MLSU BSc B.Ed. Mathematics Previous Year Question Papers

Symmetric, skew-symmetric, Hermitian & skew-Hermitian matrices; row/column rank; Eigen values, Eigen vectors, Cayley-Hamilton theorem; consistency of linear equations. Roots & coefficients of polynomial equations; Descarte's rule, Cardan method (cubic), Ferrari's method (biquadratic). Groups: order, permutation & cyclic groups, subgroups, cosets, Lagrange's theorem; normal subgroups, quotient groups, class equation. Homomorphism, isomorphism, Cayley's theorem, three isomorphism theorems, automorphisms.

Polar coordinates, derivatives of arc, pedal equation; Rolle's theorem, MVT, Taylor's theorem with proofs & applications. Asymptotes, curvature, concavity/convexity, inflexion, multiple points, curve tracing (Cartesian & polar). Beta & Gamma functions, Quadrature, Rectification. 1st order DEs: separable, homogeneous, linear, exact; 1st order higher degree (solvable for x, y, p), Clairaut's form. Linear DEs with constant coefficients; homogeneous linear DEs.

General equation of 2nd degree; ellipse: tangent, normal, pole & polar, conjugate diameters, director circle; hyperbola: asymptotes, rectangular hyperbola; polar equations of conics. 3D: plane and line equations, projections, volume; skew lines, shortest distance, intersection of planes. Sphere: tangent plane, pole & polar, radical plane, coaxial spheres; Cone: enveloping cone, reciprocal cone; Cylinder: right circular. General 2nd degree in 3D, conicoids: tangent, diametral plane, principal planes.

Continuity (Cauchy definition), limit & continuity of functions of two variables, discontinuities; derivability, Darboux theorem. Partial differentiation, envelopes & evolutes, maxima & minima of two/more variables, Lagrange's method. Double & triple integrals, Dirichlet's theorem & Liouville's extension, change of order, volume & surface of revolution. Jacobians, change of variables; Vector Calculus: gradient, divergence, curl; Gauss's, Stokes', Green's theorems.

Exact DEs of special forms; simultaneous DEs; total differential equations. 2nd order linear DEs: integral of CF by inspection, change of variables, removal of 1st derivative, operational factors, undetermined coefficients, variation of parameters. Linear PDEs of 1st order (Lagrange's method); non-linear PDEs of 1st order (standard forms). Charpit's method; Monge's method (Rr + Ss + Tt = V); higher-order homogeneous linear PDEs. Numerical solutions: Euler's method, Picard's method, simultaneous equations.

Equilibrium of bodies under three or more forces; Friction; common catenary. Virtual work; projectile on inclined plane; impact. Velocity & acceleration (tangential, normal, radial, transversal); rectilinear motion; Hooke's law, horizontal & vertical strings. Constrained motion (circular & cycloidal); motion in resisting medium. Fluid pressure; thrust on immersed plane surfaces; centre of pressure.

Real number system: field, ordered field, completeness, Archimedean property; open/closed sets, Bolzano-Weierstrass, Heine-Borel theorems, compact & connected sets. Sequences: convergence, monotonic, Cauchy sequences, subsequences; Series: convergence tests, alternating series (Leibnitz), absolute/conditional convergence. Riemann integration: Darboux sums, integrability, fundamental theorem. Uniform convergence of sequences/series, Mn-test, Weierstrass M-test; Fourier series (even, odd, full & half interval). Improper integrals; equivalent sets, countable/uncountable sets, denumerability.

Rings: integral domain, division ring, field, characteristic, subring, subfield; left/right ideals, principal ideal rings; maximal, prime, principal ideals. Quotient rings, homomorphism, kernel, fundamental & three isomorphism theorems; quotient field of integral domain. Vector spaces: subspaces, intersection, sum, direct sum, linear span, linear dependence/independence. Basis, dimension, finite-dimensional spaces, extension theorem, complementary subspaces, quotient space. Linear transformations: rank, nullity, Sylvester law; matrix of LT; algebra of LTs; dual space, bidual space, reflexivity.

Sets, propositions, cardinality, mathematical induction, inclusion-exclusion; formal languages, phrase-structure grammars; permutations & combinations. Binary relations, equivalence & partial order relations, lattices, Pigeon Hole principle; graphs, planar graphs, Eulerian paths, shortest paths. Trees: rooted, binary, decision, spanning, minimal spanning; finite state machines; time & problem complexity. Recurrence relations: linear with constant coefficients, homogeneous/particular/total solutions, generating functions. Boolean algebras, lattice structures, Boolean functions, propositional calculus, switching circuits.

Differences & derivatives; Newton-Gregory (forward/backward), divided differences, Lagrange's interpolation. Gauss, Stirling, Bessel formulae; inverse interpolation; numerical differentiation; Newton-Cotes, Trapezoidal, Simpson's ⅓ & ⅜ rules. Gauss quadrature, estimation of errors; root-finding: Descarte's rule, Bisection, Regula-Falsi methods. LPP: graphical method, Simplex method, convex set theory. Duality; assignment & transportation problems and their optimum solutions.

Probability: definitions, addition & multiplication laws, conditional probability, independent events, Baye's theorem. Random variables, distribution functions, probability mass & density functions; joint, marginal & conditional distributions. Mathematical expectation, moments, covariance, moment generating & characteristic functions. Theoretical distributions: Binomial, Poisson, Normal — definitions & properties. Curve fitting (least squares), straight line & parabola; bivariate linear correlation & regression.