, Logarithmic test and Gauss’s test. Alternative series, Leibnitz,s test. Absolute and conditional convergent series. Rearrangement of series through examples. |

MTMH3 | 2 | Differentiation | Definition of differentiability. Meaning of sign of derivative. Chain rule. |

MTMH3 | 2 | Successive derivative | Successive differentiation: Leibnitz theorem and its application. |

MTMH3 | 2 | Indeterminate form | Statement of L. Hospital’s rule and its application. |

MTMH3 | 2 | Functions of several variables | Limit and continuity (definition and example only). Statement of Taylor’s theorem for functions of two variables. |

MTMH3 | 3 | Maximum and Minimum | Point of local extremum (maximum, minimum) of a function in an interval. Sufficient condition for the existence of a local maximum/minimum of a function at a poi nt (statement only). Determination of local extremum using first order derivative. Application of the principle of maximum/minimum in geometrical problems. |

MTMH3 | 3 | Jacobian | Jacobian |

MTMH3 | 3 | Maximum and Minimum for functions of two variables | Maxima, minima, saddle points of functions of two variables (example only) |

MTMH3 | 3 | Tangent and Normal | Tangent, normal, sub-tangent and sub-normal. Length of tangent and normal. Angle of intersection of curves. |

MTMH3 | 3 | Pedal equations | Pedal equation of a curve, pedal of a curve. |

MTMH3 | 3 | Differential of arc length | Differential of arc length. |

MTMH3 | 3 | Curvature | Curvature-Radius of curvature, center of curvature, chord of curvature, evolute of a curve |

MTMH3 | 3 | Asymptotes | Rectilinear asymptotes for Cartesian and polar curves. |

MTMH3 | 3 | Envelopes | Envelopes of families of straight lines and curves (Cartesian an parametric equations only). |

MTMH3 | 3 | Reduction formulae | Reduction formulae |

MTMH3 | 4 | Vector analysis | 1. Vector function, Iimit and continuity, derivative of vector. derivative of sum and product of vector functions. A necessary and sufficient condition that a proper vector a (i) has constant length is that a . da/dt = 0, (ii) always remains parallel is that a x da/dt =0. 2 Vector integration. scalar and vector fields. directional derivatives. gradient of a scalar point function, nable operator, divergence. curl and Laplacian. 3 Line, surface and volume integral. Gauss's, Stoke's theorem and problem based on these. |

MTMH3 | 2 | Expansion of functions | Darboux theorem, Rolle's theorem, Mean value theorems of Lagrange and Cauchy - as an application of Rolle's theorem. Taylor's theorem on closed and bounded interval with Lagrange's and Cauchy's form of remainder deducted from Lagrange's and Cauchy's mean value theorem respectively. Maclaurin's theorem as a consequence of Taylor's theorem. Statement of Maclaurin's theorem on infinite series expansion. Expansion of $e^x, \log (1+x), (1+x)^m, \sin x, \cos x$ with their range of validity. |

MTMH4 | 1 | Differential Equations-I | 1. Significance of ordinary differential equation. Geometrical and physical consideration. Formation of differential equation by elimination of arbitrary constant. Meaning of the solution of ordinary differential equation . Concept of linear and non-linear differential equations. Equations of first order and first degree : Statement of existence theorem. Separable, Homogeneous and Exact equation . Condition of exactness, Integrating factor. Rules of finding integrating factor, (statement of relevant results only), Equations reducible to first order linear equations. 2. Equations of first order but not of first degree, Clairauts equation. Singular solution. Applications : Geometric applications, Orthogonal trajectories. Higher order linear equations with constant coefficients : Complementary function, Particular Integral. Symbolic operator D. 3. Method of undetermined coefficients, Method of variation of parameters. Eulers homogeneous equation and Reduction to an equation of constant coefficients. |

MTMH4 | 2 | Differential Equations-II | 1. Exact differential equations of higher order, method of solution. Non-linear exact equations. linear equations of some special forms 2. Second order linear equation with variable coefficients, reduction of order when one solution of the homogeneous part is known . Complete solution, method of variation of parameters. 3. Reduction to Normal form. Change of independent variable, Operational factors, Simple eigen value problems, Simultaneous linear differential equations. |

MTMH4 | 3 | Linear Programming Problem-I | 1. What is LPP? Mathematical form of LPP formulation. LPP in matrix notation. Graphical solution of LPP. Basic solution, Basic feasible solution, degenerate and non-degenerate BFS. 2. Euclidean space, hyperplane, convex set, extreme points, convex functions and concave functions, the hyperplane in convex set. Intersection of two convex sets is convex set, the collection of all feasible solution of a LPP constitutes a convex set. A BFS to a LPP corresponds to an extreme point of convex set of feasible solutions . 3. Slack, surplus and artificial variables, standard form of LPP, Fundamental theorem pf LPP and their applications. |

MTMH4 | 4 | Linear Programming Problem-II | Assignment problem. Solution of AP [(Maximization, unbalanced, negative cost and impossible assignment. |

MTMH4 | 4 | Transportation Problem | Transportation problem. TP in LPP form, Balanced TP. Optimality test of BFS. |

MTMH4 | 4 | Travelling Salesman Problem | Traveling salesman problem. (Problem should be set on simplex and Charne's method, two phase method in such a way that it may contain at most three or four tableau with approximate marks.) |

MTMH4 | 3 | Simplex / Big M Method | Theory and application of the simplex method of solution of LPP. Charne's M-technique. |

MTMH4 | 4 | Two phase method | The two phase method. |

MTMH4 | 4 | Degeneracy | Degeneracy |

MTMH4 | 4 | Duality | Duality theory. The dual of the dual is primal, relation between the objective function value of dual and primal problems. Relation between their optimal values. Statement of fundamental theorem of duality. |

MTMH4 | 4 | Dual simplex method
| Dual simplex method. |

MTMH5 | 1 | Analysis-I | 1. Bounded subset of R, L.U.B. (supremum) and G.L.B. (infimum) of a set. Least upper bound axiom . Characterization of R as a complete ordered field. Definition of an Archimedean ordered field. Archimedean property of R. Q is Archimedean ordered field but nor ordered complete. Neighbourhood of a point. Interior point. Open set. Union, intersection of open sets. Limit point and isolated point of a set. Bolzano-Weierstrass. Complement of open set and closed set. Dense set in R 2. Covering and compactness, Heine Borel theorem. Sequences of real number : Bounded sequence. Convergence and divergence. Examples. Every convergent sequence is bounded and limit is unique. 3. Monotone sequences and their convergence. Sandwich rule. Nested interval theorem . Cauchy's first and second limit theorems. Subsequence. Subsequential limits. Lim sup upper (limit) and lim inf (lower limit) of a sequence using inequalities. Balzano-Weierstrass theorem for sequence. Cauchy's general principle of convergence. |

MTMH5 | 2 | Analysis-II | 1. Riemann integration on [ a,b]. Riemann approach Riemann sum and Riemann integrability. Darboux's approach: upper sum U(P,f) and lower sum L(P,f), upper and lower integral, Darboux's theorem, necessary and sufficient condition of Riemann integrability. Equality of Riemann and Darboux's approach. 2 R-integrability of sum, product and quotient. R-integrability of f => R-integrability of | f | Integrability of monotone functions, continuous functions, piece wise continuous functions, function having (i) finite number of point of discontinuities, (ii) having finite number of limit points of discontinuities. 3 Function defined by the definite integral int_{a_x} f (t) dt and its properties. Primitives or indefinite integrals. First mean value theorem of integral calculus. Second mean value theorem of integral calculus (both Bonet's and Weierstrass's forms). |

MTMH5 | 3 | Analysis-III | 1. Improper integrals and their convergence, absolute and non-absolute convergence. Tests of convergence: Comparison test, m-test. Abel's and Dirichlet's test for convergence of integral of a product. 2. Beta and Gamma functions and their convergence, their properties and inter relation. 3. Geometric interpretation of definite integral, Fundamental theorem of integral area enclosed by plane curves. Rectification of plane curves. |

MTMH5 | 4 | Sequence of functions | Sequence and sequence of functions, point wise and uniform convergence, boundness and continuity, integrability and differentiability of limit functions in case of uniform convergence, Weierstrass M-test of uniform and absolute convergence. |

MTMH5 | 4 | Power series | Power series, radius of convergence using upper limit, uniform convergence of power series, properties, term by term integration and differentiation, uniqueness of power series. |

MTMH5 | 4 | Fourier series | Fourier series, Dirichlet's condition of convergence, calculation of Fourier coefficients, Fourier theorem, half range series, sine series, cosine series, Fourier series in arbitrary interval, Parseval's identity, basic theorems. |

MTMH5 | 4 | Multiple integrals | Evaluation of double and triple integrals, Dirichlet's integrals, change of order of integration in double integrals. |

MTMH5 | 4 | Differentiation under the sign of integration | Differentiability and integrability of an integral of a function of a parameter, Differentiation under the sign of integration. |

MTMH5 | 3 | Volume and Surface area of solid | Volume and surface area of solid formed by revolution of plane curves and areas about x-axis and y-axis. |

MTMH6 | 1 | Probability | 1.1 Frequency and Axiomatic definition of probability. Random Variable, distribution function, discrete and continuous distribution. Binomial, Poisson, Beta, Gama, Uniform and normal distribution. Poisson process. Transformation of random variables. 1.2 Two dimensional probability distributions, discrete and continuous distribution in two dimensions, Uniform distribution and two dimensional normal distribution. Conditional distribution. Transformation of random variables in two dimensions. i .3 Mathematical expectation, mean, variance, moment, central moments, measures of dispersion, skewness and curtosis, median, mode, quartiles. Moment generating function, characteristic function, statement of their uniqueness. Two dimensional expectation, covariance, correlation coefficient, joint characteristic function, multiplication rule for expectation, conditional expectation. |

MTMH6 | 2 | Statistics | 3.1 Random sample, concept of sampling and various types of sampling, sample and population. Collection, tabulation and graphical representation, grouping of data, sample characteristic and their computation, sampling distribution of statistic. 3.2 Estimates of population characteristic or parameter, point estimation and interval estimation, criterion of a good point estimate, maximum likelihood estimate. lnterval estimation of population proportion, interval estimation of a Normal population parameters, estimate of population parameters with large samples when distribution of the population is unknown. 3.3 Testing of hypothesis, null hypothesis and alternative hypothesis, Type one and type two error, testing of hypothesis for a population proportion and Normal population parameters and large sample test for population with unknown distribution, Chi-square test of goodness of fit. |

MTMH6 | 3 | Tensor Calculus | 1. Summation Convention. Kronecker symbol. n-dimensional space, Transformation of coordinates in S_n. Invariants. covariant and contravariant vectors. Covariant contravariant and mixed tensor . Algebra of tensors. Symmetric and skew-symmetric tensor, Contraction, outer and inner product of tensors. Quotient law, reciprocal tensor. Riemann space, the line element and metric tensor, raising and lowering of indices, associate tensor. magnitude of a vector, inclination of two vectors, orthogonal vectors. ChristoffeI symbols and their properties, transformation law of Christoffel symbols. 2 Covariant differentiation of tensors, covariant differentiation of sum, difference and product of tensors. Gradient, divergence, curl and Laplacian. Curvilinear coordinate system in E_3: line element, length of vector, angle between two vectors in E_3 in a curvilinear coordinate system. Basis in a curvilinear coordinate system, reciprocal base, covariant and contravariant components of a vector in E_3, partial derivative of a vector. Spherical and cylindrical coordinate system. 3 Curves in E_3. Parallel vector fields along a curve in E_3, parallel vector field in E_3, parallel vector space in a Riemannian space, parallel vector field in a surface of a Riemannian space. Serret-Frenet formulas. |

MTMH6 | 4 | Dynamics | 1 Simple Harmonic Motion. Tangent and normal acceleration. Velocity and acceleration along radial and transverse directions. 2 Central orbits, central forces, motion of a particle under central force. Differential equation in polar and pedal coordinates, velocity under central force. Apse, apsidal distance and apsidal angle. 3 Kepler's laws of planetary motion, artificial satellites, Escape velocity, Geo stationary satellite, Disturbed orbits. |

MTMH7 | 1 | Numerical analysis-1 | 1. Error in numerical analysis, Gross error, rounding off error, truncation error, approximate numbers, significant figure. Absolute, relative and percentage error, General formula for error, Delta, nabla, E, delta, mu operators, their properties and interrelation. Equispaced argument. difference table, propagation of error in difference table. 2. Interpolation: Statement of Weierstrass' approximation theorem, polynomial interpolation and error term in polynomial interpolation, deduction of Lagrange's interpolation formula, inverse interpolation. finding root of a equation by interpolation method. Deduction of Newton 's forward and backward interpolation formula. Statement of Gauss· forward and backward interpolation formula. Starling's and Bessel's interpolation formulae. Error terms. Divided difference , General interpolation formulae, deduction of Lagrange's, Newton's forward and backward's interpolation formula. 3. Numerical Differentiation based on Newtons's forward, Newton's backward and Lagrange's interpolation formula. Error terms. Numerical integration: Integration of Newton's interpolation formula.Newton-Cotes formula. Deduction of Trapezoidal rule and Simpson's 1/3 rule. statement of Weddle's rule. Statements of error terms. Euler Maclaurin's sum formula. |

MTMH7 | 2 | Numerical analysis-2 | 2.2 Numerical solution of system of linear equations: Gauss elimination , Gauss-Jordan method, Pivoting strategy in Gauss elimination. LU-Decomposition. Inversion of 3-3 non singular matrices by Gauss elimination and Gauss-Jordan method. Gauss-Siedal iteration method for system of linear equation. 2.3 Numerical solution of ordinary differential equation of first order: Euler's method, modified· Euler's method, Picard's method, Taylor's series method , Runge-Kutta method, Milne's method. |

MTMH7 | 3 | C Programming-1 | Algorithm and flow charts with simple examples, Branching and looping, Introduction to ANSI-C: Character set in ANSI-C. Key words: int, char, float, white etc. Constant and variables, expressions, assignment statements, formatting source files, Header files, Data types, declarations, different types of integers, different kinds of integer constants, floating-point types, initialization, mixing types, the void data type, Type defs. standard input outputs, finding address of an object, Operations and expressions, precedence and associativity, unary minus and plus operators, binary arithmetic operators, arithmetic assignment operators, increment and decrement operators, comma operators, relational operators, logical operators. |

MTMH7 | 4 | C Programming-2 | Control flow, conditional and unconditional branching, looping, nested loops, if-else, do-while, for, switch, break, continue, goto statements etc, Infinite loops, Functions, Arrays and pointers. |

MTMH7 | 2 | Numerical Solution of non-linear equations | Location of a real roots by tabular method , Bisection method, secant / Regula-Falsi, fixed point iteration and Newton-Raphson method, their geometrical significance and convergency, order of convergence. Newtons method for multiple roots. |

MTMH8 | Numerical analysis practical | 1. Problems on Newton's forward and Backward interpolation, Lagrange interpolation formula. Inverse interpolation. Finding root of a equation by interpolation method. 2. Differentiation formula based on Newton's forward and backward interpolation formula. 3. Numerical integration by Trapezoidal, Simpson's l /3 rule and Weddle's rule. 4. Finding roots of an equation by Bisection method, Regula Falsi method, fixed point iteration method, Newton-Raphson method. 5. Solution of linear equation by Gauss elimination method, Gauss-Jordan method and Gauss Siedel method. 6. Finding inverse of a third order matrix without finding its determinant. 7. Runge-Kutta Method | |

MTMH8 | C Programming practical | 1. Ascending / Descending order. Finding Largest / smallest. 2. Sum of finite series. 3. Sum of Convergent series. 4. Bisection method. 5. Checking whether a number is prime or not. Generation of prime numbers. 6. Solution of Quadratic equation 7. Newton's forward and Backward interpolation. Lagrange interpolation. 8. Bisection method. Newton-Raphson method. Regula-Falsi method. 9. Trapezoidal Rule. Simpson 's 1/3 rule. 10. Value of Determinant. 1 I . Matrix sum, subtraction, product, transposition. 12. Cramer's Rule (upto three variable). 13. Solution of linear equation by Gauss elimination method, Gauss-Jordan method. 14. Runge-Kutta Method. 15. Mean, variance, correlation coefficient, equation of regression line. |

MTMG1 | 1 | Inequalities | I. Inequalities : A. M. >= G.M >= H.M. Their generalization like the theorem of weighted mean and m-th power theorem. Statement of Cauchy-Schwartz inequality. Weierstrass inequality and their application. |

MTMG1 | 1 | Complex number | De-Moivre's theorem and its application. Exponential sine, cosine and logarithm of complex number. Direct and inverse circular and hyperbolic functions. Expansion of trigonometry functions. Gregory's series. Summation of series. |

MTMG1 | 1 | Vector algebra | Revision of definition of vectors and its algebra. Rectangular solution of vector. linear dependent and independent of vectors. Two vectors are linear dependent iff one is scalar multiple of other. Every super set of linearly dependent set of vectors is linearly dependent. The set of non-zero vectors are linearly iff one of them is scalar combination of others. Scalar and vector product of two vectors. Scalar and vector triple product. Product of four vectors. Reciprocal vectors. Simple applications to geometry. Vector equations of straight line, plane and circle. Applications to mechanics: work done, torque. |

MTMG1 | 2 | Set theory | 1 . Revision of set theory, relation and mapping. Equivalence relation , partition of a set, ·equivalence classes, composition of functions. Congruence modulo n. |

MTMG1 | 2 | Abstract algebra | Binary operation. Group Theory: Group, Abelian group, identity and inverse element in a group is unique. Subgroups, necessary and sufficient condition of a non-empty subset of a group is a subgroup, cyclic group, order of a group and order of an element. Rings and Fields: Properties of Rings directly following from the definition, unitary and commutative rings. Divisors of zero, Integral domain, Every field is an integral domain , every finite integrals domain is a field. Definitions of Sub-ring and sub-field. Statement of Necessary of sufficient condition for a subset of a ring (field) to be sub-ring (resp. subfield). |

MTMG1 | 2 | Matrix | Matrix: Matrices of real and complex numbers : Algebra of matrices. Symmetric and skew-symmetric matrices, Solution of linear equation with not more than three unknown by matrix method. Rank of a matrix. Characteristics polynomial, characteristics equations, Eigen value & Eigen Vector. Cayley Hamilton theorem (statement only). |

MTMG1 | 2 | Linear algebra | Vector space/Linear space (Def. and examples), Linear combination , independence and dependence, linear span, basis and dimension (Def. and examples). Subspace (Def. and examples), intersection and union of subspaces, linear sum of two subspaces, direct sum of subspaces, dimension of sum and subspaces. |

MTMG1 | 2 | Linear transformation | Linear transformation and their representation as matrices, kernal and range of a linear transformation , the algebra of linear transformation s, the rank nullity theorem (statement only). |

MTMG1 | 2 | Eigen values and eigen vectors | Characteristics polynomial, characteristics equations, Eigen value & Eigen Vector. Cayley Hamilton theorem (statement only). |

MTMG2 | 1 | Limit & Continuity | Idea of epsilon-delta definition of limit and continuity of a function. |

MTMG2 | 1 | Indeterminate form | Indeterminate forms. statement of L'Hospital rule and its applications. |

MTMG2 | 1 | Successive derivative | Successive differentiation, Leibnitz's theorem and its applications. |

MTMG2 | 1 | Expansion of functions | Rolle's theorem and its geometric interpretation . Mean value theorem of Lagrange and Cauchy. Geometric interpretation of Lagrange's mean value theorem, Statement of Taylor's and Maclaurin's theorem with Lagrange 's and Cauchy's form of remainder. Taylor's and Maclaurin's series (Statement only). Expansions of functions in finite and infinite series like sin(X), Cos(X), exp(X), a,, (1+x)^n, log(l +x) (with restrictions whenever necessary). |

MTMG2 | 1 | Sequence | Limit of sequence. Convergent and non convergent Cauchy sequence. |

MTMG2 | 1 | Infinite series | Convergence of infinite. Statement and use of different tests for convergence of series of non- negative terms. |

MTMG2 | 1 | Functions of several variables | Functions of several variables: Limits and continuity (definition and examples only), Partial derivative. Total differentials . Statement of Schwartz's and Young's theorem on commutative property of mixed derivative. |

MTMG2 | 1 | Partial derivative | Euler's theorem of homogeneous functions of two variables. Statement of Taylor's theorem for functions of two variables. |

MTMG2 | 1 | Jacobian | Jacobian |

MTMG2 | 1 | Extreme values | Maxima, minima, saddle points of functions of two points (examples only). |

MTMG2 | 1 | Tangent and Normal | Tangent normal sub tangent and sub normal. Length of tangent and normal. Differential of arc length. |

MTMG2 | 1 | Curvature | Curvature for Cartesian and polar curve. |

MTMG2 | 1 | Asymptotes | Rectilinear asymptote for Cartesian and polar curve. |

MTMG2 | 2 | Improper integral | Definition of improper integrals, example. |

MTMG2 | 2 | Beta & Gamma functions | Definition and simple properties of beta & Gamma functions & their uses (convergence and important relations being assumed) |

MTMG2 | 2 | Reduction formulae | Reduction formulae such as sin^n x dx , cos^n xdx ,tan^n xdx , sec^n xdx , sin^n x cos^m xdx , sin^n x cos mxdx etc where m and n are non-negative integers. |

MTMG2 | 2 | Length of plane curves | Rectification of plane curves. |

MTMG2 | 2 | Volume and Surface area of revolution | Volume and surface ·area of solid formed by revolution of plane curves and areas about x-axis and y-axis. |

MTMG2 | 2 | Multiple integrals | Working knowledge of double and triple integrals, change of order of integration. |

MTMG2 | 2 | Determination of plane area | Plane area |

MTMG2 | 2 | Differentiation under the sign of integration | Differentiability and integrability of an integral of a function of a parameter. Differentiability under the sign of integration, statements of necessary theorem. |

MTMG2 | 2 | Centroid | Centroid, Centroid of arc. |

MTMG3 | 1 | 2D-Transformation of axes | Transformation of rectangular axes, translation. rotation and their combinations. theory of invariants. |

MTMG3 | 1 | General equation of 2nd degree | General equation of second degree in two variables. reduction into canonical form. lengths and position of the axes. |

MTMG3 | 1 | Pair of Straight lines | Pair of straight lines: Condition that the general equation of second degree i n two variables may represent a pair of straight lines. Point of intersection of two intersecting straight lines, angle between two lines given by ax2+2hxy+by2. equation of bisectors of the angle between the pair of straight lines, equation of two lines joining the origin to the point in which two curves meet. |

MTMG3 | 1 | 2D-Polar coordinate | Polar coordinates, polar equation of straight lines, circles and conic referred to a focus as pole, equation of chord, tangent and normal. |

MTMG3 | 1 | 3D-Basics | Rectangular Cartesian co-ordinates in space, concept of geometric vector (directed line segment), projection of vector on a co-ordinate axis, inclination of a vector with an axis, coordinates of a vector, direction ratio and direction cosine of a vector. Distance between two points, division of directed line segment in given ratio. |

MTMG3 | 1 | Plane | Equation of a plane in general form, intercept and normal form, signed distance of a point from a plane, equation plane passing through the intersection of two planes, angle between two intersecting planes, parallel and perpendicularity of two planes. |

MTMG3 | 1 | Straight lines in three dimension | Straight lines in space, equation in symmetric and par\metric form, canonical equation of line of intersection of two intersecting planes, angle between two lines, distance of a point from a line, condition of coplanarity of two lines. |

MTMG3 | 1 | Sphere | General equation of sphere, circle, sphere through the intersection of two sphere, radical plane, tangent, normal. |

MTMG3 | 2 | Differential Equations | 1. Significance of ordinary differential equation. Geometrical and physical consideration. Formation of differential equation by elimination of arbitrary constant. Meaning of the solution of ordinary differential equation. Concept of linear and non-linear differential equations. Equations of first order and first degree: Statement of existence theorem. Separable, Homogeneous and Exact equation. Condition of exactness, Integrating factor. Rules of finding integrating factor, (statement of relevant results only), Equations reducible to first order linear equations. 2. Equations of first order but not of first degree, Clairaut's equation, Singular solution. Applications : Geometric applications, Orthogonal trajectories. Higher order linear equations with constant coefficients : Complementary function, Particular Integral, Symbolic operator D. 3. Method of undetermined coefficients, Method of variation of parameters. Euler's homogeneous equation and Reduction to an equation of constant coefficients. Ordinary simultaneous differential equations. |

MTMG4 | 1 | Linear Programming Problem | 1. What is LPP ? Mathematical form of LPP formulation. LPP in matrix notation. Graphical solution of LPP. Basic solution. Basic feasible solution. degenerate and non-degenerate BFS. Euclidean space. hyperplane, convex set, extreme points. convex functions and concave functions. the hyperplane in convex set. Intersection of two convex sets is convex set, the collection of all feasible solution of a LPP constitutes a convex set. A BFS to a LPP corresponds to an extreme point of convex set of feasible solution. 2. Slack, surplus and artificial variables, standard form of LPP, Fundamental theorem of LPP and their applications, |

MTMG4 | 1 | Simplex / Big M | Theory and application of the simplex method of solution of LPP. Charne's M-technique. |

MTMG4 | 1 | Duality | Duality. |

MTMG4 | 1 | Transportation Problem | Transportation problem. TP in LPP form, Balanced TP. Optimality test of BFS. |

MTMG4 | 1 | Assignment Problem | Assignment problem. Solution of AP (Maximization , unbalanced, negative cost and impossible assignment). |

MTMG4 | 1 | Travelling Salesman Problem | Traveling salesman problem. (Problem should be set on simplex and Charne's method, two phase method in such a way that it may contain at most three or four tableau with approximate marks.) |

MTMG4 | 2 | Probability | Frequency and Axiomatic definition of probability. Random variables. Probability Distribution function. Discrete and continuous random variable, probability mass function and probability density function, mathematical expectation, mean and variance (simple problems only). Binomial, Poission, uniform , Normal , Beta and Gamma Distributions. Moments of a probability distribution, skewness and kurtosis of a probability distribution , moment generating function. Transformation of one dimensional random variable (simple applications). |

MTMG4 | 2 | Vector analysis | Vector function, limit and continuity , derivative of vector, derivative of sums and product of vector functions. A necessary and sufficient condition that a proper vector a (i) has a constant length that a . da/dt = 0. (ii) always remains parallel is that a x da/dt=0. Vector integration, scalar and vector fields, directional derivatives, gradient of a scalar point function, nebla operator, divergence, curl and Laplacian . Line, surface and volume integral. Statement of Gauss's, Stoke's theorem and problem based on these. |

MTMG5 | 1 | Numerical analysis | 1. Approximate numbers and significant figures, rounding off numbers. Error and Absolute, relative and percentage errors. Linear operation, Difference, finite difference interpolation. Lagrange interpolation. Newton's forward and backward difference formula. Differentiation formula based on Newton's forward and backward difference formula, Numerical integration, deduction of Trapezoidal, Simpson 's 1/3 rule from Newton 's forward difference formula. 2. Solution of linear equations: Gauss elimination, Gauss-Jordan method. LU Decomposition. Inversion of 3 x 3 non-singular matrices by Gauss elimination and Gauss-Jordan method. |

MTMG5 | 2 | C Programming | 1. Introduction to ANSI-C : Character set in ANSI-C. Key words: int, char, float, while etc. .Constant and Variables, expressions, assignment statements, formatting source files. Header files.Data types, declarations, different types of integers, different kinds of integer constants, floating-point types, initialization, standard input/output. finding address of an object. 2. Operations and expressions, precedence and associatively, unary plus and minus operators, binary arithmetic operators, arithmetic assignment operators, increment and decrement operators, comma operator, relational operators, logical operators. 3. Control flow, conditional and unconditional bracing, looping, nested loops. if-else, do while, for, switch, break, continue, goto statements etc., Infinite loops. Arrays and Pointers |

MTMG5 | C Programming practical | 1. Ascending / Descending order. Finding Largest / smallest. 2. Sum of finite series. Mean and variance . 3. Conversion of binary to decimal and decimal to bi nary. 4. Checking whether a number i s prime or not. Generation prime numbers. 5. Solution of Quadratic equation . Newtone-Raphson's method. Lagrange interpolation. 6. Bisection method. Newton-Raphson method. 7. Trapezoidal Rule. Simpson 's 1/3 rule. 8. Value of Determinant. 9. Cramer's Rule ( for two variables). 10. Matrix addition, subtraction, transposition. | |

MTMG5 | 1 | Numerical solution of algebraic and transcendental equations | Bisection, Secant / Regula Falsi, Newton's-Raphson method, iteration method. |